WHOLE TIME-DOMAIN APPROACH TO THE INVERSE NATURAL CONVECTION PROBLEM

Abstract

A numerical solution is presented for the inverse natural convection problem, based on the optimization approach, without specifying any representation of the unknown boundary condition. In this context the minimization of the error, which is at the heart of the solution algorithm, is achieved using conjugate gradients in infinite-dimensional function space. The gradients of the object functional associated with the error are obtained from a system of adjoint equations. The direct, sensitivity, and adjoint equation systems are salved numerically, using a implicit, control volume discretization procedure. Results are presented for two-dimensional flow in a square cavity heated from the side, with the remaining walls adiabatic, at several Rayleigh numbers, for both an unsteady uniform flux and a steady nonuniform flux. It is found that the sensitivity, and thereby the accuracy and stability of the method, are affected by convection and thus depend on the Rayleigh number as well as the type of boundary conditions. At the present state of the art the optimization approach by conjugate gradients in infinite-dimensional function space was shown to provide satisfactory results for inverse convection at Rayleigh numbers <104 with an imposed heat flux of the form q = -sin ( π t) cos ( π y).

Notes

Address correspondence to Professor Michel Prud'homme, Department of Mechanical Engineering, Ecole Polytechnique de Montreal, P.O. Box 6079, Montreal, Quebec IDe 3A7, Canada. E-mail: [email protected]