Moment of fluid method for multimaterial flows

Transcription

1 Moment of fluid method for multimaterial flows M. Jemison, M. Sussman, M. Shashkov Department of Mathematics, Florida State University October 1, 2012

2 Recent Motivating Work Volume preserving Moment-of-Fluid reconstruction for multiple materials. Moment of Fluid method for deforming boundary problems (Ahn and Shashkov, JCP 2007; Ahn and Shashkov, JCP 2009). variable density cell centered pressure projection (Kwatra, Su, Gretarsson, Fedkiw 2009)

3 Laminar jet - density ratio 819:1 X. Li (UTRC), M. Arienti (UTRC), M. Soteriou (UTRC), M. Sussman (FSU).

4 Turbulent jet spray - density ratio 819:1 X. Li (UTRC), M. Arienti (UTRC), M. Soteriou (UTRC), M. Sussman (FSU).

5 Impinging jets - density ratio 819:1 Arienti (UTRC), Soteriou (UTRC), Sussman (FSU).

6 Flow through diesel nozzle with moving valve

7 Flow through diesel nozzle with moving valve

8 Simulation of Flow through diesel nozzle with moving valve

9 AMR grid

10 Six hole diesel nozzle Pressure (bar)! Figure: Snapshot of a simulation of liquid injection in a Bosch six-hole vertical diesel nozzle. Effective fine grid resolution 640x640x256.

11 Microfluidics Sussman (FSU), Jemison (FSU), Duffy (FSU), Roper (FSU)

12 Vortex Rings Sussman (FSU), Ohta (Tokushima)

13 Vortex Rings Sussman (FSU), Ohta (Tokushima)

14 Background Shock capturing (ENO) - Shu, Osher, JCP 1989 Front capturing (LS) - Sussman, Smereka, Osher, JCP 1994 Conservative LS - Olsson, Kreiss, Zahedi, JCP 2007 Front tracking - Unverdi, Tryggvason, JCP 1992 VOF - Kothe, Brackbill, Zemach, JCP 1992 CLSVOF - Sussman, Puckett; Sussman, Smith, Hussaini, et al. CLSVOF - Stern Particle LS - Enright, Fedkiw Refined LS - Herrmann, Pitsch

15 Grid

16 We propose Semi-Lagrangian technique for multimaterial problems, in contrast to a finite volume technique which is (in 1D, u is constant): ρ i = 1 xi+1/2 ρ(x)dx x x i 1/2 d ρ i (t) dt = f (ρ i+1/2, ρ+ i+1/2 ) f (ρ i 1/2, ρ+ i 1/2 ) x where d/dt discretized using high order TVD preserving RK and ρ i+1/2 derived from a high order ENO or WENO reconstruction.

17 Results motivating SL instead of Finite volume

18 Results motivating MOF/VOF instead of ENO/WENO reconstruction

19 Results motivating MOF/VOF instead of ENO/WENO reconstruction

20 Governing Equations ρ(φ) Du = p + (2µ(φ)D) σκ(φ) H(φ) + ρ(φ)gẑ Dt u = 0 Dφ Dt = 0 { 1 φ 0 H(φ) = 0 φ < 0 ρ(φ) = ρ L H(φ) + ρ G (1 H(φ)), µ(φ) = µ L H(φ) + µ G (1 H(φ)), κ(φ) = φ φ, D = u + ( u)t, 2

21 Splitting of advection terms from the pressure terms Kwatra, Su, Gretarsson (JCP, 2009) ρ t + (ρu) = 0 (ρu) t + (ρuu) = p (ρe) t + (ρue) = (up)

22 Splitting of advection terms from the pressure terms Kwatra, Su, Gretarsson (JCP, 2009) solve the following equations to get ρ n+1, u, and E. These equations are solved using a directionally split method in which ρ at material boundaries is derived from the multimaterial MOF reconstruction. ρ t + (ρu) = 0 (ρu) t + (ρuu) = 0 (ρe) t + (ρue) = 0

23 Splitting of advection terms from the pressure terms p n+1 p = ρc 2 u t u n+1 = u t p ρ n+1 p n+1 ρc 2 t 2 pn+1 ρ n+1 = p ρc 2 t 2 1 t u

24 Splitting of advection terms from the pressure terms u n+1 i+1/2 = u i+1/2 u n+1 i pn+1 i+1 t pn+1 i ρ i+1/2 x = u i t pn+1 i+1/2 pn+1 i 1/2 ρ i x (ρe) n+1 = (ρe) t (u n+1 p n+1 )

25 CLSMOF (illustration of Piecewise linear reconstruction)

26 Multimaterial MOF reconstruction For cell i 1, slope of Red (solid) cut determined first, then slope of green (fluid 1) cut determined second. Blue material occupies remaining unfilled region. For cell i, slope of blue (fluid 2) cut determined first. Green material occupies remaining unfilled region.

