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Verification Study of Sliding and Overset Grid Methods
An Application to Wind Turbine CFD Simulations
Tiago João das Neves Gomes
Thesis to obtain the Master of Science Degree in
Aerospace Engineering
Supervisors: Dr. Guilherme Nuno Vasconcelos Beleza Vaz
Prof. Fernando José Parracho Lau
Examination Committee
Chairperson: Prof. Filipe Szolnoky Ramos Pinto Cunha
Supervisor: Prof. Fernando José Parracho Lau
Member of the Committee: Dr. Auke van der Ploeg
November 2021
ii
Dedicado aos meus pais, Jo˜
ao e Susana.
iii
iv
Acknowledgments
First of all, I would like to thank S´
ebastien Lemaire for his close supervision and guidance in the
development of this work, whose knowledge and insight were essential. My special thanks also to Gui-
lherme Vaz, for having received me within his company, and having introduced me to the CFD research
community, whose expert advice and insight were equally critical.
I also wish to acknowledge and thank WavEC and blueOASIS, for having provided their facilities
and computational resources for the completion of this work. Particularly to blueOASIS, for its financial
support, which enabled me to participate at the NuTTS 2021 conference, to present and share this work.
Furthermore, the supervision of professor Fernando Lau was also important in the preparation of
this document, to whom I thank; and extend it to all professors in the Aerospace department of Instituto
Superior T ´
ecnico, whose contribution to my academic path led me to this point, of an ending cycle.
I am also particularly grateful to my parents, Jo ˜
ao and Susana, to whom I dedicate this thesis, and
my brother, Sim˜
ao, whose never-ending support contributed as well to the completion of this work.
Moreover, my special thanks to Louis, for his close support.
Finally, to all my friends, my colleagues, and every person who somehow contributed to this, thank
you.
v
vi
Resumo
As turbinas e´
olicas s˜
ao sistemas dinˆ
amicos complexos, envolvendo m´
ultiplos efeitos acoplados.
Para simulac¸ ˜
oes de MFC precisas ´
e necess´
ario capturar o movimento das componentes da sua estru-
tura, o que pode ser impratic´
avel com uma ´
unica malha. Nesse aspeto Sliding e Overset Grids s˜
ao
m´
etodos conhecidos, que permitem o uso de m ´
ultiplas malhas na discretizac¸ ˜
ao. Como estas podem
mover-se independentemente, os movimentos dos componentes da estrutura podem ser capturados,
existindo j´
a v´
arias implementac¸ ˜
oes de sucesso. Contudo, n ˜
ao existe na literatura qualquer estudo do
impacto destas t´
ecnicas nos erros de uma simulac¸ ˜
ao destes escoamentos. Portanto, o objetivo ´
e aplicar
procedimentos de Verificac¸ ˜
ao em trˆ
es casos de teste, para que os erros de discretizac¸ ˜
ao possam ser
avaliados e os m´
etodos comparados. O primeiro ´
e um escoamento de Poiseuille, com soluc¸ ˜
ao exata,
que permite testar rapidamente m ´
ultiplos parˆ
ametros. O segundo ´
e uma soluc¸ ˜
ao anal´
ıtica in´
edita de um
escoamento de turbina e´
olica, produzido pelo M´
etodo das Soluc¸ ˜
oes Manufaturadas. O ´
ultimo caso ´
e a
turbina e´
olica NREL 5MW, para testar os procedimentos num caso pr´
atico. Em cada teste um estudo
de sensibilidade de parˆ
ametros ´
e realizado, para compilar uma lista de boas pr´
aticas. Os resultados
sugerem que as Sliding Grids s ˜
ao adequadas para a simulac¸ ˜
ao de MFC de rotores de turbinas e´
olicas,
enquanto o Overset apresenta uma propens˜
ao a dificuldades de implementac¸ ˜
ao e erros mais elevados.
Tamb ´
em se conclui que as oscilac¸ ˜
oes de press˜
ao nas interfaces, pela n˜
ao-conservac¸ ˜
ao de massa in-
troduzida por estes m´
etodos, podem ser minimizadas por esquemas de interpolac¸ ˜
ao de pelo menos
segunda ordem.
Palavras-chave: Sliding Grids, Overset Grids, Verificac¸ ˜
ao, Turbinas E´
olicas, MFC
vii
viii
Abstract
Wind turbines are complex dynamic systems, with coupled effects from multiple fields. More accu-
rate CFD simulations require the motion of the different parts of the structure to be captured, which can
be impracticable using only a single grid. On that matter Sliding and Overset Grids are two well-known
methods that enable the discretization of the domain with multiple meshes. Since the grids can move
relatively to one another, body motion can be incorporated, with successful implementations already
existent. Nevertheless, no literature can be found assessing their impact on the solution’s accuracy
in these flows. Therefore, the goal is to apply Verification procedures to three test cases, so that dis-
cretization errors can be probed in isolation and the methods compared. The first is a Poiseuille flow,
with known exact solution, yet inexpensive to test multiple parameters. The second is a novel analytical
solution of a wind turbine flow, designed with the Method of Manufactured Solutions. The final test case
is the NREL 5MW wind turbine, to test the procedures in a practical scenario. Each test case has a
parameter sensitivity test performed, with the respective conclusions used as inputs to the next one.
The results suggest that Sliding Grids are suitable to be applied in the CFD simulation of wind turbine
rotors, whereas Overset tends to present additional implementation difficulties and higher errors. More-
over, pressure oscillations at the interfaces, due to the mass imbalance introduced by these methods,
are found to be minimized with interpolation schemes of at least second order.
