Minimal stabilization techniques for incompressible flows

Transcription

1 Minimal stabilization tecniques for incompressible flows G. Lube 1, L. Röe 1 and T. Knopp 2 1 Numerical and Applied Matematics Georg-August-University of Göttingen D Göttingen, Germany 2 German Aerospace Center Göttingen VMS-Symposium 2008 Saarbrücken, June 23-24, 2008 G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

2 Outline 1 Introduction 2 Reduced RBS-sceme for laminar flows 3 Local projection stabilization (LPS) for laminar flows 4 Minimal stabilization for LES of turbulent flows 5 Summary. Outlook F. Brezzi/ M. Fortin: A minimal stabilisation procedure for mixed finite element metods, Numer. Mat. 89 (2001), Goal: Critical review of stabilization tecniques for inf-sup stable pairs (in view of VMS-metods) Acknowledgments: Tanks to G. Matties, J. Löwe, T. Heister and X. Zang. G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

3 Introduction Treatment of nonstationary laminar Navier-Stokes problem Find U = (u, p) V Q := ( H0 1(Ω)) d L 2 0 (Ω) suc tat B ( U, V ) = ( f, v) V = ( v, q ) V Q (1) B ( U, V ) := ν( u, v) + ( (u )u, v ) (p, div v) + (q, div u). Semidiscretise (1) first in time (e.g., BDF(q) or SDIRK metods). Newton-type iteration in eac time step leads to Oseen type problem: Find U = (u, p) V Q suc tat a ( U, V ) = ( f, v ) V = ( v, q ) V Q. a ( U, V ) := ν( u, v) + ( (b )u + σu, v ) (p, div v) + (q, div u) wit given b H(div, Ω) ( L (Ω) ) d, div b = 0, ν > 0 and 1 t σ 0 G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

4 Introduction Galerkin finite element discretization T sape-regular decomposition of polyedral domain Ω Y r T := {v C( Ω) v K P r (K) or Q r (K) K T }, r N FE spaces for velocity/ pressure: V r := [ Y r T H 1 0(Ω) ] d, Q r 1 := Y r 1 T L 2 0(Ω) wit discrete inf-sup compatibility condition Galerkin FEM: Find U = (u, p) W r,r 1 a(u, V) = (f, v) := V r Q r 1, s.t. V = (v, q) W r,r 1 Goal: Robustness w.r.t. ν, σ,. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

5 Introduction Classical residual-based stabilisation (RBS) Residual-based sceme: Find U = (u, p) W r,r 1 a rbs (U, V) = l rbs (V) = V r Q r 1, s.t. V = (v, q) W r,r 1 a rbs (U, V) := l rbs (V) := (f, v) Ω + a(u, V) + X (L Os(u, p), τ K((b )v + q)) K + X (γ K ( u), v) K K T K T {z } SUPG and PSPG stabilisation z X } { K T (f, τ K((b )v + q)) K {z } (div )div stabilisation Oter variants: Galerkin/ Least-squares metod (GaLS): Test wit τ KL Os(v, q) Algebraic subgrid-scale metod ( unusual GaLS): Test wit τ KL Os(v, q) G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

6 Introduction Drawbacks of classical RBS-scemes A-priori analysis: wit empasis on r-dependence see: GL/ G. Rapin M 3 AS 16 (2006) 7 Con s of RBS-scemes Basic drawback: Strong velocity-pressure coupling in SUPG-terms! (Rater) expensive implementation in 3D! Sensitive design of parameters τ K and γ K Non-symmetric form of stabilisation terms Construction of efficient preconditioners for mixed algebraic problem! Problem: Is PSPG-stabilization necessary for div-stable pairs? G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

7 Reduced RBS-sceme for laminar flows Reduced RBS-sceme Reduced RBS sceme: Joint work wit G. Matties, L. Röe (2008) Find U = (u, p) W r,r 1 a red (U, V) = l red (V) = V r Q r 1, s.t. V = (v, q) W r,r 1 a red (U, V) := l red (V) := (f, v) Ω + wit Oseen operator a(u, V) + X (L Os(u, p), τ K((b )v)) K + X (γ K ( u), v) K K T K T {z } SUPG stabilisation z X } { K T (f, τ K((b )v)) K L Os(u, p) := ν u + (b )u + σu + p {z } (div )div stabilisation G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

