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日 本 機 械 学 会 論 文 集 (B 編 ) 77 巻 773 号 (2011-1) 論 文 No.10-0610 Ghost Fluid 1, 2, 2 Application of Multigrid Ghost Fluid Method to the Interaction of Shock Waves with Bubbles in Liquids Kazumichi KOBAYASHI 1, Yoshinori JINBO and Hiroyuki TAKAHIRA 1 Division of Mechanical and Space Engineering, Hokkaido University Kita13 Nishi8, Kita-ku, Sapporo, Hokkaido, 060-8628, Japan A new numerical method based on the ghost fluid method was developed for compressible two-phase flows; the idea of adaptive mesh refinement with multigrid was implemented in the method. In the method, interpolation techniques between multiple grids near interfaces were also proposed. The present techniques are effective in diminishing the numerical instability caused by the discontinuity of physical variables across the interfaces. The bubble collapse induced by the interaction of an incident shock with a gas bubble in water was simulated with the present multigrid ghost fluid method. We have succeeded in capturing the fine interface and vortex structure during the collapsing and rebounding stage. The mass conservation is improved with the adaptive mesh refinement combined with the hybrid particle level set method. Also, the interaction of an incident shock wave with a gas bubble near a deformable boundary was simulated successfully with the method in which the motions of three phases, i.e., the gas inside the bubble, the ambient liquid, and the wall material, are taken into consideration. Key Words : Bubble, Shock Wave, Gas-Liquid Two-Phase Flow, Numerical Simulation, Adaptive Mesh Refinement, Multigrid Ghost Fluid Method. 1. (1) (ESWL) (2) Ghost Fluid (3) (4),(5) (4),(5) Lagrange 2010 8 6 1 ( 060-8628 13 8 ) 2 ( 599-8531 1-1) Email: kobakazu@eng.hokudai.ac.jp

(a) r (b) r H z H z W Water Shock Bubble L s L sb W Water Shock Bubble L s L sb L wb R 0 R 0 Wall Fig. 1 Schematic of bubble arrangement. (6),(7) Lagrange Euler () (8) Quirk & Karni (9) Niederhaus (10) Haas & Sturtevant (11) Riemann (12) Takahira (3),(4) ( Ghost Fluid ) ( ( ) ) 2 1 2. 1 (a) ( ) (b) z R 0 (a) W = 16R 0 H = 8R 0 L sb =1.5R 0 L s =6.5R 0 (b) W = 38.2R 0 H = 4R 0 L sb =1.4R 0 L s =25.4R 0 L wb =1.2R 0 ( ) Euler Q t + F r + G = S, (1) z ρ ρu r ρw ρu r Q = ρu r ρw, F = ρu 2 r + p ρu r w, G = ρu r w ρw 2 + p, S = 1 ρu 2 r r ρu r w. (2) E (E + p)u r (E + p)w (E + p)u r t r z ρ u r w r z p E e : E = ρ(e + (u 2 r + w 2 )/2) ( ) ( )

Stiffened Gas (13) p = (γ 1)ρe γπ, (3) γ Π γ = 1.4 Π = 0 Pa γ = 4.4 Π = 6 10 8 Pa γ = 1.459 Π = 112 10 8 Pa Level Set φ (14) Level Set φ t + u φ r r + w φ = 0. (4) z Level Set Level Set (14) φ τ + S(φ 0)( φ 1) = 0, (5) S(φ 0 ) = φ 0 / φ0 2 + δx2 δx ( r = z ) τ φ 0 τ = 0 Hybrid Particle Level Set (HPLS) (15) HPLS 32 Euler 3rd order TVD Runge-Kutta 3rd order ENO-LLF (16) Level Set 3rd order TVD Runge-Kutta 5th order WENO (17) 2 2 Ghost Fluid 2 2 1 Ghost Fluid Ghost Fluid (18) Ghost Fluid Ghost Fluid Ghost Fluid (3),(18) Level Set (Ghost Fluid) φ > 0 1 φ < 0 2 φ > 0 φ < 0 (Ghost Fluid 2) φ < 0 φ > 0 (Ghost Fluid 1) Level Set φ > 0 1 2 φ < 0 2 1 Ghost Fluid Takahira (3) Ghost Fluid Fast Marching (19) Riemann (12) (3) (4) 2 2 2 Ghost Fluid 2(a) Layer 1 Layer 1 Layer 2 Layer 1 Layer 2 Layer 1 1 Layer 2 4 Layer 1 r 1 = z 1 Layer 2 r 2 = z 2 = 0.5 r 1 CFL Layer 1 t 1 Layer 2 0.5 t 1 2(b)

