FEM-Level Set Techniques for Multiphase Flow --- Some recent results

Transcription

1 FEM-Level Set Techniques for Multiphase Flow --- Some recent results ENUMATH09, Uppsala Stefan Turek, Otto Mierka, Dmitri Kuzmin, Shuren Hysing Institut für Angewandte Mathematik, TU Dortmund Page

2 Motivation Accurate, robust, flexible and efficient simulation of multiphase problems, particularly in 3D, is still a challenge Specific Application: Monodisperse laminar droplets Problem: Results are due to many parameters, i.e., density (ratio), viscosity (ratio), rheological behaviour, surface tension, flow conditions, Efficient CFD techniques (implicit FEM-Multigrid-Level Set solver in FEATFLOW, dynamically adapted meshes, parallel/hpc techniques) Benchmarking/Validation via experiments Vorarbeiten mit FEATFLOW Page 2

3 Governing Equations The incompressible Navier Stokes equation v T ρ + v v ( μ[ v + ( unknown v) ]) + p t interphase v = 0 location Interphase tension force μ = μ D v f, ST Γ, = σκ ρ n, = ρ Γ κ = n on ( ( ) ) ( ) Dependency of physical quantities μ = μ ( D( v) ) Γ = f ST + ρg Page 3

4 Required: Efficient Flow Solver Discretization: a) Navier-Stokes: FEM Q 2 /Q /P b) Levelset: DG-FEM P Crank-Nicholson scheme in time in space + stabilization (TVD,EO-FEM) no stabilization! Main features of the FeatFlow approach: Parallelization based on domain decomposition High order discretization schemes Use of unstructured meshes Multigrid linear solver FCT & EO stabilization techniques Adaptive grid deformation UCHPC Page 4

5 Required: Efficient Interphase Tracking Levelset method ( smooth distance function) φ + v φ = 0 t Benefits: Provides an accurate representation of the interphase Provides other auxiliary quantities (normal, curvature) Allows topology changes Treatment of viscosity, density and surface tension without explicit representation of the interphase Adaptive grid advantageous, but not necessary Problems: It is not conservative mass loss Needs to be reinitialized to maintain its distance property Higher order discretization? Page 5

6 Problems and Challenges Steep gradients of the velocity field and of physical properties near the interphase (oscillations!) Reinitialization (smoothed sign function, artificial movement of the interphase ( mass loss), how often to perform?) Mass conservation (during levelset advection and reinitialization) Representation of interphacial tension: CSF, Line Integral, Laplace-Beltrami, Phasefield, etc., explicit or implicit treatment? Page 6

7 Stabilization Techniques Steep changes of physical quantities: ) Elementwise averaging of the physical properties (prevents oscillations): ( x) ρ 2, μe = xμ + ( x) μ 2 ρ = xρ x is the volume fraction e + 2) Flow part: Extension of nonlinear stabilization schemes (AFC, TVD) for the momentum equation for LBB stable element pairs. 3) Interphase tracking part with DG-FEM: Flux limiters satisfying LED requirements. Page 7

8 Reinitialisation Alternatives Brute force (introducing new points at the zero level surfaces) Fast sweeping (applying advancing front upwind type formulas) Fast marching Algebraic Newton method Hyperbolic PDE approach many more.. Globally defined normal vectors Maintaining the signed distance function by PDE reinitialization φ φ + u φ = S( φ) u = S( φ) τ φ φ = Problems: What to do with the sign function at the interphase? (smoothing?) How often to perform? (expensive steady state) Page 8

9 Problems and solutions Our reinitialization is performed in combination of 2 ingredients: ) Elements intersected by the interphase are modified w.r.t. the slope of the distance distribution ( Parolini trick ) such that φ = 2) Far field reinitialization: realization is based on the PDE approach ( FBM ), but it does not require smoothening of the distance function! In addition: continuous projection of the interphase (smoothening of the discontinuous P based distance function) L 2 projection L2 projection φ φ φ P Q P Page 9

10 Problems and solutions Mass conservation Must be satisfied on the continuous and discrete level as well! (In work) Replacement of: ( φ ) φ ρ + v φ = 0, by + = t t ( ρ( φ) v) 0 Nonlinear PDE! Remark: No stabilization! (to avoid numerical diffusion) distance function should be smooth anyway Good agreement with experimental measurements (in terms of droplet size, frequency) Page 0

11 Benefits of DG-FEM for Levelset possibility of reduction of the computational domain (only for levelset) mass preserving (interphase preserving!) reinitialization exact localization of the interphase (polygons) exact evaluation of volume fractions relatively simple parallelization Layering concept Parallelization based on domain decomposition Interphase representation by polygons Page

12 Surface Tension Continuum Surface Force (CSF): Transformation of the surface integrals to volume integrals with the help of a regularized Dirac delta function δ Requires globally defined normals and curvature Reduction of spurious oscillations L2 projection n n continuous normal field ( x ε ) f = σκ nδ, ST κ P Q = n Q dx dx Q continuous curvature field Levelset distribution Distribution of the smoothed σκδ interphacial tension force ( ) Q Resulting pressure distribution Page 2

13 Surface Tension - Alternative Treatments Phase Field (PF) approach f ST = σ ( φ φ ) PF No reconstruction of normals and curvature needed Fully implicit treatment is possible Also possible for LS (?) PF Page 3

14 Surface Tension - Alternative Treatments Semi implicit CSF formulation based on Laplace-Beltrami f ST = = σκnˆ vδ ( Γ, x) dx σ x Γ = ( vδ ( Γ, x) ) dx = σ ( Δx ) ( vδ ( Γ, x) ) σ x Application of the semi-implicit time integration yields Γ Γ dx vδ ( Γ, x) dx n+ n x n+ Γ = x n Γ + Δt u fst = σ δε ( dist( Γ, x) ) Δt=.00 Δt=0.50 Δt=0.25 Δt n+ σ δ ε ~ x n Γ n ( dist( Γ, x) ) v dx u n+ v dx CSF Advantages Relaxes capillary time step restriction Optimal for FEM-Levelset approach CSF LBI Page 4

15 Bubble Benchmarks t=0 t=3.0 t=.0 t=0.5 t=2.5 t=2.0 t=.5 CFX TP2D FreeLIFE MooNMD Comsol Fluent Comsol TP2D FreeLIFE MooNMD Comsol Fluent Fluent 0 FreeLIFE MooNMD TP2D 3 Benchmark quantities Center of mass Mean rise velocity x Circularity c/ = c U c = = P P a b 2 x dx 2 dx u dx 2 2 dx πd = P b a Hysing, S.; Turek, S.; Kuzmin, D.; Parolini, N.; Burman, E.; Ganesan, S.; Tobiska, L.: Quantitative benchmark computations of two-dimensional bubble dynamics, International Journal for Numerical Methods in Fluids, in press, DOI: 0.002/fld.934, 2009 Page 5

16 Adaptive HPC Techniques CELL processor (PS3), 28 GFLOP/s, 3.2 GHz GPU (NVIDIA GTX 285): 240 GHz,.242 GHz memory bus (60 GB/s).06 TFLOP/s 40 GFLOP/s, 40 GB/s on GeForce GTX (.4) GFLOP/s on core of Xeon E5450 Page 6

17 Droplet Jetting Application Next step: Extension to liquid-gas systems Page 7

18 Outlook Page 8