Une méthode de pénalisation par face pour l approximation des équations de Navier-Stokes à nombre de Reynolds élevé

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1 Une méthode de pénalisation par face pour l approximation des équations de Navier-Stokes à nombre de Reynolds élevé CMCS/IACS Ecole Polytechnique Federale de Lausanne Erik.Burman@epfl.ch Méthodes Numériques pour les Fluides, Paris 19 Décembre 2005

2 Outline 1 Introduction 2 Brief history 3 Stabilized methods by scale separation 4 A priori error estimation 5 Monitoring artificial diffusion 6 Numerical results 7 Consclusions

3 State of the art: face penalty methods The addition of a term penalizing the jump of the gradient over element edges J(u h, v h ) = γh K α [ u h n][ v h n] ds K K\ Ω to the standard Galerkin formulation may be used to stabilize transport operators Stokes like systems symmetric Friedrichs systems Error analysis for linear problems leads to (quasi) optimal apriori error estimates for continuous finite element spaces V h = {v : v C 0 (Ω); v K P k (K), K T h }.

4 State of the art: face penalty methods The addition of a term penalizing the jump of the gradient over element edges J(u h, v h ) = γh K α [ u h n][ v h n] ds K K\ Ω to the standard Galerkin formulation may be used to stabilize transport operators Stokes like systems symmetric Friedrichs systems Theorem (Burman-Fernández-Hansbo (2004)) There exists an interpolation operator πh on V h k such that ( ) h 1 2 I π h vh 2 0,Ω γ hk 2 v h n 2, K T h with v def = v + v if K Ω and v def = 0 if K Ω. K

5 Brief history Babuska & Zlamal, biharmonic operator, (1972). Douglas & Dupont, second order elliptic and parabolic problems, (1976). Burman & Hansbo, convection dominated limit, (2003). Burman & Ern, discrete maximum principle, (2004). Burman & Fernández & Hansbo, the Oseen s problem, (2004). Burman & Ern, hp-fem for transport operators, (2005). Burman & Fernández, the incompressible Navier-Stokes equations, semidiscretization in space (2005). Important related work: Guermond, subgrid viscosity, Codina, orthogonal subscale stabilization, Brezzi & Fortin, A minimal stabilisation procedure, Becker & Braack, local projection stabilization, 2001.

6 Variational multiscale methods and projection based LES Knowing the exact solution (u, p) we could compute the ideal projection (π h u, π h p) [V h ] d V h. Since the exact solution is unknown we have to do with a working projection given by a discrete scheme (typically Galerkin FEM). The working projection should be stable and accurate uniformly in the Reynolds number: standard Galerkin has to be modified. Assumption: the Bernoulli hypothesis: all fine to coarse interaction is dissipative. We choose π h, (the ideal projection) to be the L 2 -projection.

7 Stabilized methods based on scale separation, the Euler equations 1 Let W = H(div) L 2 0, U = (u, p), L(w)U = (w )u + p, π = (I πh ). Assume f [V h] d. Find U W such that ( t u + L(u)U, v) + ( u, q) = (f, v), (v, q) W.

8 Stabilized methods based on scale separation, the Euler equations 1 Find U W such that ( t u + L(u)U, v) + ( u, q) = (f, v), (v, q) W. 2 Scale separation U = U h + Ũ, U h = π h U π h U is the L 2 -projection of U onto W h = [V h ] d V h Ũ orthogonal to the finite element space (c.f. Codina).

9 Stabilized methods based on scale separation, the Euler equations 1 Find U W such that ( t u + L(u)U, v) + ( u, q) = (f, v), (v, q) W. 2 Scale separation U = U h + Ũ, U h = π h U 3 Inserting U h + Ũ yields the formulation ( t u h +L(u h )U h, v h ) + ( u h, q h ) = (f, v h ) +(T 1 (π L(u h )U h, π u h ), (π L(u h )V h, π v h )) +((ũ )u, v h ) V = (v, q) W. 4 T 1 is the solution operator for the fine scale equation ( t ũ + (u )ũ + (ũ )u h + p, ṽ) + ( ũ, q) = (π L(u h )U h, ṽ) + (π u h, q).

