Integral Transform Benchmarks of Diffusion, Convection–Diffusion, and Conjugated Problems in Complex Domains

The Generalized Integral Transform Technique (GITT) is reviewed as a hybrid numerical–analytical approach for linear or nonlinear diffusive and convective–diffusive partial differential formulations, including an important class of conjugated problems in heat transfer and fluid flow analyses. This chapter focus is on the handling of irregular regions and heterogeneous domains, as a tribute to Prof. D. B. Spalding, who stimulated this research direction in a private communication with the first author, back in 1994. First, formal solutions for nonlinear diffusion and convection–diffusion formulations are reviewed, including the alternatives of adopting nonlinear and/or convective eigenvalue problems, either on total or partial transformation schemes. Next, the GITT itself is formalized in the solution of linear and nonlinear eigenvalue problems, including the direct integral transformation of problems defined in irregular domains, based on simpler auxiliary eigenvalue problems written for the same geometry. Then, a single domain reformulation strategy is discussed, which accounts for heterogeneities on either physical properties or geometrical forms, by rewriting the different media transitions as space variable equation coefficients and source terms. The two complementary strategies are then illustrated through representative examples in convection and conjugated conduction–convection problems, confirming the excellent convergence characteristics of the proposed eigenfunction expansions, toward the establishment of sets of benchmark reference results. The present hybrid solutions are also co-verified against results from purely numerical general-purpose CFD codes.