(a) Multigrid Layer 1 r z Layer 2 (b) n step (t=t n) Layer 1 Layer 2 n step (t=t n) Time integration B. C. B. C. Time integration n+1/2 step (t=t n+ t/2) Time integration n+1 step n (t=t + t) Interpolation of physical variables n+1 step n (t=t + t) Fig. 2 Sequence of computations in two layers. 1. t n Layer 1 t n+1 Layer 1 2. Layer 1 t n+1 Layer 1 Layer 2 Layer 2 t n+1/2 3. 2 t n+1 Layer 2 4. Layer 1 Layer 2 Layer 2 Layer 1 Layer 1 5. 1 4 Layer 1 Layer 2 Layer 2 Layer 1 3 4 Ghost Fluid φ > 0 1 φ < 0 2 Layer 2 ( 3 (i, j),(i + 1, j),(i, j + 1),(i + 1, j + 1)) Layer 1 (I,J) 1. Ghost Fluid (Ghost Fluid) ( 10 ) Ghost Fluid (i, j) (i, j + 1) 2 Ghost Fluid 2 (i + 1, j) (i + 1, j + 1) 1 Ghost Fluid 1 Layer 2 ( 1 2) 2. Layer 1 (I,J) Layer 2 (I,J) Layer 1 (I,J) A I,J Layer 2 4 a i, j, a i+1, j, a i, j+1, a i+1, j+1 A I,J = (a i, j + a i+1, j + a i, j+1 + a i+1, j+1 )/4, (6)

(1) Fluid 1 Fluid 2 (2) (3) Fluid 1 Fluid 1 φ > 0 φ < 0 φ > 0 (i, j+1) (i+1, j+1) (i, j+1) (i+1, j+1) Fluid 2 φ < 0 (I, J) (I, J) (i, j) (i+1, j) Interface φ = 0 (i, j) Fluid 2 (i+1, j) Interface φ = 0 (i, j+1) (i+1, j+1) (I, J) (i, j) (i+1, j) Fig. 3 Interpolation from Layer 2 to Layer 1 with ghost fluids. (1) Fluid 1 φ > 0 (I, J+1) Fluid 2 φ < 0 (I+1, J+1) (2) Fluid 1 (I, J+1) (I+1, J+1) (i, j+1) (i+1, j+1) (i, j) (i+1, j) (3) Fluid 1 φ > 0 (i, j+1) Fluid 2 φ < 0 (i+1, j+1) (I, J) (I+1, J) Interface φ = 0 (I, J) (I, J+1) (I+1, J) Fluid 2 (I+1, J+1) (i, j) (i+1, j) Interface φ = 0 (i, j+1) (i+1, j+1) (i, j) (i+1, j) (I, J) (I+1, J) Fig. 4 Interpolation from Layer 1 to Layer 2 with ghost fluids. 3. Layer 1 (I,J) φ (I,J) φ < 0 2 4 Layer 1 Layer 2 Ghost Fluid Layer 1 Layer 1 A Layer 2 (i, j) a i, j a i, j = (9A I,J + 3(A I+1,J + A I,J+1 ) + A I+1,J+1 )/16, (7) Layer 2 (i, j) φ Ghost Fluid 2 2 3 Adaptive Mesh Refinement ( AMR ) (9),(10) Layer 2 Layer 1

