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CFD Open Series/Patch 2.30
Elements of Fluid
Dynamics
Adapted & Edited By : Ideen Sadrehaghighi
AN N A P O L I S , MD
Flow Instability Great Wave by Kanagawa
1
Contents
List of Figures ...................................................................................................................................................................... 6
1 Introduction ................................................................................................................................ 11
1.1 Flow Analysis and Model Decomposition ................................................................................................. 11
1.2 Hierarchy of Model Equations ....................................................................................................................... 12
1.3 Conservation Laws ............................................................................................................................................. 14
2 Some Preliminary Concepts in Fluid Mechanics ............................................................ 16
2.1 Linear and Non-Linear Systems .................................................................................................................... 16
2.1.1 Mathematical Definition .................................................................................................... 16
2.1.1.1 Linear Algebraic Equation ............................................................................................................ 16
2.1.1.2 Nonlinear Algebraic Equations ................................................................................................... 16
2.1.2 Differential Equation ......................................................................................................... 17
2.1.2.1 Ordinary Differential Equation ................................................................................................... 17
2.1.2.2 Partial Differential Equation ........................................................................................................ 17
2.1.3 Weak vs. Strong Solutions ................................................................................................. 18
2.2 Total Differential ................................................................................................................................................. 18
2.3 Lagrangian vs. Eulerian Description ........................................................................................................... 18
2.4 Fluid Properties ................................................................................................................................................... 19
2.4.1 Kinematic Properties ......................................................................................................... 19
2.4.2 Thermodynamic Properties ............................................................................................... 19
2.4.3 Transport Properties .......................................................................................................... 20
2.4.4 Other Misc. Properties ...................................................................................................... 20
2.4.5 Compressible vs. Incompressible Flows ............................................................................ 20
2.5 Stream Lines .......................................................................................................................................................... 21
2.6 Viscosity .................................................................................................................................................................. 22
2.7 Vorticity ................................................................................................................................................................... 22
2.7.1 Vorticity vs. Circulation ..................................................................................................... 23
2.7.2 Kármán Vortex Street ........................................................................................................ 24
2.8 Conservative and Non-Conservative forms of PDE ............................................................................... 24
2.8.1 Physical .............................................................................................................................. 25
2.8.2 Mathematical..................................................................................................................... 25
2.8.3 How to choose which one to use? ..................................................................................... 26
2.9 Divergence Theorem - Control Volume Formulation ........................................................................... 26
2.10 General Transport Equation .................................................................................................................... 26
2.11 Newtonian vs. non-Newtonian Fluid .................................................................................................... 27
2.12 Some Flow Field Phenomena .................................................................................................................. 28
2.12.1 Viscous Dissipation ............................................................................................................ 28
2.12.2 Diffusion............................................................................................................................. 29
2.12.3 Convection ......................................................................................................................... 29
2.12.4 Dispersion .......................................................................................................................... 29
2.12.5 Advection ........................................................................................................................... 29
2.13 Inviscid vs. Viscous ...................................................................................................................................... 29
2.14 Steady-State vs. Transient ........................................................................................................................ 30
2.15 Flow Field Classification ............................................................................................................................ 30
2
3 Brief Review of Thermodynamics ....................................................................................... 32
3.1 Pressure .................................................................................................................................................................. 32
3.2 Perfect (Ideal) Gas............................................................................................................................................... 32
3.3 Total Energy .......................................................................................................................................................... 32
3.4 Thermodynamic Process .................................................................................................................................. 32
3.5 First Law of Thermodynamics ....................................................................................................................... 32
3.6 Second Law of Thermodynamics .................................................................................................................. 33
3.6.1 Case Study - Heat Balance and Entropy Maximum ........................................................... 33
3.7 Isentropic Relation ............................................................................................................................................. 34
3.8 Static (Local) Condition .................................................................................................................................... 34
3.9 Stagnation (Total) Condition .......................................................................................................................... 35
3.10 Total Pressure (Incompressible) ........................................................................................................... 35
3.11 Pressure Coefficient .................................................................................................................................... 36
3.12 Application of 1st Law to Turbomachinery ........................................................................................ 36
3.12.1 Moment of Momentum .................................................................................................... 37
3.12.1.1 Euler‘s Pump & Turbine Equations ..................................................................................... 37
3.12.1.2 Case Study - Application of 1st and 2nd Laws of Thermodynamics to Single Stage
Turbo Machines ................................................................................................................................................... 38
4 Viscous Flow ............................................................................................................................... 40
4.1 Qualitative Aspects of Viscous Flow ............................................................................................................ 40
4.1.1 Drag Definition and its Types ............................................................................................. 40
4.1.2 No-Slip Wall Condition ....................................................................................................... 41
4.1.3 Flow Separation ................................................................................................................. 41
4.1.3.1 Supersonic Laminar Flow .............................................................................................................. 42
4.1.4 Skin Friction and Skin Friction Coefficient ......................................................................... 43
4.1.4.1 Case Study - Image-Based Modelling of the Skin-Friction Coefficient ........................ 43
4.1.4.2 Inclined Structures and Drag Production ............................................................................... 44
4.1.4.3 Reference ............................................................................................................................................. 45
4.1.5 Aerodynamic Heating ........................................................................................................ 46
4.1.6 Reynolds Number .............................................................................................................. 46
4.1.7 Reynolds Number Effects in Reduced Model .................................................................... 47
4.1.8 Case Study 1 - Scaling and Skin Friction Estimation in Flight using Reynold Number ....... 48
4.1.8.1 Interaction Between Shock Wave and Boundary Layer ................................................... 48
4.1.8.2 Reynolds Number Scaling ............................................................................................................. 49
4.1.8.3 Discrepancy in Flight Performance and Wind Tunnel Testing ...................................... 50
4.1.8.4 Flow Separation Type (A - B) ....................................................................................................... 51
4.1.8.5 Over-Sensitive Prediction in Flight Performance ................................................................ 52
4.1.8.6 Aerodynamic Prediction ................................................................................................................ 52
4.1.8.7 Skin Friction Estimation ................................................................................................................ 53
4.1.9 Case Study 2 - Reynolds Number Effects Compared To Semi-Empirical Methods............ 54
4.1.9.1 Scaling Effects Due to Reynolds Number ................................................................................ 54
4.1.9.2 Direct and Indirect Reynolds Number ..................................................................................... 56
4.1.9.3 CFD Calculations ............................................................................................................................... 56
4.1.9.4 Description of the CFD Code ........................................................................................................ 56
4.1.9.5 Mesh Generation ............................................................................................................................... 56
4.1.9.6 Residual & Mesh Dependence ..................................................................................................... 57
4.1.9.7 Results and Discussion ................................................................................................................... 58
4.1.9.8 Reynolds Number Scaling ............................................................................................................. 59
3
4.1.9.9 Reynolds Number Scaling using Semi Empirical Skin Friction Methods ................... 60
4.1.9.10 Inspection of Local Boundary Layer Properties for Varying Reynolds Number
64
4.1.9.11 Concluding Remarks ................................................................................................................. 65
5 Conservation Laws and Governing Equations ................................................................ 67
5.1 Control Volume Approach ............................................................................................................................... 67
5.2 Integral Forms of Conservation Equations ............................................................................................... 67
5.3 Mathematical Operators ................................................................................................................................... 68
5.4 Conservation of Mass (Continuity Equation) .......................................................................................... 69
5.5 Centrifugal and Coriolis Forces ..................................................................................................................... 69
5.6 Conservation of Momentum (Newton 2nd Law) ..................................................................................... 70
5.7 Conservation of Energy (1st Law of Thermodynamics) ...................................................................... 70
5.8 Scalar Transport Equation .............................................................................................................................. 71
5.9 Vector Form of N-S Equations ........................................................................................................................ 71
5.10 Orthogonal Curvilinear Coordinate ...................................................................................................... 72
5.10.1 Cylindrical Coordinate for Governing Equation ................................................................. 73
5.11 Generalized Transformation to N-S Equation .................................................................................. 74
5.12 Coupled and Uncoupled (Segregated) Flows .................................................................................... 77
5.13 Simplification to N-S Equations (Parabolized) ................................................................................ 77
5.14 Non-dimensional Numbers in Fluid Dynamics ................................................................................ 78
5.14.1 Prandtl Number ................................................................................................................. 79
5.14.2 Nusselt Number ................................................................................................................. 79
5.14.3 Rayleigh Number ............................................................................................................... 79
5.14.4 Other Dimensionless Number ........................................................................................... 80
5.14.5 Non-Dimensionalizing (Scaling) of Governing Equations .................................................. 80
5.15 Incompressible Navier-Stokes Equation ............................................................................................ 82
5.16 Vorticity Consideration in Incompressible Flow ............................................................................ 83
5.17 Euler Equation ............................................................................................................................................... 84
5.17.1 Steady-Inviscid–Adiabatic Compressible Equations .......................................................... 85
5.17.2 Velocity Potential Equation ............................................................................................... 85
6 Boundary Layer Theory .......................................................................................................... 90
6.1 Definitions .............................................................................................................................................................. 90
6.2 Scaling Analysis for Boundary Layer Equation (2-D)........................................................................... 91
6.3 3D Boundary Layer ............................................................................................................................................. 91
6.3.1 Thermal Boundary Layer ................................................................................................... 92
6.4 Boundary Layer Separation ............................................................................................................................ 93
6.5 Adverse Pressure Gradient ............................................................................................................................. 93
6.5.1 Influencing Parameters...................................................................................................... 93
6.6 Internal Separation ............................................................................................................................................. 94
6.7 Effects of Boundary Layer Separation ........................................................................................................ 94
6.8 Shock wave/Boundary layer Interactions (SWBLIs) ........................................................................... 95
6.8.1 Case Study 1 - Shock Wave/Boundary Layer Interaction (SWBLI) ..................................... 95
6.8.1.1 Background and Introduction ..................................................................................................... 95
6.8.1.2 Basic Properties of Shock Induced Phenomena ................................................................... 96
6.8.1.3 Case of Interaction with the Ramp ............................................................................................ 99
6.8.1.4 Mechanisms For Control ............................................................................................................. 100
4
6.8.1.5 Objectives and Application of Some Control Actions ...................................................... 102
6.8.1.6 Finishing Remarks ......................................................................................................................... 106
6.8.1.7 References ........................................................................................................................................ 107
6.8.2 Case Study 2 - Effect of Freestream Parameters on the Laminar Separation in Hypersonic
Shock Wave Boundary Layer Interaction ........................................................................................ 108
6.8.2.1 Governing Equations and the Numerical Schemes .......................................................... 108
6.8.2.2 Problem Statement and the Boundary Conditions .......................................................... 109
6.8.2.3 Results and Discussion ................................................................................................................ 110
6.8.2.4 Concluding Remarks ..................................................................................................................... 111
6.8.2.5 References ........................................................................................................................................ 111
7 Boundary Conditions Types ............................................................................................... 113
7.1 Introduction ....................................................................................................................................................... 113
7.2 Naming Convention for Different Types of Boundaries ................................................................... 113
7.2.1 Dirichlet Boundary Condition .......................................................................................... 113
7.2.2 Von Neumann Boundary Condition ................................................................................. 113
7.2.3 Mixed or Combination of Dirichlet and von Neumann Boundary Condition .................. 113
7.2.4 Robin Boundary Condition .............................................................................................. 114
7.2.5 Cauchy Boundary Condition ............................................................................................ 114
7.2.6 Periodic (Cyclic Symmetry) Boundary Condition ............................................................. 114
7.2.7 Generic Boundary Conditions .......................................................................................... 114
7.3 Wall Boundary Conditions ........................................................................................................................... 115
7.3.1 Velocity Field ................................................................................................................... 115
7.3.2 Pressure ........................................................................................................................... 115
7.3.3 Scalars/Temperature ....................................................................................................... 116
7.3.3.1 Common Inputs for Wall Boundary Condition .................................................................. 116
7.4 Symmetry Planes ............................................................................................................................................ 116
7.5 Inflow Boundaries ............................................................................................................................................ 117
7.5.1 Velocity Inlet .................................................................................................................... 117
7.5.2 Pressure Inlet ................................................................................................................... 117
7.5.3 Mass Flow Inlet ................................................................................................................ 117
7.5.4 Inlet Vent ......................................................................................................................... 118
7.6 Outflow Boundaries ........................................................................................................................................ 118
7.6.1 Pressure Outlet ................................................................................................................ 118
7.6.2 Pressure Far-Field ............................................................................................................ 119
7.6.3 Outflow ............................................................................................................................ 119
7.6.4 Outlet Vent ...................................................................................................................... 119
7.6.5 Exhaust Fan ...................................................................................................................... 119
7.7 Free Surface Boundaries ............................................................................................................................... 120
7.7.1 Velocity Field and Pressure.............................................................................................. 120
7.7.2 Scalars/Temperature ....................................................................................................... 120
7.8 Pole (Axis) Boundaries .................................................................................................................................. 120
7.9 Periodic Flow Boundaries............................................................................................................................. 120
7.10 Non-Reflecting Boundary Conditions (NRBCs) ............................................................................ 120
7.10.1 Case Study 1 - Turbomachinery Application of 2-D Subsonic Cascade ........................... 121
7.10.2 Case Study 2 - CAA Application of Airfoil Turbulence Interaction Noise Simulation ...... 123
7.11 Turbulence Intensity Boundaries ....................................................................................................... 123
7.11.1 Turbulence Intensity ........................................................................................................ 123
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7.12 Immersed Boundaries ............................................................................................................................. 124
7.13 Free Surface Boundary ......................................................................................................................... 124
7.13.1 The Kinematic Boundary Condition ................................................................................. 124
7.13.2 The Dynamic Boundary Condition ................................................................................... 125
7.14 Other Boundary Conditions .................................................................................................................. 125
7.15 Further Remarks ....................................................................................................................................... 126
8 Linear PDEs and Model Equations ................................................................................... 127
8.1 Mathematical Character of Basic Equations.......................................................................................... 127
8.1.1 Nonsingular Transformation............................................................................................ 128
8.1.2 The 'Par-Elliptic' problem ................................................................................................ 128
8.2 Exact (Closed Form) Solution Methods to Model Equations .......................................................... 129
8.2.1 Linear Wave Equation (1st Order) .................................................................................... 129
8.2.2 Inviscid Burgers Equation ................................................................................................ 129
8.2.3 Diffusion (Heat) Equation ................................................................................................ 130
8.2.4 Viscous Burgers Equation ................................................................................................ 130
8.2.5 Tricomi Equation .............................................................................................................. 131
8.2.6 2D Laplace Equation ........................................................................................................ 131
8.2.6.1 Boundary Conditions ................................................................................................................... 131
8.2.7 Poisson’s Equation ........................................................................................................... 132
8.2.8 The Advection-Diffusion Equation ................................................................................... 132
8.2.9 The Korteweg-De Vries Equation..................................................................................... 132
8.2.10 Helmholtz Equation ......................................................................................................... 132
8.2.11 Exact Solution Methods ................................................................................................... 133
8.3 Solution Methods for In-Viscid (Euler) Equations ............................................................................. 133
8.3.1 Method of Characteristics ............................................................................................... 133
8.3.2 Linear Systems ................................................................................................................. 134
8.3.3 Non-Linear Systems ......................................................................................................... 135
9 Lattice Boltzmann Method (LBM) and Its Siblings ..................................................... 138
9.1 Preliminaries and Background ................................................................................................................... 138
9.1.1 Approaches ...................................................................................................................... 138
9.1.1.1 Dilute Gas Regimes ........................................................................................................................ 139
9.1.2 Book Keeping ................................................................................................................... 140
9.1.3 Kinetic Theory .................................................................................................................. 140
9.1.4 Maxwell Distribution Function ........................................................................................ 141
9.1.5 Boltzmann Transport Equation ........................................................................................ 142
9.1.6 The BGKW Approximation ............................................................................................... 143
9.1.6.1 Relaxation Time.............................................................................................................................. 144
9.1.6.2 Choice of Relaxation Time in the LBM .................................................................................. 144
9.1.7 Lattices & DnQm Classification ........................................................................................ 144
9.1.8 Lattice Arrangements ...................................................................................................... 145
9.1.8.1 1D Lattice Boltzmann Method (D1O2) ................................................................................. 145
9.1.8.2 2D Lattice Boltzmann Method (D2Q9) ................................................................................. 147
9.1.8.3 Case Study 1 - Lid-Driven Cavity Flow .................................................................................. 148
9.1.8.4 Some Observations and Stability Consideration Regarding SRT, MRT and TRT 149
9.1.8.5 References ........................................................................................................................................ 150
9.1.8.6 Case Study 2 - Flow Past A Circular Cylinder ..................................................................... 151
6
9.2 Differences Between Finite Volume Method (FVM) and Lattice Boltzmann Methods (LBM)
152
9.2.1 Advantages ...................................................................................................................... 153
9.2.2 Limitations ....................................................................................................................... 153
9.3 Unified Flow Solver (UFS) ............................................................................................................................ 153
9.3.1 Description of Kinetic Schemes of Direct Methods ......................................................... 154
9.3.2 Continuum Switching Parameter..................................................................................... 155
9.4 The Navier-Stokes Equations (NS) ............................................................................................................ 157
9.4.1 N-S Equation in Non-Inertial Frame of Reference ........................................................... 157
9.4.2 Some Basic Functional Analysis ....................................................................................... 158
9.4.2.1 Fourier Series and Hilbert Spaces........................................................................................... 158
9.4.2.2 Weak vs. Genuine (Strong) Solution ...................................................................................... 158
10 Porous Media ........................................................................................................................... 160
10.1 Introduction and Background .............................................................................................................. 160
10.2 Porous Modeling ........................................................................................................................................ 160
10.3 Porous Medium .......................................................................................................................................... 162
10.3.1 Literature Survey ............................................................................................................. 163
10.3.2 Some Insight into Physical Consideration of Porous Medium......................................... 164
10.3.3 Velocity–Pressure Formulation ....................................................................................... 165
10.3.4 Derivation of Volume Average N-S Equations (VAN-S) ................................................... 166
10.3.5 Discussion ........................................................................................................................ 167
List of Tables
Table 4.1.1 Mesh and Residual Dependence on CD in Drag counts relative to the baseline mesh
with a Residual of -5.5 - (Courtesy of Pettersson and Rizzi) ............................................................................... 57
Table 4.1.2 Comparison of the Extrapolated Data and CFD in Drag Counts at Reynolds Number 56
M ................................................................................................................................................................................................... 62
Table 6.8.1 Laminar Separation and Re-Attachment vs Freestream Pressure for Given Freestream
Temperatures – Courtesy of [Kumar et al] .............................................................................................................. 110
Table 6.8.2 Laminar Separation and Re-Attachment vs Freestream Temperature for give
Freestream Pressure ......................................................................................................................................................... 110
Table 6.8.3 Laminar Separation and Re-Attachment vs Mach number ................................................... 111
Table 8.3.1 Classification of the Euler Equation on Different Regimes .................................................... 133
List of Figures
Figure 1.1.1 The von Karman vortex street generated by the Rishiri Island of Hokkaido, Japan
(top, photo from NASA, 2001; STS-100). This wake produced at high Reynolds number shares great
similarity with the cylinder wake at low Reynolds number (bottom) ............................................................ 11
Figure 1.2.1 Hierarchy of Basic Fluid Flow ............................................................................................................. 13
Figure 2.3.1 Description of Flow: Lagrangian (left) and Eulerian (right) ................................................. 19
Figure 2.4.1 Control Volume Variation for Compressible vs Incompressible Flows ............................. 21
Figure 2.5.1 Stream Lines around an Airfoil & Cylinder ................................................................................... 22
Figure 2.6.1 Viscosity effects in parallel plate ....................................................................................................... 22
Figure 2.7.1 A Sink Vortex Fow Over a Drain and History of a Rolle Up of a Vortex Over Tme ........ 23
Figure 2.7.2 Circulation (Right) vs. Vorticity (Left) ............................................................................................ 23
7
Figure 2.7.3 The same cylinder, now with a fin, suppressing the vortex street by reducing the
region in which the side eddies can interact .............................................................................................................. 24
Figure 2.7.4 Air Forming a Vortex Street Behind a Circular Cylinder ........................................................ 24
Figure 2.8.1 A region V bounded by the surface S = ∂V with the surface normal n ............................... 26
Figure 2.11.1 Variation of shear stress with the rate of deformation for Newtonian and non-
Newtonian fluids (the slope of a curve at a point is the apparent viscosity of the fluid at that point)
....................................................................................................................................................................................................... 28
Figure 2.12.1 Diffusion Process in Physics ............................................................................................................. 29
Figure 2.14.1 Left-Schematic Diagram of Flow and Right- Flow Structure in Wake ............................. 30
Figure 2.15.1 Physical Aspects of a Typical Flow Field ..................................................................................... 31
Figure 3.6.1 Physical Model .......................................................................................................................................... 33
Figure 3.11.1 Control Volume showing sign convention for heat and work transfer ........................... 36
Figure 3.12.1 Control Volume for a Generalized Turbomachine ................................................................... 37
Figure 3.12.2 Schematic section of Single Stage Turbomachine .................................................................... 38
Figure 4.1.1 Boundary Layer Flow along a Wall ................................................................................................... 40
Figure 4.1.2 Airflow Separating from a Wing at a High Angle of Attack ..................................................... 41
Figure 4.1.3 Drag on Slender & Blunt Bodies ........................................................................................................ 41
Figure 4.1.4 The Porous Titanium LFC Glove is Clearly .................................................................................... 42
Figure 4.1.5 Illustrating the calculation of Skin Friction ................................................................................... 43
Figure 4.1.6 Evolutionary geometry of vortical or scalar structures, sketched by the ellipses with
different scales and inclination angles, in the boundary-layer transition, along with the rise of the
skin-friction coefficient Cf .................................................................................................................................................. 44
Figure 4.1.7 Diagram of the geometry of material surfaces and typical vortex lines near the
surfaces, along with the rise of cf . Solid lines denote vortex lines, and solid vectors n_ denote the
normal of material surfaces. ............................................................................................................................................. 44
Figure 4.1.8 The contour of the Lagrangian wall-normal displacement ΔY and contour lines for the
strong shear layers on the x–y plane in the transitional region at M∞ = 6. .................................................... 45
Figure 4.1.9 Quantitate Aspects of Viscous Flow ................................................................................................. 46
Figure 4.1.10 Effects of Reynolds Number in Inertia vs Viscosity ................................................................ 46
Figure 4.1.11 Drag Coefficient versus Reynolds Number for a 1:5 Model and a Car (Courtesy of 35)
....................................................................................................................................................................................................... 47
Figure 4.1.12 Flow features sensitive to Reynolds number for a cruise condition on a wing section
....................................................................................................................................................................................................... 49
Figure 4.1.13 Schematic representation of direct and indirect Reynolds number effects ................. 50
Figure 4.1.14 Comparison of C-141 Wing Pressure Distributions Between Wind Tunnel and Flight
....................................................................................................................................................................................................... 50
Figure 4.1.15 Flat plate Skin Friction Correlations Comparison ................................................................... 55
Figure 4.1.16 Cut of the Volume Mesh along the Sweep ................................................................................... 57
Figure 4.1.17 Convergence history for the adapted mesh at Reynolds number 20 M - (Courtesy of
Pettersson and Rizzi) ........................................................................................................................................................... 58
Figure 4.1.18 Wing Colored by Cp Contours - (Courtesy of Pettersson and Rizzi) ................................ 58
Figure 4.1.19 Simulation Criterion as a Function of Reynolds Number for a Recrit at Reynolds
Number 50 M - (Courtesy of Pettersson and Rizzi) ................................................................................................. 60
Figure 4.1.20 Skin Friction Estimated with Karman-Shoenherr and Sommer-Short Methods
anchored to CFD data at Reynolds Number 38 M - (Courtesy of Pettersson and Rizzi) .......................... 61
Figure 4.1.21 Numerical Fit of Drag Due to Pressure - (Courtesy of Pettersson and Rizzi) ............... 63
Figure 4.1.22 HTP seen from above, positions where ....................................................................................... 65
Figure 5.1.1 Control Volume Bond Corresponding Control Surface S ......................................................... 67
Figure 5.5.1 Centrifugal and Coriolis force ............................................................................................................. 70
Figure 5.10.1 Relation between Cartesian and Cylindrical; coordinate ..................................................... 73
8
Figure 5.13.1 Conditions and Mathematical Character of N-S and its variation ..................................... 78
Figure 5.14.2 Some Methods for Simplifying Governing Equations............................................................. 81
Figure 5.16.1 Evolution of a Vortex Tube in pyroclastic flows ....................................................................... 83
Figure 5.17.1 Condition and Mathematical Character of Inviscid Flow .................................................... 87
Figure 5.17.2 From N-S to Linearized Equation ................................................................................................... 87
Figure 5.17.3 Hierarchy of Flow Equations ............................................................................................................ 88
Figure 6.1.1 The Development of the Boundary Layer for Flow Over a Flat Plate ................................. 90
Figure 6.1.1 Definition of Boundary Layer Thickness: (a) Standard Boundary Layer (u = 99%U),
(b) ................................................................................................................................................................................................. 91
Figure 6.3.1 3-D Boundary Layer Velocity Profile ............................................................................................... 92
Figure 6.4.1 Representation Boundary Later Velocity Profile (Courtesy of Sturm et al.) .................. 93
Figure 6.5.1 Physical Features of a Ramp Flow with Boundary Layer Separation - Courtesy of
[Delery & Bur] ......................................................................................................................................................................... 94
Figure 6.8.1 Physical features of an oblique shock reflection with boundary layer separation -
Courtesy of Delery & Bur .................................................................................................................................................... 97
Figure 6.8.2 2D High Speed Flow Over a Compression Corner Involving SWBL Interaction –
Courtesy of Kumar ................................................................................................................................................................ 98
Figure 6.8.3 Physical Features of an Oblique Shock Reflection Without Boundary Layer
Separation. The Upstream Propagation Mechanism. Courtesy of Delery & Bur ....................................... 99
Figure 6.8.4 Schematic Representation of the Flow in a Separated Bubble. a - basic case, b - with
Mass Suction, c - with Mass Injection - Courtesy of Delery & Bur .................................................................. 100
Figure 6.8.5 Mach Number Contours of Transonic Interaction with Suction of the Boundary Layer
Through a Slot - Courtesy of Delery & Bur .............................................................................................................. 101
Figure 6.8.6 Principle of Passive Control of a Transonic Interaction - Courtesy of Delery & Bur 103
Figure 6.8.7 Interaction Control by a Local Deformation of the Wall or the Bump Concept -
Courtesy of Delery & Bur ................................................................................................................................................. 104
Figure 6.8.8 Transonic Interaction Control by a Bump - Mach Number Contours ............................. 104
Figure 6.8.9 Principle of Hybrid Control of a Transonic Interaction - Courtesy of Delery & Bur . 105
Figure 6.8.10 Mach Numbers Contours of Different Control Actions on a Transonic Interaction -
Courtesy of [Delery & Bur] ............................................................................................................................................. 106
Figure 6.8.11 Schematic diagram representing 2D high speed flow over a compression corner
involving SWBL interaction............................................................................................................................................ 107
Figure 7.2.1 Mixed Boundary Conditions ............................................................................................................. 113
Figure 7.3.1 Symmetry Plane to Model one Quarter of a 3D Duct ............................................................. 116
Figure 7.7.1 Pole (Axis) Boundary .......................................................................................................................... 120
Figure 7.10.1 Periodic Boundary ............................................................................................................................. 121
Figure 7.10.2 Pressure contours plot for 2nd order spatial discretization scheme ............................. 122
Figure 7.10.3 Aero-Acoustics Application for NRBC’....................................................................................... 123
Figure 7.12.1 Immersed Boundaries ...................................................................................................................... 124
Figure 7.13.2 Sketch Exemplifying the conditions at a Free Surface Formed by the Interface
Between Two Fluids .......................................................................................................................................................... 125
Figure 8.1.1 Two-way interchange of information between Parabolic and Elliptic flows ............... 128
Figure 8.2.1 Solution of linear Wave Equation................................................................................................... 129
Figure 8.2.2 Formulation of discontinuities in non-linear Burgers (wave) equation ........................ 129
Figure 8.2.3 Rate of Decay of solution to diffusion Equation ....................................................................... 130
Figure 8.2.4 Solution to Laplace equation ............................................................................................................ 131
Figure 8.2.5 Solution to Poisson's equation ........................................................................................................ 132
Figure 8.3.1 Characteristics of Linear Equation ................................................................................................ 134
Figure 8.3.2 Characteristics of nonlinear solution point ............................................................................... 136
Figure 9.1.1 The hierarchy of conservation laws .............................................................................................. 138
9
Figure 9.1.2 Flow Regimes for Diluted Gas .......................................................................................................... 140
Figure 9.1.3 Position and velocity vector for a particle after and before applying a force, F ......... 142
Figure 9.1.4 Real Molecules vs. LB Particles ...................................................................................................... 144
Figure 9.1.5 Lattice Arrangements for Velocity Vectors for Typical 1D, 2D and 3D Discretization
.................................................................................................................................................................................................... 146
Figure 9.1.6 Schematics of solving 2D Lattice Boltzmann Model ............................................................... 148
Figure 9.1.7 Lid-Driven Cavity, Streamlines Pattern For Different Reynolds Numbers ................... 149
Figure 9.1.1 Flow Past Cylinder, Velocity Magnitude Contours .................................................................. 152
Figure 9.3.1 UFS Key Components .......................................................................................................................... 154
Figure 9.3.2 Gas flow Around a Cylinder for M = 3 for Different Kn Numbers (0.5, 0.005). ............ 155
Figure 9.3.3 Distribution of Normal Force Over the Cylinder Surface for Different Values of the
Continuum Breakdown Parameter S .......................................................................................................................... 156
Figure 9.3.4 Computational Mesh (top) and Gas Density Contours (bottom) ...................................... 156
Figure 10.1.1 Velocity Magnitude color plot with a red arrow plot representing the orientation of
the flow through the pore space (Courtesy of Comsol) ...................................................................................... 160
Figure 10.3.1 Earliest Forms of Porous ................................................................................................................. 162
Figure 10.3.2 Effect of surface machining on the same numerically generated porous sample: .. 164
Figure 10.3.3 Sketch of a Porous Medium, with l*f and l*s the Characteristic Lengths of the ......... 165
10
Preface
This note is intended for all undergraduate, graduate, and scholars of Fluid Mechanics. It is not
completed and never claims to be as such. Therefore, all the comments are greatly appreciated. In
assembling that, I was influenced with sources from my textbooks, papers, and materials that I
deemed to be important. At best, it could be used as a reference. I also would like to express my
appreciation to several people who have given thoughts and time to the development of this article.
Special thanks should be forwarded to the authors whose papers seemed relevant to topics, and
consequently, it appears here©. Finally I would like to thank my wife, Sudabeh for her understanding
and the hours she relinquished to me. Their continuous support and encouragement are greatly
appreciated.
Ideen Sadrehaghighi
June 2018
11
1 Introduction
1.1 Flow Analysis and Model Decomposition
Fluid mechanics is the study of fluids either in motion (fluid dynamics) or at rest (fluid statics) and
the subsequent effects of the fluid upon the boundaries, which may be either solid surfaces or
interfaces with other fluids. Both gases and liquids are classified as fluids, and the number of fluids
engineering applications is enormous: breathing, blood flow, swimming, pumps, fans, turbines,
airplanes, ships, rivers, windmills, pipes, missiles, icebergs, engines, filters, jets, and sprinklers, to
name a few. When you think about it, almost everything on this planet either is a fluid or moves within
or near a fluid
1
.
According to [Taira et al.]
2
, the AIAA Discussion Group on Modal Decomposition Methods for
Aerodynamic Flows (2015-2018), the field of fluid mechanics involves a range of rich and vibrant
problems with complex dynamics
stemming from instabilities,
nonlinearities, and turbulence. The
analysis of these flows benefits
from having access to high-
resolution spatial-temporal data
that capture the intricate physics.
With the rapid advancement in
computational hardware and
experimental measurement
techniques over the past few
decades, studies of ever more
complex fluid flows have become
possible.
While these analyses provide great
details of complex unsteady fluid
flows, we are now faced with the
challenge of analyzing vast and
growing data and high-dimensional
nonlinear dynamics representing
increasingly complex flows.
Although the analysis of these
complex flows may appear
daunting, the fact that common flow
features emerge across a wide
spectrum of fluid flows or over a
large range of non-dimensional
flow parameters suggests that there
are key underlying phenomena that
serve as the foundation of many flows. The emergence of these prominent features, including the von
Karman vortex shedding and the Kelvin-Helmholtz instability, provides hope that a lot of the
flows we encounter share low-dimensional features embedded in high-dimensional dynamics.
Shown as an example in Figure 1.1.1 is a photograph taken from the Space Shuttle (STS-100) of the
1
Frank M. White, “Fluid Mechanics”, 4th Edition, McGraw Hill Company.
2
Kunihiko Taira, Maziar S. Hemati, Steven L. Bruntonz, Yiyang Sun, Karthik Duraisamy, Scott T. M. Dawson, and
Chi-An Yeh, “Modal Analysis of Fluid Flows: Applications and Outlook”, AIAA, 2019.
Figure 1.1.1 The von Karman vortex street generated by the
Rishiri Island of Hokkaido, Japan (top, photo from NASA, 2001;
STS-100). This wake produced at high Reynolds number shares
great similarity with the cylinder wake at low Reynolds number
(bottom)
12
von Karman vortex street generated by the Rishiri Island of Japan, whose wake is visualized by the
clouds. Let us compare this image with the two-dimensional low Reynolds number flow over a
circular cylinder shown in the same figure. The striking similarity between these two flows suggests
the existence of spatial features that capture the essence of the flow physics. In this work, we present
modal analysis techniques to mathematically extract the underlying flow features from flow field
data or the flow evolution operators.
In addition to flow analysis, modal decomposition techniques can also be used to facilitate
reduced-order flow modeling and control. Indeed, modal decomposition techniques offer a
powerful means of identifying an effective low-dimensional coordinate system for capturing
dominant flow mechanisms. The reduction of the system order corresponds to the choice of an
appropriate (reduced basis) coordinate system to represent the fluid flow. This concept has
implications for nearly every ensuing modeling and control decision. A linear subspace to describe
the flow, for example, obtained via proper orthogonal decomposition, is the most common choice for
a low-dimensional basis. After the choice of coordinate system, there are two main distinctions in
modeling procedures: depending on
➢ whether or not the model is physics-based or data-driven,
➢ whether or not the model is linear or nonlinear.
1.2 Hierarchy of Model Equations
There is a clear hierarchy of physical models to choose from. The most general model under routine
use is at the level of the fluid molecule where the motion of individual molecules is tracked and inter-
molecular interactions are simulated. For example, this level of approximation is required for the
rarefied gases encountered during the reentry of spacecraft in the upper atmosphere. Although this
model can be certainly used at lower speeds and altitudes, it becomes prohibitively expensive to track
individual molecules under non-rarefied conditions. Thus, another mathematical model is needed. In
moving from the molecular description to the continuum model we basically performed an averaging
process over the molecules to obtain bulk quantities such as temperature and pressure. It turns out
that averaging is one of the primary means of simplifying our mathematical model. For example, if
we average the Navier–Stokes equations in one spatial dimension, then we are left with the two-
dimensional Navier–Stokes equations. As long as the process that we wish to simulate is
approximately two-dimensional then this will be an adequate model. Of course, we can continue by
averaging over two spatial dimensions or even over all three directions if we are only interested in
the variation of mean quantities. The 1D, 2D analyses are discussed in details later on.
Given the hierarchy of mathematical models, and the selection in Figure 1.2.1, it is possible, under
certain circumstances, to make further approximations that take into account special physical
characteristics of the flow under consideration. For example, Prandlt’s landmark discovery that
viscous effects are primarily limited to a boundary layer near a solid surface has led to the boundary
layer equations which are a special form of the Navier–Stokes equations that are considerably easier
to solve numerically. Outside of the boundary layer, which means most of the flow in the case of an
aircraft, the flow is generally inviscid and the viscous terms in the Navier–Stokes equations can be
dropped leading to the Euler equations. If there are no shock waves in the flow, then further
simplification can be obtained by using the potential flow equations, the compressible Navier-
Stokes equations. Many of the most important aspects of these relations are nonlinear and, as a
consequence, often have no analytic solution
3
-
4
.
3
Collis,, S,, “An Introduction to Numerical Analysis for Computational Fluid Mechanics”, Sandia National
Laboratories Albuquerque, New Mexico 87185 and Livermore, California 94550.
4
Lomax, H., and Pulliam, T.,H., “Fundamentals of Computational Fluid Dynamics”, NASA Ames Research Center,
1999.
13
The ultimate goal of fluid dynamics is to understand the physical events that occur in the flow of
fluids around and within designated objects. These events are related to the action and interaction of
phenomena such as dissipation, diffusion, convection, shock waves, slip surfaces, boundary layers,
and turbulence. In the field of aerodynamics, all of these phenomena are governed by the
compressible Navier-Stokes equations. Since there is no analytical solutions, therefore, the idea of
Computational Fluid Dynamics (CFD) comes to mind where the flow equation being discretized and
solved with appropriate simplification. For fluids which are sufficiently dense to be a continuum, do
not contain ionized species, and have flow velocities small in relation to the speed of light, the
momentum equations for Newtonian fluids are the Navier–Stokes equations, which is a non-linear
set of differential equations that describes the flow of a fluid whose stress depends linearly on flow
velocity gradients and pressure.
The simplified equations do not have a general closed-form solution, so they are primarily of use in
Computational Fluid Dynamics (CFD). The equations can be simplified in a number of ways, all of
Figure 1.2.1 Hierarchy of Basic Fluid Flow
Continuous
Rarefied Gas Dynamics
Boltzmans Linear Theory
Conservation Laws
Aerodynamics
Hydrodynamicse
rodynamics
Gas Dynamics
Inviscid
Viscous
Bernoulli’s
Eqs.
Euler Eqs.
Potential
Eqs.
Navier-Stokes
Eqs.
Boundary
Layer Eqs.
No
Yes
Fluid Dynamics
14
which make them easier to solve. In some cases, further simplification is allowed to appropriate fluid
dynamics problems to be solved in closed form.
1.3 Conservation Laws
The conservation laws are used to solve fluid dynamics problems, and may be written in integral or
differential form. It is ironic that modern numerical methods for solving compressible fluid flows,
and in particular those high Reynolds number flows that feature shocks and/or turbulence, are based
on finite volume (FV) methods which also solve the conservation laws in integral form. In other
words, the discrete variables in a FV method are the volume-averaged fields as opposed to the
field variables evaluated at a point within the computational cell.
Mathematical formulations of these conservation laws may be interpreted by considering the
concept of a control volume. A control volume is a specified volume in space through which air can
flow in and out. Integral formulations of the conservation laws consider the change in mass,
momentum, or energy within the control volume. Differential formulations of the conservation laws
applies Stokes' theorem to yield an expression which may be interpreted as the integral form of law
applied to an infinitesimal volume at a point within the flow.
Mass continuity (conservation of mass) is the rate of change of fluid mass inside a control volume
must be equal to the net rate of fluid flow into the volume
5
. Physically, this statement requires that
mass is neither created nor destroyed in the control volume, and can be translated into the integral
form of the continuity equation. All fluids are compressible to some extent, that is, changes in
pressure or temperature will result in changes in density. However, in many situations the changes
in pressure and temperature are sufficiently small that the changes in density are negligible. In this
case the flow can be modeled as an incompressible flow. Otherwise the more general compressible
flow equations must be used. For conservation Momentum, Newton’s famous 2nd law was applied,
and Energy make use of 1st law of Thermodynamics (Energy Conservation).
Consequently, the assumptions inherent to a fluid mechanical treatment of a physical system can be
expressed in terms of mathematical equations
6
. Fundamentally, every fluid mechanical system is
assumed to obey:
• Conservation of mass
• Conservation of energy
• Conservation of momentum
• The continuum assumption
The continuum assumption is an idealization of continuum mechanics under which fluids can be
treated as continuous, even though, on a microscopic scale, they are composed of molecules. is
mostly applying in the continuum gas domain which is limited to the negligible Knudson number
7
;
Nn = λ/l << 1.0. In this physical domain, the mean-free-path (λ) of particle collisions is negligible in
comparison with the characteristic length (l) of the flow field considered.
For flow of gases, to determine whether to use compressible or incompressible fluid dynamics, the
Mach number of the flow is to be evaluated. As a rough guide, compressible effects can be ignored at
Mach numbers below approximately 0.3. Therefore, it is safe to assume incompressible flow. For
liquids, whether the incompressible assumption is valid depends on the fluid properties (specifically
the critical pressure and temperature of the fluid) and the flow conditions (how close to the critical
pressure the actual flow pressure becomes). Acoustic problems always require allowing
5
Wikipedia, “Fluid Dynamics“.
6
Wikipedia, ”Fluid Mechanics”.
7
Joseph J. S. Shang, “Landmarks and new frontiers of computational fluid dynamics”, Shang Advances in
Aerodynamics (2019) 1.5 https://doi.org/10.1186/s42774-019-0003-x
15
compressibility, since sound waves are compression waves involving changes in pressure and
density of the medium through which they propagate.
16
2 Some Preliminary Concepts in Fluid Mechanics
2.1 Linear and Non-Linear Systems
In physical sciences, a nonlinear system is a system in which the change of the output is not
proportional to the change of the input
8
. Nonlinear problems are of interest to engineers, physicists
9
and mathematicians and many other scientists because most systems are inherently nonlinear in
nature. Nonlinear systems may appear chaotic, unpredictable or counterintuitive, contrasting with
the much simpler linear systems. Typically, the behavior of a nonlinear system is described in
mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which
the unknowns (or the unknown functions in the case of differential equations) appear as variables of
a polynomial of degree higher than one or in the argument of a function which is not a polynomial of
degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot
be written as a linear combination of the unknown variables or functions that appear in them.
Systems can be defined as non-linear, regardless of whether or not known linear functions appear in
the equations. In particular, a differential equation is linear if it is linear in terms of the unknown
function and its derivatives, even if nonlinear in terms of the other variables appearing in it. As
nonlinear equations are difficult to solve, nonlinear systems are commonly approximated by linear
equations (linearization). This works well up to some accuracy and some range for the input values,
but some interesting phenomena such as solitons, chaos and singularities are hidden by linearization.
It follows that some aspects of the behavior of a nonlinear system appear commonly to be
counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble
random behavior, it is absolutely not random. For example, some aspects of the weather are seen to
be chaotic, where simple changes in one part of the system produce complex effects throughout. This
nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current
technology.
2.1.1 Mathematical Definition
2.1.1.1 Linear Algebraic Equation
In mathematics, a linear function (or map) f(x) is one which satisfies both of the following properties:
• Additivity or Superposition: f(x + y) = f(x) + f(y)
• Homogeneity: f (αx ) = αf (x)
Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a
complex α, homogeneity does not follow from additivity. For example, an ant linear map is additive
but not homogeneous. The conditions of additivity and homogeneity are often combined in the
superposition principle f (αx + βy) = αf(x) + βf (y). An equation written as f (x) = C is called linear if
f (x ) is a linear map (as defined above) and nonlinear otherwise. The equation is called
homogeneous if C = 0. The definition f(x) = C is very general in that x can be any sensible
mathematical object (number, vector, function, etc.), and the function f(x) can literally be any
mapping, including integration or differentiation with associated constraints (such as boundary
values). Condition f(x) contains differentiation with respect to x , the result will be a differential
equation.
2.1.1.2 Nonlinear Algebraic Equations
Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating
polynomials to zero. For example, x2 + x −1 = 0. For a single polynomial equation, root-finding
algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that
8
From Wikipedia, the free encyclopedia.
9
Gintautas, V. "Resonant forcing of nonlinear systems of differential equations". Chaos. 18, 2008.
17
satisfy the equation). However, systems of algebraic equations are more complicated; their study is
one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is
even difficult to decide whether a given algebraic system has complex solutions. Nevertheless, in the
case of the systems with a finite number of complex solutions, these systems of polynomial equations
are now well understood and efficient methods exist for solving them
10
.
2.1.2 Differential Equation
A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving
nonlinear differential equations are extremely diverse, and methods of solution or analysis are
problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in
fluid dynamics and the Lotka–Volterra equations in biology. One of the greatest difficulties of
nonlinear problems is that it is not generally possible to combine known solutions into new solutions.
In linear problems, for example, a family of linearly independent solutions can be used to construct
general solutions through the superposition principle. A good example of this is one-dimensional
heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-
dependent linear combination of sinusoids of differing frequencies; this makes solutions very
flexible. It is often possible to find several very specific solutions to nonlinear equations, however the
lack of a superposition principle prevents the construction of new solutions.
2.1.2.1 Ordinary Differential Equation
First order ordinary differential equations are often exactly solvable by separation of variables,
especially for autonomous equations. For example, the nonlinear equation du/d x = − u2 has u = 1/x
+C as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the
general solution when C tends to infinity). The equation is nonlinear because it may be written as d
u/d x + u2 = 0 and the left-hand side of the equation is not a linear function of u and its derivatives.
