Suitability of LPS for Laminar and Turbulent Flow

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Numerical and Applied Mathematics VMS 2015 10th International

1 Suitability of LPS for Laminar and Turbulent Flow Daniel Arndt Helene Dallmann Georg-August-Universität Göttingen Institute for Numerical and Applied Mathematics VMS th International Workshop on Variational Multiscale and Stabilized Finite Elements February

2 Overview 1 Introduction 2 Stabilized Spatial Discretization 3 Numerical Results Convergence Results LPS for Laminar Flow LPS for Turbulent Flow 4 Summary Daniel Arndt LPS for Laminar and Turbulent Flow 2

3 Overview 1 Introduction 2 Stabilized Spatial Discretization 3 Numerical Results Convergence Results LPS for Laminar Flow LPS for Turbulent Flow 4 Summary Daniel Arndt LPS for Laminar and Turbulent Flow 3

4 Setting Navier Stokes Equations u t + (u )u ν u + p = f in Ω (0, T ] Ω R d bounded polyhedral domain Averaged Navier Stokes Equations u t u = 0 in Ω (0, T ] + (u )u τ ν u + p = f in Ω (0, T ] u = 0 in Ω (0, T ] Reynolds subgrid tensor τ = uu T uu T = (u u)(u u) T Daniel Arndt LPS for Laminar and Turbulent Flow 4

5 Overview 1 Introduction 2 Stabilized Spatial Discretization 3 Numerical Results Convergence Results LPS for Laminar Flow LPS for Turbulent Flow 4 Summary Daniel Arndt LPS for Laminar and Turbulent Flow 5

6 Weak Formulation Find U = (u, p) : (0, T ) V Q = [H0 1(Ω)]d L 2 0 (Ω), such that where ( t u, v) + A G (u, U, V) = (f, v) V = (v, q) V Q A G (w; U, V) := a G (U, V) + c(w; u, v) a G (U, V) := ν( u, v) (p, v) + (q, u) c(w, u, v) := ((w )u, v) ((w )v, u) 2 Daniel Arndt LPS for Laminar and Turbulent Flow 6

7 Local Projection Stabilization Idea Separate discrete function spaces into small and large scales Add stabilization terms only on small scales. Notations and prerequisites Family of shape-regular macro decompositions {M h } Let D M [L (M)] d denote a FE space on M M h for u h. For each M M h, let π M : [L 2 (M)] d D M be the orthogonal L 2 -projection. κ M = Id πh u fluctuation operator Averaged streamline direction u M R d : u M C u L (M), u u M L (M) Ch M u W 1, (M) Daniel Arndt LPS for Laminar and Turbulent Flow 7

8 Assumptions Assumption (A.1) Consider FE spaces (V h, Q h ) satisfying a discrete inf-sup-condition: inf sup q Q h \{0} v V h \{0} ( v, q) v L 2 (Ω) q L 2 (Ω) β > 0 V h div := {v h V h ( v h, q h ) = 0 q h Q h } {0} Assumption (A.2) The fluctuation operator κ M = id π M provides the approximation property (depending on D M and s {0,, k}): κ M w L 2 (M) Ch l M w W l,2 (M), w W l,2 (M), M M h, l s. A sufficient condition for (A.2) is P s 1 D M. Daniel Arndt LPS for Laminar and Turbulent Flow 8

9 Assumptions Assumption (A.3) Let the FE space V h satisfy the local inverse inequality v h L2 (M) Ch 1 M v h L2 (M) v h V h, M M h. Assumption (A.4) There are (quasi-)interpolation operators j u : V V h and j p : Q Q h such that for all M M h, for all w V [W l,2 (Ω)] d with 2 l k + 1: w j u w L 2 (M) + h M (w j u w) L 2 (M) Ch l M w W l,2 (ω M ) and for all q Q H l (M) with 2 l k on a suitable patch ω M M: q j p q L2 (M) + h M (q j p q) L 2 (M) Ch l M q W l,2 (ω M ) v j u v L (M) Ch M v W 1, (M) v [W 1, (M)] d. Daniel Arndt LPS for Laminar and Turbulent Flow 9

