A fluid-mixture type algorithm for barotropic two-fluid flow problems. (English) Zbl 1115.76344
Summary: Our goal is to present a simple interface-capturing approach for barotropic two-fluid flow problems in more than one space dimension. We use the compressible Euler equations in isentropic form as a model system with the thermodynamic property of each fluid component characterized by the Tait equation of state. The algorithm uses a non-isentropic form of the Tait equation of state as a basis to the modeling of the numerically induced mixing between two different barotropic fluid components within a grid cell. Similar to our previous work for multicomponent problems, see [J. Comput. Phys. 171, No. 2, 678–707 (2001; Zbl 1047.76573)] and references cited therein, we introduce a mixture type of the model system that consists of the full Euler equations for the basic conserved variables and an additional set of evolution equations for the problem-dependent material quantities and also the approximate location of the interfaces. A standard high-resolution method based on a wave-propagation formulation is employed to solve the proposed model system with the dimensional-splitting technique incorporated in the method for multidimensional problems. Several numerical results are presented in one, two, and three space dimensions that show the feasibility of the method as applied to a reasonable class of practical problems without introducing any spurious oscillations in the pressure near the smeared material interfaces.
MSC:
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
Barotropic flow; Tait equation of state; Stiffened gas equation of state; Fluid-mixture model; Wave propagation methodCitations:
Zbl 1047.76573References:
| [1] | Abgrall, R., How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach, J. Comput. Phys, 125, 150-160 (1996) · Zbl 0847.76060 |
| [2] | Abgrall, R.; Karni, S., Computations of compressible multifluids, J. Comput. Phys, 169, 594-623 (2001) · Zbl 1033.76029 |
| [3] | Allaire, G.; Clerc, S.; Kokh, S., A five-equation model for the simulation of interface between compressible fluids, J. Comput. Phys, 181, 577-616 (2002) · Zbl 1169.76407 |
| [4] | T. Barberon, P. Helluy, Finite volume simulation of cavitating flows, Comput. Fluids (2004), to appear; T. Barberon, P. Helluy, Finite volume simulation of cavitating flows, Comput. Fluids (2004), to appear · Zbl 1134.76392 |
| [5] | Barberon, T.; Helluy, P.; Rouy, S., Practical computation of axisymmetrical multifluid flows, Int. J. Finite Volumes, 1, 1-34 (2003) · Zbl 1490.76158 |
| [6] | Batten, P.; Clarke, N.; Lambert, C.; Causon, D., On the choice of wave speeds for the HLLC Riemann solvers, SIAM J. Sci. Comput, 18, 1553-1570 (1997) · Zbl 0992.65088 |
| [7] | Colella, P., Glimm’s method for gas dynamics, SIAM J. Sci. Stat. Comput, 3, 1, 76-110 (1982) · Zbl 0502.76073 |
| [8] | Colella, P.; Glaz, H. M., Efficient solution algorithms for the Riemann problem for real gases, J. Comput. Phys, 59, 264-289 (1985) · Zbl 0581.76079 |
| [9] | Courant, R.; Friedrichs, K. O., Supersonic Flow and Shock waves (1948), Wiley-Interscience: Wiley-Interscience New York · Zbl 0041.11302 |
| [10] | Davis, S. F., Simplified second-order Godunov-type methods, SIAM J. Sci. Stat. Comput, 9, 445-473 (1988) · Zbl 0645.65050 |
| [11] | Edwards, J. R.; Franklin, R. K.; Liou, M.-S., Low-diffusion flux-splitting methods for real fluid flows with phase transition, AIAA J, 38, 1624-1633 (2000) |
| [12] | Fedkiw, R. P.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys, 152, 457-492 (1999) · Zbl 0957.76052 |
| [13] | Fermi, E., Thermodynamics (1956), Dover: Dover New York |
