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 v vi 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 vii 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 viii 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 ix x