Numerical solution of optimal magnetic suppression of natural convection in magneto-hydrodynamic flows by empirical reduction of modes

摘要

The control of natural convection in magneto-hydrodynamic (MHD) flow is investigated by means of the Karhunen-Loeve Galerkin procedure [Int J Numer Meth Engng 1998;41:1133-51]. The Karhunen-Loeve Galerkin procedure, which is a type of Galerkin methods that employs the empirical eigenfunctions of the Karhunen-Loeve decomposition as basis functions, can reduce non-linear 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. In the present investigation, it is demonstrated that the Karhunen-Loeve Galerkin procedure is well suited for the problems of control or optimization, where one has to solve the governing equations repeatedly but one can also estimate the approximate solution space from the range of control variable. The performance of the Karhunen-Loeve Galerkin procedure for solving the optimal control problem of natural convection is assessed in comparison with the traditional technique employing the Boussinesq equation, and is found to be very accurate as well as efficient.