Some remarks on grad-div stabilization of incompressible flow simulations

Transcription

1 Some remarks on grad-div stabilization of incompressible flow simulations Gert Lube Institute for Numerical and Applied Mathematics Georg-August-University Göttingen M. Stynes Workshop Numerical Analysis for Singularly Perturbed Problems Dresden University of Technology, November 16-18, 2011 Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

2 Outline 1 Incompressible Navier-Stokes model 2 Numerical analysis of grad-div stabilized Oseen problem 3 Some recent result on limit case γ 4 A (potentially) new approach to parameter design Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

3 Incompressible Navier-Stokes model Navier-Stokes problem Incompressible Navier-Stokes model: Find velocity u, pressure p t u (2νDu) + (u )u + p = f in (0, T] Ω u = 0 in [0, T] Ω u t=0 = u 0 in Ω R d no-slip boundary conditions u = 0 (for simplicity) deformation tensor Du = 1 2 ( u + ( u)t ) viscosity ν (Reynolds number Re = UL ν ). Claude Louis Marie Henri Navier George Gabriel Stokes Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

4 Incompressible Navier-Stokes model Finite element approximation V = [H0(Ω)] 1 3, Q = L0(Ω) 2 := {q L 2 (Ω) : q dx = 0} Ω T h admissible (possibly anisotropic) mesh Ω = K Th K Conforming finite element spaces: V h V, Q h Q Basic Galerkin FE method: find (u h, p h ): [0, T] V h Q h s.t. (v h, q h ) V h Q h ( tu h, v h ) + (2νDu h, Dv h ) + b S(u h, u h, v h ) (p h, v h ) = (f, v h ) (q h, u h ) = 0 with skew-symmetric convective term b S(u, v, w) := 1 [((u )v, w) ((u )w, v)] 2 Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

5 Incompressible Navier-Stokes model Examples of inf-sup stable approximations Inf-sup stable velocity-pressure FE spaces V h Q h V Q β β(h) s.t. inf q h Q h (q h, v h) sup β > 0 v h V h q h 0 v h 0 No additional pressure stabilization required (at least for laminar flows) Taylor-Hood elements: for k N Vh TH Vh TH Q TH h = [P k+1 (T h ) H 1 0(Ω)] d [P k (T h ) C(Ω)] or Q TH h = [Q k+1 (T h ) H 1 0(Ω)] d [Q k (T h ) C(Ω)], k N problems with mass conservation with increasing order k Scott-Vogelius elements: on barycenter refined tetrahedral meshes T h V SV h Q SV h = [P k+1 (T h ) H 1 0(Ω)] d [P disc k (T h ) L 2 0(Ω)], for k d 1 with property [P k+1 (T h )] d P disc k (T h ) strong (pointwise) conservation of mass Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

6 Incompressible Navier-Stokes model Grad-div stabilized Galerkin methods Galerkin FE method with grad-div stabilization: find (u h, p h ): [0, T] V h Q h s.t. (v h, q h ) V h Q h ( tu h, v h ) + (2νDu h, Dv h ) + b S(u h, u h, v h ) (p h, v h ) + (q h, u h ) +(γ u h, v h ) = (f, v h ) classical augmented Lagrangian approach, see e.g. FORTIN/GLOWINSKI [1983] also applied to Maxwell problem introduced by HUGHES/FRANCA [1986] for equal-order interpolation with γ 0(h) Numerical analysis for inf-sup stable interpolation by GELHARD ET AL. [2005] and OLSHANSKII ET AL. [2009]: Choice γ 0(h) is not correct in general case! Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

7 Numerical analysis of grad-div stabilized Oseen problem Grad-div stabilized Oseen model Oseen model: For given b with b = 0, find velocity u, pressure p ν u + (b )u + p + σu = f u = 0 in Ω R d C.W. Oseen Grad-div stabilized Oseen problem: Find (u h, p h ) V h Q h V Q s.t. (v, q) V h Q h : a γ(u h, p h ; v h, q h ) := (ν u h, v h ) + b S(b, u h, v h ) + (σ u h, v h ) (p, v) + (q, u) + K γ K( u h, v h ) K = (f, v h ) Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

