Turbulent thermal convection in a finite domain: Part I. Theory

Abstract

Determination of the empirical eigenfunctions for turbulent flows, which result from the Karhunen–Loève procedure, is considered in some generality for fully inhomogeneous flows. Group theoretical considerations are shown to lead to considerable increases in an available database. In addition, group representation procedures are shown to lead to substantial simplification. In fact, for the application considered here, a nonmanageable problem is reduced to one that is solvable. The general methods and techniques presented here are applied to the case of Rayleigh–Bénard convection in a finite box. In addition, indication is made of how to apply the procedures to several other cases. Some results of applying the method of empirical eigenfunctions to a numerical simulation of this particular flow [H. Park and L. Sirovich, Phys. Fluids A 2, 1659 (1990)] are presented here.