计算流体力学讲义(任玉新)清华大学 基础篇

200371§1312×2/ComputationalFluidDynamicsCFDCFD/meshcellcontrolvolumegridgridpointCFDCFDCFDCFDCFDCFDCFD2NavierStokesEulerNSEulerNSNSNSReynoldsNSCFDCFDCFDPostprocessing16872050171819EulerLagrangeNavierStokes20PrantlVonKarman3“”CFD2060CFDCFDCFDCFDCFD“”CFDCFDCFDCFDCFDCFD“”CFDCFDCFDCFDCFDReynoldsNavierStokes10CFDCFDCFDCFDCFDCFDCFD1CFD423“”“”4CFDCFDCFDCFDCFDCFDCFDCFDCFDCFDCFD“”Reynolds“”CFDCFDCFDCFDCFDCFD1.CFDCFDCFD“”CFD5CFDCFD“”CFDCFDCFDCampbellMuellerKimMoinCFD2.CFDCFD20CFDEuler/ReynoldsNavier-StokesCFDCFD5-10CFD40-50%15-20%50%CFDCFDCFDFiniteDifferenceFiniteVolume6§2NavierStokesCFDNavierStokes10Sddtρρ∂S+=∂∫∫∫∫∫VnVVi()1a()t0ρρ∂+∇⋅=∂V1b2SSddSdtρρρτ⇒∂+=+∂∫∫∫∫∫∫∫∫∫∫VVVnFVVVViidSn2a()()tρρρ⇒∂+∇⋅=+∇⋅∂VVVFτ2b*p⇒⇒=−+τIτ⇒⇒*τ*()SSdpdSdtρρρ⇒⇒∂++=+∂∫∫∫∫∫∫∫∫∫∫VVVInFτnVVVViidS2c()()pt*ρρρ⇒⇒∂+∇⋅+=+∇⋅∂VVVIFτ2d3()SSEdEdSdtρρρ⇒∂+=+⋅−∂∫∫∫∫∫∫∫∫∫∫VnFVτVqnVVVViidSi3a()(EEt)ρρρ⇒∂+∇⋅=⋅−∇⋅+∇⋅⋅∂VFVqτV3b22EeVρρρ=+eE7()()(*p⇒⇒∇⋅⋅=−∇⋅+∇⋅⋅τVVτV)()kT−∇⋅=∇⋅∇q*()()SSEdEpdSdkTtρρρ⇒∂++=+⋅+∇∂∫∫∫∫∫∫∫∫∫∫VnFVτVnVVVViidSi3c[()]()(*)EEpkTtρρρ⇒∂+∇⋅+=⋅+∇⋅∇+∇⋅⋅∂VFVτV3d1pRTpeρργ==−EulerEulerGaussLagrange0DdDtρ∫∫∫VV=0tρρρ∂+⋅∇+∇⋅=∂VVGaussGauss8CFD1.NavierStokesEulerNS111()()()0txyz∂−∂−∂−∂+++∂∂∂∂FFGGHHU=4uvwEρρρρρ=U2()uupuvuwEpuρρρρρ+=+F2()vvuvpvwEpvρρρρρ=++G2()wuwvwwpEpwρρρρρ=++H10xxxyxzxxxyxzTuvwkxττττττ=∂+++∂F10xyyyyzxyyyyzTuvwkyττττττ=∂+++∂G10xzzyzzxzzyzzTuvwkzττττττ=∂+++∂HNSEuler0txyz∂∂∂∂+++=∂∂∂∂UFGH545FGflux11,,,,,,UFGHFGH111,FGH11U,,,,H,,FGH11,,FGH945450t⇒∂+∇=∂UEi611()()(⇒=−+−+−EFFiGGjHH1)k⇒=++EFiGjHk6Gauss0Sddt⇒∂S+=∂∫∫∫∫∫UEnVVi774562.20CFDCFD45shockcapturingshockfitting100w=VTw()wwqTnk∂=−∂()wTn0∂=∂0w=Vni§3EulerNavierStokesBAtx∂∂+=∂∂UUC1Um,C,BAmm×1,BAU1Euler0tx∂∂+=∂∂UF211U,F,umEρρρρε=U1222232(1)()2()[(1)()]2mfummupfEpufmmρργερρρεγερρ=++−−++−−F111222333fffmfffmxxmxxmfffmρερερερρε∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂=++=∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂FFFFxεU111222333fffmfffAmfffmρερερε∂∂∂∂∂∂∂∂∂∂=∂∂∂∂∂∂∂∂∂∂FU3Euler0Atx∂∂+=∂∂UU4aJacobiA23201(3)(3)123(1)(1)2uAuuuEuEuγγγγγγγ=−−−−−−−+0γ4bLaplace22220xy∂Φ∂Φ+=∂∂5a12,uvxy∂Φ∂==Φ∂∂500uvxyvuxy∂∂+=∂∂∂∂−=∂∂0Axy∂∂+=∂∂UU01,10uv==−UA5bNavierStokes1,,11(1,2,...,)mmjjijijijjuubacitx==∂∂+==∂∂∑∑m66,tx(,)xt:((,dxUxtdtλΓ=))7ΓDdxDttxdttxφφφφλ∂∂∂∂=+=+∂∂∂∂φ86ΓΓΓ0()uuaaconsttx∂∂+==∂∂913ΓDuuuDttλx∂∂=+∂∂aλ=0DuuuaDttx∂∂=+=∂∂109:dxadt=Γ0DuDt=:dxadtΓ=90DuDt=0DuDt=uc9|onsΓ=t0(,0)()uxux=u9