8 Reduced RBS-sceme for laminar flows Stability analysis Seminorm/ norm: [v] red := ( ν v σ v γ v K τ K b v 2 ) 1 2 0,K V red := ( [v] 2 red + α q 2 ) Conditional stability: γ K γ 0, 0 τ K β µ 2 K ϕ 2 wit ϕ 2 := ν + σc 2 F + b 2 L (Ω) min ( 1 σ ; C2 F ν ) + γ Set: β2 0 ϕ 2 α β2 0 ϕ 2 β S β S (ν, σ, ) : a red (V, W ) inf sup β S > 0 V W V red W red G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

9 Reduced RBS-sceme for laminar flows Convergence result. Parameter design Preliminary a-priori estimate: Let (u, p) [V H r+1 (Ω)] d [Q H r (Ω)]. Stability assumption implies τ K C2 K γ+ν+σc 2 F. U U 2 red C X K + 2 K ν + γ 2(r 1) K p 2 r,k γ + ν + σ 2 K + τ K b 2,K + b 2,K 2 K τ K b 2,K + ν + σ2 K i 2r K u 2 r+1,ω(k) Conclusions for stabilization: SUPG not required if: ν b 2,K 2 K, i.e. max K Re K 1 ν and/ or σ b 2,K time step restriction: 2 K δt 1 σ 1 Div-stabilization useful if: p r,k (ν + γ) u r+1,ω(k) b 2 G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

10 Reduced RBS-sceme for laminar flows Upper bound of critical Galerkin terms Upper bound of a red (, ) requires sarp estimates of Galerkin terms: Discrete-divergence preserving interpolant I GIRAULT/SCOTT [ 03] Standard Lagrangian interpolant J For all W = (w, r ) V Q : (r, (u I u)) = 0 avoids negative power of pressure weigt α X 2 (p J p, w ) C ν + γ 2r K p r,k W red X (b (u I u), w ) C K K 3 b 2,K 2 K τ K b 2,K + ν + σ2 K 2r K u 2 r+1,ω(k) 1 2 W red Action of div- resp. SUPG-stabilization avoid negative powers of ν resp. ν, σ G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

11 Reduced RBS-sceme for laminar flows Can SUPG be avoided for laminar flows? Example: Driven cavity wit stationary solutions SUPG is not required up to Re = Non-stationary approac wit moderately large time steps Driven-cavity problem wit Re = 5, 000: Cross-sections of te solutions for Q 2 /Q 1 witout SUPG/PSPG and Q 2 /Q 2 wit SUPG/PSPG G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

12 Reduced RBS-sceme for laminar flows Role of grad-div stabilization I Examples wit p r,k u r+1,ω(k) Poiseuille flow: p = ν u div-stabilization superfluous Stationary flow around cylinder at ν = (corresponds to Re = 20) 10 1 µ = 0 µ = µ= µ = 0 µ = µ=1 relerr c l (lift) relerr c d (drag) grid level grid level Convergence plots wit Q 2 /Q 1 for lift and drag coefficients G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

13 Reduced RBS-sceme for laminar flows Role of grad-div stabilization II Examples wit p r,k u r+1,ω(k) γ 1 ν ν u + (b )u + p = f wit solution u = b = (sin(πx 1 ), πx 2 cos(πx 1 )) T, p = sin(πx 1 ) cos(πx 2 ) Table: Comparison of different variants of stabilization wit Q 2 /Q 1 and ν = 10 6, σ = 1, = 1 64 SUPG: τ 0 div: γ 0 PSPG: α 0 u u 1 u u 0 u 0 p p E E E E E E E E E E E E E E E E E E E E-4 Problem: General criterion for div-div stabilization or a-posteriori approac?! G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