(a) Layer i (b) Layer 6 r Hi Hi+1 z L L i,i+1 Layer i+1 W i+1 R L i,i+1 H6 L L B6 Bubble R L B6 W i W 6 Fig. 5 Schematic of multiple computational layers. (i) t/t 0 = 0.0000 (ii) t/t 0 = 0.2240 (iii) t/t 0 = 0.8000 (iv) t/t 0 = 0.9760 (v) t/t 0 = 1.088 (vi) t/t 0 = 1.142 (vii) t/t 0 = 1.158 (viii) t/t 0 = 1.264 (ix) t/t 0 = 1.600 Thin gas layer Fig. 6 Schlieren images for the bubble collapse in Case 5. 5(b) Layer 6 z [z L B 60 z 6 z R B + 40 z 6] r [0, 0.9R 0 ] ( z 6 Layer 6 ) z L B ( ) zr B ( ) z Layer 6 z R B Layer 6 zr 6 LR B6 = zr 6 zr B z L B Layer 6 zl 6 LL B6 = zl B zl 6 Layer 6 ( 5(b) ) 1. LB6 R < 44 z 6 Layer 6 z R 6 4 z 6 LB6 R 44 z 6 48 z 6 2. LB6 L > 36 z 6 Layer 6 z L 6 4 z 6 LB6 L 32 z 6 36 z 6 Layer 6 Layer i (i = 3,4,5) Layer i + 1 Li,i+1 R Layer i Layer i+1 Li,i+1 L LR i,i+1 = 24 z i 28 z i Li,i+1 L = 36 z i 38 z i (i = 3), 32 z i 36 z i (i = 4,5)

2.0 Table 1 Grid spacing z i and increment of time t i for Fig. 1(a). z i ti Wi Hi Layer 1 0.08 8.0 10 4 16 8 Layer 2 0.04 4.0 10 4 10 3.2 Layer 3 0.02 2.0 10 4 6 2 Layer 4 0.01 1.0 10 4 3 1.5 Layer 5 0.005 5.0 10 5 2.5 1.2 Layer 6 0.0025 2.5 10 5 - - 2.0 1.0 layer 4 layer 5 layer 6 1.0 0 0 7.0 8.0 9.0 10.0 7.0 8.0 9.0 10.0 (i) t/t 0 = 0.7648 (ii) t/t 0 = 1.056 2.0 2.0 1.0 1.0 0 0 7.0 8.0 9.0 10.0 7.0 8.0 9.0 10.0 (iii) t/t 0 = 1.142 (iv) t/t 0 = 1.400 Fig. 7 Adaptive mesh for the bubble collapse in Case 5. 3 1 3. R 0 1.0 mm p s =10 8 Pa p 0 =1.013 10 5 Pa Rankine-Hugoniot 1.2 kg/m 3 998.6 kg/m 3 5 1 Layer i W i (Wi = W i /R 0 ) H i (Hi = H i /R 0 ) Layer 1 Layer 1 r1 (= r 1/R 0 ) = z 1 (= z 1/R 0 ) = 0.08 t1 = ( t 1/t 0 ) = 8.0 10 4 t 0 t 0 = R 0 / p/ρ s p = p s p 0 ρ s Layer 2 5 () 1 5(b) Layer 6 r 0.8R 0 5 Case Case 1: Layer 3 Case 2: Layer 1 3 Case 3: Layer 1 4 Case 4: Layer 1 5 Case 5: Layer 1 6 Case 3 5 HPLS (Case 5-WOP (Without Particles) )

(i) t/t 0 = 0.9920 Case 1 (ii) t/t 0 = 1.088 (iii) t/t 0 = 1.152 (iv) t/t 0 = 1.184 (v) t/t 0 = 1.232 Case 2 Case 3 Case 4 Case 5 Case 5 - WOP Fig. 8 Comparison of bubble shapes for Cases 1 5. Case 5 () 6 7 6 ( 6(ii)) ( 6(iii)) ( 6(iv)) ( 6(v)) 6(vi) ( 6(vii)-(ix)) ( 7 8 ) 7 8 9(a) 9(b) Γ R R = (3V /4π) (1/3) V = H(φ)dV Heaviside H 0 for φ < 0, (liquid phase), H(φ) = (8) 1 for φ > 0, (gas phase).