10 Simplifications leading to edge oriented stabilization 1 We drop the fine to coarse interaction terms ((ũ )u, v h ) 2 Bernoulli hypothesis: approximate T 1 with a scaled diagonal matrix. 3 Stabilized FEM based on the projected residual: Find U h W h such that ( t u h +L(u h )U h, v h ) + ( u h, q h ) = (f, v h ) (δ u π L(u h )U h, π L(u h )V h )) (δ div π u h, π v h )) V h = (v h, q h ) W h. 4 Equivalent dissipation (recall π = (I π h )): π L(u h )U h 2 K e E(K) e E(K) e e γh K [L(u h )U h ] 2 ds γh K { [(uh )u h ] 2 + [ p h ] 2} ds

11 Edge Stabilized FE, Navier-Stokes: Space Semi-Discretization For all t (0, T ), find (u h (t), p h (t)) [V h ] d V h such that ( t u h, v h ) + a(u h ; u h, v h ) + b(p h, v h ) = (f, v h ), b(q h, u h ) = 0, u h (0) = π h u 0, for all (v h, q h ) [V h ] d V h, with a(u h ; u h, v h ) def = (u h u h, v h ) + (ν u h, v h ) ( u h, u h v h ) + bd terms b(p h, v h ) def = (p h, v h ) + bd terms

12 Edge Stabilized FE, Navier-Stokes: Space Semi-Discretization For all t (0, T ), find (u h (t), p h (t)) [V h ] d V h such that ( t u h, v h ) + a(u h ; u h, v h ) + b(p h, v h ) + j uh (u h, v h ) = (f, v h ), b(q h, u h )+j(p h, q h ) = 0, u h (0) = π h u 0, for all (v h, q h ) [V h ] d V h, with j uh (u h, v h ) def = γhk 2 (1 + u h n 2 ) u h : v h, K T h j(p h, q h ) def = K T h K K γh 2 K p h q h.

13 Convergence (selected results) J [ u h ; (u h, p h ), (v h, q h ) ] = j uh (u h, v h ) + j(p h, q h ) Triple-norm: (v h, q h ) 2 w h def = ν 1 2 v h 2 0,Ω + J[w h; (v h, q h ), (v h, q h )]. Theorem (Velocity convergence, Burman & Fernández, 2005) The following estimates hold (when ν < h) ( 0 π h u u h L ((0,T );L 2 (Ω)) h 3 2 C(u, p)e c(u)t, ) 1 2 (π h u u h, π h p p h ) 2 u h dt h 3 2 C(u, p, T )e c(u)t, 0 J[u h, (u h, p h ), (u h, p h )] dt h 3 C(u, p)e c(u)t with c(u) depending on u L (0,T ;W 1, (Ω)) and C(u, p) depending on u L2 (0,T ;H 2 (Ω)), p L2 (0,T ;H 2 (Ω)), u L (0,T ;W 1, (Ω)).

14 Energy consistency: monitoring artificial dissipation For the Navier-Stokes equations there holds u(t ) 2 + ν 1 2 u 2 dt = u(0) 2 + (f, u). 0 Any reasonable numerical method will satisfy { } u h (T ) 2 + ν 1 2 uh 2 + S(u h, p h ) dt = u h (0) 2 + (f, u h ). 0 S(u h, p h ) the artificial dissipation added for the method to remain stable. Define D = 0 S(u h,p h ) dt : T 0 ν 1 2 u h 2 dt u h (T ) 2 + (1 + D) ν 1 2 uh 2 dt = u h (0) 2 + (f, u h ). 0

15 Some remarks on face penalty stabilization 1 The interior penalty operator can be seen as a subgrid viscosity: starting from polynomial order 3 and onward the kernel is a C 1 space with approximation properties. 2 Scale separation by polynomial order instead of hierarchic meshes. 3 The dissipation ratio D measures the energy consistency and is (related to) an a posteriori error estimator. 4 For high Reynolds number flow theory predicts (P1 elements and sufficiently regular solution): computational error numerical dissipation = stabilization Ch 3

16 Scale separation and the energy inequality Let us now assume f = 0 and consider the projection on mesh T h of the exact solution π h u(t ) 2 + (I π h )u(t ) ν 1 2 u 2 dt = u(0) 2, but (I π h )u(t ) represents the unresolved scales and hence (I π h )u(t ) 2 ξ h E(ξ) dξ (where E(ξ) denotes the energy distribution over the wave numbers)

