Projection‐based variational multiscale method for incompressible Navier–Stokes equations in time‐dependent domains

Abstract

Summary A variational multiscale method for computations of incompressible Navier–Stokes equations in time‐dependent domains is presented. The proposed scheme is a three‐scale variational multiscale method with a projection‐based scale separation that uses an additional tensor valued space for the large scales. The resolved large and small scales are computed in a coupled way with the effects of unresolved scales confined to the resolved small scales. In particular, the Smagorinsky eddy viscosity model is used to model the effects of unresolved scales. The deforming domain is handled by the arbitrary Lagrangian–Eulerian approach and by using an elastic mesh update technique with a mesh‐dependent stiffness. Further, the choice of orthogonal finite element basis function for the resolved large scale leads to a computationally efficient scheme. Simulations of flow around a static beam attached to a square base, around an oscillating beam and around a plunging aerofoil are presented. Copyright © 2016 John Wiley & Sons, Ltd.

Stability analysis and error estimates of a projection based variational multiscale method for Oseen equations in moving domains

“…In this paper we shall discuss the mathematical analysis of the projection based variational multiscale (VMS) scheme that we developed for the Navier-Stokes equations (NSE) in arbitrary Lagrangian Eulerian (ALE) formulation [16]. Stability and error estimates for NSE in stationary domains have been given in a series of papers by Heywood and Rannacher in the 1980s [8,9,10,11].…”