The effect of the virtual mass term on the stability of the two-fluid model against perturbations

Abstract

The effect of the virtual mass term on the stability of the two-fluid model against perturbations is studied. Three types of virtual mass term in the momentum equation are discussed: two types of objective form and a simplified form. The differential equation system with no virtual mass term is ill-posed and the solution is unstable against perturbations. By introducing an objective form of the virtual mass term derived by Drew et al., it is shown that the equation system is rendered to be well-posed. The equation system is shown to be ill-posed, however, when a more recent definition of virtual mass acceleration of Drew and Lahey is applied. With a simplified form of the virtual mass term, which is composed only of temporal acceleration terms, the equation system is well-posed or ill-posed depending on velocities. A linear stability analysis is also performed for the implicit upwind finite difference scheme. A hypothetical accelerated flow problem is then numerically simulated by solving the discretized equation systems. It is shown that the solution can be numerically unstable even for the cases when the differential equation system is well-posed. The numerical stability of the solution must therefore be judged based on the spectral radius of the discretized equation system.