| [14] | Godlewski, E.; Raviart, P.-A., Numerical Approximation of Hyperbolic Systems of Conservation Laws. Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Science, vol. 118 (1996), Springer-Verlag · Zbl 0860.65075 |
| [15] | Hankin, R. K.S., The Euler equations for multiphase compressible flow in conservation form: Simulation of shock-bubble interactions, J. Comput. Phys, 172, 808-826 (2001) · Zbl 1028.76050 |
| [16] | Harten, A.; Lax, P. D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev, 25, 35-61 (1983) · Zbl 0565.65051 |
| [17] | Ivings, M. J.; Causon, D. M.; Toro, E. F., On Riemann solvers for compressible liquids, Int J. Numer. Meth Fluids, 28, 395-418 (1998) · Zbl 0918.76047 |
| [18] | Jenny, P.; Mueller, B.; Thomann, H., Correction of conservative Euler solvers for gas mixtures, J. Comput. Phys, 132, 91-107 (1997) · Zbl 0879.76059 |
| [19] | Kameda, M.; Matsumoto, Y., Shock waves in a liquid containing small gas bubbles, Phys. Fluids, 8, 322-335 (1996) |
| [20] | Kameda, M.; Shimaura, N.; Higashino, F., Shock waves in a uniform bubbly flow, Phys. Fluids, 10, 10, 2661-2668 (1998) |
| [21] | Karni, S., Multicomponent flow calculations by a consistent primitive algorithm, J. Comput. Phys, 112, 31-43 (1994) · Zbl 0811.76044 |
| [22] | D. Kincaid, W. Cheney, Numerical Analysis Brooks/Cole, 1990; D. Kincaid, W. Cheney, Numerical Analysis Brooks/Cole, 1990 |
| [23] | Koren, B.; Lewis, M. R.; van Brummelen, E. H.; van Leer, B., Riemann-problem and level-set approaches for homentropic two-fluid flow computations, J. Comput. Phys, 185, 654-674 (2002) · Zbl 1178.76281 |
| [24] | Langseth, J. O.; LeVeque, R. J., A wave propagation method for three-dimensional hyperbolic conservation laws, J. Comput. Phys, 165, 126-166 (2000) · Zbl 0967.65095 |
| [25] | LeVeque, R. J., High resolution finite volume methods on arbitrary grids via wave propagation, J. Comput. Phys, 78, 36-63 (1988) · Zbl 0649.65050 |
| [26] | LeVeque, R. J., Numerical Methods for Conservation Laws (1992), Birkhäuser-Verlag · Zbl 0847.65053 |
| [27] | LeVeque, R. J., Wave propagation algorithms for multi-dimensional hyperbolic systems, J. Comput. Phys, 131, 327-353 (1997) · Zbl 0872.76075 |
| [28] | LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems (2002), Cambridge University Press · Zbl 1010.65040 |
| [29] | LeVeque, R. J.; Shyue, K.-M., One-dimensional front tracking based on high resolution wave propagation methods, SIAM J. Sci. Comput, 16, 348-377 (1995) · Zbl 0824.65094 |
| [30] | Marquina, A.; Mulet, P., A flux-split algorithm applied to conservative models for multicomponent compressible flows, J. Comput. Phys, 185, 120-138 (2003) · Zbl 1064.76078 |
| [31] | Marsh, S. P., LASL Shock Hugoniot Data (1980), University of California Press: University of California Press Berkeley |
| [32] | Miller, G. H.; Colella, P., A high-order Eulerian Godunov method for elastic-plastic flow in solids, J. Comput. Phys, 167, 131-176 (2001) · Zbl 0997.74078 |
| [33] | Miller, G. H.; Puckett, E. G., A high order Godunov method for multiple condensed phases, J. Comput. Phys, 128, 134-164 (1996) · Zbl 0861.65117 |
| [34] | Murnaghan, F. D., Finite Deformation of an Elastic Solid (1951), Wiley: Wiley New York · Zbl 0045.26504 |
| [35] | Nagoya, H.; Obara, T.; Takayama, K., Underwater shock wave propagation and focusing in inhomogeneous media, (Brun, R.; Dumitrescu, L. Z., Proceedings of the 19th International Symposium on Shock Waves, Marseille (1995), Springer-Verlag: Springer-Verlag Berlin), 439-444 |
| [36] | Picone, J. M.; Boris, J., Vorticity generation by shock propagation through bubbles in gas, J. Fluid Mech, 189, 23-51 (1988) |
| [37] | Puckett, E. G.; Saltzman, J. S., A 3D adaptive mesh refinement algorithm for multimaterial gas dynamics, Physica D, 60, 84-93 (1992) · Zbl 0779.76059 |
| [38] | Quirk, J. J.; Karni, S., On the dynamics of a shock-bubble interaction, J. Fluid Mech, 318, 129-163 (1996) · Zbl 0877.76046 |
