A Domain Decomposition Method for Large Scale Simulations of - PowerPoint PPT Presentation

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion A Domain Decomposition Method for Large Scale Simulations of Two-phase Flows with Moving Contact Lines Li Luo 1 , Qian Zhang 1 , Xiao-Ping Wang 1 , Xiao-Chuan Cai 2 1 Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong 2 Department of Computer Science, University of Colorado Boulder, Boulder, USA DD23 July 8, 2015 Jeju Island, Korea 1 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Outline Introduction 1 Model 2 Numerical scheme 3 Domain decomposition 4 Numerical experiments 5 Parallel performance 6 Conclusion 7 2 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Outline Introduction 1 Model 2 Numerical scheme 3 Domain decomposition 4 Numerical experiments 5 Parallel performance 6 Conclusion 7 2 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Outline Introduction 1 Model 2 Numerical scheme 3 Domain decomposition 4 Numerical experiments 5 Parallel performance 6 Conclusion 7 2 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Outline Introduction 1 Model 2 Numerical scheme 3 Domain decomposition 4 Numerical experiments 5 Parallel performance 6 Conclusion 7 2 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Outline Introduction 1 Model 2 Numerical scheme 3 Domain decomposition 4 Numerical experiments 5 Parallel performance 6 Conclusion 7 2 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Outline Introduction 1 Model 2 Numerical scheme 3 Domain decomposition 4 Numerical experiments 5 Parallel performance 6 Conclusion 7 2 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Outline Introduction 1 Model 2 Numerical scheme 3 Domain decomposition 4 Numerical experiments 5 Parallel performance 6 Conclusion 7 2 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Outline Introduction 1 Model 2 Numerical scheme 3 Domain decomposition 4 Numerical experiments 5 Parallel performance 6 Conclusion 7 3 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Two-Phase Flow Background of two-phase flow Liquid-vapor interface Liquid-liquid interface 4 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Two-Phase Flow Moving contact line problems When the fluid-fluid interface intersects the solid wall, it creates a moving contact line. 5 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Phase-field Consider two different fluids with densities ρ 1 and ρ 2 . Define the phase-field  1 , for fluid 1 φ ( x ) = ρ 1 − ρ 2  = 0 , at interface ρ 1 + ρ 2 − 1 , for fluid 2  6 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Cahn-Hilliard Theory Free energy functional: Equilibrium in 1D: ǫφ zz + 1 � [ 1 2 ǫ ( ∇ φ ) 2 + 1 ǫ ( φ − φ 3 ) = 0 , F Ω ( φ ) = ǫ f ( φ )] d Ω , Ω � z � f ( φ ) = − 1 2 φ 2 + 1 4 φ 4 φ ( z ) = tanh √ 2 ǫ 0.4 1 0.35 0.8 0.3 0.6 0.4 0.25 0.2 f( φ ) 0.2 φ 0 −0.2 0.15 −0.4 0.1 −0.6 −0.8 0.05 −1 0 −1.5 −1 −0.5 0 0.5 1 1.