Comparison of implicit, explicit, center and upwind schemes for the simulation of internal vortex flow at low Mach number

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Randolph Hawkins
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Lionel Le Penven 2 Caherine Le Ribaul 2 UCB Lyon I,

1 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Comparison of implici, explici, cener and upwind schemes for he simulaion of inernal vorex flow a low Mach number Marc Buffa Anne Cadiou 2 Lionel Le Penven 2 Caherine Le Ribaul 2 UCB Lyon I, LMFA UMR CNRS, LMFA UMR 559 Low Mach Conference (Porquerolles) June 24

2 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Oulines Inroducion 2 Numerical mehods 3 Inviscid Taylor vorex a low Mach 4 Viscous Taylor vorex a low Mach 5 Compression of he Taylor vorex 6 Perurbed 3D compression 7 Conclusion

3 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Moivaion Model flow Moivaion Moivaion: umble flow in reciprocaing engine Numerical difficulies Low Mach wih differen ime scales (slow and fas) Differen scaling of he sae variables: ρ = θ(), u = θ(), p = θ(ma 2 ) CFL sabiliy condiion based on c (celeriy), no u (velociy)

4 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Moivaion Model flow Model flow Model Taylor Vorex in a square caviy Inernal vorex flows a Low Mach Inviscid case: asympoic analysis, slow ime densiy flucuaion θ(ma 2 ) fas ime pressure flucuaion θ(ma 3 ) 2 Viscous decay: numerical dissipaion (order) 3 Compression effec: 2D flow 3D perurbed flow (ellipical insabiliy)

5 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Schemes NadiaLES NadiaDF NadiaVF Low Mach prec. Numerical schemes [ Conservaion equaions for W = ρ,ρ ] U,E = p + γ 2 ρu2 W + div (A(W)W) = div (R (W)) + S(W) }{{}}{{}}{{} Euler flux viscous flux source Numerical difficulies: Euler flux F (W) = A(W)W Explici scheme wih a sabiliy condiion CFL < Explici 2nd order cenered scheme: unsable Upwind explici scheme (using he eigenvalues of A) High order explici scheme (i.e. >2) Implici scheme CFL > Implici non linear cenered scheme

6 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Schemes NadiaLES NadiaDF NadiaVF Low Mach prec. NadiaLES code Explici mixed FE/FV wih LES model on unsrucured 3D mesh (Duchamp 999) FV/FE on unsrucured mesh (Dervieux 998) finie elemen mesh G J IJ I Precision O(d 4,h 2 ) wih Runge Kua 4 Domain decomposiion ( //e wih MPI) G2 conrol volume Low Mach Riemann solver Roe-Turkel (Vioza 997) Conrol of dissipaion and dispersion

7 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Schemes NadiaLES NadiaDF NadiaVF Low Mach prec. NadiaDF code 2D/3D explici F.D. code wih 2 high order schemes on srucured mesh PADE: Lele 992 reconsrucion of he derivaive a order 4 αf i + F i + αf i+ = a F i+ F i 2 x WENO: Jiang & Shu 996 reconsrucion of he fluxes a order 5 F i+ 2 = 2 r= ω r F +,r i r= ω 2 r F,r i+ 2 wih α = 4, a = 4 3 ri-diagonal sysem (Thomas) F +,r i /2 r= r=2 r= i 2 i i i+/2 i+ i+2

8 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Schemes NadiaLES NadiaDF NadiaVF Low Mach prec. NadiaVF code 2D/3D parallel implici FV on unsrucured mesh (wrien in C++) Non linear implici BDF 3W n+ 4W n + W n = F(W n+ ) 2 Newon G(W n+ ) = ( ) G ( W n+ k+ W ) W n+ k = G(W n+ k ) k Cenered FV of order 2 W = W n i dγ x i V k Γ k LibMesh (Kirk 22) FE mesh in //e (C++) wih non conform grid adapaion (AMR) PETSC (Barry 2) //e soluion of linear sysem Krilov, Muli-Grid METIS (Karypis 996) domain pariioning

9 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Schemes NadiaLES NadiaDF NadiaVF Low Mach prec. Precondiioning a low Mach Mach dependance of he sae variables ρ = θ() and u = θ() bu E = θ(ma 2 ) and p = θ(ma 2 ) NadiaLES Roe-Turkel (Vioza 999) β Ma precondiioning of p (β 2 ) enropic variables [p, u,s ]: NadiaVF decomposiion of E and p E(x,) = γ P () + E (x,) p(x,) = P () + p (x,) equaion for E E (x,) = θ() and p ( ) (x,) = θ() γ wih P () = V() V() V() p(x,)dx