Keywords: Sliding Grids, Overset Grids, Verification, Wind Turbines, CFD
ix
x
Contents
Acknowledgments ........................................... v
Resumo................................................. vii
Abstract................................................. ix
ListofTables .............................................. xv
ListofFigures ............................................. xvii
Nomenclature.............................................. xxi
Acronyms................................................ xxv
1 Introduction 1
1.1 Motivation............................................. 1
1.2 TopicOverview .......................................... 2
1.3 ObjectivesandDeliverables................................... 3
1.4 ThesisOutline .......................................... 5
1.5 Publications............................................ 5
2 Literature Review 6
2.1 SlidingGrids ........................................... 6
2.2 OversetGrids........................................... 11
2.3 Comparison of Sliding and Overset Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Applications on Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Numerical Methods 21
3.1 GoverningEquations....................................... 21
3.2 FiniteVolumeMethod ...................................... 23
3.2.1 UnstructuredGrids.................................... 23
3.2.2 GradientComputation .................................. 25
3.3 Mass-MomentumSolver..................................... 26
3.3.1 SIMPLE.......................................... 26
3.3.2 Pressure Weighted Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 SlidingGrids ........................................... 28
3.5 OversetGrids........................................... 29
3.6 InterpolationSchemes...................................... 30
xi
3.7 VerificationandValidation.................................... 30
3.7.1 TypesofErrors...................................... 31
3.7.2 Verification ........................................ 32
3.7.3 Method of Manufactured Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Test Cases 36
4.1 PoiseuilleFlow .......................................... 36
4.1.1 AnalyticalFormulation.................................. 36
4.1.2 GeneratedGrids ..................................... 38
4.1.3 NumericalSetup ..................................... 39
4.2 Wind Turbine - Manufactured Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.1 AnalyticalFormulation.................................. 41
4.2.2 GeneratedGrids ..................................... 45
4.2.3 NumericalSetup ..................................... 46
4.3 WindTurbine-NREL5MW ................................... 47
4.3.1 Geometry and Operating Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.2 GeneratedGrids ..................................... 47
4.3.3 NumericalSetup ..................................... 49
5 Results 50
5.1 PoiseuilleFlow .......................................... 50
5.1.1 StaticGrid ........................................ 50
5.1.2 CF LRDefinition ..................................... 52
5.1.3 RotationSpeed...................................... 53
5.1.4 TimeStep......................................... 55
5.1.5 InterpolationSchemes.................................. 56
5.1.6 UnequalGridRefinement ................................ 57
5.1.7 MassImbalance ..................................... 58
5.1.8 HighGridVelocities ................................... 60
5.1.9 Preliminary Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Wind Turbine - Manufactured Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.1 Baseline.......................................... 63
5.2.2 TimeStep......................................... 64
5.2.3 InterpolationSchemes.................................. 66
5.2.4 MassImbalance ..................................... 68
5.2.5 Preliminary Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3 WindTurbine-NREL5MW ................................... 70
xii
6 Conclusions 74
6.1 BestPractices .......................................... 74
6.2 Achievements........................................... 75
6.3 FutureWork............................................ 76
Bibliography 78
A NuTTS 2021 - Conference Paper 88
B Interpolation Schemes 95
B.1 NearestCell............................................ 95
B.2 InverseDistance ......................................... 95
B.3 NearestCellGradient ...................................... 96
B.4 LeastSquares .......................................... 96
xiii
xiv
List of Tables
2.1 Compilation of some of the most relevant papers about Sliding Grids found in the litera-
ture. Topology: Structured (S) and Unstructured (U). Inter-grid Communication: Interface
Intersection Fluxes (IIF) and Halo Cells (HC). Framework: Finite Differences (FD) and
FiniteVolume(FV)......................................... 11
2.2 Compilation of some of the most relevant papers about Overset Grids found in the liter-
ature. Topology: Structured (S) and Unstructured (U). Interpolation: Polynomial (PLY),
Nearest Cell (NC), Least Squares (LS), Nearest Cell Gradient (NCG), Inverse Distance
(ID) and Barycentric (BRC). Last digit represents order of accuracy of the scheme. Frame-
work: Finite Differences (FD) and Finite Volume (FV). . . . . . . . . . . . . . . . . . . . . 17
4.1 Poiseuille flow: Overview of flow quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Poiseuille flow: Sub-grid refinements description. Baseline: Cartesian grid only. Sliding
Grids: Rectangular w/ hole mesh fitted with Circular grid. Overset Grids: Cartesian and
Circular grids overlapped. Grid refinement hicalculated with cell count of Baseline test
case(Cartesianmesh). ..................................... 39
4.3 Wind Turbine MMS: Blending functions’ parameters. . . . . . . . . . . . . . . . . . . . . . 43
4.4 Wind Turbine MMS: Sub-grid refinements description. Baseline: Domain grid only. Sliding
Grids: Domain w/ hole mesh fitted with Rotor grid. Overset Grids: Domain and Rotor grids
overlapped. ............................................ 46
4.5 Wind Turbine NREL 5MW: Rotor geometry, operating conditions and aerodynamic prop-
ertiesat70%ofbladeradius. .................................. 47
4.6 Wind Turbine NREL 5MW: Sub-grid refinements description. Baseline: Domain grid only.
Sliding Grids: Domain w/ hole mesh fitted with Rotor grid. Overset Grids: Domain and
