Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-21T05:50:34.696Z Has data issue: false hasContentIssue false

Spatio-temporal patterns in inclined layer convection

Published online by Cambridge University Press:  06 April 2016

Priya Subramanian*
Affiliation:
Max-Planck Institute for Dynamics and Self-Organization, Göttingen 37077, Germany School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
Oliver Brausch
Affiliation:
Universität Bayreuth, Theoretische Physik I, Bayreuth 95447, Germany
Karen E. Daniels
Affiliation:
Department of Physics, North Carolina State University, NC 27695, USA
Eberhard Bodenschatz
Affiliation:
Max-Planck Institute for Dynamics and Self-Organization, Göttingen 37077, Germany
Tobias M. Schneider
Affiliation:
Max-Planck Institute for Dynamics and Self-Organization, Göttingen 37077, Germany Emergent Complexity in Physical Systems Laboratory (ECPS), École Polytechnique Fédérale de Lausanne, CH-1015, Switzerland
Werner Pesch
Affiliation:
Universität Bayreuth, Theoretische Physik I, Bayreuth 95447, Germany
*
Email address for correspondence: P.Subramanian@leeds.ac.uk

Abstract

This paper reports on a theoretical analysis of the rich variety of spatio-temporal patterns observed recently in inclined layer convection at medium Prandtl number when varying the inclination angle ${\it\gamma}$ and the Rayleigh number $R$. The present numerical investigation of the inclined layer convection system is based on the standard Oberbeck–Boussinesq equations. The patterns are shown to originate from a complicated competition of buoyancy driven and shear-flow driven pattern forming mechanisms. The former are expressed as longitudinal convection rolls with their axes oriented parallel to the incline, the latter as perpendicular transverse rolls. Along with conventional methods to study roll patterns and their stability, we employ direct numerical simulations in large spatial domains, comparable with the experimental ones. As a result, we determine the phase diagram of the characteristic complex 3-D convection patterns above onset of convection in the ${\it\gamma}{-}R$ plane, and find that it compares very well with the experiments. In particular we demonstrate that interactions of specific Fourier modes, characterized by a resonant interaction of their wavevectors in the layer plane, are key to understanding the pattern morphologies.

