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Ben-Gurion University of the Negev
Faculty of Engineering Sciences
Department of Mechanical Engineering
Development of methods for thermo-
hydraulic simulation of nuclear reactors
and similar systems in normal working
conditions and in transient processes
Thesis submitted in partial fulfillment of the requirements for the M.Sc.
degree
Submitted by: Dmitry Zviaga
Supervised by: Dr. Yuri Feldman
September 2021
1
Ben-Gurion University of the Negev
Faculty of Engineering Sciences
Department of Mechanical Engineering
Development of methods for thermo-
hydraulic simulation of nuclear reactors
and similar systems in normal working
conditions and in transient processes
Thesis submitted in partial fulfillment of the requirements for the M.Sc.
degree
Submitted by: Dmitry Zviaga
Supervised by: Dr. Yuri Feldman
Author: …………………….…. Date: 30/09/21
Supervisor: …………………… Date: 30/09/21
Chairman of Graduate Studies Committee: ………..…… Date: 30/09/21
September 2021
2.1.22
2
Abstract
The goal of this report is to present the final project conducted in order to fulfill the
requirements of the M.Sc. degree at the Department of Mechanical Engineering, Ben-Gurion
University (BGU) of the Negev. The project comprises theoretical research investigating
natural convection compressible flow with high temperature differences and with complex
geometries. The research motivation comes from long-term research investigating and
simulating the steady state and transient multiphase flow regimes existing in the reactor core,
that was established by the Soreq Nuclear Center.
The main objective of this project is to develop a comprehensive numerical methodology that
is capable of theoretical modeling of natural convection compressible flow with high
temperature differences and with complex geometries, using standard techniques of
computational fluid dynamics (CFD) – pressure-based solution algorithms and immersed
boundary methods.
This report contains:
• A comprehensive literature review surveying methods for the simulation of natural
convection flow and immersed boundary methods.
• An extended outline of the objectives of the performed research.
• A comprehensive physical model, including the governing equations, definitions,
constitutive laws, and dimensional analysis.
• A verification study by favorable comparison with corresponding independent
numerical data available in the literature for incompressible, and non-Bossinesq
compressible flows, without complex geometry.
• A comparison between results obtained in the present study and results from previous
studies for configurations with low temperature difference and complex geometry.
• A solution and analysis of the configurations with high temperature differences and
complex geometry.
• A summary, conclusions , and recommendations for possible future work.
KEYWORDS: Compressible flow; Natural convection; Non-Boussinesq; Immersed boundary
methods; Complex geometry; Single-phase flow.
3
Acknowledgements
I would like to take this opportunity to express my immense gratitude to all those who have
given their invaluable support and assistance. In particular, I am profoundly indebted to my
supervisor, Dr. Yuri Feldman, for his professional and knowledgeable support in the physical,
computational, and numerical fields. His great contribution to this research, from his laboratory
computing equipment, through the collaboration process with the Soreq Nucelar Center, and
to the countless hours of meetings and advice at any time, is remarkable. Especially, I would
like to thank him for his close supervision during the writing of a scientific manuscript.
Additionally, I would like to thank Dr. Ido Silberman who generously contributed his time and
assisted me with his theoretical background.
The research for this thesis was financially supported by Ministry of Energy and Infrastructure
under grant contract number 218-11-038, in addition to a student scholarship provided by the
Faculty of Engineering Sciences of Ben-Gurion University of the Negev. These sources of
support contributed much to the success of the study.
In this section, I would like to include a mention of my colleagues from the Group of
Computational Physics Lab founded by my supervisor, Dr. Yuri Feldman, for their discussions
and technical support.
It is also appropriate to thank for spiritual support, so I would also like to thank my family and
friends for their interest and encouragement along the way.
4
Contents
1. Introduction and literature survey .................................................................................... 14
1.1 Motivation of the study ............................................................................................. 14
1.2 Computational approaches ........................................................................................ 14
1.2.1 Pressure-based algorithms ................................................................................. 15
1.2.2 Isothermal compressible single-phase flow ....................................................... 18
1.2.3 Thermally driven compressible single-phase flow ............................................ 19
1.2.4 Immersed boundary method for thermal compressible flow ............................. 23
1.3 Objectives of the study .............................................................................................. 24
2. Theoretical background ................................................................................................... 26
2.1 Governing equations ................................................................................................. 27
2.1.1 Continuity, momentum, and energy equations .................................................. 27
2.1.2 Determining viscosity and thermal conductivity ............................................... 29
2.2 Semi-implicit fractional-step method for thermally driven compressible flow ........ 30
2.2.1 Predictor step ..................................................................................................... 30
2.2.2 Momentum corrector ......................................................................................... 30
2.2.3 Projection step for the pressure, velocity, and density fields ............................. 30
2.2.4 Solution of the energy equation ......................................................................... 31
2.2.5 Updating thermophysical properties .................................................................. 31
2.2.6 Summary of the correction loop ........................................................................ 31
2.3 Immersed boundary formulation ............................................................................... 32
2.3.1 Immersed boundary method for compressible flows ......................................... 32
2.3.2 Communication between the Eulerian and Lagrangian systems ....................... 34
2.4 Semi-implicit fractional-step method for thermally driven compressible flow with
implemented IBM ................................................................................................................ 35
2.4.1 Predictor step ..................................................................................................... 35
2.4.2 First momentum corrector.................................................................................. 36
5
2.4.3 Applying IBM for velocity to enforce a given velocity on the surface of the
immersed body ................................................................................................................. 36
2.4.4 Solution of momentum equation with the impact of the immersed body .......... 36
2.4.5 Second momentum corrector ............................................................................. 37
2.4.6 Solution of the energy equation without the impact of the immersed body ...... 37
2.4.7 Applying IBM to enforce a given temperature on the surface of the immersed
body ………………………………………………………………………………... 37
2.4.8 Solution of the energy equation taking into account the impact of the immersed
body ………………………………………………………………………………... 38
2.4.9 Updating thermophysical properties .................................................................. 38
2.4.10 Summary of the correction loop ........................................................................ 38
2.4.11 Boundary conditions .......................................................................................... 39
3. Verification study............................................................................................................. 41
3.1 Test case I - Isothermal compressible flow in narrow channel ................................. 41
3.1.1 Test case overview ............................................................................................. 41
3.1.2 Test results and comparison to the benchmark .................................................. 42
3.2 Test case II - Natural convection flow in a differentially heated cavity ................... 50
3.2.1 Test case overview ............................................................................................. 50
3.2.2 Test results and comparison to the benchmark .................................................. 51
4. Results and discussion ..................................................................................................... 59
4.1 System’s overview .................................................................................................... 59
4.2 Results and discussion ............................................................................................... 60
4.2.1 Qualitative observations..................................................................................... 60
4.2.2 Calculation of the Nusselt number ..................................................................... 66
4.2.3 Comparison between the results of the lowest-temperature-difference cases and
previous studies ................................................................................................................ 66
4.2.4 Analysis of the heat fluxes in the flow domain.................................................. 68
4.2.5 Approximation to the power law ....................................................... 71
6
4.2.6 Multiple steady-state regimes ............................................................................ 74
5. Summary and conclusions ............................................................................................... 78
6. Bibliography .................................................................................................................... 80
7
List of Figures
Figure 2.1: Graphical representation of the calculation steps at each time step for semi-implicit
FSM for thermally driven compressible flow .......................................................................... 32
Figure 2.2: Staggered grid discretization of a two-dimensional computational domain and
immersed boundary formulation for a body , depicted by a shaded object. The horizontal and
vertical arrows represent the discrete and velocity locations, respectively. Pressure
, temperature , density , viscosity and thermal conductivity are applied at the center
of each cell . Lagrangian points along are shown as filled squares where
boundary forces or boundary heat sources are applied . ...................... 33
Figure 2.3: Graphical representation of the calculation steps at each time step for semi-implicit
FSM for thermally driven compressible flow with implemented IBM ................................... 39
Figure 3.1: Schematic of the flow within an extended channel ............................................... 41
Figure 3.2: Comparison between the distribution of streamlines: (a) the current study; (b) the
independent study [22]. Grid ............................................................................ 47
Figure 3.3: Comparison between the distribution of pressure: (a) the current study; (b) the
independent study [22]. Grid ............................................................................ 47
Figure 3.4: Comparison between the distribution of vorticity: (a) the current study; (b) the
independent study [22]. Grid ............................................................................ 47
Figure 3.5: Comparison between distribution of the velocity magnitude: (a) the current study;
(b) the independent study [22]. Grid .................................................................. 48
Figure 3.6: Distribution of the density field. Grid ............................................. 48
Figure 3.7: Distribution of: (a) pressure and (b) shear stress fields along upper and lower
channel walls. Grid ............................................................................................ 49
Figure 3.8: Distribution of: (a) horizontal velocity component; (b) vertical velocity component;
(c) pressure; (d) vorticity; (e) horizontal pressure gradient; (f) normal stress at along vertical
lines passing through and locations. Grid ............................... 49
Figure 3.9: The differentially heated cavity: geometry and boundary conditions. .................. 50
Figure 3.10: Comparison of contours for : (a) velocity, current study; (b)
temperature, current study; (c) velocity, independent study [29]; (d) temperature, independent
study [29]. ................................................................................................................................ 52
8
Figure 3.11: Comparison of contours for : (a) velocity, current study; (b)
temperature, current study; (c) velocity, independent study [29]; (d) temperature, independent
study [29]. ................................................................................................................................ 53
Figure 3.12: Comparison of contours for : (a) velocity, current study; (b)
temperature, current study; (c) velocity, independent study [29]; (d) temperature, independent
study [29]. ................................................................................................................................ 53
Figure 3.13: Comparison of contours for : (a) velocity, current study; (b)
temperature, current study; (c) velocity, independent study [29]; (d) temperature, independent
study [29]. ................................................................................................................................ 54
Figure 3.14: Comparison of contours for : (a) velocity, current study; (b)
temperature, current study; (c) velocity, independent study [29]; (d) temperature, independent
study [29]. ................................................................................................................................ 55
Figure 3.15: Comparison of contours for : (a) velocity, current study; (b)
temperature, current study; (c) velocity, independent study [29]; (d) temperature, independent
study [29]. ................................................................................................................................ 55
Figure 3.16: Comparison of contours for : (a) velocity, current study; (b)
temperature, current study; (c) velocity, independent study [29]; (d) temperature, independent
study [29]. ................................................................................................................................ 56
Figure 3.17: Comparison of contours for : (a) velocity, current study; (b)
temperature, current study; (c) velocity, independent study [29]; (d) temperature, independent
study [29]. ................................................................................................................................ 56
Figure 3.18: Vertical temperature stratification parameter vs Rayleigh number..................... 58
Figure 3.19: Nusselt number vs Rayleigh number................................................................... 58
Figure 4.1: The cold cavity with a hot cylinder at the center: geometry and boundary conditions.