Note that if the u2 term were replaced with u, the problem would be linear (the exponential decay
problem). Second and higher order ordinary differential equations (more generally, systems of
nonlinear equations) rarely yield closed form solutions, though implicit solutions and solutions
involving non-elementary integrals are encountered. Common methods for the qualitative analysis
of nonlinear ordinary differential equations include:
• Examination of any conserved quantities, especially in Hamiltonian systems.
• Examination of dissipative quantities analogous to conserved quantities.
• Linearization via Taylor expansion.
• Change of variables into something easier to study.
• Bifurcation theory.
• Perturbation methods (can be applied to algebraic equations too).
2.1.2.2 Partial Differential Equation
The most common basic approach to studying nonlinear partial differential equations is to change
the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly
even linear). Sometimes, the equation may be transformed into one or more ordinary differential
equations, as seen in separation of variables, which is always useful whether or not the resulting
ordinary differential equation(s) is solvable. Another common (though less mathematic) tactic, often
seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a
certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations
can be simplified into one linear partial differential equation in the case of transient, laminar, one
dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is
10
Lazard, D. (2009). "Thirty years of Polynomial System Solving, and now?". Journal of Symbolic Computation.
44 (3): 222–231.
18
laminar and one dimensional and also yields the simplified equation.
2.1.3 Weak vs. Strong Solutions
According to [Anderson et al.]
11
, a genuine (strong, differential) solution is the one which the
solution is continuous but bounded discontinuities in the derivatives of solution may occur. A weak
(i.e., control volume) solution is the solution which is genuine except along a surface in space across
which the solution across may be discourteous. A more elaborate definition can be provided by CFD
online as “a strong form of the governing equations along with boundary conditions states the
conditions at every point over a domain that a solution must satisfy. On the other hand a weak form
states the conditions that the solution must satisfy in an integral sense. A weak form does not imply
"inaccuracy" or "inferiority". Examples of weak forms are variational formulations or weighted
residual formulations.
Other definition by [Fox and McDonald’s]
12
states a comparison in terms of (Differential vs. Control
Volume). In the first case the resulting equations are differential equations. Solution of the
differential equations of motion provides a means of determining the detailed behavior of the flow.
An example might be the pressure distribution on a wing surface. Frequently the information sought
does not require a detailed knowledge of the flow. We often are interested in the gross behavior of a
device; in such cases it is more appropriate to use integral formulations of the basic laws. An example
might be the overall lift a wing produces. Integral formulations, using finite systems or control
volumes, usually are easier to treat analytically. The basic laws of mechanics and thermodynamics,
formulated in terms of finite systems, are the basis for deriving the control volume equations.
2.2 Total Differential
Let Q(x, y, z, t) represent any property of fluid. If Dx, Dy, Dz and Dt represent arbitrary changes in
four independent variables, then total differential change in
Eq. 2.2.1
2.3 Lagrangian vs. Eulerian Description
The Lagrangian specification of flow field is a way of looking a fluid motion where observer follows
an individual parcel as it moves through space and time. This could be analog as sitting in a boat and
drifting down the river. The equations of motion that arise from this approach are relatively simple
because they result from direct application of Newton’s second law. But their solutions consist
merely of the fluid particle spatial location at each instant of time, as depicted in Figure 2.3.1 (left).
This figure shows two different fluid particles and their particle paths for a short period of time.
Notice that it is the location of the fluid parcel at each time that is given, and this can be obtained
directly by solving the corresponding equations
13
. The notation X1(0) represents particle #1 at time
t = 0, with X denoting the position vector (x, y, z) T.
Alternatively, the Eulerian is a way of looking at fluid motion that focuses at specific locations in space
and time. This could be visualized as sitting on the bank of river and watching the parcel pass the
11
Dale A. Anderson, John C. Tannehil, and Richard H. Pletcher, “Computational Fluid Mechanics and Heat
transfer”, Hemishere Publishing Co., New York 198.4, ISBN 089116-71-5.
12
Philip J. Pritchard, John C. Leylegian, “ Fox and McDonald’s, “Introduction To Fluid Mechanics”, 8th Edition,
Copyright © 2011 John Wiley & Sons, Inc.
13
J. M. McDonough, “Lectures In Elementary Fluid Dynamics: Physics, Mathematics and Applications”,
Departments of Mechanical Engineering and Mathematics University of Kentucky, 2009.
19
fixed locations. As noted above, this corresponds to a coordinate system fixed in space, and within
which fluid properties are monitored as functions of time as the flow passes fixed spatial locations.
Figure 2.3.1 (right) is a simple representation of this situation. It is evident that in this case we
need not be explicitly concerned with individual fluid parcels or their trajectories. Moreover, the flow
velocity will now be measured directly at these locations rather than being deduced from the time
rate-of-change of fluid parcel location in a neighborhood of the desired measurement points
14
. Within
fluid mechanics, one’s first interest is the fluid velocity where Eulerian description would be more
suitable. On the other hand, for solid mechanics where particle displacement is the interest,
Lagrangian description would be more appropriate. The Eulerian operation of fluid particles could
be best depicted by Total/Substantial Derivative and derived easily with aid of Error! Reference s
ource not found. as
Eq. 2.3.1
Where the terms on the RHS are called the local and conservative derivatives respectfully
15
. The
conservative term has the unfortunate distinction of being the non-linear term and source of great
mathematical difficulties. Complete knowledge of Eq. 2.3.1 is often the solution to problem of fluid
mechanics of interest.
2.4 Fluid Properties
2.4.1 Kinematic Properties
These could include (Linear Velocity, Angular Velocity, Vorticity, Acceleration, and Strain Rate).
Strictly speaking these are properties of flow field itself rather than fluid, and are related to fluid
motion.
2.4.2 Thermodynamic Properties
Includes (Pressure, Density, Temperature, Enthalpy, Entropy, Specific Heat, Prantle Number Pr, Bulk
Modulus, and Coefficient of Thermal Expansion)
16
. Within thermodynamics, a physical property is
14
See above.
15
White, Frank M. 1974: “Viscous Fluid Flow”, McGraw-Hill Inc.
16
White, Frank M. 1974: “Viscous Fluid Flow”, McGraw-Hill Inc.
Figure 2.3.1 Description of Flow: Lagrangian (left) and Eulerian (right)
20
any property that is measurable and whose value describes a state of a physical system. Physical
properties can often be categorized as being either intensive or extensive quantities, according to
how the property changes when the size (or extent) of the system changes. Accordingly, an intensive
property is one whose magnitude is independent of the size of the system. An extensive property is
one whose magnitude is additive for subsystems
17
.
2.4.3 Transport Properties
These includes (Viscosity, Thermal Conductivity, and Mass Diffusivity). They properties that bear to
movement or transport of momentum, heat, and mass respectively. Each of three coefficients relates
flux or transport to the gradient of property. Viscosity relates momentum flux to velocity gradient,
Thermal Conductivity relates heat flux to temperature, gradient, and diffusion coefficients related the
mass transport to the concentration gradient.
2.4.4 Other Misc. Properties
Those could include (surface tension, vapor pressure, eddy diffusion coefficients, surface
accommodation coefficient.
2.4.5 Compressible vs. Incompressible Flows
18
Incompressible flow refers to the fluid flow in which the fluid's density is constant. For a density to
remain constant, the control volume
Eq. 2.4.1
has to remain constant. Even though the pressure changes, the density will be constant for an
incompressible flow. Incompressible flow means flow with variation of density due to pressure
changes is negligible or infinitesimal. All the liquids at constant temperature are incompressible.
17
McNaught, A. D.; Wilkinson, A.; Nic, M.; Jirat, J.; Kosata, B.; Jenkins, A. (2014). IUPAC. Compendium of Chemical
Terminology, 2nd ed. (the "Gold Book"). 2.3.3. Oxford: Blackwell Scientific Publications.
18
Chegg Study Blog
21
Compressible flow means a flow that undergoes a notable variation in density with trending
pressure. Density ρ (x, y, z) is considered as a field variable for the flow dynamics. When the value of
Mach number crosses above 0.3, density begins to vary and the amplitude of variation spikes when
Mach number reaches and exceeds unity. The behavior of control volume (CV) [to be defined later]
for incompressible and compressible flow is depicted in Figure 2.4.1.
It can be seen that the CV remains constant for a flow that is incompressible and CV is squeezed for
compressible flow. Bernoulli's equation is applicable only when flow is assumed to be
incompressible. In case of compressible flow, Bernoulli's equation becomes invalid since the very
basic assumption for Bernoulli's equation is density ρ is constant
For compressible flow,
Eq. 2.4.2
2.5 Stream Lines
An important concept in the study of aerodynamics concerns the idea of streamlines. According to
NASA, a streamline is a path traced out by a massless particle as it moves with the flow. It is easiest
to visualize a streamline if we move along with the body (as opposed to moving with the flow).
Figure 2.4.1 Control Volume Variation for Compressible vs Incompressible Flows
22
Figure 2.5.1 shows the computed streamlines
around an airfoil and around a cylinder. In both
cases, we move with the object and the flow proceeds
from left to right. Since the streamline is traced out by
a moving particle, at every point along the path the
velocity is tangent to the path. Since there is no
normal component of the velocity along the path,
mass cannot cross a streamline. The mass contained
between any two streamlines remains the same
throughout the flow field. We can use Bernoulli's
equation to relate the pressure and velocity along the
streamline. Since no mass passes through the surface
of the airfoil (or cylinder), the surface of the object is
a streamline.
2.6 Viscosity
A measure of the importance of friction in fluid flow.
Viscosity is a fluid property by virtue of which a fluid
offers resistance to shear stresses. Consider a fluid in
2D steady shear between two infinite plates h
apart, as shown in the Figure 2.6.1. The bottom
plate is fixed, while the upper plate is moving at
a steady speed of U. It turns out that the velocity
profile, u(y) is linear, i.e. u(y) = U y/h. Also
notice that the velocity of the fluid matches that
of the wall at both the top and bottom walls.
This is known as the no slip condition. The
coefficient of Viscosity (μ) is often considered
constant, but in reality is a function of both
Pressure and Temperature, or μ = μ (T, P). A
widely used approximation resulted from kinetic theory by Sutherland (1893) using the formula
μ
μ
Eq. 2.6.1
Where S is an effective temperature, called Sutherland’s Constant and subscripts 0 refer as to
reference values.
2.7 Vorticity
The definition of a vortex is a topic of much discussion in fluid mechanics
19
. The common intuitive
features of a vortex are a pressure minimum, closed or spiraling streamlines, and iso-surfaces of
constant vorticity. [Jeong & Hussain]
20
have proposed a definition of a vortex as a pressure minimum
19
Xuerui Mao, “Vortex Instability and Transient Growth”, Thesis submitted for the degree of Doctor of
Philosophy of Imperial College London, 2010.
20
Jeong, J. & Hussain, F. “On the identification of a vortex”, J. Fluid Mech, 1995.
Figure 2.5.1 Stream Lines around an
Airfoil & Cylinder
Figure 2.6.1 Viscosity effects in parallel plate
23
in the absence of unsteady straining and viscous effects. According to [Majumdar]
21
, decreasing μ
and increasing the fluid velocity, disintegrate the fluid parcel moving along U. into smaller parcels
moving in arbitrary directions with random velocities. This is where the vortices are generated.
Vorticity ω, being twice the angular velocity, is a measure of local spin of fluid element given by curl
of velocity as ω
Eq. 2.7.1
In 3D flow, vorticity (ω) is in plane of flow and perpendicular to stream lines as depicted in Figure
2.7.1. By definition, if ω = 0, then the flow labeled irrotational. By Croce’s theorem, the gradient of
stagnation pressure is normal to both velocity vector and vorticity vector; thus it lies in the plane of
the paper and normal to V. Consequently, the stagnation pressure, P0, is constant along each
streamline and varies between streamlines only if vorticity is present
22
.
2.7.1 Vorticity vs. Circulation
The fluid circulation defined as the line integral
of the velocity V around any closed curve C.
There are distinct differences in circulation and
vorticity. Circulation is a macroscopic measure
of the rotation of a fluid element is defined as line
integral of velocity field along a fluid element,
therefore, it is a scalar quantity. Vorticity on the
other hand, is microscopic measure of the
rotation of a fluid element at any point is defined
as the curl of velocity vector. It is a vector
quantity. As far as the physical meaning is
concerned, circulation can be thought as the
amount of 'push' one feels while moving along a closed boundary or path. Vorticity however has
21
Kaushik Majumdar, “An investigation into the vortex formation in a turbulent fluid with an application in
tropical storm generation”, Electronics & Communication Sciences Unit, Indian Statistical Institute, India.
22
A., S., Shapiro, “Film Notes for Vorticity”, MIT.
Figure 2.7.1 A Sink Vortex Fow Over a Drain and History of a Rolle Up of a Vortex Over Tme
Figure 2.7.2 Circulation (Right) vs. Vorticity
(Left)
24
nothing to do with a path, it is defined at a point and would indicate the rotation in the flow field
at that point (see Figure 2.7.2). So, if an infinitesimal paddle wheel is imagined in the flow, it would
rotate due to non-zero vorticity.
2.7.2 Kármán Vortex Street
In fluid dynamics, a Kármán vortex street (or a von Kármán vortex street) is a repeating pattern of
swirling vortices, caused by a
process known as vortex shedding,
which is responsible for the
unsteady separation of flow of
a fluid around blunt bodies. (see
Figure 2.7.4). It is named after the
engineer and fluid
dynamists Theodore von
Kármán, and is responsible for
such phenomena as the "singing" of
suspended telephone or power
lines and the vibration of a car
antenna at certain speeds. In low
turbulence, tall buildings can
produce a Kármán street, so long as the structure is uniform along its height. In urban areas where
there are many other tall structures nearby, the turbulence produced by these prevents the
formation of coherent vortices [4]. See also an interesting discussion regarding matter by Phillip
Oberdorfer in COMSOL blog - https://www.comsol.com/blogs/the-beauty-of-vortex-streets/.
Periodic crosswind forces set up by vortices
along object's sides can be highly undesirable,
and hence it is important for engineers to
account for the possible effects of vortex
shedding when designing a wide range of
structures, from submarine periscopes to
industrial chimneys and skyscrapers.
In order to prevent the unwanted vibration of
such cylindrical bodies, a longitudinal fin can
be fitted on the downstream side, which,
provided it is longer than the diameter of the
cylinder, will prevent the eddies from
interacting, and consequently they remain
attached ( see Figure 2.7.3). Obviously, for a tall building or mast, the relative wind could come
from any direction. For this reason, helical projections that look like large screw threads are
sometimes placed at the top, which effectively create asymmetric three-dimensional flow, thereby
discouraging the alternate shedding of vortices; this is also found in some car antennas. Another
countermeasure with tall buildings is using variation in the diameter with height, such as tapering -
that prevents the entire building being driven at the same frequency.
2.8 Conservative and Non-Conservative forms of PDE
There are two folds to the question of differences between Conservative vs Non-Conservative
forms; namely physical and mathematical
23
.
23
Physics Stack Exchange.
Figure 2.7.4 Air Forming a Vortex Street Behind a Circular
Cylinder
Figure 2.7.3 The same cylinder, now with a fin,
suppressing the vortex street by reducing the region
in which the side eddies can interact
25
2.8.1 Physical
We drive the governing equations by considering a finite control volume. This control volume may
be fixed in space with the fluid moving through it or the control volume may be moving with the fluid
in a sense that same fluid particles are always remain inside the control volume. If the first case is
taken then the governing equations will be in conservation form else these will be in non-
conservation form. The difference between the conservative and non-conservative forms is related
to the movement of the control volume in the fluid flow. While deriving the equations of motion if we
keep the control volume fixed and write the flow equations, they are called the equations in
conservation forms. For example the continuity equation for incompressible flow in rectangular
coordinates. On the other hand, if we focus on the same particles in motion and keep the control
volume moving with them, the equations are called non-conservation equations, here, the same
particles remain in the control volume. Prime examples of conservative and non-conservative forces
are Gravity and Friction forces, respectively.
2.8.2 Mathematical
Splitting the partial derivatives for the purpose of discretization. For example, consider the term ∂
(ρu)/∂x in conservative form
Δx
u)(u)(
x
u)(
1ii −
−
=
Eq. 2.8.1
The non-conservative form of the same can be written a
Δx
ρρ
u
Δx
uu
ρ
x
ρ
u
x
u
ρ
x
u)(
1ii
i
1ii
i−− −
+
−
=
+
=
Eq. 2.8.2
The difference is obvious. While the original derivative is mathematically the same, the discrete form
is not. To demonstrate this, consider a 4 point grid for conservative one (Eq. 2.8.1)
Δx
u)(u)(
Δx
u)(u)(
Δx
u)(u)(
23
12
01
−
+
−
+
−
Eq. 2.8.3
And corresponding non-conservative (Eq. 2.8.2):
Δx
ρρ
u
Δx
uu
ρ
Δx
ρρ
u
Δx
uu
ρ
Δx
ρρ
u
Δx
uu
ρ23
3
23
3
12
2
12
2
01
1
01
1
−
+
−
+
−
+
−
+
−
+
−
Eq. 2.8.4
Those arguments just show that the non-conservative form is different, and in some ways harder. But
why is it called non-conservative? For a derivative to be conservative, it must form a telescoping
series. In other words, when you add up the terms over a grid, only the boundary terms should
remain and the artificial interior points should cancel out. Now let's look at the non-conservative
form: So now, you end up with no terms canceling! Every time you add a new grid point, you are
adding in a new term and the number of terms in the sum grows. In other words, what comes in does
not balance what goes out, so it's non-conservative.
26
2.8.3 How to choose which one to use?
Now, more to the point, when do you want to use each scheme? If your solution is expected to be
smooth, then non-conservative may work. For fluids, this is shock-free flows. If you have shocks, or
chemical reactions, or any other sharp interfaces, then you want to use the conservative form.
Overall, if there is PDE which represents a physical conservative statement, this means that the
divergence of a physical quantity can be identified in the equation, as the case in general conservation
equations later.
2.9 Divergence Theorem - Control Volume
Formulation
In vector calculus, the divergence theorem, also
known as Green Gauss's theorem, is a result that
relates the flow (that is, flux) of a vector field through a
surface to the behavior of the vector field inside the
surface, (see Figure 2.8.1). On fluid, using the integral
relations to calculate the net fluxes of mass, momentum
and energy passing through a finite region of flow. The
rate of change of any property Q within control volume
could be defined as
Eq. 2.9.1
Which could be applied to any property such as mass, momentum and energy and dQ/dm being the
amount of Q per unit mass of particle. With the aid of divergence theorem, , the surface integral could
be converted to volume integral, and the result could be integrated over a fixed volume
24
.
2.10 General Transport Equation
Part of the transport process attributed to the fluid motion alone or simply, the transport of a
property by fluid movement. In relation to general transport process of a variable Q, this could be
envisioned as Eq. 2.10.1. Thus, conservation principles can be expressed in terms of differential
equations that describe all relevant transport mechanisms, such as convection (also called
advection), diffusion, and dispersion. Each terms described below as:
24
White, Frank M. 1974: “Viscous Fluid Flow”, McGraw-Hill Inc.
Figure 2.8.1 A region V bounded by the
surface S = ∂V with the surface normal n
27
Eq. 2.10.1
2.11 Newtonian vs. non-Newtonian Fluid
In continuum mechanics, a Newtonian fluid is a fluid in which the viscous stresses arising from its
flow, at every point, are linearly proportional to the local strain rate, the rate of change of its
deformation over time. That is equivalent to saying those forces are proportional to the rates of
change of the fluid's velocity vector as one moves away from the point in question in various
directions. More precisely, a fluid is Newtonian only if the tensors that describe the viscous stress
and the strain rate are related by a constant viscosity tensor that does not depend on the stress state
and velocity of the flow. If the fluid is also isotropic (that is, its mechanical properties are the same
along any direction), the viscosity tensor reduces to two real coefficients, describing the fluid's
resistance to continuous shear deformation and continuous compression or expansion, respectively.
One also defines a total stress tensor σ that combines the shear stress τ, with conventional
(thermodynamic) pressure p . The stress-shear equation then becomes :
Eq. 2.11.1
In 1-D shear flow of Newtonian fluids, shear stress can be expressed by the linear relationship :
Eq. 2.11.2
On the other hand, a non-Newtonian fluid is a fluid that does not follow Newton's Law of Viscosity.
Most commonly, the viscosity (the gradual deformation by shear or tensile stresses) of non-
Newtonian fluids is dependent on shear rate or shear rate history. Some non-Newtonian fluids with
shear-independent viscosity, however, still exhibit normal stress-differences or other non-
28
Newtonian behavior. Many salt solutions
and molten polymers are non-Newtonian
fluids, as are many commonly found
substances such as ketchup, custard,
toothpaste, starch suspension honey,
paint, blood, and shampoo
25
.
For non-Newtonian fluids, the
relationship between shear stress and
rate of deformation is not linear, as shown
in Figure 2.11.1. The slope of the curve
on the τ versus du/dy chart is referred to
as the apparent viscosity of the fluid.
Fluids for which the apparent viscosity
increases with the rate of deformation
(such as solutions with suspended starch
or sand) are referred to as dilatant or
shear thickening fluids, and those that
exhibit the opposite behavior (the fluid
becoming less viscous as it is sheared
harder, such as some paints, polymer
solutions, and fluids with suspended
particles) are referred to as
pseudoplastic or shear thinning fluids. (Refer to Eq. 2.11.3-Eq. 2.11.4). Some materials such as
toothpaste can resist a finite shear stress and thus behave as a solid, but deform continuously when
the shear stress exceeds the yield stress and thus behave as a fluid. Such materials are referred to as
Bingham plastics after E. C. Bingham, who did pioneering work on fluid viscosity for the U.S. National
Bureau of Standards in the early twentieth century
26
. A simple but often effective analytical approach
to non-Newtonian behavior is the power-law approximation of Ostwald and de-Waele
Eq. 2.11.3
Where K and n are material parameters which in general vary with pressure and temperature. The
exponent n delineates three cases as
27
Eq. 2.11.4
2.12 Some Flow Field Phenomena
2.12.1 Viscous Dissipation
Embodies the concept of a dynamical system where important mechanical models such as waves or
oscillations, loss energy over time, typically from friction or turbulence. The lost energy converted to
heat. For a viscous flow over a body, the kinetic energy decreased under influence of friction. This
25
From Wikipedia.
26
Yunus A. Çengel, John M. Cimbala, “Fluid Mechanics: Fundamentals And Applications”, Published By McGraw-
Hill , ISBN 0–07–247236–7, 2006.
27
White, Frank. M., “Viscous Fluid Flow”, Mc Graw Hill , Inc., 1974.
Figure 2.11.1 Variation of shear stress with the rate of
deformation for Newtonian and non-Newtonian fluids
(the slope of a curve at a point is the apparent viscosity of
the fluid at that point)
29
lost kinetic energy reappears in the form of internal energy of the fluid, hence causing the
temperature to rise. This phenomenon is called viscous dissipation within fluid
28
.
2.12.2 Diffusion
This is a physical process that occurs in a flow of gas
in which some property is transported by the random
motion of the molecules of the gas. Diffusion is related
to the stress tensor and to the viscosity of the gas. Heat
conduction, turbulence, and the generation of
boundary layer are the result of diffusion in the flow.
Diffusion is an equal exchange of species where the
final state would be a uniform mixture. An ordinary
example would be pouring cream into coffee until
diffusion produces a uniform mixture. Another
example could be release of a gas mixture in room.
Standing on one side of room as gas released on the
other side, soon we notice the odor diffused to our side replacing some of our air which diffuses to
the other side. In other word, diffusion is the transport of mass, energy and momentum as the result
of molecular movement, express in mathematical language by multiplying some constant by the first
gradient of quantity of interest. Therefore, a distinguishing feature of diffusion is that it results in
mixing or mass transport, without requiring bulk motion or bulk flow (see Figure 2.12.1). Heat
transfer and viscous flow are both diffusive phenomena.
2.12.3 Convection
Refers to the fluid motion that results from forces acting upon or within it (pressure, viscosity,
gravity, etc.).
2.12.4 Dispersion
Is the combined effects of convention and diffusion? We talked about smoke dispersion from the
chimney, which is result of convective (the wind blowing it), diffusive (smoke diffusive in the air),
the bouncy forces (hot air rises).
29
2.12.5 Advection
Refers to the convection of a scalar concentration and very significant. Examples include 1st order
linear wave equation. In other word, advection is the transport mechanism of a fluid from one
location to another, and is dependent on motion and momentum of that fluid.
2.13 Inviscid vs. Viscous
A major facet of a gas or liquid is the ability of the molecules to move rather freely. As molecules
move, they transport their mass, momentum, and energy from one location to another. This
transportation on a molecular scale gives rise to the phenomena of mass diffusion, viscosity (friction),
and thermal conduction. All real flows exhibit such phenomena and such flows call viscous flows and
to be discussed in detail later. In contrast, a flow which doesn’t experience any of these called In-
viscid flow. In-viscid flows do not truly exist in nature, however, many practical aerodynamic flows
where the influence of transport phenomena is small, could be modeled as Inviscid.
28
Anderson, John D. 1984: “Fundamentals Of Aerodynamics”, Mcgraw Hills Inc.
29
CFD online forum, 2006.
Figure 2.12.1 Diffusion Process in
Physics
30
2.14 Steady-State vs. Transient
An important factor in fluid flow analysis is its dependence to time. Simply put, Steady flow is a flow
when field variables are independent of time, where for transient, they are. This dependence, or lack
of it, could change the mathematical character of governing equation, as to be discussed later,
therefore, altering the solution method. The question to be asked is when a flow could be classified
as a transient flow? This is not easy as it sounds since most depend on their expertise and problem
in hand. Nevertheless, most agree that majority of the flows are transient by nature (turbulent flows)
unless proven otherwise. To that end, a useful, but time sensitive method would be to run the flow
in steady-state and check the converging residuals. If there are large oscillations in outputs, then
there is good chance that flow is Transient and not steady. But if the residuals exhibit a relatively
smooth convergence rate, then the flow is steady.
An simplified case, investigated by [Arif and Hasanb]
30
, would be an unsteady numerical simulations
are performed to investigate the vortex shedding suppression phenomenon for mixed convective
flows past a square cylinder in the large-scale heating regime. (See Figure 2.14.1).
2.15 Flow Field Classification
In general, the fluid flows equations could be classified in terms of its Physical and Mathematical
aspects of it. Mathematically, they can classified as Elliptic, Hyperbolic, or Parabolic, depending on
flow as being Subsonic, Transonic or Supersonic, or any combination of two. This will be dealt in
details later on. Physically, they can be classified via Figure 2.15.1.
30
Md. Reyaz Arif a) and Nadeem Hasanb, “Vortex shedding suppression in mixed convective flow past a square
cylinder subjected to large-scale heating using a non-Boussinesq model”, Cite as: Phys. Fluids 31, 023602 (2019);
https://doi.org/10.1063/1.5079516.
Figure 2.14.1 Left-Schematic Diagram of Flow and Right- Flow Structure in Wake
31
Figure 2.15.1 Physical Aspects of a Typical Flow Field
32
3 Brief Review of Thermodynamics
3.1 Pressure
Pressure is the limiting form of force per unit area
31
Eq. 3.1.1
3.2 Perfect (Ideal) Gas
Cases when gas particles are far enough to be able to neglect the influence of intermolecular forces.
For an ideal gas the equation of state is:
volumespecific
ρ
1
ve wherRT Pvor T R ρ P ===
Eq. 3.2.1
3.3 Total Energy
The total energy of a system E consists of internal energy (e), kinetic energy (KE), and potential
energy (PE) as
mgz PE , mV
2
1
KE , PEKEe E 2==++=
Eq. 3.3.1
Where internal energy (e), specific enthalpy (h), are related as
(T)h h & (T) ee pveh ==+=
Eq. 3.3.2
3.4 Thermodynamic Process
The thermodynamic process is divided to three categories of :
➢ Adiabatic - one in which no heat is added to or taken away from the system
➢ Reversible - one in which no dissipative phenomena occur, i.e., where the effects of
viscosity, thermal conductivity, and mass diffusion are absent
➢ Isentropic - one which is both adiabatic and reversible
3.5 First Law of Thermodynamics
Due to molecular motion of a gas, the heat added or work done on the system causes a change in
energy
Eq. 3.5.1
which is also called the steady-state energy equation.
31
Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc.
33
3.6 Second Law of Thermodynamics
In an essence the 2nd law of thermodynamics complements the 1st law by ascertaining the proper
direction of a process by defining a new state variable called entropy,
T
δq
ds
Eq. 3.6.1
For an adiabatic process (δq = 0), Eq. 3.6.1 becomes,
0 ds
Eq. 3.6.2
Eq. 3.6.1 and Eq. 3.6.2 are forms of the second law of thermodynamics. The second law tells us in
what direction a process will take place. A process will proceed in a direction such that the entropy
of the system always increases or, at best, stay the same. For a reversible process where δw = -pdv
and δq = Tds, the more utilitarian form of 2nd law could be devised as
vdpdhTds
vdppdvdedh if pdvdeTds
−=
++=+=
Eq.
3.6.3
Integrating for a calorically perfect gas, both R and CP constant,
p
p
ln R
T
T
lncss
1
2
1
2
p12 −=−
Eq. 3.6.4
3.6.1 Case Study - Heat Balance and
Entropy Maximum
32
The container in Figure 3.6.1 is divided
into two parts by the piston: V1, V2, with
real gas 1, real gas 2, the pressure and
temperature are the same. Because of the
difference of internal energy to volume,
the system is not in the state of maximum
entropy. According to the second law of
thermodynamics, the system does not
satisfy the thermal equilibrium and is not
in conformity with the experiment. In
Figure 3.6.1, the container is divided into
two parts by the piston： V1 V2 ，There
are two kinds of real gases in it. Pressure and temperature are the same. Next, calculate the entropy
change of the system when the piston moves slightly.
32
Bo Miao, Shenzhen Byd, “Heat Balance and Entropy Maximum”.
Figure 3.6.1 Physical Model
34
Eq. 3.6.5
Since δV1 = -δV2 we get,
Eq. 3.6.6
Because there are two different kinds of real gases, the internal energy has different differentiation
to volume, so
Eq. 3.6.7
The system is not in the state of maximum entropy. According to the second law of thermodynamics,
the system is in a non-equilibrium state and the piston will move, which will lead to different pressure
on both sides, which does not satisfy the hydrostatic equilibrium. The problem is that the criterion of
thermal equilibrium of the second law of thermodynamics, "Maximum entropy" is wrong, and the
whole second law of thermodynamics cannot stand simple physical investigation. Therefore, it can
argued that entropy is not a physical quantity
33
.
3.7 Isentropic Relation
For an isentropic process which is both adiabatic and reversible, Eq. 3.6.4, ds = 0, could be
manipulated to
ρ
ργ
γ
γ
Eq. 3.7.1
The above mentioned equation (Eq. 3.7.1) is very important as it relates pressure, density, and
temperature for an isentropic process
34
. It stems from the 1st law of thermodynamics and definition
of entropy and basically is an energy relation for an isentropic process. Why so important or why is
it frequently used? When it seems so restrictive requiring both adiabatic and reversible conditions?
The answer rests in the fact that large number of practical compressible flow problems could be
assumed isentropic as dissipative effects are confined to a thin boundary layer, where outside, the
flow could be assumed to be isentropic
35
.
3.8 Static (Local) Condition
These are quantities when riding along with the gas at the local flow velocity.
33
Shufeng-Zhang, “Entropy: A concept that is not a physical quantity”, PHYSICS ESSAYS ( Volume 25, Issue 2
(June 2012) ) P172~176.
34
Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc.
35
Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc.
35
3.9 Stagnation (Total) Condition
Represents quantities when fluid elements are brought to rest adiabatically. The values are
expected to change. In particular, the value of temperature, denoted by T0 where the corresponding
enthalpy h0=CpT0 for a calorically perfect gas. From an steady, in-viscid, adiabatic energy equation
36
,
hconst
2
V
h 0
2==+
Eq. 3.9.1
Along a stream line. Analog to this, since h0 = CpT0, thus,
const T TC h 00p0 ==
Eq. 3.9.2
Using Eq. 3.9.1 and Eq. 3.9.2 as an special case of energy equation, and defining Cp in terms of
velocity, for a calorically perfect gas the ratio of total temperature to static temperature could be
express in terms of Mach number as Eq. 3.9.3. Similarly, total pressure p0 and total density ρ0 are
defined as the properties if the fluid elements brought to rest isentropically by using:
1γ
1
2
0
1γ
γ
2
0
2
0
M
2
1γ
1
ρ
ρ
M
2
1-γ
1
P
P
M
2
1γ
1
T
T
−
−
−
+=
+=
−
+=
Eq. 3.9.3
3.10 Total Pressure (Incompressible)
For incompressible flow from Bernoulli’s Equation, without any body force, the pressure is the sum
of static and dynamic pressures as,
Eq. 3.10.1
Note – Although the stagnation and total terminology are been used indiscriminately for pressure, in
general the total pressure also dependent on another identity call gravitational head (ρgz). Therefore,
for completeness, the total pressure could be represented is
head nalgravitatiodynamicstatictotal PPPP ++=
Eq. 3.10.2
Where for most cases the P gravitatio nal head is ignored, therefore, the total and stagnation pressures are
assumed to be the same. In essence, total pressure is the constant in Bernoulli’s equations.
36
Same as above.
36
3.11 Pressure Coefficient
The non-dimensional pressure coefficient could be derived with the aid of Eq. 3.10.1 as
Vρ
2
1
q where
q
pp
C p
=
−
=
Eq. 3.11.1
For incompressible flow, the, Cp could be
expressed in terms of velocities a
V
V
1C
2
p
−=
Eq. 3.11.2
3.12 Application of 1st Law to
Turbomachinery
Figure 3.11.1 shows a control volume
representing a turbomachine, through which fluid passes at a steady rate of mass flow ṁ, entering at
position 1 and leaving at position 2. Energy is transferred from the fluid to the blades of the
turbomachines, positive work being done (via shaft) at the rate Ẇx. In the general case, positive heat
transfer takes place at the rate Ǭ from the surrounding to the control volume
37
. Thus,
Eq. 3.12.1
where h is the specific enthalpy, 1/2c2 the kinetic energy per unit mass, and gz is potential energy
per unit mass. Apart from hydraulic machines, the contribution of the last term in Eq. 3.12.1 is small
and usually ignored. Defining the stagnation enthalpy h = h0+1/2c2 and assuming g(z2 - z1) is
negligible, it becomes
Eq. 3.12.2
Most turbomachinery flow processes are adiabatic (or very nearly so), and it is permissible to write
Ǭ = 0. Therefore, for work producing machines (turbines) Wx > 0 so that
Eq. 3.12.3
and work for absorbing machines (compressor) is negative of that.
37
S.L. Dixon and C.A. Hal, “Fluid Mechanics and Thermodynamics of Turbomachinery”, 6th edition, ISBN: 978-1-
85617-793-1.
Figure 3.11.1 Control Volume showing sign
convention for heat and work transfer
37
3.12.1 Moment of Momentum
In dynamics much useful information is obtained by employing Newton’s second law in the form
where it applies to the moments of forces. This form is of central importance in the analysis of the
energy transfer process in turbomachines. For a system of mass m, the vector sum of the moments
of all external forces acting on the system about some arbitrary axis A-A fixed in space is equal to the
time rate of change of angular momentum of the system about that axis, i.e.
Eq. 3.12.4
where r is distance of the mass center from the axis of rotation measured along the normal to the axis
and cθ the velocity component mutually perpendicular to both the axis and radius vector r. For a
control volume the law of moment of momentum can be obtained. Figure 3.12.1 shows the control
volume enclosing the rotor of a generalized turbomachine. Swirling fluid enters the control volume
at radius r1 with tangential velocity cθ1 and leaves at radius r2 with tangential velocity cθ2. For one-
dimensional steady flow
Eq. 3.12.5
which states that, the sum of
the moments of the external
forces acting on fluid
temporarily occupying the
control volume is equal to
the net time rate of efflux of
angular momentum from
the control volume.
3.12.1.1 Euler‘s Pump &
Turbine
Equations
For a pump or compressor
rotor running at angular
velocity 0, the rate at which
the rotor does work on the
fluid is
Eq. 3.12.6
where the blade speed U = Ω r. Thus the work done on the fluid per unit mass or specific work, is
Eq. 3.12.7
This equation is referred to as Euler’s pump equation.
Figure 3.12.1 Control Volume for a Generalized Turbomachine
38
Eq. 3.12.8
For a turbine the fluid does work on the rotor and the sign for work is then Eq. 3.12.8 will be referred
to as Euler’s turbine equation.
3.12.1.2 Case Study - Application of 1st and 2nd Laws of Thermodynamics to Single Stage Turbo
Machines
Fluid flow in turbo machines always varies in time, though it is assumed to be steady when a constant
rate of power generation occurs on an average. This is due to small load fluctuations, unsteady flow
at blade tips, the entry and the exit, separation in some regions of flow etc., which cannot avoided, no
matter how good the machine and load stabilization may be. This assumption permits the analysis of
energy and mass transfer by using the steady state control volume equations. Assuming further that
there is a single inlet (1) and a single outlet (2) for the turbo machine across the sections of which
the velocities, pressures, temperatures and other relevant properties are uniform, one writes the
steady flow equation of the First Law of Thermodynamics within a control volume for turbo
machinery in the form:
Eq. 3.12.9
Where Q is the rate of energy transfer as heat cross the CV, Power(out) is the power output, and ṁ is
the mass flow rate. In cooperating the total enthalpy relation h0 = h + V2/2 + gz, and assuming an
adiabatic process (Q = 0). Eq. 3.12.9 could be rearranged per unit mass flow as
Eq. 3.12.10
Therefore, the energy transfer as work is numerically equal to the change in stagnation (total)
enthalpy of the fluid between the inlet and the outlet of the turbo machine. In a turbo machine, the
energy transfer between the fluid and the
blades can occur only by dynamic action,
i.e., through an exchange of momentum
between the rotating blades (Figure
3.12.2, location 3) and the flowing fluid. It
thus follows that all the work is done when
the fluid flows over the rotor-blades and
not when it flows over the stator-blades. As
an example, considering a turbo machine
with a single stator-rotor combination
shown schematically in Figure 3.12.2. Let
points 1 and 2 represent respectively the
inlet and the exit of the stator. Similarly,
points 3 and 4 represent the corresponding
positions for the rotor blades. Then ideally
for flow between points 1and 2, there
should be no stagnation enthalpy changes
since no energy transfer as heat or work occurs in the stator. Thus, ho1 = ho2. For flow between points
Figure 3.12.2 Schematic section of Single Stage
Turbomachine
39
3 and 4 however, the stagnation enthalpy change may be negative or positive, depending upon
whether the machine is power- generating or power-absorbing. Hence, ho3 > ho4 if the machines
develops power (compressor), and if ho3 < ho4, the machine needs a driver and absorbs power
(turbine). If the system is perfectly reversible and adiabatic with no energy transfer as work, no
changes can occur in the stagnation properties (enthalpy, pressure and temperature) between the
inlet and the outlet of the machine. But all turbo machines exchange work with the fluid and also
suffer from frictional as well as other losses. The effect of the losses in a power-generating machine
is to reduce the stagnation pressure and to increase entropy so that the network output is less than
that in an ideal process. The corresponding work input is higher in a power-absorbing machine as
compared with that in an ideal process. In order to understand how this happens, consider the
Second Law equation of state,
vdp)dhTds (i.e.,dpvdhdsT o0000 −=−=
Eq. 3.12.11
When applied to stagnation properties. Hence,
dsTdpvδw 0000 +=−
Eq. 3.12.12
In a power-generating machine, dpo is negative since the flowing fluid undergoes a pressure drop
when mechanical energy output is obtained. However, the 2nd law requires that Todso ≥ δq, but as
δq = 0, then To dso ≥ 0. The sign of equality applies only to a reversible process which has a work
output w = – vodpo > 0. In a real machine, Todso > 0, and represents the decrease in work output due
to the irreversibility in the machine.
40
4 Viscous Flow
4.1 Qualitative Aspects of Viscous Flow
Viscous flow could be defined as a flow where the effects of viscous dissipation, thermal conductivity,
and mass diffusion are important and could not be ignored
38
. All are consequence of assuming a
viscous surface where the effects of friction, creating shear stress, on the surface are pronounced.
There are number of interesting and important conditions associated with viscous effect that should
be analyzed separately. In general, two regions to
consider, even the divisions between not very sharp:
➢ A very thin layer in the intermediate
neighborhood of the body, δ, in which the
velocity gradient normal to the wall, ∂u/∂y,
is very large (Boundary Layer). In this
region the very small viscosity of μ of the
fluid exerts an essential influence in so far as
the shearing stress τ = μ (∂u/∂y) may
assume large value.
➢ In the remaining region no such a large
velocity gradient occurs and the influence of
viscosity is unimportant. In this region the
flow is frictionless and potential.
The general form on boundary layer equations,
shown in Figure 4.1.1, and their characteristic will
be discussed later.
4.1.1 Drag Definition and its Types
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another
type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving
with respect to a surrounding fluid
39
. Drag force is proportional to the velocity for low-speed flow
and the squared velocity for high speed flow, where the distinction between low and high speed is
measured by the Reynolds number. Even though the ultimate cause of a drag is viscous friction, the
turbulent drag is independent of viscosity
40
. Drag forces always tend to decrease fluid velocity
relative to the solid object in the fluid's path. Types of drag are generally divided into the following
categories:
➢ form drag or pressure drag due to the size and shape of a body (Dp)
➢ skin friction drag or viscous drag due to the friction between the fluid and a surface which
may be the outside of an object or inside such as the bore of a pipe (Df)
The effect of streamlining on the relative proportions of skin friction and form drag is shown for two
different body sections, an airfoil, which is a streamlined body, and a cylinder, which is a bluff body.
(see Figure 4.1.3). For aircraft, pressure and friction drag are included in the definition of parasitic
drag.
➢ lift-induced drag appears with wings or a lifting body in aviation and with semi-planning
38
White, Frank M. 1974: Viscous Fluid Flow, McGraw-Hill Inc.
39
Wikipedia
40
G. Falkovich (2011). Fluid Mechanics (A short course for physicists). Cambridge University Press. ISBN 978-
1-107-00575-4.
Figure 4.1.1 Boundary Layer Flow along a
Wall
41
or planning for watercraft
➢ wave drag (aerodynamics) is caused by the presence of shockwaves and first appears at
subsonic aircraft speeds when local flow velocities become supersonic.
4.1.2 No-Slip Wall Condition
Due to influence of friction, the velocity approaches zero on the surface and this is dominant factor
in viscous flows which could easily be observed. Or more precisely:
Eq. 4.1.1
4.1.3 Flow Separation
Another contribution due to friction and shear
stress is the effects of flow separation or adverse
pressure gradient. If the flow over a body is
turbulent, it is less likely to separate from the
body surface, and if flow separation does occur,
the separated region will be smaller (see Figure
4.1.2). As a result, the pressure drag due to flow
separation Dp will be smaller for turbulent flow.
This discussion points out one of the great
compromises in aerodynamics. For the flow
over a body, is laminar or turbulent flow preferable? There is no pat answer; it depends on the shape
of the body. In general, if the body is slender, as sketched in Figure 4.1.3(a), the friction drag Df is
much greater than Dp. For this case, because Df is smaller for laminar than for turbulent flow, laminar
flow is desirable for slender bodies. In contrast, if the body is blunt, as sketched in Figure 4.1.3(b),
Dp is much greater than Df . For this case, because Dp is smaller for turbulent than for laminar flow,
turbulent flow is desirable for blunt bodies.
The above comments are not all-inclusive; they simply state general trends, and for any given body,
the aerodynamic virtues of
laminar versus turbulent
flow must always be
assessed. Although, from
the above discussion,
laminar flow is preferable
for some cases, and
turbulent flow for other
cases, in reality we have
little control over what
actually happens. Nature
makes the ultimate
decision as to whether a
flow when left to itself, will
always move toward its
state of maximum
disorder. To bring order to the system, we generally have to exert some work on the system or
expend energy in some manner. (This analogy can be carried over to daily life; a room will soon
become cluttered and disordered unless we exert some effort to keep it clean.) Since turbulent flow
is much more “disordered” than laminar flow, nature will always favor the occurrence of turbulent
flow. Indeed, in the vast majority of practical aerodynamic problems, turbulent flow is usually
Figure 4.1.3 Drag on Slender & Blunt Bodies
Figure 4.1.2 Airflow Separating from a Wing at a
High Angle of Attack
42
present
41
.
4.1.3.1 Supersonic Laminar Flow
In recent years, there was a tendency to achieve laminar flow in supersonic speeds, without violating
Figure 4.1.4. The goal was to investigate active and passive Laminar Flow Control (LFC) as a
potential technology for a future High Speed Civil Transport (HSCT)
42
. The computation of
boundary-layer properties and laminar-to-turbulent transition location is a complex problem
generally not undertaken in the context of
aircraft design
43
. Yet this is just what must
be done if an aircraft designer is to exploit
the advantages of laminar flow while
making the proper trade-offs between
inviscid drag, structural weight and skin
friction. Potential benefits of laminar flow
over an aircraft’s wings include increased
range, improved fuel economy, and
reduced aircraft weight. These benefits
add up to improved economic conditions,
while also reducing the impact of exhaust
emissions in the upper atmosphere where
a supersonic transport would normally
operate.
