Local POD Plus Galerkin Projection in the Unsteady Lid-Driven Cavity Problem

Abstract

A local proper orthogonal decomposition (POD) plus Galerkin projection method is applied to the unsteady lid-driven cavity problem, namely the incompressible fluid flow in a two-dimensional box whose upper wall is moved back and forth at moderately large values of the Reynolds number. Such a method was recently developed for one-dimensional parabolic problems. Its extension to fluid dynamics problems is nontrivial (especially if rough CFD codes are used) and consists of using a computational fluid dynamics (CFD) code and a Galerkin system (GS) in a sequence of interspersed intervals $I_{CFD}$ and $I_{GS}$, respectively. The POD manifold is calculated retaining the most energetic POD modes resulting from the snapshots computed in the $I_{CFD}$ intervals; in fact, the POD manifold is completely calculated in the first $I_{CFD}$ interval but only updated in subsequent $I_{CFD}$ intervals. Intended to mimic industrial solvers, the CFD code contains unphysical terms that are introduced for purely numerical reasons to accelerate runs. The use of such a CFD code poses the question of which equations should be Galerkin projected onto the POD manifold, the exact governing equations or the CFD numerical scheme. Also, Galerkin projection can be made using either the standard $L_2$ inner product or a nonstandard one, based on a limited number of mesh points. After addressing these issues, a method is constructed that is able to accelerate the CFD code by a factor of the order of 5–15, depending on the Reynolds number and the nature (steady, periodic, or quasi-periodic) of the forcing velocity.