Numerical simulations of compressible flows using multi-fluid models

Abstract Numerical simulations of two-fluid flow models based on the full Navier–Stokes equations are presented. The models include six and seven partial differential equations, namely, six- and seven-equation models. The seven-equation model consists of a non-conservative equation for volume fraction evolution of one of the fluids and two sets of balance equations. Each set describes the motion of the corresponding fluid, which has its own pressure, velocity, and temperature. The closure is achieved by two stiffened gas equations of state. Instantaneous relaxation towards equilibrium is achieved by velocity and pressure relaxation terms. The six-equation model is deduced from the seven-equation model by assuming an infinite rate of velocity relaxation. In this model, a single velocity is used for both fluids. The numerical solutions are obtained by applying the Strang splitting technique. The numerical solutions are examined in a set of one, two, and three dimensions for both the six- and seven-equation models. The results indicate very good agreement with the experimental results. There is an insignificant difference between the results of the two models, but the six-equation model is much more economical compared to the seven-equation model.

[1]  Derek M. Causon,et al.  A bore-capturing finite volume method for open-channel flows , 1998 .

[2]  C. Merkle,et al.  Computation of Multiphase Mixture Flows with Compressibility Effects , 2002 .

[3]  S. F. Davis.  Simplified second-order Godunov-type methods , 1988 .

[4]  W. K. Anderson,et al.  Comparison of Finite Volume Flux Vector Splittings for the Euler Equations , 1985 .

[5]  Frédéric Coquel,et al.  A Numerical Method Using Upwind Schemes for the Resolution of Two-Phase Flows , 1997 .

[6]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[7]  M. Baer,et al.  A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials , 1986 .

[8]  R. Abgrall,et al.  A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows , 1999 .

[9]  D. Stewart,et al.  Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations , 2001 .

[10]  G. D. van Albada,et al.  A comparative study of computational methods in cosmic gas dynamics , 1982 .

[11]  Hervé Guillard,et al.  A five equation reduced model for compressible two phase flow problems , 2005 .

[12]  C. Arcoumanis,et al.  Modelling of cavitation in diesel injector nozzles , 2008, Journal of Fluid Mechanics.

[13]  Richard Saurel,et al.  Treatment of interface problems with Godunov-type schemes , 1996 .

[14]  Charles L. Merkle,et al.  Multi-Disciplinary Computational Analysis in Propulsion , 2006 .

[15]  O. Le Métayer,et al.  Quantitative numerical and experimental studies of the shock accelerated heterogeneous bubbles motion , 2007 .

[16]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[17]  M. Berger,et al.  Analysis of Slope Limiters on Irregular Grids , 2005 .

[18]  Ali H. Nayfeh,et al.  Computations of the Compressible Multiphase Flow Over the Cavitating High-Speed Torpedo , 2003 .

[19]  Richard Saurel,et al.  Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures , 2009, J. Comput. Phys..

[20]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[21]  Ann S. Almgren,et al.  A Numerical Study of Methods for Moist Atmospheric Flows: Compressible Equations , 2013, 1311.4265.

[22]  M. Lallemand,et al.  Pressure relaxation procedures for multiphase compressible flows , 2005 .

[23]  Cong-Tu Ha,et al.  SENSITIVITY EVALUATION OF EMPIRICAL COEFFICIENTS IN THE CAVITATION MODELS , 2012 .