Development of an efficient Navier-Stokes/LES solver on unstructured grids for high-order accurate

246 Pages·2010·29.26 MB·English

by  Matteo Parsani

FACULTYOFENGINEERING
DepartmentofMechanicalEngineering
Development of an efficient
Navier-Stokes/LES solver on
unstructured grids for
high-order accurate schemes
Thesissubmittedinfulfillmentoftherequirementsforthe
awardofthedegreeofDoctorindeIngenieurswetenschappen
(DoctorinEngineering)by
Matteo Parsani
November2010
Advisor: Prof. Dr. Ir. ChrisLacor

Abstract
Researchersareattemptingtotackle problemswhichwereconsideredtoo
ambitious just a few years ago. Multidisciplinary analysis and design
(MAD), computational aeroacoustics (CAA), large eddy simulation (LES)
anddirectnumericalsimulation(DNS)ofturbulenceareexamplesofwhat
isbeingattemptedtoday. Improvementstotheefficiencyofthesesolutions
arenecessaryduetothecomplexityofsuchproblems.
In the field of the computational fluid dynamics (CFD), the use of higher-
orderaccuratespatialdiscretizationsforunstructuredgridsoffersapossi-
ble avenue for improving the predictive simulation capabilities for many
modernapplications. Thisisduetothefactthathigher-ordermethodsex-
hibitafasterasymptoticconvergencerate inthediscretizationerrorthan
lower(second)-orderaccurate finitevolume (FV) and finite difference(FD)
methods. The expectation is that an efficient higher-order discretization
may provide an alternate path for achieving high accuracy in a flow with
a wide disparity of length scales at reduced cost, by avoiding the use of
excessivegridresolution.
Although the formulation of compact discretization strategies for higher-
ordermethodssuch as discontinuousGalerkin(DG),spectralvolume(SV)
and spectral difference (SD) methods are now fairly well understood, the
development of techniques for efficiently solving the discrete equations
arisingfromthesemethodshasgenerallybeenlagging. Thisispartlydue
to the complex structure of the discrete equations originating from fairly
sophisticated discretization strategies, as well as the current application
of higher-order methods to problems where simple explicit time-stepping
schemes are thought to be adequate solution mechanisms such as acous-
tic phenomena. Therefore, the development of optimal, or near optimal
solutionstrategies for higher-orderdiscretizations, includingsteady-state
solutionsmethodologies,andimplicittimeintegrationstrategies,remains
i
thenoneofthekeydeterminingfactorsindevisinghigher-ordermethods.
ThemaingoalofthepresentPhDresearchistodevelopanefficientNavier-
Stokes/LES solver on unstructured grids for high-order accurate spatial
discretizations,andbuildupthenecessaryknow-howtomakeahigh-order
accuratesolverforindustrialpurposes.Inordertoachievethat,thepresent
research has been carried out in two parts. In the first part, two im-
plicit time integration schemes, namely backwardEuler (BE) schemeand
second-orderbackwarddifferenceformula(BDF2),arecoupledwithanon-
linearlower-upperGauss-Seidel(LU-SGS)algorithmforefficientlysolving
thediscreteequationsarisingfromthespatialdiscretizationwithaSVora
SDmethod.Thenon-linearLU-SGSalgorithmwiththeBEschemeiseval-
uated both with analysis and computation for both spatial operators and
steady flow problems. In addition, the capabilities and the advantages of
theSDmethodincombinationwiththeimplicittimeintegration/algebraic
solver technique is demonstrated by solving several unsteady reference
test cases. Goodagreementbetweenthepresentresults andreferenceso-
lutions is found, demonstrating the potential benefits of high-order accu-
ratespatialmethods.
