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Linear biglobal analysis of Rayleigh–Bénard instabilities in binary fluids with and without throughflow

Published online by Cambridge University Press:  19 October 2012

Jun Hu ,
Daniel Henry ,
Xie-Yuan Yin  and
Hamda BenHadid
Jun Hu*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
Daniel Henry
Affiliation:
Laboratoire de Mécanique des Fluides et d’Acoustique, CNRS/Université de Lyon, Ecole Centrale de Lyon/Université Lyon 1/INSA de Lyon, ECL, 36 avenue Guy de Collongue, 69134 Ecully CEDEX, France
Xie-Yuan Yin
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China
Hamda BenHadid
Affiliation:
Laboratoire de Mécanique des Fluides et d’Acoustique, CNRS/Université de Lyon, Ecole Centrale de Lyon/Université Lyon 1/INSA de Lyon, ECL, 36 avenue Guy de Collongue, 69134 Ecully CEDEX, France
*
†Email address for correspondence: [email protected]
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Abstract

Three-dimensional Rayleigh–Bénard instabilities in binary fluids with Soret effect are studied by linear biglobal stability analysis. The fluid is confined transversally in a duct and a longitudinal throughflow may exist or not. A negative separation factor $\psi = \ensuremath{-} 0. 01$, giving rise to oscillatory transitions, has been considered. The numerical dispersion relation associated with this stability problem is obtained with a two-dimensional Chebyshev collocation method. Symmetry considerations are used in the analysis of the results, which allow the classification of the perturbation modes as ${S}_{l} $ modes (those which keep the left–right symmetry) or ${R}_{x} $ modes (those which keep the symmetry of rotation of $\lrm{\pi} $ about the longitudinal mid-axis). Without throughflow, four dominant pairs of travelling transverse modes with finite wavenumbers $k$ have been found. Each pair corresponds to two symmetry degenerate left and right travelling modes which have the same critical Rayleigh number ${\mathit{Ra}}_{c} $. With the increase of the duct aspect ratio $A$, the critical Rayleigh numbers for these four pairs of modes decrease and closely approach the critical value ${\mathit{Ra}}_{c} = 1743. 894$ obtained in a two-dimensional situation, one of the mode (a ${S}_{l} $ mode called mode A) always remaining the dominant mode. Oscillatory longitudinal instabilities ($k\approx 0$) corresponding to either ${S}_{l} $ or ${R}_{x} $ modes have also been found. Their critical curves, globally decreasing, present oscillatory variations when the duct aspect ratio $A$ is increased, associated with an increasing number of longitudinal rolls. When a throughflow is applied, the symmetry degeneracy of the pairs of travelling transverse modes is broken, giving distinct upstream and downstream modes. For small and moderate aspect ratios $A$, the overall critical Rayleigh number in the small Reynolds number range studied is only determined by the upstream transverse mode A. In contrast, for larger aspect ratios as $A= 7$, different modes are successively dominant as the Reynolds number is increased, involving both upstream and downstream transverse modes A and even the longitudinal mode.

JFM classification

Convection: Buoyancy-driven instability Convection: Double diffusive convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 713 , 25 December 2012 , pp. 216 - 242
Copyright
©2012 Cambridge University Press

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References

Büchel, P. & Lücke, M. 2000a Influence of through flow on binary fluid convection. Phys. Rev. E 61, 3793–3810.Google Scholar
Büchel, P. & Lücke, M. 2000b Localized perturbations in binary fluid convection with and without throughflow. Phys. Rev. E 63, 016307.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 2006 Spectral Methods: Fundamentals in Single Domains. Springer.Google Scholar
Carrière, Ph. & Monkewitz, P. A. 1999 Convective versus absolute instability in mixed Rayleigh–Bénard–Poiseuille convection. J. Fluid Mech. 384, 243–262.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.CrossRefGoogle Scholar
Gage, K. S. & Reid, W. H. 1968 The stability of thermally stratified plane Poiseuille flow. J. Fluid Mech. 33, 21–32.Google Scholar
Hu, J., Ben Hadid, H. & Henry, D. 2007 Linear stability analysis of Poiseuille–Rayleigh–Bénard flows in binary fluids with Soret effect. Phys. Fluids 19, 034101.Google Scholar
Hu, J., Yin, X. Y., Ben Hadid, H. & Henry, D. 2009 Spatiotemporal evolution of Poiseuille–Rayleigh–Bénard flows in binary fluids with Soret effect under initial pulselike disturbances. Phys. Rev. E 80, 026312.Google Scholar
Jung, Ch., Lücke, M. & Büchel, P. 1996 Influence of through-flow on linear pattern formation properties in binary mixture convection. Phys. Rev. E 54, 1510–1529.CrossRefGoogle ScholarPubMed
Jung, D. & Lücke, M. 2005 Traveling wave fronts and localized traveling wave convection in binary fluid mixtures. Phys. Rev. E 72, 026307.Google Scholar
Kato, Y. & Fujimura, K. 2000 Prediction of pattern selection due to an interaction between longitudinal rolls and transverse modes in a flow through a rectangular channel heated from below. Phys. Rev. E 62, 601–611.CrossRefGoogle Scholar
Lehoucq, R. B. & Sorensen, D. C. 1996 Deflation techniques for an implicitly re-started Arnoldi iteration. SIAM J. Matrix Anal. Appl. 17, 789–821.Google Scholar
Luijkx, J. M. & Platten, J. K. 1981 On the onset of free convection in a rectangular channel. J. Non-Equilib. Thermodyn. 6, 141–158.Google Scholar
Luijkx, J. M., Platten, J. K. & Legros, J. C. 1981 On the existence of thermoconvective rolls, transverse to a superimposed mean Poiseuille flow. Intl J. Heat Mass Transfer 24, 1287–1291.Google Scholar
Müller, H. W., Lücke, M. & Kamps, M. 1992 Transversal convection patterns in horizontal shear flow. Phys. Rev. A 45, 3714–3726.Google Scholar
Müller, H. W., Tveitereid, M. & Trainoff, S. 1993 Rayleigh–Bénard problem with imposed weak through-flow: two coupled Ginzburg–Landau equations. Phys. Rev. E 48, 263–272.Google Scholar
Nicolas, X., Luijkx, J. M. & Platten, J. K. 2000 Linear stability of mixed convection flows in horizontal rectangular channels of finite transversal extension heated from below. Intl J. Heat Mass Transfer 43, 589–610.Google Scholar
Nicolas, X., Mojtabi, A. & Platten, J. K. 1997 Two-dimensional numerical analysis of the Poiseuille–Bénard flow in a rectangular channel heated from below. Phys. Fluids 9, 337–348.Google Scholar
Ouazzani, M. T., Platten, J. K. & Mojtabi, A. 1990 Etude expérimentale de la convection mixte entre deux plans horizontaux à températures différentes – II. Intl J. Heat Mass Transfer 33, 1417–1427.CrossRefGoogle Scholar
Ouazzani, M. T., Platten, J. K., Müller, H. W. & Lücke, M. 1995 Etude de la convection mixte entre deux plans horizontaux à températures différentes – III. Intl J. Heat Mass Transfer 38, 875–886.Google Scholar
Platten, J. K. & Legros, J. C. 1984 Convection in Liquids. Springer.Google Scholar
Rosenhead, L. 1963 Laminar Boundary Layers. Oxford University Press.Google Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39, 249–315.Google Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.Google Scholar