Part IV: Numerical schemes for the phase-filed model

Transcription

1 Part IV: Numerical schemes for the phase-filed model Jie Shen Department of Mathematics Purdue University IMS, Singapore July 29-3, 29

2 The complete set of governing equations Find u, p, (φ, ξ) such that ρ(u t + (u )u) + p = µ( u + t u) λ ( φ φ); u = ; φ t + (u )φ = γ( φ f(φ) + ξ(t)), d φdx = ; dt Ω (ρ, µ) = + φ 2 (ρ, µ ) + φ 2 (ρ 2, µ 2 ). Remark: No energy law for this system. 2

3 Time discretization of the full system ρ nun+ u n µ n D(u n+ ) + p n = expl. nonl. terms; δt ( ρ n ψn+, q) = ( un+ u n, q), δt p n+ = ψ n+ + p n + ν u n+ ; φ n+ φ n γ( C s δt η 2 I)φn+ = γξ k+ + expl. nonl. terms, φ k+ dx = φ k dx; Ω Ω (ρ n+, µ n+ ) = + φn+ 2 (ρ, µ ) + φn+ (ρ 2, µ 2 ). 2 3

4 Summary: Only a sequence of elliptic equations needs to be solved at each time step: The variable-coefficient term (µ n u n+ ) can be approximated by µ u n+ (µ n µ) u n, where µ is some average of µ(φ); 4

5 the same trick can not be applied to ( ρ n ψn+, q) = ( un+ u n, q). δt For problems with relatively small density ratio, one can use the so called Boussinesq approximation: replacing ρ by a constant and accounting the density difference through a gravity force. Volume of φ is conserved for all time. Only limited stability and error analysis were available for the constant density case. 5

6 Second-order accurate schemes can be constructed easily. The above scheme is very easy to implement as the main building blocks are elliptic/fast Poisson solvers. However, two challenges remains: The elliptic solve for the pressure becomes very expensive as the density ratio becomes large. The time step is mainly constraintly by the explict treatment of the surface tension term. 6

7 The new formulation with energy law Let σ = ρ. Find u, p, (φ, ξ) such that σ(σu) t + (ρu )u + (ρu)u + p 2 = µ( u + t u) λ ( φ φ); u = ; φ t + (u )φ = γ( φ f(φ) + ξ(t)), d φdx = ; dt Ω (ρ, µ) = + φ 2 (ρ, µ ) + φ 2 (ρ 2, µ 2 ). 7

8 The gauge-uzawa scheme for the new model σ n+σn+ ũ n+ σ n u n + ρ n (u n )ũ n+ + δt 2 ( (ρn u n ))ũ n+ µ ũ n+ + µ s n + λ ρ n+ ρ n+ =, ũ n+ Ω = ; ( ρ n+ ψn+ ) = ũ n+, n ψ n+ = ; u n+ = ũ n+ + ρ n+ ψn+, s n+ = s n ũ n+ ; 8

9 ρ n+ ρ n + (ũ n+ )ρ n+ γ( ρ n+ g(ρ n+ )) =, δt n ρ n+ Ω =. Thanks to the energy law for the density based phase-field model, we are able to prove that the above scheme is unconditionally stable. In practice, we treat the nonlinear terms explicitly. Second-order version can be constructed. Still need to solve a pressure equation with variable coefficients. 9

10 To avoid solving a pressure equation with variable coefficients, we consider a pressure-stabilized formulation of NSE (Rannacher 92, S. 93): ρ ɛ (u ɛ t + (u ɛ )u ɛ ) = µ u ɛ p ɛ + ρ ɛ f, ρ ɛ t + (u ɛ )ρ ɛ =, with or u ɛ ɛ p ɛ =, u ɛ ɛ p ɛ t =, p ɛ n Ω =, p ɛ t n Ω =. Notice that only a pressure Poisson equation is involved.

11 A new scheme for the density based phase-field model: σ n+σn+ u n+ σ n u n + ρ n (u n )u n+ δt µ n u n+ + p n + λ ρ n+ ρ n+ =, u n+ Ω = ; p n+ = ρ min δt un+, n p n+ Ω = ; ρ n+ ρ n + (u n+ )ρ n+ γ( ρ n+ g(ρ n+ )) =, δt n ρ n+ Ω =.

12 An improved scheme based on pressure-correction: ρ n+ u n+ ρ n u n + ρ n (u n )u n+ + δt 2 ( (ρn u n ))u n+ µ n u n+ + (p n + ψ n ) + λ ρ n+ ρ n+ =, u n+ Ω = ; ψ n+ = χ δt un+, n ψ n+ Ω =, p n+ = p n + ψ n+ ; ρ n+ ρ n δt + (u n+ )ρ n+ γ( ρ n+ g(ρ n+ )) =, n ρ n+ Ω =. 2

13 Several remarks: Second-order schemes based on pressure-correction can be constructed. For problems with large density ratios, the schemes based on pressure-correction are as accurate and much more efficient, when compared with the gauge-uzawa method. 3

