1 Proceedings of the 2 nd International Conference on Computational Heat and Mass Transfer COPPE/UFRJ – Federal University of Rio de Janeiro, Brazil, October 22-26, 2001 INTEGRAL TRANSFORM ALGORITHM FOR HEAT AND FLUID FLOW IN THREE DIMENSIONAL POROUS MEDIA H. Luz Neto  , J. N. N. Quaresma ♣ , and R. M. Cotta  Abstract. A hybrid numerical-analytical algorithm, based on the generalized integral transform technique, is developed to handle transient three-dimensional heat and fluid flow in cavities filled with a porous material. A general formulation and solution methodology for vertical cavities - insulated vertical walls with differential horizontal wall temperatures - is developed. To illustrate the algorithm computational behavior, this geometric configuration is more closely considered, under the Darcy model for natural convection in porous medium filled cavities, using a vorticity-vector potential formulation. Several computational aspects in the algorithm are analyzed, including the reordering scheme, a proposed modulation procedure, the initial condition perturbation effects, and the influence of intermediate error targets in the final converged solution. Results for cubic cavities are presented to demonstrate the convergence behavior of the proposed eigenfunction expansion solution and comparisons with previously reported steady-state solutions are critically performed. 1. Introduction. The technical literature is quite rich in terms of two-dimensional, steady or transient, treatments of natural convection in porous media, and the classical numerical techniques are quite developed in the solution of such mathematical formulations, under different flow models [1-3]. On the other hand, the analysis of three-dimensional problems is much more rare, especially for fully transient situations, mainly due to the sometimes prohibitive increase in computational effort associated with such purely discrete approaches. Some of the most relevant contributions to the objectives of the present analysis, in dealing with three-dimensional natural convection within porous cavities, are listed below [4-19]. The Darcy flow model is the most frequently adopted in these available works, together with the assumptions of constant and isotropic physical properties and linear variation with temperature of the buoyancy term (Boussinesq approximation). Also, the cubic geometry is the most commonly treated one and the stable situation of a vertical enclosure, i.e. with a heated base and thermally insulated vertical walls, is frequently employed as the test-case for the covalidation of solution methodologies. Although most of the available contributions in three-dimensional heat and fluid flow simulation deal with the primitive variables formulation, the vorticity-vector potential approach has been receiving increasing attention. AZIZ & HELLUMS [20], in a pioneering work, have shown that the vorticity-vector potential formulation could lead to more stable and fast simulations of three-dimensional flows. Later, HIRASAKI & HELLUMS [21] provided a set of consistent boundary conditions for the vector potential, which were reasonably simple for confined flows. This contribution was then complemented by HIRASAKI & HELLUMS [22], who employing the Helmholtz decomposition theorem, and introducing the concept of a scalar potential to the formulation, were able to reach simpler boundary conditions for the vector potential in every each situation. This formulation was then itself originally applied to three- dimensional natural convection in porous media [4]. On the other hand, along the last two decades, a hybrid numerical-analytical approach for diffusion and convection-diffusion problems has been progressively advanced, towards the automatic error-controlled solution of such partial differential problems. This approach, known as the Generalized Integral Transform Technique (GITT) [23-25], derives its basis from the classical integral transform method for the exact solution of linear transformable diffusion problems. To illustrate some of the contributions on this method for the specific class of problems of interest here, we can mention the general solution for nonlinear convection-diffusion [26] and its extension to petroleum reservoir analysis [27], followed by the solution of a number of natural convection problems in cavities under steady and transient regimen, for both porous media [28-29] or just fluid filled [30-33] two-dimensional enclosures. In all such contributions on natural convection, the streamfunction-only formulation was preferred, due to the inherent advantages in its combined use with this hybrid approach, as more closely discussed in [34]. Later on, this hybrid solution scheme was advanced to handle the three-dimensional Navier-Stokes  National Institute of Technology – INT/MCT, Ministério da Ciência e Tecnologia, Rio de Janeiro RJ, Brazil, heitorlu@int.gov.br ♣ Chemical Engineering Department – DEQ/UFPA, Universidade Federal do Pará, Belém, PA, Brazil, quaresma@ufpa.br  Mechanical Engineering Department – EE/COPPE/UFRJ, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil, cotta@lttc.coppe.ufrj.br