Liquid and liquid–gas flows at all speeds

All speed flows and in particular low Mach number flow algorithms are addressed for the numerical approximation of the Kapila et al. [1] 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 [2]. Many applications occurring with liquid–gas interfaces and cavitating flows involve a very wide range of Mach number, from 10−3 to supersonic (and even hypersonic) conditions with respect to the mixture sound speed. It is thus important to address numerical methods free of restrictions regarding the Mach number. To do this, 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, first in one dimension and second in the frame of 2D unstructured meshes. Two-phase flow preconditioning is then addressed in the frame of the Saurel et al. [3] 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 single phase and two-phase steady nozzle flow solutions. Last, the method is illustrated by the computation of real cavitating flows in Venturi nozzles. Vapour pocket size and instability frequencies are reproduced by the model and method without using any adjustable parameter