Rotorgridsoverlapped. ..................................... 48
5.1 CF LRcombinantions for ∆t= 0.01s............................... 53
5.2 Wind Turbine MMS: Time steps tested, based on Rotor mesh angular speed, ω, of 1.2698 rad s−1 .
Correspondent range of CF LRvalues are also presented for G1 and G4 grid refinements. 65
xv
xvi
List of Figures
1.1 New and total (cumulative) installed power capacity of wind turbine installations in Europe
- WindEurope’s Realistic Expectations Scenario [1]. . . . . . . . . . . . . . . . . . . . . . 2
2.1 Example of types of Patched Grids in Hessenius [23]. . . . . . . . . . . . . . . . . . . . . 7
2.2 Illustration of grid communication for Sliding Grids in Ram´
ırez et al. [33]........... 8
2.3 Example of Overset Grids in Hessenius [23]. . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Illustration of combined use of Overset and Sliding Grids for the simulation of rotor-stator
interactioninRai[12]. ...................................... 18
2.5 System of Overset Grids for the NREL Phase VI wind turbine used by Quallen and Xing
[81]. ................................................ 19
2.6 System of Sliding Grids for the NREL 5MW wind turbine used by Liu et al. [83]. . . . . . . 20
2.7 System of Sliding and Overset Grids for the NREL 5MW floating wind turbine used by
TranandKim[3].......................................... 20
3.1 Example of non-orthogonality correction. Method used to decompose the sfvector is
Over-Relaxed approach, where tfis orthogonal to sf..................... 25
3.2 Example of eccentricity correction. Interpolated value at the face point f0is extrapolated
in the direction of the arrow to the face’s centroid f....................... 25
3.3 Sliding Grid example. Circle mesh (black) is fitted in the Domain with hole grid (red). . . . 28
3.4 Schematic of the Halo Cell inter-grid communication method in Sliding Grids. . . . . . . . 28
3.5 Schematic of Halo Cell projection mechanisms available in ReFRESCO. . . . . . . . . . . 29
3.6 Overset Grid example. Circle mesh (black) is overlapped in the Domain grid (red). . . . . 30
3.7 Schematic of the inter-grid communication method in Overset Grids. . . . . . . . . . . . . 30
4.1 Poiseuille Flow - Domain characteristics and boundary conditions. . . . . . . . . . . . . . 37
4.2 Poiseuille flow: Example of grid refinement G2 for each type of sub-grid created. . . . . . 39
4.3 Wind Turbine MMS: CFD simulation of Actuator Disk, based on NREL 5MW character-
istics, to serve as baseline in the design of the manufactured solution. Flow quantity
presented is axial velocity, vz. Grid refinement used is G4, as described in Table 4.4. . . . 40
4.4 Wind Turbine MMS: Example of interpolated axial velocity field, vz, based on Actuator
Disk solution. While presenting good similarity, the manufactured solution was discarded
since it did not converge well due to lack of mass conservation. . . . . . . . . . . . . . . . 41
xvii
4.5 Wind Turbine MMS: Domain characteristics and boundary conditions. Slice view of the
cylindricaldomain. ........................................ 42
4.6 Wind Turbine MMS: Manufactured solution of vzand vrfields................. 43
4.7 Wind Turbine MMS: Manufactured solution of Hand pfields. ................ 45
4.8 Wind Turbine MMS: Manufactured solution momentum equation’s source fields in cylin-
dricalcoordinates. ........................................ 46
4.9 Wind Turbine MMS: Example of G1 grid refinement for SG and OG grid configurations. . . 46
4.10 Wind Turbine NREL 5MW: Domain characteristics and boundary conditions. Inner box
surrounding the rotor represents the limit of the sub-grid encompassing the rotor for the
use of Sliding and Overset Grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.11 Wind Turbine NREL 5MW: Front and side view of SG configuration. Grid G1. . . . . . . . 49
5.1 Poiseuille flow: Flow field of the Baseline test case with grid G5. Sliding and Overset
Grids results are not presented since the differences are imperceptible to the naked eye. . 51
5.2 Poiseuille flow: Grid refinement vs. Errors of velocity and pressure. Column order: Base-
line, SG and OG cases. Top row: L2norms. Bottom row: L∞norms. The value of pac
represents the observed order of accuracy, based on Equation (3.26). . . . . . . . . . . . 51
5.3 Poiseuille flow: Field of the axial velocity error in log scale. Grid G5. . . . . . . . . . . . . 52
5.4 Poiseuille flow: Field of the pressure error in log scale. Grid G5. . . . . . . . . . . . . . . . 52
5.5 Poiseuille flow: Error quantities over last five rotations using various grid refinements (G1
to G5). Angular speed is fixed at 4 rad s−1 and time step is 0.01 s. . . . . . . . . . . . . . . 54
5.6 Poiseuille flow: Average error quantities vs. C F LRusing various grid refinements (G1 to
G5). Rotation speed ranges from 1 to 64 rad s−1 along each curve, varying also C F LR.
Timestepis0.01s......................................... 55
5.7 Poiseuille flow: Average error quantities vs. grid refinement (G1 to G5). Rotation speed
ranges from 1 to 64 rad s−1.Timestepis0.01s......................... 55
5.8 Poiseuille flow: Influence of time scheme and time step (varying C F LR) on vxError - L∞
Norm. Fixed rotation speed of 4 rad s−1. ............................ 56
5.9 Poiseuille flow: Average error quantities vs. grid refinement plots with different interpo-
lation schemes. C F LRwas kept constant at 1.53 through time step refinement, while
rotation speed was kept fixed at 4rad s−1 . ........................... 56
5.10 Poiseuille flow: Average error quantities vs. Domain and Domain with Hole (outer) grid re-
finements (OGX). Each curve represents a different Circular (inner) grid refinement (IGX).
