Suitability of LPS for Laminar and Turbulent Flow
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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
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