| [39] | Roe, P. L., Fluctuations, signals- A framework for numerical evolution problems, (Morton, K. W.; Baines, M. J., Numerical Methods for Fluid Dynamics (1982), Academic Press), 219-257 · Zbl 0569.76072 |
| [40] | Roe, P. L., Upwind schemes using various formulations of the Euler equations, (Angrand, F.; Dervieux, A.; Desideri, J. A.; Glowinski, R., Numerical Methods for the Euler Equations of Fluid Dynamics (1985), SIAM), 14-31 · Zbl 0609.76075 |
| [41] | Saito, T.; Marumoto, M.; Yamashita, H.; Hosseini, S. H.R.; Nakagawa, A.; Hirano, T.; Takayama, K., Experimental and numerical studies of underwater shock wave attenuation, Shock Waves, 13, 139-148 (2003) |
| [42] | Saurel, R.; Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys, 150, 425-467 (1999) · Zbl 0937.76053 |
| [43] | Saurel, R.; Abgrall, R., A simple method for compressible multifluid flows, SIAM J. Sci. Comput, 21, 1115-1145 (1999) · Zbl 0957.76057 |
| [44] | Saurel, R.; Cocchi, P.; Butler, P. B., Numerical study of cavitation in the wake of hyperbolic underwater projectile, J. Propulsion and Power, 15, 513-522 (1999) |
| [45] | Shyue, K.-M., An efficient shock-capturing algorithm for compressible multicomponent problems, J. Comput. Phys, 142, 208-242 (1998) · Zbl 0934.76062 |
| [46] | Shyue, K.-M., A fluid-mixture type algorithm for compressible multicomponent flow with van der Waals equation of state, J. Comput. Phys, 156, 43-88 (1999) · Zbl 0957.76039 |
| [47] | Shyue, K.-M., A volume-of-fluid type algorithm for compressible two-phase flows, (Hyperbolic Problems: Theory, Numerics, Applications. Hyperbolic Problems: Theory, Numerics, Applications, International Series of Numerical Mathematics, vol. 130 (1999), Birkhäuser-Verlag), 895-904 · Zbl 0937.76064 |
| [48] | Shyue, K.-M., A fluid-mixture type algorithm for compressible multicomponent flow with Mie-Grueneisen equation of state, J. Comput. Phys, 171, 678-707 (2001) · Zbl 1047.76573 |
| [49] | Stewart, H. B.; Wendroff, B., Two-phase flows: models and methods, J. Comput. Phys, 56, 363-409 (1984) · Zbl 0596.76103 |
| [50] | Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal, 5, 506-517 (1968) · Zbl 0184.38503 |
| [51] | Sweby, P. K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer Anal, 21, 995-1011 (1984) · Zbl 0565.65048 |
| [52] | Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (1999), Springer-Verlag · Zbl 0923.76004 |
| [53] | Toro, E. F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, 4, 25-34 (1994) · Zbl 0811.76053 |
| [54] | van Brummelen, E. H.; Koren, B., A pressure-invariant conservative Godunov-type method for barotropic two-fluid flows, J. Comput. Phys, 185, 289-308 (2003) · Zbl 1047.76063 |
| [55] | van der Heul, D. R.; Vuik, C.; Wesseling, P., A staggered scheme for hyperbolic conservation laws applied to unsteady sheet cavitation, Comput. Visual Sci, 2, 63-68 (1999) · Zbl 1067.76583 |
| [56] | Xiao, F.; Yabe, T., A method to trace sharp interface of two fluids in calculations involving shocks, Shock Waves, 4, 101-108 (1994) · Zbl 0820.76063 |
| [57] | Yamada, K.; Nagoya, H.; Takayama, K., Shock wave reflection and refraction over a two-fluid interface, (Brun, R.; Dumitrescu, L. Z., Proceedings of the 19th International Symposuum on Shock Waves, Marseille (1995), Springer-Verlag: Springer-Verlag Berlin), 299-304 |
| [58] | Youngs, D. L., Time-dependent multi-material flow with large fluid distortion, (Morton, K. W.; Baines, M. J., Numerical Methods for Fluid Dynamics (1982), Academic Press), 273-285 · Zbl 0537.76071 |
| [59] | Zel’dovich, Y. B.; Raizer, Y. P., Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, vols. I & II (1967), Academic Press |
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