14 Local projection stabilization (LPS) for laminar flows Two-grid setting and local projection Primal grid T wit FE spaces V r for velocity and pressure Macro grid M = T 2 and Qr 1 wit discontinuous FE spaces D u := {v [L2 (Ω)] d : v M Y r 1 M, M M } D p := {v L2 (Ω) : v M Y k 1 M, M M }, k {0,..., r 2} Local L 2 -projection: π u/p M : L2 (M) D u/p M Global projection: π u/p : L 2 (Ω) D u/p, (π u/p w) M := π u/p M (w M) G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

15 Local projection stabilization (LPS) for laminar flows Local projection stabilisation Fluctuation operators: κ u/p : [L 2 (Ω)] [L 2 (Ω)], κ u/p := id π u/p κ u/p : [L 2 (Ω)] d [L 2 (Ω)] d, κ u/p w := ( (id π u/p )w i ) d i=1 Discrete LPS-problem: Subgrid stabilization as minimal stabilization Find U = (u, p ) W r,r 1 : a lps (U, V) = (f, v) V = (v, q) W r,r 1 a lps (U, V) = a(u, V) + s (U, V). s (U, V) = X M α M( κ u p, κ u q) M {z } pressure stab. + τ M( κ u b u, κ u b v M {z } advection stab. + γ M(κ p u, κp {z v)m } divergence stab. G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

16 Local projection stabilization (LPS) for laminar flows Stability Comparison of LPS to RBS-scemes: Symmetric, non-consistent form of stabilization terms Stabilization (or: subgrid viscosity) term acts only on fine scales (!) V lps := ( [V ] 2 lps + α q 2 ) 1 2 0, α = α(ν, σ) > 0 [V ] lps := ( ν v σ v s (V, V ) ) 1 2 Unconditional (!) stability: = Existence / uniqueness β S β S (ν, ) : inf V W r,r 1 inf V W r,r 1 sup W W r,r 1 sup W W r,r 1 (a + s )(V, W ) [V ] lps [W ] lps 1 (a + s )(V, W ) V lps W lps β S > 0 G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

17 Local projection stabilization (LPS) for laminar flows A-priori error estimate Tecnical ingredient: see talk of G. Matties Construction of special interpolation operator j u s.t. v j u v is L2 -ortogonal to D u for all v V wic preserves te discrete divergence constraint. Preliminary a-priori estimate: Let u [H 1 0 (Ω) Hr+1 (Ω)] d, p L 2 0 (Ω) Hr (Ω). C C(ν, σ, ) : [U U ] 2 lps C M M ( (α M + 2 M ) 2(r 1) M p 2 r,ω γ M + τ M 2r M b u 2 r,ω M M + (ν + σ 2 M + γ M + 2 M τ M + b 2,Mτ M ) 2r M u 2 r+1,ω M ) G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

18 Local projection stabilization (LPS) for laminar flows Parameter design Conclusions for parameter design: PSPG-type term: α M = 0 is possible! SUPG-type term: τ M M b,m Careful analysis SUPG avoidable if σ b 2 or max M Re M 1 ν Div-type term: Equilibration of red velocity and pressure terms wit α M = 0 yields γ M u r+1,ωm p r,ωm Same problem (and strategies) as for reduced RBS-sceme! G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

19 Minimal stabilization for LES of turbulent flows VMS-decomposition of Navier-Stokes problem Navier-Stokes problem : Find U = (u, p) V s.t. u(0) = u 0 and B(U, V) = f, v Decomposition of trial and test spaces: Two-level setting wit FE spaces: V H V V, W H W W V = (v, q) W V = V H (id Π)V }{{} ˆV U = U H + (id Π)U }{{} =:V =:U +Û W = W H (id Π)W }{{} Ŵ V = V H + (id Π)V +ˆV }{{} =:V =:U G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

20 Minimal stabilization for LES of turbulent flows Discrete VMS problem on FEM-level VMS assumptions: A.1 Scale separation: No direct influence of Û on U H A.2 Unresolved scales dissipate energy from small resolved scales via subgrid viscosity model S : V W Discrete VMS problem: B(U, V H ) = f, v H V H V H B(U, Ṽ ) + S (Ũ, Ṽ ) = f, ṽ Ṽ (id Π)W Assumption: S (, V H ) = 0 V H V H W H Compact discrete VMS problem: Find U V : B(U, V) + S (U, V) = f, v V W G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

