Reduction of modes for the solution of inverse natural convection problems

Abstract

The inverse natural convection problem of determining heat flux at the bottom wall of a two-dimensional cavity from temperature measurement in the domain is investigated by means of the Karhunen–Loève Galerkin procedure. The Karhunen–Loève Galerkin procedure, which is a type of Galerkin method that employs the empirical eigenfunctions of the Karhunen–Loève decomposition as basis functions, can reduce nonlinear partial differential equations to sets of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena. Previously, it had been demonstrated that the problems of optimal control of Burgers equation [H.M. Park, M.W. Lee, Y.D. Jang, Comput. Methods Appl. Mech. Engrg. 166 (1998) 289–308] and the Navier–Stokes equation [H.M. Park, M.W. Lee, Comput. Methods Appl. Mech. Engrg., 1999 (in press)] can be solved very efficiently through the reduction of modes based on the Karhunen–Loève Galerkin procedure. In the present investigation, this technique is applied to the solution of inverse natural convection problem of estimating unknown wall heat flux. The performance of the present technique of inverse analysis using the Karhunen–Loève Galerkin procedure is assessed in comparison with a traditional technique employing the Boussinesq equation, and is found to be very accurate as well as efficient.