Laminar conditions are hard to achieve
and maintain. There are two basic
techniques to achieve laminar conditions:
passive (without mechanical devices), and
active (using suction devices). Passive
laminar flow can be achieved in the wing
design process, but the laminar condition is normally very small in relation to the wing’s cord and is
usually confined to the leading edge region.
Passive laminar flow can also be created on an existing wing by altering the cross-sectional contour
of the lifting surface to change the pressure gradient. Both of these laminar conditions are called
natural laminar flow. Active control LFC must be used to achieve laminar flow across larger distances
from the leading edge. The main means of achieving active LFC is to remove a portion of the turbulent
boundary layer with a suction mechanism that uses porous material, slots in the wing, or tiny
perforations in the wing skin. Figure 4.1.4 displays the active mode of LFC with a suction system
beneath the wing’s surface was used to achieve laminar flow over 46 percent of the glove’s surface
while flying at a speed of Mach 2 in a successful demonstration of laminar flow at supersonic speeds.
Other methods include the boundary-layer analyses which are computationally inexpensive, as well
as, sufficiently accurate to provide guidance for advanced design studies. The boundary-layer solver
could be based on an enhanced quasi-3D sweep/taper theory which is revealed to agree well with
3D Navier-Stokes results
44
. The transition calculation scheme is implemented within the boundary-
layer solver and automatically triggers a turbulence model at the predicted transition front.
transition for a supersonic flight test.
41
John D. Anderson, Jr., “Fundamentals of Aerodynamics”, 5th Edition, McGraw-Hill Companies, 2007.
42
NASA TF-2004-12 DFRC.
43
Peter Sturdza, “An Aerodynamic Design Method For Supersonic Natural Laminar Flow Aircraft”, a dissertation
submitted to the department of aeronautics and astronautics and the committee on graduate studies of
Stanford university, December 2003.
44
See Previous.
Figure 4.1.4 The Porous Titanium LFC Glove is Clearly
seen on the Left Wing of test aircraft - NASA Photo
EC96-43548-7
43
4.1.4 Skin Friction and Skin Friction Coefficient
When the boundary layer equations are integrated, the velocity distribution can be deduced, and
point of separation can be determined. This in turn, permits us to calculate the viscous drag (skin
friction) around a surface by a simple
process of integrating the shearing stress at
the wall and viscous drag for a 2D flow
becomes:
Eq. 4.1.2
Where b denotes the height of cylindrical
body, φ is the angle between tangent to the
surface and the free-stream velocity U∞, and
s is the coordinate measured along the surface, as shown in Figure 4.1.5. The dimensionless friction
coefficient Cf, is commonly referred to the free-stream dynamic pressure as:
Eq. 4.1.3
4.1.4.1 Case Study - Image-Based Modelling of the Skin-Friction Coefficient
We develop a model of the skin-friction coefficient based on scalar images in the compressible,
spatially evolving boundary-layer transition [Zheng et al.]
45
. The multi-scale and multi-directional
geometric analysis is applied to characterize the averaged inclination angle of spatially evolving
filtered component fields at different scales ranging from a boundary-layer thickness to several
viscous length scales. The prediction of the skin-friction coefficient Cf in compressible boundary
layers is critically important for the design of high-speed vehicles and propulsion systems. The
boundary-layer transition has a strong influence on aerodynamic drag and heating, because much
higher friction and heating can be generated on the surface of aerospace vehicles in turbulent flows
than those in laminar flows. Despite considerable efforts in theoretical, experimental and numerical
studies, the reliable prediction of the skin friction coefficient in compressible boundary layers is still
very challenging [Zhong & Wang].
The theoretical study of the empirical formula of Cf in compressible boundary layers, in general, is
restricted to the laminar or fully developed turbulent state. The empirical formulae of Cf for
compressible laminar and turbulent boundary layers are transformed from their counterparts in
incompressible boundary layers.
45
Wenjie Zheng1, Shanxin Ruan, Yue Yang, Lin He and Shiyi Chen, “Image-based modelling of the skin-friction
coefficient in compressible boundary-layer transition”, J. Fluid Mech. (2019), vol. 875, pp. 1175_1203.
Figure 4.1.5 Illustrating the calculation of Skin
Friction
44
In general, the averaged inclination angles increase along the streamwise direction, and the variation
of the angles for large-scale structures is smaller than that for small-scale structures. Inspired by the
coincidence of the increasing averaged inclination angle and the rise of the skin-friction coefficient,
The evolutionary geometry of coherent structures with different scales and inclination angles in the
laminar–turbulent transition is sketched in Figure 4.1.6, along with the rise of Cf . The superposition
of hierarchies of attached and inclined vortical structures is also suggested by the models (e.g. Perry
& Chong 1982; Marusic & Monty) based on the attached eddy hypothesis (Townsend).
4.1.4.2 Inclined Structures and Drag Production
The relatively accurate prediction of Cf in the image-based model indicates that the generation of
inclined small-scale flow structures is closely related to the drag production. As sketched in Figure
4.1.7, one possible reason is that the lifts of material surfaces during the transition, which are good
surrogates of vortex surfaces
consisting of vortex lines, can
generate strong inclined shear
layers [Zhao et al.] to increase Cf .
Assume the flow field is filled with
wall-parallel material surfaces in
the laminar state with all the
surface normal nϕ=ϕ/|ϕ|
pointing to the wall-normal
direction.
In the transitional region, the near-
wall material surfaces are lifted due
to the growing streamwise
vorticity. This elevation event is
quantified by the wall-normal
Lagrangian displacement where Y
is the wall-normal location of a
fluid particle on a material surface.
The displacement ΔY quantifies the scalar transport in the wall-normal direction within a time
Figure 4.1.6 Evolutionary geometry of vortical or scalar structures, sketched by the ellipses with
different scales and inclination angles, in the boundary-layer transition, along with the rise of the skin-
friction coefficient Cf
Figure 4.1.7 Diagram of the geometry of material surfaces and
typical vortex lines near the surfaces, along with the rise of cf .
Solid lines denote vortex lines, and solid vectors n_ denote the
normal of material surfaces.
45
interval of interest, and [Zhao et al.] define the Lagrangian events ‘elevation’ with ΔY > 0 and ‘descent’
with 1Y < 0. The contour of ΔY and the contour lines of high shear ∂u/∂y in the transitional region
are shown in Figure 4.1.8.
In general, the inclined high shear layers cover the region with ΔY > 0, which is similar to the
observation in an incompressible temporal transitional channel flow in [Zhao et al.], because the
strong shear layer can be generated between the elevated low-speed fluid and the surrounding high-
speed fluid. Furthermore, the region with ΔY > 0, which also corresponds to the inclined scalar
structure with nϕ, deviates from the wall-normal direction as sketched in Figure 4.1.7, which can be
characterized as a finite < α > in the multi-directional analysis. Thus the inclined high shear layers
accelerate the momentum transport and produce the large Reynolds shear stress [Zhao et al.], which
can increase Cf implied by the relation between the Reynolds shear stress and Cf [Fukagata et al.;
Gomez, Flutet & Sagaut].
4.1.4.3 Reference
• Zhong, X. & Wang, X. 2012 Direct numerical simulation on the receptivity, instability, and
transition of hypersonic boundary layers. Annual Rev. Fluid Mech. 44, 527–561.
• Perry, A. E. & Chong, M. S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173–
217.
• Marusic, I. & Monty, J. P. 2019 Attached eddy model of wall turbulence. Annual Rev. Fluid Mech.
51, 49–74.
• Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, 2nd ed. Cambridge University
Press.
• Zhao, Y., Xiong, S., Yang, Y. & Chen, S. 2018 Sinuous distortion of vortex surfaces in the lateral
growth of turbulent spots. Phys. Rev. Fluids 3, 074701.
• Zhao, Y., Yang, Y. & Chen, S. 2016 Evolution of material surfaces in the temporal transition in
channel flow. J. Fluid Mech. 793, 840–876.
• Fukagata, K., Iwamoto, K. & Kasagi, N. 2002 Contribution of Reynolds stress distribution to the
skin friction in wall-bounded flows. Phys. Fluids 14, L73–76.
• Gomez, T., Flutet, V. & Sagaut, P. 2009 Contribution of Reynolds stress distribution to the skin
friction in compressible turbulent channel flows. Phys. Rev. E 79, 035301.
Figure 4.1.8 The contour of the Lagrangian wall-normal displacement ΔY and contour lines for the
strong shear layers on the x–y plane in the transitional region at M∞ = 6.
46
4.1.5 Aerodynamic Heating
Another overall physical aspect of viscous flow is the influence of thermal conduction. On a fluid over
a surface, the moving fluid elements have certain amount of kinetic energy. As the flow velocity
decreases under influence of friction, the kinetic energy decreases
46
. This lost kinetic energy
reappears in the form on internal energy of the fluid, hence, causing temperature to rise. This
phenomenon is called viscous dissipation within the fluid. This temperature gradient between fluid
and surface would cause the transfer of heat from fluid to surface. This is called Aerodynamic
Heating of a body. Aerodynamic heating becomes more severe as the flow velocity increase, because
more kinetic energy is dissipated by friction, and hence, the temperature gradient increases. In fact
it is one of the dominant aspects of hypersonic flows. The block diagram of Figure 4.1.10,
summarizes these finding for viscous
flow.
4.1.6 Reynolds Number
The Reynolds number is a measure of
ratio of inertia forces to viscous forces,
ν
UL
μ
ρUL
Re ==
Eq. 4.1.4
Where U and L are local velocity and
characteristic length. This is a very
important scaling tool for fluid flow
equations as to be seen later.
Additionally, it could be represents
using dynamic viscosity ν = μ/ρ. This
46
Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc.
Figure 4.1.10 Quantitate Aspects of Viscous Flow
Reynolds Number (Re)
0.05 10.0 200.0 3000.0
Figure 4.1.9 Effects of Reynolds Number in Inertia vs
Viscosity
47
is a really is measure or scaling of inertia vs viscous forces as shown in Figure 4.1.9 and has great
importance in Fluid Mechanics. It can be used to evaluate whether viscous or inviscid equations are
appropriate to the problem. The Reynolds Number is also valuable tool and guide to the in a
particular flow situation, and for the scaling of similar but different-sized flow situations, such as
between an aircraft model in a wind tunnel and the full size version
47
.
4.1.7 Reynolds Number Effects in Reduced Model
The kinematic similarity between full scale and scaled tests has to be maintained for reduced model
testing (wind-tunnels). In order to maintain this kinematic similarity, all forces determining a flow
field must be the same for both cases. For incompressible flow, only the forces from inertia and
friction need to be considered (i.e., Reynold Number). Two flow fields are kinematically similar if the
following condition is met
ν
LU
ν
LU
2
22
1
11 =
Eq. 4.1.5
To recognize Reynolds number effects a dependency test should be done
48
. Results from such a
dependency study are presented in Figure 4.1.11. At high Reynolds numbers, the drag coefficient is
almost constant, and the values for the full scale vehicle are slightly lower than those for the scaled
model. Below a certain Reynolds number, however, the drag coefficient from the scaled test
noticeably deviates from the full scale results. That is due to the fact, that in this range, individual
components of the car go through their critical Reynolds number. Violating Reynolds’ law of
similarity can cause considerable error. On the other hand, for small scales, sometimes it is hard to
maintain the same Reynolds number. That is for two main reasons. Wind tunnels have limited top
speed. At the same time, increasing speed in model testing also has its limits in another perspective.
47
From Wikipedia, the free encyclopedia.
48
Bc. Lukáš Fryšták, “Formula SAE Aerodynamic Optimization”, Master's Thesis, BRNO 2016.
Figure 4.1.11 Drag Coefficient versus Reynolds Number for a 1:5 Model and a Car (Courtesy of 35)
48
4.1.8 Case Study 1 - Scaling and Skin Friction Estimation in Flight using Reynold Number
Now that we familiar ourselves with some concepts if viscous flow, such as Reynolds Number,
separation, boundary layer and skin friction, it is time to see their effects in real life situation. The
purpose here is to conduct a brief review of skin-friction estimation over a range of Reynolds
numbers, as this is one of the key parameters in performance estimation and Reynolds number
scaling. These are among the most important in Aerodynamic performance. The flow around modern
aircraft can be highly sensitive to Reynolds number and its effects when they move significantly the
design of an aircraft as derived from sub-scale wind tunnel testing as investigated by [Crook ]
49
. For
a transport aircraft, the wing is the component most sensitive to Reynolds number change. Figure
4.1.12 shows the flow typically responsible for such sensitivity, which includes boundary layer
transition, shock/boundary layer interaction and trailing-edge boundary layer.
4.1.8.1 Interaction Between Shock Wave and Boundary Layer
The nature of the interaction between a shock wave and an attached boundary layer depends largely
upon whether the boundary layer is laminar or turbulent at the foot of the shock. For a laminar
boundary layer, separation of the boundary layer will occur for a relatively weak shock and upstream
of the freestream position of the shock. The majority of the pressure rise in this type of shock
/boundary layer interaction, generally described as a ¸ shock, occurs in the rear leg. The interaction
of the rear leg with the separated boundary layer causes a fan of expansion waves that tend to turn
the flow toward the wall, and hence re-attach the separated boundary layer. This is in contrast to the
interaction between a turbulent boundary layer and a shock wave, in which the majority of the
pressure rise occurs in the front leg of the shock wave. The expansion fan that causes reattachment
of the laminar separated boundary layer is therefore not present, and the turbulent boundary layer
has little tendency to re-attach. Here lies the problem of predicting the flight performance of an
aircraft when the methods used to design the aircraft have historically relied upon wind tunnels
operating below flight Reynolds number, together with other tools such as (CFD), empirical and semi-
empirical methods and previous experience of similar design aircraft. Industrial wind tunnels can
only achieve a maximum chord Reynolds number of between 3 x 106 < Rec <16 x 106, compared with
a typical value of 45 x 106 for cruise conditions. Therefore historically, results from wind tunnels have
to be extrapolated to flight conditions in a process known as Reynolds Number Scaling. Wind
tunnel models are generally supported free flying. As flow around them is constrained by the tunnel
walls, and therefore support and wall interference must be accounted for correctly. The freestream
flow may also have a different turbulent length scale, turbulence intensity and spectrum to that
occurring in the atmosphere. Other effects which can be wrongly interpreted as Reynolds number
49
A. Crook, “Skin-friction estimation at high Reynolds numbers and Reynolds-number effects for transport
aircraft”, Center for Turbulence Research Annual Research Briefs, 2002.
49
effects include the tunnel calibration, buoyancy effects, thermal equilibrium and humidity, as
discussed by [Haines]
50
4.1.8.2 Reynolds Number Scaling
Rendering to [Haines & Elsenaar]
51
, there are two types of scale effect: Direct and Indirect, which is
based upon the definition by Hall
52
of scale effects being the complex of interactions between the
boundary layer development and the external inviscid flow. Direct and Indirect Reynolds number
effects are represented schematically in Figure 4.1.13 and defined as:
1. Direct Reynolds Number effects occur as a consequence of a change in the boundary layer
development for a fixed (frozen) pressure distribution. Examples of direct effects range from
the well-known variation of skin friction with Reynolds number for a given transition
position to complex issues such as changes in the length of a shock-induced separation bubble
for a given pressure rise through a shock.
2. Indirect Reynolds number properties are associated with changes in the pressure
distribution arising from changes with Reynolds number in the boundary layer and wake
development. An example of an indirect effect is when changes in the boundary layer
displacement thickness with Reynolds number lead to changes in the development of
supercritical flow, and hence in shock position and shock strength. Therefore, a change in
50
Haines, A. B., “Scale Effects On Aircraft and Weapon Aerodynamics”, AGARDograph AG-323, 1994.
51
Haines, A. B. & Elsenaar, A., “An outline of the methodology Boundary layer simulation and control in wind
tunnels”, AGARD Advisory Report AR-224, 96-110-1988.
52
Hall, M. G., “Scale Effects In Flows Over Swept Wings”, AGARD CP 83-71, 1971.
Figure 4.1.12 Flow features sensitive to Reynolds number for a cruise condition on a wing section
50
wave drag with Reynolds
number at a given CL or
incidence, can appear as an
indirect Reynolds number
effect
53
.
4.1.8.3 Discrepancy in Flight
Performance and Wind
Tunnel Testing
[Haines]
54
provides a historical
review of scale effects and gives
examples of aircraft where direct
properties dominated the wing
flow, and indirect effects were
probably small. The examples given
are those of the VC-10 and X-1
aircraft, and correlation between
wing pressure distributions in the
wind tunnel and in flight are good. It
is observed that the shock position
in flight is slightly aft of that found
in the tunnel test for these test
conditions, when the flow is
attached, with little or no trailing
edge separation, and is turbulent.
The reason for this behavior in
53
A. Crook, “Skin-friction estimation at high Reynolds numbers and Reynolds-number effects for transport
aircraft”, Center for Turbulence Research Annual Research Briefs, 2002.
54
Haines, A. B., “Lanchester Memorial Lecture: Scale Effect in Transonic Flow”, Aeronautical J., August-September
1987, 291-313, 1987.
Figure 4.1.13 Schematic representation of direct and indirect
Reynolds number effects
Figure 4.1.14 Comparison of C-141 Wing Pressure Distributions Between Wind Tunnel and Flight
51
these two cases is the thinning of the boundary layer with increasing Reynolds number, with the
displacement thickness being roughly proportional to Re-1/5. The effective thickness of the wing
therefore decreases and the effective camber increases with increasing Reynolds number. The shock
wave will move downstream with reduced viscous effects until the limiting case of inviscid flow is
reached. If however, CL is kept constant for a given Mach number, and the Reynolds number varied,
the increased aft loading must be compensated by a decrease in the load over the front of the airfoil.
This is generally accomplished by a decrease in the angle of incidence, which normally results in the
forward movement of the shock wave. The final outcome of these opposing efforts will depend upon
their relative strength. When the flow is attached or mostly attached, indirect Reynolds-number
effects appear to be small. However, when the flow is separated large variations in the pressure
distribution can result with varying Reynolds number i.e. indirect effects can be large as
demonstrated in Figure 4.1.14 where the comparison of C-141 wing pressure distributions
between wind tunnel and flight for regions of subcritical (a) and Supercritical flow (b) is made. Aside
from the separation that can occur due to an adverse pressure gradient at the trailing edge, shock-
boundary layer interaction is one of the primary causes of separation in transonic flight.
4.1.8.4 Flow Separation Type (A - B)
Following the work of [Pearcey]
55
such flow separations are classed as either type A or B.
[Elsenaar]
56
describes the differences between type A and type B separation, and states that the final
state is the same both, namely a boundary-layer separation from the shock to the trailing edge.
However, the mechanism by which this final state is achieved, differs for the two. For a type A
separation, the bubble that forms underneath the foot of the shock grows until it reaches the trailing-
edge. The type B separation has three variants, with the common feature being a trailing edge
separation that is present before the final state is reached. The final state is reached when the
separation bubble and Trailing-edge separation merge. The type B separation is considered to be
more sensitive to Reynolds number than type A. This is partly because the trailing-edge separation
is dependent upon the boundary layer parameters such as its thickness and displacement thickness.
Furthermore, it was shown by [Pearcey & Holder]
57
that the supersonic tongue that exists in a shock-
boundary interaction is the dominant factor in the development of the separation bubble, and that
the incoming boundary layer is less important. Moreover, the local shock Mach number that causes
shock-induced separation is a weak function of the freestream Mach number. Relevant to wind
tunnel-to-flight scaling is the possibility that at sufficiently high Reynolds numbers, the trailing edge
separation will disappear and the type B flow that is observed in wind tunnels becomes a type A
separation at flight conditions.
The behavior of the trailing-edge separation and that of the separation bubble are highly coupled,
with the trailing-edge separation amplified by the upstream effects of the shock-boundary layer
interaction. The trailing-edge separation will modify the pressure distribution in a Reynolds-
number-dependent manner, and this in turn will alter the shock strength and the conditions for
separation at the foot of the shock. This will then affect the boundary layer at the trailing edge. The
sensitivity to Reynolds number of this interaction process will be dependent upon the pressure
distribution and hence the type of airfoil. It is also argued that most pre-1960 airfoils show a rapid
increase in shock strength with increasing Mach number and angle of incidence. By implication
viscous effects would be small, and the dominant effect would be lengthening of the shock-induced
separation bubble. By contrast, modern supercritical airfoils are designed to limit the variation in
55
Pearcey, H. H., Osborne, J. & Haines, A. B.,” The interaction between local effects at the shock and rear separation
- a source of significant scale effects in wind tunnel tests on airfoils and wings”, AGARD CP 35, 1968.
56
Elsenaar, A. Introduction. Elsenaar, A., Binion, T. W. & Stanewsky, E.,”Reynolds number effects in transonic
flow”, AG-303, 1-6, 1988.
57
Pearcey, H. H. & Holder, D. W., “Examples of shock induced boundary layer separation in Transonic flight”,
Aeronautical Research Council Technical Report R & M No. 3012, 1954.
52
shock-wave strength and have higher aft loading and hence greater pressure gradients over the rear
of the airfoil. Viscous effects will therefore be more important for these airfoils and there
performance more sensitive to Reynolds number.
4.1.8.5 Over-Sensitive Prediction in Flight Performance
As demonstrated by Figure 4.1.14, estimation of aircraft performance and characteristics based
upon data from wind-tunnel tests at low Reynolds number can lead to flight performance that is
worse than that predicted. In the case of the C-141, the wing pressure distribution in flight shows
that the shock is further aft than predicted by the wind tunnel tests. This increased aft loading meant
that the pitch characteristics of the wing were very different in flight to that predicted and this
necessitated a complete re-design of the wing. There are many examples of where flight performance
is worse than predicted using wind tunnel tests at lower Reynolds numbers. Examples include higher
than expected interference drag of the F-111 airframe, the lack of performance benefit for the DC-10
and recently the wing- drop phenomenon of the F/A-18E/F Super Hornet. The flight performance
need not be worse than predicted from wind tunnel data. The fact that the flight performance is better
than predicted means that the design point was calculated incorrectly and raises the possibility that
the design is overly conservative. The financial incentives for designing and predicting the flight than
predicted using wind tunnel tests at lower Reynolds numbers. Examples include higher than
performance of an aircraft at high Reynolds numbers are large. [Mack & McMasters]
58
reported that
a 1% reduction in drag equates to several million dollars in savings per year for a typical aircraft.
[Bocci]
59
examined what performance might be lost by designing an airfoil at a typical test Reynolds
number of 6 x 106 instead of a typical full-scale Reynolds number of 35 x 106. The results were gained
by calculating the 2D transonic flow over an airfoil section, and it was found that:
• The CL for the section designed (using CFD) to operate at Re = 6 x 106, but simulated at Rec =
35x106 is 4% higher for the same Mach number and shock strength on the upper surface.
• For the airfoil section designed (using CFD) for a Reynolds number of 35x106, the
improvement in CL is 13% over the section designed and simulated at a Rec = 6 x 106.
The accurate prediction of flight performance would also save time in the development process by
reducing the number of wind-tunnel hours, flight-test hours and design iterations. The use of CFD
has helped reduce the upward trend in the number of wind-tunnel hours required to develop an
aircraft, although approximately 20,000 wind tunnel hours were still required to develop the Boeing
777-200. Differences between predicted and flight performance have led to many different methods
of simulating the flight Reynolds number flow using low Reynolds number testing facilities. In flight,
transition normally occurs near the leading edge of the wing, and the boundary layer interacting with
the shock wave is therefore turbulent. In wind tunnels, it is possible for the boundary layer to remain
laminar over a large percentage of the chord, and therefore a laminar boundary layer-shock
interaction may occur. These two types of interaction are vastly different in their nature, and
therefore the flow is generally tripped.
4.1.8.6 Aerodynamic Prediction
The current status of Reynolds-number scaling can be assessed from a number of recent publications.
The full details are too long to discuss in this brief, but an attempt at a summary is provided herein.
• Angle of incidence at cruise, drag-rise Mach number, CL and CM are all functions of Reynolds
Number.
58
Mack, M. D. & McMasters, J. H.,” High Reynolds number testing in support of transport airplane development”,
AIAA Paper 92-3982.
59
Bocci, A. J., “Airfoil design for full scale Reynolds number”, ARA Memo 211, 1979.
53
• The effect of Reynolds Number on drag can be predicted if the empirical relationship is
matched to drag measured at a Reynolds number of 8-10 M or above.
• The shape of drag polar varies with Reynolds number up to flight Reynolds numbers of
approximately 40 million, although vortex generators reduce the variation slightly.
• Drag-rise Mach number is increased with increasing Reynolds number, indicating that higher
Reynolds number testing would predict a higher cruise Mach number than that achieved
using a tunnel such as the Boeing Transonic Wind Tunnel (BTWT).
• The effect of vortex generators on drag at cruise varies with Reynolds number, causing a
higher drag at low Reynolds numbers and having very little or a slightly beneficial effect at
flight Reynolds numbers. Vortex generators also have little effect on span wise loading at
flight Reynolds numbers, compared with a large effect at low Reynolds numbers. This
indicates that if wing loads were developed from low Reynolds number data, an unnecessary
structural weight penalty would be paid.
• Buffet onset is very difficult to predict, and is often difficult to measure in a wind tunnel
because the model dynamics and that of the aircraft are very different.
As Reynolds number scaling remains a topic that receives a great deal of attention 50 years after such
effects were first observed. The advent of high Reynolds number tunnels such as the NTF and ETW
has not lessened the need for good Reynolds number scaling techniques, but has provided the
facilities in which to test new methods and aircraft designs before their first flight, helping to reduce
risk. Comparison of flight data with that taken in such tunnels is good for cruise conditions. However,
buffet onset is still very difficult to predict, due primarily to the fact that the wind tunnel model and
support dynamics are very different to the real aircraft.
4.1.8.7 Skin Friction Estimation
Drag estimation is an important part of the design process, and involves the prediction of wave drag,
vortex-induced drag and viscous drag, with the latter contributing approximately 50% to the total
drag during cruise [Thibert]
60
. A simple estimate of the scaled viscous drag is often gained by using
a combination of formula and flat plate skin friction formulae once the transition location is known.
This method relies upon an accurate description of the skin friction coefficient, Cf from low Reynolds
numbers found in wind tunnels to flight Reynolds numbers. The accurate prediction of drag at flight
Reynolds number using low Reynolds number wind tunnels remains a challenge, and it appears that
a Re = 8 -10 x 106 or above is required if empirical methods are to be used for extrapolation to flight
conditions. The error in the extrapolation is likely to be higher than the variation of Cf with Reynolds
number predicted by the best empirical methods discussed. It is therefore concluded that the
measurements of skin friction taken in the NTF over a very large range of Reynolds number match
the predictions of [Spalding]
61
and [Karman-Schoenherr]
62
well enough for skin friction
extrapolation purposes.
The direct and accurate measurement of skin friction however remains very challenging, although
micro fabricated skin friction devices are promising. The relationships of Spalding and Karman-
Schoenherr34 are used for comparison with the data taken in the National Transonic Facility (NTF) at
NASA Langley in 1996. Although a flat-plate experiment was originally proposed by [Saric &
Peterson]
63
, it posed too many problems in the high-dynamic, environment of the NTF. An
axisymmetric body, 17ft long, for which transverse-curvature effects are small (δ/R = 0.25) was
60
Thibert, J. J., Reneaux, J. & Schmitt, R. V., “ONERA activities on drag reduction”, Proceedings of the 17th
Congress of the ICAS. 1053-1059, 1990.
61
Spalding, D. B.,”A new analytical expression for the drag of a °at plate valid for both turbulent and laminar
regimes”, Int. Journal Heat and Mass Transf. 5, 1133-1138, 1962.
62
Schoenherr, K. E.,”Resistance of flat surfaces moving through a fluid Trans”, SNAME. 40, 279-313, 1932.
63
Saric, W. S. & Peterson, J. B., Jr.,” Design of high Reynolds number flat plate experiments in the NTF”, AIAA, 1984.
54
therefore tested at Mach numbers between 0.2 and 0.85 and unit 6x106 < Re < 94 x106 per foot. Skin
friction was measured using three different techniques: a skin friction balance, Preston tubes and
velocity profiles from which the skin friction was inferred by the Clauser method. The last method
relies upon the validity of the logarithmic law and the constants used, which have been a subject of
debate over the last decade, and one that is still not settled. [Hites et al.]
64
compared the skin friction
velocity uτ measured by a near-wall hot wire, a micro fabricated hot wire on the wall, and a
conventional hot wire on the wall to that obtained by measuring the velocity profile using a hot wire
and applying the Clauser technique. In all cases, the measured uτ is higher than that predicted by the
Clauser technique. The prediction of uτ is also sensitive to the values of log-law. The comparison of
the measured values of uτ to that predicted by the Clauser method should however be treated with
care as significant errors can occur, even for micro fabricated devices, due to thermal conduction to
the substrate and connecting wires. More recently, Watson
65
carried out a comparison of the semi-
empirical relationships of [Ludwieg & Tillmann]
66
, [Spalding]
67
, [Schoenherr]
68
and [Fernholz]
69
. The
methods of Karman-Schoenherr and Spalding show opposite trends at low and high Reynolds
numbers with the inter section point at 6000 < Reϴ < 7000. The relationship of [Fernholz]41
consistently under-predicts the skin friction compared to the other methods. The skin friction
predicted by [Ludwieg-Tillmann]38 matches that of Karman-Schoenherr for 3000 <Reϴ < 20000. Both
the methods of [Spalding and Fernholz]41 rely upon the logarithmic law and hence the von Karman
constant κ and the additive constant, B. Watson report that the method of Spalding incorrectly
predicts the skin friction if the usual value of κ is used. This is because the relationship relies upon
Spalding's sub layer-buffer-log profile which does not take the wake region into account correctly.
Despite this, the relationships of [Karman-Schoenherr]40 and [Spalding]39 are observed to be the best
fit to the data of [Coles]
70
and [Gaudet]
71
shown in Figure 4.1.15 (courtesy of Watson et al.).
4.1.9 Case Study 2 - Reynolds Number Effects Compared To Semi-Empirical Methods
In order to estimate Reynolds number effects on a transonic transport aircraft CFD, calculations have
been performed on investigation by [Pettersson and Rizzi]
72
. The CFD calculations have been done
solving the RANS equations on an unstructured grid for varying Reynolds number at transonic
conditions. Low Reynolds number data have been extrapolated to a higher Reynolds number
condition with different scaling methodologies in order to evaluate each methods strengths and
weaknesses.
4.1.9.1 Scaling Effects Due to Reynolds Number
According to [Barlow]
73
, time is seldom available to back up and correlate free flight data with wind
tunnel data. If this is done the results are generally considered company proprietary because of the
64
Hites, M., Nagib, H. & Wark, C.,” Velocity and wall shear stress measurements in high Reynolds number turbulent
boundary layers”, AIAA Paper 97-1873, 1997.
65
Watson, R. D., Hall, R. M. & Anders, J. B., ”Review of skin friction measurements including recent high Reynolds
number results from NASA Langley NTF”, AIAA Paper 2000-2392, 2000.
66
Ludwieg, H. & Tilmann, W., “Investigations of the wall shearing stress in turbulent boundary layers”, NACA TM-
1285. National Advisory Committee for Aeronautics, 1950.
67
See 48.
68
Schoenherr, K. E., “Resistance of °at surfaces moving through a fluid”, Trans. SNAME. 40, 279-13, 1932.
69
Fernholz, H. H., Ein halbempirisches Gesetz fÄur die Wandreibung in kompressiblen turbulenten
renzschichten bei isothermer and Adiabater Wand. ZAMM. 51, 149-149-1971.
70
Coles, D.,”The turbulent boundary layer in a compressible fluid”, R-403-PR, Rand Corp, 1962.
71
Gaudet, L.,”Experimental investigation of the turbulent boundary layer at high Reynolds Number and a Mach
number of 0.8”, TR 84094, Royal Aircraft Establishment, 1984.
72
Karl Pettersson and Arthur Rizzi, “Reynolds Number Effects Identified with CFD Methods Compared to Semi-
Empirical Methods”, 25th International Congress of the Aeronautical Sciences.
73
Barlow J. B, Rae W. H, and Pope A. “Low-Speed Wind Tunnel Testing”. 3rd edition, Wiley- Inter science, 1999
55
high cost associated with obtaining them. Some of the differences between wind tunnel estimates
and free flight performance are due to scale effects. Some of these scale effects might be due to
differences in Reynolds Number between the wind tunnel and the free flight condition. One of the
objectives has been to evaluate the Reynolds Number scaling effects and to do so CFD calculations
have been performed. When comparing wind tunnel results with CFD or free flight results, some
scaling effects could be anticipated. Some of these effects are; having various gaps in the wind tunnel
model which are not present in the CFD model, wall blockage effects and wind tunnel mounting
effects, also called geometrical differences. Changing Reynolds number by increasing pressure might
influence the aero-elastic effects of the wind tunnel model. The wind tunnel model might have a
different stiffness from the free flight aircraft and it differs for sure when comparing it to the infinitely
stiff CFD model. Wind tunnel turbulence might also change with varying Reynolds number and effects
like these which initially may appear to be due to changes in Reynolds number might actually depend
on some other wind tunnel variable changing as well. There might actually be one or more variables
that might change with varying Reynolds number and these effects, called pseudo-Reynolds effects,
could easily be interpreted as Reynolds number effects. Artificial dissipation or turbulence modeling
could be categorized as a pseudo-Reynolds number effect when dealing with CFD. Before conclusions
can be drawn about the turbulence model used or the results obtained one has to assure that the
influence of mesh- and residual dependence and round off and truncation errors have been
minimized. Some of the wind tunnel pseudo-Reynolds number effects, like those associated with wall
interference and tunnel calibration, which may vary in a non-linear way with Reynolds number, could
however easily be excluded in the CFD calculations and an investigation of the true Reynolds number
effects minimizing pseudo-Reynolds number effects could be done.
Figure 4.1.15 Flat plate Skin Friction Correlations Comparison
56
4.1.9.2 Direct and Indirect Reynolds Number
Reynolds number effects in large can be divided into direct and indirect effects. The direct effects
are the ones that occur when the pressure distribution is frozen for varying Reynolds number e.g.
skin friction on a flat plate with zero pressure gradient. The indirect effects are Reynolds number
effects that involves a change in pressure distribution for varying Reynolds number. The order of
magnitude of the indirect Reynolds number effects might have an impact on the feasibility of scaling
methods which incorporates skin friction estimates based on flat plate flow to account for Reynolds
number variation. One could always argue whether or not to trust absolute numbers from CFD
calculations when having results is to see [Hemsch]
74
amongst others. The results in CD between
partners could differ with as much as 40 drag counts (where one count is 1/10000) for calculations
with the same mesh but with different solvers and turbulence models. The outlook compared to the
first drag prediction workshop however is brighter. The scatter in computed CD for partners in the
second workshop has been decreased with almost a factor of three compared to the first workshop
and the mean of the CFD results differed with approximately three drag counts compared to wind
tunnel data. Scaling methodology has according to [Bushnell]
75
improved compared to earlier
experience due to the increased use of CFD and the availability of new high Reynolds number
transonic facilities.
4.1.9.3 CFD Calculations
The following sections describes the CFD code, mesh generation and CFD calculations. The first part
describes the CFD code and turbulence modeling used. The second part describes how a mesh can be
produced with different programs using the best features of each program in order to produce a high
quality mesh and the last part describes mesh and residual dependence. In order to evaluate
Reynolds number scaling effects three different Reynolds numbers (20, 38 and 56 M) have been
evaluated. The free stream Mach number has been held constant at 0.85 at an angle of attack of 0
degrees.
4.1.9.4 Description of the CFD Code
The CFD code is an unstructured, edge-based, finite volume code developed by the Swedish Defense
Research Agency (FOI). When the time independent problem is solved, a local time stepping
approach is used with an explicit three-stage Runge-Kutta scheme. The spatial discretization utilizes
a second order central difference scheme with artificial dissipation. The turbulent quantities are
discretized with a second order up-wind scheme. In order to speed up convergence four multigrid
levels with implicit residual smoothing have been used. Turbulence has been modeled with a κ-ω
model, coupled with an explicit algebraic Reynolds stress model. It is possible to assign regions with
laminar flow but these calculations have been done using a fully turbulent flow. The laminar regions
of the flow on the aircraft are assumed to be small at these Reynolds numbers (20, 38 and 56 M) and
the assumption of fully turbulent flow were judged to be reasonable. The far-field boundary condition
for turbulence intensity was set to 0.1%.
4.1.9.5 Mesh Generation
74
Hemsch M. J and Morrison J. H. “Statistical analysis of CFD solutions “, from 2nd drag prediction workshop.
Proc AIAAd-2004-0556, 2004.
75
Bushnell D. M. Scaling: “Wind tunnel to flight”. Annual Review of Fluid Mechanics, Vol. 38, pp 111–128, 2006.
57
Generating a mesh can be tedious and time consuming work when the underlying CAD geometry is
corrupt. The most unstructured mesh programs will have problems with bad CAD geometries, such
as overlapping patches and holes. A structured Mesher has the advantage of additional user defined
information such as curve to edge and surface
to face association (where curves and surfaces
belongs to the CAD topology and edges and
faces to the mesh topology). This extra
information usually makes the Mesher using
the commercial Mesher ICEM CFD. The
unstructured surface mesh was first
generated mesh from ICEM as a “background”
grid, a starting point in the Advancing Front
Technique. In addition, with the “patch
independent” triangle Mesher for the fuselage,
wing and Vertical Tail Plane (VTP). The
Horizontal Tail Plane (HTP) was meshed with
a structured surface mesh in order to get
quads with the appropriate spacing and
stretching in flow dependent directions. This
surface mesh was converted into triangles and
merged together with the rest of the mesh
using the program TGRID, For the Reynolds
number 38 M case, a first cell height of 2×10−5
meters was chosen. The nodes in the wall
normal direction were distributed with an
exponential growth function. A total prism
height of 0.3 meters was assigned in order to
capture the thick boundary layer at the rear
end of the fuselage. The use of 40 prism layers
resulted in a ratio of spacing of 1.23. A cut of
the resulting volume mesh is shown in
Figure 4.1.16.
4.1.9.6 Residual & Mesh Dependence
To assure a grid and residual independent solution, the Reynolds number 20 M case was first
computed and investigated. The baseline mesh consisted of 3.4 M tetrahedral and 15.9 M pentas (5
sided). For the baseline mesh the change in CD is less than one drag count when comparing the results
for 10-5 and 10-6 for scaled density residual. The converged base line mesh was then adapted (h-
adapted, i.e. dividing cells) with respect to density, velocity and pressure gradients which resulted in
a mesh consisting of 4.7 M tetrahedral and 16.6 M pentas. A change in CD which was less than one
drag count was observed when comparing the results for 10-5 and 10-6 for scaled density residual.
The converged base line mesh and the converged adapted mesh differed with approximately three
drag counts when comparing results with a scaled density residual of 10-6. A summary of the mesh
and residual dependence is given in Table 4.1.1 where the difference in drag are measured in drag
counts and relative to the baseline mesh with a residual of 10-5. A multigrid approach was used
Figure 4.1.17. Mesh and residual dependence on CD in drag counts relative to the baseline mesh
with a residual of 10-5. Figure 4.1.17 shows a typical convergence history. In this case four multigrid
levels have been used with the adapted mesh at a Reynolds number of 20 M. Each tick on the y axis
in the force graph represent one drag count and the x axis starts and stops at positions corresponding
Figure 4.1.16 Cut of the Volume Mesh along the
Sweep
of the wing (Courtesy of Pettersson and Rizzi)
Mesh
Residual
10-5
Residual
10 -6
Baseline
0
0
Adapted
-3
-3
Table 4.1.1 Mesh and Residual Dependence on CD
in Drag counts relative to the baseline mesh with a
Residual of -5.5 - (Courtesy of Pettersson and Rizzi)
58
to scaled density residuals of 10-5.and 10-6 respectively.
The curve has flattened out for the last 200 iterations
and there is a relatively small change in CD at the end of
the calculations. The relatively small change in CD when
comparing residuals of 10-5 and 10-6 concluded that the
10-6 residual convergence criteria would be enough to
ensure a residual independent solution. In order to
resolve the boundary layer accurately a y+ less than
four and at least five to ten mesh points between the
wall and a y+ of 20 are required. An inspection of y+
was done for all Reynolds number cases. The typical
value of y+ was in the range of 0.5 and 2 for all wall
boundary conditions and Reynolds numbers. There
was typically seven mesh points for a y+ less than or
equal to 20.
4.1.9.7 Results and Discussion
After the investigation of mesh and residual dependence and an inspection of y+, the Reynolds
number 20, 38 and 56 M cases were computed. The free stream Mach number was held constant at
0.85 with an angle of attack of 0 degrees. The pressure distribution and Skin Friction was examined
for the wing for varying Reynolds number. The pressure coefficient, Cp, is plotted in Figure 4.1.18
together. The range of the of Cp contours are plotted for varying Reynolds number. No large change
in pressure distribution is seen. There are some influences in the pressure distribution on the wing
tip and leading edge area of the wing, but these effects are relatively small and are assumed to have
a small influence on drag due to pressure. Variation in Reynolds number is however assumed to have
an influence on skin friction. In order to estimate changes in the flow topology, the stream lines of
skin friction has been visualized in [Pettersson & Rizzi]
76
. The surface has been plotted with the x
component of skin friction and has an upper limit of zero in order to easily establish where reversed
flow exists (flow which has a negative component of skin friction in x direction). The stream lines of
skin friction over the wing has been visualized “seeding” the streams with the same points for all
Reynolds number. This was done in order to ease the inspection of the flow topology and be able to
compare the same set of stream lines for varying Reynolds number.
76
Karl Pettersson and Arthur Rizzi, “Reynolds Number Effects Identified With CFD Methods Compared to Semi-
Empirical Methods”, 25th International Congress of The Aeronautical Sciences.
Figure 4.1.17 Convergence history for the
adapted mesh at Reynolds number 20 M -
(Courtesy of Pettersson and Rizzi)
Figure 4.1.18 Wing Colored by Cp Contours - (Courtesy of Pettersson and Rizzi)
(a) Iso-lines of Cp
at Reynolds 20 M
(b) Iso-lines of Cp
at Reynolds 38 M
(c) Iso-lines of Cp
at Reynolds 56 M
59
A relatively small change in flow topology is seen over the Reynolds number range. It is also noted
that the flow stays attached over the wing for varying Reynolds number. One region where the flow
over the fuselage is detached is seen in [Pettersson & Rizzi]
77
. The x component of skin friction is
visualized together with stream lines of skin friction. The color range of CF varies between −10−6 and
10−6. This makes it easier to identify the region where the flow has a negative value of skin friction in
the x direction. The flow is reversed in the front part of the fuselage junction. The flow developed
along the fuselage is heavily dependent on Reynolds number and the length of the separation bubble
will hence probably be Reynolds number dependent. A visual inspection of the length of the
separation bubble were done for all Reynolds numbers. The length of the separation bubble
decreased with approximately 3% when comparing the 20 with the 38 M Reynolds number case.
When comparing the 38 to the 56 M Reynolds number case the length of the separation bubble
decreased once again with approximately 3%.
4.1.9.8 Reynolds Number Scaling
Scaling methodology is not synonymous to Reynolds number effects only, but could also include
effects of wind tunnel wall corrections for example as well. In this section, only Reynolds number
effects will be addressed, in order to give an overview of a few Reynolds number scaling approaches.
The Reynolds number effects identified with CFD methods will hopefully minimize the effects of
pseudo-Reynolds number effects. This will enable comparisons between corrected wind tunnel data
and CFD data in order to estimate free flight conditions. The drag force in general consists of one
normal component and one tangential to the surface. The tangential component is due to friction
forces while the normal is due to pressure forces. The pressure force is dependent on parameters
such as wall curvature and compressibility effects amongst others. This effect is hence dependent on
geometry and temperature. Wall shear stress or friction will also be dependent on temperature, since
viscosity varies with temperature, but it will also be dependent on whether the flow is laminar or
turbulent. A need for estimating the influence of transition position and changes in compressible
effects is now obvious in order to scale drag for varying Reynolds number. The problem at hand is to
predict the change in drag due to friction and pressure for varying Reynolds number in an accurate
way, incorporating effects of topological changes and pressure variations in the flow. For a flat plate
flow the change in CF for varying Reynolds number is relatively smooth and the pressure gradient is
unchanged and there is only direct Reynolds number effects present. This smooth effect and constant
pressure distribution might no longer exist when the flow passes a curved surface, for example an
airfoil.