Rotation speed and time step were kept constant, at 4 rad s−1 and 0.01 s, respectively. . . 57
5.11 Poiseuille flow: Detail of unequal grid refinement G1G4 (Circle mesh is G1, while Domain
with Hole mesh is G4). The refinement of the Domain mesh was so fine that the halo cells
could not penetrate the Circular sub-grid, halting the simulation. . . . . . . . . . . . . . . . 58
5.12 Poiseuille flow: Mass imbalance vs. grid refinement (G1 to G5). Rotation speed is ω=
4 rad s−1, with a time step of 0.01 s (yet, Baseline is static). . . . . . . . . . . . . . . . . . . 59
xviii
5.13 Poiseuille flow: Mass Imbalance over a Circular grid rotation for various interpolation
schemes. With angular speed of ω= 4 rad s−1, grid refinement G5 and timestep such
that CF LR= 1.53. ........................................ 60
5.14 Poiseuille flow: L2norm of pressure error vs. Circle mesh rotations. Rotation speed is
fixed at 64 rad s−1 and grid refinement is G4. C F LRwas decreased through time step
refinement. Dotted line represents average error of equivalent C F LRcase, equal to 2.04,
but with rotation speed of only 4 rad s−1. ............................ 60
5.15 Wind Turbine MMS: Velocity and pressure distribution in slice over xz plane. Grid G1. . . 63
5.16 Wind Turbine MMS: Error distribution of velocity and pressure in slice over xz plane. Grid
G1.................................................. 63
5.17 Wind Turbine MMS: Errors of velocity and pressure vs. grid refinement. Baseline test case. 64
5.18 Wind Turbine MMS: Numerical vs. exact axial velocity distribution at the outlet. Grid G1. . 65
5.19 Wind Turbine MMS: Errors of radial components of velocity, vx(and vy), vs. grid refine-
ment. Time step varied such that the rotor advances between 6 to 24 degrees per time
step................................................. 66
5.20 Wind Turbine MMS: Order of convergence of errors of radial components of velocity, vx
(and vy) vs. time step. Time step varied such that the rotor advances between 6 to 24
degreespertimestep....................................... 66
5.21 Wind Turbine MMS: Error distribution of axial velocity, vz, nearby the rotor region in log
scale. Slice over yz plane. Grid G4. Black line encloses rotor region. Top row: Sliding
Grids. Bottom row: Overset Grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.22 Wind Turbine MMS: Errors of pressure vs. grid refinement. Interpolation schemes tested
are ID1, NCG2 and LS3. The time step is such that the rotor advances 8 degrees per
eachtimestep. .......................................... 68
5.23 Wind Turbine MMS: Mass imbalance vs. Rotor mesh rotations. Grid refinement G4. Mass
imbalance is presented as percentage of the total mass flow rate going through the Rotor
grid. ................................................ 69
5.24 Wind Turbine MMS: Pressure errors vs. Rotor mesh rotations. Grid refinement G4. . . . . 69
5.25 Wind Turbine NREL 5MW: L2norm residuals of velocity, pressure and turbulence quan-
tities. Residuals were reduced until stagnation at each time step. Note that OG was only
able to converge with grid G1 by using half of the time step of SG. . . . . . . . . . . . . . 71
5.26 Wind Turbine NREL 5MW: Velocity magnitude field after 36 rotor rotations (180 seconds
of simulation time). Thick black box encloses zoom-in of Rotor mesh. Thiner black line
inside represents the Rotor grid’s limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.27 Wind Turbine NREL 5MW: Coefficient of power, CP, and coefficient of thrust, CT, vs. grid
refinement. Reference values extracted from Make and Vaz [19], correspondent to the
CFD simulation of the NREL 5MW rotor geometry, with the same operating conditions. . . 72
xix
5.28 Wind Turbine NREL 5MW: Mass imbalance vs. Rotor mesh rotations. Mass imbalance as
a percentage of mass flow rate passing through the rotor. Results for all grid refinements
withSGandmeshG1withOG.................................. 72
5.29 Wind Turbine NREL 5MW: Pressure history at different points in space for SG configu-
ration, except for dotted black line in lower left plot, which concerns the OG result. Blue
curve, with coordinate x= 0.25D, represents a point inside the Rotor grid, while all others
concern points in the Domain mesh. Black points are at z= 0.59D, while white ones are
at z= 0D.............................................. 73
5.30 Wind Turbine NREL 5MW: Qcriterion = 0.3iso-surfaces. Grid G1. Black line encloses
Rotormesh............................................. 73
xx
Nomenclature
Greek symbols
αError Level.
δRE Error estimation based on Richardson extrapolation.
∆Data range.
∆HTotal pressure drop.
∆PHarvested power.
∆SCell’s face area.
∆tTime step.
∆VCell’s volume.
∆xTypical cell size.
Discretization error.
φGeneral flow scalar quantity.
ΓDiffusion coefficient.
γInterpolation factor.
λBulk viscosity.
µDynamic viscosity.
νKinematic viscosity.
ρDensity.
σStandard deviation of fit.
τStress tensor.
ωAngular velocity.
ωiInterpolation weight.
xxi
Roman symbols
AMatrix of coefficients of linear system.
AArea.
bVector of coefficients of linear system.
BArbitrary fluid property.
bIntensive property of B.
CStabilization matrix of Pressure Weighted Interpolation.
CIntegration constant.
cChord.
CLLift coefficient.
CF L Courant number.
CF LRNumber of cells that an interface cell slides through at each time step.
CPCoefficient of power.
CTCoefficient of thrust.
DDivergence matrix.
DDiameter.
dDistance.
eError vector.
fForces per unit volume acting on fluid.
GGradient matrix.
gGravitational constant.
HTotal pressure.
hHeight.
hiGrid refinement.
IIdentity matrix.
kTurbulent kinectic energy.
LLower triangular matrix.
LIntegral length scale.
xxii
Lop Laplacian operator.
mMass.
MxAxial momentum.
nUnitary normal vector.
Nint Number of cells at interface.
NiNumber of cells of i-th grid.
NMNumber of cells of most refined grid.
PPower.
pStatic pressure.
pac Observed order of accuracy.
pdDecay constant.
QAdvection-diffusion matrix.
QFlow rate.
QsGeneral source term.
Qcriterion Q-Criterion.
RPressure correction matrix or Schur complement.
rDistance vector.
RaRadius of Actuator Disk.
Re Reynolds number.
SControl surface.
tTime.
TxAxial thrust.
UUpper triangular matrix.
UUncertainty.
vVelocity vector.
VControl volume.
vVelocity magnitude.
xVector of variables of linear system.
xxiii
y+Dimensionless nominal distance to the wall.
Subscripts
,Derivative.
0Free-stream condition.
αSpatial direction.
avg Average.
bBody.
cCell center.
disk Actuator Disk.
fFace center.
f0Face point.
gGrid.
i, j General indexes.
int Interface.
out Outlet.
DDonor cell.
FNeighbour cell.
RReceiving cell.
r, θ, z Cylindrical coordinates components.
sSurface.
tTip.
x, y, z Cartesian coordinates components.
Superscripts
φScalar flow quantity.
kIteration number.
T Transpose.
xxiv
Acronyms
1D One Dimensional.
2D Two Dimensional.
3D Three Dimensional.
AFM Absolute Formulation Method.
BC Boundary Condition.
BEMT Blade Element Momentum Theory.
BRC Barycentric.
CD Central Differences.
CFD Computational Fluid Mechanics.
DCI Domain Connectivity Information.
DoF Degrees of Freedom.
FD Finite Differences.