8 Numerical analysis of grad-div stabilized Oseen problem Numerical analysis of grad-div stabilized Oseen problem: M. Olshanskii, G. Lube, T. Heister, J. Löwe: Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations, CMAME 198 (2009), pp Well-posedness: a γ(v h, q h ; v h, q h ) 1 2 [v h, q h ] 2 b 1 2 A-priori estimate: [u u h, p p h ] 2 b K T h h 2k K [( ( ν v h γ v h ν + γ K + h2 K b 2 L (K) ν Equilibration of error terms: leads to dynamic parameter version ( ) p H γ K max 0; k (K) ν u H k+1 (K) ) c p q h 2 0 ν + γ max + ν 1 b 2 L (Ω) ] u 2 H k+1 (K) + 1 p 2 H ν + γ k (K) K ) Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

9 Numerical analysis of grad-div stabilized Oseen problem Example 1: Vortex pairs Oseen problem on Ω = (0, 1) 2 with ν = 10 6, σ = 0 and b = u pairs of vortices strong variation of mesh Reynolds number Re K := u,k h K ν [0, h ν ] H 1 - and L 2 -errors vs. scaling parameter γ 0 of grad-div stabilization for Example 1 with σ = 0, h 1 64 Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

10 Numerical analysis of grad-div stabilized Oseen problem Example 1 Dynamic vs. constant parameter design H 1 - and L 2 -errors vs. scaling parameter γ 0 of dynamic grad-div stabilization for Example 1 H 1 - and L 2 -errors vs. scaling constant parameter γ 0 of grad-div stabilization for Example 1 with ν = 10 6, σ = 0 Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

11 Numerical analysis of grad-div stabilized Oseen problem Example 2: Vortex in boundary layer BERRONE [2001] Oseen problem on Ω = (0, 1) 2 with b = u counter-clockwise vortex in boundary layer ν-dependent solution with u 0 ν 0.35 and p 0 ν Errors in H 1 -seminorm and L 2 -norm vs. scaling parameter γ 0 of grad-div stabilization for Example 2 with ν = 10 4 Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

12 Numerical analysis of grad-div stabilized Oseen problem Example 3: Beltrami flow EITHIER ET AL. [1994] Time-dependent Navier-Stokes flow in Ω = ( 1, 1) 3 with ν = 10 6 Series of counter-rotating vortices intersecting one another at oblique angles Diagonally implicit Runge-Kutta method of order 2 with time step t = 1 64 L 2 -error vs. t [0, 1] without stabilization for different h (left) and with grad-div stabilization for fixed h for ν = 10 6 Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

13 Numerical analysis of grad-div stabilized Oseen problem Example 3 (continued) L 2 (Ω)-error (as function of t) for different values of Re = 1 for the Galerkin scheme, i.e. ν without grad-div stabilization (left) and with grad-div stabilization (right) Obvious improvement even with (time-independent) grad-div stabilization γ K = γ 0 Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

14 Numerical analysis of grad-div stabilized Oseen problem Some conclusions Constant value of grad-div parameter γ gives very often improvements of mass conservation and of other relevant norms ( ) p Dynamic design of γ K max 0; H k (K) ν is not feasible u H k+1 (K) Grad-div stabilization is not necessary in case of e.g. in shear flows! p H k (K) ν u H k+1 (K), Poiseuille-type flow: No improvement with grad-div stabilization! Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

15 Some recent result on limit case γ Stationary Navier-Stokes problem on SV-stable meshes M. Case, V. Ervin, A. Lincke, L. Rebholz: A connection between Scott-Vogelius and grad-div stabilized Taylor-Hood FE approximations of the Navier-Stokes equations, SINUM 49 (2011) 4, pp Stationary case: Let T h s.t. SV-elements are inf-sup stable, e.g. barycenter refined meshes. Theorem: CASE ET AL. Theorem mesh and barycenter refined mesh For any sequence (u h ) γi of TH-elements solutions with constant grad-div stabilization, there is a subsequence which converges to a SV-solution as the grad-div parameter γ i, i. The sequences of TH-modified pressure solutions (p h γ i u h ) γi converges to the corresponding SV pressure. Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

16 Some recent result on limit case γ Nonstationary Navier-Stokes on SV-stable meshes Time-dependent Navier-Stokes flow Remark: A similar result is valid for the time-dependent Navier-Stokes problem with Crank-Nicolson semidiscretization in time. Result from CASE ET AL. SINUM 49 (2011), Fig. 3: Flow around a 2D-cylinder with time-dependent inflow, s.t. Re [0, 100] Behavior of u n u 0 in time Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

