Towards a Strictly Conservative Hybrid Level-Set Volume-of-Fluid Finite Volume Method for Zero Mach Number Two-Phase Flow

Abstract

Two-phase flow plays a significant role in multiple technical applications and natural phenomena. Therefore there is an increasing interest in numerical simulation of such flows for both prediction and analysis purposes. Many of these processes can be modeled as incompressible or zero Mach number flows. While there are many methods for simulation of incompressible two-phase flow at constant density, only few methods can be found, that allow for a variable density and solve the governing equations in conservative form. In principle there is no method, which consequently applies discretely conservative approximations only wherever appropriate, while remaining extendable to other flow regimes, such as the compressible or weakly-compressible low Mach number flow regime, in a conceptually consistent way. The present work is meant to serve as starting point for a Finite Volume method that satisfies these requirements while remaining extendable to equations of state beyond the assumption of a perfect gas. Within this generalized framework two key features of a numeri- cal method for simulation of two-phase flow are focused on after deriving the zero Mach number equations for immiscible chemically reacting two-phase flow at arbitrary equation of state and presenting the underlying single-phase solver as basic building block of the numerical method in detail: On the one hand an approach for coupling of the discrete representation of the interface, sharply separating the different fluid phases, and the conserved quantities representing the fluid flow is extended, analyzed and adapted to the present framework for keeping the method stable and discretely conserving the mass of each of the fluid phases. On the other hand an approach for approximation of the influence of surface tension, which is singular at the interface, is proposed, that allows for discretely conservative treatment of these effects as well. The underlying numerical scheme for solving the resulting system of differential equations is a generalized projection method, which imposes an elliptic constraint on a hyperbolic-parabolic predictor solution in each time step. Due to the fact that projection schemes – except for the solution of linear systems for individual scalars – are iteration-free, the different building blocks presented in this work are kept iteration-free as well.