Inverse determination of a heat source from natural convection in a porous cavity

The classical/spectral conjugate gradient methods with adjoint equations are applied to the natural convection problem in a porous medium for the determination of an unknown heat source. The direct, sensitivity and adjoint equations associated with the Darcy and the Forchheimer terms are given for a Boussinesq fluid, over a square porous medium in two dimensions. Inverse solutions that were determined by a second-order scheme in space and in time and a mixed finite element method are presented for a square enclosure under known temperature boundary conditions. Questions regarding the numerical accuracy of the proposed schemes for porous flow models for recovering the strength of the unknown heat source have been addressed.

► The direct, sensitivity and adjoint equations associated with the Darcy and the Forchheimer terms are investigated.
► A second-order scheme for solving the INCP in porous media using a mixed FE method is proposed.
► An alternative algorithm for finding the profile of a time-varying heat source is examined.