17 Scale separation and the energy inequality Let us now assume f = 0 and consider the projection on mesh T h of the exact solution π h u(t ) 2 + (I π h )u(t ) ν 1 2 u 2 dt = u(0) 2, but (I π h )u(t ) represents the unresolved scales and hence (I π h )u(t ) 2 ξ h E(ξ) dξ (where E(ξ) denotes the energy distribution over the wave numbers) leading to u(0) 2 π h u(t ) 2 0 ν 1 2 u 2 dt ξ 1 + h E(ξ) dξ 0 ν 1 2 u 2 dt

18 Scale separation and the energy inequality The continuous case: The discrete case: u(0) 2 π h u(t ) 2 0 ν 1 2 u 2 dt u h (0) 2 u h (T ) 2 0 ν 1 2 u h 2 dt ξ 1 + h E(ξ) dξ 0 ν 1 2 u 2 dt = 1 + D

19 Scale separation and the energy inequality The continuous case: The discrete case: u(0) 2 π h u(t ) 2 0 ν 1 2 u 2 dt u h (0) 2 u h (T ) 2 0 ν 1 2 u h 2 dt We conclude that if u h π h u is to hold then ξ 1 + h E(ξ) dξ 0 ν 1 2 u 2 dt ξ D h E(ξ) dξ 0 ν 1 2 u 2 dt = 1 + D

20 Scale separation and the energy inequality The continuous case: The discrete case: u(0) 2 π h u(t ) 2 0 ν 1 2 u 2 dt u h (0) 2 u h (T ) 2 0 ν 1 2 u h 2 dt We conclude that if u h π h u is to hold then ξ 1 + h E(ξ) dξ 0 ν 1 2 u 2 dt ξ D h E(ξ) dξ 0 ν 1 2 u 2 dt = 1 + D Remark: for standard Galerkin D = 0!

21 Scale separation and the energy inequality Definition in 2D: E(ξ) ξ û(ξ) 2 In 2D there holds for isotropic decaying turbulence:e(ξ) ξ 3 (Kraichnan, 1967). If ξ h h 1 is in the inertial range where E(ξ) ξ 3 then Assuming ξ 2 v D C ξv ξ 3 dξ 2 0 ν 1 2 u 2 dt C ξh ξv 2 0 ν 1 2 u 2 dt. ξ h negligeable we expect D h 2

22 Numerical Results: Turek benchmark Re = 100 flow around a cylinder, P1/P1 NoDOFs dt C Dmax C Lmax St P D O(h α ) lower upper

23 Numerical Results: Re = mixing layer SLIP CONDITION PERIODIC BOUNDARY σ PERIODIC BOUNDARY SLIP CONDITION Unit square, u = 1, σ = 1 28, ν = Re σ = Lesieur et al. proposed this problem as a model case for decaying 2D turbulence. They showed numerically that E(ξ) decays between ξ 4 and ξ 3 for the streamwise velocity component (Fourier transform only in the x-variable). we expect: c 1 h 3 < D < c 2 h 2 to be consistent with Lesieur and D h 2 to be consistent with Kraichnan.

24 Numerical Results: Re = mixing layer P1 el. D O(h α D ) J(u h, p h ) P2 el. D O(h α D ) J(u h, p h ) E E E E E E-6 The convergence of D implies E(ξ) ξ 3 coherent with the scaling law of Kraichnan and with the numerical results of Lesieur.

25 Conclusions and outlook Face oriented interior penalty methods work for incompressible flow at high Reynolds number. Interaction with turbulence? Future work focuses on complex flow problems such as: Incompressible flow in 3D at high Reynolds number (turbulence) Viscoelastic flow Freesurface flow Compressible flow

26 Mixing layer, Reynolds 10000, P1/80 80, P2/32 32, P2/ , t=50,80,100

27 Mixing layer, Reynolds 10000, P1/ , P2/32 32, P2/ , t=50,80,100

28 P2/P2, 32 32, t=20,30,50,70,80,100,120,140,200

29 P2/P2, , t=20,30,50,70,80,100,120,140,200

30 P1/P1, 40 40, t=20,30,50,70,80,100,120,140,200

31 P1/P1, 80 80, t=20,30,50,70,80,100,120,140,200

32 P1/P1, , t=20,30,50,70,80,100,120,140,200