QUELQUES APPLICATIONS D UN SCHEMA SCHEMA MIXTE-ELEMENT-VOLUME A LA LES, A L ACOUSTIQUE, AUX INTERFACES

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1 1 QUELQUES APPLICATIONS D UN SCHEMA SCHEMA MIXTE-ELEMENT-VOLUME A LA LES, A L ACOUSTIQUE, AUX INTERFACES O. ALLAIN(*), A. DERVIEUX(**), I. ABALAKIN(***), S. CAMARRI(****), H. GUILLARD(**), B. KOOBUS(*****), T. KOZUBSKAYA(***), A.-C. LESAGE(*), A. MURRONE(**), M.-V. SALVETTI(****), S. WORNOM(**) (*)Societe Lemma, La Roquette sur Siagne, France, (**)INRIA, route des Lucioles, Sophia-Antipolis, France, (***)Institute for athematical Modeling, Moscou, Russie, (****)U. de Pise, Italie, (*****)U. de Montpellier IMFT, Toulouse, 9 mars

2 2 Overview Ingredients of the Mixed Element Volume (MEV) method: Finite Element: Finite Volume: P1 exactness, H1 consistency. conservations, positivity. IMFT, Toulouse, 9 mars

3 3 Outline of the talk 1. Basic options of the numerical method 2. Multiple levels/scales 3. Conservations 4. Positivity IMFT, Toulouse, 9 mars

4 4 1. BASIC OPTIONS OF THE MEV METHOD degrees of freedom at vertices i median dual cells C i variational or integral formulation with two types of test functions: φ i, continuous-p 1, and χ i, characteristic of C i. Finite-Element evaluation of second derivatives Finite-Element/Finite-Volume evaluation of first derivatives IMFT, Toulouse, 9 mars

5 5 First-order MEV W t + divf = divr Vol(i) n+1 (Wi Wi n ) = Σ j N(i) Φ ij t R(W ). φ i dx Upwind numerical integration (Roe): Φ ij = 1 2 (F(W i).n ij + F(W j ).n ij ) A (W j W i ) A = d dw F(W ).n ij IMFT, Toulouse, 9 mars

6 6 First extension: MUSCL Edge-based reconstructions using upwind elements of each edge. ( W ) ij = 1 W 3 F EM (T ij ) + 2 W 3 F EM (T ijk ) W ij = W i + ( W ) ij. ij Φ ij = 0.5(Φ(W ij ) + Φ(W ji )) δ A (W ji (W ij ). IMFT, Toulouse, 9 mars

7 7 Upwind-element reconstruction: 3D Edge-upwind tetrahedra IMFT, Toulouse, 9 mars

8 8 Arbitrary Lagrangian-Eulerian formulation t (V (x,t)w(t)) + F c (w(t),x,ẋ) = R(w(t),x) Ω n+1 i W n+1 i = Ω n i W n i t j V (i) Ω ij Φ ( W n+1 i,w n+1 j, ν ij, xn+1 ij t x n ij ) Φ(W,W,ν,κ) = F (W ) ν x + G(W ) ν y + H(W ) ν z κw. IMFT, Toulouse, 9 mars

9 9 Back to the fluid numerics : V6 ( W ) V6 ij. ij = 1 3 ( W ) Tij. ij (W j W i ) + ξ a ( ( W ) Tij. ij 2(W j W i ) + ( W ) Tji. ij) + ξ b ( ( W ) D ij. ij 2 ( W ) i. ij + ( W ) j. ij ) ( W ) D ij : linear interpolation of nodal gradients in nodes m and n. - Acoustics: (Abalakin-Dervieux-Kozubskaya, IJ Acoustics, 3:2, ,2004) IMFT, Toulouse, 9 mars

10 10 V6 : Tam s case for acoustics IMFT, Toulouse, 9 mars

11 11 2. MULTIPLES LEVELS/SCALES Agglomeration: several neighboring cells a coarse one. inconsistency for second-order problems. Agglomeration MG: corrector terms in the assembly of coarse diffusion terms Additive Multi-Level: smoothed basis functions Variational Multi-Scale (VMS) models IMFT, Toulouse, 9 mars

12 12 MEV implementation of Navier-Stokes + + Ω Ω Ω ρ t X i dω + SuppX i ρu.n dγ = 0 ρu t X i dω + ρ uu.n dγ + SuppX i σ φ i dω = 0 Ω E t X i dω + (E + P )u.n dγ + SuppX i λ T. φ i dω = 0 Ω SuppX i P n dγ Ω σu. φ i dω IMFT, Toulouse, 9 mars