.................................................................................................................................................. 59
Figure 4.2: Streamlines distribution for and ........... 62
Figure 4.3: Streamlines distribution for and ............... 62
Figure 4.4: Streamlines distribution for and ........... 62
Figure 4.5: Streamlines distribution for and ................ 62
Figure 4.6: Streamlines distribution for and ........... 63
Figure 4.7: Streamlines distribution for and ............... 63
Figure 4.8: Streamlines distribution for and . ........... 63
Figure 4.9: Streamlines distribution for and ............... 63
9
Figure 4.10: Temperature distribution for and . ....... 64
Figure 4.11: Temperature distribution for and ........... 64
Figure 4.12: Temperature distribution for and . ...... 64
Figure 4.13: Temperature distribution for and ........... 64
Figure 4.14: Temperature distribution for and ...... 65
Figure 4.15: Temperature distribution for and ........... 65
Figure 4.16: Temperature distribution for and ...... 65
Figure 4.17: Temperature distribution for and ........... 65
Figure 4.18: Physical model of a hot cylinder inside a cold tube in [72] and [58]. ................. 67
Figure 4.19: Nusselt vs Rayleigh for .................................................................... 72
Figure 4.20: Nusselt vs Rayleigh for .................................................................... 72
Figure 4.21: Nusselt vs Rayleigh for .................................................................... 73
Figure 4.22: Nusselt vs Rayleigh for . .................................................................... 73
Figure 4.23: Flow and temperature patterns corresponding to two different steady state
branches obtained for , and .................................................... 74
Figure 4.24: Flow and temperature patterns corresponding to two different steady state
branches obtained for , and .................................................... 75
Figure 4.25: Flow and temperature patterns corresponding to two different steady state
branches obtained for , and ................................................... 76
Figure 4.26: Flow and temperature patterns corresponding to two different steady state
branches obtained for and . .................................................... 77
10
List of Tables
Table 3.1: Values of velocity component currently obtained on different grids along vertical
centerline passing through ............................................................................................ 43
Table 3.2: Values of velocity component currently obtained on different grids along vertical
centerline passing through ......................................................................................... 44
Table 3.3: Values of velocity component currently obtained on different grids along
vertical centerline passing through ............................................................................... 45
Table 3.4: Values of velocity component currently obtained on different grids along
vertical centerline passing through ............................................................................ 46
Table 4.1: Comparison between the present and the previously published
values averaged
over the surface of a hot cylinder placed within a cold cube for . .......................... 67
Table 4.2: Comparison between the present and the previously published
values averaged
over the surface of a hot cylinder placed within a cold cube for . .......................... 68
Table 4.3: on the left wall ................................................................................................ 69
Table 4.4: on the right wall .............................................................................................. 69
Table 4.5: on the bottom wall .......................................................................................... 70
Table 4.6: on the top wall ................................................................................................ 70
11
Nomenclature
Initials
Description
Units
Specific heat capacity at constant pressure
Specific heat capacity at constant volume
Internal energy
Immersed forces on Eulerian grid points
Immersed forces on Lagrangian grid points
Gravity acceleration
Enthalpy
Domain height
Thermal conductivity
Domain length
Diffusion fluxes of the dimensionless
momentum equations (linear term)
Mass flow rate
Mass flow imbalance after the prediction
step
Direction
Convection fluxes of the dimensionless
momentum equations (non-linear term)
Pressure
Immersed heat fluxes on Eulerian grid
points
Immersed heat fluxes on Lagrangian grid
points
Specific gas context
Time
Time interval
Sutherland temperature for thermal
conduction
Sutherland temperature for viscosity
12
Temperature
Velocity
Volume
Coordinates on Eulerian grid
Coordinates of Lagrangian grid points
Greek Letters
Initials
Description
Units
Thermal diffusivity
Discrete delta function
Normalized temperature difference parameter
Ratio of specific heat capacities
Viscosity
Density
Stress tensor
Dimensionless variables
Symbol
Description
Formulation
Sutherland temperature for thermal
conductivity
Sutherland temperature for viscosity
Density coefficient
Mach number
Prandtl number
Rayleigh number
13
Subscripts
Symbol
Description
Symbol
Description
Characteristic value
Value at long distance
from the immersed body
Hot surface
Cold surface
Horizontal component
Vertical component
Eulerian grid
Lagrangian grid
Normal to surface
Tangential to surface
Average value
Wall
Superscripts
Symbol
Description
Symbol
Description
Dimensional value
Vector
Unit vector
Matrix/Tensor
Current time step
Next time step
Previous time step
Current iteration
Previous iteration
Corrected value after first
correction
Corrected value after second
correction
First correction value
Second correction value
Desired value
Operators
Symbol
Description
Symbol
Description
Addition
Subtraction
Scalar multiplication
Vector multiplication
Tensor multiplication
Derivative by variable
Gradient
Divergence
14
1. Introduction and literature survey
1.1 Motivation of the study
The thermal-hydraulic behaviour of nuclear reactors and similar systems, both in normal
working conditions and in transient operation, is a sophisticated physical process for which in-
depth investigation is essential in order to promote further scientific research and engineering
development. Different regimes in the reactor core host several coupled phenomena. The most
important of them is the multiphase flow coupled with nuclear reactions and neutron
distribution in the reactor core. Physical modelling of this phenomenon is essential for
enhancing the reliability and the engineering maintenance of nuclear reactors. For this reason,
a long-term research project has been established, that aims to develop a state-of-the-art solver
capable of the simulation of the steady state and transient multiphase flow regimes existing in
the reactor core. These flow regimes may contain both incompressible and compressible
phases, involve heat and mass transfer and host different phase change phenomena, such as
boiling and condensation. The final version of this advanced solver should employ state-of-
the-art numerical methods to handle fully coupled steady state and transient continuity,
momentum, and energy equations by simultaneously employing the equation of state, and
thermodynamic correlations necessary to obtain thermo-physical properties of the flow. In
addition, the solver should be capable of addressing the complex geometry of the reactor core,
as well as the moving liquid-vapor interface in loss-of-coolant-accidents (LOCA). The current
study constitutes a first, but very important, step within the long-term research, while focussing
on the development of a single-phase compressible solver for the simulation of non-Boussinesq
flows in the presence of complex geometries.
1.2 Computational approaches
Numerical approaches typically utilized in Computational Fluid Dynamics can be classified
into two general families, namely, density-based and pressure-based approaches, both aimed
to ensure continuity constraints of the simulated flows. The decision regarding the choice of
specific approach should be made by carefully, considering the physical characteristics of the
specific problem. Traditionally, pressure-based solvers were developed for low-speed
incompressible flows in which the energy equation (if it exists) is typically written solely in
terms of temperature, while all the thermo-physical flow properties are assumed to be constant
15
(i.e., there is no need for the equation of state). In this case, the pressure field plays the role of
a distributed Lagrange multiplier that aims to ensure the incompressibility constraint of the
flow. The density-based schemes were typically used to solve high-speed compressible flows,
in which the viscous effects can be neglected, and the non-viscid momentum (Euler) equations
are derived from the full Navier-Stokes equations.