10 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Asympoic Slow ime Fas ime Numerical simulaion Asympoic analysis Iniial condiion Taylor vorex wih uniform densiy ρ = : U = U 2 (x) = [sinπx cosπy, cosπx sinπy], P = P + P 2 (x) = + (cos2πx + cos2πy) γma 2 4 asympoic soluion ρ(x,,τ) = + Maρ 2 2 (x,τ) + Maρ 3 3 (x,,τ) U(x,,τ) = U 2 (x) + Ma 2 U 4(x,,τ) P(x,,τ) = γ M 2 a + P 2 (x) + M a P 3 (x,,τ) Physical soluion seady incompressible par + slow ime (τ) par + fas ime () par

11 rho_ rho x Inro Numeric Inviscid Viscous Comp. 3D Conclusion Asympoic Slow ime Fas ime Numerical simulaion Slow ime soluion (convecion) A =, sreamlines isobars Flucuaion of densiy ρ 2 Conservaion of enropy s = s + M 2 a s 2 along a rajecory X(τ,X ) ρ 2 (X) + M2 a 2 U2 2 (X) = cse = M2 a 2 U2 2 (X ) ρ 2 (X) periodic wih T = T p 4 (T p(x )= ravel ime along he rajecory) (a) ρ 2 (X) (b) ρ 2 (X ) (c) U 2 2 (x,y) (d) k 4 T p Figure: X = [.2,.] (T p = 3.95) and X = [.5,.25]

12 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Asympoic Slow ime Fas ime Numerical simulaion Slow ime ρ2 (X, τ) field Simulaion wih FEMLAB (EF P 2 ) of he ranspor equaion ρ2 ~ ~ M2 ~ 2 ~ 2 U2 + U2 ρ2 = a U τ 2 (a) τ =. (b) τ =.5 (c) τ = 2 (d) τ = 4 Figure: ρ2 (x, y, τ) a differen imes (ANIMATION)

13 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Asympoic Slow ime Fas ime Numerical simulaion Fas ime soluion (acousic) The pressure P 3 (x,y,) is soluion of a wave equaion 2 P 3 2 M 2 a P 3 = wih P 3 n = a = P 3 =, and P 3 = M a f(u 2. U 2 2 ) P 3 (x,y,) is a combinaison of he acousic modes of he caviy P 3 (x,y,) = M a p,q= P pq cospπx cosqπy e I M a p 2 +q 2 π Firs exied acousic mode P 3, P,3 wih frequency f 3, = 2 M a

14 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Asympoic Slow ime Fas ime Numerical simulaion Behaviour of he numerical schemes Every scheme is able o capure he incompressible soluion M a NadiaVF NadiaVF Padé Weno NadiaLES CFL ρ ρ < 4. < < 5 6 < 4 ρu ρ U Table: deparure from he incompressible soluion ρ = ρ, U(x,y) = U 2 (x,y) a ime = 8.72 wih 8 2 Bu differen behaviours for he predicion of he slow and fas soluion

predicion of he slow par (NadiaVF) 4 x 3 3 2.7 < ρ 2 <.

15 ρ 2 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Asympoic Slow ime Fas ime Numerical simulaion Numerical predicion of he slow par (NadiaVF) 4 x < ρ 2 < ρ 2 (τ) a X = [.8,.5].2. P P < P 3 < F P 3 ()a X = [.8,.5]

16 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Asympoic Slow ime Fas ime Numerical simulaion Comparison of he slow par predicions (a) NadiaVF (b) CFL= (c) Padé (d) NadiaVF Figure: Slow par ρ 2 (x,y,τ) a τ = 2 and τ = 8.7 (mesh 8 2 )

17 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Asympoic Slow ime Fas ime Numerical simulaion Comparison of he slow par predicions.2 CFL=.5 CFL=..2 CFL=.5.2 6x6 8x rho(.5,.5).998 rho(.5,.5) rho(.5,.5) PADE WENO.2. PADE.2. 8x8 6x6 32x rho(.5,.5).998 rho(.5,.5) rho(.5,.5) Figure: Slow par ρ 2 (.5,.5,τ): influence of M a and mesh size wih NadiaVF and NadiaDF

18 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Asympoic Slow ime Fas ime Numerical simulaion Numerical predicion of he acousic par P3 P P F P3 8 x F 6 x x 3 2 x 3 x 3 P3 P P x F P3.4 x F 8 x (a) VF CFL=.5 (b) VF CFL= (c) PADE CFL=.6 Figure: Acousic pressure P 3 (.66,.5,) a M a =. and M a =.

Inro Numeric Inviscid Viscous Comp. 3D Conclusion Asympoic Slow ime Fas ime Numerical simulaion Influence of he precondiioning wih prec. wihou prec.