17 Some recent result on limit case γ Stationary Navier-Stokes problem on regular meshes Consider the following steady state problem on regular meshes T h : ν( z h, v h ) + ((z h )z h, v h ) = (f, v h ) v h Vh 0 (r h, v h ) = ν( z h, v h ) + ((z h )z h, v h ) (f, v h ) v h (Vh TH ) with V 0 h := {v h V h : v h K = 0 K T h } V TH h := {v h V h : ( v h, q h ) = 0 q h Q TH h } Theorem: CASE ET AL., Theorem 4.1 For any sequence (u h, p h ) γi of TH-elements solutions with grad-div stabilization parameters γ i, a subsequence of (u h, p h γ iρ h ) γi with (ρ h,i, v h ) := ( u h,i, v h ) v h (Vh TH ) converges to (z h, r h ) as the grad-div parameter γ i, i. Remark: A similar result is valid for the time-dependent Navier-Stokes problem with Crank-Nicolson semidiscretization in time. Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

18 Some recent result on limit case γ Some further conclusions Easy implementation of grad-div stabilization into existing code with Taylor-Hood elements! Moderate grad-div stabilization may lead to considerable improvement of mass conservation without deterioration of other solution properties! Besides interesting results for constant γ in CASE ET AL. [2011] results for 3D driven cavity show a deterioration of solution properties for γ on regular meshes Optimal choice of γ:... the search of an optimal γ as a trade-off between mass conservation and energy balance in the FE system... OLSHANSKII ET AL. [2009] Potential remedy: Adaptive choice of γ K based on information from velocity gradients, see next chapter Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

19 A (potentially) new approach to parameter design Eddy-viscosity closure of Reynolds-averaged Navier-Stokes model tu i + 3 j(u iu j) + i(p + τ 3 kk 3 ) ν j 2 u i j=1 ν e := { j=1 3 ju j = 0 j=1 3 (2ν ed ij) = f i, i = 1,..., 3 j=1 c 2 B(u), if u 2 u 2 F := 3 i,j=1 ( iuj)2 0 F 0, if u F = 0 B(u) := β 11β 22 β β 11β 33 β β 22β 33 β 2 23, β ij := 3 mu i mu j m=1 Vreman, A.W.: An eddy-viscosity subgrid-scale model for turbulent shear flows: algebraic theory and application. Phys. Fluids 16 (2004), pp Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

20 A (potentially) new approach to parameter design Some observations: Design of eddy viscosity requires information from velocity gradient (similar to classical Smagorinsky model ν e = (C S ) 2 D(u) F) Data can be easily derived within time discretization and linearization Design rotationally invariant for isotropic filter width Careful algebraic considerations of 320 possible cases of local behavior of flow field u: identifies 13 types of laminar flow (shear-type flow) with zero energy transfer to subgrid scales ν e = 0 Such cases may be identified as coherent structures Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

21 A (potentially) new approach to parameter design Application to low-turbulent channel flow at Re τ = 360 Results taken from VREMAN [2004], Fig. 4: Left: Mean streamwise velocity (normalized by u τ Right: Subgrid eddy viscosity ν e (normalized by u τ H) DNS (circles), Vreman (solid), dynamic Smagorinsky (dashed), standard Smagorinsky (dotted), 0-model (triangles) Important: Vreman s eddy viscosity tends to zero in the viscous sublayer at the wall No van Driest damping in boundary layer necessary! Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

22 A (potentially) new approach to parameter design A potentially new parameter design Application to grad-div stabilization Vreman s eddy viscosity ν e models influence of unresolved velocity fluctuations u = u u Grad-div stabilization may be identified as model of unresolved pressure fluctuations p = p p Application of Vreman s eddy viscosity ν e for parameter γ K (eventually with another scaling) would deactivate grad-div stabilization for shear-type flow To be done: Idea can be transfered to variational multiscale approach with local projection stabilization!! Application of VMS framework of RÖHE/ LUBE CMAME [2010] Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27

23 A (potentially) new approach to parameter design Summary. Outlook Improvement of mass conservation with grad-div stabilization of Taylor-Hood elements Further consideration of parameter choice γ Analysis and test of new non-constant parameter design THANKS FOR YOUR ATTENTION! ALL THE BEST TO YOU, DEAR MARTIN! Gert Lube (University of Göttingen) Some remarks on grad-div stabilization Dresden, November 16-18, / 27