13 13 VMS projection C M m(k) : macro-cell containing the cell C k. (V ol(c j ): volume of cell C j ) I k = {j / C j C M m(k) }. φ k = V ol(c k) φ j. V ol(c j ) j I k j I k W = k φ k W k ; W = W W IMFT, Toulouse, 9 mars

14 14 Smagorinski Large Eddy Simulation model S ij = 1 2 ( u i x j + u j x i ), u = (u 1,u 2,u 3), small scale velocity, S = 2S ij S ij, C s = 0.1, l = V ol(t l) 1 3,. µ t = ρ(c s ) 2 S, τ ij = µ t(2s ij 2 3 S kk δ ij). IMFT, Toulouse, 9 mars

15 15 MEV implementation of the VMS LES model (ended) + + Ω Ω Ω ρ t X i dω + SuppX i ρu.nx i dγ = 0 ρu t X i dω + ρ u u nx i dγ + SuppX i σ φ i dω + τ φ i dω = 0 Ω Ω E t X i dω + (E + P )u.nx i dγ + SuppX i C p µ t λ T. φ i dω + T. φ i dω = 0 P r t Ω Ω SuppX i P nx i dγ Ω σu. φ i dω IMFT, Toulouse, 9 mars

16 16 Results: flow past a square cylinder Reynolds=20,000. From Farhat-Koobus, IMFT, Toulouse, 9 mars

17 17 Results: flow past a square cylinder, end d LES C d C d C l S t l r C pb Classical LES VMS-LES Experiments C d C d C l S t l r C pb Lyn et al Luo et al Bearman et al Tab. 1 Bulk coefficients (*): LES simulations and experimental data (*) l r is the recirculation length behind the square cylinder, and C pb is the mean pressure coefficient on rear face at y = 0. IMFT, Toulouse, 9 mars

18 18 AND NEXT... IMFT, Toulouse, 9 mars

19 19 3. CONSERVATIONS IN ALE FORMULATION - Conservation of extensive quantities. - Geometric Conservation Law (GCL). - Energy budget. IMFT, Toulouse, 9 mars

20 20 Geometric Conservation law A ALE-GCL scheme computes exactly a uniform flow field t Ω n+1 i U n+1 i = Ω n i U n i j V (i) Ω ij Φ ( U n+1 i,u n+1 j, ν ij, xn+1 ij t x n ij ν ij = 0.5 (ν ij (x(t 1 + α 1 (t 2 t 1 ))) + ν ij (x(t 1 + α 2 (t 2 t 1 )))) Ω n+1 i Ω n i = ẋ i n i dγ Ω h (t) - Sufficient condition for 1st-order accuracy (Guillard-Farhat). - Practical stability and accuracy improvements. ) IMFT, Toulouse, 9 mars

21 21 Energy budget of a deforming fluid (1) Work transfers : pressure work in moment equation: M t 2. (x n+1 ij x n ij) = t 1 t ( ) Ω h,i Φ M Ω W n+1 i, ν ij, xn+1 ij x n ij. (x n+1 ij t i Ω h W ork t 2 = t Ω h,i p i ν i. (x n+1 ij x n ij) t 1 i Ω h x n ij) IMFT, Toulouse, 9 mars

22 22 Energy budget of a deforming fluid (2) The energy equation must satisfy the Geometric Conservation Law and this is obtained by an adhoc time integration: Ω n i E n i t Ω n+1 i E n+1 i = j V (i) Ω ij and ν ij specified by GCL. ( Ω ij Φ E U n+1 i,u n+1 j, ν ij, xn+1 ij t x n ij ) IMFT, Toulouse, 9 mars

23 23 Energy budget of a deforming fluid (3) Total energy variation: E 2 t 1 = t Ω h,i Φ E Ω i Ω h Slip condition: E 2 t 1 = t Ω h,i i Ω h ( ( W n+1 i, ν ij, xn+1 ij t x n ij Ω h,i (pu) i. ν i dγ ) ) IMFT, Toulouse, 9 mars

24 24 Energy budget of a deforming fluid (4) A way of integrating the above is: E t 2 = t Ω h,i p i ν i. (x n+1 ij t 1 i Ω h x n ij) Lemma: By replacing the energy flux by a product of boundary pressure times the GCL-preserving integration of mesh motion, we can derive a scheme that is conservative, satisfies GCL and have an exact energy budget: Work of pressure = Loss of total energy. Vàzquez-Koobus-Dervieux-Farhat, Spatial discretization issues for the energy conservation in compressible flow problems on moving grids, INRIA-RR4742 IMFT, Toulouse, 9 mars

