Numerical method for simulating rarefaction shocks in the approximation of phase-flip hydrodynamics

doi:10.1007/s10483-021-2734-6

Abstract

Abstract: A finite-difference algorithm is proposed for numerical modeling of hydrodynamic flows with rarefaction shocks, in which the fluid undergoes a jump-like liquid-gas phase transition. This new type of flow discontinuity, unexplored so far in computational fluid dynamics, arises in the approximation of phase-flip (PF) hydrodynamics, where a highly dynamic fluid is allowed to reach the innermost limit of metastability at the spinodal, upon which an instantaneous relaxation to the full phase equilibrium (EQ) is assumed. A new element in the proposed method is artificial kinetics of the phase transition, represented by an artificial relaxation term in the energy equation for a "hidden" component of the internal energy, temporarily withdrawn from the fluid at the moment of the PF transition. When combined with an appropriate variant of artificial viscosity in the Lagrangian framework, the latter ensures convergence to exact discontinuous solutions, which is demonstrated with several test cases.

Key words: fluid dynamics with phase transitions, two-phase flow, rarefaction shock, Lagrangian scheme, artificial viscosity, artificial kinetics

2010 MSC Number:

M. M. BASKO. Numerical method for simulating rarefaction shocks in the approximation of phase-flip hydrodynamics. Applied Mathematics and Mechanics (English Edition), 2021, 42(6): 871-884.

