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

Pierre, Charles and Plouraboué, Franck and Quintard, Michel Convergence of the Generalized Volume Averaging Method on a Convection-Diffusion Problem: A Spectral Perspective. (2005) SIAM Journal on Applied Mathematics , 66 (1). 122-152. ISSN 0036-1399

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Official URL: http://dx.doi.org/10.1137/040610015

## Abstract

A mixed formulation is proposed and analyzed mathematically for coupled convection-diﬀusion in heterogeneous medias. Transfer in solid parts driven by pure diﬀusion is coupled with convection-diﬀusion transfer in ﬂuid parts. This study is carried out for translation-invariant geometries (general inﬁnite cylinders) and unidirectional ﬂows. This formulation brings to the fore a new convection-diﬀusion operator, the properties of which are mathematically studied: its symmetry is ﬁrst shown using a suitable scalar product. It is proved to be self-adjoint with compact resolvent 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-diﬀusion 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 inﬁnite 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.

Item Type: Article Thanks to SIAM editor. The original PDF is available at http://epubs.siam.org/siap/ International peer-reviewed journal French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)Other partners > Ecole Centrale de Nantes (FRANCE)Université de Toulouse > Institut National Polytechnique de Toulouse - INPT (FRANCE)Université de Toulouse > Université Toulouse III - Paul Sabatier - UPS (FRANCE)Other partners > Université de Nantes (FRANCE) download Franck PLOURABOUE 03 May 2012 09:02

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