27 Multimaterial MOF reconstruction Slope of Red (solid) cut determined first, then slope of green (fluid 1) cut determined second. Blue material occupies remaining unfilled region.

28 MOF optimization problem ˆn points into the material being reconstructed. The starting guess is ˆn = x ref x unfilled x ref x unfilled. {x R 3 ˆn (x x i,j,k ) + b = 0} F ref (ˆn, b) F A (ˆn, b) = 0 E MOF = x ref x A (ˆn, b) 2 E MOF (Φ, Θ ) = f (Φ, Θ ) 2 = min (Φ,Θ) f (Φ, Θ) 2, f : R 2 R 3, f (Φ, Θ) = (x ref x A (Φ, Θ)) Constrained optimization problem solved using the Gauss-Newton method.

29 Gauss-Newton minimization algorithm 0. initial angles: (Φ 0, Θ 0 ). Tolerance: tol = 10 8 x. while not converged 1. find b k (Φ k, Θ k ). 2. find the centroid x k (b k, Φ k, Θ k ) 3. find the Jacobian matrix J k of f evaluated in (Φ k, Θ k ) and f k = f (Φ k, Θ k ) 4. stop if one of the following three conditions is fulfilled: J T k f k tol 10 2 x fk < tol k = 11 else continue 5. solve the linear least squares problem: find s k R 2 such that J k s k + f k 2 = min s R 2 J ks + f k 2 by means of the normal equations. (J T k J ks k = J T k f k) 6. update the angles: (Φ k+1, Θ k+1 ) = (Φ k, Θ k ) + s k 7. k := k + 1

30 Property of MOF reconstruction x ref x A Figure: The reference centroid, x ref, does not coincide with the exact centroid, x A, for a parabolic interface cutting the cell. The difference between the two centroids is proportional to the curvature.

31 Backwards Tracing A = x i 1/2 u i 1/2 t B = x i 1/2 C = x i+1/2 u i+1/2 t

32 Backwards Tracing (before and after) A = x i 1/2 u i 1/2 t B = x i 1/2 C = x i+1/2 u i+1/2 t D = x i 1/2 E = J i (x i 1/2 ) F = x i+1/2

33 Backwards Tracing - conserved variables AKA Eulerian Implicit ρ i,j,k = 1 x y z xi+1/2 yj+1/2 zk+1/2 x i 1/2 WLOG, drop j and k subscripts. ρ n+1 i = y j 1/2 z k 1/2 i+1 i =i 1 Ω D i Ω i ρn i (x)dx x ρ(x, y, z)dzdydx

34 Backwards Tracing - volume fraction AKA Eulerian Implicit (Scardovelli and Zaleski, Le Chenedac and Pitsch) 1 xi+1/2 yj+1/2 zk+1/2 F i,j,k = H(φ(x, y, z))dzdydx x y z x i 1/2 y j 1/2 z k 1/2 H(φ) = 1 if φ 0 (i.e. (x, y, z) inside the material being advected). φ n+1 i = φ n i (J 1 i (x), y, z) F n+1 i = i+1 i =i 1 J i (Ω Di Ω H(φn+1 i ) i x y z (x, y, z))dxdydz

35 Backwards Tracing - centroid AKA Eulerian Implicit x act i,j,k = 1 F i,j,k x y z xi+1/2 yj+1/2 zk+1/2 x i 1/2 y j 1/2 z k 1/2 H(φ) = 1 if φ 0 (i.e. (x, y, z) inside the material being advected). φ n+1 i = φ n i (J 1 i (x), y, z) xh(φ(x, y, z))dzdydx x n+1 i = i+1 i =i 1 J i (Ω Di Ω xh(φn+1 i ) i x y z F n+1 i (x, y, z))dxdydz