In the second part, the SD method coupled with a LES approach is in-
vestigated. Thewall-adaptedlocaleddy-viscosity(WALE)modelischosen
as a subgrid-scale modeland a new idea is presented for the definition of
thefilterwidthintheclosureoftheLESequations. Themethodissuccess-
fullyappliedtocomputetwo-andthree-dimensionalturbulentcases. Good
agreementbetweenthepresentnumericalresultsandreferencesolutions
is observed, showing the capability and the quality of the new coupling
approach.
ii
Jury members
President Prof. HugoSOL
VrijeUniversiteitBrussel
Vice-president Prof. RikPINTELON
VrijeUniversiteitBrussel
Secretary Prof. PatrickGUILLAUME
VrijeUniversiteitBrussel
Internalmembers Prof. GertDESMET
VrijeUniversiteitBrussel
Externalmembers Prof. WimDESMET
KatholiekeUniversiteitLeuven
Prof. EliTURKEL
TelAvivUniversity
Advisor Prof. ChrisLACOR
VrijeUniversiteitBrussel
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Contents
1 Introduction 1
2 Literaturesurvey 7
2.1 Spectralvolumemethod . . . . . . . . . . . . . . . . . . . . . 8
2.2 Spectraldifferencemethod . . . . . . . . . . . . . . . . . . . . 9
2.3 Timeintegrationschemes . . . . . . . . . . . . . . . . . . . . 10
2.4 Geometricandp-multigridmethods . . . . . . . . . . . . . . 11
3 Governingequations 13
3.1 CompressibleNavier-Stokesequations . . . . . . . . . . . . . 14
3.1.1 Newtonianfluid . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 Thermalconductivity . . . . . . . . . . . . . . . . . . . 17
3.1.3 Thermodynamicproperties: idealgasmodel . . . . . 17
3.1.4 FormulationinCartesianspace . . . . . . . . . . . . . 20
3.1.5 Dimensionlessnumbers . . . . . . . . . . . . . . . . . 23
3.2 Largeeddysimulation . . . . . . . . . . . . . . . . . . . . . . 24
3.2.1 FormulationinCartesianspace . . . . . . . . . . . . . 26
3.2.2 Thewall-adaptedlocaleddy-viscositymodel . . . . . 28
3.3 Boundaryconditions . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Farfield. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.2 Inletmassdensityandvelocity . . . . . . . . . . . . . 32
3.3.3 Pressureoutlet . . . . . . . . . . . . . . . . . . . . . . 33
3.3.4 Solidwall . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Aerodynamiccoefficients . . . . . . . . . . . . . . . . . . . . . 36
3.5 Linearconvectionequation . . . . . . . . . . . . . . . . . . . 37
4 Spatialdiscretization 39
4.1 Spectralvolumemethod . . . . . . . . . . . . . . . . . . . . . 40
4.1.1 Discretizationofconvectiveterm . . . . . . . . . . . . 40
4.1.2 SVbasispolynomials . . . . . . . . . . . . . . . . . . . 43
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4.1.3 Discretizationofdiffusiveterms . . . . . . . . . . . . 45
4.1.4 Spectralvolumepartition . . . . . . . . . . . . . . . . 47
4.2 Spectraldifferencemethod . . . . . . . . . . . . . . . . . . . . 52
4.2.1 Discretizationofconvectiveterm . . . . . . . . . . . . 52
4.2.2 SDbasispolynomials . . . . . . . . . . . . . . . . . . . 55
4.2.3 Discretizationofdiffusiveterms . . . . . . . . . . . . 56
4.2.4 Component-wisefluxpointdistribution . . . . . . . . 58
4.2.5 Solutionandfluxpointsdistribution . . . . . . . . . . 59
4.2.6 Gridfilterwidthforthesubgrid-scalemodel . . . . . 62
4.3 Concludingremarks. . . . . . . . . . . . . . . . . . . . . . . . 64
5 Timediscretization 67
5.1 BackwardEulerscheme . . . . . . . . . . . . . . . . . . . . . 70