The pressure-based algorithms use velocity components and pressure (or pressure-correction)
as their primary variables. In the first step, the pressure field is taken as known from the
previous time step and the momentum equations are solved by yielding the velocity field, which
does not satisfy the continuity equation. In the second stage, the pressure-correction equation
(which is derived from the continuity equation) is solved. Afterwards the pressure and the
velocity components are corrected to meet the continuity constraint. If the solver is
compressible, then, after the solution of the pressure-correction equation the energy equation
should be solved, followed by updating the density and the viscosity fields by relying on the
equations of state and Sutherland equations. Typically, outer iterations are needed to achieve
convergence of the velocity, the temperature, and the density fields and to satisfy the continuity
constraint. In density-based algorithms, the density and all the velocity component fields play
the role of primary variables. Continuity, momentum, and all other transport equations are first
solved in a fully coupled manner, and then the pressure field is derived from the equation of
state. Both pressure-based and density-based formulations have been extensively used to
simulate a broad range of flows, but the density-based formulation is typically expected to be
more accurate for simulating high-speed compressible flows, while the pressure-based
formulation is advantageous for simulating low-speed viscous compressible flows.
The main criteria that characterize flow regimes in the nuclear reactor core are: (i) low Mach
numbers, (ii) high temperature differences and, therefore, large deviations in density, viscosity,
and thermal conductivity, (iii) the presence and significance of gravity and (iv) an enclosed
expanse. As a result, in the framework of the current study the choice was made in favor of
employing the pressure-based approach. We next briefly survey the state-of-the-art of the
numerical techniques that were adopted in the current study when analyzing isothermal and
thermally driven non-Boussinesq flows.
1.2.1 Pressure-based algorithms
Pressure-based algorithms belong to the family of pressure-velocity segregated approaches, the
vast majority of which are either based on the SIMPLE method or on its derivatives (e.g., the
16
SIMPLER, SIMPLEC, SIMPLEST, SIMPLEX, SIMPLEM, PISO and PRIME approaches) as
reviewed in [1], or on the fractional-step method (FSM) [2], [3], [4], [5], [6], [7]. The two
families can be classified as marching methods, consisting of four major steps: (i) predictor
step, constituting solution of the momentum equations to obtain intermediate velocity using
velocity and pressure fields from the previous iteration/time step, (ii) solution of the Poisson
equation for pressure-correction, (iii) correction-projection step, constituting correction of the
pressure and projection of velocity fields to fulfill the continuity constraint and (iv) solution of
other transport equations, if required. Depending on the time discretization (fully- or semi-
implicit) of non-linear terms, numerical solution of the governing equations may (or may not)
contain outer iterations [8]. It is also noteworthy that in FSMs the velocity correction only
appears in the transient term, while in the SIMPLE and related methods the corrections appear
in all the velocity entries contributing to the main diagonal of the transient, convection and
diffusion terms. The theoretical basis of all state-of-the-art pressure-correction approaches
from the SIMPLE and the FSM families is given in the monograph [8]. This book can also be
consulted for gaining in-depth understanding of the ways of extending the SIMPLE and the
FSM approaches for the simulation of compressible flows, including the properties of the
pressure-correction equation and implementation of specific boundary conditions (e.g.,
prescribed total pressure, total temperature, static pressure for outflow boundary and supersonic
outflow boundary).
We next briefly review the major developments of the pressure-based algorithms for the
SIMPLE and the FSM families, in chronological order.
The FSM approach
A numerical solution of the incompressible Navier-Stokes equations based on the fractional-
step approach was presented in the seminal study of Kim and Moin in [2]. The solution was
performed on a staggered grid by calculating the pressure correction in the corrector rather than
in the predictor step. The study was then extended to collocated grids and curvilinear
coordinates [3]. Both works employed an approximate factorization ADI scheme in the
predictor step. The importance of these formulations is the implicit treatment of the viscous (or
turbulent) terms, which enabled avoiding the time-step limitations typical of their explicit
discretization. According to Gresho [4], the FSM can be classified into P1, P2 and P3 methods.
In the P1 method, the pressure field is set to zero in the momentum equation, whose solution
is further used as a predictor for the velocity field. The Poisson equation is then solved to obtain
17
the pressure and to project non-solenoidal velocity on the divergence free subspace. Following
this classification, it can be concluded that studies [2] and [3] adopted the P1 method. In the
P2 method, the pressure term is present in the momentum equation, where its value is taken
from the previous time step. The pressure-correction Poisson equation is then solved, followed
by a standard correction-projection step. The P3 method is very similar to the P2 method with
the only difference being that the pressure used in the momentum equation is extrapolated with
a second order of accuracy by utilizing the pressure field values from two previous time steps.
Further contributions in the assessment of the numerical accuracy of different formulations of
FSM is due to the works of [9], [10], [5], [6], [7], [11]. It was shown that for both staggered
[9] and collocated [10] grids the solution of the Navier-Stokes equations obtained by employing
a simplified formulation of the pressure-correction equation not including elliptic pressure
coupling is superior in terms of computational time, and is characterized by the same accuracy
compared to its counterpart obtained by employing the full pressure-correction equation. High
efficiency of the methods based on the solution of a simplified formulation of the pressure-
correction equation, not requiring outer iterations, was further demonstrated in [5] where the
second order accuracy of the results was achieved by modifying boundary conditions for the
predicted velocity. In their next study [6], the authors confirmed that the standard P1 and P2
methods provide the first and the second order accuracy in time of the pressure field,
respectively. It was also shown that the second order accuracy in time could be achieved for
the P1 method by utilizing the full pressure correction. Third order accuracy in time of the P3
method was reported in [7]. In the same study, the authors also proposed a new approach, which
they called the pressure method. The key idea was to solve the Poisson pressure correction
equation prior to the solution of the momentum equations. Both alternatives were shown to
reduce the overall error and to increase the efficiency as compared to the standard P2 method.
The SIMPLE and related approaches
The SIMPLE approach was first formulated by Patankar and Spalding in [12] as a marching
procedure to calculate the transport processes in three-dimensional incompressible flows. The
algorithm consists of three major steps: (i) implicit solution of the momentum equations to
obtain the intermediate velocity field based on the estimated pressure field, (ii) obtaining the
pressure correction by solution of the Poisson equation and calculation of the corrected pressure
and velocity fields to fulfill the continuity constraint and (iii) solution of other transport
equations, if required. If the continuity constraint is not satisfied up to a given precision, the
18
pressure and velocity fields should be updated and steps (i) and (ii) are reiterated. In the
SIMPLE algorithm, velocity corrections of the convection and the diffusion terms that are not
in the main diagonal of the coefficient matrix in the corrector step are neglected. As stated by
the authors of [12], the SIMPLE algorithm can overestimate the pressure term, which may
negatively affect the convergence rate. For this reason, Patankar recommended to apply
underrelaxation for pressure-correction calculation in order to improve the convergence rate
and to avoid divergence [13], [14]. In an attempt to improve the convergence rate, a SIMPLER
algorithm [13], [14] was developed. A key idea was not to correct the velocity field after the
pressure field had been corrected. Instead, the momentum equations are solved again, but this
time by using an updated pressure field. Then, a second pressure corrector is employed, and
pressure and velocity fields are updated. The time step is completed by proceeding to the
solution of other transport equations, if required. Similarly to the SIMPLE algorithm, the
SIMPLER algorithm only takes into account the velocity corrections entering the main
diagonal of transient, convection and diffusion terms. Van Doormal and Raithby introduced a
SIMPLEC method [15] as a different development of SIMPLE, in which the definitions of
SIMPLE’s coefficients in the pressure-correction equation are modified so that there is no need
for underrelaxation of pressure. Maliska and Raithby presented the PRIME algorithm [16], in
which the momentum equations in the predictor step are solved explicitly, and then the solution
procedure continues as in SIMPLE. Issa presented the PISO method [17], which, similarly to
the SIMPLER algorithm, is based on two corrector steps. The differences are that velocity
corrections are omitted in the convective and diffusive terms of the second corrector, and the
energy equation is solved twice, namely, after the first and the second correctors. We finish
this chapter by mentioning the developments in the field of implementation of the boundary
conditions for incompressible and compressible flows in the variety of pressure-based solvers
based on the SIMPLE algorithm, which can be found in Moulkalled et al. [18].
