Hybrid Eigenfunction-Discretization Methodology for Solving Convective Heat Transfer Problems

This paper presents a novel methodology for the solution of problems that include diffusion and advection effects, as naturally occur in convective heat transfer problems. The methodology is based on writing the unknown temperature field in terms of eigenfunction expansions, as traditionally carried-out with the Generalized Integral Transform Technique (GITT). However, a different approach is used for handling advective derivatives. Rather than transforming the advection terms as done in traditional GITT solutions, upwind discretization schemes (UDS) are used prior to the integral transformation. With the introduction of upwind approximations, numerical diffusion is introduced, which can be used to reduce unwanted oscillations that arise at higher Péclet values. This combined methodology is termed the GITT-UDS for convective problems. The procedure is illustrated for a simple case of one-dimensional Burgers’ equation with temperature-dependent velocities. Numerical results are calculated, showing that augmenting the upwind approximation parameter can effectively reduce solution oscillations for higher Péclet values.