Discretization Techniques for Surface Tension Forces in Fully Conservative Two-Phase Flow Finite Volume Methods

Transcription

1 Discretization Techniques for Surface Tension Forces in Fully Conservative Two-Phase Flow Finite Volume Methods Matthias Waidmann 1, Stephan Gerber 2 Michael Oevermann 3, Rupert Klein 1 1 Freie Universität Berlin, Institut für Mathematik AG Geophysical Fluid Dynamics 2 Federal Institute for Materials Research and Testing 3 Chalmers University of Technology, Gothenburg, Sweden Department of Applied Mechanics

2 Outline Basic Concepts (Waidmann) Conservative Continuous Surface Stress Approach (Waidmann) Conservative Well-Balanced Sharp Interface Approach (Gerber) Summary (Gerber) SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 2

3 Basic Concepts fully conservative Finite Volume DNS solving space-time integral conservation laws conservative discretization Cartesian grid p q (2d & 3d) sharp interface p q r s projection type predictor-corrector method C++ implementation built on SAMRAI (AMR, parallel), LLNL p q ñ cut grid cells at the sharp interface r s ñ jumps (ρ, p, µ, D...) at the sharp interfaces SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 3

4 Basic Concepts governing equations, leading order system r1s, zero Mach number limit ˆ d 1 mass : ρ dv pρθ u nq ds dt θ Ω ˆ d Ys spec : ρy s dv pρθ u nq ds j s n ds ` 9σ s dv dt θ Ω Ω ˆ d u mom : ρu dv pρθ u nq ds ` Tpuq n ds ` ρg dv p p2q n ds ` σt Γ dl dt θ Ω Ω XΓptq en : 0 pρθ u nq ds ` ρθd dv Ω with ρθd dv dpp0q θ θ dt c 2 dv ` c 2 q ` ÿ j s h s ` ÿ ı h s 9σ s dv Ξ s s Ω Ω Ω vof : d d ρθα dv pρθ u nq α ds ls : dt dt G u G Ω surf : d χ ds χ u n dl D Γptq Γptq χ t Γptq dl ` dt j χ n Γptq ds ΩXΓptq XΓptq XΓptq ΩXΓptq r2s [1] Klein; Theor. Comput. Fluid Dyn. (2009) [2] Waidmann et. al.; Proceedings in Mathematics & Statistics 77 (2014) SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 4

5 Continuous Surface Stress Approach space-time integral momentum equation at zero velocity and gravity t n`1 t n`1 ρu n`1 ρu n p p2q n ds dt ` σt Γ dl dt t n t n XΓ σt Γ dl T σ Γ t Γ dl Γ T σ Γ ds pt σ Γ δ Γq dv T σ Γ δpd Γp x, tqq n da XΓ XΓ ΩXΓ Ω T σ Γ : σ pid n Γ n Γ q [2] Lafaurie et. al.; Journal of Computational Physics (1994) [4] Tryggvason et. al.; Camb. Univ.Press (2011) [3] Jacqmin; AIAA (1996) [5] Yeoh et. al.; Butterworth-Heinemann (2010) SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 5

6 Continuous Surface Stress Approach space-time integral momentum equation at zero velocity and gravity SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 6

7 Continuous Surface Stress Approach Continuous Surface Stress r2s r3s r4s r5s ρ u n`1 ρ u n t n`1 t n`1 p p2q n da dt ` fσ dv dt t n t n Ω Advantages: ρ u n`1 ρ u n t n`1 t n p p2q Id T σ Γ δpd Γp x, tqq n da dt (1) T σ Γ : σ `Id n Γ n Γ pressure and surface stress tensor treated as a single unit r9s no explicit curvature evaluation (no discrete second derivatives) fully conservative momentum equation Ñ fluxes arbitrary locally varying surface tension coefficient σ [9] Francois et al., A balanced-force algorithm..., JCP 213, 2006 SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 7

8 Continuous Surface Stress Approach Discretization t n`1 t n`1 t n`1 T σ t n Γ δ d n da dt fa δ d da dt «fa G δ r1s t n t n ɛ G pgq da dt (2) force on control volume boundary: fa : T σ Γ n σ `I n Γ n Γ n σ ` n nγ ` nγ n ˆ σ n G ˆ G G G n Approximation of Dirac distribution for non-distance functions G: δ d δ ` x ˆ G x Γ δ r1s pdq «δ r1s G ˆ «δɛ r1s G G (3) (4) ˆ δɛ r1s G G : 1 ˆ G ɛ Ψɛ G G ɛ G Ψ ɛ G pgq G δ r1s pgq (5) ɛ G " 1 ξ 1D linear hat function : Ψ ɛ pξq : ɛ 1 sgnpξq ξ ξ ă ɛ ɛ 0 ξ ě ɛ (6) SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 8

9 Continuous Surface Stress Approach space-time integral momentum equation at zero velocity and gravity SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 9

10 Continuous Surface Stress Approach Discretization, 2D t n`1 t n`1 T σ t n Γ δ Γ n da dt t n t n`1 ˆ 1 fa G 1 sgnpgq G da dt t n ɛ G ɛ G t n`1 lomon F A E ɛ `1 sgnpgq lomon G Eɛ da dt t n : F : G F G,E,E ÿ ÿs τ nτ uτ,sptq 2D 1ÿ 1ÿ pfq k ij t i y j 1ÿ 1ÿ τ s 0 o τ l τ,sptq i 0 j 0 i 0 j 0 fa G δ r1s pgq da dt ɛ G ŝ pτ,sq ij t i y j dy dt (7) t t t SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 10

