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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.

4

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