FOW First Order Upwind.
FOWT Floating Offshore Wind Turbine.
FSI Fluid Structure Interaction.
FV Finite Volume.
FVM Finite Volume Method.
HC Halo Cells.
HPC High Performance Computing.
ID Inverse Distance.
IIF Interface Intersection Fluxes.
LHS Left-Hand Side.
LS Least Squares.
MFBI Mass Flux Based Interpolation.
MMS Method of Manufactured Solutions.
MPI Message Passing Interface.
NC Nearest Cell.
NCG Nearest Cell Gradient.
OG Overset Grids.
PLY Polynomial.
xxv
PWI Pressure Weighted Interpolation.
QUICK Quadratic Upstream Interpolation for Convective Kinematics.
RANS Reynolds-Averaged Navier-Stokes.
RHS Right-Hand Side.
SG Sliding Grids.
SIMPLE Semi-Implicit Method for Pressure-Linked Equations.
TSR Tip Speed Ratio.
TVD Total Variation Diminishing.
V&V Verification and Validation.
xxvi
Chapter 1
Introduction
The present Chapter introduces the topic under analysis in this Thesis. It starts with an outlook of the
wind turbine industry, some usual difficulties faced on the numerical simulation of the flow around the
wind turbines and how the Sliding and Overset Grid methods are relevant to solve them. Additionally,
the goals and deliverables established for this Thesis are stated and the test cases planned to achieve
them are briefly described.
1.1 Motivation
Wind turbines comprise an important element of the on-going energetical transition to a greener
society, in the attempt to reduce greenhouse gas emissions and limit climate change. These devices
are already well established in Europe, representing 16% of its electricity demand in 2020, with a total
installed power of around 220 GW according to WindEurope [1].
However, demand for renewable sources of energy is still high and critical, and wind power encloses
great potential. In fact, the same report [1] predicts an addition of 105 GW in registered wind turbine
installations, with a positive net balance in installed wind power production capacity of 98GW, until 2025
based on a Realistic Expectations Scenario, as it can be seen in Figure 1.1, practically a 45% increase of
the current numbers. These predictions are in line with the European Commission’s objective of Europe
becoming carbon neutral until 2050, in a total investment estimated at EUR 800 million in new offshore
renewable sources of energy, including wind turbines [2].
This process will be partially fueled by two key technical aspects in the wind turbine industry: 1)
increasing popularity of offshore installations; 2) further improvements in efficiency achieved through
the development of new designs. Both depend heavily on Computational Fluid Dynamics (CFD), to
simulate and iterate through different designs, potentially reducing associated costs and optimizing the
whole device. CFD in turn relies on the numerical methods available to provide accurate, but efficient
results given the limited computational resources available. This is where Sliding Grids (SG) and Overset
Grids (OG) are presented as two important discretization methods to capture the complex motion of the
various structural parts of a wind turbine during operation, which already started to have a positive
1
Figure 1.1: New and total (cumulative) installed power capacity of wind turbine installations in Europe -
WindEurope’s Realistic Expectations Scenario [1].
impact on their research field [3].
Yet, information in the literature regarding practical applications of SG and OG in wind turbine simu-
lations usually lack of enough detail, or miss entirely, on the reasoning behind the choice of parameters
selected and their impact. Not only this has the potential of hindering a more widespread adoption of
the methodologies, giving the overwhelming number of combinations that might exist in setting up a sim-
ulation with either Sliding or Overset methods, but also lead to sub-optimal setups that can negatively
impact the solution’s accuracy.
Additionally, both SG and OG have been known and used for many years in various other industries,
therefore any improvements or insights on these methods have the potential of positively impacting other
research areas:
•Overset Grids:
– Aerospace Industry: general spacecraft [4], store separation [5], helicopters [6]. Some
NASA particular applications, which first developed OG, include the Space Shuttle and com-
plex aircraft configurations [7, 8];
– Maritime Industry: ship motion [9] and maneuvering [10];
– Turbomachinery Industry: cooling of turbine blades [11] and rotor-stator interaction [12].
•Sliding Grids:
– Aerospace Industry: 3D complex aircraft configurations [13], deflection of control surfaces
[14], helicopters [15];
– Maritime Industry: ship-propeller interaction [16];
– Turbomachinery Industry: large scale turbomachinery computations [17].
1.2 Topic Overview
The CFD simulation of a wind turbine presents various challenges, since it is a complex dynamic
system with coupled aerodynamic, structural and even hydrodynamic effects in the case of floating
2
offshore installations. A major one lies in the domain discretization with a single grid, since the rotation
of the rotor geometry can only be captured by remeshing at every time step. This procedure can quickly
become prohibitively expensive as finer time steps are used, not to mention the difficulty in ensuring
consistent grid quality.
A possible turn-around is to not account for the geometry of the rotor in the discretization, such
that a single (but static) grid could be easily obtained. Instead, the effect of the rotor on the flow is
mimicked by body force terms distributed spatially in the mesh. These can be implemented as in the
Actuator Disk [18], with the related body-force field being given via empirical relations, Blade Element
Momentum Theory (BEMT), Lifting-line/Surface or even Panel methods. In general, these provide good
results, are relatively inexpensive and can capture some unsteady aerodynamic effects, specially at
medium/far-field locations of the rotor. However, they are based on several simplifications, which can
prevent reliable results at the near-field and in off-design conditions [19].
Another alternative is the use of the Absolute-Formulation (AFM) [19], in which the governing flow
equations are solved in a moving reference frame, but using variables written in terms of absolute earth-
fixed quantities. This way the rotor geometry can be static, facilitating the grid generation. Nonetheless,
it fails to capture the unsteadiness of the flow if the transient term is dropped and only the rotor can be
accounted for in the simulation, severely limiting its application scope.