11 Continuous Surface Stress Approach Discretization, 2D L T σ t n Γ δ d n da dt ÿ ÿs τ nτ uτ,sptq 2ÿ 2ÿ ˆ q ij pfq k ij, ŝ pτ,sq ij t i y j dy dt τ s 0 o τ l τ,sptq i 0 j 0 ÿ pn τ o τ q 1 ÿs τ 2ÿ 2 i ÿ w pτ,sq ij nτ i τ 2 oj τ (8) s 0 i 0 j 0 t n`1 y U G>e G>0 O G<0 G<-e G = G< o n q pτ,sq pa`cqpb`dq : 1 4 e<g u(n) u(o) l(n) l(o) N ˆ 2 a ` c G = 0 t ˆ 2 ˆ b ` d w pτ,sq ij : 1 rψ pi`jq w pτ,sq ija : 1 rζ pi`jqa 2ÿ a 0 a`1 ÿ q pτ,sq pi`jqa w pτ,sq ija rϕpa`1 bqi rϕ bj U ab b 0 U ab : upnq b a`1 b upoq b lpnq a`1 b lpoq ŝ pτ,sq ab k p Fq cd ` ŝ pτ,sq cd pfq k ab ` ŝ pτ,sq ad k p Fq cb ` ŝ pτ,sq cb j pfq k ad SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 11

12 Continuous Surface Stress Approach Discretization, 2D ˆ2 ` m rψ m : m rϕ mn : ˆm ` n n ˆ2 ` m ` n rζ mn : n ~ ζ ~ ζ 2a ~ ζ 1a 0a φ ~ i^ 0 φ ~ i^1 φ ~ i^ ζ ~ 4a ζ ~ 3a ζ ~ 2a ~ ζ 0a ζ ~ 1a φ ~ i^ 0 φ ~ i^ φ ~ i^2 φ ~ i^3 φ ~ i^ SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 12

13 Continuous Surface Stress Approach First Results, 2D SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 13

14 Continuous Surface Stress Approach First Results, 2D SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 14

15 Continuous Surface Stress Approach First Results, 2D vpw σκ σ 1 R (9) R : 3 2, σp x, tq σ 0 1 Ñ vpw 2 3 SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 15

16 Continuous Surface Stress Approach Upcoming Challenges integration into flow solver for density ratios 1 integral conservation law sharp interface conservative finite volume discretization cut Cartesian grid cells continuous approach reducing/minimizing spurious currents due to pressure - surface force balance due to large density ration between the two fluids 3D SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 16

17 Sharp Balanced Surface Stress Approach Conservative Well-Balanced Sharp Interface Approach, 2D momentum balance: d ρu dv p p2q n ds ` dt Ω XΓptq σt Γ dl aim: local second order scheme which handles interfaces forces well balanced AND conservative idea: re-interpretation of the phase wise pressure data with pressure jump as continuous pressure jump field at old time level σtδ Γ da σt ds σ 0 XBΓ s B s A κ n Γ ds ΩXBΓ vpw da SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 17

18 Sharp Balanced Surface Stress Approach Proof of concept for arbitrary and under-resolved cases static case SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 18

19 Sharp Balanced Surface Stress Approach Conservative Well-Balanced Sharp Interface Approach, 2D - dynamic case predictor pressure gradient term: exact space-time integration of double bilinear pressure pressure ansatz functions on cell faces movement of the levelset with possible change of curvature Ñ geometrical usage of information on curvature change neglect the impact of curvature change on pressure data (old time pressure data (implicitly gives vpw σκ)) local extrapolation of pressure data for newly cut cells (care for symmetry) naturally forces of opposed extrapolation in corrector step SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 19

20 Sharp Balanced Surface Stress Approach Conservative Well-Balanced Sharp Interface Approach, 2D - dynamic case predictor surface force term: projection of surface force on cell faces (neglect bilinear information) integration of temporally changed normal forces consider temporal change of interface normal as change of projection on cell face usage of old time pressure data on temporal changing interface (analogous to pressure gradient term handling) pitfall: artificial time level breaks well balancing (earlier separation of two types of in-cell interface movement) remark: No usage of explicit curvature data in predictor step! Ñ fitting procedure instead of finite difference approximation in corrector step SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 20

21 Sharp Balanced Surface Stress Approach projection of surface force term usage of ONE machinery x j`1{2(t) j ` 1{2 t0 t1 y t1 xj`1{2 t 0 x j 1{2 vpw n y dl dt x j 1{2(t) t1 t 0 x xj`1{2 j 1{2 x 0 vpw dx xj 1{2 x 0 vpw dx j dt x j 1{2(t1) x j`1{2(t1) t t x j 1{2(t0) x x j`1{2(t0) x SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 21

22 Sharp Balanced Surface Stress Approach Proof of concept for arbitrary and under-resolved cases dynamic case (1) SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 22

23 Sharp Balanced Surface Stress Approach Proof of concept for arbitrary and under-resolved cases dynamic case (2) SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 23

24 Sharp Balanced Surface Stress Approach Cut-cell Poisson extension / Implementation of space-time integration time using of midpoint rule for spatial integration mid point value and interval length as linear time functions space t2 x2 t 1 ppx, tqdt dx «x 1 p m ptq xptq t dt two opportunities: either difference of old time / new time curvature data as jump cond. OR old time pressure space time integral on rhs and new time curvature data as jump SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 24

25 Summary - well balance scheme currently: loc. sec. order scheme for a well balanced and conservative surface force handling (predictor) loc. sec. order scheme for time dep. cut cell solver upcoming: frictionless case for soap bubble oscillations usage of extrapolated pressure node / inexact levelsets splitting of correction fluxes for velocity corrections incremental vs. total corrector comparison tangential force handling (dynamic) investigation of damping behavior of the scheme finally: case with friction and coefficient jumps (density etc.) SPP1506, Darmstadt; October 6, 2015; Waidmann, Gerber, Oevermann, Klein (FU Berlin) Page 25