On proper orthogonal decomposition of randomly perturbed fields with applications to flow past a cylinder and natural convection over a horizontal plate

Abstract

The connections between the random elements of a discrete random flow field and the uncertainty in the hierarchical set of its spatio-temporal scales, obtained by the symmetric version of the proper orthogonal decomposition (POD) method, are investigated. It is shown that the relevant statistics for the energy levels, the temporal modes and the spatial modes can be expressed in an explicit form as power series of the flow field standard deviation. Such expansions characterize accurately interesting phenomena of mixing between different flow scales. The basis of the present work is the assumption that the randomness is characterized by a Gaussian uncorrelated random field. Two applications of the theory developed are proposed: to the incompressible flow past a cylinder at Reynolds number $\mbox{\textit{Re}}\,{=}\,100$ and to the natural convective flow over an isothermal horizontal plate at Rayleigh number $\mbox{\textit{Ra}}\,{=}\,4.75\,{\times}\,10^6$. The theoretical predictions are confirmed well by Monte Carlo simulations and interesting relations between the random flows and the relevant statistics of their POD spatio-temporal scales are determined and discussed.