Type
Papers
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bergholz, R. F. 1977 Instability of steady natural convection in a vertical slot. J. Fluid Mech. 94, 743768.Google Scholar
Birikh, R. V., Gershuni, G. Z., Zhukhovitzkii, E. M. & Rudakov, R. N. 1972 On oscillatory instability of plane parallel convective motion in a vertical channel. Prikl. Mat. Mekh. 36, 745748.Google Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709778.CrossRefGoogle Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Dover.Google Scholar
de Bruyn, J. R., Bodenschatz, E., Morris, S. W., Trainoff, S. P., Hu, Y., Cannell, D. S. & Ahlers, G. 1996 Apparatus for the study of Rayleigh–Bénard convection in gases under pressure. Rev. Sci. Instrum. 67, 20432067.Google Scholar
Busse, F. H. 1989 Fundamentals of thermal convection. In Mantle Convection: Plate Tectonics and Global Dynamics (ed. Peltier, W. H.), Gordon and Breach.Google Scholar
Busse, F. H. & Clever, R. M. 1979 Instabilities of convection rolls in a fluid of moderate Prandtl number. J. Fluid Mech. 91, 319335.Google Scholar
Busse, F. H. & Clever, R. M. 1992 Three-dimensional convection in an inclined layer heated from below. J. Engng Maths 26, 149.Google Scholar
Busse, F. H. & Clever, R. M. 1996 The sequence-of-bifurcations approach towards an understanding of complex flows. In Mathematical Modelling and Simulation in Hydrodynamic Stability (ed. Riahi, D. N.), World Scientific.Google Scholar
Busse, F. H. & Clever, R. M. 2000 Bursts in inclined layer convection. Phys. Fluids 12, 21372140.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Chen, Y. M. & Pearlstein, A. J. 1989 Stability of free-convection flows of variable-viscosity fluids in vertical and inclined slots. J. Fluid Mech. 198, 513541; note that the inclination angle ( ${\it\delta}$ in this work) is measured with respect to the vertical direction.Google Scholar
Clever, R. M. & Busse, F. H. 1977 Instabilities of longitudinal convection rolls in an inclined layer. J. Fluid Mech. 81, 107127.Google Scholar
Clever, R. M. & Busse, F. H. 1995 Tertiary and Quarternary solutions for convection in a vertical fluid layer heated from the side. Chaos, Solitons Fractals 5, 17951803.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 8521111.CrossRefGoogle Scholar
Daniels, K.2002 Pattern formation and dynamics in inclined layer convection. PhD thesis, Cornell University, USA.Google Scholar
Daniels, K. E. & Bodenschatz, E. 2002 Defect turbulence in inclined layer convection. Phys. Rev. Lett. 88, 034501.Google Scholar
Daniels, K. E., Brausch, O., Pesch, W. & Bodenschatz, E. 2008 Competition and bistability of ordered undulations and undulation chaos in inclined layer convection. J. Fluid Mech. 597, 261282.Google Scholar
Daniels, K. E., Plapp, B. B. & Bodenschatz, E. 2000 Pattern formation in inclined layer convection. Phys. Rev. Lett. 84, 53205323.Google Scholar
Daniels, K. E., Wiener, R. J. & Bodenschatz, E. 2003 Localized transverse bursts in inclined layer convection. Phys. Rev. Lett. 91, 114501.CrossRefGoogle ScholarPubMed
Dominguez-Lerma, M. A., Ahlers, G. & Cannell, D. S. 1984 Marginal stability curve and linear growth rate for rotating Couette–Taylor flow and Rayleigh–Bénard convection. Phys. Fluids 27, 856860.Google Scholar
Egolf, D., Melnikov, I. V., Pesch, W. & Ecke, R. 2000 Extensive spatiotemporal chaos in Rayleigh–Bénard convection. Nature 404, 733736.Google Scholar
Fujimura, K. & Kelly, R. E. 1992 Mixed mode convection in an inclined slot. J. Fluid Mech. 246, 545568.Google Scholar
Gershuni, G. Z. & Zhukhovitzkii, E. M. 1969 Stability of plane-parallel convective motion with respect to spatial perturbations. Prikl. Mat. Mekh. 33, 855860.Google Scholar
Hart, J. E. 1971 Stability of flow in a differentially heated inclined box. J. Fluid Mech. 91, 319335.Google Scholar
Koikari, S. 2009 Planar measurements of differential diffusion in turbulent jets. ACM Trans. Math. Softw. 36, 12,1–20.Google Scholar
Lappa, M. 2009 Thermal Convection, Patterns, Evolution and Stability. Wiley.Google Scholar
Lemoult, G., Gumowski, K., Aider, J.-L. & Wesfreid, J. E. 2014 Turbulent spots in channel flow: an experimental study. Eur. Phys. J. E 37 (4), 25.Google Scholar
Pesch, W. 1996 Complex spatiotemporal convection patterns. Chaos 6, 348357.Google Scholar
Rudakov, R. N. 1967 Spectrum of perturbations and stability of convective motion between vertical planes. Prikl. Mat. Mekh. 31, 349355.Google Scholar
Ruth, D. W., Hollands, K. G. T. & Raithby, G. D. 1980 On free convection experiments in inclined air layers heated from below. J. Fluid Mech. 96, 461479.Google Scholar
Swinney, H. L. & Gollub, J. P. 1985 Hydrodynamic Instabilities and the Transition to Turbulence, 2nd edn. Springer.Google Scholar
Trainoff, S. P. & Canell, D. S. 2002 Physical optics treatment of the shadowgraph. Phys. Fluids 14, 13401363.Google Scholar
Tuckerman, L. S., Kreilos, T., Schrobsdorff, H., Schneider, T. M. & Gibson, J. F. 2014 Turbulent–laminar patterns in plane Poiseuille flow. Phys. Fluids 26 (11), 114103.Google Scholar
Vest, C. M & Arpaci, V. S. 1969 Stability of natural convection in a vertical slot. J. Fluid Mech. 36, 115.Google Scholar