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The Generalized Graetz Problem in Finite Domains

Fehrenbach, Jérôme and De Gournay, Frédéric and Pierre, Charles and Plouraboué, Franck The Generalized Graetz Problem in Finite Domains. (2012) SIAM Journal on Applied Mathematics, vol. 72 (n° 1). pp. 99-123. ISSN 0036-1399

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Official URL: http://dx.doi.org/10.1137/11082542X

Abstract

We consider the generalized Graetz problem associated with stationary convection-diffusion inside a domain having any regular three-dimensional translationally invariant section and finite or semi-infinite extent. Our framework encompasses any previous “extended” and “conjugated” Graetz generalizations and provides theoretical bases for computing the orthogonal set of generalized two-dimensional Graetz modes. The theoretical framework includes both heterogeneous and possibly anisotropic diffusion tensors. In the case of semi-infinite domains, the existence of a bounded solution is shown from the analysis of two-dimensional operator eigenvectors which form a basis of L2 . In the case of finite domains a similar basis can be exhibited, and the mode’s amplitudes can be obtained from the inversion of newly defined finite domain operator. Our analysis includes both the theoretical and practical issues associated with this finite domain operator inversion as well as its interpretation as a multireflection image method. Error estimates are provided when numerically truncating the spectrum to a finite number of modes. Numerical examples are validated for reference configurations and provided in nontrivial cases. Our methodology shows how to map the solution of stationary convection-diffusion problems in finite three-dimensional domains into a two-dimensional operator spectrum, which leads to a drastic reduction in computational cost.

Item Type:Article
Additional Information:Thanks to SIAM editor. The definitive version is available at http://epubs.siam.org
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS
Université de Toulouse > Institut National Polytechnique de Toulouse - INPT
Université de Toulouse > Institut National des Sciences Appliquées de Toulouse - INSA
Université de Toulouse > Université Paul Sabatier-Toulouse III - UPS
Université de Toulouse > Université de Toulouse II-Le Mirail - UTM
Université de Toulouse > Université de Toulouse I-Sciences Sociales - UT1
Other partners > Université de Versailles Saint-Quentin-en-Yvelines -UVSQ (FRANCE)
Other partners > Université de Pau et des Pays de l'Adour - UPPA (FRANCE)
Laboratory name:
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Deposited By: Franck PLOURABOUE

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