1.2.2 Isothermal compressible single-phase flow
When developing a novel solver for the compressible Navier Stokes equations, it is common
practice to verify its capability to successfully handle incompressible (or slightly compressible)
isothermal flows. In recent decades, a significant number of studies have been dedicated to
developing pressure-based solvers, adopting various prediction-projection algorithms for the
simulation of both incompressible and compressible flows typical of a broad range of
configurations. Armaly et al. [19] utilized an iterative finite-difference numerical scheme
developed in [20] for the investigation of laminar, transitional, and turbulent flows of air in an
19
extended channel for a Reynolds number’s range of . The authors assumed
isothermal flow when solving compressible equations of the conservation of mass and
momentum. An algorithm capable of the simulation of unsteady/transient viscous stratified
compressible flows, characterized by a broad range of subsonic Mach numbers, including
nearly incompressible flows, was developed by Mary et al. [21]. In their work the authors
solved the fully compressible Navier-Stokes equations and verified their results by comparison
with the corresponding data reported in [19] and in [22]. To remove the stiffness of the
numerical problem due to the large disparity between the flow and the acoustic wave speeds at
low Mach numbers, an approximate Newton method, based on artificial compressibility, was
used by the authors in [21]. Hauke et al. [23] developed computational techniques employing
segregated stabilized methods with standard pressure boundary conditions capable of the
simulation of both incompressible and isothermal compressible flows. Their goal was to
establish robust segregated methods characterized by superior computational performance
compared to more common coupled methods typically used for the simulation of compressible
flows. In their next work [24] the authors successfully extended their methodology to non-
isothermal compressible flows. Frehse et al. [25] addressed isothermal compressible flow from
the mathematical point of view. They validated the existence of a weak solution for the case of
mixed boundary conditions by the means of the stream function analysis.
1.2.3 Thermally driven compressible single-phase flow
Another important step in the development of a general compressible solver is to prove its
capability to handle compressible natural convection flows characterized by low values of
Mach number. For years, natural convection flows were addressed by applying the Boussinesq
approximation, i.e., by assuming incompressible flow, while introducing the buoyancy effects
in the form of a temperature dependent source in the momentum equations. However, the
Boussinesq approximation is only valid for small temperature differences (not more than 30
℃). To address the configurations characterized by high temperature differences, fully
compressible Navier Stokes equations must be solved. An extensive overview of both
compressible and incompressible approximations when simulating natural convection flows
was given by Mayeli and Sheard in [26]. Regarding incompressible approximations, the
Oberbeck-Boussinesq, the thermodynamic Boussinesq and the Gay-Lussac approximations are
typically utilized in numerical simulations. Regarding compressible approximations, the fully
compressible and the weakly compressible approaches are typically employed.
20
Weakly compressible approximations
The majority of weakly compressible approximations developed for the simulation of low
Mach number compressible flows are based on the asymptotic model, in which the total
pressure is split into the thermodynamic and the hydrodynamic pressures. In the literature, this
model is often referred to as a “classical low Mach number model”. The asymptotic model for
the simulation of thermally driven natural convection flow was introduced for the first time by
Rehm and Baum [27]. The key idea was to split the pressure field into the large, time-dependent
thermodynamic part, and the stationary part including extremely small spatial deviations. Such
a decomposition was found to be applicable for the simulation of low Mach number thermally
driven flows, and allows the pressure terms to be of the same order of magnitude as other terms
in the momentum and the energy equations. Further progress in this field is due to the work of
Paolucci [28], who formulated and solved anelastic transient equations, allowing to avoid the
appearance of acoustic waves in natural convection non-Boussinesq flows. The study presented
in [29] extended the methodology described in [28] by applying it to the simulation of two-
dimensional compressible natural convection flow in a vertical slot with large horizontal
temperature differences. Le Quere et al. [30] approached the non-Boussinesq natural
convection confined flow by extending a pseudo-spectral algorithm from [31] that was
originally developed by them for Boussinesq natural convection flow. The algorithm developed
in [29] was further successfully extended for three-dimensional configurations, as presented in
[32]. Paillere et al. [33] investigated two numerical methods to solve low Mach number
compressible flows by simulating natural convection flow in a differentially heated cavity with
a high temperature difference. Elmo and Cioni [34] compared the Boussinesq approximation
and quasi-compressible model, based on a classical low Mach number model, when applying
it to the simulation of the pebble bed reactor. Schall et al. [35] studied steady state and transient
thermal compressible flow by applying Turkel preconditioned Roe splitting, which is more
likely to apply to the fully compressible hyperbolic solvers (see below), and to flows modelled
by the classical low Mach number model. Le Maitre et al. [36] developed a stochastic
projection method (SPM) for the quantitative propagation of uncertainty in compressible low
Mach number flows (which they referred to as “zero Mach number flows”) under non-
Boussinesq conditions. Beccantini et al. [37] investigated a transient injection flow in a low
Mach number regime, employing three different approaches where two of them are based on
asymptotic models of the Navier-Stokes equations. Reddy et al. [38] studied conjugate natural
convection in a vertical annulus with a centrally located vertical heat generating rod, employing
21
compressible fluid transport equations and an asymptotic model for low Mach numbers. Lappa
[39] employed low Mach number asymptotics and developed a flexible and modular solver
that takes into account all the molecular (translational, rotational and vibrational) degrees of
freedom and their effective excitation, and guarantees adequate interplay between molecular
and macroscopic-level entities and processes. Armengol et al. [40] studied the effects of air
variable properties in the transient case of the classical differentially heated square cavity
problem, employing the SIMPLE algorithm, a low Mach number approximation and the finite
volume method.
Some authors approached the weakly compressible approximation and addressed thermal
systems with mixed natural convection-radiation thermal mechanisms, considering
compressible flow. Darbandi and Abrar [41] developed a hybrid incompressible-compressible
method to solve the combined natural convection-radiation heat transfer in a participating
medium at steady state without addressing the Boussinesq approximation or employing the low
Mach number asymptotic assumption. Their study showed that compressibility effects become
more dominant in combined natural convection-radiation problems than in the pure natural-
convection problem. Parmananda et al. [42] presented a computational framework for non-
Oberbeck-Boussinesq buoyancy driven turbulent transient convection coupled with thermal
radiation at large temperature differences. They used a low Mach number model based on the
Favre-averaged (Navier-Stokes and energy) equations, with the standard model
presented using an unstructured finite volume method. The same research group as in [42]
continued their work, and in [43] presented the development of a non-Boussinesq flow solver
employing low Mach number asymptotics and a fractional-step method to simulate combined
radiative-convective heat transfer, which is suitable for arbitrary polygonal meshes. They also
continued their work in [44], where they investigated three different algorithms for the
numerical simulation of non-Boussinesq convection with thermal radiative heat transfer based
on a low Mach number formulation. The first algorithm (Algorithm A) uses conservation of
mass and energy equations to compute density and temperature. The other two algorithms
(Algorithm B and Algorithm C) calculate temperature and density from the equation of state,
respectively, and solve a conservative form of the continuity and energy equations to obtain
density and temperature, respectively.
22
Fully compressible approximation
Compared to the weakly compressible approximations, fully compressible approximations for
low Mach number flows adopted several different approaches. Some fully compressible
approximations for low Mach number compressible flows employ density-based solvers with
splitting schemes and preconditioning. Paillere et al. [33] employed a hyperbolic solver based
on the resolution of the compressible Euler equations, with a Roe’s flux difference splitting
scheme and Turkel preconditioning. Beccantini et al. [37] used the fully compressible Navier-
Stokes equations, employing a density-based solver, and using a HLLC computation scheme
and a ILUTP preconditioner. Fu et al. [45] studied natural convection in a channel under a high
temperature difference with a fully compressible approximation using Roe preconditioning and
dual time stepping. In order to resolve reflections induced by acoustic waves at the boundaries
of the channel, non-reflection conditions at the boundaries of the channel were derived. El-
Gendi and Aly in [46] simulated natural convection compressible flow using a Roe scheme and
a dual time method, in square and sinusoidal cavities without applying the Boussinesq
approximation. Li in [47] studied the laminar-turbulent transition induced by natural
convection with high temperature differences, employing compressible transport equations,
Roe preconditioning and an implicit large eddy simulation.
Other fully compressible approximations employ pressure-based solvers. Sewall and Tafti in
[48] developed a variable property algorithm for time-dependent resolution of flows with large
temperature differences without using the low Mach number assumption; in this algorithm, the
momentum and energy equations are integrated in time using an implicit Crank-Nicolson
method, the Helmholtz equation for pressure was solved at each inner iteration, and local
density changes were coupled with variations of both local temperature and pressure fields,
whereas other properties are only coupled with temperature variations. Barrios-Pina et al. [49]
adapted the method from [48] to conduct thermodynamic analysis to determine the contribution
of each term in the total energy equation.