10 Stabilization Terms LPS Streamline upwind Petrov-Galerkin (SUPG) s h (w h ; u, v) := τ M (w M )(κ M ((w M )u), κ M ((w M )v)) M M M h grad-div t h (w h ; u, v) := M M h γ M (w M )( u, v) M Daniel Arndt LPS for Laminar and Turbulent Flow 10

11 Stability and Existence Stabilized Problem Find U h = (u h, p h ) : (0, T ) V h div Q h, s.t. V h = (v h, q h ) V h div Q h ( tu h, v h ) + A G (u h ; U h, V h ) + (s h + t h )(u h ; u h, v h ) = (f, v h ) Stability Define for V V Q the norm V 2 LPS := ν v 2 + (s h + t h )(V, V). Then the following stability result holds: u h (t) 2 L 2 (Ω) + t 0 U h (s) 2 LPS ds u h (0) 2 L 2 (Ω) + 3 f 2 L 2 (0,T ;L 2 (Ω)) Corollary (using the generalized Peano theorem) discrete solution u h : [0, T ] V h div for the LPS model. Daniel Arndt LPS for Laminar and Turbulent Flow 11

12 Theorem (A., D., Lube 2014) Assume a solution according to u [L (0, T ; W 1, (Ω)) L 2 (0, T ; [W k+1,2 (Ω))] d, tu [L 2 (0, T ; W k,2 (Ω))] d, p L 2 (0, T ; W k,2 (Ω)). Then we obtain for e h = u h j uu: e h 2 L (0,t);L 2 (Ω)) + C M h 2k M t 0 t e C G (u)(t τ) 0 e h (τ) 2 LPSdτ [ min ( d ν, 1 ) p(τ) 2 W γ k,2 (ω M ) M + (1 + νre 2 M + τ M u M 2 + dγ M ) u(τ) 2 W k+1,2 (ω M ) + τ M u M 2 h 2(s k) M u(τ) 2 W s+1,2 (ω M ) + tu(τ) 2 W k,2 (ω M ) with Re M := h M u L (M), s {0,, k} and the Gronwall constant ν C G (u) = 1 + C u L (0,T ;W 1, (Ω)) + Ch u 2 L (0,T ;W 1, (Ω)) ] dτ Daniel Arndt LPS for Laminar and Turbulent Flow 12

13 Theorem (A., D., Lube 2014) Assume a solution according to u [L (0, T ; W 1, (Ω)) L 2 (0, T ; [W k+1,2 (Ω))] d, tu [L 2 (0, T ; W k,2 (Ω))] d, p L 2 (0, T ; W k,2 (Ω)). Then we obtain for e h = u h j uu: e h 2 L (0,t);L 2 (Ω)) + C M h 2k M t 0 t e C G (u)(t τ) 0 e h (τ) 2 LPSdτ [ min ( d ν, 1 ) p(τ) 2 W γ k,2 (ω M ) M + (1 + νre 2 M + τ M u M 2 + dγ M ) u(τ) 2 W k+1,2 (ω M ) + τ M u M 2 h 2(s k) M u(τ) 2 W s+1,2 (ω M ) + tu(τ) 2 W k,2 (ω M ) with Re M := h M u L (M), s {0,, k} and the Gronwall constant ν C G (u) = 1 + C u L (0,T ;W 1, (Ω)) + Ch u 2 L (0,T ;W 1, (Ω)) ] dτ Daniel Arndt LPS for Laminar and Turbulent Flow 12

14 Theorem (A., D., Lube 2014) Assume a solution according to u [L (0, T ; W 1, (Ω)) L 2 (0, T ; [W k+1,2 (Ω))] d, tu [L 2 (0, T ; W k,2 (Ω))] d, p L 2 (0, T ; W k,2 (Ω)). Then we obtain for e h = u h j uu: e h 2 L (0,t);L 2 (Ω)) + C M h 2k M t 0 t e C G (u)(t τ) 0 e h (τ) 2 LPSdτ [ min ( d ν, 1 ) p(τ) 2 W γ k,2 (ω M ) M + (1 + νre 2 M + τ M u M 2 + dγ M ) u(τ) 2 W k+1,2 (ω M ) + τ M u M 2 h 2(s k) M u(τ) 2 W s+1,2 (ω M ) + tu(τ) 2 W k,2 (ω M ) with Re M := h M u L (M), s {0,, k} and the Gronwall constant ν C G (u) = 1 + C u L (0,T ;W 1, (Ω)) + Ch u 2 L (0,T ;W 1, (Ω)) ] dτ Daniel Arndt LPS for Laminar and Turbulent Flow 12

