An unconditionally stable finite element-finite volume pressure correction scheme for the drift-flux model

Abstract

We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of a Darcy-like relation, the drift term becomes dissipative. Finally, the present algorithm preserves a constant pressure and a constant velocity through moving interfaces between phases. To ensure the stability as well as to obtain this latter property, a key ingredient is to couple the mass balance and the transport equation for the dispersed phase in an original pressure correction step. The existence of a solution to each step of the algorithm is proven; in particular, the existence of a solution to the pressure correction step is derived as a consequence of a more general existence result for discrete problems associated to the drift-flux model. Numerical tests show a near-first-order convergence rate for the scheme, both in time and space, and confirm its stability.

Pressure correction staggered schemes for barotropic one-phase and two-phase flows

“…This scheme was already known to be unconditionally stable [8,10]. Numerical tests performed here show that, provided that a sufficient numerical dissipation is introduced in the scheme, it converges to the (weak and entropic) solution to the continuous problem; in addition, it shows a satisfactory behaviour up to large CFL numbers.…”

“…Note that, for a given space discretization, this equation must be established at the algebraic level [10], with the discrete equivalent manipulations which were necessary to derive it at the continuous level (i.e. multiplying the first equation by ρ n+1 /ρ n , taking its divergence and subtracting to the second relation).…”

“…before the derivation of the elliptic problem for the pressure, which is thoroughly described in [10]. Indeed, this latter computation is purely algebraic, in the sense that it transforms a non-linear algebraic system into another non-linear algebraic system which is strictly equivalent, and thus has no impact on the properties of the scheme (besides, of course, the efficiency issue).…”

“…Extensions to multi-phase flows are scarcer and seem to be restricted to iterative algorithms, often similar in spirit to the usual SIMPLE algorithm for incompressible flows [27,22,20]. In this paper, we perform a numerical study of a non-iterative pressurecorrection scheme introduced in [10], based on a low order finite element and a finite volume discretization, which enjoys the following properties:…”

“…Finally, the scheme boils down to the usual projection scheme for incompressible flows (obtained in the present framework when y = 0 or, asymptotically, when the function g becomes constant), and is indeed routinely used for the computation of low Mach number flows, as, for instance, classical bubble columns of chemical engineering processes or two-phase flows encountered in nuclear safety studies. Its accuracy was assessed for smooth solutions in [10], and the aim of the present paper is to check its convergence and accuracy in non-diffusive cases, for weak solutions with discontinuities.…”