Another fully compressible approximation employed a mesoscopic computational method
based on a model Boltzmann equation. Wen et al. [50], [51] recovered the fully compressible
Navier–Stokes equations by employing the Boltzmann equation with the Bhatnagar–Gross–
Krook (BGK) model used by Guo et al. [52]. A standard differentially heated cavity with a
large temperature difference was chosen as a testbed for validating the mesoscopic method
when applied to the simulation of Boussinesq natural convection flow, as well as for addressing
23
the non-Boussinesq flow at near-turbulent and turbulent steady states [50] and for transient
[51] flows.
1.2.4 Immersed boundary method for thermal compressible flow
In order to acquire the ability to simulate thermal compressible flows in the presence of
complex boundaries it is common to extend the existing solvers by incorporating capabilities
of the immersed boundary method (IBM) within them. IBM, initially developed by Peskin [53],
can simulate flow in the presence of complex, movable, and deformable boundaries. The
simulations take advantage of solvers utilizing compact and simple stencils of discretized
differential operators that can be efficiently employed on structured grids. The boundary
conditions of no-slip and prescribed values of temperature (or heat flux) on each immersed
boundary are enforced by introducing forces and heat fluxes as additional unknowns of the
problem. A closure of the overall system is achieved by including additional equations in the
form of kinematic constraints for all the unknowns.
In the last decade IBM has been widely utilized for investigating natural convection within
enclosures with embedded discrete thermally active sources (or sinks) of various geometries.
Interest in this field has been motivated by its addressing a broad spectrum of engineering
applications based on gas-solid heat exchangers and a fundamental understanding of the
instability of highly separated confined flows. It is worth mentioning in this context the works
of [54], [55], [56], [57] and the studies of [58], [59], [60], [61], [62], which addressed natural
convection confined flows in the presence of complex two-dimensional and three-dimensional
geometries, respectively.
Studies utilizing IBM for the analysis of thermal compressible flows are relatively scarce. Most
of the works in this field addressed high Mach number compressible flows, focusing on
transonic/supersonic transitions, or comparing between characteristics of subsonic and
supersonic flows. For high Mach number flows the impact of viscosity and the thermal
behavior of the flow is negligible compared to the compressibility effects, and thus in these
studies both phenomena are typically neglected. In contrast, when simulating low Mach
number thermal flows, which is the focus of the current study, both effects play a significant
role and should be carefully addressed.
It should be noted that accurate implementation of IBM forcing in low Mach number
compressible flows is still the subject of active research. In a recently developed IBM scheme
[63] the authors introduced a novel pressure-based correction of the IBM forcing (in addition
24
to the classical one based on the time derivative of velocity)
1
and applied it to the analysis of
three-dimensional low and high Mach number pressure driven flows. Comparison of the
obtained results with the corresponding data obtained by body-fitted DNS revealed that
pressure-correction of the IBM forcing significantly improves the accuracy of the IBM
procedure for low Mach number flows. An additional contribution to the application of IBM
for the simulation of compressible thermally driven confined flow is due to the work of Kumar
and Natarajan [64]. The authors developed a diffuse immersed boundary approach for
thermally driven non-Boussinesq flows, which, however, relies on a quasi-incompressible
formulation of the governing equations and therefore cannot be considered to be a fully
compressible approach.
1.3 Objectives of the study
The present study aims to develop and extensively verify a general transient pressure-based
solver for the simulation of thermally driven non-Boussinesq flows within complex geometries
typical of modern nuclear reactors. A set of governing equations, including the continuity, the
momentum and the energy equations along with the equation of state, are solved, while the
thermophysical properties of the fluid are determined at each time step. The solver employs
backward second-order finite difference and standard second-order finite volume methods for
the temporal and the spatial discretizations, respectively. A semi-implicit fractional-step
method for all the flow speeds is implemented for the pressure-velocity coupling. The IBM
formulation that has been successfully applied for compressible pressure driven flows [63] was
chosen, and further extended for the simulation of thermally driven non-Boussinesq flows.
The present study is accomplished in three stages: (i) the development of a generic solver for
the simulation of non-Boussinesq thermally driven flows in rectangular containers, (ii) the
development of a novel formulation of the IBM suitable for non-Boussinesq thermally driven
flows and (iii) implementation of the developed IBM by incorporating it into the generic solver
developed in stage (i).
In the first stage, the results reported in [19] and [22] are carefully replicated with regard to
the simulation of isothermal compressible flow. The methodology utilized in the current study
1
We note in passing that pressure-correction of the direct forcing IBM must not be confused with a pressure-
correction equation of the fractional-step and the SIMPLE related methods
25
differs from that reported in [19] and [22] in the sense that: (i) the flow is assumed to be
compressible a priori, which implies solution of the compressible continuity, momentum and
energy equations, (ii) the governing equations are unsteady and therefore time integration is
performed until the flow converges to its steady state up to the predefined convergence
criterion. We next carefully replicate the results reported in [29] with respect to both Bousinesq
and non-Bousinesq flows in differentially heated square cavities. Finally, the current study
extends the IBM formulation reported in [63] to the non-Boussinesq thermally driven flows
and presents in-depth analysis of the characteristics of non-Boussinesq natural convection flow
typical of the configuration consisting of a hot cylinder placed within a square cold cavity.
26
2. Theoretical background
Natural convection flow is ubiquitous in a vast variety of engineering applications. The flow is
governed by the system of continuity, Navier-Stokes, and energy equations, which, due to their
non-linearity, must be solved numerically. The methods developed over the years for numerical
simulation of natural convection phenomena are classified into coupled and segregated
approaches to provide full pressure-velocity coupling. The current study utilizes the segregated
pressure–velocity coupling approach, namely the fractional-step method. When utilized in the
context of the Boussinesq approximation, (i.e., flow is assumed incompressible) the method
consists of a number of basic steps: (i) predictor that aims to accurately estimate the velocity
field by utilizing the pressure field from the previous time step, (ii) corrector that aims to obtain
the pressure correction for the current time step, (iii) projection using the pressure-correction
values to update the pressure field and to project the velocity field on the divergence free
subspace and (iv) solution of the energy equation. For non-Boussinesq natural convection flows
(i.e., the flow is assumed to be compressible), the procedure is more sophisticated, since in this
case the pressure is a thermodynamic rather than simply a hydrodynamic property. An
additional difficulty is that for compressible flow the density, the viscosity and the thermal
conductivity fields are not constant anymore. Rather, the density is coupled with the pressure
by the equation of state, while the viscosity and thermal conductivity are coupled with the
temperature field by the Sutherland laws. To resolve this, step (iii) is modified. Namely, at step
(iii) the pressure corrections are further utilized to update both the pressure and the velocity
fields at the current time step and to calculate intermediate density by utilizing the equation of
state. At step (iv) the energy equation is solved, and the current time-step temperature is
obtained. At the final step, step (v), the viscosity and thermal conductivity are updated by
utilizing the Sutherland equations. Note that steps (iii) – (v) are repeated within the outer
iteration until the continuity equation is satisfied up to a given precision.
Most engineering applications of fluid dynamics and heat transfer involve complex geometries
whose modeling by standard body conformed grids is often a challenge. The problem exists
regardless of the heat transfer mechanism governing the system. There are several strategies to
overcome this problem, including stepwise approximation of orthogonal grids, overlapping
grids and boundary-fitted non-orthogonal grids, and utilizing IBM.
IBM utilizes regular (typically Cartesian) grids and resolves the boundary irregularity by
locally inducing additional volumetric sources and heat fluxes to impose kinematic constraints
27
on the immersed surface. Details on the immersed boundary formulation utilized in the current
study are given in the following.
The current study focuses on the solution of compressible natural convection flows in the
presence of complex geometries by combining implicit fractional-step and immersed boundary
methods. A detailed description of the methodology and the numerical formulation utilized in
the current study is presented below.
2.1 Governing equations
2.1.1 Continuity, momentum, and energy equations
The dimensional form of the continuity, momentum, and total energy equations is given by:
(2.1)
(2.2)
(2.3)
where
is the velocity, is the density, is the pressure, is the internal energy, is the
dynamic viscosity,
is the thermal conductivity, is the standard gravity, and is time. The
generic subscript symbol corresponds to a dimensional variable.
The energy equation can be modified slightly. First, the kinetic and potential energies are
relatively small in natural convection flows. Second, the dissipation term is typically neglected.
Finally, the internal energy, , is a thermodynamic state variable that is rarely used in practical
engineering applications, while the more commonly used quantity is the enthalpy,
, that is
related to the internal energy. For ideal gases, enthalpy is equal to the temperature multiplied
by the specific heat capacity at constant pressure. Then, the energy equation is rendered as:
(2.4)
28
where
are the specific heat capacity at constant pressure and temperature, respectively.
For an ideal gas, the equation of state is:
(2.5)
where is the specific gas constant.
The governing equations were rendered dimensionless by utilizing the scaling proposed in [29]:
(2.6)
where are geometric coordinates, is a geometric characteristic length, is the thermal
diffusivity, is the characteristic pressure, is the characteristic density, is the
characteristic temperature, is the characteristic specific heat capacity for constant pressure,
is the characteristic dynamic viscosity and is the characteristic thermal conductivity.