This is where Sliding Grids and Overset Grids are presented as viable alternatives to obtain more
complete and accurate simulations of wind turbines. At their core these methods allow for the use of
multiple independent grids, e.g. one for the rotor and another for the domain, coupling them afterwards
during the simulation by interpolating the flow information. The great advantage in this specific example
is that the rotor grid can rotate relative to the rest of the domain, embedding into the simulation the
rotation effect without the need for expensive remeshing. In the case of Sliding Grids each grid is
generated to fit into one another, sharing a common interface. The sub-grids might be capable of simple
movements, as unidirectional translation or rotation over cylindrical/conical surfaces of revolution. As for
Overset Grids, their generation is more flexible, since the only requirement is that they overlap. Because
of this consideration, grid motion is virtually unlimited when compared with Sliding Grids.
As it was already stated, their immediate advantage to the aerodynamic simulation of wind turbines
is the more natural approach to introduce the rotor motion effect. However, it does not limit itself in that
aspect. These methods open a wide range of more complex simulations, namely with the possibility to
simulate and capture fluid-structure interaction (FSI) induced motions and also the possibility to simulate
floating offshore wind turbines (FOWT) in more accurate conditions, by taking into account the motion
induced by the sea waves and the mooring cables’ influence on the floating platform.
1.3 Objectives and Deliverables
As elicited in the Thesis title, the general goal of the present investigation is to analyze and compare
both Sliding and Overset Grid methods using Verification procedures [20], something scarce in the liter-
ature despite their versatility and interchangeability. Therefore, the following deliverables are planned:
3
•Input parameter sensitivity test. These include interpolation schemes, time steps, grid refine-
ment, grid velocity and other parameters specific to each method. Impact on solution’s accuracy
to be evaluated using Verification procedures;
•Design of a wind turbine flow manufactured solution. Novel type of manufactured solution,
capturing the main features of a wind turbine flow, to be used as the second test case in the
present work. The analytical solution of the flow is forced to exist, so discretization errors can be
probed in isolation.
•Investigation of possible implementation errors and improvements to be included in the
next version of the CFD solver. The current implementation is still relatively unexplored and
recent in the case of Overset Grids, for example.
•Compilation of good practices and default parameters to be used. Given the large number
of possible combinations of input parameters in Sliding and Overset Grids, it can be difficult for a
user to start adopting these methods and obtain optimal combinations. This compilation will be
based on the parameter sensitivity tests performed.
These deliverables will in turn be achieved through the development and application of three distinct
test cases, related to wind turbine applications:
•Poiseuille Flow. Simple, low Reynolds number, 2D and fully developed flow through two plates. It
is one of the few analytical solutions of the Navier-Stokes equations, therefore discretization errors
can be assessed in isolation through the use of Verification theory;
•Wind Turbine - Manufactured Solution. Novel type of manufactured solution in the literature.
Equations describing the pressure and velocity fields are designed to resemble a wind turbine
flow. Afterwards, they are enforced to become the exact solution of the Navier-Stokes equations
through the Method of Manufactured Solutions. Similarly to the Poiseuille flow, discretization errors
can be assessed in isolation with Verification theory;
•Wind Turbine - NREL 5MW. Final test case, using the rotor geometry of the NREL 5MW wind
turbine [21]. Enables the application of the methods in an industry relevant scenario, to further test
and compare SG and OG in the aerodynamic simulation of the rotor.
The above-mentioned test cases have increasing computational complexity, therefore the Poiseuille
flow allows to quickly test through a large amount of parameter combinations in a first phase. After
each test case, the preliminary conclusions obtained will serve as a basis to define the parameters
to be tested in the next one. This enables the number of combinations to be narrowed down, hence
rationalizing the use of computational resources.
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1.4 Thesis Outline
In Chapter 2, Literature Review, a general overview of the literature existent about the development
and application of Sliding Grids and Overset Grids is presented, followed by an analysis of publications
comparing directly both methods. Finally, a section is dedicated to the study of applications of both
methods on wind turbines.
Afterwards, in Chapter 3, a review of the CFD methods and frameworks used in the present work
is performed. Emphasis is provided not only to the Sliding and Overset Grid methods, but also to
Verification theory, which is fundamental to enable the comparison of the methods and to evaluate their
impact on the simulation’s accuracy.
Chapter 4 follows through the Test Cases, explaining in more detail each one: Poiseuille flow, wind
turbine Manufactured Solution and the full-scale wind turbine NREL 5MW. Their theoretical basis is
detailed, also with the generated grids and the numerical setup adopted.
Then, the results are presented and discussed in Chapter 5, separated by each test case in the same
order as they were presented in Chapter 4. It is aimed at providing a continuous line of thought, actively
establishing comparisons and correlations with previous results, not only in the present document, but
also within the literature.
Finally, Chapter 6 wraps up the Thesis with an overview of the fulfilled achievements and a set of
recommended good practices when using SG and OG in CFD simulations of wind turbines. Also, an
outlook to future work is provided.
1.5 Publications
During the preparation of this Thesis a conference paper [22] was submitted and presented at the
23rd Numerical Towing Tank Symposium (NuTTS 2021), in M¨
ulheim an der Ruhr, Germany. It focused
on the second test case, the Manufactured Solution of a wind turbine flow, and is presented in Appendix
A.
A participation at the 41st International Conference on Ocean, Offshore & Arctic Engineering (OMAE
2022) is also planned, having an Extended Abstract already been submitted and accepted, focusing on
the last two test cases of the present work.
5
Chapter 2
Literature Review
The present chapter is a literature review of the Sliding and Overset Grid methods. Firstly, an individ-
ual investigation of the origins and key developments of each technique is presented. Considering that
the literature is quite extensive, only areas relevant to this thesis are subject to a more in-depth review.
These include, among others, interpolation schemes, concerns with conservation and applications with
unstructured grids. Secondly, a section is dedicated to comparisons between both methods. Finally,
applications of Sliding and Overset Grids within the wind turbine industry are presented.