5.2 Second-orderbackwarddifferenceformula . . . . . . . . . . 74
5.3 Timestep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6 Analysisofthenon-linearLU-SGSalgorithm 77
6.1 Summaryofthemethodology . . . . . . . . . . . . . . . . . . 77
6.2 SVmethodfortriangularcells . . . . . . . . . . . . . . . . . . 82
6.2.1 Second-orderSVmethod . . . . . . . . . . . . . . . . . 83
6.2.2 Third-orderSVmethod . . . . . . . . . . . . . . . . . . 92
6.2.3 Fourth-orderSVmethod . . . . . . . . . . . . . . . . . 94
6.3 SDmethodforquadrilateralcells . . . . . . . . . . . . . . . . 95
6.3.1 Second-orderSDmethod . . . . . . . . . . . . . . . . . 95
6.3.2 Third-orderSDmethod . . . . . . . . . . . . . . . . . . 96
6.3.3 Fourth-orderSDmethod . . . . . . . . . . . . . . . . . 98
6.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7 ApplicationI:spectralvolumemethod 101
7.1 Two-dimensionallaminarsteadyflowsimulations . . . . . . 102
7.1.1 Flowoveracircularcylinder. . . . . . . . . . . . . . . 104
7.1.2 FlowoveraNACA0012airfoil . . . . . . . . . . . . . . 110
7.1.3 Flowinachannelwithabackward-facingstep . . . . 116
8 ApplicationII:spectraldifferencemethod 121
8.1 Steadylaminarflowsimulations . . . . . . . . . . . . . . . . 122
8.1.1 FlowoveraNACA0012airfoil . . . . . . . . . . . . . . 122
8.1.2 Flowthrougha90 bendingsquareduct . . . . . . . . 127
◦
8.2 Unsteadylaminarflowsimulations . . . . . . . . . . . . . . . 133
8.2.1 Flowoveranopencavity . . . . . . . . . . . . . . . . . 133
8.2.2 Flowpastasquarecylinder . . . . . . . . . . . . . . . 137
8.2.3 Flowpastacircularcylinder. . . . . . . . . . . . . . . 141
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8.3 Largeeddysimulations . . . . . . . . . . . . . . . . . . . . . . 145
8.3.1 FlowaroundaNACA0012airfoil . . . . . . . . . . . . 146
8.3.2 FlowaroundasquarecylinderatRe=104 . . . . . . 149
8.3.3 FlowaroundasquarecylinderatRe=2.2 104 . . . 158
×
8.3.4 Flowinamuffler . . . . . . . . . . . . . . . . . . . . . 162
9 Conclusionsandfuturedirections 169
9.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9.2 Futurework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9.2.1 Compacthigh-orderaccuratespatialmethods . . . . 173
9.2.2 Timeintegration/solutioniterativeapproaches . . . . 175
A Timeintegrationmethodsforspace-discretizedequations 177
A.1 Stabilityofspatialdiscretizations. . . . . . . . . . . . . . . . 178
A.2 Stabilityoftimediscretizations . . . . . . . . . . . . . . . . . 182
A.2.1 ForwardEulerscheme . . . . . . . . . . . . . . . . . . 183
A.2.2 BackwardEulerscheme . . . . . . . . . . . . . . . . . 185
A.2.3 Second-orderbackwarddifferenceformula . . . . . . 186
A.2.4 Higher-orderbackwarddifferenceformulae . . . . . . 188
A.3 Methodfortheanalysisofthenon-linearLU-SGSalgorithm 190
A.3.1 Directinversionmethod . . . . . . . . . . . . . . . . . 191
A.3.2 Non-linearLU-SGSalgorithm. . . . . . . . . . . . . . 191
A.3.3 Eigenvaluespectrumoftheamplificationmatrix . . . 195
B p-Multigrid 197
B.1 Fullapproximationscheme . . . . . . . . . . . . . . . . . . . 198
B.2 Transferoperators . . . . . . . . . . . . . . . . . . . . . . . . 199
C Newton-RaphsonGMRESsolver 201
C.1 Newton-Raphsonalgorithm . . . . . . . . . . . . . . . . . . . 202
C.2 GMRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
D ESDIRKschemes 205
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