A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM+-up scheme

In this paper, we propose a new approach to compute compressible multifluid equations. Firstly, a single-pressure compressible multifluid model based on the stratified flow model is proposed. The stratified flow model, which defines different fluids in separated regions, is shown to be amenable to the finite volume method. We can apply the conservation law to each subregion and obtain a set of balance equations. Secondly, the AUSM^+ scheme, which is originally designed for the compressible gas flow, is extended to solve compressible liquid flows. By introducing additional dissipation terms into the numerical flux, the new scheme, called AUSM^+-up, can be applied to both liquid and gas flows. Thirdly, the contribution to the numerical flux due to interactions between different phases is taken into account and solved by the exact Riemann solver. We will show that the proposed approach yields an accurate and robust method for computing compressible multiphase flows involving discontinuities, such as shock waves and fluid interfaces. Several one-dimensional test problems are used to demonstrate the capability of our method, including the Ransom's water faucet problem and the air-water shock tube problem. Finally, several two dimensional problems will show the capability to capture enormous details and complicated wave patterns in flows having large disparities in the fluid density and velocities, such as interactions between water shock wave and air bubble, between air shock wave and water column(s), and underwater explosion.

[1]  J. Edwards,et al.  Low-Diffusion Flux-Splitting Methods for Flows at All Speeds , 1997 .

[2]  Richard Saurel,et al.  A multiphase model with internal degrees of freedom: application to shock–bubble interaction , 2003, Journal of Fluid Mechanics.

[3]  Meng-Sing Liou,et al.  A New Approach to the Simulation of Compressible Multifluid Flows with AUSM+ Scheme , 2003 .

[4]  Rémi Abgrall,et al.  Discrete equations for physical and numerical compressible multiphase mixtures , 2003 .

[5]  G. Wallis.  One Dimensional Two-Phase Flow , 1969 .

[6]  R. E. Nieman,et al.  One-dimensional models for transient gas-liquid flows in ducts , 1980 .

[7]  Christophe Eric Corre,et al.  On the extension of the AUSM+ scheme to compressible two-fluid models , 2003 .

[8]  Meng-Sing Liou,et al.  Hyperbolicity, Discontinuities, and Numerics of Two-Fluid Models , 2007 .

[9]  J. H. Stuhmiller,et al.  The influence of interfacial pressure forces on the character of two-phase flow model equations , 1977 .

[10]  D. Drew,et al.  THE ANALYSIS OF VIRTUAL MASS EFFECTS IN TWO-PHASE FLOW , 1979 .

[11]  Robin K. S. Hankin,et al.  The Euler equations for multiphase compressible flow in conservation form: simulation of shock-bubble interactions , 2001 .

[12]  Meng-Sing Liou,et al.  Numerical speed of sound and its application to schemes for all speeds , 1999 .

[13]  Rémi Abgrall,et al.  A Simple Method for Compressible Multifluid Flows , 1999, SIAM J. Sci. Comput..

[14]  H. Bruce Stewart,et al.  Two-phase flow: Models and methods , 1984 .

[15]  M. Ishii.  Thermo-fluid dynamic theory of two-phase flow , 1975 .

[16]  S. Osher,et al.  Computing with high-resolution upwind schemes for hyperbolic equations , 1985 .

[17]  G. J. Li,et al.  Aerobreakup in Rarefied Supersonic Gas Flows , 2004 .

[18]  Meng-Sing Liou,et al.  A sequel to AUSM, Part II: AUSM+-up for all speeds , 2006, J. Comput. Phys..

[19]  H. Paillere,et al.  Approximate riemann solvers and flux vector splitting schemes for two-phase flow , 1999 .

[20]  Richard Saurel,et al.  Modelling evaporation fronts with reactive Riemann solvers , 2005 .

[21]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[22]  B. Leer,et al.  Flux-vector splitting for the Euler equations , 1997 .

[23]  Meng-Sing Liou,et al.  Simulation of Multifluid Multiphase Flows with AUSM+-up Scheme , 2006 .

[24]  M. Liou.  A Sequel to AUSM , 1996 .

[25]  Yu. A. Buyevich,et al.  Statistical hydromechanics of disperse systems Part 1. Physical background and general equations , 1971, Journal of Fluid Mechanics.

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

[27]  I. Toumi,et al.  An upwind numerical method for two-fluid two-phase flow models , 1996 .