One example of the topological change of the flow for varying Reynolds number is given by
[Schewe]
78
. When the Reynolds number is below a critical Reynolds number (sub-critical flow), the
incompressible flow is relatively uniform over the two dimensional wing. When the Reynolds
number is larger than the critical Reynolds (super-critical flow) number however, the flow changes
and the oil flow now shows an “owl eyes” topology instead. The effects encountered here, changes in
the appearance of the oil flow and transition position, are likely to have an impact on skin friction.
Another example, including changes in the compressible effects, for varying Reynolds number is also
given in order to illustrate some typical Reynolds number effects that could be encountered. The
effect of a thinner boundary layer for a higher Reynolds number implies a higher effective camber of
a wing and this might move potential shock waves in chord-wise direction. The change in position
and strength of the shock wave might separate the flow and these effects combined, could have a
large impact on the pressure distribution over the wing.
These Reynolds number effects implies a significant change in pressure distribution and these effects
are said to be; indirect Reynolds number effects. A classic example of this effect is the observation of
77
See Previous.
78
Schewe G. “Reynolds-number effects in flow around more-or-less bluff bodies”. Journal of Wind Engineering
and Industrial Aerodynamics, Vol. 89, pp 1267–1289, 2001.
60
movement of shock position comparing wind tunnel and free flight data for the Lockheed C-141
transport aircraft. This investigation, done in the 1960s, is an example of the drastic effects of
pressure distribution change due to a disproportionate boundary layer thickness. This lead to the
establishment of a boundary layer scaling criterion for airfoils with different pressure distributions
79
.
Estimating Reynolds number scaling effects in a wind tunnel might be conducted using either a
Reynolds number or a transition sweep. In the second case a simulation criterion has to be chosen in
order to give the same viscous flow behavior in the wind tunnel with its forced transition at low
Reynolds number as the free flight condition with its high Reynolds number and natural transition.
A suitable choice of simulation criterion, in order to achieve comparisons between free flight or CFD
results to wind tunnel results, has to be determined. The AGARD methodology proposes that CFD
calculations should be computed prior to the wind tunnel measurements in order to determine the
critical Reynolds number for a given simulation criterion. Suggestions for the simulation criterion
could be:
➢ A zero-level criterion such as the boundary layer momentum thickness at the trailing edge of
the equivalent flat plate, or
➢ A “first-order” criterion such as shock position, shock strength or the boundary layer
momentum or displacement thickness at the trailing edge of the real wing, or
➢ A “second-order” or local criterion such as the boundary layer shape factor near the trailing
edge of the non-dimensional length of a shock-induced separation bubble.
Typical appearance of the simulation
criterion as a function of Reynolds
number is shown in Figure 4.1.19.
When the critical Reynolds number has
been determined one could then scale
the wind tunnel data as:
➢ Follow the measured trend from
Re test to Re crit (extrapolate to Re
crit if Retest < Re crit )
➢ Move parallel to the computed
trend for Reynolds numbers
ranging from Re crit to Re free-flight.
A few different ways to perform the
Reynolds Number Scaling is presented
below.
4.1.9.9 Reynolds Number Scaling
using Semi Empirical Skin
Friction Methods
A fairly frequent approach of scaling aerodynamic data with varying Reynolds number is the
approach given by [Wahls]
80
amongst others. This approach implies that skin friction drag is
estimated with equivalent flat plate theory, plus form factors, using the (Blasius and Karman-
Shoenherr) incompressible skin friction correlations for laminar and turbulent boundary layers
79
Schewe G. “Reynolds-number effects in flow around more-or-less bluff bodies”. Journal of Wind Engineering
and Industrial Aerodynamics, Vol. 89, pp 1267–1289, 2001
80
Wahls R, Owens L, and Rivers S. “Reynolds number effects on a supersonic transport at transonic conditions”,
AIAA, 2001.
Figure 4.1.19 Simulation Criterion as a Function of
Reynolds Number for a Recrit at Reynolds Number 50 M -
(Courtesy of Pettersson and Rizzi)
61
respectively, with compressibility effects accounted for with the reference temperature method by
[Sommer and Short]
81
. How the shape factors could be constructed is given by [Covert]
82
or
[Paterson]
83
for example. Many companies do however have their own definitions of shape factors
and scaling methodologies, and this is part of their competitive edge. The extensive use of shape
factors or other semi empirical methods which heavily relies on data from previously designed
aircraft contradicts the idea that “understand the flow” is a better maxim than “use the numbers from
the last aircraft”. As a complement to shape factors based on previously designed aircraft and semi
empirical methods is the usage of CFD methods. When CFD is used the skin friction and pressure drag
is available for each component of the aircraft and an alternative way of constructing shape factors
or drag due to camber etc. is possible. The Eq. 4.1.6 has been modified from its original form in
[Paterson et al.]
84
where the free flight term on the right hand side has been replaced with CFD.
Eq. 4.1.6
When CFD calculations have been performed
the ratio of drag due to pressure and skin
friction is known. Using Eq. 4.1.6 gives the
opportunity to scale the isolated skin friction of
the corrected wind tunnel results to free flight
Reynolds number. The semi-empirical skin
friction method used here is the (Karman-
Schoenherr Equation) and compressibility
effects are corrected with the Sommer-Short
estimate using a recovery factor of 0.89. Figure
4.1.20 illustrates the (Karman-Schoenherr)
and Sommer-Short estimate has been anchored
to the CFD data. The shift of the skin friction
estimate was done by multiplication with a
constant factor. The Reynolds number which
the experimental data is anchored to, is typically
the highest available Reynolds number data. The factor which anchors the skin friction estimate to
the experimental data includes the effects of flow over curved surfaces and interference effects. Using
a constant factor to shift the skin friction estimate implies a constant influence of the effects due to
flow over curved surfaces and interference effects for varying Reynolds number. Whether this is true
or not needs to be determined. Here the data was anchored to the 38 M Reynolds number case in
order to compare the extrapolated results with the computed CFD data at a higher Reynolds number.
If the drag of the wings, for example, are assumed to have the same proportion to the total drag for
both wind tunnel and CFD results, skin friction could also be scaled part by part. The idea is that the
extra information of using the characteristic length of each part will make the scaling more accurate.
The shorter length scale of the VTP might not be the appropriate length scale for scaling the fuselage
81
Sommer S. C and Short B. J. “Free-flight measurements of turbulent-boundary-layer skin friction in the presence
of severe aerodynamic heating at Mach numbers from 2.8 to 7.0”. NACA-TN-3391, March 1955.
82
Covert E. E. “Thrust and drag: Its prediction and verification”, Progress in Astronautics and Aeronautics Series,
Vol. 98. AIAA, 1985.
83
Paterson J, MacWilkinson D, and Blackerby W. “A survey of drag prediction techniques applicable to subsonic
and transonic aircraft design”. AGARD-CP-124, pp 1–38, 1973.
84
Same as Previous.
Figure 4.1.20 Skin Friction Estimated with
Karman-Shoenherr and Sommer-Short Methods
anchored to CFD data at Reynolds Number 38 M -
(Courtesy of Pettersson and Rizzi)
62
which is much longer. Eq. 4.1.7 follows the idea of Eq. 4.1.6 and ratios between the corrected wind
tunnel data is assumed to correlate to the CFD data. If this would not be true, some discrepancies in
drag due to pressure or skin friction is present, or pseudo-Reynolds number effects has not been
accounted for in the corrected wind tunnel data.
Eq. 4.1.7
If Eq. 4.1.7 holds the results from the CFD calculations could be used to determine each components
drag due to skin friction. Each components skin friction could then be anchored to the highest
available Reynolds number data and an appropriate method of extrapolating skin friction with
Reynolds number could be used in order to estimate the free flight Reynolds number drag. In Figure
4.1.20 the Karman-Shoenherr skin friction estimate with the Sommer-Short method has been used
to scale each component of the aircraft separately.
The extrapolated drag of each component has
been anchored to its drag predicted by the CFD
calculation at Reynolds number 38 M. Skin friction
extrapolated compared to CFD. Skin friction
estimated with (Karman-Shoenherr and Sommer-
Short) methods for each part of the aircraft
separately, anchored to CFD data at Reynolds
Number 38 M. Combining the use of scaling the
whole aircraft or each part separately and scaling skin friction or total Drag with a semi empirical
method results in four different results. These four results are shown in Table 4.1.2 where the data
is given as the extrapolated data minus the CFD data at Reynolds number 56 M in Drag counts. The
data has been rounded towards the closest integer.
It is noticeable that scaling the aircraft skin friction part by part does not seem to imply an increased
accuracy of the extrapolated results at Reynolds number 56 M compared to scaling the whole aircraft
at once. The error in extrapolating data is however lower when the skin friction is extrapolated
compared to when the total drag is extrapolated. The drag due to pressure varies with Reynolds
number. Following the idea of [Reichenbach]
85
, where an aerodynamic quantity is fitted to an
interpolation curve, is utilized here for scaling drag due to pressure with varying Reynolds number.
Reichenbach fits CL to a function with Reynolds number as the variable and three constants. These
three constants are then determined by three measurement points; one from CFD and two from wind
tunnel tests. Here we fit drag due to pressure using the same type of function, see Eq. 4.1.7.
Eq. 4.1.8
The three Reynolds numbers 20, 38 and 56 M for the CFD results were used to determine the
constants. An extra calculation at Reynolds number 74 M were performed in order to evaluate the
numerical fit of the CFD data. The results of the fitted data and CFD calculations are shown in Figure
4.1.21. The difference at the Reynolds number 74 M case between the numerically fitted data and
the CFD calculation was approximately -0.02 drag counts. The root mean square (RMS) of the errors
for the numerical fit at the three lower Reynolds number compared to the CFD calculations was
1.9×10−9. The condition number, see Eq. 4.1.9, of the problem was 0.05.
85
Reichenbach S and McMasters J. “A Semi-empirical interpolation technique for predicting full scale flight
characteristics”. AIAA-87-0427, January 1987.
Forces
Skin
Friction
Total
Drag
Part by Part
2
-3
Aircraft
2
3
Table 4.1.2 Comparison of the Extrapolated
Data and CFD in Drag Counts at Reynolds
Number 56 M
63
Eq. 4.1.9
If Eq. 4.1.9 holds, Eq. 4.1.8 could be used to extrapolate wind tunnel drag due to pressure. The
change in drag due to assure for varying Reynolds number might give insight of when the critical
Reynolds Number has been reached and the only change in drag for an increase in Reynolds Number
is due to changes in skin friction. One alternative way of determining the critical Reynolds Number
could be when the drag due to pressure varies with one percent, comparing the critical and free
stream Reynolds number conditions. This parameter is just a proposal and it might have to be
corrected when other types of flow conditions are encountered. One proposed strategy of
extrapolating Reynolds number data is;
➢ Estimate from previous experience the critical Reynolds number.
➢ Perform CFD calculations for at least three Reynolds numbers, some above and some below
the estimated critical Reynolds number.
➢ Calculate the drag due to pressure and fit the results to Eq. 4.1.8.
➢ Determine the accuracy which might be expected in the CFD and wind tunnel measurements.
➢ Establish the critical Reynolds
number as when the change in drag
due to pressure is one percent of
free flight conditions.
➢ Examine trailing edge criterion
behavior at the trailing edges of the
wing, HTP, VTP and fuselage in
order to investigate additional local
Reynolds number dependence
(these local effects on drag might
cancel each other when global drag
is examined).
➢ Determine Recrit ε for both wind
tunnel and CFD.
➢ Investigate if there is any difference
between critical Reynolds number
from the CFD and wind tunnel
measurements.
➢ Visually inspect and determine
whether there are any changes in the topological structure of the oil flow in both CFD and
wind tunnel results or not. No large differences in oil flow topology should be visible when
comparing Reynolds number data which are super critical.
➢ Scale the wind tunnel data with CFD trends up to the critical Reynolds number and then
further on to free flight conditions.
If no clear critical Reynolds Number could be established this could be due to at least two factors; no
clear critical Reynolds Number exist for this specific type of flow, or the Reynolds Number to which
the extrapolation is done is below a critical Reynolds number. Regarding the last point of the
proposed strategy, the (Karman-Shoenherr and Sommer-Short) methods have been used to evaluate
the compressible skin friction variation and a numerical fit of the drag due to pressure in order to
scale the data from Reynolds number 38 M to free flight conditions. The accuracy of the CFD method
Figure 4.1.21 Numerical Fit of Drag Due to Pressure -
(Courtesy of Pettersson and Rizzi)
64
would not be a function of how much it would differ from the wind tunnel measurements, since this
would regard the wind tunnel results to be the correct answer in an absolute sense. The accuracy in
the CFD method would rather depend on the residual, mesh and turbulence modeling sensitivity.
This would typically be known from experience in using a specific CFD code and turbulence model.
It is not the absolute numbers that are important when determining the critical Reynolds number
but the trends; the change, in some parameter, between the corrected wind tunnel data and CFD
results will hopefully be constant for varying Reynolds number. The accuracy in the wind tunnel
would on the other hand depend on repeatability and accuracy in the measuring devices etc. The
extra effort in investigating critical Reynolds number, trailing edge criterion and oil flow topology
will hopefully increase the understanding of the specific flow case at hand, revealing potential
differences between CFD and wind tunnel measurements and improve scaling wind tunnel data to
free flight conditions.
4.1.9.10 Inspection of Local Boundary Layer Properties for Varying Reynolds Number
When comparisons between either free flight, CFD or wind tunnel measurements and Reynolds
number sensitivity are to be evaluated, some parameter of interest has to be chosen. When drag due
to Reynolds number variation is the subject of the investigation, the local parameter CF , seems to be
a natural candidate for investigation. An accurate and robust (if it will be done at free flight
conditions) way of measuring skin friction has to be determined. Measuring Skin Friction could be
done in either a direct or an indirect manner. The direct measurement of skin friction could be done
by using either small floating balances or through the use of surface-imaging interferometry. The
former has the disadvantage of being susceptible to damage and thermal shifts and the latter is
usually limited to laboratory settings. Preston tubes and boundary layer rakes are the two most
common tools to indirectly estimate skin friction on vehicles. In the work by [Whitmore et al.]
86
a
combination of boundary layer rakes and Preston tubes are used to evaluate viscous fore body drag.
A set of equations consisting of normalized boundary layer thickness, Reynolds number and Clauses
parameter are used in order to estimate the local skin friction, momentum thickness, displacement
thickness and boundary layer shape factor. These equations are solved in order to account for the
entire viscous fore body drag including; effects of skin friction, fore body separation and fore body
wakes and parasite drag due to protuberances. There exists a variety of empirical and semi-empirical
methods to predict skin friction for flat plate flow. Methods such as those based upon the 1/7th power
law and the logarithmic law relates CF to Rex and suffer from the difficulty of estimating the unknown
origin, according to [Crook]
87
. A variety of methods trying to compensate for the effect of an unknown
transition position and pressure gradients are based on integrated boundary layer properties, like
displacement and momentum thickness instead of the local Reynolds number. Two different
methods of determining skin friction have been chosen in order to evaluate their accuracy in
predicting trends of Reynolds number scale effects on local skin friction. One method from [White]
88
,
utilizes the local Reynolds number (Rex) and the correlation from [Watson]
89
as well as Reynolds
number based on local momentum thickness (Reθ).
86
Whitmore S. A, Hurtado M, and Naughton J. W. A real-time method for estimating viscous fore body drag
coefficients. Technical Report NASA/TM-2000-209015, NASA, January 2000.
87
Crook A. “Skin-friction estimation at high Reynolds numbers and Reynolds-number effects for transport
aircraft”. Annual Research Briefs, pp 427–438. Center for Turbulence Research, 2002.
88
White F.M. Viscous Fluid Flow. 2nd edition, Mechanical Engineering, McGraw-Hill, 1991.
89
Watson R, Hall R, and Anders J. “Review of skin friction measurements including recent high Reynolds number
results from NASA Langley NTF”, AIAA-2000-2392, 2000.
65
Eq. 4.1.10
The evaluation of the two methods have
been performed on the HTP at positions
corresponding to the dots in Figure 4.1.22.
The HTP has been divided in twelve parts in
total, three major parts: inboard, outboard
and middle part which in turn has been
divided into a front, an upper, a lower and a
trailing edge part. The major parts are
approximately 25, 65 and 10% of the total
wetted area of the HTP, for the inboard,
middle and outboard parts respectively.
The measuring positions in chord-wise
direction are located at 5 and 50% of the
local chord length and at half of the trailing
edge length in span wise direction for each
major part respectively. Rex and Reθ were
evaluated at 20 and 38 M Reynolds number.
The data were then linearly extrapolated to
the 56 M Reynolds number case and compared to CFD calculations.
Eq. 4.1.10 shows the variation of Reθ as function of free stream Reynolds number and the
extrapolated data for the middle top part of the HTP. A least square fit of the linear approximation
of Reθ resulted in a RMS of the error of 20 for the front position and 30 for the middle position. Note
that the order of magnitude of Reθ is of 104, so the fit of the linear extrapolation is judged as a very
good approximation. The Reθ and Rex from the 20 and 38 M cases using CFD calculations and the data
extrapolated to 56 M Reynolds number were used to calculate CF. The Skin Friction were then
anchored to the 38 M Reynolds number CFD case in order to evaluate each methods capability to
predict skin friction trends. A RMS of the error for the twelve extrapolated results of skin friction
compared to CFD were calculated. The method using Eq. 4.1.10 resulted in a RMS of the errors of
0.95 drag counts as well as in a RMS of the errors of 0.88 drag counts comparing extrapolated skin
friction with CFD calculations at Reynolds number 56 M. Note that if any of these points shown in
Figure 4.1.22 would have been chosen as trailing edge criterion, neither of them would have
revealed any major deviations with increasing Reynolds number which would have implied a critical
Reynolds number (compare to the kink in Figure 4.1.19) implying a typical flow topology change
or a shock wave moving from an up or down stream position).
4.1.9.11 Concluding Remarks
An aerodynamic assessment of a modern transonic transport aircraft has been evaluated with CFD
methods at Reynolds number 20, 38 and 56 M with a free stream Mach number of 0.85 at an angle of
attack of 0 degrees. Reynolds number scaling of global Drag and local Skin Friction has been evaluated.
Scaling the aircraft skin friction with the (Karman-Shoenherr) estimate and correcting compressible
effects with the Sommer-Short method was accurate within 2 drag counts when comparing scaled
results with CFD results at Reynolds number 56 M. Fitting the drag due to pressure at 20, 38 and 56
M Reynolds Number and comparing the fitted data with CFD data at 74 M Reynolds Number resulted
in a deviation of approximately 0.02 drag counts. An investigation of scaling local Skin Friction was
Figure 4.1.22 HTP seen from above, positions where
Skin Friction Evaluated Marked with Dots - (Courtesy
of Pettersson and Rizzi)
66
performed with two different methods in order to evaluate their capabilities of predicting changes in
local skin friction due to a Reynolds number change. Marginal improvements were shown when skin
friction were scaled with a method based on Reθ compared to a more classical estimate for turbulent
flat plate flow. Both methods predicted variations in local skin friction fairly well. Readers are
encouraged to consult [Pettersson and Rizzi]
90
for further information.
90
Karl Pettersson and Arthur Rizzi, “Reynolds Number Effects Identified with CFD Methods Compared to Semi-
Empirical Methods”, 25th International Congress of the Aeronautical Sciences.
67
5 Conservation Laws and Governing Equations
5.1 Control Volume Approach
We dwell on the numerical treatment of differential equations that govern the evolution of scalar
fluid properties
91
. The derivation of these equations is usually based on certain conservation
principles, as applied to an arbitrary control volume V ϲ Rd, where d = 1, 2 or 3 is the number of
space dimensions. If the fluid is in motion, it may flow in
and out across the control surface S which forms the
boundary of V, see Figure 5.1.1. Individual molecules may
travel across the interface even if the fluid is at rest.
Therefore, the physical and chemical properties of the
fluid inside V are influenced by those of the surrounding
medium. Some quantities, such as mass, momentum, and
energy, are conserved. That is, they may move from one
place to another but cannot emerge out of nothing or
disappear spontaneously
92
. The physical forces that
transport, produce or destroy these quantities are well-
known, and reliable mathematical models are available.
Thus, conservation principles can be expressed in terms of
differential equations that describe all relevant transport
mechanisms, such as convection (also called advection), diffusion, and dispersion. The three
basic equations considered for physical systems are:
• Conservation of mass (Continuity Equation)
• Conservation of momentum (Newton’s 2nd law)
• Conservation of energy (1st law of thermodynamics)
The six unknowns which must be solved are velocity V, density ρ, pressure p and temperature T.
These 5 equations with 6 unknowns, for dependent flow and transport properties the system could
be enclosed with the aids of equations of states,
Eq. 5.1.1
The 5 scalar equation, continuity, 3 momentum, energy, contain six unknowns, ρ, T, p, u, v, and w. To
enclose the system, one more equations are needed. That could be accomplished by using the ideal
gas law, described previously. Further, although the transport quantities of μ, k is constant for most
cases, but in reality there are functions of temperature, specifically in high speed flows. On those
cases, empirical relations such as Sutherland’s law could be used, relating them to local temperature.
For multi-component consideration, two extra basic relations is warranted as:
• Conservation of species
• Laws of chemical reaction
5.2 Integral Forms of Conservation Equations
The integral forms of mass conservation equation states that mass cannot be created nor it can
disappear, and is expressed as
91
Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc.
92
Kuzmin, Dmitri: “A Guide to Numerical Methods for Transport Equations”, 2010.
Figure 5.1.1 Control Volume Bond
Corresponding Control Surface S
68
Eq. 5.2.1
where V is the cell control volume, ρ is the flow density, is the velocity of the flow, n is the unit
normal vector to the cell surface, and S is the area of the face. The first term of Eq. 5.2.1 represents
the time rate of change of the total mass inside the cell. The second term represents the mass flow of
the fluid through the surface of the cell. The linear momentum conservation equation is expressed as
Eq. 5.2.2
where ğ is the body force per unit mass, p is pressure imposed by the fluid on the boundary, and τ
are the shear and normal stresses, resulting from the friction between the fluid and the boundary
surface. The first term in Eq. 5.2.2 represent the time variation of linear momentum. The second
term represents the convective momentum flux or the transfer of momentum across the boundary
of the control volume. The third term represents the volume (or body) forces. The last term
represents the surface forces. The energy conservation equation is expressed as
Eq. 5.2.3
where E is the total energy per unit of mass, q is the heat source, and ğ is the body force per unit mass.
The first term in Eq. 5.2.3 represent the time variation of the total energy of the fluid in the control
volume. The second term represents the convective energy flux or the transfer of energy across the
boundary of the control volume. The third term represents the heat sources and the work done by
the body forces. The fourth term is called the diffusive or dissipative flux, and it represents the
diffusion of heat due to molecular thermal conduction. The last two terms represent the rate of work
done by the pressure as well as the shear and normal stresses on the fluid element. The total energy
E per unit mass is expressed as
Eq. 5.2.4
where e is the internal energy, and u, v, and w are the x, y and z-components of the velocity vector V.
5.3 Mathematical Operators
Some of the most frequently used math operators in Cartesian coordinate, related to flow equation
are displayed below. For example, for velocity V = (u, v, w), and scalar φ = φ(x, y, z):
69
ωV
2
V
)V(V. (vector)identity Vorticity
VV.
t
V
Dt
DV
(scalar) derivetiveTotat
zyx
(scalar)Laplacian
z
w
y
v
x
uV. (scalar) div
y
u
x
v
,
x
w
z
u
,
z
v
y
w
ωV (vector) Curl
z
w
y
v
x
u
.V (scalar) Divergence
z
,
y
,
x
(vector)Gradient
2
2
2
2
2
2
2
2
−=
+
=
+
+
=
+
+
=
−
−
−
==
+
+
=
=
Eq. 5.3.1
5.4 Conservation of Mass (Continuity Equation)
The principle of conservation of mass states that mass cannot be created nor destroyed. Therefore, if
we consider a volume fixed in space, V , then the change of mass inside this volume can only take
place if mass flows in or out through the boundary of this volume. Since the volume is fixed in space
we can take the derivative inside the integral, and by applying the divergence theorem to the
boundary fluxes on the right hand side we have in initial notation:
Eq. 5.4.1
5.5 Centrifugal and Coriolis Forces
From elementary dynamics, the centrifugal force is an inertial force (also called a 'fictitious' or
'pseudo' force) directed away from the axis of rotation that appears to act on all objects when
viewed in a rotating reference frame (i.e., ω = Curl v). The Coriolis force is an inertial force (also called
a fictitious force) that acts on objects that are in motion relative to a rotating reference frame
93
. In
reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one
with anticlockwise rotation, the force acts to the right (see Figure 5.5.1)
93
David Apsley, “Fluid-Flow Equations”, spring 2017.
70
5.6 Conservation of Momentum (Newton 2nd Law)
94
The equation of motion is derived by assuming that Newton's 2nd law of motion is valid for any
arbitrary volume cut out of the fluid. Thus, the rate of change of momentum of a fixed volume is the
net momentum flux across the boundaries of the volume plus the net forces acting on the volume. It
gives: body surface
Dt
ρ
ρττμ
μδ
ρ
ρδτρ
Eq. 5.6.1
5.7 Conservation of Energy (1st Law of Thermodynamics)
The first law of thermodynamic states that the sum of work and heat added to the system will result
in increase of energy, (kinetic + internal + gravitational) = work done on system+ heat added, or
94
White, Frank M. 1974: Viscous Fluid Flow, McGraw-Hill Inc.
Figure 5.5.1 Centrifugal and Coriolis force
ω
71
x
u
τ
x
p
u
t
p
x
T
kTuρc
xt
T)c(
or
x
u
τT)k.(
Dt
Dp
Dt
DT
ρc T)(k
Dt
DQ
and g.r)V
2
1
ρ(eE re whe
Dt
DW
Dt
DQ
Dt
DE
or dW dQdE
j
i
ij
j
j
j
jp
j
p
j
i
ijp
2
t
t
t
+
+
=
−
+
++=→=
−+=+=+=
Eq. 5.7.1
5.8 Scalar Transport Equation
The scalar Transport or better known as convection–diffusion equation is a combination of the
diffusion and convection (advection) equations, and describes physical phenomena where particles,
energy, or other physical quantities are transferred inside a physical system due to two processes:
diffusion and convection. Depending on context, the same equation can be called the advection–
diffusion equation, drift–diffusion equation, or (generic) scalar transport equation. This is probably
one of the most important equations of fluid mechanics, which the term are:
• Transient Term – the energy accumulated.
• Convection Term – the flux of energy leaving Control Volume due to velocity U of fluid.
• Diffusion Term – describe the energy flux leaving the C.V. due to molecule diffusion. This
process the transport energy from high to low concentration is independent of velocity field U.
• Source Term – Describes the generation of energy in Control Volume.
Eq. 5.8.1
5.9 Vector Form of N-S Equations
It is customary to present all 5 conservation equations in the vector form for compactness. The
compressible N-S equations in Cartesian coordinates without body force or external heat addition,
in vector notation can be written as:
72
++
+=
−=
−=
−=
+
==
+
==
+
==
−
−
=
−
−
=
−
−
=
+−−−+
−+
−
−
=
+−−−+
−
−+
−
=
+−−−+
−+
−
−+
=
=
=
+
+
+
2
wvu
eρE and
z
T
kq ,
y
T
kq ,
x
T
kq
y
w
z
v
μτ τ,
z
u
x
w
μτ τ,
x
v
y
u
μτ τ
y
v
x
u
z
w
2μ
3
2
τ,
z
w
x
u
y
v
2μ
3
2
τ,
z
w
y
v
x
u
2μ
3
2
τ
where
qwτvτuτp)w(E
τpρw
τρvw
τρuw
ρw
qwτvτuτp)v(E
τρvw
τpρv
τρuv
ρv
,
qwτvτuτp)u(E
τpρuw
τρuv
τpρu
ρu
,
E
ρw
ρv
ρu
ρ
by given are and,,, where0
zyxt
222
tzyx
zyyzzxxzyxxy
zzyyxx
zzzyzxzt
zz
2
yz
xz
yyzyyxyt
yz
yy
2
xy
xxzxyxxt
xz
xy
xx
2
t
G , F
EU
GFEU
GFEU
Eq. 5.9.1
5.10 Orthogonal Curvilinear Coordinate
Up to this point, we have been using the Cartesian coordinates. The basis governing equations of
motion is been written in tensor form and valid in any coordinate system. A straight forward
procedure is using stretching factors hi to related new curvilinear coordinate to Cartesian system.
Let the general curvilinear coordinate system (x1, y1, z1) be related to a Cartesian system (x, y, z) so
that the element of arc length ds is given by
95
x
z
x
y
x
x
h
)dx(h)dx(h)dx(h(dz)(dy)(dx)(ds)
2
1
2
1
2
1
1
2
33
2
22
2
11
2222
+
+
=
++=++=
Eq. 5.10.1
With similar expressions for h2 and h3. The scale factor gives a measure of how a change in the
coordinate changes the position of a point. For a Gradient, Divergent, Curl, and Laplacian operators
96
:
95
Workbook Level 1, “28.3: Orthogonal Curvilinear Coordinates”, 2004.
96
Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
73
( ) ( ) ( )
xh
hh
xxh
hh
xxh
hh
xhhh
1
VhVhVh xxx
hhh
hhh
1
hhV
x
hhV
x
hhV
xhhh
1
xh
1
,
xh
1
,
xh
1
33
21
322
31
211
32
1321
2
332211
321
321
321
213
3
312
2
321
1321
332211
+
+
=
=
+
+
=
=
V
V
Eq. 5.10.2
For cylindrical coordinated (h1=1, h2=r, and h3=1) and V (Vr, Vϴ, and Vz), we obtain:
z
V
θ
V
r
1
r
V
r
rr
1
V ,
z
V
θr
V
r
VV.
z
V
θ
V
r
1
(rVr)
rr
1
.V ,
z
V
,
θ
V
r
1
,
r
V
V
2z
2
2θ
2
2
r
2
z
θ
r
z
θ
z
θ
r
+
+
=
+
+
=
+
+
=
=
Eq. 5.10.3
5.10.1 Cylindrical Coordinate for Governing Equation
These coordinates (r, Ɵ, z) are related to the
Cartesian system (x, y, z) by
(3.9)
x
y
tan
x
y
arctan θ , yxr
zz , θsin r y , θ cosr x
1-22 ==+=
→===
Eq. 5.10.4
where z is the axial. θ is tangential, and r is radial
directions, respectively, are from elementary
calculus and exposed in Figure 5.10.1. The
scalar equation of motion for incompressible
flow with constant transport properties and
velocity components (Vr, VƟ, and Vz.) are drive as:
Figure 5.10.1 Relation between Cartesian
and Cylindrical; coordinate
74
( ) ( ) ( )
( )
( )
( )
z
2
zz
z
2
θ
2
r
2
θ
2
θ
θr
θ
θ
θ
22
r
r
2
r
2
θr
r
zθr
V g
z
p
ρ
1
V V.
t
V
Momentumz
r
V
θ
V
r
2
V g
θ
p
ρr
1
r
VV
V V.
t
V
Momentumθ
θ
V
r
2
r
V
V g
r
p
ρ
1
V
r
1
V V.
t
V
Momentum-r
0V
z
V
θr
1
Vr
rr
1
Continuity
++
−=+
−
−
+++
=−+
−
−−++
=−+
=
+
+
Eq. 5.10.5
As an special version of cylindrical coordinate which previously used for turbomachinery
applications in cruse conditions, are unsteady 3D Euler equations with Ω express as rotational
speed of impeller defined by [Soulis]
97
. The energy equation and the spherical coordinates can be
found at
98
. A special case of cylindrical equations is the axisymmetric flow where all references to δƟ
vanishes. Axisymmetric flow defined as a flow in which the streamlines are symmetrically located
around an axis. Every longitudinal plane through the axis would exhibit the same streamline pattern.
Alternatively, flow pattern is said to be axisymmetric when it is identical in every plane that passes
through a certain straight-line. The straight-line in question is referred to as the symmetry axis.
5.11 Generalized Transformation to N-S Equation
These equations can be expressed in terms of a generalized non-orthogonal curvilinear coordinated
system ξηηζ
ξηζ
ξηζξηζ
ξηζ
ξηζ
ξηζζηξηζζηξηζζη
ηξζζξηξζζξηξζζξ
ζξηηξζξηηξζξηηξ
Eq. 5.11.1
The generalized transformation to the compressible N-S equations can be written in vector form as
99
97
Johannes Vassiliou Soulis,” An Euler Solver For Three-Dimensional Turbomachinery Flows”, International
Journal For Numerical Methods In Fluids, Vol. 20,L-30 (1995).
98
See the above.
99
Anderson, Dale A; Tannehill, John C; Plecher Richard H; 1984:”Computational Fluid Mechanics and Heat
Transfer”, Hemisphere Publishing Corporation.
75
Eq. 5.11.2
By no means is this form of representation is conclusive. Many texts and researchers are attain their
own representation as they see to be relevant. For example, [Liao &He]
100
produced following form
of N-S equations in curvilinear coordinates as
Eq. 5.11.3
But the concepts should be the same. This is the strong conversation form of governing equation
which is best suited for finite differencing schemes. It should be noted that the vectors E1, F1 and G1
contain partial derivatives of viscous and heat transfer terms which also has to be transformed.
Alternatively, the vectors E, F, and G can be split into an inviscid (i) and a viscous parts (v). Reason
for doing this becomes evident later, and also this make it more modular and easy to handle.
100
FeiLiao, Guowei He, “High-order adapter schemes for cell-centered finite difference method”, Journal of
Computational Physics403(2020)109090.
76
( )
) -k( q , ) -k( q , ) -k( q
)()(
)()(
)()(
)()()(2
3
2
)()()(2
3
2
)()()(2
3
2
, 2/)(E
where
0
,
p)w(E
pρw
ρvw
ρuw
ρw
0
p)v(E
ρvw
pρv
ρuv
ρv
qwτvτuτ
τ
τ
τ
0
,
p)u(E
pρuw
ρuv
pρu
ρu
,
E
ρw
ρv
ρu
ρ
0)(ζ)(ζ)(ζ
1
ζ
)(η)(η)(η
1
η
)(ξ)(ξ)(ξ
1
ξ
t
(3.13) 0
z
)-(
y
)-(
x
)-(
t
zyx
222
t
v
t
2
yx
v
t
2
xxzxyxx
xz
xy
xx
t
2
t
vizviyvix
vizviyvix
vizviyvix
vivivi
TTTTTTTTT
wwwvvv
wwwuuu
vvvuuu
vvvuuuwww
wwwuuuvvv
wwwvvvuuu
wvue
qwvu
qwvu
zzzyyyxxx
yyyzzzzyyz
yxxzzzzxxz
yxxyyyyxxy
yyyyxxzzzzz
zzzyxxyyyyy
zzzyyyxxxxx
zzzyzxz
zz
zy
zx
yyzyyxy
yz
yy
++=++=++=
+++++==
+++++==
+++++==
++−++−++=
++−++−++=
++−++−++=
+++=
−++
=
+
+
=
−++
=
+
+
=
−++
=
+
+
+
=
=
=
−+−+−
+
−+−+−
+
−+−+−
+
=
+
+
+
GG
F , F
EEU
GGFFEE
J
GGFFEE
J
GGFFEE
JJ
U
GGFFEE
U
i
i
vi
77
Eq. 5.11.4
Another portrayal of the N-S equations using the generalized curvilinear coordinates are given by
[Allahyari et al.]
101
.
5.12 Coupled and Uncoupled (Segregated) Flows
In some heat transfer problems, temperature difference are small compared with the absolute (and
pressure change are small compared with absolute pressure). Therefore, the changes in density,
viscosity and conductivity produced by temperature differences are small enough to be neglected in
the momentum and thermal energy equations, i.e., assuming constant. Even though we are trying to
calculate the thermal energy change, produced by the same temperature. This is the un-coupled flow
or constant property approximation and equations could be solved separated or segregated.
Where else, the heat transfer with large temperature difference, the temperature field equations
become non-linear and are coupled to the velocity field equation, and should be solved together,
because the viscosity (and in a gas the density) depended on temperature i.e.
Eq. 5.12.1
The issue will be visited later on as it plays a significant role in heat transfer, i.e., convection heat
transfer.
5.13 Simplification to N-S Equations (Parabolized)
Due to the fact that the Navier-Stokes equation are very difficult to solve in their complete form. In
general, a very large amount of computer resources is needed to obtain a solution. This is particularly
true for the compressible N-S equations which are a mixed set of elliptic-parabolic equations, while
the unsteady incompressible N-S equations are a mixed set of hyperbolic-parabolic equations. As a
consequence, different numerical techniques must be used to solve them. The time dependent
solution is normally used when a steady-state flow is computed. That is, the unsteady N-S solutions
are integrated in time until a steady-state solution is achieved. Thus, for a three-dimensional flow
field a four-dimensional (3 space+1 time) problem must be solved. Besides being very CPU intensive,
it needs a very large amount of storage. That is why whenever possible, the complete compressible
N-S equation should be avoided. Fortunately, for many of the viscous flow problems it is possible to
solve a reduced set of equations. These reduced equation belongs to a class set of which often refers
as Thin-Layer or Parabolized N-S equations. They are:
• Thin-Layer Naver-Stokes equation
• Parabolized Naver-Stokes equation
• Partially Parabolize Naver-Stokes equation
• Conical Naver-Stokes equation
• Viscous Shock Layer equation
These sets of equation applicable to both viscous and inviscid equation, and when appropriate, used
instead of full N-S equation. There are several advantage to use these. First of all, there are few terms
in the equations which leads to some reduction in the required computation time. Secondly, for
steady flow most of these equations are mixed set of hyperbolic-parabolic equations in stream wise
direction. In other word, the N-S equation are parabolized in the stream direction. Figure 5.13.1
shows the mathematical character of each equation. It depicts summary of some of the governing
equations and their character, as well as, dependencies. This is by no means a complete set as other
101
M. Allahyari, K. Yousefi, V. Esfahanian and M. Darzi, “A Block–Interface Approach for High–Order Finite–
Difference Simulations of Compressible Flows”, Journal of Applied Fluid Mechanics, Vol. 14, No. 2, 2021.
78
situations give rise to more or different versions, but encompasses the most important and
frequently used ones.
5.14 Non-dimensional Numbers in Fluid Dynamics
The governing fluid dynamic equation are often put into non-dimensional form. The advantage of
this is to normalized variables with their values falls between (0 and 1). Non-dimensional the flow-
field parameters in a CFD code removes the necessity of converting from one system to another
within the code
102
. In addition to reducing the number of parameters, non-dimensional equation
helps to gain a greater insight into the relative size of various terms present in the equation.
103
-
104
Following appropriate selecting of scales for the non-dimensional process, this leads to identification
of small terms in the equation. Neglecting the smaller terms against the bigger ones allows for the
simplification of the situation. For the case of flow without heat transfer, the non-dimensional
Navier–Stokes equation depend only on the Reynolds Number and hence all physical realizations of
the related experiment will have the same value of non-dimensional variables for the same Reynolds
Number
105
. Scaling helps provide better understanding of the physical situation, with the variation
in dimensions of the parameters involved in the equation. This allows for experiments to be
conducted on smaller scale prototypes provided that any physical effects which are not included in
the non-dimensional equation are unimportant. We noticed that non-dimensionlizing the variables
based on free-steam conditions (∞) does not change the equation character except for heat flux
102
“Non-dimensionalization “, CFL3D User’s Manual, pp.53-64, NASA.
103
Versteeg H.K, “An introduction to computational fluid dynamics: the finite volume method”, 2007.
104
Patankar Suhas V., “Numerical heat transfer and fluid flow”, 1980, Taylor & Francis, 9780891165224.
105
Salvi Rodolfo, “The Navier Stokes equation theory and numerical methods”, 2002.
Figure 5.13.1 Conditions and Mathematical Character of N-S and its variation
79
vector and shear stress tensor
106
. So far we discussed Reynold and Mach numbers. We attempt to
discussed Prandtl and Nusselt numbers which are very useful in heat transfer problem.
5.14.1 Prandtl Number
This is an index which is proportional to the ratio of energy dissipated by friction to the energy
transported by thermal conductivity:
Eq. 5.14.1
The Prandtl number (Pr) is the ratio of fluid’s ability to diffuse momentum, to its ability to diffuse
heat and imperative in heat transfer. Small values of the Prandtl number, Pr << 1, means the thermal
diffusivity dominates. Whereas with large values, Pr >> 1, the momentum diffusivity dominates the
behavior. For example, the listed value for liquid mercury indicates that the heat conduction is more
significant compared to convection, so thermal diffusivity is dominant. However, for engine oil,
convection is very effective in transferring energy from an area in comparison to pure conduction, so
momentum diffusivity is dominant. In heat transfer problems, the Prandtl number controls the
relative thickness of the momentum and thermal boundary layers. When Pr is small, it means that
the heat diffuses quickly compared to the velocity (momentum). This means that for liquid metals
the thickness of the thermal boundary layer is much bigger than the velocity boundary layer. The
mass transfer analog of the Prandtl number is the Schmidt number.
5.14.2 Nusselt Number
In heat transfer at a boundary (surface) within a fluid, the Nusselt number (Nu) is the ratio of
convective to conductive heat transfer across (normal to) the boundary. In this context, convection
includes both advection and diffusion. The conductive component is measured under the same
conditions as the heat convection but with a (hypothetically) stagnant (or motionless) fluid. A similar
non-dimensional parameter is Biot Number, with the difference that the thermal conductivity is of
the solid body and not the fluid. A Nusselt number close to one, namely convection and conduction
of similar magnitude, is characteristic of "slug flow" or laminar flow. A larger Nusselt number
corresponds to more active convection, with turbulent flow typically in the 100–1000 range. The
convection and conduction heat flows are parallel to each other and to the surface normal of the
boundary surface, and are all perpendicular to the mean fluid flow in the simple case.
Eq. 5.14.2
where h is the convective heat transfer coefficient of the flow, L is the characteristic length, k is the
thermal conductivity of the fluid.
5.14.3 Rayleigh Number
Rayleigh number, Ra, is a dimensionless term used in the calculation of natural convection. It can be
define as
106
D. Anderson, J., Tannehill, R., Pletcher, ”Computational Fluid Mechanics and Heat Transfer”, ISBN 0-89116-
471-5 – 1984.
80
Eq. 5.14.3
where g is acceleration due to gravity, β is coefficient of thermal expansion of the fluid, ΔT is
temperature difference, x is length, ν is kinematic viscosity and k is thermal diffusivity of the fluid. Gr
is the Grashof Number and Pr is the Prandtl Number. The magnitude of the Rayleigh number is a good
indication as to whether the natural convection boundary layer is laminar or turbulent. For a vertical
surface, for example, the transition takes place when Ra ≈ 109. Various examples of the relationship
between the magnitude of natural convective heat transfer and Rayleigh number are given in the
following reference
107
.
5.14.4 Other Dimensionless Number
Other dimensionless numbers which encounter more frequently in fluid dynamics, but not limited
to, are: Eckert - Euler - Froude - Grashof - Knudsen - Laplace - Lewis - Péclet - Richardson -
Schmidt – Stanton - Stokes - Strouhal - Weber. For more info about these number and more, reader
should refer to
108
.
5.14.5 Non-Dimensionalizing (Scaling) of Governing Equations
The equations of continuity, momentum and energy could be made dimensionless by dividing each
variable to appropriate constant reference property
109
. For example x = x/L where L is the
characteristic length. Similarly for v = v/v∞ and so on. For the momentum equation revised as,
Eq. 5.14.4
Where the gravity terms being ignored for high speed gas flows and the pronounce effects of Re
number is evident. For high speed flows (Re∞ >> 1), the convective terms on the L.H.S. becomes
dominant, where for creep flows (Re∞ << 1) the viscous dissipation becomes more important. For
incompressible flow as Re∞ → ∞, all the viscous terms eliminated and becomes in-viscid Euler
relation. Now considering the energy equation:
Eq. 5.14.5
With influence of Ec, Pr, and Re numbers evident and subscript ∞ denotes free-stream quantities.
The appearance of Ec number in non-dimensional energy equation indicates that the temperature
107
Thermopedia©.
108
Wikipedia.
109
White, Frank M. 1974: Viscous Fluid Flow, McGraw-Hill Inc.
81
variations are always important in high speed gas flows; and these equations are truly coupled, and
should be solved simultaneously on high speed flow simulations. While for high speed flows all three
scaling factors are important, in low speed flows, (M < 0.3), we can usually neglect both the pressure
and dissipation terms and leaving only the Pe=RePr coefficient as the dominant contributor. The
significant effect of this assumption, beside its simplification, decouples the energy equation from
momentum and continuity, as temperature is now confined to energy equation, and allows for
separate solution. It is also noted that the continuity equation remains the same and omitted from
non-dimensional analysis. First we considering incompressible equations which has countless
importance in fluid dynamic. Then the unsteady term which causes the mathematical character of
equations to change. Viscous vs Inviscid is another prominent feature. There are in fact countless
numerous way in which governing equations can be simplified, but the user should exercise great
caution to whether it could be applied to the problem in hand or not. Figure 5.14.2 shows some of
most important simplification to be applied.