25 25 4. POSITIVITY t (V (x,t)u(t)) + F c (w(t),x,ẋ) = 0 a n+1 i U n+1 i a n i U i n t + j N (i) φ(u n ij,u n ji,ν ij,κ ij ) = 0. Φ(u,u,ν,κ) = F (u) ν x + G(u) ν y + H(u) ν z κu. with φ(u,v,ν,κ) monotone: φ u 0 and φ v 0. IMFT, Toulouse, 9 mars

26 26 Limiters U ij = W F EM (T ij ). ij ; 0 U ij = W F EM (T ijk ). ij ( U) ij. ij = L( U ij, 0 U ij, higher U ij ) From usual conditions on limiteurs, and from the position of upwind-elements : U ij = Σp k (U i U k ) ; U ij = p j (U j U i ) where all p s are positive. IMFT, Toulouse, 9 mars

27 27 POSITIVITY, cont d a n+1 i U n+1 i a n i U i n t + j N (i) φ ij (U n ij,u n ji, ν ij, κ ij ) = 0. a n+1 i U n+1 i a n i U i n t = e i + j N (i) g ij (U n ji U n i ) + h ij (U n ij U n i ) IMFT, Toulouse, 9 mars

28 28 POSITIVITY, cont d g ij = t a i φ ij (U n ij,u n ji,ν ij, κ ij ) φ ij (U n ij,u n i,ν ij, κ ij ) U n ji U n i h ij = t a i φ ij (U n ij,u n i,ν ij, κ ij ) φ ij (U n i,u n i,ν ij, κ ij ) U n ij U n i e i = an i an+1 i a n i + a n+1 i j N (i) Ω ij κ ij IMFT, Toulouse, 9 mars

29 29 POSITIVITY, cont d Lemma: Assuming the discrete GCL condition and usual limiter properties, under a CFL condition, the above scheme satisfies the maximum/minimum principle. Cournde-Koobus-Dervieux, soumis Revue EEF, 2005 Farhat-Geuzaine-Grandmont, JCP 2001 IMFT, Toulouse, 9 mars

30 30 POSITIVITY, cont d F (U) = U = (ρ,ρu,ρv,ρw,e,ρy ), ρu ρu 2 + P ρuv ρuw (e + P )u ρuy G(U) = P = (γ 1) U F (U) + + G(U) + H(U) t x y z ρv ρuv ρv 2 + P ρvw H(U) = (e + P )v ρvy ( e 1 2 (u2 + v 2 + w 2 ) ) = 0 ρw ρuw ρvw ρw 2 + P (e + P )w ρwy Lemma: Assume that the Riemann solver is ρ-positive, then under a CFL condition, the limited ALE-MEV scheme preserves ρ-positivity and satisfies the maximum/minimum principle for Y = ρy/ρ. IMFT, Toulouse, 9 mars

31 31 Cournde-Koobus-Dervieux, soumis Revue EEF, 2005 IMFT, Toulouse, 9 mars

32 32 Application to a two-phase flow Five-equation quasi conservative reduced model t α kρ k + div(α k ρ k u) = 0 ρu + div(ρu u) + p = 0 t ρe + div(ρe + p)u = 0 t t α 2 + u. α 2 = α 1 α 2 ρ 1 a 2 1 ρ 2 a k=1 α k ρ ka 2 kdivu with e = ε + u 2 /2 and ρε = 2 k=1 α kρ k ε k (p,ρ k ). IMFT, Toulouse, 9 mars

33 33 NUMERICS FOR TWO-PHASE FLOW - Acoustic Riemann solver (Murrone-Guillard, Comp.Fluids 2003, JCP 2004)) - Second order limited MEV. - Source term: upwind integration of divu. - 3D parallel extension with MEV (Wornom et al.) IMFT, Toulouse, 9 mars

34 34 Falling drop (2D) IMFT, Toulouse, 9 mars

35 35 Shock on a drop (3D) IMFT, Toulouse, 9 mars

36 36 Conclusions The choice of a very simple scheme permits the proof of a very large set of properties: - many conservation statements can be made exactly satisfied, - Maximum principle and positivity, including the moving mesh case. Today s challenge are: - showing positivity statements for the new techniques of thick interfaces. - showing the above properties for new schemes. IMFT, Toulouse, 9 mars