15 Choice of Parameters and Projection Spaces We achieve a method of order k provided τ M u M 2 h 2(s k) M νre 2 M C h M C ν u L (M) h 2(k s) M C τ M τ 0 u M 2 max{ 1 γ M, γ M } C γ M = γ 0 Examples for suitable projection spaces One-Level: Q k /Q k 1 /Q t, P k /P k 1 /P t, Q k /P (k 1) /P t t k 1 Two-Level: P k /P k 1 /P t, Q k /Q k 1 /Q t, Q k /P (k 1) /P t t k 1 Daniel Arndt LPS for Laminar and Turbulent Flow 13

16 Overview 1 Introduction 2 Stabilized Spatial Discretization 3 Numerical Results Convergence Results LPS for Laminar Flow LPS for Turbulent Flow 4 Summary Daniel Arndt LPS for Laminar and Turbulent Flow 14

17 Overview 1 Introduction 2 Stabilized Spatial Discretization 3 Numerical Results Convergence Results LPS for Laminar Flow LPS for Turbulent Flow 4 Summary Daniel Arndt LPS for Laminar and Turbulent Flow 15

18 Couzy Testcase Choose f such that the following pair is a solution in Ω = [0, 1] 2 : cos ( π 2 x) sin ( π 2 y) sin ( πz) u(x) = sin(πt) sin ( π 2 x) cos ( π 2 y) sin ( πz) sin ( π 2 x) sin ( π 2 y) cos ( πz) ( π ) ( π ) ( π ) p(x) = π sin 2 x sin 2 y sin 2 x sin ( πz) sin (πt) Daniel Arndt LPS for Laminar and Turbulent Flow 16

19 Couzy Testcase H 1 (u) error Q + 2 /Q 1/Q 1 γ M = 1,τ M = 0 Q + 2 /Q 1/Q 1 γ M = 1,τ M = 1/u 2 max,m Q 2 /Q 1 /Q 1 γ M = 1,τ M = 0 Q 2 /Q 1 /Q 1 γ M = 1,τ M = 1/u 2 max,m h 1.5 h 2 1/8 1/4 1/ h Figure: Errors for Re = 10 3 using periodic boundary conditions Daniel Arndt LPS for Laminar and Turbulent Flow 17

20 Couzy Testcase L 2 (div u) error Q + 2 /Q 1/Q 1 γ M = 1,τ M = 0 Q + 2 /Q 1/Q 1 γ M = 1,τ M = 1/u 2 max,m Q 2 /Q 1 /Q 1 γ M = 1,τ M = 0 Q 2 /Q 1 /Q 1 γ M = 1,τ M = 1/u 2 max,m h 2 1/8 1/4 1/ h Figure: Errors for Re = 10 3 using periodic boundary conditions Daniel Arndt LPS for Laminar and Turbulent Flow 18

21 Couzy Testcase H 1 (u) error Q + 2 /Q 1/Q 1 γ M = 1,τ M = 0 Q + 2 /Q 1/Q 1 γ M = 1,τ M = 1/u 2 max,m Q 2 /Q 1 /Q 1 γ M = 1,τ M = 0 Q 2 /Q 1 /Q 1 γ M = 1,τ M = 1/u 2 max,m h 1.5 h 2 1/16 1/8 1/4 1/2 1 2 h Figure: Errors for Re = 10 3 using inhomogeneous Dirichlet boundary data Daniel Arndt LPS for Laminar and Turbulent Flow 19

22 Couzy Testcase 10-2 Q + 2 /Q 1/Q 1 γ M = 1,τ M = 0 L 2 (div u) error Q + 2 /Q 1/Q 1 γ M = 1,τ M = 1/u 2 max,m Q 2 /Q 1 /Q 1 γ M = 1,τ M = 0 Q 2 /Q 1 /Q 1 γ M = 1,τ M = 1/u 2 max,m h 1.5 h /16 1/8 1/4 1/2 1 2 h Figure: Errors for Re = 10 3 using inhomogeneous Dirichlet boundary data Daniel Arndt LPS for Laminar and Turbulent Flow 20