Figure 5.14.2 Some Methods for Simplifying Governing Equations
Flowfield
Compressible
Vs
Incompressible
Viscous Vs
Inviscid
Steady Vs
Unsteady
2-D Vs 3-D
Order of
Magnitude
(OoM)
Analysis
2-D
Vorticity
function
Rotational
Vs
Irrotational
Coupled Vs
Segregated
ρ=ρ(p,T),
μ=μ(p,T),
k=k(p,T)
Body
Forces
82
5.15 Incompressible Navier-Stokes Equation
Almost all, if not all, fluid is compressible. Incompressible flow is an approximation of flow where
flow speed is insignificant everywhere compared to the speed of sound of the medium
110
. If
incompressible flow is defined in this way, the majority of the fluid and associated flow we encounter
in our daily lives belong is the incompressible category. For example. most of the flow associated with
air and water (such as flow related to automobiles, ships, submarines, the water supply through pipes
and channels, hydraulic turbines, pumps, lows peed airplanes, and trout swimming in mountain
streams) and flow of bio-fluid (such as blood) are all in the incompressible flow domain. One of the
earliest mathematical models of incompressible flow is the famous equation by Bernoulli, who in
1730 developed the model equation while investigating blood flow. It is not surprising that scientists
have been investigating incompressible flow analytically, experimentally, and computationally ever
since. Mathematically speaking the incompressible flow formulation poses unique issues not present
in compressible equations because of the incompressibility requirement. Physically, information
travels at infinite speed in an incompressible medium, which imposes stringent requirements on
computational algorithms for satisfying incompressibility as well as difficulties in designing
downstream boundary conditions. Differences in various methods of solving the incompressible flow
equations originate from differences in strategies of satisfying incompressibility. For example the
incompressibility could be assumed on most low speed flows in conjunction with constant
properties. For incompressible flow with constant properties, where full N-S equations could be
reduced as F = ma,
x
u
x
u
μ τ
x
u
τTk
Dt
DT
ρc :Energy
x
u
μpρg
Dt
Du
ρ :Momentum
0u :Continuity
i
j
j
i
ij
j
i
ij
2
v
j
i
2
i
i
+
=
+=
+−=
=
Eq. 5.15.1
These equations (1 vector, 2 scalar) are mixed set of elliptic-parabolic equations which contain the
unknowns (ui, p, T). Note that temperature only appears in Energy equation so we have essentially
uncoupled the Energy equation from the Continuity and Momentum equations.
110
Dochan Kwak, Cetin Kirk and Chang Sung Kim, “Computational Challenges of Viscous Incompressible Flows”,
NAS Applications Branch , Moffett Field, CA 94035.
83
One further note; although the incompressible flow is ideally suited for flows with M < 0.3, a
conservative finite-volume framework, for the simulation of flows at all speeds, applicable to
incompressible, ideal-gas and real-gas fluids is proposed by [Denner et al.]
111
.
5.16 Vorticity Consideration in Incompressible Flow
Taking curl of incompressible momentum equation with constant properties, the vorticity
potential equation is derived as
Eq. 5.16.1
Where the scalars such as pressure vanish. While vorticity is not a primary variable in flow analysis,
it is still extremely instructive to examine the impact of vorticity vector on N-S equation. The first
term on the right arises from the convective derivative and is called vortex stretching term. The
second term is obviously a viscous-diffusion term. In initial notation:
Eq. 5.16.2
The 1st term in the RHS represents evolution of vortex tubes, as illustrated in Figure 5.16.1 with
solid dots correspond to fluid elements. Due to the shear in the velocity field, the vortex tube is
stretched and tilted. However, as long as the fluid is inviscid and barotropic, incompressible or
111
Fabian Denner, Fabien Evrard, Berend G.M. van Wachem, “Conservative finite-volume framework and
pressure-based algorithm for flows of incompressible, ideal-gas and real-gas fluids at all speeds”, [physics.comp-
ph], Feb 2020.
Figure 5.16.1 Evolution of a Vortex Tube in pyroclastic flows
84
isobaric, Kelvin’s circularity theorem assures that the circularity is conserved with time. In addition,
since vorticity is divergence-free, the circularity along different cross sections of the same vortex-
tube is the same. Additionally, Vortex line is a line that points in the direction of the vortex vector.
Hence is vortex line is to w what a streamlines is to u. Note that a vortex line associated with a fluid
line is always perpendicular to the streamline associated with that fluid element. And Vortex tube is
a bundle of vortex lines (see Figure 5.16.1). The circularity of a curve C is proportional to the
number of vortex lines that thread the enclosed area. The 2nd term describes the diffusion pf vorticity
due to viscosity, and is obviously zero for an inviscid fluid (ν = 0). Typically, viscosity
generates/creates vorticity at a bounding surface: due to the no-slip boundary condition shear arises
giving rise to vorticity, which is subsequently diffused into the fluid by the viscosity. In the interior
of a fluid, no new vorticity is generated; rather, viscosity diffuses and dissipates vorticity. Further
information can be obtained from [white]
112
. For 2D , we have simply (ω.u=0) and Figure 5.16.1
becomes
Eq. 5.16.3
5.17 Euler Equation
Dropping the viscous terms and neglecting the body force and no external heat transfer from N-
S equations, we
2
wvu
eρE
where
p)w(E
pρw
ρvw
ρuw
ρw
p)v(E
ρvw
pρv
ρuv
ρv
,
p)u(E
pρuw
ρuv
pρu
ρu
,
E
ρw
ρv
ρu
ρ
by given are and,,, where0
zyxt
222
t
t
2
t
2
t
2
t
++
+=
+
+
=
+
+
=
+
+
+
=
=
=
+
+
+
G , FEU
GFEU
GFEU
Eq. 5.17.1
Further, using an initial notation and assuming steady flow and integrating the momentum along a
stream line:
112
Frank M. White , “Viscous Fluid Flow”, McGraw-Hill Book Company, 1974.
85
streamline a alongconstant ρV
2
1
p , wvuV re wheρVdVdp
constant) ( ibleIncompress
ρ
dp
2
V
p
ρ
1
-VV. p
Dt
DV
ρ
2222
2
==++=−=
→=→+→=−=
Eq. 5.17.2
To obtain the celebrated Bernoulli’s equation along a stream line for incompressible flow. For an
irrotational flow, the Bernoulli’s Equation holds everywhere. It emulates the relationship between
pressure and velocity on in-viscid flows and been used throughout gas dynamics. It reveals that an
increase in one quantity would cause decreasing other. For an isentropic, compressible flow, it can
be expressed as
constant
ρ
p
1-γ
γ
2
V
2=+
Eq. 5.17.3
Which is sometimes refers to as compressible Bernoulli’s equation. Similarly, for inviscid Energy
equation we obtain:
streamline a alongconstant
2
V
h 2=+
Eq. 5.17.4
5.17.1 Steady-Inviscid–Adiabatic Compressible Equations
The complete set known as Shock Relation and in being used for that purpose. The steady in-viscid
adiabatic compressible flow relations could be derived directly from continuity, momentum and
energy equations, and ignoring the body forces as
TCe ρRTp 0
2
V
h Energy
p)(ρ Momentum
0)(ρ Continuity
V
2===
+
−=
=
VV
V
Eq. 5.17.5
5.17.2 Velocity Potential Equation
Euler’s equation could be simplified by making the steady, irrotational, isentropic assumptions.
The three momentum equations now could be reduced to single velocity potential equation. The
velocity components have been replaced by
z
w,
y
v,
x
u
=
=
=→=
V
Eq. 5.17.6
On continuity equation after some manipulation via momentum equation, the velocity potential
Equation is obtained as,
86
sound of speeda , wvuV and V
2
1γ
aa ere wh
2
d
a
-dor
2
-dp :Energy &Momentum
0
a
2
a
2
a
2
a
1
a
1
a
-1 :Continuity
0
2222
0
222
2
222
yz
2
zy
xz
2zx
xy
2
yx
zz
2
2
z
yy
2
2
y
xx
2
2
x
=++=
−
−=
++
=
++
=
=−−−
−+
−+
zyxzyx
Eq. 5.17.7
Note that for an incompressible flow, a → ∞, the velocity potential reduces to linear Laplace’s
equation and could be solve with relative ease. It represents a combination of continuity, momentum
and energy equations and could be solved for velocities (i.e., Mach number) and then temperature,
pressure and density could be obtained using the isentropic relations, previously defined for local
values. It should be noted that the total quantities are known and obtained from free-stream
conditions. But Eq. 5.17.7 is till non-linear in nature for compressible applications. One technique
to overcome that is by using a velocity perturbation component in addition to uniform velocity V∞
Eq. 5.17.8
Using these transformations and some magnitude comparisons,
Eq. 5.17.9
This linear relation is obviously greatly influenced by M∞ and valid only on subsonic and supersonic
regions; but not hypersonic or transonic. The solution and its type is directly impacted by the value
of M∞ due to changing nature of governing equation, as discussed before. It is only valid for slender
or thin bodies at small angle of attacks where small perturbations are expected for velocity. The
imposed restrictions are consequence (cost) of assumptions
113
.
Figure 5.17.1 shows the conditions of inviscid flows and their mathematical characteristics. The
different numerical techniques must be used in to solve the equations depending mainly in
mathematical character of flow regions. For example, the Euler equations governing an in-viscid,
non-heat conducting gas have a different character in different flow regions. If the time dependent
terms are retained, the resulting unsteady equations are hyperbolic and solutions can be obtained
by marching procedures. The situation is different when a steady flow is assumed. In that case, the
Euler equations are elliptic when flow is subsonic and hyperbolic when it is supersonic. It could be
said that Euler’s equation is hyperbolic in temporal domain and elliptic in special domain.
Therefore, different flow regions means different characteristics and demands different solving
procedure. A major difference between subsonic and supersonic flows is that flow disturbances
propagate everywhere throughout a subsonic flow; whereas they cannot propagate upstream in
supersonic flow. Alternatively, Figure 5.17.2 also shows the same in different light. The hierarchy
of some well-established, Newtonian flow equations are depicted in Figure 5.17.3 with following
chart with some of corresponding conditions and assumptions outlined.
113
Anderson, John D. 1984: “Fundamentals of Aerodynamics”, McGraw Hills Inc.
87
Figure 5.17.1 Condition and Mathematical Character of Inviscid Flow
Figure 5.17.2 From N-S to Linearized Equation
Lineariezed Equations
Transonic Small Disturbulence Equations
Non-linear terms ignored
Potential Equations + Bernouli Law
Small Distribuence M → 1
Euler
Irrotational Isentropic
Navier-Stokes
Inviscid Adaiabatic
88
Figure 5.17.3 Hierarchy of Flow Equations
Full Navier Stokes (N-S)
Viscous
Parabolized and
variation N-S
Boundary Layer
Vorticity (Roataional and
Irrotational)
Inviscid
Euler Equation
Rrotational Irrotational
Velocity Potential
Small Disturbence Equation
Lapace Equation
Bernoulli Equation
Simple Linear Equation (Uniform flow,
Source/Sink, Doublet, etc.)
Incompressible Compressible
Steady Transient
Laminar Turbulent
89
90
6 Boundary Layer Theory
6.1 Definitions
The concept of the boundary layer was developed by [Prandtl] in 1904. It provides an important link
between ideal fluid flow and real-fluid flow. Fluids having relatively small viscosity , the effect of
internal friction in a fluid is appreciable only in a narrow region surrounding the fluid boundaries.
Since the fluid at the boundaries has zero velocity, there is a steep velocity gradient from the
boundary into the flow. This velocity gradient in a real fluid sets up shear forces near the boundary
that reduce the flow speed to that of the boundary. That fluid layer which has had its velocity affected
by the boundary shear is called the boundary layer. For smooth upstream boundaries the boundary
layer starts out as a laminar boundary layer in which the fluid particles move in smooth layers. As
the laminar boundary layer increases in thickness, it becomes unstable and finally transforms into a
turbulent boundary layer in which the fluid particles move in haphazard paths. When the boundary
layer has become turbulent, there is still a very than layer next to the boundary layer that has laminar
motion. It is called the laminar sublayer. Various definitions of boundary–layer thickness δ have
been suggested. The most basic definition refers to the displacement of the main flow due to slowing
down of particles in the boundary zone. (see Figure 6.1.1).
This thickness δ1 called the displacement thickness is expressed by
Eq. 6.1.1
The boundary layer thickness is defined also as that distance from the plate at which the fluid velocity
is within some arbitrary value of the upstream velocity. Typically, as indicated in (Figure 6.1.1-a),
d = y where u = 0.99U. Another boundary layer characteristic, called as the boundary layer
momentum thickness, Θ as
Eq. 6.1.2
All three boundary layer thickness definition δ , δ*1, Θ are use in boundary layer analysis.
Figure 6.1.1 The Development of the Boundary Layer for Flow Over a Flat Plate
91
6.2 Scaling Analysis for Boundary Layer Equation (2-D)
The first simplification is an order of magnitude analysis to derive the velocity and thermal boundary
layers
114
. We assume that the thickness of the viscous boundary (δ) and thermal boundary (δt) layers
are small relative to characteristic length (L) in the primary flow direction. Considering a steady 2D
boundary layer (ReL >>1) where there is the thin layer of flow adjacent to a surface where the flow is
retarded by influence of friction between a solid surface and fluid. (see Figure 6.1.1). Within this
thin layer, known as Boundary Layer thickness, δ, the flow variables are influenced mostly on the
normal direction to solid surface (ie., v << u , d/dx << d/dy , etc.). That is, δ/L <<1 and δt/L <<1 and
terms with same order can be ignored. An order of magnitude analysis on Eq. 2.3 for momentum
equation w.r.t. BL thickness δ, on a 2D incompressible Prandtl’s Boundary Layer results in
Eq. 6.2.1
Therefore, P = P(x) only. More importantly, this pressure is related to free stream velocity U∞ through
Bernoulli’s relation as
Eq. 6.2.2
As pressure is now readily available through free stream velocity. This parabolic equation is
relatively easier to solve than the full N-S provided that the domain of interest is close to viscous layer
(i.e., δ) close to surface.
6.3 3D Boundary Layer
The 3D compressible, unsteady, boundary layer, following the 2D assumption, it is possible to neglect
the derivatives with respect to the coordinated which are parallel to the wall. Regarding equation in
y-coordinates we again obtain the results that ∂p/∂y is very small and may be neglected. Thus, the
pressure is seen to depend on x and z alone, and is impressed on the boundary layer by the potential
114
Same as previous.
Figure 6.1.1 Definition of Boundary Layer Thickness: (a) Standard Boundary Layer (u = 99%U), (b)
Boundary Layer Displacement Thickness
92
flow. The chordwise acceleration of the potential
flow induces an inboard oriented crossflow
component inside the boundary layer
115
. An
example is given in Figure 6.3.1, where:
Eq. 6.3.1
6.3.1 Thermal Boundary Layer
Similarly, the energy equation, we obtained the
following simplified equation for 2D compressible
as:
Eq. 6.3.2
For compressible, constant viscosity, steady flow
ρ
μ
Eq. 6.3.3
And the equations are couples as evident the last term in R.H.S. Furthermore assuming no viscous
dissipation and no flow, we have
Eq. 6.3.4
This is the celebrated Laplace equation with solutions readily available.
115
P. Wassermann, M. Kloker, “DNS of Laminar Turbulent Transition in a 3-D Aerodynamics Boundary Layer
Flow”, Institut fur Aerodynamik und Gasdynamik, Universit_at Stuttgart, Germany.
Figure 6.3.1 3-D Boundary Layer Velocity
Profile
93
6.4 Boundary Layer Separation
Boundary layer separation is the detachment of a boundary layer from the surface into a broader
wake
116
. Boundary layer separation occurs when the portion of the boundary layer closest to the
wall or leading edge reverses in flow direction. The separation point is defined as the point between
the forward and backward flow, where the shear stress is zero. The overall boundary layer initially
thickens suddenly at the separation point and is then forced off the surface by the reversed flow at
its bottom (back flow) (see Figure 6.4.1)
117
.
6.5 Adverse Pressure Gradient
Graphical representation of the velocity profile in the boundary layer. The last profile represents
reverse flow which shows separated flow. The flow reversal is primarily caused by adverse pressure
gradient imposed on the boundary layer by the outer potential flow. The stream wise momentum
equation inside the boundary layer is approximately stated as
Eq. 6.5.1
where x, y are stream-wise and normal coordinates. An adverse pressure gradient is when , which
then can be seen to cause the velocity to decrease along and possibly go to zero if the adverse
pressure gradient is strong enough. Figure 6.5.1 shows graphical representation of the velocity
profile in the boundary layer, where the last profile represents reverse flow which shows separated
flow
118
.
6.5.1 Influencing Parameters
The tendency of a boundary layer to separate primarily depends on the distribution of the adverse
or negative edge velocity gradient du0/dx < 0 along the surface, which in turn is directly related to
116
White, "Fluid Mechanics", (7th ed.), 2010.
117
P. Wassermann, M. Kloker, “DNS of Laminar Turbulent Transition in a 3-D Aerodynamics Boundary Layer
Flow”, Institut fur Aerodynamik und Gasdynamik, Universit_at Stuttgart, Germany.
118
Hannes Sturm, Gerrit Dumstorff, Peter Busche, Dieter Westermann and Walter Lang, “Boundary Layer
Separation and Reattachment Detection on Airfoils by Thermal Flow Sensors”, Sensors 2012, 12, 14292-14306.
Figure 6.4.1 Representation Boundary Later Velocity Profile (Courtesy of Sturm et al.)
94
the pressure and its gradient by the differential form of the Bernoulli relation, which is the same as
the momentum equation for the outer inviscid flow.
Eq. 6.5.2
But the general magnitudes
of du0/dx required for
separation are much greater
for turbulent than
for laminar flow, the former
being able to tolerate nearly
an order of magnitude
stronger flow deceleration. A
secondary influence is
the Reynolds number. For a
given adverse du0/dx
distribution, the separation
resistance of a turbulent
boundary layer increases
slightly with increasing
Reynolds number. In contrast,
the separation resistance of a
laminar boundary layer is
independent of Reynolds
number; a somewhat counterintuitive fact.
6.6 Internal Separation
Boundary layer separation can occur for internal flows. It can result from such causes such as a
rapidly expanding duct of pipe. Separation occurs due to an adverse pressure gradient encountered
as the flow expands, causing an extended region of separated flow. The part of the flow that separates
the recirculating flow and the flow through the central region of the duct is called the dividing
streamline. The point where the dividing streamline attaches to the wall again is called the
reattachment point. As the flow goes farther downstream it eventually achieves an equilibrium state
and has no reverse flow.
6.7 Effects of Boundary Layer Separation
When the boundary layer separates, its displacement thickness increases sharply, which modifies the
outside potential flow and pressure field. In the case of airfoils, the pressure field modification results
in an increase in pressure drag, and if severe enough will also result in loss of lift and stall, all of which
are undesirable. For internal flows, flow separation produces an increase in the flow losses, and stall-
type phenomena such as compressor surge, both undesirable phenomena
119
. Another effect of
boundary layer separation is shedding vortices, known as Kármán vortex street. When the vortices
begin to shed off the bounded surface they do so at a certain frequency. The shedding of the vortices
then could cause vibrations in the structure that they are shedding off. When the frequency of the
shedding vortices reaches the resonance frequency of the structure, it could cause serious structural
failures.
119
Fielding, Suzanne. "Laminar Boundary Layer Separation", University of Manchester, 12 March 2008.
Figure 6.5.1 Physical Features of a Ramp Flow with Boundary Layer
Separation - Courtesy of [Delery & Bur]
95
6.8 Shock wave/Boundary layer Interactions (SWBLIs)
Shock wave–boundary-layer interactions (SWBLIs) occur when a shock wave and a boundary
layer converge and, since both can be found in almost every supersonic flow, these interactions are
commonplace. The most obvious way for them to arise is for an externally generated shock wave to
impinge onto a surface on which there is a boundary layer. However, these interactions also can be
produced if the slope of the body surface changes in such a way as to produce a sharp compression
of the flow near the surface – as occurs, for example, at the beginning of a ramp or a flare, or in front
of an isolated object attached to a surface such as a vertical fin. If the flow is supersonic, a
compression of this sort usually produces a shock wave that has its origin within the boundary layer.
This has the same effect on the viscous flow as an impinging wave coming from an external source.
In the transonic regime, shock waves are formed at the downstream edge of an embedded supersonic
region; where these shocks come close to the surface, an SWBLI is produced [Babinsky & Harvey,
(2011)]
120
.
6.8.1 Case Study 1 - Shock Wave/Boundary Layer Interaction (SWBLI)
Citation : Jean M. Delery and Reynald S. Bur, (2000), “The Physics of Shock Wave/Boundary Layer
Interaction Control: Last Lessons Learned”, European Congress on Computational Methods in Applied
Sciences and Engineering ECCOMAS2000, Barcelona.
6.8.1.1 Background and Introduction
In high speed flow, the existence of shock waves most often entails either drag increase or efficiency
losses (Delery and Bur, 2000)[1]. A major cause of performance degradation is the interaction of the
shock with a boundary layer. Then complex phenomena occur which contributes to increase friction
losses, especially if the shock is strong enough to separate the boundary layer. To a separated flow
are associated typical wave patterns resulting from the shocks induced by the separation and
reattachment processes and which play a major role in the production of entropy by the flow. Since
shocks cannot be avoided in most situations, control techniques have been proposed to limit their
negative effects. The mode of operation of these techniques can be well understood by a clear
identification of shock wave/boundary layer interaction properties. The control actions can be
performed by a proper manipulation of the boundary layer upstream of the interaction domain in
order to increase its resistance to the shock action (by blowing or lowering the wall temperature, or
using vortex generators) or by a local action in the shock foot region. Then active, passive or hybrid
control which combines the two previous actions can be applied. Other methods can be envisaged,
like the installation of a bump in the shock foot region to adapt the surface contour in order to weaken
the shock. None of these techniques brings the ideal answer to the problem of shock wave/boundary
layer interaction control. Thus, the definition of a solution closely depends on the objective of the
control.
In addition, the appropriateness of implementing a control device highly depends on economic issues
in terms of weight penalty, manufacturing and maintenance cost and energy consumption. Shock
waves almost inevitably occur when a flow becomes supersonic or transonic. They are provoked by
a change in the flow direction, as at the compression ramps of a supersonic air intake or at a control
surface, an increase of the downstream pressure as on a transonic wing, a pressure jump as in an
over expanded propulsive nozzle, a brutal deceleration as in front of the nose of a re-entry vehicle.
The presence of shock waves entails the existence of discontinuities and regions of high gradients
which are the shocks themselves and the shear layers resulting from the interaction with the
120
Babinsky, H., & Harvey, J. (2011). Introduction. In H. Babinsky & J. Harvey (Eds.), Shock Wave-Boundary-Layer
Interactions (Cambridge Aerospace Series, pp. 1-4). Cambridge: Cambridge University Press.
doi:10.1017/CBO9780511842757.001].
96
boundary layers developing on the vehicle surface. These gradients "activate" the viscous terms
producing entropy, which makes shock waves an important source of drag:
1 Directly by entropy generation in the thickness of the shock: this contribution is the wave
drag;
2 Indirectly, be enhancing dissipation in the boundary layers: hence an amplification of the
viscous drag.
In addition, strong interactions with the boundary layers may lead to catastrophic separation with
possible occurrence of large scale unsteadiness (wing buffeting, air intake buzz). Shock waves play a
dramatic role at hypersonic velocities because of their intensity which leads to spectacular shock-
shock interferences with specific and complex wave patterns and strong interactions with the
boundary layers [2]. In addition, the temperature rise at the crossing of the shocks most often affects
the gas thermodynamics properties (the so-called real gas effects) and is at the origin of high heat
transfer levels at the vehicle surface [3].
The idea to control shock wave/boundary layer interaction in order to avoid, or minimize, its
detrimental effects is nearly contemporaneous with the advent of studies on high-speed flows. Thus,
as early as 1941, [Regenscheit] studied the effect of suction through a slit at the surface of an airfoil
tested at transonic velocity. The same arrangement was considered by (Fage and Sargent) [4]. Since
that time, a great deal of research has been devoted to this important practical problem. In fact, the
problem of shock wave/boundary layer (SWBL) interaction control is multiple in the sense that
according to the objective looked for antagonistic mechanisms can be at work. As it will be seen, the
reduction of a profile drag can be obtained by a smearing of the shock entailing a reduction of the
wave drag, but at the price of a thickening of the boundary layer and an increase of the friction drag.
On the other hand, separation suppression or shock stabilization most often entails an intensification
of the shock, hence an increase of the losses through the shock. Thus, the target of shock
wave/boundary layer control must be clearly identified.
The purpose is to focus on the physical phenomena involved in shock wave/boundary layer
interactions first without, then with control action. The results presented and discussed here are
mostly based on a recent thorough investigation of various control actions applied to transonic
interactions executed within the framework of the European projects Euro shock I and II. In
particular, detailed flow descriptions allowed a clear identification of the flow processes at work in
different kinds of control devices giving a good status of the art on control techniques in transonic
flows [5].
6.8.1.2 Basic Properties of Shock Induced Phenomena
The consideration which follow essentially apply to turbulent flows which are more common in usual
aeronautical applications than laminar flows. When a boundary layer is submitted to the strong
retardation imparted by a shock wave, complex phenomena occur within its structure which is
basically a parallel rotational flow with the Mach number varying from the outer supersonic value to
zero at the wall. The process has been analyzed by [Lighthill]
121
in the framework of its famous triple
deck theory. The interaction resulting from the reflection of an oblique shock (C1) is sketched in
Figure 6.8.1.
A clear consequence of shock wave/boundary layer interaction is that the presence of the shock is
felt upstream of its impact point in the perfect fluid model. This upstream influence phenomenon is
in great part an inviscid mechanism, the pressure rise caused by the shock being transmitted
upstream through the subsonic part of the boundary layer. The thickness of the subsonic layer
depending of the velocity distribution, a fuller profile involves a thinner subsonic channel, hence a
shorter upstream influence length. At the same time, a boundary layer profile with a low velocity
121
Lighthill, M. J., On boundary-layer upstream influence. II Supersonic flows without separation. Proc. R. Soc., A
217, 1953, pp. 478-507.
97
deficit has a higher momentum, hence a greater resistance to the retardation imparted by an adverse
pressure gradient. The dilatation of the boundary layer subsonic region is felt by the outer supersonic
flow which includes the major part of the boundary layer as a ramp effect inducing compression
waves whose coalescence forms the reflected shock (C2). This mechanism, which is amplified by a
general retardation of the boundary layer flow, can be interpreted as a viscous ramp effect resulting
in a spreading of the shock near the surface, the incident-plus-reflected shock of the perfect fluid
theory being replaced by a continuous process. The physics involved in the interaction is in fact more
subtle since the viscous terms must play a role in the immediate vicinity of the wall because of the no
slip condition, otherwise one is confronted to inconsistencies as it was found by [Lighthill]. Thus, the
interaction results from a competition between pressure plus inertia terms belonging to the Euler
part of the Navier-Stokes equations and viscous forces tending to counteract the previous terms. This
fact explains that the Reynolds number effect is nearly non-existent is fully turbulent flow whereas
it has a strong effect in laminar flows which are viscous dominated.
Since the retardation effect is larger in the boundary layer inner part, a situation can be reached here
the flow is pushed in the upstream direction by the adverse pressure gradient so that a separated
region forms. Then, the flow adopts the structure sketched in Figure 6.8.2. The separation process
is basically a free interaction process resulting from a local self-induced interaction between the
boundary layer and the outer inviscid stream
122
. Hence, it does not depend on downstream
conditions, in particular the intensity of the incident shock, the pressure rise at separation Δp1 being
only function of the incoming flow. Downstream of the separation point exists a bubble made of a
122
Chapman, D.R., Kuehn, D. M. and Larson, H.K., Investigation of separated flow in supersonic and subsonic
streams with emphasis on the effect of transition. NACA TN 3869, 1957.
Figure 6.8.1 Physical features of an oblique shock reflection with boundary layer separation -
Courtesy of Delery & Bur
98
recirculating flow bounded by a
discriminately streamline (S)
originating at the separation
point S and ending at the
reattachment point R. Due to
the action of the strong mixing
taking place in the detached
shear layer which emanates
from S, an energy transfer is
operated from the outer high
speed flow towards the
separated region. As a
consequence, the velocity Us on
the discriminating streamline
(S) steadily increases, until the
reattachment process begins.
The transmitted shock (C3)
penetrates in the separated
viscous flow where it is
reflected as an expansion wave.
There results a deflection of the
shear layer in the direction of
the wall on which it reattaches.
At reattachment, the separated bubble vanishes, the flow on (S) being decelerated until it stagnates
at R. This process is accompanied by a compression wave ending into a reattachment shock in the
outer stream. A major consequence of the interaction is to divide the pressure jump Δ.p imparted by
the hock reflection into a first compression Δp1 at separation with the associated shock (C2) and a
second compression Δp2 at reattachment, the overall pressure rise being such that Δp = Δp1+ Δp2 .
The extent of the separated region is dictated by the ability of the shear layer issuing from the
separation point S to overcome the pressure rise at reattachment. This ability is function of the
momentum available on (S) or maximum velocity (Us)max at the starting of the reattachment
process. Since the pressure rise to separation does not depend on downstream conditions, an
increase of the overall pressure rise imparted to the boundary layer - or incident shock strength
entails a higher pressure rise at reattachment. This can only be achieved by an increase of the
maximum velocity (Us)max reached on the discriminating streamline, hence an increase of the shear
layer length allowing a greater transfer of momentum from the outer flow. Thus the length of the
separated region will grow in proportion to the pressure rise at reattachment. When there is
separation, the interaction of the shock wave with the boundary layer has deep repercussions on the
contiguous inviscid flow. As shown in Figure 6.8.2, the simple inviscid shock pattern made of an
incident plus reflected shock is replaced by a pattern involving 5 shock waves:
➢ incident shock (C1),
➢ separation shock (C2 ),
➢ transmitted shock (C3 ) emanating from the intersection point I of (C1) and (C2),
➢ second transmitted shock (C4),
➢ reattachment shock (C5 ).
The structure involving shocks (C1), (C2), (C3) and (C4) is a Type I shock/shock interference
Figure 6.8.2 2D High Speed Flow Over a Compression Corner
Involving SWBL Interaction – Courtesy of Kumar
99
pattern, according to the Edney classification
123
. If the slope, or intensity, of the incident shock is
increased, the interaction of (C1) with (C2) may be singular, a normal shock (C6) forming between
(C1) and (C2) to constitute a Mach phenomenon or Type II interference. Transonic and ramp
induced interactions lead to a similar organization for the separated flow, the boundary layer
reacting to a pressure rise, no matter the cause of this pressure rise. What changes is the shock
pattern associated with the interaction.
6.8.1.3 Case of Interaction with the Ramp
In the case of the interaction at a ramp, the shock system is a Type VI interference pattern (see
Figure 6.8.3). In transonic flow, the interaction leads to a specific flow organization which can be
considered as a variant of
the above situations. Here,
the interaction is provoked
by a normal shock (C3)
forming on a transonic
airfoil or in a channel.
Another view for a 2D case
is shown in Figure 6.8.4
by [Kumar et al]
124
. As
revealed, separation of the
boundary layer at point S
induces a deflection of the
flow giving rise to the
oblique shock (C1) as in the
previous cases, the flow
behind (C1) being still
supersonic. The two
shocks (C1) and (C3) meet
at the triple point I where a
variant of the Type VI
shock-shock interference takes place. The states 2 and 3 behind (C1 ) and (C3) not being compatible
with the Rankine-Hugoniot equations, a third shock (C2) starts from I leading to the state 4
compatible with 3. Downstream of I, the two states 3 and 4 separated by the slip line CE) coexist. A
deeper analysis of the solution shows that the shocks (C3) and (C2) are strong oblique shocks, in the
sense of the strong solution to the oblique shock equations. Downstream of (C2) the flow can still be
supersonic, the further transition to subsonic being most often isentropic. Away from the wall, the
flow contains the nearly normal shock (C3) which caused the separation. The structure represented
in Figure 6.8.4 is commonly called a lambda pattern. Thus, to a shock-induced separated flow is
attached a shock pattern in which the simple inviscid flow solution is replaced by a multi-shock
system tending to minimize entropy production through the shock. This fact is of great importance
in control techniques aiming at drag or efficiency loss reduction.
123
Edney, B., Anomalous heat transfer and pressure distributions on blunt bodies at hypersonic sp eeds in the
presence of an impinging shock. Aeronautical Research Institute of Sweden, FFA Report N° 115, 1968.
124
Vikash Kumar, Nishant , Md. Asif Hussain , Paragmoni Kalita, “Effect of Freestream Parameters on the Laminar
Separation in Hypersonic Shock Wave Boundary Layer Interaction”, ADBU-Journal of Engineering Technology.
Figure 6.8.3 Physical Features of an Oblique Shock Reflection Without
Boundary Layer Separation. The Upstream Propagation Mechanism.
Courtesy of Delery & Bur
100
6.8.1.4 Mechanisms For Control
The above physical description of
typical shock wave/boundary layer
interactions brings out the most salient
phenomena involved in an interaction,
thus providing indications on the
means which can be considered to
modify or control the interaction. The
upstream influence of the shock and
the resistance of a turbulent boundary
layer depending mainly of its
momentum, a means to restrict the
effect of the shock is to increase the
boundary layer momentum prior to its
interaction with the shock. This can be
done by proper boundary layer
manipulation:
• One can perform an injection
through one or several slots
located upstream of the shock
origin, this technique being
called boundary layer
blowing.
• A distributed suction applied
over a certain boundary layer
run before the interaction
reduces its shape parameter,
thus producing a fuller velocity
profile. On the contrary,
distributed injection
transpiration
• increases the shape
parameter, which renders the
boundary layer more sensible
to the shock. Vortex generators
can be classified in this
category of control actions,
since the vertical structures
that they create operate a momentum transfer from the outer high speed flow at the benefit
of the boundary layer.
• It is also possible to eliminate the low speed part of the boundary layer by making a strong
suction through a slot located at a well-chosen location upstream of the interaction.
• The interaction being in great part dictated by the boundary layer properties, one can
envisage to modify these properties by changing the temperature of the wall. An increase
of the wall temperature lowers the Mach number level by increasing the sound speed
whereas a lowering of the wall temperature increases the Mach number by diminishing the
sound speed. Thus upstream propagation of the shock influence will be affected by this effect,
the thickness of the boundary layer subsonic channel being augmented on a heated wall and
reduced on a cooled wall. In addition, the temperature level in the boundary layer changes
Figure 6.8.4 Schematic Representation of the Flow in a
Separated Bubble. a - basic case, b - with Mass Suction, c -
with Mass Injection - Courtesy of Delery & Bur
101
the density in the boundary layer flow. There results a change of the boundary layer
momentum, favorable in the case of cooling, unfavorable in the case of heating. The effect of
the wall temperature has been mainly studied for the cold wall situation
125
.
In this case, a strong contraction of the interaction domain is observed, with in some circumstances
a suppression of the separation bubble. The situation of a heated wall has been investigated for the
reflection of an oblique shock on a wall heated to a temperature equal to twice the recovery
temperature
126
. The iso Mach number contours plotted in Figure 6.8.5 show the extension of the
interaction occurring when the wall is heated. The interaction properties can also be affected by
performing an action in the interaction region itself. A key role being played by the velocity or
momentum - level which must be reached on the discriminating streamline to allow shear layer
reattachment, any action modifying this level will influence the interaction. Thus, if some fluid is
sucked through the wall, topological considerations lead to the flow structure sketched in Figure
6.8.5-b. The streamline (S2) stagnating at the reattachment point R comes from upstream "infinity"
and is located above (S1), the sucked-off fluid flowing between (S1) and (S2). Provided the extracted
mass flow be moderate enough so as not to entail a total alteration of the general flow structure, the
velocity profile through the shear layer is nearly the same as in the basic case of Figure 6.8.5-a. As
streamline (S2) is located at a greater distance from the wall, the velocity (Us)max on (S2) is greater
125
Kilburg, R.F. and Kotansky, D.R., Experimental investigation of the interaction of a plane oblique incident -
reflecting shock wave with a turbulent boundary layer on a cooled surface. NACA CR-66-841, 1969.
126
Delery, J. , Etude experimentale de la reflexion d'une onde de choc sur une paroi chauffee en presence d'une
couche limite turbulente. La Recherche Aerospatiale, N° 1992- 1, 1992, pp. 1-23.
Figure 6.8.5 Mach Number Contours of Transonic Interaction with Suction of the Boundary Layer
Through a Slot - Courtesy of Delery & Bur
102
than in the basic case: hence, an increase of the ability of the flow to withstand a more important
compression and a subsequent contraction of the interaction domain. The case of a fluid injection at
low velocity is sketched in Figure 6.8.5-c. Now, the velocity on (S2) is smaller, since (S2) is located
at a lower altitude on the velocity profile. The resulting effect is a lengthening of the separation
bubble. However, if the injected mass flow is increased, there will be a reversal of the effect, since the
velocities on the bottom part of the profile and in particular (Us)max will increase if the mass flow
fed into the separated region goes beyond a certain limit. A similar mechanism is at work in base
bleed aiming at the reduction of base drag through an increase of the base pressure.
6.8.1.5 Objectives and Application of Some Control Actions
When considering shock wave/boundary layer interaction control, it is essential to state the objective
which is looked for:
➢ A control action can be considered to prevent separation and/or to stabilize the shock in a
duct or a nozzle. Boundary layer blowing, suction, wall cooling can be very efficient for these
purposes.
➢ If the objective is to decrease the drag of a profile, or limit the loss of efficiency in an air intake,
the situation is more subtle since the drag has its origin both in the shock and in the boundary
layer.
Any action "strengthening" the boundary layer tends also to strengthen the shock since the
spreading caused by the interaction is reduced. This effect is illustrated in Figure 6.8.6 which
shows, by means of iso-Mach number contours (deduced from LDV measurements), a comparison
between a basic transonic interaction, without control and the flow resulting from boundary layer
suction through a slot. In the second case, the entropy production through the shock is more
important and the wave drag higher. On the other hand, since the downstream boundary layer profile
is more filled, the momentum loss in the boundary layer is much reduced which leads to a reduction
of the friction drag. In addition, this active control action reduces significantly the turbulence level
downstream of the interaction region, which contributes to produce a more stable flow, less affected
by fluctuations. The most favorable location of the slot is at a few distance downstream of the
interaction, upstream and centered locations leading to a slightly greater rise of the boundary layer
displacement and momentum thicknesses. In evaluating the benefit of local suction in terms of drag,
one must be aware of the capitation drag caused by the swallowing of a part of the incoming flow.
As seen above, when separation occurs the simple inviscid shock system is replaced by a pattern
made of continuous compression waves and multiple shocks through which the entropy production
is less compared to the inviscid solution (for identical upstream and downstream conditions). Thus,
the smearing of the shock system and splitting of the compression achieved by the interaction reduce
the wave drag or the efficiency loss - due to the shock. On the other hand, the momentum loss in a
separated boundary layer is far more important than through an attached boundary layer, so that
separation most often results in a sharp increase of drag or efficiency loss (not speaking of
unsteadiness). Since separation has a favorable effect on the wave drag, one can envisage to replace
a strong, but not separated interaction, by a separated or separated-like flow organization. The
passive control concept suggested in the early 80s, aims at combining the two effects by spreading
the shock system while reducing the boundary layer thickening. The principle of passive control
consists in replacing a part of the surface by a perforated plate installed over a closed cavity. The
plate being implemented in the shock region, there occurs a natural flow circulation via the cavity
103
from the downstream high pressure part of the interaction to the upstream low pressure part. The
resulting effect on the flow is sketched in Figure 6.8.7.
The upstream transpiration provokes a thickening of the boundary layer, hence a rapid growth of its
displacement thickness which is felt by the outer supersonic flow as a viscous ramp effect inducing
an oblique shock (C1): The meeting of (C1) with the main normal shock (C3) at the point I leads to a
lambda shock pattern with the trailing shock (C2). The situation is similar to the case of a natural
shock-induced separation, the "strong" normal shock (C3) being replaced by a two shock system in
the vicinity of the surface: hence a reduction of the wave drag. The negative effect on the boundary
layer, which would increase the friction losses, is limited by the suction operated in the downstream
part of the plate. The effect of passive control is illustrated by the schlieren photographs (see Delery
and Bur]
127
. The comparison with the reference case, without control, how the positive influence of
an increase of the displacement thickness (viscous ramp effect) and the negative consequences
corning from the dramatic rise in the momentum thickness.
The obvious merit of the above concept is the absence of an energy supply, the system being self-
operating. However, the effectiveness of passive control to reduce the total drag produced by the
shock is impaired by the high friction drag along the perforated plate, the gain being in general
modest when not negative. The concept can be improved by making some suction in order to
minimize the boundary layer growth over the control region. There are different ways to combine
127
Jean M. Delery and Reynald S. Bur, “The Physics Of Shock Wave/Boundary Layer Interaction Control: Last
Lessons Learned”, European Congress on Computational Methods in Applied Sciences and Engineering
ECCOMAS2000, Barcelona, 11-14 September 2000.
Figure 6.8.6 Principle of Passive Control of a Transonic Interaction - Courtesy of Delery & Bur
104
the passive and active control effects. One can simply perform some suction in the cavity itself. It is
also possible to perform suction downstream of the passive control cavity either in a separate cavity
or by a slot, as sketched in Figure 6.8.8. This hybrid control device could combine the advantage
of passive control to decrease the wave drag and the high effectiveness of suction to reduce friction
losses. In fact, experiment shows that it is necessary to extract a relatively high mass flow to
substantially reduce the friction losses and make the device efficient
128
.
Since friction drag production in passive or hybrid control, is frequently unacceptable, one may
consider the possibility to reproduce the separated flow structure by a local deformation of the
surface having, in
the case of the
oblique shock
reflection of
Figure 6.8.2 a
double wedge like
shape which
materializes the
viscous separated
fluid of the original
interaction, as
shown in Figure
6.8.9. Such a
contour induces a
first shock at its
origin and a second
shock at its trailing
edge, thus leading to a shock pattern similar to that of the original reflection. One can hope that this
system would not too much destabilize the boundary layer, while reducing substantially the wave
drag. This bump concept has been considered for application to transonic interactions with a view to
control the flow past civil transport aircraft wings.
128
Bur, R., Benay, R., Corbel, B., Soares-Morgadhino, R. and Soule vant, D., Study of control devices applied to a
transonic shock wave/boundary layer interaction. Onera contribution to Task 1 of the Euroshock II Project.
Onera Technical Report N° 126/7078DAFE/Y, July 1999.
Figure 6.8.7 Interaction Control by a Local Deformation of the Wall or the Bump Concept - Courtesy of
Delery & Bur
Figure 6.8.8 Transonic Interaction Control by a Bump - Mach Number Contours
105
A large, both experimental and theoretical, programmed has been recently executed in the
framework of the Euro shock II project. The bump is a local deformation of the airfoil contour located
in the shock foot region which produces a nearly isentropic compression in its upstream part thus
acting as a compression ramp. The height of such a bump is less than one per cent of the airfoil cord
length and it extends over a distance which is of the order of 0.10 to 0.20 chord length. The bump
contour must be carefully
designed, one method
consisting in adopting a
shape which reproduces the
evolution of the boundary
layer displacement
thickness in the case
without control
129
.
As shown by the iso-Mach
number contours in Figure
6.8.10, the bump is very
effective to spread the
compression produced by
the shock in the wall region,
thus to reduce the wave
drag. At the same time, as
shown by the characteristic
thicknesses, the bump has a
reasonable effect on the
boundary layer properties,
in the sense that the
boundary layer flow is not
very different from what it
is in the reference case, contrary to the passive control device. The friction losses are thus limited so
that the gain due to the wave drag reduction is not compromised. The drawback of the system is that
the effectiveness of the bump highly depends on the shock location with respect to the bump.
Considering that the location achieved in Figure 6.8.10 is nearly the optimum, a downstream shift
of the shock leads to a steepening of the compression at the wall and a more important thickening of
the boundary layer. Moreover if the shock travels to the bump trailing edge region, its strength
increases because of the expansion after the bump crest. This leads to the formation of a large
separated zone at the bump trailing edge. Thus both the wave drag and the friction drag can be
increased. A comparison of the different tested methods is shown in Figure 6.8.11 by plotting of the
iso-Mach number contours deduced from LDV measurements. One sees the deep repercussion of the
control actions both on the outer inviscid flow structure and on the boundary layer behavior.
129
Bur, R., Corbel, B., Delery, J. and Soulevant, D., Transonic shock wave/boundary layer interaction control.
Complementary experiments on suction slot and bump control. Onera contribution to Task 1 of the Euroshock II
Project. Onera Technical Report N° 131/7078DAFE/Y, Jan. 2000.
