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Experimental investigation into localized instabilities of mixed Rayleigh–Bénard–Poiseuille convection

Published online by Cambridge University Press:  02 December 2009

EMERIC GRANDJEAN  and
PETER A. MONKEWITZ
EMERIC GRANDJEAN
Affiliation:
Laboratoire de Mécanique des Fluides, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Lausanne, Switzerland
PETER A. MONKEWITZ*
Affiliation:
Laboratoire de Mécanique des Fluides, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Lausanne, Switzerland
*
†Email address for correspondence: peter.monkewitz@epfl.ch
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Abstract

The stability of the Rayleigh–Bénard–Poiseuille flow in a channel with large transverse aspect ratio (ratio of width to vertical channel height) is studied experimentally. The onset of thermal convection in the form of ‘transverse rolls’ (rolls with axes perpendicular to the Poiseuille flow direction) is determined in the Reynolds–Rayleigh number plane for two different working fluids: water and mineral oil with Prandtl numbers of approximately 6.5 and 450, respectively. By analysing experimental realizations of the system impulse response it is demonstrated that the observed onset of transverse rolls corresponds to their transition from convective to absolute instability. Finally, the system response to localized patches of supercriticality (in practice local ‘hot spots’) is observed and compared with analytical and numerical results of Martinand, Carrière & Monkewitz (J. Fluid Mech., vol. 502, 2004, p. 175 and vol. 551, 2006, p. 275). The experimentally observed two-dimensional saturated global modes associated with these patches appear to be of the ‘steep’ variety, analogous to the one-dimensional steep nonlinear modes of Pier, Huerre & Chomaz (Physica D, vol. 148, 2001, p. 49).

JFM classification

Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 640 , 10 December 2009 , pp. 401 - 419
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Boussinesq, J. 1903 Théorie analytique de la chaleur, mise en harmonie avec la thermodynamique et la théorie mécanique de la lumière, Tome II: Refroidissement et échauffement par rayonnement. Conductibilité. Courant de convection. Gauthier-Villars.Google Scholar
Busse, F. H. & Riahi, N. 1980 Nonlinear convection in a layer with nearly insulating boundaries. J. Fluid Mech. 96, 243–256.CrossRefGoogle Scholar
Carrière, P. & Monkewitz, P. A. 1999 Convective versus absolute instability in mixed Rayleigh–Bénard–Poiseuille convection. J. Fluid Mech. 384, 243–262.CrossRefGoogle Scholar
Chapman, C. J. & Proctor, M. R. E. 1980 Nonlinear Rayleigh–Bénard convection between poorly conducting boundaries. J. Fluid Mech. 101, 759–782.CrossRefGoogle Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37 (1), 357–392.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.CrossRefGoogle Scholar
Grandjean, E. 2008 Experimental investigation into localized instabilities of mixed Rayleigh–Bénard–Poiseuille convection. PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland.CrossRefGoogle Scholar
Hurle, D. T. J., Jakeman, E. & Pike, E. R. 1967 On the solution of the Bénard problem with boundaries of finite conductivity. R. Soc. Lond. Proc. A 296, 469–475.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
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.CrossRefGoogle Scholar
Martinand, D., Carrière, P. & Monkewitz, P. A. 2004 Envelope equations for the Rayleigh–Bénard–Poiseuille system. Part 2. Linear global modes in the case of two-dimensional non-uniform heating. J. Fluid Mech. 502, 175–197.CrossRefGoogle Scholar
Martinand, D., Carrière, P. & Monkewitz, P. A. 2006 Three-dimensional global instability modes associated with a localized hot spot in Rayleigh–Bénard–Poiseuille convection. J. Fluid Mech. 551, 275–301.CrossRefGoogle Scholar
Müller, H. W. 1990 Thermische Konvection in horizontaler Scherströmung. PhD thesis, Universitat des Saarlandes, Sarrebrück, Germany.Google Scholar
Müller, H. W., Lücke, M. & Kamps, M. 1992 Transversal convection patterns in horizontal shear flow. Phys. Rev. A 45, 3714–3726.CrossRefGoogle ScholarPubMed
Nicolas, X. 2002 Revue bibliographique sur les écoulements de Poiseuille–Rayleigh–Bénard. Intl J. Therm. Sci. 41, 961–1016.CrossRefGoogle 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.CrossRefGoogle Scholar
Oppenheim, A. V. & Schafer, R. W. 1989 Discrete-Time Signal Processing. Prentice-Hall.Google Scholar
Ouazzani, M. T., Caltagirone, J. P., Meyer, G. & Mojtabi, A. 1989 Etude numérique et expérimentale de la convetion mixte entre deux plans horizontaux á température différentes. Intl J. Heat Mass Transfer 32, 261–269.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.CrossRefGoogle Scholar
Pier, B. & Huerre, P. 2001 Nonlinear synchronization in open flows. J. Fluids Struct. 15 (3–4), 471–480.CrossRefGoogle Scholar
Pier, B., Huerre, P. & Chomaz, J.-M. 2001 Bifurcation to fully nonlinear synchronized structures in slowly varying media. Physica D 148, 49–96.Google Scholar
Sameen, A. & Govindarajan, R. 2007 The effect of wall heating on instability of channel flow. J. Fluid Mech. 577, 417–442.CrossRefGoogle Scholar
Settles, G. S. 2001 Schlieren and Shadowgraph Techniques. Springer.CrossRefGoogle Scholar
Silveston, P. L. 1958 Wärmedurchgang in waagerechten Flüssigkeitsschichten. Forsch. Ingenieurw. 24, 59–69.CrossRefGoogle Scholar
Stengel, K. C., Oliver, D. S. & Booker, J. R. 1982 Onset of convection in a variable-viscosity fluid. J. Fluid Mech. 120, 411–431.CrossRefGoogle Scholar
Sveen, J. K. 2004 An introduction to MatPIV v.1.6.1. Eprint no. 2, ISSN 0809-4403. Department of Mathematics, University of Oslo, Oslo, Norway. http://www.math.uio.no/~jks/matpiv.Google Scholar
Trainoff, S. P. 1997 Rayleigh–Bénard convection in the presence of a weak lateral flow. PhD thesis, University of California, Santa Barbara, CA.Google Scholar
White, F. M. 1974 Viscous Fluid Flow. McGraw-Hill.Google Scholar