36 Backwards Tracing - mapping function Ω i = [x i 1/2, x i+1/2 ] Ω D i = [x i 1/2 u i 1/2 t, x i+1/2 u i+1/2 t] Ω T i = [x i 1/2, x i+1/2 ]

37 Backwards Tracing - mapping function Let J i (x) = Ax + B be a function which maps from cell i departure region to cell i target region. For backwards tracing, J i (x i 1/2 u i 1/2 t) = A(x i 1/2 u i 1/2 t) + B = x i 1/2 J i (x i+1/2 u i+1/2 t) = A(x i+1/2 u i+1/2 t) + B = x i+1/2 A = 1 1 (u i+1/2 u i 1/2 ) t x B = x i 1/2 A(x i 1/2 u i 1/2 t) As long as u t < x/2, the mapping has an inverse.

38 Backwards Tracing - inverse mapping function Let J 1 i (x) be the inverse map which is a function which maps from cell i target region to cell i departure region. If J i (x) = Ax + B, then J 1 i (x) = 1 A x B A Let Ω 1 = J 1 i (Ω) be the inverse map which finds the region Ω 1 which maps to Ω under the definition of J i. In otherwords, Ω 1 = J 1 i (Ω) = [J 1 i Ω = [a, b] (a), J 1 i (b)]

39 Backwards Tracing - mapping function to find target region Likewise, J i (Ω) is defined as the target region that the region Ω maps to: Ω = [a, b] J i (Ω) = [J i (a), J i (b)]

40 Backwards Tracing - interface advection defined by mapping Given a local level set function, After advection, we have φ n i (x, y, z) = n (x x i ) + α, φ n+1 i (x, y, z) = φ n i (J 1 i (x), y, z) = n 1 ( x A B A x i ) + n 2(y y j ) + n 3 (z z k ) = n 1 A (x J i(x i )) + n 2 (y y j ) + n 3 (z z k ) + α

41 Forwards Tracing A = J 1 i 1 (x i 1/2) B = x i 1/2 C = J 1 i (x i+1/2 ) J i 1 and J i define the mappings of cells i 1 and i contents forward, respectively.

42 Forwards Tracing (before and after) A = J 1 i 1 (x i 1/2) B = x i 1/2 C = J 1 i (x i+1/2 ) D = x i 1/2 E = J i (x i 1/2 ) F = x i+1/2

43 Forwards Tracing - conserved variables AKA Lagrangian Explicit ρ n+1 i = ρ i = 1 x i+1 i =i 1 xi+1/2 x i 1/2 ρ(x)dx J 1 i (Ω T i Ω i ) ρn i (x)dx x

44 Forwards Tracing - volume fraction AKA Lagrangian Explicit (Scardovelli and Zaleski, Le Chenedac and Pitsch) F i,j,k = 1 x y z xi+1/2 yj+1/2 zk+1/2 x i 1/2 y j 1/2 z k 1/2 H(φ(x, y, z))dzdydx H(φ) = 1 if φ 0 (i.e. (x, y, z) inside the material being advected). φ n+1 i (x, y, z) = φ n i (J 1 i (x), y, z) F n+1 i = i+1 i =i 1 Ω i Ω Ti H(φn+1 i x y z (x, y, z))dxdydz

45 Forwards Tracing - centroid AKA Lagrangian Explicit x act i,j,k = 1 F i,j,k x y z xi+1/2 yj+1/2 zk+1/2 x i 1/2 y j 1/2 z k 1/2 H(φ) = 1 if φ 0 (i.e. (x, y, z) inside the material being advected). φ n+1 i (x, y, z) = φ n i (J 1 i (x), y, z) xh(φ(x, y, z))dzdydx x n+1 i = i+1 i =i 1 Ω i Ω Ti xh(φn+1 i F n+1 i x y z (x, y, z))dxdydz

46 Forwards Tracing - mapping function Ω i = [x i 1/2, x i+1/2 ] Ω T i = [x i 1/2 + u i 1/2 t, x i+1/2 + u i+1/2 t] Ω D i = [x i 1/2, x i+1/2 ]

47 Forwards Tracing - mapping function Let J i (x) = Ax + B be a function which maps from cell i departure region to cell i target region. For forwards tracing, J i (x i 1/2 ) = Ax i 1/2 + B = x i 1/2 + u i 1/2 t J i (x i+1/2 ) = Ax i+1/2 + B = x i+1/2 + u i+1/2 t A = 1 + (u i+1/2 u i 1/2 ) t x B = x i 1/2 + u i 1/2 t Ax i 1/2 As long as u t < x/2, A is positive.