The above scaling yields the following non-dimensional groups governing the natural
convection flow:
(2.7)
where , , and are the Rayleigh, Prandtl and Mach numbers, respectively, is the ratio
of specific heat capacities, i.e., , is a normalized temperature difference
parameter, are the maximal and minimal temperatures, respectively, whereas the relation
between these temperatures and the characteristic temperature is .
The non-dimensional mass, momentum and energy equations are next formulated as:
(2.8)
(2.9)
(2.10)
29
The term appears to be equal to unity, and therefore the non-dimensional equation
of state is:
(2.11)
where is a density coefficient equal to .
2.1.2 Determining viscosity and thermal conductivity
2.1.2.1 Dynamic viscosity
For compressible flow, viscosity must be determined using specific correlation. For ideal gases,
the Sutherland equation is typically used:
(2.12)
where is the dimensional Sutherland temperature for viscosity.
Eq. (2.12) is rendered dimensionless as:
(2.13)
where
is the dimensionless Sutherland temperature for viscosity.
2.1.2.2 Thermal conductivity
For ideal gases, Sutherland’s law can also be applied for thermal conductivity:
(2.14)
where is the dimensional Sutherland temperature for thermal conductivity.
Eq. (2.14) is rendered dimensionless as:
(2.15)
where
is the dimensionless Sutherland temperature for thermal conductivity.
30
2.2 Semi-implicit fractional-step method for thermally driven
compressible flow
2.2.1 Predictor step
When solving the system of continuity, energy and Navier-Stokes equations, outer iterations
are used to ensure the mass conservation of the flow. Specifically, at the mth iteration of the
current time step the momentum equation is first solved in order to predict the velocity field.
The 2nd-order, three-time-level scheme was utilized for the time integration, while the diffusion
term was taken in implicit/explicit form, and convective and pressure terms were taken
explicitly from the previous calculations:
(2.16)
where
is the predictor field for
,
is the convection fluxes (non-linear term) and
is the diffusion fluxes (linear term).
2.2.2 Momentum corrector
After obtaining the predicted velocity field, the pressure-correction equation developed from
the continuity equation is solved as follows:
(2.17)
where is a mass flow imbalance that exists because the predicted velocity does not
necessarily satisfy the continuity equation:
(2.18)
2.2.3 Projection step for the pressure, velocity, and density fields
The solution of Eq. (2.17) is followed by the projection step, at which pressure, density, and
velocity are updated:
(2.19)
where:
31
(2.20)
2.2.4 Solution of the energy equation
The procedure sequence is finalized by solution of the energy equation to obtain the new
temperature field:
(2.21)
where is a non-dimensional specific heat capacity which is equal to unity.
2.2.5 Updating thermophysical properties
After the energy equation has been solved, the viscosity,
, and thermal conductivity,
,
are updated using the Sutherland equations given by Eqs. (2.13) and (2.15). The coefficient
and density field are updated by using the equation of state:
(2.22)
2.2.6 Summary of the correction loop
Eqs. (2.17), (2.19)-(2.22) constitute an outer iteration loop, which is run at each time step. At
the end of the iteration, after updating the thermophysical properties, the solver performs a
mass conservation check by restricting the L-infinity norm to a value of . The fields of ,
and
are updated until predefined values of convergence criteria are achieved for each field.
After a sufficient number of iterations has been performed, corrections become negligible, the
flow can be considered as mass-conservative and the solution can proceed to the next time step
by setting
and . A graphical representation of
all the steps that are performed in each time step is shown in Fig 2.1.
32
Figure 2.1: Graphical representation of the calculation steps at each time step for semi-implicit FSM for thermally driven
compressible flow
2.3 Immersed boundary formulation
2.3.1 Immersed boundary method for compressible flows
The immersed boundary formulation utilized in the framework of the current study is briefly
introduced first, and the strategies for its application in the analysis of compressible natural
convection flow are next explained. Fig. 2.2 shows the setup of a typical spatial discretization
implemented on a staggered grid. The grid is characterized by offset velocity, temperature,
pressure, density, viscosity, and thermal conductivity fields. An arbitrary immersed object, ,
within a computational domain, , (whose geometry does not, in general, have to conform to
the underlying spatial grid), is represented by the surface, , determined by a set of
Lagrangian points, . At the Lagrangian points, appropriate surface forces, , and heat
Predictor step -
Momentum corrector -
Projection step -
Solution of the energy equation -
Updating thermophysical properties -
Checking continuity constraint
Proceed to the next time step
Continuity conserved
Continuity
is not
conserved
33
sources, , are applied to enforce the non-slip velocity and the Dirichlet temperature boundary
conditions along .
Figure 2.2: Staggered grid discretization of a
two-dimensional computational domain
and immersed boundary formulation for a
body , depicted by a shaded object. The
horizontal and vertical arrows
represent the discrete and velocity
locations, respectively. Pressure ,
temperature , density , viscosity and
thermal conductivity are applied at the
center of each cell . Lagrangian points
along are shown as filled
squares where boundary forces
or boundary heat sources
are applied .
These forces and heat fluxes appear as additional unknown variables, whose values – along
with those for the pressure, temperature, and velocity fields – are provided by solving the
Navier Stokes and the energy equations and are directly accounted for in the overall balance,
enabling direct calculation of the Nusselt number. Since the location of the Lagrangian
boundary points does not necessarily coincide with the underlying spatial discretization,
regularization and interpolation operators must be defined to convey information in both
directions of the body. The regularization operator smears singularly acting surface forces, ,
and heat sources, , on the nearby computational domain, while the interpolation operator
acts in the opposite direction and imposes non-slip/thermal boundary conditions on the points
located on the body surface. Equations governing the thermally driven, compressible natural
convection flow, along with the embedded immersed boundary formulation can be written as:
(2.23)
(2.24)
(2.25)
where the volumetric force and volumetric heat source reflect the impact of the
immersed body on the surrounding flow.
34
The immersed boundary formulation of the compressible flow utilized in the current study was
first introduced by Riahi et al. [63] and is currently extended for the simulation of thermally
driven compressible flows. This method originates from the works of Uhlmann [65] and Pinelli
et al. [66], combining advantages of both continuous [67] and discrete forcing methods [68].
2.3.2 Communication between the Eulerian and Lagrangian systems
Communication between the Lagrangian and Eulerian grids is performed in three steps, as
described below.
2.3.2.1 Interpolation step
In the interpolation step, physical quantities in the Eulerian mesh are interpolated on the
boundary ∂B. We next give a specific example for the interpolation of
, ,
quantities discretized on the Eulerian mesh. The values of
and interpolated
on any Lagrangian point are calculated by employing the interpolation operator :
(2.26)
(2.27)
(2.28)
where is a dimension of the Eulerian grid and is a discrete Dirac delta function
defined in Eq. (2.29).
Convolutions with the Dirac delta function are used to facilitate the exchange of information
from the Eulerian to the Lagrangian grid. Among the variety of discrete delta functions
available, we chose the function described by Roma et al. [69], which was specifically designed
for use on staggered grids, where even/odd de-coupling does not occur:
(2.29)
35
The chosen discrete delta function is supported only over three cells, which is advantageous
for computational efficiency. As observed by Colonius and Taira [70], no significant
differences in the results are expected when using the alternative discrete delta functions used
in previous works.
2.3.2.2 Calculation of direct forcing sources on Lagrangian grid
Following the interpolation step, the Lagrangian volumetric forces and heat sources are
calculated as was suggested in [63]:
(2.30)
(2.31)
where superscript applies for the desired boundary condition of the immersed body.
2.3.2.3 Regularization step
In the regularization step, the values of volumetric sources previously calculated on the
Lagrangian grid are regularized (spread) back to the Eulerian grid. This backward
communication is implemented by utilizing the same delta functions as for the interpolation
step. The values of the volumetric force and heat source terms evaluated on the Eulerian mesh
are given by:
(2.32)
(2.33)
2.4 Semi-implicit fractional-step method for thermally driven
compressible flow with implemented IBM
2.4.1 Predictor step
The solution of the momentum equation to predict the velocity field without taking into account
the presence of an immersed body is given by:
36
(2.34)
2.4.2 First momentum corrector
After obtaining the predicted velocity, the pressure-correction equation, which has been
developed from the continuity equation, is solved again not taking into account the presence of
the immersed body:
(2.35)
where is a mass flow imbalance that exists because the predicted velocity does not
necessarily satisfy the continuity equation:
(2.36)
Following calculation of the first pressure-correction, the intermediate pressure is calculated,
and is used later when employing immersed boundary functionality:
(2.37)
2.4.3 Applying IBM for velocity to enforce a given velocity on the
surface of the immersed body
After acquiring an intermediate pressure, IBM for velocity is implemented via Eqs. (2.26),
(2.27), (2.29), (2.30) and (2.32). Note that the term is not recalculated during the correction
loop and is determined only once at the beginning of the time step. As a result, the volume
force from the immersed body is added to the right-hand side of the momentum equation.