21 Minimal stabilization for LES of turbulent flows Simplest parametrization of subgrid models: MILES Assumption on subgrid viscosity model: (i) Symmetry: S (U, V) = S (V, U) U, V V W (ii) Coercivity: S (Ũ, Ũ) c Ũ 2 Ũ Ṽ := (id Π)V Variational multiscale metod (VMS): Scale separation: Π as L 2 -ortogonal projection of V onto V H Subgrid viscosity model: local projection stabilization (LPS) S (U, V) = ) (τmd((id H Π)u), D((id Π)v) M T H Parametrization of τm H : based on a-priori analysis for linearized models τ H M = τ 0 u L (M) M, τ 0 =? M corresponds to monotonically integrated LES (MILES) G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

22 Minimal stabilization for LES of turbulent flows Simplest parametrization of turbulence subgrid model Smagorinsky-type VMS-metod: V. Jon et al Scale separation: Π as L 2 -ortogonal projection of V onto V H Subgrid viscosity model: S (U, V) = ) (τm(u)d((id H Π)u), D((id Π)v) M T H τ H M(u) = (C S M ) 2 D(u) F, C S =? M Potential compromise: see also talk by J. L. Guermond S (U, V) = ) (τm(u)d((id H Π)u), D((id Π)v) M M T H ) τm(u) H = min (τ 0 u L (M) M ; (C S M ) 2 D(u) F G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

23 Minimal stabilization for LES of turbulent flows Experience wit low-order finite-volume code Finite-volume code Teta (DLR Göttingen): Vertex-based variant P 1 /P 1 or Q 1 /Q 1 pressure stabilization required Corin decoupling of velocity / pressure Minimal stabilization: Galerkin discretization Subgrid viscosity model: as upwind stabilization dissipates fluctuations Smagorinsky model S (U, V) = ( (C S ) 2 D(u) F D(u), D(v) ), C S =? Experiments by X. Zang (2007, 08) G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

24 Minimal stabilization for LES of turbulent flows Decaying omogeneous turbulence (DHT) Basic calibration model: Decaying omogeneous turbulence Results for turbulent kinetic energy k = 1 2 (u u )2 Careful experimental data for k (Comte/Bellot) Fourier space caracterization of k(t) via energy spectral density E(κ, t) P Cost functional: J(C S) = 1 2 P i 2 M 2 j=1 i=1 E(κ i, C S, t j) E exp(κ i, t j) Calibration of Smagorinsky constant C S and optimized energy spectrum G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

25 Minimal stabilization for LES of turbulent flows Turbulent cannel flow at Re τ = 395 Anisotropic resolution of boundary layer region First-order statistics: mean streamwise velocity U = u e 1 Second-order statistics: turbulent kinetic energy k = 1 (u 2 u )2 and teir normalized variants U + = U/u τ, k + = k/u 2 τ Grid convergence wit N 3 nodes Problem: Wall-resolved LES is almost as expensive as DNS Proper resolution of near-wall region in LES requires Re 2 τ nodes (Bagett et al. 1997) G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

26 Minimal stabilization for LES of turbulent flows DES for backward facing step at Re = Hybrid approac: LES simulation away from boundary layers RANS type universal wall-functions used to bridge near-wall region (y + 40) significant savings in grid points and allows to increase time step G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32

27 Summary. Outlook Summary. Outlook Summary PSPG-type stabilization avoidable for div-stable velocity-pressure pairs SUPG-type stabilization less important for laminar flows New caracterization of div-div stabilization LPS-type approac as minimal stabilization in LES/DES Outlook FEM wit LPS-type LES: MILES vs. turbulent subgrid model Model reduction wit DES THANKS FOR YOUR ATTENTION! G. Lube 1, L. Röe 1 and T. Knopp 2 (University of Göttingen) Minimal stabilization for incompressible flows VMS-Symposium / 32