14 Moving mesh method r-refinement Goal: redistribute points, through a smooth mapping, to resolve thin interfaces/regions with large gradients. Approach: Given a function u(x; t), it amounts to find a suitable smooth mapping ξ = ξ(x; t) such that v(ξ; t) := u(x(ξ; t), t) is well-behaved u(x) in Ω P u(x) in Ω C 4

15 An effective way to find such a mapping is to min x(ξ;t) Ω c + β 2 v 2 ξ dξ which can be approximated by solving a nonlinear moving mesh PDE (Huang, Ren & Wang, Ceniceros & Hou, Du et al. 6): x d t = τ (a i a j ) ξ i( + β 2 ξ v 2 x ξ j) with a i = x ξ i. i,j= 5

16 With the coordinate transform, the elliptic operator with constant coefficients will become one with variable coefficients, but can still be solved using a suitable problem with constant coefficients as preconditioner. Typically a moving mesh strategy can reduce the number of points needed in each direction by a factor of 3-4, leading to significantly savings, particularly in 3-D. We are currently developing an adaptive (Legendre polynomial based) multi-wavelet method which is capable to further reduce the total number of points. 6

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19 Two-phase Newtonian flow with large density ratios: air bubble in water 9

20 Figure : ρ =.6 and ρ max = 995 and homogenous viscosity µ =.86 at t =.5,.75,,.5,.75, 2. 2

21 2.5 IP PM NIP GUM 2.5 IP PM NIP GUM 2.5 IP PM NIP GUM IP PM NIP GUM 2.5 IP PM NIP GUM 2.5 IP PM NIP GUM

22 Figure 3: ρ =.6, µ =.86 and ρ 2 = 995, µ 2 =.7977 at t =.5,.75,,.5,.75, 2. 22

23 2.5 IP PM NIP GUM 2.5 IP PM NIP GUM 2.5 IP PM NIP GUM IP PM NIP GUM 2.5 IP PM NIP GUM 2.5 IP PM NIP GUM

24 Figure 5: inverted heart shape in experiment by Akers & Belmonte 6: air bubble rising in a polymeric fluid 24

25 Mixture of nematic liquid crystals and a Newtonian fluid Let φ = representing the liquid crystal drop (represented by director n) and φ = representing the Newtonian fluid. Mixing energy density: f mix (φ, φ) = λ 2 φ 2 + λ 4η 2(φ2 ) 2, Bulk energy density: f bulk = K [ 2 n : ( n)t + ( n 2 ) 2 ] 4δ 2, 25

26 Anchoring energy density: f anch = A 2 (n φ)2, Total energy: F (φ, n, φ, n) = Ω f mix + + φ 2 f bulk + f anch where ( + φ)/2 is the volume fraction of the nematic component. 26

27 Governing equations φ t + v φ = γ 2δF (Cahn-Hilliard) or, δφ φ t + v φ = γ δf (Allen-Cahn) + a Lagrange multiplier, δφ n t + v n = γ δf 2 δn, v =, ( ) v ρ t + v v = p + [µ( v + v T ) + τ e] + g(ρ), where γ is the interfacial mobility and γ 2 determines the relax- 27

28 ation time of n, with [ ] δf δφ = λ 2 φ + φ(φ2 ) η 2 δf δn = K [ ( ) + φ 2 n + 2 f bulk A [(n φ)n], + + φ 2 (n 2 ] )n δ 2 A(n φ) φ, τ e = λ( φ φ) K + φ 2 ( n) ( n)t A(n φ)n φ. 28

29 29 Figure 6: A Newtonian bubble rising in a nematic fluid (Zhou, Feng, Yue, Liu & S. 7)

30 Nematic liquid crystal drops in a Newtonian matrix under simple shear Figure 7: Newtonian drop in a Newtonian matrix: Cahn-Hilliard phase equation 3

31 3 Figure 8: Liquid crystal drop in a Newtonian matrix: Top, Cahn-Hilliard; bottom: Allen-Cahn

32 Figure 9: Liquid crystal drop in a Newtonian matrix (Cahn-Hilliard): left figure with larger viscosity 32

33 Concluding remarks The mixture of two incompressible fluids can be described by a flexible and robust energetic phase field model. An alternative phase-field model based on the density is proposed: it is particularly suitable for problems with large density ratios and it admits an energy law. Some efficient numerical approaches for solving the coupled system are proposed: a stablized semi-implicit scheme for the phase equation; 33

34 new projection type schemes which only require solving regular Poisson equation for the pressure; a moving mesh spectral discretization in space. Advantages and challenges: The proposed numerical schemes are easy to implement as they are all based only on elliptic solves with regular Poisson equation for the pressure. It can be easily extended to handle some non-newtonian flows such as liquid crystal flows and visco-elastic flows,... 34

35 It is possible to treat multiphase (> 2) flows by introducing multiple phase functions. Treating all nonlinear terms explicitly lead to a somewhat restrictive time step constraint. We are currently working on ways to efficiently treat some nonlinear terms implicitly to improve the stability. Thank you! 35