Physical aspects of the relaxation model in two-phase flow

Abstract

The paper explores the potential of the homogeneous relaxation model (HRM) as a basis for the description of adiabatic, one-dimensional, two-phase flows. To this end, a rigorous mathematical analysis highlights the similarities and differences between this and the homogeneous equilibrium model (HEM) emphasizing the physical and qualitative aspects of the problem. Special attention is placed on a study of dispersion, characteristics, choking and shock waves. The most essential features are discovered with reference to the appropriate and convenient phase space Ω for HRM, which consists of pressure P, enthalpy h, dryness fraction x, velocity w, and length coordinate z. The geometric properties of the phase space Ω enable us to sketch the topological pattern of all solutions of the model. The study of choking is intimately connected with the occurrence of singular points of the set of simultaneous first-order differential equations of the model. The very powerful centre manifold theorem allows us to reduce the study of singular points to a two-dimensional plane Π, which is tangent to the solutions at a singular point, and so to demonstrate that only three singular-point patterns can appear (excepting degenerate cases), namely saddle points, nodal points and spiral points. The analysis reveals the existence of two limiting velocities of wave propagation, the frozen velocity af and the equilibrium velocity ae. The critical velocity of choking is the frozen speed of sound. The analysis proves unequivocally that transition from ω < af to w > af can take place only via a singular point. Such a condition can also be attained at the end of a channel. The paper concludes with a short discussion of normal, fully dispersed and partly dispersed shock waves.