A Five-Equation Model for the Simulation of Interfaces between Compressible Fluids

Abstract

A diffuse-interface method is proposed for the simulation of interfaces between compressible fluids with general equations of state, including tabulated laws. The interface is allowed to diffuse on a small number of computational cells and a mixture model is given for this transition region. We write conservation equations for the mass of each fluid and for the total momentum and energy of the mixture and an advection equation for the volume fraction of one of the two fluids. The model needs an additional closure law. We study two different closure laws: isobaric and isothermal. We study the mathematical properties of the resulting models: consistency, hyperbolicity, and existence of a mathematical entropy. We also study the stability of the interfaces with respect to averaging due to the numerical diffusion, a crucial property for the simulation of interface problems by conservative schemes. We show that the isobaric closure is preferable to the isothermal closure with respect to this property. We propose a Roe-type numerical scheme for the simulation of the model and show numerical results for classical test cases. c 2002 Elsevier Science (USA)

A new formulation of Kapila’s five-equation model for compressible two-fluid flow, and its numerical treatment

“…Assumption (1) prescribes that the pressures and velocity vectors on both sides of a two-fluid interface are equal. It is the known step to reduce a seven-equation model to a five-equation model, see, e.g., [1,20,10,3,14,6]. Further, viscosity and heat conduction are neglected.…”

“…The archetype five-equation model is that of Kapila et al [10]. It has already found many applications, a non-exhaustive list of excellent references is [12,3,14,2,21,17]. (Recently, Saurel et al have even derived a six-equation model from a five-equation one [22].)…”

Numerical simulation of unsteady cavitation in liquid hydrogen flows

“…Further, these models have been investigated and applied to metastable states and evaporation front dynamics (Saurel & Metayer 2001, Ishimoto & Kamijo 2004. Another set of similar models, the five-equation model, is also derived on the basis velocity and pressure equilibrium (Allaire et al 2002, Kapila et al 2001, Kreeft & Koren 2010. This also can be expressed as a four-equation model by assuming thermal equilibrium between phases.…”

On a diphasic low mach number system

“…Moreover, with appropriate modelling hypothesis -again satisfied by a large class of generalized van der Waals equations of state -, the DLMN system degenerates (formally) toward the incompressible Navier-Stokes system for one of the two fluids. 3 The bubble Ω 1 has to be on the middle of Ω when t → +∞ since the initial conditions are such that the mass in Ω a 2 is equal to the mass in Ω b 2 . Test case 2 Iterations = 0, 450 and 10 000.…”

“…where a k , b k , π k are positive constants and where γ k > 1 is another positive constant (see [3,16,27] for example). And, we verify that the equation of state (38) satisfies (37) if and only if a k = 0; in that case, we find…”