Figure 6.8.9 Principle of Hybrid Control of a Transonic Interaction -
Courtesy of Delery & Bur
106
6.8.1.6 Finishing Remarks
The signification of shock wave/boundary layer interaction control is ambiguous, in the sense that
control can aim at minimizing the effect of the shock on the boundary layer properties or reducing
the overall losses caused by the interaction, the two objectives being partly contradictory. In the first
case, the main objective is most often to avoid boundary layer separation which can be achieved
either by manipulation of the boundary layer prior to its interaction with the shock (upstream
blowing, suction through a slot, wall cooling) or by suction or bleeding in the interaction region. In
the second case, one has to reduce also the wave drag which is achieved by a splitting or spreading
of the shock near the wall, which can be done by creation of a separated-like flow structure by means
of a passive control cavity or a bump.
Control techniques are more suited to internal aerodynamics applications interesting air intakes,
Figure 6.8.10 Mach Numbers Contours of Different Control Actions on a Transonic Interaction -
Courtesy of [Delery & Bur]
107
diffusers, nozzles because the size of
the region to be controlled is far more
reduced and because of the proximity
of the energy supply; thus avoiding
long exhibitory tubing. The energetic
and economic aspects of interaction
control, as also the problem raised by
the installation of a control device in
an airplane wing, have not been
examined, although their
consideration is essential. Neither we
have discussed the problems raised
by the modelling of an interaction
under control conditions, considering
the incidence of flow manipulation on
turbulence behavior and the
definition of a law to represent the
suction/transpiration velocities in
the control region. The extensive
experimental programmed executed
within the Euro shock projects has
allowed to constitute unprecedented
data banks which already contributed to improve the physical models used in the prediction of shock
wave/boundary layer interactions under control conditions. This is also a lesson learned.
6.8.1.7 References
[1] Jean M. Delery and Reynald S. Bur, “The Physics of Shock Wave/Boundary Layer Interaction
Control: Last Lessons Learned”, European Congress on Computational Methods in Applied Sciences
and Engineering ECCOMAS2000, Barcelona, 11-14 September 2000.
[2] Delery, J., Shock phenomena in high speed. aerodynamics: still a source of major concern. The
Aeronautical Journal of the Royal Aeronautical Society, Jan. 1999, pp. 1934.
[3] Delery, J., Shock interaction phenomena in hypersonic flows. Part II: Physical features of shock
wave/boundary layer interaction in hypersonic flows. AGARD Conference on Future Aerospace
Technology in the Service of the Alliance, 14-16 April 1997, Ecole Polytechnique, Palaiseau, France.
[4] Fage, A. and Sargent, R. F., Effect on aerofoil drag of boundary layer suction behind a shock wave,
ARC R&M No, 1913.
[5] Euroshock - Drag reduction by passive control. Results of the project Euroshock. AER2-CT92-0049
Supported by the European Union 1993-1995, Stanewsky, E., Delery, J., Fulker, J. and Geissler, W.
(Eds), Notes on Numerical Fluid Mechanics, 56, Vieweg, 1997.
Figure 6.8.11 Schematic diagram representing 2D high
speed flow over a compression corner involving SWBL
interaction
108
6.8.2 Case Study 2 - Effect of Freestream Parameters on the Laminar Separation in Hypersonic
Shock Wave Boundary Layer Interaction
Citation : Kumar, Vikash et al. “Effect of Freestream Parameters on the Laminar Separation in
Hypersonic Shock Wave Boundary Layer Interaction.” ADBU Journal of Engineering Technology
(AJET) 4 (2016).
Two-dimensional hypersonic flow over a ramped passage is computed on a finite volume framework
using an in-house solver [1]. Van Leer’s Flux Vector Splitting (FVS) scheme is used to compute the
inviscid fluxes. The gradients in the viscous flux terms are computed using the Green’s theorem.
The effects of freestream parameters on the interaction between the boundary layer and the ramp-
induced shock are investigated. For a given Reynolds number, the effects of freestream pressure and
temperature on the laminar boundary layer separation are studied. It is seen that increase in
freestream pressure reduces the flow separation; however increase in freestream temperature shifts
the separation point upstream and the reattachment point downstream. Additionally the effect of
Mach number at a given Reynolds number and freestream temperature on the boundary layer
separation is discussed.
6.8.2.1 Governing Equations and the Numerical Schemes
The flow is governed by 2D Navier-Stokes equations. The inviscid fluxes are computed by using the
van Leer‘s FVS scheme, which splits the flux at the cell interface based upon the sign of the
eigenvalues of the flux Jacobian matrices. The mathematical formulation of the scheme is shown in
sub-section B. The gradients in the viscous flux terms are computed by using the Green‘s theorem
which is briefly discussed in sub-section C.
(A) The Navier-Stokes equations for 2D flow are [2]:
Eq. 6.8.1
In these equations, U is the vector of conserved variables, Fi and Gi are the inviscid or convective flux
vectors, Fv and Gv are the viscous flux vectors, where,
Eq. 6.8.2
such that, E is the total fluid energy per unit mass and rest of the symbols have their usual meanings.
These equations are solved by time-marching to obtain the steady state solutions. The first order
Euler explicit technique is used for the time integration. The stress-tensor can be written using the
indicial notation due to Einstein as follows:
109
Eq. 6.8.3
The dynamic coefficient of viscosity μ is calculated by using the Sutherland‘s law [3].
(B) The Van Leer‘s Flux Vector Splitting Scheme
van Leer split the flux vector into two parts based upon the split Mach Number as [4]:
Eq. 6.8.4
where Q is the flux normal to the cell face and subscripts L and R represent the cells on the upstream
and downstream sides of the cell face respectively. The Mach No. at the cell interface was obtained
as,
Eq. 6.8.5
Additional information regarding the method of solution can be obtained from [5].
6.8.2.2 Problem Statement and the Boundary Conditions
Hypersonic flow of air over a ramped surface is considered. As shown in Figure 6.8.11, a weak
shock emanates from the sharp leading-edge. This is named as the leading-edge shock. Due to strong
viscous effects prevailing in hypersonic flows a boundary layer develops over the solid surface. Due
to the ramp an oblique shock is developed. In case of inviscid flow, the oblique shock would have
emanated from the compression corner itself. However in viscous flows, the oblique shock is formed
upstream of the compression corner owing to viscous interactions. This shock is also called
separation shock. Due to adverse gradients generated across the separation shock, and a ramp angle
greater than the incipient separation angle, the boundary layer separates at the foot of the separation
shock. The boundary layer reattaches downstream of the compression corner. A reattachment shock
emanates from this location. This shock intersects with the separation shock further downstream.
The length of the laminar separation bubble is used as a measure of the severity of the boundary layer
separation. The length of the flat surface from the ramp up to the compression corner is taken as 0.05
m. The total length of the ramped passage along the x-direction is 0.12 m. The ramp angle is 15 deg.
The parameter Re is taken as 8 x 105 . The wall temperature is 300 K. The computations are done at
varying freestream static pressure and temperature. From earlier reporting , the results are
considered to be grid-independent for the 300x360 mesh.
The boundary conditions for the problem have to be implemented in conjunction with a careful
choice of the computational domain. At the inlet, the freestream stagnation temperature, freestream
Mach number, the parameter Re and the v-velocity are specified. Since the freestream is parallel to
the x-axis, so the v-velocity at inlet is set as zero. At the wall, the velocity components in the dummy
cell adjacent to the solid surface are computed by using the no slip boundary condition. The pressure
in the dummy cell is set equal to the value at the interior cell adjacent to the solid wall. The
temperature at the dummy cell is set equal to the specified wall temperature. The density at the
dummy cell is calculated from the values of pressure and temperature in that cell by using the
equation of state. At the outlet, all the variables are extrapolated from within the computational
domain.
110
6.8.2.3 Results and Discussion
The computations are done on a structured grid. The effects of freestream pressure on the laminar
separation for given freestream temperature and Reynolds number are studied. At Re∞ = 8 x105, the
freestream temperature is kept constant at 120 K, 130, 140 and 150 K. For every freestream
temperature the freestream pressure is varied as 150, 200, 250, 300 and 350 N/m2. Additionally, the
effect of Mach number on the separation and re-attachment is investigated for the stagnation
temperature of 1080 K and Re∞ = 8 x 105 at Mach numbers 5, 6, 7 and 8. Indications are that the skin
friction coefficient increases with decrease in freestream temperature. This can be explained by the
fact that higher freestream temperature increases the viscosity as per Sutherland‘s law, thereby
increasing the boundary layer thickness. As the boundary layer thickens the velocity gradient at the
solid surface decreases and hence the skin friction coefficient also decreases. The effects of
freestream temperature and pressure on the separation bubble are studied from the point of view of
points of separation and reattachment. Table 6.8.1 presents these locations as well as the laminar
separation bubble (LSB) size at varying freestream pressures for given freestream temperatures of
120 K and 150 K. The effects of varying freestream temperatures for given freestream pressures of
Freestream
Temperature
(K)
Freestream
pressure
(N/m2)
Effect on LSB
Location of
Separation
(mm)
Location of Re-
Attachment
(mm)
LSB
size(mm)
120
150
44.2
73.4
29.2
200
44.6
65.0
20.4
250
44.6
61.4
16.8
300
44.6
59.8
15.2
350
44.6
59.0
14.4
150
150
44.6
84.2
39.6
200
43.8
73.8
30.0
250
43.8
67.4
23.6
300
44.2
64.4
20.2
350
44.2
63.8
19.6
Table 6.8.1 Laminar Separation and Re-Attachment vs Freestream Pressure for Given Freestream
Temperatures – Courtesy of [Kumar et al]
Freestream
Pressure
(N/m2)
Freestream
Temperature
(K)
Effect on LSB
Location of
Separation
(mm)
Location of Re-
Attachment
(mm)
LSB
size(mm)
150
120
44.2
73.4
29.2
130
44.2
77.4
32.2
140
44.2
81.0
36.8
150
44.6
84.2
39.6
350
120
44.6
59.0
14.4
130
43.8
60.6
16.0
140
43.8
62.2
18.0
150
44.2
63.8
19.6
Table 6.8.2 Laminar Separation and Re-Attachment vs Freestream Temperature for give Freestream
Pressure
111
150 N/m2 and 350 N/m2 on the separation and reattachment points along-with the LSB size are
summarized in Table 6.8.2.
Table 6.8.1 reveals that at a given freestream temperature, the location of the point of separation
remains almost unaltered, but the point of reattachment advances with the increase in the freestream
pressure, thereby decreasing the laminar bubble size. Thus it can be inferred that freestream
pressure suppresses the laminar separation. From Table 6.8.2, it can be observed that for given
freestream pressure also, the location of the point of separation does not vary with the freestream
temperature. But increase in the freestream temperature shifts the point of re-attachment further
downstream, thus increasing the LSB size. This means that increase in freestream temperature raises
the separation tendency. However at higher pressure, the LSB size is relatively less affected by
temperature than at low
pressure.
Table 6.8.3 shows the
variation of the points of
separation and
reattachment and the LSB
size with Mach number. It is
evident that with increase in
the Mach number the
separation delays and the
re-attachment advances,
thus the LSB size decreases.
In other words, the Mach number influences locations of both the separation as well as re-attachment
points.
6.8.2.4 Concluding Remarks
Hypersonic shock-wave boundary layer interaction over a ramped surface is computed using the van
Leer‘s FVS scheme. It is seen that the location of the point of separation does not change appreciably
with freestream pressure and temperature. However, the re-attachment point advances upstream
when the freestream pressure is increased at a given freestream temperature. On the other hand,
with the increase in freestream temperature at a given freestream pressure, the re-attachment point
shifts further downstream, thereby increasing the separation length bubble size. In other words,
increase in freestream pressure and decrease in freestream temperature lowers the separation
tendency. The pressure coefficient increased with freestream pressure but decreases with
freestream temperature. Mach number influences the locations of both separation and re-attachment
points. Increase in Mach number delays the separation and advances the re-attachment. For further
info, please consult the work by [Kumar et al]
130
.
6.8.2.5 References
[1] Vikash Kumar, Nishan , Md. Asif Hussain , Paragmoni Kalita, “Effect of Freestream Parameters on
the Laminar Separation in Hypersonic Shock Wave Boundary Layer Interaction”, ADBU-Journal of
Engineering Technology.
[2] M. Delanaye, Polynomial reconstruction finite volume schemes for the compressible Euler and
Navier-Stokes equations on unstructured adaptive grids, Ph. D. Thesis, University De Liege, 1996.
[3] J. D. Anderson Jr., Hypersonic and High Temperature Gas Dynamics, McGraw Hill, pp. 13-24, 1989.
[4] W.K. Anderson, J.L. Thomas and B. van Leer, Comparison of finite volume flux vector splitting for
the Euler equations, AIAA J., vol. 24, No. 9, pp. 1435-1460, 1986.
130
Vikash Kumar, Nishan , Md. Asif Hussain , Paragmoni Kalita, “Effect of Freestream Parameters on the Laminar
Separation in Hypersonic Shock Wave Boundary Layer Interaction”, ADBU-Journal of Engineering Technology.
Mach
Number
(M)
Effect on LSB
Location of
Separation
(mm)
Location of Re-
Attachment
(mm)
LSB
size(mm)
5
44.2
60.8
16.6
6
44.5
60.8
16.3
7
45.2
61.2
16.0
8
46.2
60.5
14.3
Table 6.8.3 Laminar Separation and Re-Attachment vs Mach number
112
[5] Vikash Kumar, Nishan , Md. Asif Hussain , Paragmoni Kalita, “Effect of Freestream Parameters on
the Laminar Separation in Hypersonic Shock Wave Boundary Layer Interaction”, ADBU-Journal of
Engineering Technology.
113
7 Boundary Conditions Types
7.1 Introduction
According to Wikipedia, in the field of differential equations, a boundary value problem (BVP) is a
differential equation together with a set of additional constraints, called the boundary conditions. A
solution to a boundary value problem is a solution to the differential equation which also satisfies the
boundary conditions. To be useful in applications, a boundary value problem should be well posed.
This means that given the input to the problem there exists a unique solution, which depends
continuously on the input. The specification of proper initial conditions (IC) and boundary
conditions (BC) for a PDE is essential, and they are:
➢ If too many IC/BC are specified, there will be no solution.
➢ If too few IC/BC are specified, the solution will not be unique.
➢ If the number of IC/BC is right, but they are specified at the wrong place or time, the solution
will be unique, but it will not depend smoothly on the IC/BC.
➢ This means that small errors in the IC/BC will produce huge errors in the solution.
In any of these cases we have an ill-posed problem
131
. As an example, [Huang et al.]
132
studied the
well possess of inflow/outflow boundary conditions on a subsonic flow reigns for a flat plate. They
demonstrate that some conventional ways of posing subsonic inflow/outflow boundary conditions
are ill-posed.
7.2 Naming Convention for Different Types of Boundaries
Boundary conditions and their correct implementation are among the most critical aspects of a
correct CFD simulation
133
. Mathematically, there are four types of Dirichlet, Von Neumann, Mixed,
Robin, Cauchy, and Periodic.
7.2.1 Dirichlet Boundary Condition
Direct specification of the variable value at the boundary. E.g.
setting the distribution of a racer ϕi at a west boundary to
zero: ϕw = 0.
7.2.2 Von Neumann Boundary Condition
Specification of the (normal) gradient of the variable at the
boundary. E.g., setting a zero gradient ∂ϕ i /∂n=0 at a
symmetry boundary.
7.2.3 Mixed or Combination of Dirichlet and von Neumann
Boundary Condition
Direct specification of the variable value as well as its
gradient. It is required to satisfy different boundary
conditions on disjoint parts of the boundary of the domain
where the condition is stated. (see Figure 7.2.1).
131
Macintosh HD, Documents, AOSC614-DOCS:PPTClasses, ch3.1: PDEs Well Posed IC&BC. doc Created on
September 26, 2007.
132
Arthur C. Huang, Steven R. Allmaras, Marshall C. Galbraith and David L. Darmofal, “Well-Posed Subsonic
Inflow-Outflow Boundary Conditions for the Navier-Stokes Equations”, 2018 AIAA Aerospace Sciences Meeting.
133
Bakker André, Applied Computational Fluid Dynamics; Solution Methods; 2002.
Figure 7.2.1 Mixed Boundary
Conditions
114
7.2.4 Robin Boundary Condition
It is similar to Mixed conditions except that a specification is a linear combination of the values of a
function and the values of its derivative on the boundary of the domain
134
. Robin boundary conditions
are a weighted combination of Dirichlet boundary conditions and Neumann boundary conditions.
This contrasts to mixed boundary conditions, which are boundary conditions of different types
specified on different subsets of the boundary. Robin boundary conditions are also called impedance
boundary conditions, from their application in electromagnetic problems, or convective
boundary conditions, from their application in heat transfer problems. If Ω is the domain on which
the given equation is to be solved and ∂Ω denotes its boundary, the Robin boundary condition is:
Eq. 7.2.1
for some non-zero constants a and b and a given function g defined on ∂Ω. Here, u is the unknown
solution defined on Ω and ∂u/∂n denotes the normal derivative at the boundary. More
generally, a and b are allowed to be (given) functions, rather than constants
135
.
7.2.5 Cauchy Boundary Condition
In mathematics, a Cauchy boundary conditions augments an ordinary differential equation or a
partial differential equation with conditions that the solution must satisfy on the boundary; ideally
so to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function
value and normal derivative on the boundary of the domain. This corresponds to imposing both a
Dirichlet and a Neumann boundary conditions. It is named after the prolific 19th-century French
mathematical analyst Augustin Louis Cauchy
136
.
7.2.6 Periodic (Cyclic Symmetry) Boundary Condition
Two opposite boundaries are connected and their values are set equal when the physical flow
problem can be considered to be periodic in space. They could be either physical or non-physical in
nature. Among non-physical conditions, inflow, outflow, symmetry plane, pressure and for physical
the wall (fixed, moving, impermeable, adiabatic, etc.). Some vendors choose their boundary to be
reflected by above description, (OpenFOAM®); and some (i.e., CD-Adapco® and Fluent®) to use their
own particular naming, depending to application in hand.
7.2.7 Generic Boundary Conditions
The most widely used generic B.C’s are:
• Walls (fixed, moving, impermeable, adiabatic etc.)
• Symmetry planes
• Inflow
• Outflow
• Free surface
• Pressure
• Scalars (Temperature, Heat flux)
• Velocity
• Internal
134
Gustafson, K., (1998). Domain Decomposition, Operator Trigonometry, Robin Condition, Contemporary
Mathematics, 218. 432–437.
135
Wikipedia,
136
From Wikipedia, the free encyclopedia.
115
• Pole
• Periodic
• Porous media
• Free-Stream
• Non-Reflecting
• Turbulence-Intensity
• Immersed
• Free Surface
Among others and excellent descriptive available through literature for each.
7.3 Wall Boundary Conditions
All practically relevant flows situations are wall-bounded and near walls the exchange of
mass, momentum and scalar quantities is largest. At a solid wall Stokes flow theory is valid
i.e. the fluid adheres to the wall and moves with the wall velocity. Different treatment for the
different variables in the Navier-Stokes equations is required.
7.3.1 Velocity Field
The fluid velocity components equal the velocity of the wall. The normal and tangential velocity
components at an impermeable, non-moving wall are:
Eq. 7.3.1
Mass fluxes are zero and hence convective fluxes are zero.
Eq. 7.3.2
Diffusive fluxes are non-zero and result in wall-shear stresses.
Eq. 7.3.3
7.3.2 Pressure
The specification of wall boundary conditions for the pressure depends on the flow situation. In a
parabolic or convection dominated flow a von Neumann boundary condition is used at the wall:
Eq. 7.3.4
In a flow with complex curvilinear boundaries, at moving walls, or in flows with considerably large
external forces there may exist large pressure gradients towards the walls. The most common
treatment of such boundaries is a linear extrapolation form the inner flow region. If the exact value
of the pressure at the boundaries is not of interest no boundary conditions are needed when a
staggered grid is used. When a pressure correction method is used, wall boundary conditions are
also needed for pressure correction variable p’. Conservation of mass is only ensured when p’=0 at
the walls. For the purpose of stability this is usually accomplished by a zero gradient condition. The
boundary conditions for the pressure and for the velocity components are valid for both laminar and
116
turbulent flows. In the case of a turbulent flow near wall gradients are significantly larger and a very
high resolution is required particularly for high Reynolds number flows. Therefore, wall functions
were invented that bridge the near wall flow with adequate (mostly empirical) relationships.
7.3.3 Scalars/Temperature
Direct specification of the scalar/temperature at the wall boundary (Dirichlet Boundary condition)
Eq. 7.3.5
Specification of a scalar/temperature gradient i.e. specification of a scalar/temperature flux (von
Neumann Boundary condition):
Eq. 7.3.6
7.3.3.1 Common Inputs for Wall Boundary Condition
• Thermal boundary conditions (for heat transfer calculations).
• Wall motion conditions (for moving or rotating walls).
• Shear conditions (for slip walls, optional).
• Wall roughness (for turbulent flows, optional).
• Species boundary conditions (for species calculations).
• Chemical reaction boundary conditions (for surface reactions).
• Radiation boundary conditions.
• Discrete phase boundary conditions (for discrete phase calculations).
• Wall adhesion contact angle (for VOF calculations, optional).
7.4 Symmetry Planes
Used at the centerline (y = 0) of a 2D
axisymmetric grid. Can also be used where
multiple grid lines meet at a point in a 3D O type
grid. They used in CFD simulations to reduce the
numerical effort (see Figure 7.3.1). Must be used
carefully and only when both geometry and flow are
symmetrical. Unsteady flows around symmetrical
obstacles are always asymmetric: e.g. flow around a
square obstacle. Steady flows in symmetrical
diffusers or channel expansions can be asymmetric
and symmetry conditions should only be used when
an asymmetric flow can be excluded a priori. At a
symmetry boundary the following conditions apply:
➢ The boundary normal component of the velocity disappears and the flux through the
boundary is zero:
Eq. 7.4.1
➢ Scalars have all zero gradients. Consequently the diffusive fluxes of the scalars are also
zero:
Figure 7.3.1 Symmetry Plane to Model one
Quarter of a 3D Duct
117
Eq. 7.4.2
➢ The boundary normal gradient of tangential velocity components is also zero. As a result,
the shear stresses disappear
7.5 Inflow Boundaries
An inflow boundary is an artificial boundary that is used in CFD simulations because the
computational domain must be finite. Proper use of inflow boundary conditions can reduce the
numerical effort and need to be selected carefully so that the flow physics is not altered. At the inflow
usually variables are specified directly i.e. Dirichlet condition. The convective fluxes can be computed
and are added to source term. Diffusive fluxes are computed and added to the central coefficient AP.
Common inflow boundaries are: Pressure inlet, Velocity inlet, Mass flow inlet, among others.
7.5.1 Velocity Inlet
This types of boundary conditions are used to define the velocity and scalar properties of the flow at
inlet boundaries. The contribution inputs usually includes:
• Velocity magnitude and direction or velocity components
• Rotating (Swirl) velocity (for 2D axisymmetric problems with swirl)
• Temperature (for energy calculations)
• Turbulence parameters (for turbulent calculations)
• Radiation parameters
• Chemical species mass fractions (for species calculations)
• Mixture fraction and variance (for non-premixed or partially premixed combustion
calculations)
• Discrete phase boundary conditions (for discrete phase calculations)
• Multiphase boundary conditions (for general multiphase calculations)
7.5.2 Pressure Inlet
These boundary conditions are used to define the total pressure and other scalar quantities at flow
inlets. Required inputs are:
• Total (stagnation) Pressure
• Total (stagnation) Temperature
• Flow direction
• Static pressure
• Turbulence parameters (for turbulent calculations)
• Radiation parameters
• Chemical species mass fractions (for species calculations)
• Mixture fraction and variance (for non-premixed or partially premixed combustion
calculations)
7.5.3 Mass Flow Inlet
These boundary conditions are used in compressible flows to prescribe a mass flow rate at an inlet.
It is not necessary to use mass flow inlets in incompressible flows because when density is constant,
velocity inlet boundary conditions will fix the mass flow. Some of the common inputs are:
• Mass flow rate, mass flux, or (primarily for the mixing plane model) mass flux with average
mass flux
118
• Total (stagnation) temperature
• Static pressure
• Flow direction
• Turbulence parameters (for turbulent calculations)
• Radiation parameters
• Chemical species mass fractions (for species calculations)
• Mixture fraction and variance (for non-premixed or partially premixed combustion
calculations)
• Discrete phase boundary conditions (for discrete phase calculations)
• Open channel flow parameters (for open channel flow calculations using the VOF multiphase
model)
7.5.4 Inlet Vent
boundary conditions are used to model an inlet vent with a specified loss coefficient, flow direction,
and ambient (inlet) total pressure and temperature.
7.6 Outflow Boundaries
An outflow boundary is also an artificial boundary that is used in CFD simulations because the
computational domain must be finite. The location of the outflow boundary must be sufficiently
downstream of the region of interest. At the outlet boundary recirculation zones may not be present
and streamlines should be parallel. The mathematical formulation of the boundary condition may not
influence the flow in the inner part of the domain. Zero gradient conditions are most widely used for
all variables. The outlet boundary is usually used to check global mass conservation during an
iterative process. Commonly used outflow boundaries include: Pressure outlet, Pressure far-field,
Outlet vent, and Exhaust fan.
7.6.1 Pressure Outlet
These boundary conditions are used to define the static pressure at flow outlets (and also other scalar
variables, in case of back flow). The use of a pressure outlet boundary condition instead of an out
flow condition often results in a better rate of convergence when back flow occurs during iteration.
The contributions inputs requires are:
• Static pressure
• Backflow conditions
• Total (stagnation) Temperature (for energy calculations)
• Backflow direction specification method
• Turbulence parameters (for turbulent calculations)
• Chemical species mass fractions (for species calculations)
• Mixture fraction and variance (for non-premixed or partially premixed combustion
calculations)
• Multiphase boundary conditions (for general multiphase calculations)
• Radiation parameters
• Discrete phase boundary conditions (for discrete phase calculations)
• Open channel flow parameters (for open channel ow calculations using the VOF
• multiphase model)
• Non-reflecting boundary (for compressible density-based solver)
• Target mass flow rate (not available for multiphase flows)
119
7.6.2 Pressure Far-Field
boundary conditions are used to model a free-stream compressible flow at in unity, with free-stream
Mach number and static conditions specified. This boundary type is available only for compressible
flows. Inputs are:
• Static pressure.
• Mach number.
• Temperature.
• Flow direction.
• Turbulence parameters (for turbulent calculations).
• Radiation parameters.
• Chemical species mass fractions (for species calculations).
• Discrete phase boundary conditions (for discrete phase calculations).
7.6.3 Outflow
Boundary conditions are used to model flow exits where the details of the flow velocity and pressure
are not known prior to solution of the flow problem. They are appropriate where the exit flow is close
to a fully developed condition, as the outflow boundary condition assumes a zero normal gradient
for all flow variables except pressure. They are not appropriate for compressible flow calculations.
7.6.4 Outlet Vent
boundary conditions are used to model an outlet vent with a specified loss coefficient and ambient
(discharge) static pressure and temperature. The inputs are:
• Static pressure
• Backflow conditions
• Total (stagnation) temperature (for energy calculations)
• Turbulence parameters (for turbulent calculations)
• Chemical species mass fractions (for species calculations)
• Mixture fraction and variance (for non-premixed or partially premixed combustion
calculations)
• Multiphase boundary conditions (for general multiphase calculations)
• Radiation parameters
• Discrete phase boundary conditions (for discrete phase calculations)
• Loss coefficient
• Open channel flow parameters (for open channel flow calculations using the VOF
multiphase model)
7.6.5 Exhaust Fan
Boundary conditions are used to model an external exhaust fan with a specified pressure jump and
ambient (discharge) static pressure.
• Static pressure
• Backflow conditions
• Total (stagnation) temperature (for energy calculations)
• Turbulence parameters (for turbulent calculations)
• Chemical species mass fractions (for species calculations)
• Mixture fraction and variance (for non-premixed or partially premixed combustion
calculations)
• Multiphase boundary conditions (for general multiphase calculations)
120
• User-defined scalar boundary conditions (for user-defined scalar calculations)
• Radiation parameters
• Discrete phase boundary conditions (for discrete phase calculations)
• Pressure jump
• Open channel flow parameters (for open channel ow calculations using the VOF
multiphase model).
7.7 Free Surface Boundaries
7.7.1 Velocity Field and Pressure
Free surface boundaries can be rather complex and the location of the free surface is usually not
known a-priori. E.g. the swash of a fluid in a tank, the pouring of liquid into a glass. Only at the initial
time the position of the free surface is known and in the following an additional transport equation
to determine the location of the free surface is needed. Two boundary conditions apply at the free
surface boundary:
• Kinematic boundary condition - Fluid cannot flow through the boundary. i.e. the normal
component is equal to the surface velocity.
• Dynamic boundary condition - All forces that are acting on the free surface have to be in
equilibrium. These include shear stresses from the fluid below the surface and possibly
from a second fluid on the other side fluid and surface tension
137
.
In many CFD applications the free surface is treated as a flat plane where the symmetry condition is
applied.
7.7.2 Scalars/Temperature
Treated in an analogue manner as
the wall boundary condition. Direct
specification i.e. Dirichlet boundary
conditions or von Neumann
boundary conditions or a
combination of both.
7.8 Pole (Axis) Boundaries
Used at the centerline (y = 0) of a 2-D axisymmetric grid (see Figure 7.7.1). It can also be used where
multiple grid lines meet at a point in a 3-D O-type grid. No other inputs are required.
7.9 Periodic Flow Boundaries
Periodicity simply corresponds to matching conditions on the two boundaries. The velocity field is
periodic BUT the pressure field is not. The pressure gradient drives the flow and is periodic. A
pressure JUMP condition on the boundary must be specified
138
. Used when physical geometry of
interest and expected flow pattern and the thermal solution are of a periodically repeating nature
(see Figure 7.10.1).
7.10 Non-Reflecting Boundary Conditions (NRBCs)
Many problems in computational fluid dynamics occur within a limited portion of a very large or
infinite domain. Difficulties immediately arise when one attempts to define the boundary condition
137
Georgia Tech Computational Fluid Dynamics Graduate Course; spring 2007.
138
Solution methods for the Incompressible Navier-Stokes Equations.
Figure 7.7.1 Pole (Axis) Boundary
121
for such a system. Such boundary conditions are necessary for the problem to be well-posed, but the
physical system under consideration has no boundary to model. One needs to define an artificial
boundary whose behavior models the open edge of the physical system. Such a boundary definition
is often called a non-reflecting boundary condition (NRBC), as its primary function is to permit wave
phenomena to pass through the open boundary without reflection. The standard pressure boundary
condition, imposed on the boundaries of artificially truncated domain, results in the reflection of the
outgoing waves. As a consequence, the interior domain will contain spurious wave reflections. Many
applications require precise control of the wave reflections from the domain boundaries to obtain
accurate flow solutions. Non-reflecting boundary conditions provide a special treatment to the
domain boundaries to control these spurious wave reflections. The method is based on the Fourier
transformation of solution variables at the non-reflecting boundary
139
. The solution is rearranged as
a sum of terms corresponding to different frequencies, and their contributions are calculated
independently. While the method was originally designed for axial turbomachinery, it has been
extended for use with radial turbomachinery. In many applications of CFD such as Turbomachinery
because of close approximately of blades and the physical conditions, it is warranted to use NRBC’s.
Another prime candidate is Computational Aero-Acoustics (CAA) which is concerns with propagation
of traveling sound waves. In other word, by restricting our area of interest, we effectively create a
boundary where none exists physically, dividing our computational domain from the rest of the
physical domain. The challenge we must overcome, then, is defining this boundary in such a way that
it behaves computationally as if there were no physical boundary
140
.
7.10.1 Case Study 1 - Turbomachinery Application of 2-D Subsonic Cascade
The first test case is an axial turbine blade where both the in- and outflow are subsonic and the NRBC
will be compared to the Riemann boundary conditions. In the short flow-field simulations the in- and
outflow boundaries are positioned at 0.4 times the chord from the airfoil. For the long flow-field
simulation this distance becomes 1.5 times the chord. Figure 7.10.2 shows contour plot of the
pressure of the flow. The field of interest is the flow-field close to the boundary
141
. To give a detailed
look at that part of the flow, the pressure contours are put in close proximity. Unfortunately this
139
M. Giles, “Non-Reflecting Boundary Conditions for the Euler Equations.”, Technical Report TR 88-1-1988,
Computational Fluid Dynamics Laboratory, Massachusetts Institute of Technology, Cambridge, MA.
140
John R. Dea, “High-Order Non-Reflecting Boundary Conditions for the Linearized Euler Equations”, Monterey,
California, 2008.
141
F. De Raedt, “Non-Reflecting Boundary Conditions for non-ideal compressible fluid flows”, Master of Science at
the Delft University of Technology, defended publicly on December 2015.
Figure 7.10.1 Periodic Boundary
122
means the flow-field at the suction side becomes less clear. The subsonic flow means that any
reflections diffuse fairly quickly. Therefore there are almost no observable differences when the long
flow-field is considered. For the short flow-field the reflections become more apparent when
Riemann boundary conditions are used. At the outflow the pressure contours are clearly deflected
away from the boundary and never cross it. At the inflow the opposite happens and the pressure
contours are bend towards the boundary. This behavior is not observed when looking at the NRBC.
Clearly these boundary conditions are successful in removing the reflections from the flow. One can
have a closer look at the boundary itself to further clarify this comparison. The pressure at the
outflow boundary presented in Figure 7.10.2, where we notice that the NRBC do a better job of
simulating the pressure at the outflow, although it should be noted that on the absolute scale, all the
differences are very small.
Figure 7.10.2 Pressure contours plot for 2nd order spatial discretization scheme
123
7.10.2 Case Study 2 - CAA Application of Airfoil Turbulence Interaction Noise Simulation
The instantaneous contours of the non-dimensional pressure that is radiated from the airfoil due to
the turbulence interaction mechanism (see Figure 7.10.3). In each case, the entire simulated
domain is shown. It is qualitatively displays that the acoustic pressure waves do not appear to be
acted by the edges of the domain, and are not acted by the changes in domain size between the two
simulations. An exception to this is at the domain edge directly downstream of the airfoil. In this
region, unphysical pressure disturbances can be seen that correspond to the vortical turbulence
encountering the NRBC’s region. However, because these pressure disturbances appear inside the
zonal NRBC region, they are contained and do not radiate back into the domain
142
.
7.11 Turbulence Intensity Boundaries
When turbulent flow enters domain at inlet, outlet, or at a far-field boundary, boundary values are
required for
143
:
➢ Turbulent kinetic energy k.
➢ Turbulence dissipation rate ε.
Four methods available for specifying turbulence parameters:
➢ Set k and ε explicitly.
➢ Set turbulence intensity and turbulence length scale.
➢ Set turbulence intensity and turbulent viscosity ratio.
➢ Set turbulence intensity and hydraulic diameter.
7.11.1 Turbulence Intensity
The turbulence intensity I defined as:
u
2/3k
I =
Eq. 7.11.1
Here k is the turbulent kinetic energy and u is the local velocity magnitude. Intensity and length scale
depend on conditions upstream:
142
James Gill, Ryu Fattah, and Xin Zhangz, “Evaluation and Development of Non-Reactive Boundary Conditions
for Aeroacoustics Simulations”, University of Southampton, Hampshire, SO16 7QF, UK.
143
Bakker, Andre, ”Applied Computational Fluid Dynamics; Lecture 6 - Boundary Conditions”, 2002.
Figure 7.10.3 Aero-Acoustics Application for NRBC’
Domain X Domain X/2
124
➢ Exhaust of a turbine. (Intensity = 20%.
Length scale=1-10 % of blade span).
➢ Downstream of perforated plate or
screen (intensity=10%. Length scale =
screen/hole size).
➢ Fully-developed flow in a duct or pipe
(intensity= 5%. Length scale =
hydraulic diameter).
7.12 Immersed Boundaries
The immersed boundaries (IB) method allows
one to greatly simplify the grid generation and
even to automate it completely. The governing
equations are solved directly on a grid in their
simplest form by means of very efficient
numerical schemes. The grid generator detects
the cell faces that are cut by the body surface
and divides the cells into three types: solid and fluid cells, whose centers lie within the body and
within the fluid, respectively; and fluid/solid interface cells, which have at least one of their neighbors
inside the body/fluid. Then, the centers of the fluid and solid-interface cells are projected onto the
body surface along its normal direction, so as to obtain fluid-cells projection points and solid-cell
projection points, (see Figure 7.12.1).
7.13 Free Surface Boundary
Free surfaces occur at the interface between two fluids. Such interfaces require two boundary
conditions to be applied
144
:
• A kinematic condition which relates the motion of the free interface to the fluid velocities at
the free surface and
• A dynamic condition which is concerned with the force balance at the free surface.
7.13.1 The Kinematic Boundary Condition
The position of a free surface can always be given in implicit form as F(xj , t) = 0. For instance, in
Figure 7.13.2 the height of the free surface above the x-axis is specified as y = h(x, t) and an
appropriate function F(x, y, t) would be given by F(x, y, t) = h(x, t) − y. Fluid particles on the free
surface always remain part of the free surface, therefore we must have:
Eq. 7.13.1
144
“Viscous Fluid Flow: Boundary and initial conditions”, Lecture Series, Manchester, UK.
Figure 7.12.1 Immersed Boundaries
Fluid Cells
Solid Cells
Interface Cells
125
This is the kinematic boundary condition. For
surfaces whose position is described in the
form z = h(x, y, t), the kinematic boundary
condition becomes
Eq. 7.13.2
where u, v, w, are the velocities in the x, y, z
directions, respectively. For steady problems,
we have DF/Dt = 0 and the kinematic boundary
condition can be written as uini = 0 or
symbolically u · n = 0, where n is the outer unit
normal on the free surface. This condition
implies that there is no flow through the free
surface (but there can be a flow tangential to
it!).
7.13.2 The Dynamic Boundary Condition
The dynamic boundary condition requires the stress to be continuous across the free surface which
separates the two fluids (e.g., air and water). The traction exerted by fluid (1) onto fluid (2) is equal
and opposite to the traction exerted by fluid (2) on fluid (1). Therefore we must have t(1) = −t(2).
Since n(1) = −n(2) (see Figure 7.13.2) we obtain the dynamic boundary condition as
Eq. 7.13.3
where we can use either n(1) or n(2) as the unit normal. On curved surfaces, surface tension can
create a pressure jump across the free surface. The surface tension induced pressure jump is given
by
Eq. 7.13.4
In this expression σ is the surface tension of the fluid and κ is equal to twice the mean curvature of
the free surface, where, R1 and R2 are the principal radii of curvature of the surface (for instance, κ
= 2/a for a spherical drop of radius a and κ = 1/a for a circular jet of radius a). Surface tension acts
like a tensioned membrane at the free surface and tries to minimize the surface area. Hence the
pressure inside a spherical drop (or inside a circular liquid jet) tends to be higher than the pressure
in the surrounding medium. If surface tension is important, the dynamic boundary condition has to
be modified to
Eq. 7.13.5
where κ > 0 if the centers of curvature lie inside fluid (1).
7.14 Other Boundary Conditions
Other boundary conditions can occur in special applications. For instance, the presence of an elastic
boundary leads to fluid-structure interaction problems in which the fluid velocity has to be equal to
the velocity of the elastic wall, while the elastic wall deforms in response to the traction that the fluid
Figure 7.13.2 Sketch Exemplifying the
conditions at a Free Surface Formed by the
Interface Between Two Fluids
126
exerts on it. At porous walls, the no-penetration condition no longer holds: the volume flux into the
wall is often proportional to the pressure gradient at the porous surface. Non-uniformly distributed
surfactants (substances which reduce the surface tension) can induce tangential stresses at free
surfaces, etc.
7.15 Further Remarks
For an incompressible fluid, the boundary conditions need to fulfill the overall consistency condition
Eq. 7.15.1
where ∂V is the surface of the spatially fixed volume in which the equations are solved. If there are
no free surfaces (and associated dynamic boundary conditions), the pressure is only defined up to an
arbitrary constant as only the pressure gradient (but not the pressure itself) appears in the Navier-
Stokes equations. For initial value problems, the initial velocity field (at t = 0) already has to fulfill the
incompressibility constraint. These remarks are particularly important for the numerical solution of
the Navier-Stokes equations
145
.
145
“Viscous Fluid Flow: Boundary and initial conditions”, Lecture Series, Manchester, UK.
127
8 Linear PDEs and Model Equations
8.1 Mathematical Character of Basic Equations
The general theory concerning the character of PDE has been developed based on following relation
where A, B, C, D, E, F are assumed to be function of (x , y) only for the time being; making it linear in
nature
Eq. 8.1.1
It is found that character of Eq. 8.1.1 depends upon the sign of determinate function B2 - 4AC as the
flow dependencies for each case shown by solid line. In summary,
0
yx
0
yx
0
yx 2
2
2
2
2
2
2
2
2
2=
−
=
−
=
+
In reality, the viscous flow equations are simply too complicated to fit into a single mode. They can
be elliptic, parabolic, and hyperbolic or mixture of all three, depending to specific flow, geometry
and time dependencies. Some examples of these model equations will be dealt extensively later. For
example, the unsteady compressible N-S equations are a mixed set of hyperbolic-parabolic equations,
while, for 2D unsteady incompressible N-S for x-momentum is mixed set of elliptic-parabolic-
hyperbolic equations as depicted below:
Eq. 8.1.2
Consequently, different numerical techniques must be used in to solve the N-S equations in
compressible and incompressible flow regions. Similarly the Euler equations governing an in-viscid,
B2- 4AC < 0
Elliptic
Boundary Value
problem
complete contour
boundary specification
Laplace equation
No real charateristics
B2- 4AC = 0
Parabolic
mixed initial and
Boundary value problem
boundary conditions must be
closed at one end but remain
open at other
Heat conduction equation
One real charateristics
B2- 4AC > 0
Hyperbolic
initial value problem
specifying boundary
conditions at one end but
remain open at others
Wave equation
Two real charateristics
128
non-heat conducting gas have a different character in different flow regions. If the time dependent
terms are retained, the resulting unsteady equations are hyperbolic and solutions can be obtained by
marching procedures. The situation is different when a steady flow is assumed. In that case, the Euler
equations are elliptic when flow is subsonic and hyperbolic when it is supersonic. It could be said
that Euler’s equation is hyperbolic in temporal domain and elliptic in special domain. Therefore,
different flow regions means different characteristics and demands different solving procedure. A
major difference between subsonic and supersonic flows is that flow disturbances propagate
everywhere throughout a subsonic flow; whereas they cannot propagate upstream in supersonic
flow.
8.1.1 Nonsingular Transformation
It implies that x and y are transformed into new independent variables ξ and η. We also require that
this transformation be nonsingular which provides that a one to one relationship exists between (x,
y) and (ξ, η)
146
. We are also assumed of a non-singled mapping provided that the Jacobian of
Transformation is non-zero.
0ηξηξ
y
η
x
η
y
ξ
x
ξ
y)(x,
),(
xyyx −=
=
=
J
Eq. 8.1.3
Therefore, any real nonsingular transformation does not change the type of PDE
147
.
8.1.2 The 'Par-Elliptic' problem
An important practical use of the parabolic solution procedure is to refine an elliptic-flow solution of
the region outside the boundary layer. In this case it is from the elliptic-flow solution that the
pressure must be extracted: pressures for the
nearest-to-surface cells of the elliptic grid are
transferred to all the cells in the parabolic grid,
at the same z location. In general, because the
cells of the parabolic and elliptic grids are likely
to be of different sizes, as shown below,
interpolation is needed. The above sketch
contains a reminder that a two-way
interchange of information may take place
between the elliptic and parabolic calculations
(see Figure 8.1.1). Thus the elliptic calculation
may take place at first, with the assumption
that friction at the solid surfaces is absent. Its
predicted pressure distribution is for that
reason not quite correct. However the ensuing
parabolic calculation takes detailed account of
friction and can report the so-called
'displacement thickness' of the boundary layer.
If this is transmitted back to the elliptic solver, that can repeat its calculation on the assumption that
146
Anderson, Dale A; Tannehill, John C; Plecher Richard H; 1984:”Computational Fluid Mechanics and Heat
Transfer”, Hemisphere Publishing Corporation.
147
Taylor, A., E., “Advanced Calculus”, Ginn and Company, Boston, 1955.
Figure 8.1.1 Two-way interchange of information
between Parabolic and Elliptic flows
129
the effective size of the solid object is larger than it first supposed. Its second flow prediction will be
correspondingly more accurate
148
.
8.2 Exact (Closed Form) Solution Methods to Model Equations
Since the exact solutions of Naver-Stokes , Boundary Layer, or Euler methods is not available yet, we
resort to model equation with reduced order to find a closed form solution. For example the viscous
Burger equation can be modelled as a reduced NS equation. The discussion here has centered on the
2nd order equation given by Wave Equation, Heat Equation, and Laplace Equation. In addition, system
of 1st order equations were examined. A number of other very important equations should be
mentioned since they govern common physical phenomena or they are used as simple models for
more complex problems. In many cases, exact analytical solutions for these equations exist.
