Effects of mesh motion on the stability and convergence of ALE based formulations for moving boundary flows

This paper investigates the effects of mesh motion on the stability of fluid-flow equations when written in an Arbitrary Lagrangian-Eulerian frame for solving moving boundary flow problems. Employing the advection-diffusion equation as a model problem we present a mathematical proof of the destabilizing effects induced by an arbitrary mesh motion on the stability and convergence of an otherwise stable scheme. We show that the satisfaction of the so-called geometric conservation laws is essential to the development of an identity that plays a crucial role in establishing stability. We explicitly show that the advection dominated case is susceptible to growth in error because of the motion of the computational grid. To retain the bound on the growth in error, the mesh motion techniques need to account for a domain based constraint that minimizes the relative mesh velocity. Analysis presented in this work can also be extended to the Navier-Stokes equations when written in an ALE frame for FSI problems.