A KARHUNEN-LOÈVE-BASED APPROACH TO NUMERICAL SIMULATION OF TRANSITION IN RAYLEIGH-BÈNARD CONVECTION

Abstract

A Karhunen-Loève (K-L) basis is generated empirically, using a database obtained by numerical integration of Boussinesq equations representing Rayleigh-Benard convection in a weakly turbulent state in a periodic convective box with free upper and lower surfaces. This basis is then used to reduce the governing partial differential equation (PDE) into a truncated system of amplitude equations under Galerkin projection. In the generation and implementation of the basis, the symmetries of the PDE and the geometry are fully exploited. The resulting amplitude equations are integrated numerically and it is shown that, with the use of the K-L basis in the present formulation, the known dynamics of the flow in the transition region is completely captured.