Comptes Rendus
Analytical development of disturbed matrix eigenvalue problem applied to mixed convection stability analysis in Darcy media
Comptes Rendus. Mécanique, Volume 336 (2008) no. 8, pp. 656-663.

This work consists in evaluating algebraically and numerically the influence of a disturbance on the spectral values of a diagonalizable matrix. Thus, two approaches will be possible; to use the theorem of disturbances of a matrix depending on a parameter, due to Lidskii and primarily based on the structure of Jordan of the no disturbed matrix. The second approach consists in factorizing the matrix system, and then carrying out a numerical calculation of the roots of the disturbances matrix characteristic polynomial. This problem can be a standard model in the equations of the continuous media mechanics. During this work, we chose to use the second approach and in order to illustrate the application, we choose the Rayleigh–Bénard problem in Darcy media, disturbed by a filtering through flow. The matrix form of the problem is calculated starting from a linear stability analysis by a finite elements method. We show that it is possible to break up the general phenomenon into other elementary ones described respectively by a disturbed matrix and a disturbance. A good agreement between the two methods was seen.

Ce travail consiste à évaluer algébriquement et numériquement l'influence d'une perturbation sur les valeurs spectrales d'une matrice diagonalisable. Ainsi, deux approches seront possibles ; utiliser le théorème de perturbations d'une matrice dépendant d'un paramètre, dû à Lidskii et essentiellement basé sur la structure de Jordan de la matrice perturbée. La seconde approche consiste à factoriser le système matriciel puis procéder à un calcul numérique des racines du polynôme caractéristique de la matrice des perturbations. Ce problème peut être un modèle type dans les équations de la mécanique des milieux continus. Au cours de ce travail, nous avons choisi d'utiliser la seconde approche et d'utiliser comme application illustrative, le problème de la convection de Rayleigh–Bénard en milieu de Darcy, perturbée par un débit filtrant. La forme matricielle du problème est calculée à partir d'une analyse de stabilité linéaire par une méthode d'éléments finis. Nous démontrons qu'il est possible de décomposer le phénomène général en d'autres élémentaires décrits respectivement par une matrice perturbée et une perturbation. Un bon accord entre les deux méthodes a été relevé.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2008.06.002
Keywords: Computational fluid mechanics, Eigenvalue problem, Perturbed matrices, Computing Fluid Dynamics: CFD, Mixed convection, Linear stability, Algebraic development
Mot clés : Mécanique des fluides numérique, Problème à valeurs propres, Matrice perturbée, Convection mixte, Stabilité linéaire, Développement algébrique

Haikel Ben Hamed 1; Rachid Bennacer 1

1 Université de Cergy-Pontoise, LEEVAM, 5, mail Gay-Lussac, 95031 Neuville-sur-Oise, France
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Haikel Ben Hamed; Rachid Bennacer. Analytical development of disturbed matrix eigenvalue problem applied to mixed convection stability analysis in Darcy media. Comptes Rendus. Mécanique, Volume 336 (2008) no. 8, pp. 656-663. doi : 10.1016/j.crme.2008.06.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2008.06.002/

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