|constΓ=0(,)()uxtuxat=−6,,111[]mmmjjiijijiijjuulbactx===∂∂0+−=∂∂∑∑∑10li10(1,2,...,)im=,,1111[()()]mmmmjjiijiijiijiiiuulblalctx====∂∂+=∂∂∑∑∑∑∀j,1,1miijimiijilalbλ===∑∑11,111[()()]mmmjjiijiijiiuulblctxλ===∂∂+=∂∂∑∑∑14,111[()]mmmjiijiijiiDulblcDt====∑∑∑12:dxdtλΓ12=j∀11,,1()0,1,2,...,miijijilabjmλ=−==∑131(,,...,)mmlll=l13()AB0λ−=l14a()TTTABλ0−=l14bllml14l,T1(,,...,)mmlll=0ABλ−=15λmm(1,2,...,)kkλ=mkλUkλ14kλkl()|()kkkkDBBkDttxλ∂∂=+=∂∂UUUlllC1615mmmmm0ABλ−=m150ABλ−=m0ABλ−=mkkkkm0A=X17Xm171712,,...nXXX12,...nX,XX12,,...,nXXX0AX=nmrr−=AArAr1r+λ15kλkABλ−rmk=−10ABλ−=mm10ABλ−=kk110ABλ−=10ABλ−=10ABλ−=1161.Euler4BI=23201(3)(3)123(1)(1)2uAuuuEuEuγγγγγγγ=−−−−−−−+0γ0AIλ−=A123,,uauuaλλλ=−==+/paγρ=Euler2.Laplace501,10BIA==−0AIλ−=12,iλλ==ii−Laplace173.Eulerργρρρρρρρpayvxuaypvxpuypyvvxvuxpyuvxuuyvxuyvxu==∂∂+∂∂−∂∂+∂∂∂∂−=∂∂+∂∂∂∂−=∂∂+∂∂=∂∂+∂∂+∂∂+∂∂220)(110)(0ABxy∂∂+=∂∂UU0Cxy∂∂+=∂∂UU=pvuρU=upuuuA0000010000γρρ=vpvvvBγρρ0010000000222222222222212222222()0()1000vvuvuuauauuauvavuauauaCABvuuvauuvuauauaρρρρργρ−−−−−−−−−−==−−−−0CIλ−={}22222224()[()]()vuvuaauvauλλ−−−−++0=C1,2223,422vuuvauvauaλλ=2±+−=−1822201(uvaM+−⇔超音速）22201uvaM+−⇔（亚音速）22201uvaM+−⇔＝＝（音速）0CIλ−=3,422uvuaλ=−EulerEuler70MurmanColeEuler4.Euler0ABtxy∂∂∂++=∂∂∂UUUxt−A1234,,,uauuuaλλλλ=−===+xt−EulerEulerEulerEuleryt−195Navier–Stokes)(1)(1022222222yvxvypyvvxvuyuxuxpyuvxuuyvxu∂∂+∂∂+∂∂−=∂∂+∂∂∂∂+∂∂+∂∂−=∂∂+∂∂=∂∂+∂∂ρµρρµρxvhyugyvxuf∂∂=∂∂=∂∂−=∂∂=1()1()00pfgufvgxxyphfuhvfyxyfgyxfhxyµρρµρρ∂∂∂−++=+∂∂∂∂∂∂−+−=−∂∂∂∂∂−=∂∂∂∂+=∂∂Navier–Stokes6.1)Navier–Stokes2)Navier–Stokes3)Navier–Stokes4)5)N–SN–SX202222yx∂∂∂∂Navier–StokesN-SCFD1.Eulert2uλ=1uaλ=−3uaλ=+1uaλ=−2uλ=3uaλ=+(,)PPPxtA1A2A3A4xxaxbC1[,]abxxEuler12P(1,2,3kkλ=0))(ptdtdt−P21,PPtttPtuaλ=−,uaλ=+,abxx,0txx===P3t0=Ct0t0t=01λua=−(,)PPCC,C3aλ,2u=+(1k=,3kλ)xtPttP221t211ax=1Att0()const∂+==ttttt1xA1EulerA1A2A2A3A4EulerxxxEulerEulerEuleruuaatx∂∂∂22Burgers0uuutx∂∂+=∂∂2.2PtPt(,)PPPxttxxaxb2NavierStokesNavier–Stokes22(0uutxγγ∂∂)=∂∂2322(,uuuaaconstconsttxxγγ∂∂∂0)+===∂∂∂Burgers22(uuuuconsttxxγγ∂∂∂+==∂∂∂0)3.EulerNavier–StokesNavier–StokesLaplace22220uuxy∂∂+=∂∂Poisson2222(,)uusxyxy∂∂+=∂∂§12422(0uutxγγ∂∂=∂∂)1(,)[0,1][0,]xt∈×∞(,0)()(0,)()(1,)()uxfxutatutbt===2Dirichletalgebraicdifferencequotientmethodoffinitedifference11discretization1.M1M+01,,...,Mxxxkxkx=∆1/xM∆=25(xk,tn)t2t1t0x0=1x2x3xm-1xm=112.t1NT=012,,,...Nttttntnt=∆∆=/tTN3.1u(,)(,)(0,1,...,,0,1,...,)knuxtukxntkMn=∆∆==NCFDuxu