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Rayleigh–Bénard convection in the presence of spatial temperature modulations

Published online by Cambridge University Press:  24 February 2011

G. FREUND ,
W. PESCH  and
W. ZIMMERMANN
G. FREUND
Affiliation:
Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany
W. PESCH*
Affiliation:
Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany
W. ZIMMERMANN
Affiliation:
Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany
*
†Email address for correspondence: [email protected]
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Abstract

Motivated by recent experiments, we study a rich variation of the familiar Rayleigh–Bénard convection (RBC), where the temperature at the lower boundary varies sinusoidally about a mean value. As usual the Rayleigh number R measures the average temperature gradient, while the additional spatial modulation is characterized by a (small) amplitude δm and a wavevector qm. Our analysis relies on precise numerical solutions of suitably adapted Oberbeck–Boussinesq equations (OBE). In the absence of forcing (δm = 0), convection rolls with wavenumber qc bifurcate only for R above the critical Rayleigh number Rc. In contrast, for δm≠0, convection is unavoidable for any finite R; in the most simple case in the form of ‘forced rolls’ with wavevector qm. According to our first comprehensive stability diagram of these forced rolls in the qm – R plane, they develop instabilities against resonant oblique modes at R ≲ Rc in a wide range of qm/qc. Only for qm in the vicinity of qc, the forced rolls remain stable up to fairly large R > Rc. Direct numerical simulations of the OBE support and extend the findings of the stability analysis. Moreover, we are in line with the experimental results and also with some earlier theoretical results on this problem, based on asymptotic expansions in the limit δm → 0 and R → Rc. It is satisfying that in many cases the numerical results can be directly interpreted in terms of suitably constructed amplitude and generalized Swift–Hohenberg equations.

JFM classification

Instability: Nonlinear instability Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 673 , 25 April 2011 , pp. 318 - 348
Copyright
Copyright © Cambridge University Press 2011

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