The Second Gradient Method for the Direct Numerical Simulation of Liquid–Vapor Flows with Phase Change

Abstract

This paper presents a new method to simulate liquid–vapor flows with phase change using a phase-field-like approach. In this method, the liquid–vapor interface is described as a three-dimensional continuous medium across which physical properties have strong but continuous variations. This continuous variation is made possible by imposing that the internal energy of the fluid depends on its density gradient. This description, called the second gradient theory, is numerically attractive since a single system of partial differential equations (PDEs) is necessary to determine the flow in the entire two-phase system, the phase change, the displacement of the interfaces, and their change in topology being a part of their solution. However, to solve these PDEs using a reasonable number of grid points on a fixed grid, the interfaces need to be artificially enlarged. It is shown that this artificial enlargement can be thermodynamically consistent if the thermodynamic behavior of the fluid is modified within the binodal curve. The consequences of this thermodynamic modification are studied in detail. In particular it is shown that, within the frame of the second gradient theory, the interface thickness and the surface tension vary with the mass and heat fluxes across the interface and that these variations increase with the thickness of the interface. As a consequence, for a given accuracy, an upper bound exists for the interfacial heat and mass fluxes that can be simulated. Examples of applications in one and two dimensions show the potentialities of the method presented, in particular to deal with moving contact lines, the description of which is a part of the second gradient theory.