(a) R/R 0 Case 2 Case 3 Case 4 Case 5 Case 5 - WOP t/t 0 Case 2 Case 3 Case 4 Case 5 Case 5 - WOP (b) Γ t/t 0 Fig. 9 Time histories of (a) equivalent bubble radii and (b) circulation inside the bubble. baroclinic vorticity () 8 Case 5-WOP Case 1 5 Case 1 Case 2 Case 2 Case 1 1/16 9(a) 9(b) HPLS Case 5-WOP 8 Level Set Case 10 Case 2 5 () 11 HPLS Case 3 5 δm δm = M M 0 M M = ρh(φ)dv M 0 8 11 HPLS

δm/m 0 Case 2 Case 3 Case 4 Case 5 Case 5 - WOP t/t 0 Fig. 10 Comparison of relative error of mass. δm/m 0 Case 3 - WOP Case 4 - WOP Case 5 - WOP t/t 0 Fig. 11 Comparison of relative error of mass (Cases 3 5 without HPLS). HPLS ( 10) HPLS Case 5 Case 4 Case 5 12 Case 5 HPLS 32 16 8 Case 5 8 HPLS HPLS 3 2 1(b) Ghost Fluid 1703 kg/m 3 3098 m/s 3.2 (4), (5) 13 2 L 12 /R 0 = 20 L 23 /R 0 = 4 Layer 1 t/t 0 = 0.688 Layer 2 t/t 0 = 0.86 Layer 3

8 particles per cell 16 particles per cell δm/m 0 32 particles per cell Fig. 12 Influence of the number of marker particles per cell on mass conservation for Case 5. t/t 0 Layer 1 Water Wall Layer 2 r H 1 H 3 H 2 Layer 3 z Bubble L 12 L 23 W 3 W 2 W 1 Fig. 13 Multiple computational layers across a deformable wall boundary. Table 2 Grid spacing z i and increment of time t i for Fig. 1(b). z i ti Wi Hi Layer 1 0.02 t1 = 8.288 10 4 38.2 4 Layer 2 0.01 0.5 t 1 16 2 Layer 3 0.005 0.25 t 1 9 1.5 13 Layer 2 Layer 3 Layer 1 Layer 2 Layer 2 Layer 3 Ghost Fluid 14 R 0 1.0 mm p s =10 8 Pa p 0 =1.013 10 5 Pa 6 (4),(5) 4. Ghost Fluid ( Ghost Fluid ) Ghost Fluid