8.2.1 Linear Wave Equation (1st Order)
The first and widely used is the 1st order linear
wave equation which governs the propagation
of a wave moving to the right at a constant speed
a. This is also called the advection equation and
(see Figure 8.2.1). A classic example of
hyperbolic equation also requires an initial
condition, u(x, 0) = u0(x). The question of what
boundary conditions are appropriate for this
equation can be more easily be answered after
considering its solution. It is easy to verify that
the solution is given by u(x, t) = u0(x − at). This
describes the propagation of the quantity u(x, t)
moving with speed “a” in the x-direction. The
solution is constant along the characteristic
line x − at = c with u(x, t) = u0(c). From the
knowledge of the solution, we can appreciate
that for a > 0 a boundary condition should be
prescribed at x = 0, (e.g., u (0) = α0) where
information is being fed into the solution
domain. The value of the solution at x = 1 is
determined by the initial conditions or the
boundary condition at x = 0 and cannot,
therefore, be prescribed. This simple argument
shows that, in a hyperbolic problem, the
selection of appropriate conditions at a
boundary point depends on the solution at that
point. If the velocity is negative, the previous
treatment of the boundary conditions is
reversed.
8.2.2 Inviscid Burgers Equation
This is also called non-linear 1st order wave
equation and governs the propagation of
nonlinear wave for 1-D case. An analogous
analysis to that used for the advection equation
148
Parabolic Flows by “PHOENICS”.
xkeAtxu
x
u
u
t
u
m
t
mmmsin),(
0
=
=
+
Figure 8.2.2 Formulation of discontinuities in
non-linear Burgers (wave) equation
0=
+
x
u
a
t
u
Figure 8.2.1 Solution of linear Wave Equation
130
shows that u(x, t) is constant if we are moving with a local velocity also given by u(x, t). This means
that some regions of the solution advance faster than other regions leading to the formation of sharp
gradients. This is illustrated in Figure 8.2.2. The initial velocity is represented by a triangular “zig-
zag” wave. Peaks and troughs in the solution will advance, in opposite directions, with maximum
speed. This will eventually lead to an overlap as depicted by the dotted line. This also results in a
non-uniqueness of the solution which is obviously non-physical and to resolve this problem we must
allow for the formation and propagation of discontinuities when two characteristics intersect.
8.2.3 Diffusion (Heat) Equation
Which is parabolic in nature and in addition to appropriate boundary conditions of the form used for
elliptic equations, we also require an initial condition at t = 0 of the form u(x, 0) = u0(x) where u0 is a
given function. If b is constant, this equation admits solutions of the form u(x, t) = Aeβt sin kx if β +
k2a = 0. A notable feature of the solution is that it decays when b is positive as the exponent β < 0.
The rate of decay is a function of a. The more diffusive the equation (i.e., larger a) the faster the decay
of the solution is. In general the solution can be made up of many sine waves of different frequencies,
i.e., a Fourier expansion of the form
x sinkeAt)u(x, m
tβ
mmm
=
Eq. 8.2.1
Where Am and km represent the amplitude
and the frequency of a Fourier mode,
respectively. The decay of the solution
depends on the Fourier contents of the
initial data since βm = −k2m b. High
frequencies decay at a faster rate than the
low frequencies which physically means
that the solution is being smoothed. This is
illustrated in Figure 8.2.3 which shows
the time evolution of u(x, t) for an initial
condition u0(x) = 20x for 0 ≤ x ≤ 1/2 and
u0(x) = 20(1 − x) for 1/2 ≤ x ≤ 1. The
solution shows a rapid smoothing of the
slope discontinuity of the initial condition
at x = 1/2. The presence of a positive
diffusion (a > 0) physically results in a
smoothing of the solution which stabilizes
it. On the other hand, negative diffusion (a
< 0) is de-stabilizing but most physical
problems have positive diffusion.
8.2.4 Viscous Burgers Equation
This is the nonlinear equation with diffusion added. This particular form is very similar to the
equations of governing fluid flow and can be used as a simple nonlinear model for numeric.
0
2
2=
−
x
u
a
t
u
Figure 8.2.3 Rate of Decay of solution to diffusion
Equation
131
xtaxξ where
)/)aexp((11
)/)a1)exp((1(2a1
u solution with
x
u
υ
x
u
u
t
u
00
0
00
2
2
−−=
−+
−−+
=
=
+
Eq. 8.2.2
8.2.5 Tricomi Equation
This equation governs problems of the mixed type such as inviscid transonic flows. The properties of
Tricomi equations include a change of form from elliptic to hyperbolic character depending upon sign
of y.
0
y
u
x
u
y 2
2
2
2=
+
Eq. 8.2.3
8.2.6 2D Laplace Equation
The linear Elliptic 2D Laplace equation (see Figure 8.2.4) has following solutions
)sinh(
x)y)sin(sinh(y)x)sin(sinh(
[0,1] , constant k sin(kx)e ,
yx)(1
2y
, yx ,xy
:solution a have 0
kx
22
22
2
2
2
2
+
=
==
++
=−==
=
+
yx
Eq. 8.2.4
8.2.6.1 Boundary Conditions
The Dirichlet problem for Laplace's equation consists of finding a solution φ on some domain D such
that φ on the boundary of D is equal to some given function. Since the Laplace operator appears in
the heat equation, one physical interpretation of this
problem is as follows: fix the temperature on the
boundary of the domain according to the given
specification of the boundary condition. Allow heat
to flow until a stationary state is reached in which
the temperature at each point on the domain doesn't
change anymore. The temperature distribution in
the interior will then be given by the solution to the
corresponding Dirichlet problem
149
.
The Neumann boundary conditions for Laplace's
equation specify not the function φ itself on the
boundary of D, but its normal derivative. Physically,
this corresponds to the construction of a potential
for a vector field whose effect is known at the
boundary of D alone. Solutions of Laplace's equation
149
From Wikipedia, the free encyclopedia.
0
2
2
2
2=
+
yx
Figure 8.2.4 Solution to Laplace equation
132
are called harmonic functions; they are all analytic within the domain where the equation is satisfied.
If any two functions are solutions to Laplace's equation (or any linear homogeneous differential
equation), their sum (or any linear combination) is also a solution. This property, called the principle
of superposition, is very useful, e.g., solutions to complex problems can be constructed by summing
simple solutions
150
-
151
.
8.2.7 Poisson’s Equation
This elliptic equation governs the temperature
distribution in the solid with heat source described
by f(x, y). Poisson’s equation (see Figure 8.2.5) also
determines the electric field in a region containing a
charge density f(x, y). The f(x,y) in this case is
sin(πx)Sin(πy) and the B.C. is u(x,y)=0 subject to 0 ≤
x , 1 ≤ y, ux(0,y)=-sin(yπ)/2π. The solution is
obtained in bottom of Figure 8.2.5.
8.2.8 The Advection-Diffusion Equation
This particular expression represents the advection
of a quantity ξ in a region with velocity u. The
quantity υ is a diffusion or viscosity coefficient and
a is a constant > 0.
[0,1]u sin(kx)u(x) andconstant k
at)xkυυxt)sin(exp(t)u(x,
: is solution exact the
(4.6)
x
ξ
υ
x
ξ
a
t
ξ
2
2
==
−−=
=
+
Eq. 8.2.5
8.2.9 The Korteweg-De Vries Equation
The motion of nonlinear dispersive wave is governed by this example.
0
x
u
x
u
u
t
u
3
3=
+
+
Eq. 8.2.6
8.2.10 Helmholtz Equation
This equation governs the motion of time dependent harmonic waves where k is a frequency
parameter. Application includes the propagation of acoustics waves.
150
Hazewinkel, Michiel, ed. (2001), "Laplace equation", Encyclopedia of Mathematics, Springer, ISBN 978-1-
55608-010-4.
151
Example initial-boundary value problems using Laplace's equation from exampleproblems.com.
y) sin(π x)sin(πy)f(x,
y
u
x
u
2
2
2
2==
+
Figure 8.2.5 Solution to Poisson's equation
2
2
)()(
),(
−
=ySinxSin
yxu
133
0uk
y
u
x
u
2
2
2
2
2=+
+
Eq. 8.2.7
8.2.11 Exact Solution Methods
The solution is obtained from the list provided below. This list by no means exclusive and many more
exists in literature.
1. Method of Characteristics
2. Shock Capturing Methods
3. Similarity Solutions
4. SCM (Split Coefficient Method)
5. Methods for solving Potential Equation
6. Methods for solving Laplace equation
7. Separation of Variable
8. Complex Variables
9. Superposition of Non-Linear Equation
10. Transformation of Variables
11. Manufacturing Solutions
8.3 Solution Methods for In-Viscid (Euler) Equations
The interest in Euler equations arises from the fact that in many primary design the information
about the pressure alone is needed. In boundary layer where the skin friction and heat transfer is
required, the outer edge condition using the Euler. The Euler equation is also of interest because of
interest in major flow internal discontinuities such as shock wave or contact surfaces. Solutions
relating to Rankine-Hugonist equations are embedded in Euler equation. The Euler equations
govern the motion of an Inviscid, non-heat-conducting flow have different character in different
regions. If the flow is time-dependent, the flow regimes is hyperbolic for all the Mach numbers and
solution can be obtained using
marching procedures. The
situation is very different
when a steady flow is
assumed. In this case, Euler
equations are elliptic when
the flow is subsonic, and
hyperbolic when the flow is
supersonic. For transonic flows, has required research and development for many years. Table
8.3.1 shows the deferent flow regimes and corresponding mathematical character of the equations.
8.3.1 Method of Characteristics
Closed form solutions of non-linear hyperbolic partial differential equation do not exists for general
cases. In order to obtain the solution to such an equations we are required to resort to numerical
methods. The method of characteristics is the oldest and most nearly exact method in use to solve
hyperbolic PDEs. Even though this technique is been replaced by newer finite difference method. A
background in characteristic theory and its application is essential. The method of a characteristics
is a technique which utilizes the known physical behavior of the solution in each point in the flow.
Flow
Subsonic
M<1
Sonic M=1
Supersonic
M>1
Steady
Elliptic
Parabolic
Hyperbolic
Unsteady
Hyperbolic
Hyperbolic
Hyperbolic
Table 8.3.1 Classification of the Euler Equation on Different Regimes
134
8.3.2 Linear Systems
Consider Steady Supersonic of Inviscid, Non-heat conducting of small perturbation for 2D perfect
gas
152
.
−
−
=
=
=
+
=
−
=
−
=
==−=+−
01 β
1
0
][ and
v
u
where
0
y
][ formin vector e writ0
y
u
x
v
, 0
y
v
x
u
β
y
v,
x
u and β)M(1 denoting 0)M(1
2
2
2
yyxx
Aw
w
A
w
x
Eq. 8.3.1
The eigenvalues of this system are the eigenvalues of [A]. These are obtained by extracting the roots
of characteristics equation of [A] as
β
1
λ ,
β
1
λ , 0
β
1
λ
, 0
λ1β
1
λ
or 0]λ[][
21
2
2
2
−===−
=
−−
−−
=− IA
Eq. 8.3.2
This is pair of roots from the differential equation of
characteristics (see Figure 8.3.1). Next we
determine the compatibility equation. These
equations are obtained by pre-multiplying the
system of equations by left eigenvectors of [A]. This
effectively provides a method for writing the
equations along the characteristics. Let L1
represents the left eigenvectors of [A]
corresponding to λ1 and L2 represents the left
eigenvectors corresponding to λ2. Drive the eigenvectors of [A]:
1
β
L ,
1
β
L 0
β
1
1
β
1
β
1
l,l
l
l
L
0IλAL
21
A
2
L
21
2
1
1
i
T
i
T
=
−
==
−−
−−
→
=
=−
Eq. 8.3.3
152
D. Anderson, J., Tannehill, R., Pletcher,”Computational Fluid Mechanics and Heat Transfer”, ISBN 0-89116-
471-5 – 1984.
Figure 8.3.1 Characteristics of Linear
Equation
135
The compatibility equations along λ1 is obtained from
( ) ( ) ( ) ( )
vβu
yβ
1
vβu
x
manar similar in 0vβu
yβ
1
vβu
x
0
v
β
1
v
u
β
1
u
1] [-β obtained is λ alongty compatabil
(7.4) 0wλwLor 0[A]wwL
yx
yx
1
yix
T
i
yx
iT
+
−+
=−
+−
=
+
+
=+=+
Eq. 8.3.4
It is expressed the fact that quantity (βu-v) is constant along λ1, and (βu+v) is constant along λ2. The
quantities are called Riemann Invariants. Since these two quantities are constant and opposite pair
of characteristics, it is easy to determine u and v at a point. If at a point we know (βu-v) and (βu+v),
we can immediately compute both u and v.
8.3.3 Non-Linear Systems
The development presented so far is for a system linear equations for simplicity. In more complex
nonlinear settings, the results are not as easily obtained. In the general case, the characteristics slopes
are not constant and vary with fluid properties
153
. For a general nonlinear problem, the
characteristics equation must be integrated numerically to obtain a complete flow field solutions.
Consider a 2D supersonics flow of a perfect gas over a flat surface. The Euler equation governing this
inviscid flow as a matrix form
( )
( )
au
u
v
u
v
ρuρv
0uvρuaρva
0
ρu
au
au
u
v
0
0
p
v
auv
au
1
and
e
p
v
u
where
(7.5) 0
y
][
x
22
22
22
22
2
22
−−
−
−
−
−−
−
=
=
=
+
[A]w
w
A
w
Eq. 8.3.5
The eigenvalues of [A] determine the characteristics direction and are
154
153
See previous.
154
D. Anderson, J., Tannehill, R., Pletcher,”Computational Fluid Mechanics and Heat Transfer”, ISBN 0-89116-
471-5 – 1984.
136
au
avuauv
λ ,
au
avuauv
λ ,
u
v
λ ,
u
v
λ 22
222
4
22
222
321 −
−+−
=
−
−++
===
Eq. 8.3.6
The matrix of left eigenvectors associated with these values of λ may be written as
0
ρva
1
avu
1
v
u
v
avu
1
0
ρva
1
avu
1
v
u
avu
101ρvρu
10
a
ρv
a
ρu
][
222222
222222
22
1
−+
−
−+
−+
+
−+
−
=
−
T
Eq. 8.3.7
We obtain the compatibility relations by pre-multiplying the original system by [T]-1. These relations
along the wave fronts are given by:
4
444
3
333
λ
dx
dy
along 0
ds
dp
ρ
β
ds
dv
u
ds
du
v
λ
dx
dy
along 0
ds
dp
ρ
β
ds
dv
u
ds
du
v
==+−
==++−
Eq. 8.3.8
These are an ordinary differential equations which holds along the characteristic with slope λ3, λ4,
while arc length along this characteristics is denoted by s3, s4. In contrast to linear example, the
analytical solution for characteristics is not known for the general nonlinear problem. It is clear that
we must numerically integrate to
determine the shape of the
characteristics in step by step manner.
Consider the characteristic defined by
λ3. Stating at an initial data surface, the
expression can be integrated to obtain
the coordinates of next point at the
curve. At the same time, the
differentials equation defining the
other wave front characteristics can
be integrated. For a simple first-order
integration this provide us with two
equations for wave front
characteristics. From this expressions,
we determine the coordinate of their
intersection, point A. Once the point A
is known, the compatibility relations,
are integrated along the
characteristics to this point. This
provide a system of equations at point
Figure 8.3.2 Characteristics of nonlinear solution point
137
A. This is a first-order estimate of the both the location of point A and the associated flow variables.
In the next step, the new intersection point B can be calculated which now includes the nonlinear
nature of the characteristic curve. In a similar manner, the dependent variables at point B are
computed. Since the problem is nonlinear, the final intersection point B does not necessary appear at
the same value of x for all solution points. Consequently, the solution is usually interpolated onto an
x=constant surface before the next integration step. This requires additional logic and added
considerably to the difficulty in turning an accurate solution (see Figure 8.3.2)
155
.
155
D. Anderson, J., Tannehill, R., Pletcher,”Computational Fluid Mechanics and Heat Transfer”, ISBN 0-89116-
471-5 – 1984.
138
9 Lattice Boltzmann Method (LBM) and Its Siblings
9.1 Preliminaries and Background
Lattice Boltzmann Methods (LBM) (or Thermal Lattice Boltzmann Methods (TLBM)) is a CFD
methods for fluid simulation. Instead of solving the Navier–Stokes equations, the discrete Boltzmann
equation is solved to simulate the flow of a Newtonian fluid with collision models such as Bhatnagar–
Gross–Krook (BGK). By simulating streaming and collision processes across a limited number of
particles, the intrinsic particle interactions evince a microcosm of viscous flow behavior applicable
across the greater mass
156
. It is a modern approach in Computational Fluid Dynamics and often used
to solve the incompressible, time-dependent Navier-Stokes equations numerically. Its strength lies
however in the ability to easily represent complex physical phenomena, ranging from multiphase
flows to chemical interactions between the fluid and the surroundings. The method finds its origin in
a molecular description of a fluid and can directly incorporate physical terms stemming from a
knowledge of the interaction between molecules.
For this reason, it is an invaluable tool in fundamental research, as it keeps the cycle between the
elaboration of a theory and the formulation of a corresponding numerical model short. At the same
time, it has proven to be an efficient and convenient alternative to traditional solvers for a large
variety of industrial problems
157
. In LBM, the fluid is replaced by fractious particles. These particles
stream along given directions (lattice links) and collide at the lattice sites. The LBM can be considered
as an explicit method. The collision and streaming processes are local. Hence, it can be programmed
naturally for parallel processing machines. Another beauty of the LBM is handling complex
phenomena such as moving boundaries (multiphase, solidification, and melting problems), naturally,
without a need for
face tracing
method as it is in
the traditional
CFD.
The link between
the microscopic
and macroscopic
description of gas
dynamics can be
established by the
method of
moments, but the
most successful
approach is the
Enskog’s infinite
series expansion
[3]. Under the
collision equilibrium condition, the Boltzmann equations transform directly to the Euler equations,
which are essentially the Navier-Stokes equations but containing only the inviscid terms. The
hierarchy of fluid dynamics governing equation is depicting in Figure 9.1.1.
9.1.1 Approaches
There are two main approaches in simulating the transport equations (heat, mass, and momentum),
156
CFD online
157
NIST is an agency of the U.S. Commerce Department, “Lattice Boltzmann Methods”, 2002.
Figure 9.1.1 The hierarchy of conservation laws
139
continuum and discrete
158
. In continuum approach, ordinary or partial differential equations can be
achieved by applying conservation of energy, mass, and momentum for an infinitesimal control
volume. Since it is difficult to solve the governing differential equations for many reasons
(nonlinearity, complex boundary conditions, complex geometry, etc.), therefore finite difference,
finite volume, finite element, etc., schemes are used to convert the differential equations with a given
boundary and initial conditions into a system of algebraic equations. The algebraic equations can be
solved iteratively until convergence is insured. Let us discuss the procedure in more detail, first the
governing equations are identified (mainly partial differential equation). The next step is to discretize
the domain into volume, girds, or elements depending on the method of solution. We can look at this
step as each volume or node or element contains a collection of particles (huge number, order of
1016). The scale is macroscopic. The velocity, pressure, temperature of all those particles represented
by a nodal value, or averaged over a finite volume, or simply assumed linearly or bi-linearly varied
from one node to another.
The properties such as viscosity, thermal conductivity, heat capacity, etc. are in general known
parameters (input parameters, except for inverse problems). For inverse problems, one or more
thermos-physical properties may be unknown. On the other extreme, the medium can be considered
made of small particles (atom, molecule) and these particles collide with each other. This scale is
microscale. Hence, we need to identify the inter-particle (inter-molecular) forces and solve ordinary
differential equation of Newton’s second law (momentum conservation). At each time step, we need
to identify location and velocity of each particle, i.e, trajectory of the particles. At this level, there is
no definition of temperature, pressure, and thermo-physical properties, such as viscosity, thermal
conductivity, heat capacity, etc. For instance, temperature and pressure are related to the kinetic
energy of the particles (mass and velocity) and frequency of particles bombardment on the
boundaries, respectively. This method is called Molecular Dynamics (MD) simulations.
9.1.1.1 Dilute Gas Regimes
The rarefied gas dynamics can be best classified by Knudsen Number (Kn) where it is a measure of
collisions in a gas flow, and equals to mean free path of a molecule λ divided by L the flow length
scale. For rarefied gas, the Monte Carlo simulation has been used where it described as: Direct
simulation Monte Carlo (DSMC) method is the Monte Carlo method for simulation of dilute gas
flows on molecular level, i.e. on the level of individual molecules. To date DSMC is the basic
numerical method in the kinetic theory of gases and rarefied gas dynamics. It is basically a
generic numerical method for a variety of mathematical problems based on computer
generation of random numbers. Applications of DSMC simulations would be:
➢ Satellites and space crafts on LEO and in deep space
➢ Re-entry vehicles in upper atmosphere
➢ Nozzles and jets in space environment
➢ Dynamics of upper planetary atmospheres
➢ Fast, non-equilibrium gas flows (laser ablation, evaporation, deposition)
➢ Flows on microscale, microfluidics
to name a few
159
. The flow regimes of diluted gas, with the aid of Knudsen number, is demonstrated
in Figure 9.1.2.
158
A. A. Mohamad, “Lattice Boltzmann Method”, Fundamentals and Engineering Applications with Computer
Codes, ISBN 978-0-85729-454-8.
159
G.A. Bird, “Molecular gas dynamics and the direct simulation of gas flows”. Clarendon Press, 1994
140
9.1.2 Book Keeping
As usual, in the process of , the book keeping plays a major role. We need to identify location (x, y, z)
and velocity (cx, cy, and cz are velocity components in x; y and z direction, respectively) of each
particle. Also, the simulation time step should be less than the particles collision time, which is in the
order of fero-seconds (10-12 s). Hence, it is impossible to solve large size problems (order of cm) by
MD method. At this scale, there is no definition of viscosity, thermal conductivity, temperature,
pressure, and other phenomenological properties. Statistical mechanics need to be used as a
translator between the molecular world and the macroscopic world. The question is, is the velocity
and location of each particle important for us? For instance, in this room there are billions of
molecules traveling at high speed order of 400 m/s; like rockets, hitting us. But, we do not feel them,
because their mass (momentum) is so small. The resultant effect of such a ‘‘chaotic’’ motion is almost
nil, where the air in the room is almost stagnant (i.e., velocity in the room is almost zero). Hence, the
behavior of the individual particles is not an important issue on the macroscopic scale, the important
thing is the resultant effects. What about a middle man, sitting at the middle of both mentioned
techniques? the lattice Boltzmann method (LBM). The main idea of Boltzmann is to bridge the gap
between micro-scale and macro-scale by not considering each particle behavior alone but
behavior of a collection of particles as a unit. The property of the collection of particles is
represented by a distribution function. The keyword is the distribution function. The distribution
function acts as a representative for collection of particles. This scale is called meso-scale. The
mentioned methods are illustrated in Figure 9.1.2. LBM enjoys advantages of both macroscopic and
microscopic approaches, with manageable computer resources. LBM has many advantages. It is easy
to apply for complex domains, easy to treat multi-phase and multi-component flows without a need
to trace the interfaces between different phases. Furthermore, it can be naturally adapted to parallel
processes computing. Moreover, there is no need to solve Laplace equation at each time step to satisfy
continuity equation of incompressible, unsteady flows, as it is in solving Navier–Stokes (NS) equation.
However, it needs more computer memory compared with NS solver, which is not a big constraint.
Also, it can handle a problem in micro and macro scales with reliable accuracy.
9.1.3 Kinetic Theory
It is necessary to be familiar with the concepts and terminology of kinetic theory before proceeding
to LBM. The following sections are intended to introduce the reader to the basics and fundamentals
of kinetic theory of particles. To avoid the detail of mathematics; however more emphasis is given to
Figure 9.1.2 Flow Regimes for Diluted Gas
Free molecular
(collisionless) Flow
•Non-equilibrium
flows
•Kn >> 1 (Kn > 10)
Transitional Flow
•Non-equilibrium
flows
•Kn ~ 1
Continuum Flow
•Local equilibrium
•Kn << 1 (Kn < 0.01)
141
the physics. For detail information, see [A. A. Mohamad]
160
, [Bird]
161
, and [Golse]
162
.
9.1.4 Maxwell Distribution Function
In 1859, Maxwell (1831–1879) recognized that dealing with a huge number of molecules is difficult
to formulate, even though the governing equation (Newton’s second law) is known. As mentioned
before, tracing the trajectory of each molecule is out of hand for a macroscopic system. Then, the idea
of averaging came into picture. The idea of Maxwell is that the knowledge of velocity and position of
each molecule at every instant of time is not important. The distribution function is the important
parameter to characterize the effect of the molecules; what percentage of the molecules in a certain
location of a container have velocities within a certain range, at a given instant of time. The molecules
of a gas have a wide range of velocities colliding with each other’s, the fast molecules transfer
momentum to the slow molecule. The result of the collision is that the momentum is conserved. For
a gas in thermal equilibrium, the distribution function is not a function of time, where the gas is
distributed uniformly in the container; the only unknown is the velocity distribution function. For a
gas of N particles, the number of particles having velocities in the x-direction between cx and cx + dcx
is Nf (cx)dcx. The function f(dcx) is the fraction of the particles having velocities in the interval cx and
cx dcx; in the x-direction. Similarly, for other directions, the probability distribution function can be
defined as before. Then, the probability for the velocity to lie down between cx and cxdcx; cy and cy dcy;
and cz and cz dcz will be N f(cx) f(cy) f(cz) dcx dcy dcz: It is important to mention that if the above
equation is integrated (summed) over all possible values of the velocities, yields the total number of
particles to be N,
1dcdcdc )f(c )f(c )f(c zyxzyx
=
Eq. 9.1.1
Since any direction can be x, or y or z, the distribution function should not depend on the direction,
but only on the speed of the particles. Therefore,
)cc(c)f(c )f(c )f(c 2
z
2
y
2
xzyx ++=
Eq. 9.1.2
where φ is another unknown function, that need to be determined. The value of distribution function
should be positive (between zero and unity). Hence, in Eq. 9.1.2, velocity is squared to avoid negative
magnitude. The possible function that has property of Eq. 9.1.2 is logarithmic or exponential
function. After some manipulation and math, as well as help from kinetic theory, the final form of
distribution is
ec
2ππk
m
4π f(c) 2kT
mc
2
2
32
−
=
Eq. 9.1.3
Note that this function increases parabolically from zero for low speeds, reaches a maximum value
and then decreases exponentially. As the temperature increases, the position of the maximum shifts
to the right. The total area under the curve is always one, by definition. This equation called Maxwell
or Maxwell–Boltzmann Distribution function. For detail information, see 21.
160
A. A. Mohamad, “Lattice Boltzmann Method”, Fundamentals and Engineering Applications with Computer
Codes, ISBN 978-0-85729-454-8.
161
G.A. Bird, “Molecular gas dynamics and the direct simulation of gas flows”. Clarendon Press, 1994
162
François Golse, “The Boltzmann Equation and Its Hydrodynamic Limits”, Handbook of Differential Equations
Evolutionary Equations, 2005.
142
9.1.5 Boltzmann Transport Equation
A statistical description of a
system can be explained by
distribution function f (r, c, t) ;
where f (c, r, t) ; c is the number
of molecules at time t positioned
between r and dr which have
velocities between c and cdc; as
mentioned before. An external
force F acting on a gas molecule
of unit mass will change the
velocity of the molecule from c to
c +F dt and its position from r to r
+c dt. (see Figure 9.1.3). The
number of molecules, f (r, c, t)
before applying the external
force is equal to the number of
molecules after the disturbance, f
(r + cdt, c + Fdt, t + dt); if no
collisions take place between the molecules. Hence,
0dcdr t)c,f(r,dcdr dt) t,Fdt c ,cdt f(r =−+++
Eq. 9.1.4
However, if collisions take place between the molecules there will be a net difference between the
numbers of molecules in the interval drdc: The rate of change between final and initial status of the
distribution function is called collision operator, Ώ. Hence, the equation for evolution of the number
of the molecules can be written as,
dt dcdr Ω(f)dcdr t)c,f(r,dcdr dt) t,Fdt c ,cdt f(r =−+++
Eq. 9.1.5
Dividing the above equation by dt dr dc and as the limit dt → 0; yields
Ω(f)
dt
t)c,(r, df
=
Eq. 9.1.6
The above equation states that the total rate of change of the distribution function is equal to the rate
of the collision. The Ώ is a function of f and need to be determined to solve the Boltzmann equation.
For system without an external force, the Boltzmann equation can be written as,
vectorsare f and c ere whΩ(f)fc.
t
t)c,(r, f
=+
Eq. 9.1.7
Eq. 9.1.7 is an advection equation with a source term (Ώ), or advection with a reaction term, which
can be solved exactly along the characteristic lines that is tangent to the vector c, if Ώ is explicitly
known. The problem is that Ώ is a function of f and Eq. 9.1.7 is an integral-differential equation,
which is difficult to solve. Therefore, there are several approximations available for Ώ. The relation
between the above equation and macroscopic quantities such as fluid density, q; fluid velocity vector
Figure 9.1.3 Position and velocity vector for a particle after and
before applying a force, F
143
u, and internal energy e, is as follows
Tk
2m
3
e and dc t)c,f(r,u m
2
1
t)e(r, t)ρ(r,
dc t)c,f(r, c mt)u(r, t)ρ(r,
dc t)c,f(r, mt)ρ(r,
B
2
a==
=
=
Eq. 9.1.8
where m is the molecular mass and ua the particle velocity relative to the fluid velocity, the peculiar
velocity, ua = c – u. The conservation of mass, momentum, and energy are shown Eq. 9.1.8
respectively.
9.1.6 The BGKW Approximation
It is difficult to solve Boltzmann equation because the collision term is very complicated. The outcome
of two body collisions is not likely to influence significantly, the values of many measured quantities.
Hence, it is possible to approximate the collision operator with simple operator without introducing
significant error to the outcome of the solution. Bhatnagar, Gross and Krook (BGK) in 1954
introduced a simplified model for collision operator. At the same time, independently, introduced
similar operator. The collision operator is replaced as,
f)-(f
τ
1
f)-ω(fΩ(f) eqeq ==
Eq. 9.1.9
The coefficient ω is called the collision frequency and τ is called relaxation factor (ω =1/τ).
The local equilibrium distribution function is denoted by feq; which is Maxwell–Boltzmann
distribution function. After introducing BGKW approximation, the Boltzmann equation (Eq. 9.1.9),
without external forces can be approximated as,
t)(r,ft)(r,f
τ
Δt
t)(r,fΔt) t,Δt c(rf
f)-(f
τ
1
fc.
t
f
i
eq
iiii
eq
−+=++
=+
Eq. 9.1.10
The above equation is the working horse of the Lattice Boltzmann Method and replaces Navier–
Stokes equation in CFD simulations. It is possible to derive Navier–Stokes equation from Boltzmann
equation. The local equilibrium distribution function with a relaxation time determine the
type of problem needed to be solved. The beauty of this equation lies in its simplicity and can be
applied for many physics by simply specifying a different equilibrium distribution function and
source term (external force). Adding a source term (force term) to the above equation is
straightforward. However, there are a few concerns, which will be discussed in the following
chapters. Also, the details
of implementing the above equation for different problems, such as momentum, heat and mass
diffusion, advection–diffusion without and with external forces. It is possible to use finite difference
or finite volume to solve partial differential. Some authors used this approach to solve fluid dynamic
problems on non-uniform grids. The main emphasis is to solve Eq. 9.1.10 in two steps, collision and
144
streaming.
9.1.6.1 Relaxation Time
The lattice Boltzmann method (LBM) based on single-relaxation-time (SRT) or multiple-relaxation-
time (MRT) collision operators is widely used in simulating flow and transport phenomena. The LBM
based on two-relaxation-time (TRT) collision operators possesses strengths from the SRT and MRT
LBMs, such as its simple implementation and good numerical stability, although tedious
mathematical derivations and presentations of the TRT LBM hinder its application to a broad range
of flow and transport phenomena. This paper describes the TRT LBM clearly and provides a
pseudocode for easy implementation. Various transport phenomena were simulated using the TRT
LBM to illustrate its applications in subsurface environments. These phenomena include advection-
diffusion in uniform flow, Taylor dispersion in a pipe, solute transport in a packed column, reactive
transport in uniform flow, and bacterial chemotaxis in porous media. The TRT LBM demonstrated
good numerical performance in terms of accuracy and stability in predicting these transport
phenomena. Therefore, the TRT LBM is a powerful tool to simulate various geophysical and
biogeochemical processes in subsurface environments
163
.
9.1.6.2 Choice of Relaxation Time in the LBM
The dimensionless relaxation time τ should be chosen as to obtain the correct fluid viscosity.
That is :
Eq. 9.1.11
where c is the lattice celerity, Δt the time step and ρ the density. Generally ρ is chosen as unity in
LBM. Therefore, generally, you know the physical property you want to simulate (μ and ρ) and you
use a value of τ that you choose. Then, with this choice of value of τ (which is generally 1 or lower),
you obtain the value of the time step Δt
164
.
9.1.7 Lattices & DnQm Classification
Lattice Boltzmann models can be operated on a number of different lattices, both cubic and
163
ZhifengYana, Xiaofan Yangb, Siliang Lia, Markus Hilpert, “Two-relaxation-time lattice Boltzmann method and
its application to advective-diffusive-reactive transport “, Published by Elsevier Ltd, 2017.
164
Stack Exchange Blog.
Figure 9.1.4 Real Molecules vs. LB Particles
145
triangular, and with or without rest particles in the discrete distribution function
165
.A popular way
of classifying the different methods by lattice is the DnQm scheme. Here "Dn" stands for "n
dimensions", while "Qm" stands for "m speeds". For example, D3Q15 is a 3Dimensional Lattice
Boltzmann model on a cubic grid, with rest particles present. Each node has a crystal shape and can
deliver particles to 15 nodes: each of the 6 neighboring nodes that share a surface, the 8 neighboring
nodes sharing a corner, and itself (The D3Q15 model does not contain particles moving to the 12
neighboring nodes that share an edge; adding those would create a "D3Q27" model). Real quantities
as space and time need to be converted to lattice units prior to simulation. Non-dimensional
quantities, like the Reynolds number, remain the same. (see Figure 9.1.4).
9.1.8 Lattice Arrangements
In general, two models can be used for lattice arrangements, called D1Q3Q and D1Q5Q, as shown in
Figure 9.1.5. D1Q3 is the most popular one. The black nodes are the central node, while the gray
nodes are neighboring nodes. The factitious particles stream from the central node to neighboring
nodes through linkages with a specified speed, called lattice speed.
9.1.8.1 1D Lattice Boltzmann Method (D1O2)
The kinetic equation for the distribution function (temperature distribution, species distributions,
12, etc.), fk(x,t) can be written as
t
x
c Ω
x
t)(r,f
.c
t
t)(r, f
kk
k
k
k
==
+
Eq. 9.1.12
165
From Wikipedia, the free encyclopedia.
146
where k = 1,2 (for one dimensional problem, D1Q2). The left-hand side terms represent the
streaming process, where the distribution function streams (advects) along the lattice link. The right-
hand term, Xk; represents the rate of change of distribution function, fk; in the collision process. BGK
approximation for the collision operator can be approximated by
t)(x,f-t)(x,(f
τ
1
Ω eq
kkk =
Eq. 9.1.13
The term s represents a relaxation time toward the equilibrium distribution (f eq k ), which is related
to the diffusion coefficient on the macroscopic scale. The above equation is the working horse for the
diffusion problem in one dimensional space, which can be reformulated as,
ΔΔωω
Eq. 9.1.14
where ω = Δt/τ is the relaxation time. Eq. 9.1.14 represents a number of equations for different k
values (k = ¼, 1 and 2), in each direction. The schematic diagram for three nodes with necessary
linkages, central and neighboring nodes, where c1 = c, c2 = -c. The dependent variable can be related
to the distribution function fi, as,
Figure 9.1.5 Lattice Arrangements for Velocity Vectors for Typical 1D, 2D and 3D Discretization
147
ϕ
ϕ
Eq. 9.1.15
9.1.8.2 2D Lattice Boltzmann Method (D2Q9)
The process can be modelled using the Boltzmann transport equation, which is
Ωf.
t
t),( f
=+
u
x
Eq. 9.1.16
where f(x, t) is the particle distribution function, u is the particle velocity, and is the collision
operator
166
. The LBM simplifies Boltzmann's original idea of gas dynamics by reducing the number
of particles and confining them to the nodes of a lattice. For a two dimensional model, a particle is
restricted to stream in a possible of 9 directions, including the one staying at rest. These velocities
are referred to as the microscopic velocities and denoted by ei, where i = 0, , , , 8. This model is
commonly known as the D2Q9 model as it is two dimensional and involves 9 velocity vectors. For
each particle on the lattice, we associate a discrete probability distribution function fi(x, ei, t) or
simply fi(x, t), i = 0 , , , 8, which describes the probability of streaming in one particular direction. The
macroscopic fluid density and velocity can be defined as a summation of microscopic particle
distribution function,
==
== 8
0i i
8
0i i t),(f
1
t),( , t),(ft),( xxuxx
Eq. 9.1.17
The key steps in LBM are the streaming and collision processes which are given by
Eq. 9.1.18
In the actual implementation of the model, streaming and collision are computed separately, and
special attention is given to these when dealing with boundary lattice nodes. In the collision term of
(Eq. 9.1.18), feq i (x ; t) is the equilibrium distribution, and _ is considered as the relaxation time
towards local equilibrium. For simulating single phase owes, it success to use Bhatnagar-Gross-Krook
166
Yuanxun Bill Bao & Justin Meskas,” Lattice Boltzmann Method for Fluid Simulations”, April 14, 2011.
148
(BGK) collision, whose equilibrium distribution feqi is defined by
speed lattice theis
Δt
Δx
c ,
5,6,7,8i
1,2,3,4i
0i
1/36
1/9
4/9
w
c
.
2
3
c
).(
2
9
c
.
3w)s( , t)),((ρsρwt),(f
i
22
2
ii
iii
eq
i
=
=
=
=
=
−+=+= uu
ueue
uxux
Eq. 9.1.19
The fluid kinematic viscosity in the D2Q9 model is relate d to the relaxation time by
x)(
6
1-2
2
t
=
Eq. 9.1.20
The algorithm can be summarized in Figure 9.1.6. Notice that numerical issues can arise as τ → 1/2.
During the streaming and collision step, the boundary nodes require some special treatments on the
distribution functions in order to satisfy the imposed macroscopic boundary conditions.
9.1.8.3 Case Study 1 - Lid-Driven Cavity Flow
In this simulation performed by [Mužík]
167
, have a 2D fluid flow that is driven by a lid at the top which
moves at a speed of u = 1.0 in the right direction. The other three walls have no-slip boundary
conditions for the velocity, u = 0 and v = 0. The initial state is described by the zero velocity field and
the initial values of the distribution function is determined using the weights fi = wi. This results in
an initial condition that ρ = 1 (see Figure 9.1.7). The only exception is the velocity of the fluid on the
top is set to be u = 1.0. Simulations were done with a 600 x 600 lattice grid with Re = 400 and 1000.
167
Juraj Mužík, “Lattice Boltzmann Method For Two-Dimensional Unsteady Incompressible Flow”, Civil and
Environmental Engineering · January 2016.
Figure 9.1.6 Schematics of solving 2D Lattice Boltzmann Model
1 - Initialize ρ,
u, fiand feq
2 - Streaming
step: move fi→
fi* in the
direction of ei
3 - Compute
macroscopic ρ
and u from f*i
using Eq. (4.29)
4 - Compute feq i
using Eq. (4.20)
5 - Collision step:
calculate the updated
distribution function
fi = f*i -1/τ(f*i-feqi)
using Eq. (4.20)
6 - Repeat step
2 to 5
149
9.1.8.4 Some Observations and Stability Consideration Regarding SRT, MRT and TRT
Observation 1 by [Perumal & Dass]
168
- The lid-driven cavity flow problem consisting in an
incompressible viscous flow in a cavity whose top wall moves with a uniform velocity in its own plane
has long been used for evaluating numerical techniques for the solution of incompressible viscous
flows [1]. From the literature, it is found that most of the work deals with lattice Boltzmann method
with single-relaxation-time (LBM-SRT) and bounce-back boundary condition to study the cavity flow
field. There appears to be very little work done on rectangular cavities (deep and shallow cavity) by
continuum based methods and lattice Boltzmann method with multi-relaxation-time (LBM–MRT)
model, although they are of theoretical interest.
The main objective is to describe a detailed implementation of the LBM models and to demonstrate
the validity of the LBM models at high Reynolds numbers. Therefore, the lid-driven cavity problem
with different aspect ratios is studied in an effort to evaluate the performance of the LBM and to
produce accurate steady state solutions for different values of the Reynolds number.
Observation 2 by [Aslan et al.]
169
- The simplest LBE is the Lattice Bhatnagar-Groos-Krook (LBGK)
equation, based on a Single Relaxation Time (LBM-SRT) approximation [3]. Due to extreme
simplicity, the LBGK equation has become most popular Lattice Boltzmann equation in spite of its
well-known deficiencies, for example, flow simulation at high Reynolds numbers [4].
Flow simulation at high Reynolds numbers, collision frequency (ω ) which is the main ingredient of
the LBM-SRT, exhibits a theoretical upper bound (ω < 2 ) that is related with the positiveness of the
molecular kinematic viscosity [5]. Thus, stability problems arise as the collision frequency
approaches to this limiting value [6]. For incompressible flows, the flow velocities are limited, since
the model immanent Mach number needs to be kept sufficiently small. Therefore, a lowering
kinematic viscosity, for achieving high Reynolds numbers for a given geometry, pushes the collision
frequency towards the above-mentioned stability limit. It is possible to increase the value of ω by
168
D. Arumuga Perumal, Anoop K. Dass, “Application of lattice Boltzmann method for incompressible viscous
flows”, Applied Mathematical Modelling 37 (2013) 4075–4092.
169
E. Aslan, I. Taymaz, and A. C. Benim, “Investigation of the Lattice Boltzmann SRT and MRT Stability for Lid
Driven Cavity Flow”, International Journal of Materials, Mechanics and Manufacturing, Vol. 2, No. 4, 2014.
Figure 9.1.7 Lid-Driven Cavity, Streamlines Pattern For Different Reynolds Numbers
150
decreasing the size of lattices, however, it needs more computer resources [7]. Alternatively, using
LBM-MRT increases stability limit and resolve the mentioned issue [8].
In the literature, there are comparative studies of the LBM-SRT and the LBM-MRT for lid driven cavity
flows [2]. Those studies find that, the LBM-MRT is superior to the LBM-SRT at higher Reynolds
number flow simulations, especially for numerical stability. Also, the LBM-SRT and the LBM-MRT
produces accurate results for all Reynolds numbers. In addition that, the code using the LBM-MRT
takes only 15% more CPU time than using the LBM-SRT.
Observation 3 by [Kuzmin et al.]
170
– There is a well-established hierarchy of linear lattice
Boltzmann relaxation operators for both hydrodynamic and anisotropic advection–diffusion
equations (AADE); from the minimal single-relaxation-time BGK operator [10] to the richest
ancestor: multiple-relaxation-time MRT operators [11–14], via the two-relaxation-time (TRT)
operator [15,16]. A MRT operator offers d independent relaxation rates for d anisotropic diagonal
components of the diffusion tensor, against only one available for the BGK and TRT operators. The
alternative link wise operators, the BGK-type operator [17,18] and the L-operator [15,19] with,
respectively, one and two relaxation rates per velocity axis (called links), offer a number of
independent times equal to the number of links for describing the full anisotropic tensors with the
cross-diagonal diffusion elements. Since BGK TRT MRT and BGK TRT L, the TRT operator is
the largest common subclass of the MRT and L operators. The BGK and TRT models can also solve
the AADE, but only with the help of anisotropic equilibrium distributions [15,20,21].
Of course, any practitioner will give a preference to a more conceptually complicated operator only
if this ‘‘complexity’’ is justified, even when the two operators have the same computational cost as
the BGK and TRT. To date there has been some evidence, e.g., in the works [14,22,23], of the possible
gain in stability coming from ‘‘ghost’’ or ‘‘kinetic’’ relaxation times, but the whole scenario is not well
understood, even for the simplest linear isotropic one-dimensional advection–diffusion equation.
9.1.8.5 References
[1] U. Ghia, K.N. Ghia, C.T. Shin, High-Re solutions for incompressible flow using the Navier–Stokes
equations and a multigrid method, J. Comp. Phys. 48 (1983) 387–411.
[2] J. S. Wu and Y. L. Shao, “Assessment of SRT and MRT scheme in parallel lattice Boltzmann method
for lid-driven cavity flows,” in Proc. the 10th National Computational Fluid Dynamics Conference, Hua-
Lien, 2003.
[3] P. L. Bhatnagar, E. P. Groos, and M. Krook, “A model for collision process in gases I. small amplitude
process in charged and neutral one component system,” Physical Review, vol. 94, pp. 511-525, 1954.
