Proper orthogonal decomposition and low-dimensional models for driven cavity flows

Abstract

A proper orthogonal decomposition (POD) of the flow in a square lid-driven cavity at Re=22,000 is computed to educe the coherent structures in this flow and to construct a low-dimensional model for driven cavity flows. Among all linear decompositions, the POD is the most efficient in the sense that it captures the largest possible amount of kinetic energy (for any given number of modes). The first 80 POD modes of the driven cavity flow are computed from 700 snapshots that are taken from a direct numerical simulation (DNS). The first 80 spatial POD modes capture (on average) 95% of the fluctuating kinetic energy. From the snapshots a motion picture of the coherent structures is made by projecting the Navier–Stokes equation on a space spanned by the first 80 spatial POD modes. We have evaluated how well the dynamics of this 80-dimensional model mimics the dynamics given by the Navier–Stokes equations. The results can be summarized as follows. A closure model is needed to integrate the 80-dimensional system at Re=22,000 over long times. With a simple closure the energy spectrum of the DNS is recovered. A linear stability analysis shows that the first (Hopf) bifurcation of the 80-dimensional dynamical system takes place at Re=7,819. This number lies about 0.7% above the critical Reynolds number given in Poliashenko and Aidun [J. Comput. Phys. 121, 246 (1995)] and differs by about 2% from the first instability found with DNS. In addition to that, the unstable eigenvector displays the correct mechanism: a centrifugal instability of the primary eddy, however, the frequency of the periodic solution after the first bifurcation differs from that of the DNS. The stability of periodic solutions of the 80-dimensional system is analyzed by means of Floquet multipliers. For Re=11,188−11,500 the ratio of the two periods of the stable 2-periodic solution of the 80-dimensional system is approximately the same as the ratio of the two periods of the 2-periodic solution of the DNS at Re=11,000. For slightly higher Reynolds numbers both solutions lose one period. The periodic solutions of the dynamical system at Re=11,800 and the DNS at Re=12,000 have approximately the same period and have qualitatively the same behavior.