(i) t/t 0 = 0.0000 (ii) t/t 0 = 0.7460 (iii) t/t 0 = 0.9946 (iv) t/t 0 = 1.094 (v) t/t 0 = 1.160 (vi) t/t 0 = 1.326 (vii) t/t 0 = 1.492 (viii) t/t 0 = 1.724 (ix) t/t 0 = 1.956 Fig. 14 Successive bubble shapes for the shock-bubble interaction near a deformable wall. Level Set Hybrid Particle Level Set Ghost Fluid (No. 21360086) (1) Futakawa, M., Kogawa, H., Hasegawa, S., Naoe, T., Ida, M., Haga, K., Wakui, T., Tanaka, N., Matsumoto, Y. and Ikeda, Y., Mitigation Technologies for Damage Induced by Pressure Waves in High-Power Mercury Spallation Neutron Sources (II) (Bubbling Effect to Reduce Pressure Wave), Journal of Nuclear Science and Technology, Vol. 45, No. 10 (2008), pp. 1041-1048. (2) Kodama, T. and Takayama, K., Dynamic Behavior of Bubbles During Extracorporeal Shock-Wave Lithotripsy, Ultrasound in Medicine & Biology, Vol. 24, No. 5 (1998), pp. 723-738. (3),, Ghost Fluid ( 2 ), B, Vol. 72, No. 723 (2006), pp. 2643-2651. (4) Takahira, H. and Matsuno, T. and Shuto K., Numerical investigations of shock-bubble interactions in mercury, Fluid Dynamics Research, Vol. 40, Nos. 7-8 (2008), pp. 510-520. (5) Takahira, H., Kobayashi, K. and Matsuno, T., Direct Numerical Simulations of Interaction of Strong Shock Waves with Nonspherical Gas Bubbles near Glass Boundaries in Mercury, International Journal of Emerging Multidisciplinary Fluid Science, Vol. 1, No. 2 (2009), pp. 85-99. (6) Unverdi, S. O. and Tryggvason, G., A Front-Tracking Method for Viscous, Incompressible, Multi-fluid Flows, Journal of Computational Physics, Vol. 100 (1992), pp. 25-37.

(7) Zhang, Y. L., Yeo, K. S. and Wang, C., 3D Jet Impact and Toroidal Bubbles, Journal of Computational Physics, Vol. 166 (2001), pp. 336-360. (8) Hyman, J. M. and Li, S., Interactive and Dynamic Control of Adaptive Mesh Refinement with Nested Hierarchical Grids, Los Alamos National Laboratory Report (LA-UR-98-5462), (1998). (9) Quirk, J. J. and Karni, S., On the Dynamics of a Shock-Bubble Interaction, Journal of Fluid Mechanics, Vol. 318 (1996), pp. 129-163. (10) Niederhaus, J. H. J., Greenough, J. A., Oakley, J. G., Ranjan, D., Anderson, M. H. and Bonazza, R., A Computational Parameter Study for the Three-Dimensional Shock-Bubble Interaction, Journal of Fluid Mechanics, Vol. 594 (2008), pp. 85-124. (11) Haas, J.-F. and Sturtevant, B., Interaction of Weak Shock Waves with Cylindrical and Spherical Gas Inhomogeneities, Journal of Fluid Mechanics, Vol. 181 (1987), pp. 41-76. (12) Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, (1997), Springer. (13) Saurel, R. and Abgrall, R., A Simple Method for Compressible Multifluid Flows, SIAM Journal on Scientific Computing, Vol. 21 (1999), pp. 1115-1145. (14) Sussman, M., Peter, S. and Osher, S., A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow, Journal of Computational Physics, Vol. 114 (1994), pp. 146-159. (15) Enright, D., Fedkiw, R., Ferziger, J. and Mitchell I., A Hybrid Particle Level Set Method for Improved Interface Capturing, Journal of Computational Physics, Vol. 183 (2002), pp. 83-116. (16) Shu, C.-W. and Osher, S., Efficient Implementation of Essentially Non-Oscillatory Shock Capturing Schemes II, Journal of Computational Physics, Vol. 83 (1989), pp. 32-78. (17) Jiang, G.-S. and Peng, D., Weighted ENO Schemes for Hamilton-Jacobi Equations, SIAM Journal on Scientific Computing, Vol. 21 (2000), pp. 2126-2143. (18) Fedkiw, R., Aslam, T. and Xu, S., The Ghost Fluid Method for Deflagration and Detonation Discontinuities, Journal of Computational Physics, Vol. 154 (1999), pp. 393-427. (19) Adalsteinsson, D. and Sethian, J. A., The Fast Construction of Extension Velocities in Level Set Methods, Journal of Computational Physics, Vol. 148 (1999), pp. 2-22.

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