[5] Y. Qian, D. D’Humières, and P. Lallemand, “Lattice BGK models for Navier-stokes Equation,” Euro
physics Letter, vol. 17, pp. 479-484, 1992.
[6] S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford, Calderon Press,
2001.
[7] M. Fink, “Simulation von Nasenströmungen mit Lattice-BGK-Methoden,” Ph.D. dissertation, Essen-
Duisburg University, Germany, 2007.
[8] A. C. Benim, E. Aslan, and I. Taymaz, “Investigation into LBM analyses of incompressible laminar
flows at high Reynolds numbers,” WSEAS Transactions on Fluid Mechanics, vol. 4, no. 4, 2009.
[9] D. d’Humières, “Generalized lattice Boltzmann equation,” AIAA Rarefied Gas Dynamics: Theory
and simulations, vol. 159, pp. 450-458, 1992.
[10] Y. Qian, D. d’Humières, P. Lallemand, Lattice BGK models for Navier–Stokes equation, Europhys.
Lett. 17 (1992) 479–484.
170
A. Kuzmin, I. Ginzburg , A.A. Mohamad, “The role of the kinetic parameter in the stability of two-relaxation-
time advection–diffusion lattice Boltzmann schemes”, Computers and Mathematics with Applications 61 (2011)
3417–3442.
151
[11] F.J. Higuera, J. Jiménez, Boltzmann approach to lattice gas simulations, Europhys. Lett. 9 (1989)
663–668.
[12] F.J. Higuera, S. Succi, R. Benzi, Lattice gas dynamics with enhanced collisions, Europhys. Lett. 9
(1989) 345–349.
[13] D. d’Humières, Generalized lattice-Boltzmann Equations. AIAA Rarefied Gas Dynamics: Theory
and Simulations, in: Progress in Astronautics and Aeronautics, vol. 59, 1992, pp. 450–548.
[14] D. d’Humières, I. Ginzburg, M. Krafczyk, P. Lallemand, L.-S. Luo, Multiple-relaxation-time lattice
Boltzmann models in three dimensions, Philos. Trans. R. Soc. Lond. A 360 (2002) 437–451.
[15] I. Ginzburg, Equilibrium-type and link-type lattice Boltzmann models for generic advection and
anisotropic-dispersion equation, Adv. Water Res. 28 (2005) 1171–1195.
[16] I. Ginzburg, Lattice Boltzmann modeling with discontinuous collision components. Hydrodynamic
and advection–diffusion equations, J. Stat. Phys. 126 (2007) 157–203.
[17] X. Zhang, A.G. Bengough, L.K. Deeks, J.W. Crawford, I.M. Young, A lattice BGK model for advection
and anisotropic dispersion equation, Adv. Water Res. 25 (2002) 1–8.
[18] X. Zhang, A.G. Bengough, L.K. Deeks, J.W. Crawford, I.M. Young, A novel three-dimensional lattice
Boltzmann model for solute transport in variably saturated porous media, Water Res. 38 (2002) 1167–
1177.
[19] Y. Li, P. Huang, A coupled lattice Boltzmann model for advection and anisotropic dispersion
problem in shallow water, Adv. Water Res. 31 (12) (2008).
[20] I. Ginzburg, D. d’Humières, Lattice Boltzmann and analytical modeling of flow processes in
anisotropic and heterogeneous stratified aquifers, Adv. Water Res. 30 (2007) 2202–2234.
[21] B. Servan-Camas, F.T.C. Tsai, Saltwater intrusion modeling in heterogeneous confined aquifers
using two-relaxation-time lattice Boltzmann method, J. Comp. Phys. 228 (2009) 236–256.
[22] R.G.M. van der Sman, M.H. Ernst, Diffusion lattice Boltzmann scheme on an orthorhombic lattice,
J. Stat. Phys. 94 (1–2) (1999) 203–217.
[23] P. Lallemand, L.-S. Luo, Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy,
Galilean invariance, and stability, Phys. Rev. E 61(2000) 6546–6562.
9.1.8.6 Case Study 2 - Flow Past A Circular Cylinder
The study of flow past an object dates its history back to ships design. Many investigators were
interested in the new ship shapes and describing the behavior of the flow past the ship body. One of
the most important results of these investigations is that Reynolds number plays an important role
in characterizing the behavior of the flow. This numerical study consists of the 2D channel with a
steady Poiseuille flow profile, with circular cylinder immersed into the flow at the start of the
simulation. On lattice grid the no-slip boundary conditions applied to the boundary of the cylinder as
well as the top and the bottom boundaries of the channel. Velocity and pressure (density) boundary
conditions are applied to the inlet and the outlet side of the channel. Numerical simulation with Re =
400 have been carried out on the 2000 x 400 lattice grid and the results shows the generation of the
Karman vortex street in the area past the cylinder (Figure 9.1.1)
171
.
171
Juraj Mužík, “Lattice Boltzmann Method For Two-Dimensional Unsteady Incompressible Flow”, Civil and
Environmental Engineering · January 2016.
152
9.2 Differences Between Finite Volume Method (FVM) and Lattice Boltzmann
Methods (LBM)
Several studied investigated comparing between the FVM and LBM. Chief among them are [Goodarzi
et al.]
172
using different discretization methods, and arrived at the following conclusion:
1 The finite volume method results are more accurate compared to those of LBM, especially at
the corners.
2 LBM needs a 4-5-fold CPU usage time and 8-9 times more iterations compared to the finite
volume method to solve the problem considered here.
3 Among the studied discretization/pressure-velocity linking algorithms, the 1st order
upwind/SIMPLEC provides the most precise results against experimental benchmark data,
especially in the boundary layers.
4 The numbers of iterations for all FVM discretization/pressure-velocity linking methods are
nearly equal.
5 The higher-order accurate schemes are more time consuming.
One, however, notes that the above observations are valid within the limits of the parameters and
problem considered in this study and could not be generalized to other cases without further
investigations. Other researcher such as [Liu et al.]
173
at Penn State, the variance are summarized
below:
Wikipedia provides a more conservative explanations as considers pro and cons of LBM as:
172
M. Goodarzi, M. R. Safaei, A. Karimipour, K. Hooman, M. Dahari, S. N. Kazi, and E. Sadeghinezhad,” Comparison
of the Finite Volume and Lattice Boltzmann Methods for Solving Natural Convection Heat Transfer Problems inside
Cavities and Enclosures”, Hindawi Publishing Corporation Abstract and Applied Analysis, Volume 2014, Article
ID 762184, 15 pages, http://dx.doi.org/10.1155/2014/762184.
173
Rui Liu, Chengheng Lu, Junjun Li, “Lattice Boltzmann Method”, Penn State.
Figure 9.1.1 Flow Past Cylinder, Velocity Magnitude Contours
153
9.2.1 Advantages
The LBM was designed from scratch to run efficiently on massively parallel architectures, ranging
from inexpensive embedded FPGAs and DSPs up to GPUs and heterogeneous clusters and
supercomputers (even with a slow interconnection network). It enables complex physics and
sophisticated algorithms. Efficiency leads to a qualitatively new level of understanding since it allows
solving problems that previously could not be approached (or only with insufficient accuracy). The
method originates from a molecular description of a fluid and can directly incorporate physical terms
stemming from a knowledge of the interaction between molecules. Hence it is an indispensable
instrument in fundamental research, as it keeps the cycle between the elaboration of a theory and
the formulation of a corresponding numerical model short.
Automated data pre-processing and mesh generation in a time that accounts for a small fraction of
the total simulation. Parallel data analysis, post-processing and evaluation. Fully resolved multi-
phase flow with small droplets and bubbles. Fully resolved flow through complex geometries and
porous media. Complex, coupled flow with heat transfer and chemical reactions.
9.2.2 Limitations
Despite the increasing popularity of LBM in simulating complex fluid systems, this novel approach
has some limitations. At present, high-Mach number flows in aerodynamics are still difficult for LBM,
and a consistent thermo-hydrodynamic scheme is absent. However, as with Navier–Stokes based
CFD, LBM methods have been successfully coupled with thermal-specific solutions to enable heat
transfer (solids-based conduction, convection and radiation) simulation capability. For
multiphase/multicomponent models, the interface thickness is usually large and the density ratio
across the interface is small when compared with real fluids. Recently this problem has been
resolved by [Yuan and Schaefer] who improved on models by [Shan and Chen, Swift], and [He, Chen,
and Zhang]. They were able to reach density ratios of 1000:1 by simply changing the equation of
state. It has been proposed to apply Galilean Transformation to overcome the limitation of modelling
high-speed fluid flows. Nevertheless, the wide applications and fast advancements of this method
during the past twenty years have proven its potential in computational physics, including
microfluidics: LBM demonstrates promising results in the area of high Knudsen number flows. It
appears that LBM is suited mainly for incompressible flows and not for high Reynolds number
distinctive of aerodynamic applications.
9.3 Unified Flow Solver (UFS)
A variety of gas flow problems are characterized by the presence of rarefied and continuum domains.
In a rarefied domain, the mean free path of gas molecules is comparable to (or larger than) a
characteristic scale of the system. The rarefied domains are best described by particle models such
as Direct Simulation Monte Carlo (DSMC); or, they involve solution of the Boltzmann kinetic
equation for the particle distribution function. The continuum flows are best described by Euler or
Navier-Stokes equations in terms of average flow velocity, gas density, and temperature and are
solved by computational fluid dynamics (CFD) codes. The development of hybrid solvers combining
kinetic and continuum models has been an important area of research over the last decade. Potential
applications of such solvers range from high-altitude flight to gas flow in microsystems
174
.
The key parameter governing selection of the appropriate physical model is the Knudsen number
(Kn), defined as the ratio of the local mean free path to the characteristic size of the system. For
flights at high altitude, where the atmospheric density is relatively low, the Kn is essentially a
representation of gas density, which is the predominant factor that mandates model selection. For
174
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upcoming articles, late breaking industry news, hot products and design ideas, links to online resources, and
much more.
154
gas flows in microsystems, the small
dimension of the system has the most
influence on the Kn and therefore dictates
selection of a kinetic model.
Until recently, most attempts to create hybrid
gas flow solvers have involved coupling DSMC
codes with CFD codes.1 As part of a Small
Business Innovation Research project, AFRL
partnered with CFD Research Corporation
and the Russian Academy of Sciences to
develop a Unified Flow Solver (UFS) based on
the direct numerical solution (DNS) of the
Boltzmann transport equation combined with
kinetic schemes for the CFD equations.
Choosing a DNS rather than a DSMC-based
approach enabled the developers to use
similar numerical techniques for the rarefied
and continuum domains and thus facilitate
coupling of the rarefied and continuum
solvers. The UFS can automatically identify
kinetic and continuum domains using
preestablished criteria, introduce and remove
kinetic patches, and select suitable solvers to maximize the accuracy and efficiency of simulations.
Figure 9.3.1 shows the key components of the UFS. The Boltzmann kinetic solver implemented in
the UFS utilizes the numerical algorithms and computational methods described by [Aristov and
Tcheremissine]. The continuum solvers are based on kinetic schemes employing numerical
algorithms similar to those of the Boltzmann solver. The remaining UFS components include criteria
for domain decomposition into rarefied and continuum parts and coupling algorithms.
9.3.1 Description of Kinetic Schemes of Direct Methods
Let us briefly describe the direct method of solving the Boltzmann transport equation (BTE)
developed by [Aristov et al.]
175
. Boltzmann equation is the fundamental transport equation,
describing the evolution of a velocity distribution function f in phase space (r, ξ). Introducing
Cartesian grid with nodes ξβ in velocity space, the BTE is reduced to a set of equations in physical
space
Eq. 9.3.1
Here r is a position vector in physical space, n is the velocity vector and t is time. The right-hand side
of Eq. 9.3.1 contains an integral operator describing binary collisions among particles. For
explanation of elastic collisions in a monatomic gas, please refer to [Kolobov et al]
176
. We split the
solution of this system of equations into two stages: free flow and relaxation. For the free flow, we
use an explicit finite volume numerical scheme and for the relaxation stage the explicit finite-
175
V. V. Aristov, A. A. Frolova, S. A. Zabelok, V. I. Kolobov, and R. R. Arslanbekov, “Unified Flow Solver Combining
Boltzmann and Continuum Models for Simulations of Gas Flows for the Entire Range of Knudsen Numbers”,
Dorodnicyn Computing Center of the Russian Academy of Sciences, Moscow, Russia in collaboration with CFD
Research Corporation, Alabama, USA.
176
V.I. Kolobov, R.R. Arslanbekov, V.V. Aristov, A.A. Frolova, S.A. Zabelok , “Unified solver for rarefied and
continuum flows with adaptive mesh and algorithm refinement”, Journal of Computational Physics, (2006).
Figure 9.3.1 UFS Key Components
155
difference scheme is applied. Collision integrals are considered in the symmetric form of 8-fold
integrals. The quasi-Monte Carlo procedure with the Korobov sequences is used for evaluation of the
collision integrals. We currently use uniform grid in velocity space. For the continuum equations,
kinetic schemes are used. Generally, kinetic schemes are preferable for developing hybrid codes since
the BTE is used as a foundation for both algorithms. Kinetic schemes for the Euler and Navier-Stokes
equations by means of the distribution function have been proposed. Kinetic schemes using moments
of the equilibrium distribution function in the modern variant are developed. Recently, we extended
our Euler-kinetic scheme for gas mixtures with chemical reactions. We use the Gerris framework to
generate dynamically adaptive Cartesian mesh in physical space. The parallel algorithm with
dynamic load balancing among processors has been developed.
9.3.2 Continuum Switching Parameter
The main problem of unified methods is to separate kinetic and continuum regions. We have used
several switching criteria, one of them is as follows:
Eq. 9.3.2
where ρ is the density, p is the pressure, u, v, w are appropriate components of velocity (all values
are given in dimensionless form), Kn is the global Knudsen number (e.g., for a flow around a Kn = λ/R
cylinder where λ is the mean free path and R is the radius of a cylinder).
The Figure 9.3.2 illustrates some current capabilities of the UFS. In Figure 9.3.2
177
, domain
decomposition into kinetic and continuum regions is presented for supersonic gas flow around a
cylinder at Mach number M = 3, for different Knudsen numbers Kn. We have studied the influence of
the breakdown parameter on the flow characteristics calculated by the UFS. Figure 9.3.3 illustrates
the convergence of the computations with respect to a breakdown parameter is illustrated for the
case of M = 3, Kn = 0.25. One can see that all curves converge at small S numbers when the Boltzmann
region grows. At the same time, by decreasing the S number, the computation time increases.
Therefore, for quick results one can use larger S numbers if precision of the order of 10% is
satisfactory, [Aristov et al.]
178
. Another test case, using a backward step where shock capturing is the
177
The density profiles are on the left side, the computational grids with kinetic (red) and continuum (white)
domains are on the right side.
178
V. V. Aristov, A. A. Frolova, S. A. Zabelok, V. I. Kolobov, and R. R. Arslanbekov, “Unified Flow Solver Combining
Boltzmann and Continuum Models for Simulations of Gas Flows for the Entire Range of Knudsen Numbers”,
Figure 9.3.2 Gas flow Around a Cylinder for M = 3 for Different Kn Numbers (0.5, 0.005).
156
objective, as kinetic schemes of the Euler
equations for monatomic gas, were
proposed by [Kolobov et al.]
179
and the
results displayed in Figure 9.3.4.
Dorodnicyn Computing Center of the Russian Academy of Sciences, Moscow, Russia in collaboration with CFD
Research Corporation, Alabama, USA.
179
V.I. Kolobov , R.R. Arslanbekov, V.V. Aristov , A.A. Frolova , S.A. Zabelok, “Unified solver for rarefied and
continuum flows with adaptive mesh and algorithm refinement”, Journal of Computational Physics xxx (2006).
Figure 9.3.3 Distribution of Normal Force Over the
Cylinder Surface for Different Values of the Continuum
Breakdown Parameter S
Figure 9.3.4 Computational Mesh (top) and Gas Density Contours (bottom)
157
9.4 The Navier-Stokes Equations (NS)
The Navier-Stokes equations are the basic governing equations for a viscous, heat conducting fluid.
It was originally developed by the Frenchman (Claude Louis Marie Henri Navier) and Englishman
(George Gabriel Stokes) who proposed them in the early to mid-19th century. It is a vector equation
obtained by applying Newton's Law of Motion to a fluid element and is also called the momentum
equation. It is supplemented by the mass conservation equation, also called continuity equation and
the energy equation. Usually, the term Navier-Stokes equations is used to refer to all of these
equations, as illustrated by (constant properties) as:
Eq. 9.4.1
The equations can be written in initial notation as well, and will be discussed in detail later on. where
ρ is density of the fluid (taken to be a known constant); u ≡ (u, v, w)T is the velocity vector; p is fluid
pressure; μ is viscosity, and FB is a body force. D/Dt is the substantial derivative expressing the
Lagrangian, or total, acceleration of a fluid parcel in terms of a convenient laboratory-fixed Eulerian
reference frame; is the gradient operator; Δ is the Laplacian, and · is the divergence operator. We
remind the reader that the first of these equations (which is a three component vector equation) is
just Newton’s second law of motion applied to a fluid parcel; the left-hand side is mass (per unit
volume) times acceleration, while the right-hand side is the sum of forces acting on the fluid element.
While in LBM, the fluid is replaced by fractious particles, Navier–Stokes equations (NS) solve mass,
momentum and energy conservation equations on discrete nodes, elements, or volumes
180
. In other
words, the nonlinear partial differential equations convert into a set of nonlinear algebraic equations,
which are solved iteratively
181
. The primary reason why LBM can serve as a method for fluid
simulations is that the Navier-Stokes equations can be recovered from the discrete equations through
the Chapman-Enskog procedure, a multi-scaling expansion technique
182
. It is an excellent exercise
to drive the NS equation from discrete LBE equations.
9.4.1 N-S Equation in Non-Inertial Frame of Reference
Generally, for turbomachine applications, the rotating frame of reference introduces some
interesting pseudo-forces into the equations through the material derivative term. Consider a
stationary inertial frame of reference K, and a non-inertial frame of reference K', which is translating
with velocity U (t) and rotating with angular velocity Ω (t) with respect to the stationary frame. The
Navier-Stokes equation observed from the non-inertial frame then becomes:
Eq. 9.4.2
Here and are measured in the non-inertial frame. The first term in the parenthesis represents
Coriolis acceleration, the second term is due to centripetal acceleration, the third is due to the
180
Z. Guo, B. Shi, and N. Wang, “Lattice BGK Model for Incompressible Navier-Stokes Equation”, J. Computational
Phys. 165, 288-306 (2000).
181
M. Sukop and D.T. Thorne, “Lattice Boltzmann Modeling: an introduction for geoscientists and engineers ”.
Springer Verlag, 1st edition. (2006).
182
R. Begum, and M.A. Basit, “Lattice Boltzmann Method and its Applications to Fluid Flow Problems”, Euro. J. Sci.
Research 22, 216-231 (2008).
158
linear acceleration of K' with respect to K and the fourth term is due to the angular acceleration of K'
with respect to K
183
.
9.4.2 Some Basic Functional Analysis
Here, we will introduce some basic notions from fairly advanced mathematics that are essential to
anything beyond a superficial understanding of the Navier–Stokes equations. There are numerous
key definitions and basic ideas needed for analysis of the N-S equations. These include Fourier
series, Hilbert (and Sobolev) spaces, the Galerkin procedure, Weak and Strong solutions, and
various notions such as completeness, compactness and convergence
184
.
9.4.2.1 Fourier Series and Hilbert Spaces
In this subsection we will briefly introduce several topics required for the study of PDEs in general,
and the Navier–Stokes equations in particular, from a modern viewpoint. These will include Fourier
series, Hilbert spaces and some basic ideas associated with function spaces, generally. Fourier Series
We begin by noting that there has long been a tendency, especially among engineers, to believe that
only periodic functions can be represented by Fourier series. This misconception apparently arises
from insufficient understanding of convergence of series (of functions), and associated with this, the
fact that periodic functions are often used in constructing Fourier series. Thus, if we demand uniform
convergence some sense could be made of the claim that only periodic functions can be represented.
But, indeed, it is not possible to impose such a stringent requirement; if it were the case that only
periodic functions could be represented (and only uniform convergence accepted), there would be
no modern theory of PDEs. Recall for a function, say f(x), defined for x [0,L], that formally its Fourier
series is of the form
Eq. 9.4.3
where {ϕk} is a complete set of basic functions, and
Eq. 9.4.4
The integral on the right-hand side is a linear functional termed the inner product when f and ϕk are
in appropriate function spaces. The requirements on f for existence of such a representation can be
found
185
.
9.4.2.2 Weak vs. Genuine (Strong) Solution
A genuine solution is the one in which function is continuous but bounded discontinuities in the
derivative of function may occur. A weak solution is the solution which is genuine except along a
surface space across which the function may be discontinuous. A constraint is placed upon the jump
in function across the discontinuity in domain of interest. Clearly the existence of shock waves in
inviscid supersonic flow is an example of a weak solution. Therefore, genuine solution is a weak
183
Wikipedia.
184
J. M. McDonough, “Lectures in Computational Fluid Dynamics of Incompressible Flow: Mathematics,
Algorithms and Implementations”, Departments of Mechanical Engineering and Mathematics University of
Kentucky, 2007.
185
J. M. McDonough, “Lectures in Computational Fluid Dynamics of Incompressible Flow: Mathematics,
Algorithms and Implementations”, Departments of Mechanical Engineering and Mathematics University of
Kentucky, 1991, 2003, 2007.
159
solution and a weak solution which is contentious is a genuine solution
186
. In numerical solutions, the
differential form is used together with difference approximations (FDM), where integral form is used
with the finite volume method, FVM. These are equivalent in uniform grids. The differential form
does not have a solution in the classical sense in presence of discontinuities (e.g., compressible flows
with shocks), hence, one uses the weak form of the integral equations
187
.
186
Anderson, Dale A; Tannehill, John C; Plecher Richard H; 1984:”Computational Fluid Mechanics and Heat
Transfer”, Hemisphere Publishing Corporation.
187
V. Viitanen, VTT Technical Research Centre of Finland, ResearchGate.
160
10 Porous Media
10.1 Introduction and Background
A porous medium (or a porous material) is a material containing pores (voids). The skeletal portion
of the material is often called the “matrix” or “frame.” The pores are typically filled with a fluid (liquid
or gas). The skeletal material is
usually a solid, but structures like
foams are often also usefully analyzed
using concept of porous media.
Figure 10.1.1 displays velocity
magnitude with a red arrow plot
representing the orientation of the
flow through the pore space. A porous
medium is most often characterized
by its porosity. Other properties of the
medium (e.g., permeability, tensile
strength, electrical conductivity) can
sometimes be derived from the
respective properties of its
constituents (solid matrix and fluid)
and the media porosity and pores
structure, but such a derivation is usually complex. Even the concept of porosity is only
straightforward for a pyroclastic medium.
Often both the solid matrix and the pore network (also known as the pore space) are continuous, so
as to form two interpenetrating continua such as in a sponge. However, there is also a concept of
closed porosity and effective porosity, i.e., the pore space accessible to flow. Many natural substances
such as rocks and soil (e.g., aquifers, petroleum reservoirs), zeolites, biological tissues (e.g., bones,
wood, cork), and man-made materials such as cements and ceramics can be considered as porous
media. Many of their important properties can only be rationalized by considering them to be porous
media.
The concept of porous media is used in many areas of applied science and engineering: filtration,
mechanics (acoustics, geo mechanics, soil mechanics, rock mechanics), engineering (petroleum
engineering, bio-remediation, construction engineering), geosciences (hydrogeology, petroleum
geology, geophysics), biology and biophysics, material science, etc. Fluid flow through porous media
is a subject of most common interest and has emerged a separate field of study. The study of more
general behavior of porous media involving deformation of the solid frame is called poro mechanics.
10.2 Porous Modeling
Porous modeling is governed by three models [Çetin et al.].
188
The simplest model is the Darcy’s
model which is suggested by Henry Darcy (1856) during his investigations on hydrology of the water
supplies. Darcy’s equation is expressed as:
Eq. 10.2.1
where, Δp is the pressure drop, l is the pipe length, V is the average velocity, μ is the dynamic viscosity
and α is permeability of porous domain. Permeability depends on the fluid properties and the
188
Barbaros Çetin, Kadir G. Güler and Mehmet Haluk Aksel, “Computational Modeling of Vehicle Radiators Using
Porous Medium Approach”, Intech, 2017.
Figure 10.1.1 Velocity Magnitude color plot with a red
arrow plot representing the orientation of the flow through
the pore space (Courtesy of Comsol)
161
geometrical properties of the medium. The dependence of the pressure drop on velocity in the
Darcy’s equation is linear; therefore, Darcy’s equation is applicable when the flow is laminar. As the
velocity increases, the dependence of the pressure drop on velocity becomes non-linear due to drag
caused by solid obstacles. At this point, there are two extended models proposed in the literature
namely Forchheimer and Forchheimer-Brinkman model. For moderate Reynolds numbers, including
nonlinear effects, pressure drop is defined as Forchheimer’s equation
189
:
Eq. 10.2.2
where CF is the dimensionless form-drag constant and ρ is the density of the fluid. The first term
denotes the viscous characteristics of porous flow and the second term (also called Forchheimer
term) denotes the inertial characteristics. Lastly, Forchheimer-Brinkman model includes additional
Laplacian term in addition to Forchheimer’s equation. Forchheimer-Brinkman model is expressed
as:
Eq. 10.2.3
where is the effective viscosity. In general, added Laplacian term (also known as Brinkman term)
resolves effects of the flow characteristics in a thin boundary layer at the near wall regions. Strictly
speaking, the last term becomes important for large porosity (ratio of the fluid volume to the solid
volume in a porous medium) values which means the effect is negligible for many practical
applications where typically porosity value is relatively small. Eq. 10.2.3 without the quadratic term
is known as extended Darcy (or Brinkman) model. Therefore, Forchheimer-Brinkman model is the
most general model, but the inclusion of the Brinkman and Forchheimer term on the left-hand side
can be questionable since the Brinkman term is appropriate for large porosity values, yet there exists
uncertainty about the validity of the Forchheimer term at larger porosity values
190
. Velocity
definition in porous modeling is specified by using two different descriptions: superficial formulation
and physical velocity formulation. Superficial velocity formulation does not take the porosity into
account during the evaluation of the continuity, momentum and energy equations. On the other hand,
physical velocity formulation includes porosity during the calculation of transport equations
191
. The
continuity and momentum transport equation for a porous domain using Forcheimer’s model can be
written as
192
:
Eq. 10.2.4
where γ is the porosity, C2 is the inertial coefficient for porous domain and Bf is the body force term.
Besides flow modeling, heat transfer modeling for porous flow is described by using two models
which are :
189
Bejan A., Nield D.A. “Convection in Porous Media”. 3rd ed. Springer; New York 2006.
190
See Previous.
191
Kim D., Kim S.J. Compact modelling of fluid flow and heat transfer in straight fin heat sinks. ASME.J. Electronic
Packaging. 2004;126:247–255.
192
Pavel B.I., Mohamad A.A. “An experimental and numerical study on heat transfer enhancement for gas heat
exchangers fitted with porous media”. Int. J. Heat Mass Transfer. 2004;47:4939–4952.
162
➢ Equilibrium model and
➢ Non-equilibrium model.
Equilibrium model (one-equation energy model) is used when the porous medium and fluid phase
are in thermal equilibrium. However, in most cases, fluid phase and porous medium are not in
thermal equilibrium. For such cases, non-equilibrium thermal model is more realistic. Therefore, the
non-equilibrium model includes two energy equations (known as also two-equation energy model):
one is for the fluid domain and the other is for the solid domain. The coupling of these two models is
via the term which represents the heat transfer between the fluid and the solid domains. The
conservation equations for the two energy model. See [Pavel & Mohamad]
193
.
10.3 Porous Medium
In the fluid region the flow is governed by the incompressible Navier–Stokes (NS) equations, while
in the porous layers the Volume-Averaged Navier–Stokes equations (VANS) are used, which are
obtained by volume-averaging the microscopic flow field over a small volume that is larger than the
typical dimensions of the pores [Rosti et al.]
194
. In this way the porous medium has a continuum
description, and can be specified without the need of a detailed knowledge of the pore microstructure
by independently assigning permeability and porosity. At the interface between the porous material
and the fluid region, momentum-transfer conditions are
applied, in which an available coefficient related to the
unknown structure of the interface can be used as an error
estimate. Most of the simulations are carried out at Red =
180 and consider low-permeability materials; a parameter
study is used to describe the role played by permeability,
porosity, thickness of the porous material, and the
coefficient of the momentum-transfer interface conditions.
Among them permeability, even when very small, is shown
to play a major role in determining the response of the
channel flow to the permeable wall. Turbulence statistics
and instantaneous flow fields, in comparative form to the
flow over a smooth impermeable wall, are used to
understand the main changes introduced by the porous
material.
A simulation at higher Reynolds number is used to illustrate
the main scaling quantities. and from chemistry to medicine.
The main properties characterizing a porous material are
porosity and permeability. These are obviously average
properties measured on samples of porous materials large
with respect to the characteristic pore size. Porosity (or
void fraction), ε, is a dimensionless measure of the void
spaces in a material. It is expressed as a fraction of the
volume of voids over the total volume and its value varies between 0 and 1. Permeability, K*, with
dimensions of length squared, is a measure of the ease with which a fluid flows through a porous
medium (throughout this article, an asterisk denotes a dimensional quantity). K*= 0 if a medium is
impermeable, i.e., if no fluid can flow through it, while it becomes infinite if a medium offers no
resistance to a fluid flow. (see Figure 10.3.1). Typical values of porosity for porous materials made
193
Pavel B.I., Mohamad A.A. “An experimental and numerical study on heat transfer enhancement for gas heat
exchangers fitted with porous media”. Int. J. Heat Mass Transfer. 2004;47:4939–4952.
194
Marco E. Rosti, Luca Cortelezzi and Maurizio Quadrio, “Direct numerical simulation of turbulent channel flow
over porous walls”, J. Fluid Mechanics, vol. 784, pp. 396_442, Cambridge University Press 2015.
Figure 10.3.1 Earliest Forms of
Porous
Media
163
of packed particles are [Macdonald et al. 1979] 0.366 < ε < 0.64 for spherical glass beads packed in a
‘uniformly random’ manner, 0.367 < ε < 0.515 for spherical marble mixtures, sand and gravel
mixtures, and ground Blue Metal mixtures, 0.123 < ε < 0.378 for a variety of consolidated media, 0.32
< ε < 0.59 for a wide variety of cylindrical packings, and 0.682 < ε < 0.919 for cylindrical fibers.
[Beavers & Joseph (1967)] performed experiments with three types of foametals 0.78 < ε < 0.79 and
two types of aloxites 0.52 < ε < 0.58.
10.3.1 Literature Survey
The first empirical law governing Stokes flow through porous media was derived by Darcy in 1856.
More than a century later, [Beavers & Joseph]
195
presented the first interface (jump) condition
coupling a porous flow governed by Darcy’s law with an adjacent fully developed laminar channel
flow. This condition was further developed, with different degrees of success, by [Neale & Nader]
196
,
[Vafai & Thiyagaraja]
197
, [Vafai & Kim]
198
and [Hahn, Je & Choi]
199
, among others. General porous flow
equations, the so-called volume-averaged Navier–Stokes equation. The main effects of porous
materials on adjacent fluid flows are the destabilization of laminar flows and the enhancement of the
Reynolds-shear stresses, with a consequent increase in skin-friction drag in turbulent flows. The first
results showing the destabilizing effects of wall permeability were obtained experimentally by
various studies. [Sparrow et al.]
200
experimentally determined a few critical Reynolds numbers in a
channel with one porous wall, and performed a two-dimensional temporal linear stability analysis
using Darcy’s law with the interface condition introduced by [Beavers & Joseph]
201
. For full extension
of survey, readers encourage to consult with [Rosti et al.]
202
.
[Tilton & Cortelezzi]
203
reported that small amounts of wall permeability destabilize the Tollmien–
Schlichting wave and cause a substantial broadening of the unstable region. As a result, the
stabilization of boundary layers by wall suction is substantially less effective and more expensive
than what is predicted by classical boundary-layer theory. There is also a scarcity of literature
regarding the effects of permeability on turbulent flows, as well as an increase in skin friction in
boundary layers over porous walls made of sintered metals, bonded screen sheets and perforated
titanium sheets. They reported ‘significant skin-friction reductions’ at the permeable walls. This
result conflicts with the experimental evidence and is presumably due to the enforcement of a zero
normal velocity at the porous/fluid interface, a boundary condition that inhibits the correct exchange
of momentum across the interface. It is also interesting to note that [Perot & Moin]
204
performed a
DNS of a turbulent plane channel flow with infinite permeability and no-slip boundary conditions in
order to remove the wall-blocking mechanism and study the contribution of splashes on wall
195
Beavers, G. S. & Joseph, D. D., “Boundary conditions at a naturally permeable wall”. J. Fluid Mech. 30 (1), 1967.
196
Neale, G. & Nader, W. ,”Practical significance of Brinkman’s extension of Darcy’s law: coupled parallel flows
within a channel and a bounding porous medium,” Can. J. Chem. Eng. 52 (4), 475–478, 1974.
197
Vafai, K. & Thiyagaraja, R., “Analysis of flow and heat transfer at the interface region of a porous medium”. Intl
J. Heat Mass Transfer 30 (7), 1391–1405, 1987.
198
Vafai, K. & Kim, S. J., “ Fluid mechanics of the interface region between a porous medium and a fluid layer: An
exact solution”. Intl J. Heat Fluid Flow 11 (3), 254–256, 1990.
199
Hahn, S., Je, J. & Choi, H. ,”Direct numerical simulation of turbulent channel flow with permeable wall.” J. Fluid
Mech. 450, 259–285, 2002.
200
Sparrow, E. M., Beavers, G. S., Chen, T. S. & Lloyd, J. R., “Breakdown of the laminar flow regime in permeable-
walled ducts”. Trans. ASME J. Appl. Mech. 40, 337–342, 1973.
201
Beavers, G. S. & Joseph, D. D. ”Boundary conditions at a naturally permeable wall,“ J. Fluid Mechanics , 1967.
202
Marco E. Rosti, Luca Cortelezzi and Maurizio Quadrio, “Direct numerical simulation of turbulent channel flow
over porous walls”, J. Fluid Mechanics, vol. 784, pp. 396_442, Cambridge University Press 2015.
203
Tilton, N. & Cortelezzi, L., “ Stability of boundary layers over porous walls with suction,” AIAA J. 2015.
204
Perot, B. & Moin, P. ,”Shear-free turbulent boundary layers. Part I. Physical insights into near-wall turbulence,”
J. Fluid Mech. 295, 199–227.
164
turbulence.
10.3.2 Some Insight into Physical Consideration of Porous Medium
In order to describe accurately the mass and momentum transfer between a fluid-saturated porous
layer and a turbulent flow, one could, in principle, perform a DNS by solving the NS equations over
the entire domain and enforcing the no-slip and no-penetration conditions on the highly convoluted
surface representing the boundary of the porous material. In practice, however, this approach is hard
to implement because the boundary of a porous material has, in general, an extremely complex
geometry that, often, is not known in full details. Therefore, this approach has been used only in cases
in which the porous medium is highly idealized and has a simple geometry. The flow at the interface
and in the transition region within the porous layer depends, for a given porous material, mostly on
the surface machining of the interface. Figure 10.3.2 shows that even using the same (numerically
generated) porous sample and the same surface machining technique, a totally different interfacial
geometry is obtained simply by cutting the porous material half of a particle diameter deeper.
Obviously, the flow just above the interface and within the transition region for the two samples
shown in Figure 10.3.2 will be noticeably different. Therefore, in general, it is nearly impossible to
introduce a variable-porosity model that is capable of fitting all possible interface geometries, even
if the porous material used is exactly the same. On the contrary, we believe that making a choice for
a variable-porosity model reduces the generality of the model to a particular porous material with a
particular interface.
The complexity of modelling the flow at the porous/fluid interface and within the transition region
becomes less severe for porous materials of small permeability and small mean particle size. In
particular, for sufficiently low permeability’s, the fluid velocity at the interface is small; consequently
the convective effects and the drag DNS of turbulent channel flow over porous walls force
experienced by the fluid become negligible, because the dense channel-like structures of the porous
matrix impede motion between layers of fluid. In this case, the transition region (of the order of a few
pore diameters thick) and roughness (of the order of one pore diameter high) can be assumed to
have zero thickness, and porosity and permeability to have a constant value up to an interface that
has a clearly defined position.
As a consequence, porosity and permeability are effectively decoupled because they are assumed to
Figure 10.3.2 Effect of surface machining on the same numerically generated porous sample:
between-particles cut (a) and through-particles cut (b) yield highly different interfaces.
force experienced
165
be constant up to the porous/fluid interface. The zero-thickness assumption produces a jump in the
shear stresses at the interface because, at the interface, the sphere over which the volume averaging
is performed lies partly in the porous region and partly in the free fluid region (see Figure 10.3.3).
This jump in stress produces an additional boundary condition at the interface, where the magnitude
of the jump is proportional to a dimensionless parameter τ , the momentum-transfer coefficient,
which is of the order of one and can be both positive or negative. The parameter τ is decoupled from
porosity and permeability. In a recent interpretation, a negative τ quantifies the amount of stress
transferred from the free fluid to the porous matrix, while a positive τ quantifies the amount of stress
transferred from the porous matrix to the free fluid; when τ = 0 the stress carried by the free fluid is
fully transferred to the fluid saturating the porous matrix.
A porous wall of small permeability K* is often thought to behave as an effectively impermeable wall.
It is true that, in the limit K* → 0, a porous wall becomes impermeable and behaves as a solid wall.
On the other hand, very small amounts of permeability have major effects on the stability of fully
developed laminar flows in channels with one or both porous walls, and even on asymptotic suction
boundary layers. In particular, the critical Reynolds number is most sensitive to small permeability’s,
where it experiences its sharpest drop. Hence, in this paper we focus on low-permeability porous
materials. There are two main reasons for this choice. First, porous materials of small permeability’s
are common in nature and in industrial applications and, therefore, it is of interest to characterize
their effects on turbulent flows. Second, since porosity, permeability and momentum-transfer
coefficient are decoupled in the model used in the present study, our results could provide insights
for the design of novel porous materials which target specific engineering applications.
10.3.3 Velocity–Pressure Formulation
The flow of an incompressible viscous fluid through the domain can be governed by the non-
dimensional Navier–Stokes equations
Figure 10.3.3 Sketch of a Porous Medium, with l*f and l*s the Characteristic Lengths of the
Pore and Particle Diameters of the Pore-like Structures
166
Eq. 10.3.1
However, it is nearly impossible to apply this model to a flow confined by porous layers because
porous materials, in general, have very complex geometries and are characterized by a wide range of
length scales. As exemplified in Eq. 10.3.1, such scales are bounded by the smallest scales l*f (of
the fluid phase) and l*s (of the solid phase), related to the characteristic pore and particle diameters
of the pore-like structures, and the largest scale L*p, which is the characteristic thickness of the
porous layer. To overcome these difficulties, [Whitaker]
205
-
206
-
207
proposed to model only the large-
scale behavior of a flow in a porous medium by averaging the NS equations over a small sphere, of
volume V* and radius r*. This averaging procedure, which is similar to the LES decomposition, results
in the so-called volume-averaged Navier–Stokes (VANS) equations, and relies on the assumption that
the length scales of the problem are well separated, i.e., l*s l*f << r* << L*p. Under this assumption,
the volume-averaged quantities are smooth and free of small-scale fluctuations. In other words, the
fluid-saturated porous medium is described as a continuum, so that fluid quantities, such as velocity
or pressure, are defined at every point in space, regardless of their position within the fluid or solid
phase.
10.3.4 Derivation of Volume Average N-S Equations (VAN-S)
The first step in the derivation of the VANS equations consists in choosing the appropriate averaging
method. The superficial volume average <φ*>S, of a generic scalar quantity φ*, is defined as
Eq. 10.3.2
where V*f <V* is the volume of fluid contained within the averaging volume V*, while the intrinsic
volume average is defined as
Eq. 10.3.3
These two averages are related as follows
Eq. 10.3.4
where ε = V*f/V* is the porosity, or the volume-fraction of fluid contained in V* which is generally a
function of the position in a heterogeneous porous medium. The second step in the derivation of the
VANS equations consists in defining a relationship between the volume average of a derivative of a
scalar quantity and the derivative of the volume average of the same quantity, both for time and
spatial derivatives. The general transport theorem [Whitaker]
208
, a generalized formulation of the
Reynolds transport theorem, provides us with a relationship for the volume average of a time
205
Whitaker, S. , “Advances in theory of fluid motion in porous media”. Ind. Eng. Chem. 61(12), 14–28, 1969.
206
Whitaker, S. , “Flow in porous media. I: a theoretical derivation of Darcy’s law”, Transp. Porous Med. , 1986.
207
Whitaker, S. , “The Forchheimer equation: a theoretical development”. Transp. Porous Med. , 1996.
208
Whitaker, S., “Advances in theory of fluid motion in porous media”. Ind. Eng. Chem. 61(12), 14–28, 1969.
167
derivative, while the spatial averaging theorem provides us with a relationship for the volume
average of a spatial derivative. Finally, assuming the porous material to be homogeneous and
isotropic, i.e. porosity and permeability remain constant throughout the porous walls, and
permeability to be sufficiently small to neglect inertial effects, the dimensionless VANS equations
become linear because the drag term containing the Forchheimer tensor can be neglected with
respect to the Darcy drag. Since the focus of the present work are the effects of porous materials of
small permeability’s on turbulent channel flows, we assume that the fluid motion within the porous
walls is governed by the dimensionless VANS equations as:
Eq. 10.3.5
where σ=√K*/h* is the dimensionless permeability. Note that in the momentum both averages are
used. The preferred representation of the velocity is the superficial volume-average velocity, huis,
because it is always solenoidal, while the preferred representation of the pressure is the intrinsic
volume-average pressure, <p>f , because it is the pressure measured by a probe in an experimental
apparatus.
10.3.5 Discussion
To simulate accurately a turbulent flow over a porous wall, it is of crucial importance to couple
correctly the flow in the fluid region, governed by the NS equations, to the flow in the porous layers,
modelled by the VANS. In a real system, this coupling takes place in a thin layer (a few pore diameters
thick) of the porous wall, the so-called transition region, adjacent to the interface between fluid and
porous regions. Porosity and permeability change in the transition region depending on the structure
of the porous material and on how the surface of the porous layer has been machined. In general,
porosity and permeability increase rapidly from their values ε and σ within the homogeneous porous
region to unity and infinity, respectively, slightly above the interface. As a consequence of this
variation in porosity and permeability, the fluid velocity increases from the Darcy velocity in the
homogeneous porous region to its slip value just above the interface. This is achieved in the transition
region where mass and momentum transfer take place. The variations of porosity and permeability
in the transition region are difficult to model theoretically, and also their measurement is a challenge
to experimentalists.
In general, as explained by Minale
209
-
210
, at the porous/fluid interface, the total stress carried by a
fluid freely flowing over the interface equals the sum of the stresses transferred to the fluid within
the porous material and that transferred to the porous matrix. In particular, depending on the
geometry of the porous material and the machining of the interface, as the fluid flows over a saturated
porous material, in certain cases part of the momentum is transferred from the free flowing fluid to
the porous matrix, and in other cases it is the opposite. In the case of porous materials of small
permeability’s, the difficulty in modelling accurately the flow near the interface can be reduced by
assuming the transition region to have zero thickness, and porosity and permeability to have
constant values, ε and σ respectively, up to the interface. This assumption, however, produces an
error in the local averaged velocity <u>S and pressure <h>f at the interface because, at the interface,
the sphere over which the averages are computed lies partly in the porous region and partly in the
free fluid. This error is corrected by means of an additional stress jump condition at the porous/fluid
209
Minale, M. ,”Momentum transfer within a porous medium. I. Theoretical derivation of the momentum balance
on the solid skeleton”. Phys. Fluids.
210
Minale, M., “Momentum transfer within a porous medium. II. Stress boundary condition.” Phys. Fluids.
168
interface, that fully couples the NS equations to the VANS. Velocity and pressure are forced to be
continuous at the interface, while the shear stresses are in general discontinuous, with the magnitude
of this discontinuity being controlled by a dimensionless parameter τ. Over the years, the stress jump
condition proposed by researchers was further developed. Although these developments have better
explained the physical mechanisms responsible for mass and momentum transfer at the condition of
porous/fluid interface, the magnitude of the corrections with respect to the stress jump are minor.
The momentum-transfer coefficient τ, as suggested by
211
, models the transfer of stress at the
interface. This dimensionless parameter is of the order of one and can be both positive or negative,
depending on the type of porous material considered and the machining of the interface. A negative
τ quantifies the amount of stress transferred from the free fluid to the porous matrix, a positive τ
quantifies the amount of stress transferred from the porous matrix to the free fluid, and when τ = 0
the stress carried by the free fluid is fully transferred to the fluid saturating the porous matrix.
211
See previous.