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 φ z 7 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Outline Introduction 1 Model 2 Numerical scheme 3 Domain decomposition 4 Numerical experiments 5 Parallel performance 6 Conclusion 7 8 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Model A coupled Cahn-Hilliard and Navier-Stokes system is used to model the MCL problem, as follows: ∂φ ∂ t + v · ∇ φ = L d ∆ µ, in Ω , (2.1) Re ρ [ ∂ u ∂ t + ( u · ∇ ) u ] = −∇ p + ∇ · [ η D ( u )] + B µ ∇ φ, in Ω , (2.2) ∇ · u = 0 , in Ω . (2.3) Here µ = − ǫ ∆ φ − φ/ǫ + φ 3 /ǫ is the chemical potential, ǫ is the ratio between interface thickness ξ and characteristic length L ; density ρ = 1 + φ + λ ρ 1 − φ 2 , viscosity 2 η = 1 + φ + λ η 1 − φ 2 , λ ρ = ρ 2 /ρ 1 and λ η = η 2 /η 1 are density and viscosity ratios; 2 u = ( u x , u y , u z ) where u x , u y , u z are velocities along x , y , z directions, D ( u ) = ∇ u + ( ∇ u ) T is the rate of stress tensor. 9 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Model The motion of the contact line at solid boundaries can be described by the generalized Navier boundary condition (GNBC) [Qian et. al, 03, 06] which evaluates the velocity as: [ L s l s ] − 1 u slip τ 1 = B L ( φ ) ∂ τ 1 φ/η − n · D ( u ) · τ 1 , (2.4) [ L s l s ] − 1 u slip τ 2 = B L ( φ ) ∂ τ 2 φ/η − n · D ( u ) · τ 2 , (2.5) √ 3 cos θ surf 2 sin ( π here L ( φ ) = ǫ∂ n φ + ∂γ wf ( φ ) /∂φ , and γ wf ( φ ) = − 2 φ ) ; slip length s l s = 1 + φ 1 − φ + λ l s 2 . τ 1 and τ 2 are two unit tangent directions along the solid surface, 2 τ 1 · τ 2 = 0. In addition, a relaxation boundary condition is imposed on the phase function ∂φ ∂ t + u τ 1 ∂ τ 1 φ + u τ 2 ∂ τ 2 φ = −V s [ L ( φ )] , (2.6) together with the following impermeability conditions: u n = 0 , ∂ n µ = 0 . (2.7) 10 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Outline Introduction 1 Model 2 Numerical scheme 3 Domain decomposition 4 Numerical experiments 5 Parallel performance 6 Conclusion 7 11 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Discretization Discretization in time: a semi-implicit scheme 1 Cahn-Hilliard equaiton: nonlinear terms and high order derivative impose severe constrains on time step length and difficulties for finite-element discretizations — Separate into two equations of φ and µ — Convex splitting method [Eyre, 98] Navier-Stokes equations: variable density as a coefficient — A pressure stabilized scheme [Gao and Wang, 12] further decouples the velocity and pressure — A pressure Poisson equation is to be solved Discretization in space: a piecewise linear continuous finite element method 2 W h = { w h ∈ C 0 (Ω) ∩ H 1 (Ω) : w h | T ∈ P 1 ( T ) or Q 1 ( T ) , ∀ T ∈ T h } , 0 (Ω)] 3 : u h | T ∈ P 1 ( T ) 3 or Q 1 ( T ) 3 , ∀ T ∈ T h } . U h = { u h ∈ [ C 0 (Ω) ∩ H 1 12 / 35

Introduction Model Numerical scheme Domain decomposition Numerical experiments Parallel performance Conclusion Numerical scheme Step 1 : Solve the Cahn-Hilliard equation using a convex-splitting method: find ( φ n + 1 , µ n + 1 ) ∈ W h × W h , such that for ∀ w h ∈ W h , h h ( φ n + 1 − φ n  h h , w h ) + ( u n h · ∇ φ n h , w h ) = −L d ( ∇ µ n + 1 , ∇ w h ) ,   h  δ t        , ∇ w h ) + s , w h ) + 1  h ) 3 − ( 1 + s )( φ n ( µ n + 1 , w h ) = ǫ ( ∇ φ n + 1 ǫ ( φ n + 1 ǫ (( φ n (3.1) h ) , w h ) h h h √   ( φ n + 1 − φ n  + � [ 1 2 cos ( π  h h + u n τ 1 , h ∂ τ 1 φ n h + u n τ 2 , h ∂ τ 2 φ n 6 π cos θ surf 2 φ n  h ) − h )   s  V s δ t   α ( φ n + 1 − φ n  +˜ h )] , w h � . h Step 2 : Update ρ n + 1 , η n + 1 and l n + 1 : s ) = 1 + φ n + 1 + ( λ ρ , λ η , λ ls ) 1 − φ n + 1 ( ρ n + 1 , η n + 1 , l n + 1 . (3.2) s 2 2 13 / 35