Compressible two-phase flows by central and upwind schemes

Abstract

Abstract. This paper concerns numerical methods for two-phase flows. The governing equations are the compressible 2-velocity, 2-pressure flow model. Pressure and velocity relaxation are included as source terms. Results obtained by a Godunov-type central scheme and a Roe-type upwind scheme are presented. Issues of preservation of pressure equilibrium, and positivity of the partial densities are addressed.Mathematics Subject Classification. 35L65, 65M06, 76N15, 76T99.

Interface tracking method for compressible multifluids

“…Our numerical experiments indicate that the resulting method is not too sensitive to the choice of β, so in all numerical results reported below, we have used β = 0.5. We then remove the δ-source terms from the RHS of the system (2.20)-(2.22) and linearize it about some average state, again with no special regard to conservation, for example, about U av := 1 2 ( U in + U out ), and consider the linearized Riemann problem: 25) where σ av is a piecewise constant approximation of the source term σ ≡ (σ (1) , σ (2) , σ (3) ) T := (−ρq/r, 0, ρqc 2 /r) T , which is a nonsingular part of the source term in the system (2.20)-(2.22), and A is the coefficient matrix of this system: 26) whose eigenvalues and the corresponding right eigenvectors are:…”

“…Loss of strict conservation compromises numerical convergence theory, and while the aforementioned methods were demonstrated to be essentially conservative, in that conservation error is typically proportional to the grid size and goes to zero with mesh refinement, the possibility remains that computed solutions may converge to incorrect weak limits, particularly in higher dimensions. Global conservation may be respected by using more complete flow models, which solve separately for the individual species energies [42,46,47] or the full multiphase flow model [5,9,26,38]. However, the individual species equations contain nonconservative products and do not completely circumvent the nonconservation doubt.…”

A Godunov-type method for the seven-equation model of compressible two-phase flow

“…For instance in Gallouët et al [21] (see also Guillemaud [25]), the approximation of the convective terms of the system is based on the Rusanov scheme (Rusanov [36]) and the so-called VFRoe-ncv scheme (Buffard et al [8]), these strategies being adapted to the nonconservative framework. In Munkejord [34] and Karni et al [30], the author use Roe-type schemes. Many of the above mentioned methods are only devoted to the convective part of the seven-equation model, focusing in addition on a specific choice of the interfacial velocity u I naturally present in the governing equations.…”

Large Time-Step Numerical Scheme for the Seven-Equation Model of Compressible Two-Phase Flows

“…It was first proposed in Baer & Nunziato [4] and has since aroused more and more interest, see for instance Embid & Baer [7], Stewart & Wendroff [13], Abgrall & Saurel [11], Gallouët, Hérard & Seguin [8], Andrianov & Warnecke [3], Karni et al [9] Schwendeman, Wahle & Kapila [12], Munkejord [10], Tokareva & Toro [14], Ambroso, Chalons, Coquel & Galié [1], Ambroso, Chalons & Raviart [2], and the references therein. One of the main features of this model is that it is hyperbolic, at least in the context of subsonic flows.…”

“…This property will be used in the numerical strategy. Numerous papers are devoted to the numerical study of two-fluid two-pressure models, see again for instance [8], [3], [9], [12], [10], [14], [1], [2] and the references therein. Many of the proposed methods are based on time-explicit, exact or approximate, Godunov-type methods (Roe or Roe-like scheme, HLL or HLLC scheme...).…”