48 Forwards Tracing - inverse mapping function Let J 1 i (x) be the inverse map which is a function which maps from cell i target region to cell i departure region. If J i (x) = Ax + B, then J 1 i (x) = 1 A x B A Let Ω 1 = J 1 i (Ω) be the inverse map which finds the region Ω 1 which maps to Ω under the definition of J i. In otherwords, Ω 1 = J 1 i (Ω) = [J 1 i Ω = [a, b] (a), J 1 i (b)]

49 Forwards Tracing - mapping function to find target region Likewise, J i (Ω) is defined as the target region that the region Ω maps to: Ω = [a, b] J i (Ω) = [J i (a), J i (b)]

50 Forwards Tracing - advection of interface reconstruction by defined mapping Given a local level set function, φ n i (x, y, z) = n (x x i ) + α, After forward advection, we have φ n+1 i (x, y, z) = φ n i (J 1 i (x), y, z) = n 1 ( x A B A x i ) + n 2(y y j ) + n 3 (z z k ) = n 1 A (x J i (x i )) + n 2(y y j ) + n 3 (z z k ) + α

51 DS Semi-Lagrangian discretization - Strang Splitting 1. x direction; backwards projection. (Eulerian Implicit) 2. y direction; forwards projection. (Lagrangian Explicit) 3. z direction; backwards projection. (Eulerian Implicit) 4. z direction; forwards projection. (Lagrangian Explicit) 5. y direction; backwards projection. (Eulerian Implicit) 6. x direction; forwards projection. (Lagrangian Explicit)

52 2D volume conserved exactly

53 reversible single vortex

54 2D Unsplit MOF, reversible vortex T=1/2 RK2, backwards tracing: Figure: single vortex problem (T = 1/2) midway through the simulation and at the end. Shape deforms back to a circle. Two materials. At t = T = 1/2, symmetric difference error is 7.4E 5 (operator split symmetric difference error: 14.3E 5)

55 2D Unsplit multimaterial MOF, reversible vortex T=1/2 RK2, backwards tracing: Figure: single vortex problem (T = 1/2) midway through the simulation and at the end. Shape deforms back to a circle. Three materials. Initially, Material 1 (red) is the left half of the circle, Material 2 (green) is the right half, and Material 3 (blue) is the space outside the circle. At t = T = 1/2, symmetric difference error is 9.6E 5 for material 1, 10.1E 5 for material 2, and 14.8E 5 for material 3.

56 3D Unsplit MOF, reversible vortex T=3 RK2, backwards tracing: Figure: 3D single vortex problem (T = 3) midway through the simulation and at the end. Shape deforms back to a sphere. Two materials. At t = T = 3, symmetric difference error is (operator split symmetric difference error: )

57 3D Unsplit multimaterial MOF, reversible vortex T=3 RK2, backwards tracing: Figure: single vortex problem (T = 3) midway through the simulation and at the end. Shape deforms back to a sphere. Three materials. Initially, Material 1 (red) is the left half of the sphere, and Material 2 (green) is the right half. At t = T = 3, symmetric difference error is for material 1, for material 2, and for material 3.

58 Rotating Notched disk (Zalesak s problem)

59 Rotating Letter A

60 Rising gas bubble in liquid Figure: Left: condition 1 Right: condition 3 Figure: Left: condition 4 Right: condition 6

61 Surface tension driven vibrations of a drop

62 Oscillating Cylinder

63 Cylinder falling into a pool of liquid

64 Impinging jets - same material

65 Impinging jets - different materials

66 Future work The multimaterial MOF representation enables more accurate simulation of: multimaterial flows with minimal volume fluctuation - capturing corners and filaments. surface tension and contact line effects. simulating transport on deforming surfaces. predicting mass transfer on deforming surfaces. predicting boundary layer effects on underresolved grids. Improvements are still being made to the multimaterial MOF scheme in accelerating the multimaterial MOF reconstruction and unsplit multimaterial MOF advection algorithm. Adding the capability to robustly simulate compressible multiphase/multimaterial flows is one priority now.