Convergence of the Generalized Volume Averaging Method on a Convection-Diffusion Problem: A Spectral Perspective

A mixed formulation is proposed and analyzed mathematically for coupled convection-diusion in heterogeneous medias. Transfer in solid parts driven by pure diusion is coupled\udwith convection-diusion transfer in uid parts. This study is carried out for translation-invariant geometries (general innite cylinders) and unidirectional ows. This formulation brings to the fore a new convection-diusion operator, the properties of which are mathematically studied: its symmetry is rst shown using a suitable scalar product. It is proved to be self-adjoint with compact\udresolvent on a simple Hilbert space. Its spectrum is characterized as being composed of a double set of eigenvalues: one converging towards −∞ and the other towards +∞, thus resulting in a nonsectorial operator. The decomposition of the convection-diusion problem into a generalized eigenvalue problem permits the reduction of the original three-dimensional problem into a two-dimensional one. Despite the operator being nonsectorial, a complete solution on the innite cylinder, associated to a step change of the wall temperature at the origin, is exhibited with the help of the operator’s two sets of eigenvalues/eigenfunctions. On the computational point of view, a mixed variational formulation is naturally associated to the eigenvalue problem. Numerical illustrations are provided for axisymmetrical situations, the convergence of which is found to be consistent with the numerical discretization.\u