23 Overview 1 Introduction 2 Stabilized Spatial Discretization 3 Numerical Results Convergence Results LPS for Laminar Flow LPS for Turbulent Flow 4 Summary Daniel Arndt LPS for Laminar and Turbulent Flow 21

24 Blasius Flow, Boundary Layer Equation Consider the flow over an infinitesimal thin horizontal plate. Then the Navier-Stokes equations simplify to Prandtl s boundary layer equations: 2f (η) + f (η)f (η) = 0 f (0) = 0 f (0) = 0 lim f (η) = 1 x η = y u0 2νx Daniel Arndt LPS for Laminar and Turbulent Flow 22

25 Blasius Flow, ν = u/u 0.6 Re Ω = 10 3, γ = 1, τ M = 0, h = Blasius profile x=0.1 x=0.5 x= η=y (u /(ν x)) 1/2 Daniel Arndt LPS for Laminar and Turbulent Flow 23

26 Choice of the Stabilization Parameter τ M Abb: a) τ M = 0, b) τ M = h 2 / u M 2, c) τ M = h/ u M 2, d) τ M = 1/ u M 2 Daniel Arndt LPS for Laminar and Turbulent Flow 24

27 Choice of the Coarse Space u/u 0.6 Reference profile 0.4 DM = Q1 τm = 1 DM = Q0 τm = DM = Q0 τm = h DM = τm = 1 DM = τm = h η=y (u /(ν x)) 1/2 Figure: Profiles for various coarse spaces D M and τ M at x = 0.1 (left) Profile u for D M = and τ M = 1 (right) Daniel Arndt LPS for Laminar and Turbulent Flow 25

28 Blasius Flow, ν = 10 3, grad-div Refine cell K with midpoint (x, y) if y < δ Daniel Arndt LPS for Laminar and Turbulent Flow 26

29 Blasius Flow, ν = 10 3, grad-div Refine in the boundary layer Daniel Arndt LPS for Laminar and Turbulent Flow 27

30 Blasius Flow, ν = 10 3, grad-div Refine cell K if u max,k u min,k > 1/5 Daniel Arndt LPS for Laminar and Turbulent Flow 28

31 Overview 1 Introduction 2 Stabilized Spatial Discretization 3 Numerical Results Convergence Results LPS for Laminar Flow LPS for Turbulent Flow 4 Summary Daniel Arndt LPS for Laminar and Turbulent Flow 29

32 Energy Cascade E(k, t) = 1 2 k 1 2 k k+ 1 2 û(k, t) û(k, t) Kolmogorov s second hypothesis In the inertial subrange the energy is distributed like E(k, t) = αε 2/3 k 5/3 assuming locally isotropic turbulence. Daniel Arndt LPS for Laminar and Turbulent Flow 30

33 Taylor Green Vortex - Setting Flow in a perodic box [0, 2π] 3 for Re = 10 4 and initial data cos(x) sin(y) sin(z) u 0 = sin(x) cos(y) sin(z) 0 p 0 = 1 (cos(2x) + cos(2y)) (cos(2z) + 2). 16 The initial energy is concentrated on the wave numbers k = (±1, ±1, ±1) E 0 = π 3 1 k=2 Daniel Arndt LPS for Laminar and Turbulent Flow 31

34 Parameter Choice due to the Lilly argument h β τ M (κ(u M u h ), κ(u M v h )) τ M = τ M u M γ Assumption for the Lilly argument ε = τ M κ(u M u h ) 2 E(k, t) = K 0 ε 2/3 k 5/3 κ u h 2 = = α kf kf k 2 E(k) dk = α k c kf τ 2/3 M k c k c k 1/3 ε 2/3 dk κ(u M u h ) 4/3 k 1/3 dk = C h2/3β u M 2/3γ τ M 2/3 κ(u M u h ) 4/3 h 4/3 Daniel Arndt LPS for Laminar and Turbulent Flow 32

