A Proper-Orthogonal Decomposition Variational Multiscale Approximation Method for a Generalized Oseen Problem

We introduce the variational multiscale (VMS) stabilization for the reduced-order modeling of incompressible flows. It is well known that the proper orthogonal decomposition (POD) technique in reduced-order modeling experiences numerical instability when applied to complex flow problems. In this case a POD discretization naturally separates out structures which corresponding to the energy cascade on large and small scales, in order, a VMS approach is natural. In this paper, we provide the mathematical background necessary for implementing VMS to a POD-Galerkin model of a generalized Oseen problem. We provide theoretical evidence which indicates the consistency of utilizing a VMS approach in the stabilization of reduced order flows. In addition we provide numerical experiments indicating that VMS improves fidelity in reproducing the qualitative properties of the flow. 1. Introduction In this paper, we present an analysis and numerical experiments associated with the use of the variational multiscale (VMS) stabilization technique applied to reduced-order modeling (ROM) of a generalized Oseen problem. In producing error estimates and stability results regarding the simulation of incompressible flow problems, the Oseen problem is a model problem in that it possesses all of the features which may prove challenging in the full scale numerical simulation of turbulent flows, while at the same time being posed as a linear problem. In addition, when using an iterative method for resolving the nonlinearity in the incompressible Navier-Stokes equations, an Oseen problem is solved at each iteration. The generalized Oseen problem may be stated as follows. Let be a domain in ( denoting the spatial dimension), and find such that In the case of fluid flow, gives the fluid velocity, is the pressure, represents the fluid viscosity, is a solenoidal vector function, , representing the associated velocity field, and is a reaction term. One promising idea when considering the fast simulation of is the use of ROM by way of proper orthogonal decomposition (POD). The idea of applying POD to turbulent flows is nothing new. In fact, the principal components of a flow can effectively give a lot of information about flow’s coherent structure and dynamical behavior. However, what remains a mystery is the idea of using the principal components in the fast reproduction of (or variation of modeling parameters for) a fluid flow simulation, which is subject to numerical instability. The idea behind POD based modeling is nothing new in engineering and is incorporated in many areas. It is