19 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Asympoic Slow ime Fas ime Numerical simulaion Influence of he precondiioning wih prec. wihou prec. rho wih prec. wihou prec rho Figure: NadiaVF: influence of he precondiioning a M a =. CFL= (8 2 )

20 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Parameers dissipaion ν h Slow pred. Viscous Taylor vorex a low Mach Parameers Re = U max L ν = L =, U max =, ρ =, P = M 2 a γ Compressible soluion inviscid + viscous decay mean hermodynamic par θ(m 2 a ): P = M 2 a γ unseady incompressible par θ() (damped): U(x,y,) = U 2 (x,y)e 2νπ2, P(x,y,) = P 2 (x,y)e 4νπ2 slow par θ(m 2 a): ρ 2 (x,y,τ) acousic par θ(m a )(damped): P 3 (x,y,)

21 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Parameers dissipaion ν h Slow pred. Numerical dissipaion versus h and Ma NadiaVF % de viscosie numerique NadiaDF % de viscosie numerique. Ma=. Ma=.. WENO Ma=. PADE Ma=. WENO Ma=. PADE Ma=... % nuh % nuh... e-5... h e-6... h NadiaLES % de viscosie numerique Ma=. Ma=. ρu =f ρu = = e 2ν eπ 2 f.. ν e = ν + ν h. % nuh. e-5... h

22 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Parameers dissipaion ν h Slow pred. Influence of he mesh on ρ 2 (slow par) (a) VF 9 (b) Padé (c) Weno (d) EF (e) VF 9 (f) Padé 2 (g) Weno 2 (h) EF 4

.6.4.2.8.6.4 soluion Ma=.2 2 3 4 5 6 7 8 9 ρ U ρ V Inro Numeric Inviscid Viscous Comp.

23 soluion Ma= ρ U ρ V Inro Numeric Inviscid Viscous Comp. 3D Conclusion Parameers Soluion Slow par Acousic Tcpu Compression of a Taylor vorex Norme L2 model of he umble in a combusion chamber Parameers Re = U max L() ν = (sable) L() = V p ω + V p ω sinω V p =.44, ω =,.36, T c = 8.7 soluion a M a = compressed Taylor vorex ρ () = ρ L() L(), P () = ρ () γma 2 acceleraion of he v componen viscous decay

24 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Parameers Soluion Slow par Acousic Tcpu Numerical soluion Evoluion of he velociy norm a Ma =. and Ma =..55 Ma=. Ma= norme L / Velociy U 2 (x,y,) All schemes predic he righ velociy U 2 (x,y,) soluion a Ma =

25 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Parameers Soluion Slow par Acousic Tcpu ρ 2 densiy flucuaion (slow par) x 3 x Figure: ρ 2 (x,y,t c ) a Ma =. (NadiaVF (animaion), NadiaDF, NadiaLES) NadiaVF NadiaWENO NadiaPADE NadiaLES Ma = Ma = Table: maximum of he densiy flucuaion: ρ max ρ min a = T c and 4 2

26 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Parameers Soluion Slow par Acousic Tcpu Acousic U en.88,.5 a Ma=. E en.88,.5 a Ma=. CFL=.5 CFL= CFL=.5 CFL= U -.5 E Figure: flucuaion a (.88,.5) (near he pison) of ρu and E a M a =. The implici scheme damp he acousic if CFL > acousic waves are amplified by he compression pble wih high order schemes

27 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Parameers Soluion Slow par Acousic Tcpu Compuer cos of he schemes NadiaVF NadiaPADE NadiaWENO NadiaLES Ma = Ma = Table: Number of ime seps wih he mesh 4 2 CPU ime per ieraion NadiaVF (Penium IV 2.7 Ghz):.3s (Ma =.) and.7s (Ma =.) NadiaDF (Penium IV.7 Ghz) PADE:.4s and WENO:.7s NadiaLES (Penium IV.7 Ghz) maillage 3D (3 4 2 ):.6s

28 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Perurbed 3D compression Specral soluion a Ma = (Le Penven 22) Figure: compression of a perurbed vorex a ( =,3,5,8)

29 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Numerical simulaion Figure: Iso-Voriciy a = T c wih NadiaVF, NadiaDF and NadiaLES Animaions of Iso Voriciy and Iso densiy ρ 2

30 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Scheme properies (a) Scheme properies (b) Scheme properies (a) All he schemes predic he incompressible (M a = ) par NadiaVF (implici cener scheme): properies a low Mach needs precondiioning (siff linear and non linear sysem) good predicion of he slow par even wih CFL>, wih very small dissipaion bu some dispersion filering of acousic wih CFL > and good predicion of he acousic wih CFL<

31 Inro Numeric Inviscid Viscous Comp. 3D Conclusion Scheme properies (a) Scheme properies (b) Scheme properies (b) NadiaDF (high order Padé and Weno schemes): properies a low Mach predicion of he slow par wih some dissipaion, bu wihou dispersion precise schemes a M a =., bu need precondiioning a M a =. for he acousic NadiaLES (mixed FV/FE Roe scheme): properies a low Mach robus code bu leas precise code needs fine mesh o ge reasonable resuls for he slow par

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