Reduction of modes for the solution of inverse natural convection problems

The inverse natural convection problem of determining heat flux at the bott om wall of a two-dimensional cavity from temperature measurement in the dom ain is investigated by means of the Karhunen-Loeve Galerkin procedure. The Karhunen-Loeve Galerkin procedure, which is a type of Galerkin method that employs the empirical eigenfunctions of the Karhunen-Loeve decomposition as basis functions, can reduce nonlinear partial differential equations to se ts of minimal number of ordinary differential equations by limiting the sol ution space to the smallest linear subspace that is sufficient to describe the observed phenomena. Previously, it had been demonstrated that the probl ems 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-Stok es 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 b ased on the Karhunen-Loeve Galerkin procedure. In the present investigation , this technique is applied to the solution of inverse natural convection p roblem of estimating unknown wall heat flux. The performance of the present technique of inverse analysis using the Karhunen-Loeve Galerkin procedure is assessed in comparison with a traditional technique employing the Boussi nesq equation, and is found to be very accurate as well as efficient. (C) 2 000 Elsevier Science S.A. All rights reserved.