2.4.4 Solution of momentum equation with the impact of the
immersed body
After the volumetric force has been calculated, the momentum equation with an updated right-
hand side is solved to determine the new velocity field that takes into account the impact of the
immersed body:
37
(2.38)
2.4.5 Second momentum corrector
At this stage, the pressure-correction equation is solved with an updated right-hand side:
(2.39)
where is a mass flow imbalance that exists because the predicted velocity did not
necessarily satisfy the continuity equation:
(2.40)
Next, the new pressure and intermediate density are acquired:
(2.41)
2.4.6 Solution of the energy equation without the impact of the
immersed body
We next solve the energy equation without taking into account the existence of an immersed
body to obtain the intermediate temperature:
(2.42)
2.4.7 Applying IBM to enforce a given temperature on the surface of
the immersed body
At each time step in the first correction iteration, after acquiring an intermediate temperature,
IBM for temperature is implemented via Eqs. (2.28), (2.29), (2.31) and (2.33). Note the term
is not recalculated during the FSM loop and is determined only once at the beginning of the
time step. As a result, the calculated volumetric heat source is added to the right-hand side of
the energy equation.
38
2.4.8 Solution of the energy equation taking into account the impact
of the immersed body
(2.43)
2.4.9 Updating thermophysical properties
After the energy equation has been solved, viscosity
, thermal conductivity
, the
coefficient and density
, are updated by using the Sutherland equations and the equation of
state via Eqs. (2.13), (2.15) and (2.22).
2.4.10 Summary of the correction loop
Similarly to the general formulation of the compressible semi-implicit fractional-step method
presented in section 2.2, in the semi-implicit fractional-step method combined with IBM, Eqs.
(2.35), (2.37)-(2.39) and (2.41)-(2.43) constitute an outer iteration loop, which is run at each
time step. At the end of the iteration, after updating the thermophysical properties, the solver
performs a mass conservation check by restricting the L-infinity norm to a value of . The
fields of , and
are updated until predefined values of convergence criteria are achieved
for each field. After performing a sufficient number of iterations, corrections become negligible
and the flow can be considered to be mass-conservative. The solution can then proceed to the
next time step by setting
and . The graphical
representation of all the steps that are performed in each time step are shown in Fig 2.3.
39
Figure 2.3: Graphical representation of the calculation steps at each time step for semi-implicit FSM for thermally driven
compressible flow with implemented IBM
2.4.11 Boundary conditions
The numerical methodology applied in the current study was implemented by utilizing
primitive variables, i.e., velocity components, pressure, and temperature. The reason for the
above choice is the simplicity of formulating the boundary conditions, at least for standard
configurations like tube/channel flow or natural convection flow within confinements.
Determining proper boundary conditions for the velocity vector is a simple process of choosing
between a few variants: no-slip at a wall , slip at a wall
, constant inlet profile, constant outlet profile and zero gradient at the outlet
(if the pressure at the outlet is known). This is also correct with respect to determining boundary
Predictor step -
First momentum corrector -
IBM for velocity -
Solution of momentum equation with the impact of the immersed body -
Updating thermophysical properties -
Checking continuity constraint
Proceed to the next time step
Continuity conserved
Continuity
is not
conserved
Second momentum corrector -
Solution of energy equation without the impact of the immersed body -
IBM for temperature -
Solution of energy equation with the impact of the immersed body -
40
conditions for temperature which can be either constant temperature or constant heat
flux at a specified boundary. Determining boundary conditions for pressure
is not as straightforward. In confined enclosures it is common to apply a zero gradient boundary
condition normal to the wall direction. However, such an approximation requires further
clarifications. The subject was investigated thoroughly by Gresho and Sani [71] with respect
to applying physically correct boundary conditions for pressure for incompressible flow in
enclosures. The key idea was to employ locally the momentum equation near the specified
boundary, i.e., wall. In the current study we followed the same principle when simulating
compressible flow, i.e., locally employing the momentum equation Eq. (2.9). As a result of the
no-slip constraint, all the velocity components at the cavity walls are equal to zero, resulting in
and
. The momentum equation at the wall thus can be reduced
to:
(2.44)
At this point the order of every term entering Eq. (2.44) can be assessed. In the first term the
maximal value of is , the is constant and equal to , and the values of and
density are in the range of . Therefore, the maximal value of the first term is of
order . The second term multiplies the values of , the dynamic viscosity and the second
derivative of the velocity. As already mentioned, the value of is equal to 0.71; the values of
viscosity lie in the range of (for the specific values of ), which means that the
order of the second term is determined by the second derivative of velocity at the wall. Both
terms are multiplied by
, which is of order . Assuming that the order of the second
derivative of velocity is significantly lower than the order of
, the boundary condition of
zero pressure gradient normal to the wall direction can be justified. After the simulations were
performed, the above assumption was further successfully verified by calculating the second
derivative of velocity in the vicinity of all boundaries whose value was of order , which
bounds the normal pressure gradient at the cavity walls by .
41
3. Verification study
The methodology described in chapter 2 was verified by applying it to the solution of two
benchmark problems - simulating incompressible, and thermally driven compressible flows.
The flow was driven by two different mechanisms: isothermal flow in a long rectangular
narrow channel driven by a pressure gradient, and natural convection flow within a
differentially heated square cavity driven by a temperature gradient. The results acquired for
both configurations were compared with data available in the literature.
3.1 Test case I - Isothermal compressible flow in narrow
channel
3.1.1 Test case overview
This test case focusses on proving the capability to address incompressible isothermal flow by
applying the currently developed methodology for compressible flow. The flow configuration
presented in [22] simulates flow within an extended channel to minimize the effect of the
outflow boundary on the upstream recirculation zones. A schematic of the geometric properties
and boundary conditions of the considered configuration are shown in Fig. 3.1.
Figure 3.1: Schematic of the flow within an extended channel
The fluid enters the domain at the upper half of the left side, proceeds through the channel and
exits at the right side. No-slip velocity and zero-gradient pressure boundary conditions were
applied at all stationary walls. At the inlet, the vertical component of the velocity was equal to
zero, a zero-gradient boundary condition was applied to the pressure field and a parabolic
distribution with maximal and average values equal to and , respectively,
were assigned to the horizontal velocity component. At the outflow, zero values were set to the
normal stress, , and to the vertical velocity component. Considering that
42
the pressure field at the outlet is known and set to zero, the gradient of the horizontal velocity
component is also equal to zero.
The Navier-Stokes equations presented in subsection 2.1.1 governing the isothermal flow
within an extended channel were rendered dimensionless by using the scales defined in Eq.
(3.1):
(3.1)
where , is the characteristic velocity and is the Reynolds number, and all other quantities
have been defined in subsection 2.1.1. The Reynolds number was set to be 800. When
calculating the normal stress, in the postprocessing stage the viscosity was adjusted to meet the
above value of the Reynolds number.
Non-dimensional continuity and momentum equations and the equation of state are formulated
as:
(3.2)
(3.3)
.
(3.4)
3.1.2 Test results and comparison to the benchmark
The results obtained by the developed methodology were compared both qualitatively and
quantitively with the corresponding data provided in [22]. To prove grid independence of the
obtained results the simulations were performed on five uniform grids:
and . The results that are presented below
were achieved with the grid of cells. Tables 3.1-3.4 summarize the results of the
grid independence study in terms of the values obtained for the and velocity components
along two vertical lines passing through the and locations. It can be seen that all
the values obtained on the two finest grids coincide up to four significant digits, which
successfully proves grid independence of the obtained results.