2.1 Sliding Grids
The predecessor of the method of Sliding Grids started to be developed in the 1980’s. According
to Hessenius [23], until that time most of the grid generation necessary for CFD was performed in a
body-oriented fashion, i.e. with a single grid for the whole domain. While acceptable for most simple
geometries used at the time, the fast-pacing evolution of both CFD and computational power increased
demand for more complex ones. Since most grid generation software were based on structured grid
topologies, motivated by the capabilities of the CFD solvers available at the time, this led to difficulties
in assuring the quality of the generated mesh or even the impossibility of creating one in those new
situations. The same paper [23] cites several other authors for their work on the impact of certain grid
parameters, including skewness, smoothness and cell aspect ratio, on the overall accuracy of the solu-
tion and convergence rate of the methods used. All of those were usually precarious in grids generated
around complex geometries, even with the most advanced methods in mesh generation available at the
time.
These reasons motivated the development of alternative ways to simplify the grid generation, from
which resulted the Patched Grids method. By discretizing the domain with several independently gen-
erated sub-grids, each sharing common interfaces with each other and having a simpler geometry, the
overall process becomes easier, more flexible and the quality of the grids can be better controlled locally
[23, 24]. Several researchers started applying the Patched Grids method to more practical test cases
in the aeronautic industry, including 2D turbine cascades [25] and wing-canard configurations [26]; they
6
were, however, all based on point-continuous Patched Grids. While each mesh could have different
topologies, they were forced to share the same nodes at their interfaces, i.e. no hanging nodes were
allowed. This way interpolation schemes were not necessary to transfer information between grids, al-
lowing conservation properties to be maintained. Nevertheless, it was recognized that this strategy was
still too restrictive in grid generation and still troublesome in terms of quality. Therefore, the only viable
alternative was to use point-discontinuous interfaces. An example of both types of Patched Grids are
presented in Figure 2.1.
(a) Point Continuous. (b) Point Discontinuous.
Figure 2.1: Example of types of Patched Grids in Hessenius [23].
With Patched Grids with point-discontinuous interfaces, new challenges needed to be solved. These
included mostly the mechanism of information transfer at the interfaces and respective conservation
of fluxes. Cambier et al. (1984, as cited in [23]) present a characteristic boundary method for 2D
Patched Grids with point-discontinuous interfaces. It was, however, non-conservative, yielding issues
when handling discontinuities, like shock waves. The same method was also applied by Bush [27]
to a 2D flow on an external compression inlet. The first conservative method in Patched Grids with
discontinuous interfaces is usually attributed to Rai [28], which developed an explicit, first order accurate,
conservative integration scheme for Euler equations in the Finite Differences framework, to update the
nodes at the interfaces. It was extended in subsequent papers to explicit and implicit second order
accurate methods [24, 29]. The same author applied it to several test cases and proved the methods to
be accurate, stable, general to any curvilinear system of coordinates and, of course, conservative. This
last consideration yielded freely moving discontinuities in the flow, with the correct strength and position,
something that was also verified by Hessenius in [23].
At the same time, the idea of introducing relative motion to the grids also started to be developed,
originating the Sliding Grids method. Rai, for example, suggested and tested with success Patched Grids
moving relatively to each other in [29], with animated supersonic airfoils, proving that the used integration
method was also time-accurate and foreseeing possibilities of use in unsteady turbine rotor-stator and
helicopter rotor-fuselage interactions. In Usera et al. [30] an additional benefit is also mentioned, namely
the reduced need of recomputing the grid at each time step when using moving objects, improving
performance. These considerations, of course, could be extended to many other applications with similar
mechanics and yielded great potential in Sliding Grids.
More practical engineering applications started to appear more frequently during the 1990’s, with the
introduction of the SG method in commercial CFD codes. Bakker et al. [31] used ANSYS Fluent to model
7
and simulate a stirred reactor tank using Sliding Grids. Until then, most of the work developed in the
area relied on experimental data to improve simulation results in some areas of the tank, however with
this novel method that was no longer necessary, since it enabled more complex simulations. Liu and Hill
[32] used STAR-CD in some of their simulations, while comparing the use of steady-state approximation
models, like the Frozen Rotor and the Circumferential Averaged model, with the unsteady Sliding Grids
method, in centrifugal compressor systems. While the latter was more expensive, it was the only method
able to capture the unsteadiness of the flow and to match the off-design operation experimental data.
Moreover, as commercial codes started to become more common in the industry, the development
and improvement of the SG method did not stagnate. One key area concerns the mechanism by which
information is transferred through the interfaces. Two main families of methods can be distinguished:
Interface Intersection Fluxes (IIF) and Halo Cells (HC), which are illustrated in Figure 2.2.
(a) Interface Intersection Fluxes. (b) Halo Cells
Figure 2.2: Illustration of grid communication for Sliding Grids in Ram´
ırez et al. [33].
The first one determines the cell’s face intersections on each side of the interface. Based on them,
the fluxes crossing it can be calculated, which enables them to be conserved through a balance. This
method was the first developed and has been extensively used in 2D problems with structured grids.
As previously mentioned, flux conservation is a major concern as it is crucial for transonic applications
[23–26, 28, 29]. In order to capture accurately the flow discontinuities, including shock waves, both the
discretization and interpolation schemes of the solver need to be in conservative form [13]. Therefore,
many of the first efforts were based on this method, with special attention to Rai [24, 28, 29], which
devised a conservative algorithm for Patched Grids with Euler equations, and Lerat and Wu [34], which
later proposed a different approach to an also conservative method, but unconditionally stable for dissi-
pative difference schemes.
However, this method does not scale up well with grids that are unstructured or 3D: the intersection
of interfaces to calculate the fluxes relies mostly on the connectivity information of structured grids [35]
and while in 2D the intersections are lines, in 3D they potentially become arbitrarily shaped polygons
[36]. Despite this, various efforts in extending the methodology occurred. As stated in [35], other authors
used unstructured grids, but simplified them so that at each time step all of the nodes coincided and no
interpolation was necessary. This of course was not a new idea, considering earlier efforts using point-
continuous Patched Grids. Nevertheless, it was once again a method too restrictive to be used. Mathur
[36] went even further and devised a different approach, where interpolation was also unnecessary, but
8
was not as restrictive in terms of the node placement and grid connectivity on the boundaries. This was
accomplished by reconstructing the interface cell topology based on the intersections, so that the new
faces matched, in conjunction with an efficient search algorithm based on triangular (2D) or tetrahedral
(3D) meshes. That approach was also adopted in a similar fashion by Rinaldi et al. [37] to guarantee the
flux conservation on the non-conformal interface of the mesh, with the new interface grid being named by
the authors as ”super-mesh”. However, despite the little modifications to the code that were necessary
in [36], and that many operations of the CFD solver could still be performed by using the old or the
new faces, the 3D case of intersections was still characterized by the author as being a ”formidable
challenge”.