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References

[1] SOKOLOWSKI-TINTEN, K., BIALKOWSKI, J., CAVALLERI, A., VON DER LINDE, D., OPARIN, A., MEYER-TER-VEHN, J., and ANISIMOV, S. I. Transient states of matter during short pulse laser ablation. Physical Review Letters, 81, 224-227(1988)
[2] INOGAMOV, N. A., ANISIMOV, S. I., and RETHFELD, B. Rarefaction wave and gravitational equilibrium in a two-phase liquid-vapor medium. Journal of Experimental and Theoretical Physics, 88, 1143-1150(1999)
[3] AGRANAT, M. B., ANISIMOV, S. I., ASHITKOV, S. I., ZHAKHOVSKII, V. V., INOGAMOV, N. A., KOMAROV, P. S., OVCHINNIKOV, A. V., FORTOV, V. E., KHOKHLOV, V. A., and SHEPELEV, V. V. Strength properties of an aluminum melt at extremely high tension rates under the action of femtosecond laser pulses. JETP Letters, 91, 471-477(2010)
[4] BASKO, M. M., KRIVOKORYTOV, M. S., VINOKHODOV, A. Y., SIDELNIKOV, Y. V., KRIVTSUN, V. M., MEDVEDEV, V. V., KIM, D. A., KOMPANETS, V. O., LASH, A. A., and KOSHELEV, K. N. Fragmentation dynamics of liquid-metal droplets under ultra-short laser pulses. Laser Physics Letters, 14, 036001(2017)
[5] BRENNEN, C. E. Cavitation and Bubble Dynamics, Oxford University Press, New York (1995)
[6] UTKIN, A. V., SOSIKOV, V. A., BOGACH, A. A., and FORTOV, V. E. Tension of liquids by shock waves. AIP Conference Proceedings, 706, 765-770(2004)
[7] DE RESSÉGUIER, T., SIGNOR, L., DRAGON, A., BOUSTIE, M., ROY, G., and LLORCA, F. Experimental investigation of liquid spall in laser shock-loaded tin. Journal of Applied Physics, 101, 013506(2007)
[8] STAN, C. A., WILLMOTT, P. R., STONE, H. A., KOGLIN, J. E., LIANG, M., AQUILA, A. L., ROBINSON, J. S., GUMERLOCK, K. L., BLAJ, G., SIERRA, R. G., BOUTET, S., GUILLET, S. A. H., CURTIS, R. H., VETTER, S. L., LOOS, H., TURNER, J. L., and DECKER, F. J. Negative pressures and spallation in water drops subjected to nanosecond shock waves. The Journal of Physical Chemistry Letters, 7, 2055-2062(2016)
[9] COLOMBIER, J. P., COMBIS, P., BONNEAU, F., LE HARZIC, R., and AUDOUARD, E. Hydrodynamic simulations of metal ablation by femtosecond laser irradiation. Physical Review B, 71, 165406(2005)
[10] ZHAO, N., MENTRELLI, A., RUGGERI, T., and SUGIYAMA, M. Admissible shock waves and shock-induced phase transitions in a van der Waals fluid. Physics of Fluids, 23, 086101(2011)
[11] BASKO, M. M. Centered rarefaction wave with a liquid-gas phase transition in the approximation of "phase-flip" hydrodynamics. Physics of Fluids, 30, 123306(2018)
[12] BLANDER, M. and KATZ, J. L. Bubble nucleation in liquids. AIChE Journal, 21, 833-848(1975)
[13] MARTYNYUK, M. M. Phase explosion of a metastable fluid. Combustion, Explosion and Shock Waves, 13, 178-191(1977)
[14] SKRIPOV, V. P. and SKRIPOV, A. V. Spinodal decomposition (phase transitions via unstable states). Soviet Physics Uspekhi, 22, 389-410(1979)
[15] FAIK, S., BASKO, M. M., TAUSCHWITZ, A., IOSILEVSKIY, I., and MARUHN, J. A. Dynamics of volumetrically heated matter passing through the liquid-vapor metastable states. High Energy Density Physics, 8, 349-359(2012)
[16] GRADY, D. E. Spall and fragmentation in high-temperature metals. High Pressure Shock Compression of Solids II:Dynamic Fracture and Fragmentation (eds. DAVISON, L., GRADY, D. E., and SHAHINPOOR, M.), Springer, New York, 219-236(1996)
[17] SAUREL, R., PETITPAS, F., and ABGRALL, R. Modelling phase transition in metastable liquids:application to cavitating and flashing flows. Journal of Fluid Mechanics, 607, 313-350(2008)
[18] ZEIN, A., HANTKE, M., and WARNECKE, G. Modeling phase transition for compressible twophase flows applied to metastable liquids. Journal of Computational Physics, 229, 2964-2998(2010)
[19] CAI, Y., WU, H. A., and LUO, S. N. Spall strength of liquid copper and accuracy of the acoustic method. Journal of Applied Physics, 121, 105901(2017)
[20] MAYER, A. E. and MAYER, P. N. Strain rate dependence of spall strength for solid and molten lead and tin. International Journal of Fracture, 222, 171-195(2020)
[21] FARHAT, C., GERBEAU, J. F., and RALLU, A. FIVER:a finite volume method based on exact two-phase Riemann problems and sparse grids for multi-material flows with large density jumps. Journal of Computational Physics, 231, 6360-6379(2012)
[22] SHAHBAZI, K. Robust second-order scheme for multi-phase flow computations. Journal of Computational Physics, 339, 163-178(2017)
[23] ZHANG, C. and MENSHOV, I. Eulerian model for simulating multi-fluid flows with an arbitrary number of immiscible compressible components. Journal of Scientific Computing, 83, 31(2020)
[24] MARTYNYUK, M. M. Generalized van der Waals equation of state for liquids and gases. Zhurnal Fizicheskoi Khimii (Russian Journal of Physical Chemistry A), 65, 1716-1717(1991)
[25] MARTYNYUK, M. M. Transition of liquid metals into vapor in the process of pulse heating by current. International Journal of Thermophysics, 14, 457-470(1993)
[26] BASKO, M. M. Generalized van der Waals equation of state for in-line use in hydrodynamic codes. Keldysh Institute Preprints (2018) https://doi.org/10.20948/prepr-2018-112-e
[27] RICHTMYER, R. D. and MORTON, K. W. Difference Methods for Initial-Value Problems, 2nd ed., Interscience Publishers, New York (1967)
[28] LANDAU, L. D. and LIFSHITZ, E. M. Fluid Mechanics, 2nd ed., Pergamon Press, Oxford (1987)
[29] ZEL'DOVICH, Y. B. and RAIZER, Y. P. Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Dover Publications, New York (2012)
[30] MATTSSON, A. E. and RIDER, W. J. Artificial viscosity:back to the basics. International Journal for Numerical Methods in Fluids, 77, 400-417(2015)
[31] LANDSHOFF, R. A Numerical Method for Treating Fluid Flow in the Presence of Shocks, Report LA-1930, Los Alamos National Laboratory, Los Alamos (1955)
[32] VON NEUMANN, J. and RICHTMYER, R. D. A method for the numerical calculation of hydrodynamic shocks. Journal of Applied Physics, 21, 232-237(1950)
[33] BASKO, M. M., SASOROV, P. V., MURAKAMI, M., NOVIKOV, V. G., and GRUSHIN, A. S. One-dimensional study of the radiation-dominated implosion of a cylindrical tungsten plasma column. Plasma Physics and Controlled Fusion, 54(5), 055003(2012)