35 Parameter Choice due to the Lilly argument h2/3β κ u h 2 = C u M 2/3γ τ M 2/3 κ(u M u h ) 4/3 h 4/3 ( ) 3/2 κ u h 2 u M 2/3γ! τ M = C h 2/3β 4/3 κ(u M u h ) 4/3 = const. ( ) 3/2 κ u h 2 4/3 u M 2/3γ C h 2/3β 4/3 u M 4/3 Therefore γ = β = 1 u h u M γ 2 h 2 β u M γ 1 h 1 β Daniel Arndt LPS for Laminar and Turbulent Flow 33

36 Parameter Choice due to the Lilly argument The LPS-SU stabilization has to satisfy τ M c D M = Q s h s = k c u M τ M h s = k 1 c u M τ M C C u M 2 h u M 2 h u M h s < k 1 c u M τ M C hk s u M 2 C h2 u M 2 Daniel Arndt LPS for Laminar and Turbulent Flow 34

37 Energy Spectrum at t = 9, grad-div 10 0 γ = 1 h = 1 8 γ = 1 h = γ = 1 h = 1 32 k 5/ E k Daniel Arndt LPS for Laminar and Turbulent Flow 35

38 Energy Spectrum at t = 9, grad-div, LPS-SU 10 0 τ M = 1 h = 1 8 τ M = 1 h = τ M = 1 h = 1 32 k 5/ E k Daniel Arndt LPS for Laminar and Turbulent Flow 36

39 Energy Spectrum at t = 9, grad-div, Smagorinsky 10 0 c smag = h = 1 8 c smag = h = c smag = h = 1 32 k 5/ E k Daniel Arndt LPS for Laminar and Turbulent Flow 37

40 Energy Spectrum at t = 9, h = γ = 1 τ M = c smag = k 5/ E k Daniel Arndt LPS for Laminar and Turbulent Flow 38

41 Energy Spectrum at t = 9, h = γ = 1 τ M = c smag = k 5/ E k Daniel Arndt LPS for Laminar and Turbulent Flow 39

42 Energy Spectrum at t = 9, h = γ = 1 τ M = c smag = k 5/ E k Daniel Arndt LPS for Laminar and Turbulent Flow 40

43 Energy Spectrum at t = 9, h = 8 1, u 0 = τ M = 1 τ M = 1 2 u max τ M = u max 2 τ M = 1 u max c smag = k 5/3 E k Daniel Arndt LPS for Laminar and Turbulent Flow 41

44 Overview 1 Introduction 2 Stabilized Spatial Discretization 3 Numerical Results Convergence Results LPS for Laminar Flow LPS for Turbulent Flow 4 Summary Daniel Arndt LPS for Laminar and Turbulent Flow 42

45 Summary The Local Projection Stabilization approach provides Stability and Existence Quasi-optimal error estimates for standard discretizations (e.g. Taylor-Hood) Numerical results confirm analytical estimates: For analytical examples SU stabilization is not necessary, but does not hurt either For non-convex domain SU stabilization prevents development of unphysical oszillations, parameters in the maximal possible range show best results We achieve satisfying results for isotropic turbulence, comparable to Smagorinsky. Improvement of parameter bounds by the Lilly argument. Daniel Arndt LPS for Laminar and Turbulent Flow 43

46 Summary The Local Projection Stabilization approach provides Stability and Existence Quasi-optimal error estimates for standard discretizations (e.g. Taylor-Hood) Numerical results confirm analytical estimates: For analytical examples SU stabilization is not necessary, but does not hurt either For non-convex domain SU stabilization prevents development of unphysical oszillations, parameters in the maximal possible range show best results We achieve satisfying results for isotropic turbulence, comparable to Smagorinsky. Improvement of parameter bounds by the Lilly argument. Daniel Arndt LPS for Laminar and Turbulent Flow 43

47 Thank you for your attention! Daniel Arndt LPS for Laminar and Turbulent Flow 44

Projection Methods for Rotating Flow

Projection Methods for Rotating Flow Daniel Arndt Gert Lube Georg-August-Universität Göttingen Institute for Numerical and Applied Mathematics IACM - ECCOMAS 2014 Computational Modeling of Turbulent and