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Convective Eigenvalue Problems for Convergence Enhancement of Eigenfunction Expansions in Convection-Diffusion Problems

[+] Author and Article Information
Renato M. Cotta

LabMEMS − Laboratory of Nano & Microfluidics and Microsystems, Mechanical Engineering Department – PEM, POLI/COPPE, Nanoengineering Department − PENT, COPPE, Federal University of Rio de Janeiro, UFRJ, Cx. Postal 68503 – Cidade Universitária, 21945-970, Rio de Janeiro, RJ, Brazil; Interdisciplinary Nucleus for Social Development - NIDES/CT, UFRJ
cotta@mecanica.coppe.ufrj.br

Carolina P. Naveira-Cotta

LabMEMS − Laboratory of Nano & Microfluidics and Microsystems, Mechanical Engineering Department – PEM, POLI/COPPE, Nanoengineering Department − PENT, COPPE, Federal University of Rio de Janeiro, UFRJ, Cx. Postal 68503 – Cidade Universitária, 21945-970, Rio de Janeiro, RJ, Brazil
carolina@mecanica.coppe.ufrj.br

Diego Knupp

Mechanical Eng. Dept. – Polytechnic Institute, State University of Rio de Janeiro, IPRJ/UERJ, Nova Friburgo, RJ, Brazil
diegoknupp@gmail.com

1Corresponding author.

ASME doi:10.1115/1.4037576 History: Received December 18, 2016; Revised June 27, 2017

Abstract

Application of the Generalized Integral Transform Technique (GITT) in the solution of a class of linear or nonlinear convection-diffusion problems is considered, by fully or partially incorporating the convective effects into the chosen eigenvalue problem that forms the basis of the proposed eigenfunction expansion. The aim is to improve convergence of the eigenfunction expansions, especially for formulations with significant convective effects, by simultaneously accounting for the relative importance of convective and diffusive effects within the eigenfunctions themselves, in comparison against the more traditional GITT solution path, which adopts a purely diffusive eigenvalue problem and fully incorporates the convective effects into the problem source term. After identifying a characteristic convective operator, and through a straightforward algebraic transformation of the original convection-diffusion problem, basically by redefining the coefficients associated with the transient and diffusive terms, the characteristic convective term is merged into a generalized diffusion operator with a space variable diffusion coefficient. The generalized diffusion problem then naturally leads to the eigenvalue problem to be chosen in proposing the eigenfunction expansion for the linear situation, as well as for the appropriate linearized version in the case of a nonlinear application. The resulting eigenvalue problem with space variable coefficients is then solved through the GITT itself, yielding the corresponding algebraic eigenvalue problem, upon selection of a simple auxiliary eigenvalue problem of known analytical solution. The developed methodology is illustrated for linear and nonlinear applications, both in one-dimensional and multidimensional formulations, as represented by a few examples based on Burgers equation.

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