On the other hand, the use of Halo Cells to transfer information through the interfaces gained in-
creased popularity as an alternative to Interface Intersection Fluxes, especially for unstructured grids.
For example, in the CFD solver ReFRESCO the Halo Cell method is the one adopted, after the work
of Vaz et al. [38]. Instead of searching for the intersections, this method projects the cells at the inter-
face into the other contiguous sub-grid. The halo cell center value is interpolated from the neighbour
cells, with a stencil based on the interpolation scheme selected. It is then treated regularly by the
solver as a boundary cell, determining the face fluxes with the parent cell. While being computation-
ally cheaper, specially in unstructured 3D grids, it does not ensure conservation, as the global fluxes
calculated in one side may not be equal to the ones on the other side. Yet, many authors have tested
this [14, 15, 17, 33, 35], and concluded that the fact the method was not conservative was not affecting
considerably the results. Van der Weide et al. [17] for example applied this to unsteady simulations of
large-scale turbomachinery, using a Sliding Grid technique with Halo Cells. Despite perceiving some
mass imbalance, the results were satisfactory.
Blades and Marcum [35], while not directly using the Halo Cell’s terminology, adopted a similar
mechanism of cell extrusion, which was also non-conservative, and concluded that the mass imbalance
introduced by Sliding Grids was practically identical to a single grid in a diverging duct test case. On that
matter the authors state that ”local flux conservation guarantees global conservation, but global con-
servation does not guarantee local conservation”. And if local conservation is not respected in certain
parts of the domain, a flow discontinuity in those regions will necessarily be impacted. However, the
authors argue that with cell extrusion, local conservation can only be imposed in a fully connected inter-
face (point continuous). On the contrary, grids discontinuous at the interface can at most satisfy global
conservation, given that the calculation of fluxes from one domain to the other is done independently in
both directions. Therefore, either global conservation, or none at all, are expected to have similar results
when the method of cell extrusion is adopted, based on these arguments. Yet, these considerations
might be arguable, given that localized corrections, while not easy nor inexpensive, might be able to
respect local conservation in such a method.
The same authors [35] tested the method with a shock tube and concluded that the discontinuities
had little impact caused by the SG with no global flux conservation. In this paper the cells are extruded
from one domain into the other at the interface, either in the normal direction or using the information
of the adjacent domain topology. Moreover, a distance metric is calculated in both sides, to ensure that
9
the extruded distance is such that the new node ideally remains within the immediate element after the
interface, a paramount feature to ensure the correctness of the calculated flux, based on information
local to the interface. In fact, the authors demonstrated that minimizing this distance aids at minimizing
conservation errors. This is especially important with grids having very different typical cell sizes.
Also in Halo Cells, Steijl and Barakos [15] presented a detailed investigation. The research con-
cerned an helicopter rotor-fuselage interaction, an area that at the time had mostly been developed
using Overset Grids (Chimera). It is based on structured grids and the implementation details are thor-
oughly detailed in the paper. It uses two layers of halo cells and an area intersection weight-based
interpolation - once again, powered by the connectivity of structured grids. The authors noted that un-
equal grid refinements across the interface could act as a spatial filter, hence increasing the errors given
the interpolation scheme selected. Moreover, Message Passing Interface (MPI) parallel implementation
details are also presented, a very relevant subject when using High-Performance Computing (HPC),
where it was concluded that several improvements were still needed to improve its efficiency. Overall,
the Patched Grids test cases used, namely the transonic/supersonic flow around a bump and an airfoil,
did not introduce any numerical artifacts, except for shock spreading because of the non-conservative
fluxes by the Halo Cell method and respective interpolation scheme used.
Concerning the option of using either the conservative Interface Intersection Fluxes method or Halo
Cells, a comprehensive discussion was made on Fenwick and Allen [14]. The authors did an unsteady,
aeroservoelastic simulation of a realistic airfoil configuration, where Sliding Grids are used to deflect the
control surface. Both methodologies are tested in 2D and 3D with structured topology grids. In the shock
tube test case, either with a planar or curvilinear interface, both Halo Cells and conservative Interface
Intersection Fluxes present good agreement of results, leading to the conclusion that the method being
or not conservative does not significantly impact the results, similarly to [35]. The same conclusions are
drawn from the airfoil simulations. However, it is important to note that the 3D application of conservative
Interface Intersection Fluxes was done by taking advantage of the grid connectivity, whose absence
related difficulties have been already discussed by other authors, but not in this case. Furthermore,
unequal grid refinements across the interfaces were used, also without any significant variations of the
results. Nevertheless, Rai [29] obtained different conclusions on that matter and suggested a maximum
discrepancy of grid refinement ratio of 3:1 at the interface. Moreover, another finding was the number
of cells that an interface cell should slide through at each time step, which directly affected the results
of the aeroservoelastic simulation. It was found that a loss of 5% of the mean lift coefficient value,
CL, and some oscillatory behaviour of it, occurred when a cell slided through another five at each time
step. In fact, converged results were only found when that value was close or below one, in which the
authors characterized the information transfer as occurring within the cell’s zone of influence. Therefore,
this geometrical parameter might be one key factor to obtain good results when using Sliding Grids. A
similar conclusion was also obtained by Franc¸ ois et al. [39].
More recently, Ram´
ırez et al. [33] also compared Sliding Grids with conservative Interface Intersec-
tion Fluxes and Halo Cells, but in this case using unstructured grids with higher order schemes, namely
third and fourth order. In all test cases, either with inviscid compressible or viscous incompressible flows,
10
both methodologies yielded similar order of convergence and accuracy. Once again, it is pointed out the
higher flexibility of Halo Cells. The use of the higher than second or