计算流体力学课件 chap03a

Chapter3PreliminaryComputationalTechniqueNumericalMethodsAnalyticsolution–linearequation,simplegeometry,simpleinitialandboundaryconditionsAnalyticsolutiontechniques–separationofvariables,Greenfunction,Laplacetransform,Theoryofcharacteristics,…ComplexgeometryComplexequations(nonlinear,coupled)Complexinitial/boundaryconditionsNoanalyticsolutionsNumericalmethodsneeded!!NumericalMethodsObjective:Accuracyatminimumcost(notatanycost!)NumericalAccuracy(erroranalysis)NumericalStability(stabilityanalysis)NumericalEfficiency(minimizecost)Validation(model/prototypedata,fielddata,analyticsolution,theory,asymptoticsolution)ReliabilityandFlexibility(reducepreparationanddebuggingtime)FlowVisualization(graphicsandanimations)OVERVIEWOverviewofthecomputationalsolutionproceduresGoverningEquationsICS/BCSDiscretizationSystemofAlgebraicEquationsEquation(Matrix)SolverApproximateSolutionContinuousSolutionsFinite-DifferenceFinite-VolumeFinite-ElementSpectralBoundaryElementDiscreteNodalValuesTridiagonalADISORGauss-SeidelConjugategradientGaussianeliminationUi(x,y,z,t)p(x,y,z,t)T(x,y,z,t)or(,,,)3.1Discretization1.Timederivativesalmostexclusivelybyfinite-differencemethods2.SpatialderivativesFDM(Finite-DifferenceMethods)FVM(Finite-VolumeMethods)FEM(Finite-ElementMethods)SpectralMethodsBoundaryElementMethods,etc.Analyticsolution3.1.1ConvertingDerivativestoDiscreteAlgebraicEquationsHeatEquationUnsteady,one-dimensionalParabolicPDEMarchingintime,ellipticinspaceThesimplestsystemtoillustrateboththe“propagation”and“equilibrium”behaviors1x0xT0xTdt1Tbt0TxTtTo22),(),(),(,),(TParabolicEquationConvectiveTransportEquationT=temperature,=T=concentration,=DT=vorticity,=T=momentum,=T=turbulentkineticenergy,=+tT=turbulentenergydissipation,=+t/2222TTuTxTxTutTAdvection-DiffusionConvectiveTransportEquation22xTxTutTDiffusionofpolluteninastilllake(u=0)Diffusion/convectionofpolluteninariverDiscretizationChoosesuitablestepsizeandtimeincrementReplacecontinuousinformationbydiscretenodalvaluesConstructdiscretization(algebraic)equationswithsuitablenumericalmethodsSpecifyappropriateauxiliaryconditionsfordiscretizationequationsClassificationofPDEisimportantSolvethesystemof“well-posed”equationsbymatrixsolver)t,x(TT)t,x(Tnjnj:Numerical:Exact3.1.2SpatialDerivativesFinite-difference:Taylor-seriesexpansionFinite-element:low-ordershapefunctionandinterpolationfunction,continuouswithineachelementFinite-volume:integralformofPDEineachcontrolvolumeSpectralmethod:higher-orderinterpolation,continuousovertheentiredomainSpectralelement:finite-element/spectralPanelmethod,BoundaryelementmethodConvertPDEtoalgebraicequations3.1.3TimeDerivativesOne-sided(forwardorbackward)differencesTwo-levelschemesThree-levelschemesRunge-KuttamehtodsAdams-Bashforth-Moultonpredictor-correctormethodsUsuallynoadvantagesinusinghigher-orderintegrationformulaunlessthespatialdiscretizationerrorcanbeimprovedtothesameordertTTtTtTTtT1njnjnj1njorFinite-DifferenceMethodsReplacederivativesbydifferencesjj+1j+2j-1j-21j2jj1j2j1jx2jxjx1jx2312321231232()()()()()()2()3()()()26()(ooooooooooofxaaxxaxxaxxafxfxaaxxaxxafxfxaaxxafx33()()10)/2!()6()/3!()(!)(1)(1)2()()/!()ommmmomommfxaafxfxmammmaxxafxmfxa()0()()()!mmmooomfxxxxxmTaylorseriesexpansionConstructionoffinite-differenceformulaNumericalaccuracy:discretizationerrorxoxTruncationErrorsTaylorseriesTruncationerrorHowtoreducetruncationerrors?(a)Reducegridspacing,usesmallerx=xxo(b)Increaseorderofaccuracy,uselargernbxa,bxa)x(f!n)xx()x(f!3)xx()x(f!2)xx()x(f)xx()x(f)x(foo)n(noo3oo2ooooba)(f)!1n()xx(T)1n(1noE,Finite-Differencesxo=xj,x=xj+1=xj+xxox0mmmjmj1jmo0mo)m(!m)x(x)x(f)xx(f)x(f)xx(!m)x(f)x(f!)()(j0mmmm1jxTmxxT3.2ApproximationtoDerivativesPartialdifferentialequations:dx,dtFinite-differenceequations:x,tTimeandspatialderivatives(i)Taylorseriesexpansion(ii)GeneralTechnique–MethodsofundeterminedcoefficientsDiscretizationerrorsNumericalaccuracy3.2.1TaylorseriesexpansionTruncatedTaylorseries–truncationerrorxxt1njTnjTn1jTn1jTjj+1j-1nn+11M2njM0mmmmn1j1M1njM0mmmm1njx)(κxT!m)x(T)t(κtT!m)t(TTruncationerrorsFinite-DifferencesForwarddifferenceBackwarddifferenceCentraldifference(2))xO(xT!3xxT!2xxTxTxT!m)x(T(1))xO(xT!3xxT!2xxTxTxT!m)x(T4nj333nj222njnjnj0mmmmn1j4nj333nj222njnjnj0mmmmn1j)x(OxTT2TxT)x(Ox2TTxT)x(OxTTxT)x(OxTTxT22n1jnjn1jnj222n1jn1jnjn1jnjnjnjn1jnj12121+23.2.2GeneralTechniqueMethodofundeterminedcoefficientsGeneralexpressionfordiscretizationformula3-pointsymmetric3-pointasymmetric3-pointasymmetric4-pointasymmetric5-pointsymmetricj-2j-1jj+1j+2)()()()()(mn2jn1jnjn1jn2jnjmn2j