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The Euler equations for multiphase compressible flow in conservation form. Simulation of shock-bubble interactions. (English) Zbl 1028.76050
Summary: The Euler equations, together with an equation of state, govern the motion of an inviscid compressible fluid. Here, a new equation of state for volumes containing both gas and liquid is derived; this allows the Euler equations for two substances, here air and water, to be expressed in pure conservation form. This in turn allows simulation of shocks in water interacting with small bubbles of air as the meniscus no longer needs to be tracked explicitly. Extension to three space dimensions is shown to be straightforward. A test case showing how a shock wave in water interacts with a small (two-dimensional) air bubble is presented. Simulations of a shock wave interacting with two air bubbles, and a small multiphase region (comprising \(50\%\) water and \(50\%\) air by volume) are then given.

MSC:
76T10 Liquid-gas two-phase flows, bubbly flows
76N15 Gas dynamics (general theory)
76M20 Finite difference methods applied to problems in fluid mechanics
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[1] Bourne, N.K.; Field, J.E., Shock-induced collapse of single cavities in liquids, J. fluid mech., 244, 225, (1992)
[2] Campbell, A.W.; Davis, W.C.; Travis, J.R., Shock initiation of detonation in liquid explosives, Phys. fluid, 4, 498, (1961)
[3] Carrier, G.F., Shock waves in a dusty gas, J. fluid mech., 4, 376, (1958) · Zbl 0082.40305
[4] Chock, D.P., A comparison of numerical methods for solving the advection equation, part 2, Atmos. environ., 19, 571, (1985)
[5] Clarke, J.F., Numerical computation of two-dimensional unsteady detonation waves in high-energy solids, J. comput. phys., 106, 215, (1993) · Zbl 0770.76040
[6] Delius, M., Extracorporeal shock waves act by shock wave-gas bubble interaction, Ultrasound med. biol., 24, 1055, (1998)
[7] Ding, Zhong; Gracewski, S.M., The behaviour of a gas cavity impacted by a weak or strong shock wave, J. fluid mech., 309, 183, (1996) · Zbl 0870.76036
[8] Fedkiw, R.P.; Anslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. comput. phys., 152, 457, (1999) · Zbl 0957.76052
[9] Grove, J.W.; Menikoff, R., Anomalous reflection of a shock wave at a fluid interface, J. fluid mech., 219, 313, (1990)
[10] Hankin, R.K.S., Heavy gas dispersion over complex terrain, (1997)
[11] Hankin, R.K.S., LIQUIDEE: modelling of bund overtopping, (1998)
[12] Hankin, R.K.S.; Britter, R.E., TWODEE: the health and safety Laboratory’s shallow layer model for dense gas dispersion-part 1, mathematical basis and physical assumptions, J. hazardous mater., 66, 211, (1999)
[13] Hirt, C.W., Heuristic stability for finite-difference equations, J. comput. phys., 2, 39, (1968) · Zbl 0187.12101
[14] M. J. Ivings, Wave propagation through gases and liquids, Ph.D. Thesis, Manchester Metropolitan University, 1997.
[15] Jenny, P., Correction of conservative Euler solvers for gas mixtures, J. comput. phys., 132, 91, (1997) · Zbl 0879.76059
[16] Kameda, M.; Matsumoto, Y., Shock waves in a liquid containing small gas bubbles, Phys. fluids, 8, 322, (1996)
[17] Karni, S., Multicomponent flow calculations by a consistent primitive algorithm, J. comput. phys., 112, 31, (1994) · Zbl 0811.76044
[18] Karni, S., Hybrid multifluid algorithms, SIAM J. sci. comput., 17, 1019, (1996) · Zbl 0860.76056
[19] Kodama, T.; Takayhama, K., Dynamic behavior of bubbles during extracorporeal shock-wave lithotripsy, Ultrasound med. biol., 24, 723, (1998)
[20] Mazel, P.; Saurel, R.; Lorand, J.-C.; Butler, P.B., A numerical study of weak shock wave propagation in a reactive bubbly liquid, Shock waves, 6, 287, (1996) · Zbl 0894.76032
[21] Menikoff, R.; Plohr, B.J., Riemann problem for fluid flow of real materials, Rev. mod. phys., 61, 75, (1989) · Zbl 1129.35439
[22] Mulder, J.; Osher, S.; Sethian, J.A., Computing interface motion in compressible gas dynamics, J. comput. phys., 100, 209, (1992) · Zbl 0758.76044
[23] Olim, M.; Van Dongen, M.E.H.; Kitamura, T.; Takayama, K.T., Numerical simulation of the propagation of shock waves in open-cell porous foams, Int. J. multiphase flow, 20, 557, (1994) · Zbl 1134.76629
[24] Pike, J., Analytic solutions for dusty shock waves, Am. inst. aeronaut. astronaut., 32, 979, (1994) · Zbl 0805.76030
[25] Roache, P.J., Computational fluid dynamics, (1982)
[26] Saurel, R.; Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. comput. phys., 150, 425, (1999) · Zbl 0937.76053
[27] Saurel, R.; Abgrall, R., A simple method for compressible multifluid flows, SIAM J. sci. comput., 21, 1115, (1999) · Zbl 0957.76057
[28] Shyue, K.-M., An efficient shock-capturing algorithm for compressible multicomponent problems, J. comput. phys., 142, 208, (1998) · Zbl 0934.76062
[29] Sugimura, T.; Tokita, K.; Fujiwara, T., Nonsteady shock wave propagating in a bubble-liquid system, Report fac. sci. tech. meijo univ (Japan), 24, 67, (1984)
[30] Tan, M.J.; Bankoff, S.G., Propagation of pressure waves in bubbly mixtures, Phys. fluids, 27, 1362, (1984) · Zbl 0545.76084
[31] Ton, V.T., Improved shock-capturing methods for multicomponent and reacting flows, J. comput. phys., 128, 237, (1996) · Zbl 0860.76060
[32] Watanabe, M.; Prosperetti, A., Shock waves in dilute bubbly liquids, J. fluid mech., 274, 349, (1994) · Zbl 0825.76395
[33] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks (review article), J. comput. phys., 54, 115, (1984) · Zbl 0573.76057
[34] Young, J.B., The fundamental equations of gas-droplet multiphase flow, Int. J. multiphase flow, 21, 75, (1995) · Zbl 1134.76719
[35] Zalesak, S.T., Fully multidimensional flux-corrected transport algorithms for fluids, J. comput. phys., 31, 335, (1979) · Zbl 0416.76002
[36] Zalesak, S.T., High order “ZIP” differencing of convective terms, J. comput. phys., 40, 497, (1981) · Zbl 0468.76080
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