Liquid and liquid-gas flows at all speeds : Reference solutions and numerical schemes

All speed flows and in particular low Mach number flow algorithms are addressed for the numerical approximation of the Kapila et al. [19] multiphase flow model. This model is valid for fluid mixtures evolving in mechanical equilibrium but out of temperature equilibrium and is efficient for material interfaces computation separating miscible and non-miscible fluids. In this context, the interface is considered as a numerically diffused zone, captured as well as all present waves (shocks, expansion waves). The same flow model can be used to solve cavitating and boiling flows [39]. Many applications occurring with liquid-gas interfaces and cavitating flows involve a very wide range of Mach number variations, from 10−3 to 100 with respect to the mixture sound speed. It is thus important to address numerical methods free of restrictions regarding the Mach number. To assess the accuracy of such schemes, reference solutions are needed and there is a clear lack in this domain. We address here exact one-dimensional liquid and liquid-gas compressible flows solutions in nozzles. The exact solution is first derived for the compressible single liquid phase Euler equations and extends the well known ideal gas dynamic nozzle flow solutions. This reference solution is then extended to the Kapila et al. [19] model that contains two entropies and non conventional shock relations. The all Mach number scheme is then derived. A preconditioned Riemann solver is built and embedded into the Godunov explicit scheme. It is shown that this method converges to exact solutions but needs too small time steps to be efficient. An implicit version is then derived, in one dimension first and second in the frame of 3D unstructured meshes. Two-phase flow preconditioning is then addressed in the frame of the Saurel et al [38] algorithm. Modifications of the preconditioned Riemann solver are needed and detailed. Convergence of both single phase and two-phase numerical solutions are demonstrated with the help of exact ones. Last, the method is illustrated by the computation of real cavitating flows in Venturi nozzles. Vapor pocket size and instability frequencies are perfectly reproduced by the model and method without using any parameters. In particular, no turbulence model is used. Key-words: hyperbolic systems, multifluid, multiphase, Venturi, cavitation, preconditioned, unstructured meshes, HLLC. ha l-0 06 95 79 9, v er si on 1 9 M ay 2 01 2 Liquid and liquid-gas flows at all speeds : Reference solutions and numerical schemes Résumé : Mots-clés : ha l-0 06 95 79 9, v er si on 1 9 M ay 2 01 2 4 S. LeMartelot, B. Nkonga and R. Saurel