43
Table 3.1: Values of velocity component currently obtained on different grids along vertical centerline passing
through
Grid
Grid
12
Grid
15
Grid
24
Grid
30
0.5
0.0000
0.0000
0.0000
0.0000
0.0000
0.45
-0.0538
-0.0411
-0.0407
-0.0387
-0.0387
0.4
-0.0720
-0.0541
-0.0525
-0.0498
-0.0498
0.35
-0.0540
-0.0380
-0.0369
-0.0325
-0.0325
0.3
-0.0005
0.0078
0.0092
0.0143
0.0143
0.25
0.0877
0.0847
0.0851
0.0923
0.0923
0.2
0.2112
0.1950
0.1954
0.2038
0.2038
0.15
0.3705
0.3399
0.3387
0.3496
0.3496
0.1
0.5595
0.5151
0.5130
0.5242
0.5242
0.05
0.7586
0.7054
0.7021
0.7121
0.7121
0
0.9365
0.8852
0.8810
0.8886
0.8886
-0.05
1.0633
1.0267
1.0242
1.0270
1.0270
-0.1
1.1226
1.1085
1.1065
1.1068
1.1068
-0.15
1.1120
1.1209
1.1223
1.1176
1.1176
-0.2
1.0402
1.0656
1.0673
1.0606
1.0606
-0.25
0.9214
0.9538
0.9569
0.9460
0.9460
-0.3
0.7712
0.8017
0.8035
0.7903
0.7903
-0.35
0.6022
0.6272
0.6280
0.6123
0.6123
-0.4
0.4189
0.4425
0.4425
0.4276
0.4276
-0.45
0.2170
0.2399
0.2413
0.2320
0.2320
-0.5
0.0000
0.0000
0.0000
0.0000
0.0000
44
Table 3.2: Values of velocity component currently obtained on different grids along vertical centerline passing
through
Grid
Grid
12
Grid
15
Grid
24
Grid
30
0.5
0.0000
0.0000
0.0000
0.0000
0.0000
0.45
0.1090
0.1026
0.1018
0.1016
0.1016
0.4
0.2177
0.2050
0.2034
0.2030
0.2030
0.35
0.3267
0.3084
0.3061
0.3055
0.3055
0.3
0.4359
0.4132
0.4104
0.4095
0.4095
0.25
0.5430
0.5180
0.5150
0.5139
0.5139
0.2
0.6436
0.6188
0.6158
0.6147
0.6147
0.15
0.7316
0.7098
0.7075
0.7061
0.7061
0.1
0.7998
0.7837
0.7816
0.7807
0.7807
0.05
0.8418
0.8334
0.8329
0.8316
0.8316
0
0.8533
0.8539
0.8538
0.8535
0.8535
-0.05
0.8331
0.8427
0.8443
0.8437
0.8437
-0.1
0.7835
0.8007
0.8027
0.8032
0.8032
-0.15
0.7097
0.7320
0.7351
0.7355
0.7355
-0.2
0.6187
0.6430
0.6460
0.6471
0.6471
-0.25
0.5178
0.5411
0.5442
0.5451
0.5451
-0.3
0.4133
0.4330
0.4356
0.4366
0.4366
-0.35
0.3090
0.3237
0.3257
0.3264
0.3264
-0.4
0.2058
0.2153
0.2166
0.2170
0.2170
-0.45
0.1033
0.1078
0.1084
0.1086
0.1086
-0.5
0.0000
0.0000
0.0000
0.0000
0.0000
45
Table 3.3: Values of velocity component currently obtained on different grids along vertical centerline passing
through
Grid
Grid
12
Grid
15
Grid
24
Grid
30
0.5
0.0000
0.0000
0.0000
0.0000
0.0000
0.45
-0.6722
-0.3348
-0.3172
-0.2812
-0.2812
0.4
-1.6185
-0.9749
-0.9269
-0.8801
-0.8801
0.35
-1.8780
-1.4744
-1.4367
-1.4769
-1.4769
0.3
-0.7839
-1.5135
-1.5818
-1.8697
-1.8697
0.25
1.9609
-0.9820
-1.2525
-2.0565
-2.0565
0.2
6.2750
-0.0290
-0.6444
-2.2721
-2.2721
0.15
11.6588
0.9508
-0.1422
-2.9272
-2.9272
0.1
17.1599
1.4425
-0.1892
-4.4183
-4.4183
0.05
21.6296
1.0645
-1.1432
-6.8694
-6.8694
0
24.3636
-0.1940
-2.8775
-9.9872
-9.9872
-0.05
25.2470
-1.9869
-5.0369
-13.1653
-13.1653
-0.1
24.4974
-3.8318
-7.0856
-15.7391
-15.7391
-0.15
22.4985
-5.2920
-8.5360
-17.1865
-17.1865
-0.2
19.6939
-6.0502
-9.1280
-17.2092
-17.2092
-0.25
16.5009
-5.9358
-8.6368
-15.7420
-15.7420
-0.3
13.2518
-4.9144
-7.1458
-12.9113
-12.9113
-0.35
10.0899
-3.0621
-4.6952
-8.9450
-8.9450
-0.4
6.6974
-0.8907
-1.8688
-4.4132
-4.4132
-0.45
2.7300
0.2389
-0.1379
-0.9633
-0.9633
-0.5
0.0000
0.0000
0.0000
0.0000
0.0000
46
Table 3.4: Values of velocity component currently obtained on different grids along vertical centerline passing
through
Grid
Grid
12
Grid
15
Grid
24
Grid
30
0.5
0.0000
0.0000
0.0000
0.0000
0.0000
0.45
0.1878
0.1988
0.2081
0.2033
0.2033
0.4
0.5894
0.6822
0.6998
0.7098
0.7098
0.35
1.0486
1.2980
1.3398
1.3659
1.3659
0.3
1.4253
1.8968
1.9710
2.0184
2.0184
0.25
1.6079
2.3424
2.4480
2.5253
2.5253
0.2
1.5224
2.5290
2.6774
2.7747
2.7747
0.15
1.1422
2.3988
2.5758
2.7031
2.7031
0.1
0.4947
1.9522
2.1651
2.3055
2.3055
0.05
-0.3403
1.2455
1.4752
1.6332
1.6332
0
-1.2451
0.3786
0.6202
0.7817
0.7817
-0.05
-2.0861
-0.5233
-0.2850
-0.1264
-0.1264
-0.1
-2.7379
-1.3290
-1.1108
-0.9605
-0.9605
-0.15
-3.1027
-1.9216
-1.7272
-1.6007
-1.6007
-0.2
-3.1273
-2.2164
-2.0655
-1.9570
-1.9570
-0.25
-2.8152
-2.1797
-2.0630
-1.9881
-1.9881
-0.3
-2.2329
-1.8415
-1.7692
-1.7151
-1.7151
-0.35
-1.5026
-1.2976
-1.2556
-1.2258
-1.2258
-0.4
-0.7805
-0.6948
-0.6766
-0.6621
-0.6621
-0.45
-0.2294
-0.2047
-0.2050
-0.1949
-0.1949
-0.5
0.0000
0.0000
0.0000
0.0000
0.0000
Figs. 3.2-3.5 present a qualitative comparison between contours of the currently obtained flow
fields and the corresponding results reported in [22]. As can be seen from Figs. 3.3-3.5 the
values of all the currently obtained quantities lie in the same range compared to the
corresponding data reported in [22]. The distribution of streamlines, shown in Fig. 3.2, along
with the distributions of vorticity and velocity magnitude fields, shown in Figs. 3.4 and 3.5,
respectively, confirm the presence of the staggered low-speed vortices adjacent to the upper
and lower walls. The pressure field, shown in Fig. 3.3 confirms the presence of a “pressure
47
pocket” adjacent to the bottom wall between and . As can be clearly seen from
Fig. 3.6, the density variations over the domain are insignificant and the flow can be safely
considered to be incompressible.
Figure 3.2: Comparison between the distribution of streamlines: (a) the current study; (b) the independent study [22]. Grid
Figure 3.3: Comparison between the distribution of pressure: (a) the current study; (b) the independent study [22]. Grid
Figure 3.4: Comparison between the distribution of vorticity: (a) the current study; (b) the independent study [22]. Grid
48
Figure 3.5: Comparison between distribution of the velocity magnitude: (a) the current study; (b) the independent study [22].
Grid
Figure 3.6: Distribution of the density field. Grid
Figs. 3.7-3.8 present quantitative comparisons between the currently obtained flow
characteristics and the corresponding data reported in [22]. A comparison of the pressure and
the shear stress distributions along the upper and lower walls of the channel is shown in Figs.
3.7a and 3.7b, respectively. A comparison between the currently obtained and the independent
distributions of horizontal and vertical velocity components, pressure, vorticity, horizontal
velocity gradient and normal stress along two verticals passing through the and
locations is presented in Fig. 3.8. The figures demonstrate the same trends observed for all the
flow characteristics obtained in the current and in the independent studies.
Figs. 3.7a, 3.8c and 3.8f compare between the corresponding pressure and the normal stress
fields. A systematic offset between the corresponding fields observed in all the distributions
can be explained by the fact that pressure is defined by up to a constant in incompressible flow.
As such, the offset can be eliminated by simply adding a constant to the pressure field and does
not affect the precision of the obtained results.
The maximal relative deviation between the distributions of all the flow characteristics in Figs.
3.7-3.8 is bounded by 10 percent, except for the cases where large gradients were observed or
where the values of the flow characteristics are close to zero. In these cases, absolute deviation
was calculated for each quantity, and its value was found to be below 8 percent of the maximum
value of the corresponding quantity reported in the reference study.
49
In summary, acceptable qualitative and quantitative agreement exists between the currently
obtained and the independent results for the entire range of flow characteristics, which
successfully verifies the currently developed numerical methodology when applied for
simulation of almost incompressible flows.
Figure 3.7: Distribution of: (a) pressure and (b) shear stress fields along upper and lower channel walls. Grid </