Skip to main content Accessibility help
×
Home

Spatio-temporal patterns in inclined layer convection

  • Priya Subramanian (a1) (a2), Oliver Brausch (a3), Karen E. Daniels (a4), Eberhard Bodenschatz (a1), Tobias M. Schneider (a1) (a5) and Werner Pesch (a3)...

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.

Copyright

Corresponding author

Email address for correspondence: P.Subramanian@leeds.ac.uk

References

Hide All
Bergholz, R. F. 1977 Instability of steady natural convection in a vertical slot. J. Fluid Mech. 94, 743768.
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.
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709778.
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Dover.
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.
Busse, F. H. 1989 Fundamentals of thermal convection. In Mantle Convection: Plate Tectonics and Global Dynamics (ed. Peltier, W. H.), Gordon and Breach.
Busse, F. H. & Clever, R. M. 1979 Instabilities of convection rolls in a fluid of moderate Prandtl number. J. Fluid Mech. 91, 319335.
Busse, F. H. & Clever, R. M. 1992 Three-dimensional convection in an inclined layer heated from below. J. Engng Maths 26, 149.
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.
Busse, F. H. & Clever, R. M. 2000 Bursts in inclined layer convection. Phys. Fluids 12, 21372140.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
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.
Clever, R. M. & Busse, F. H. 1977 Instabilities of longitudinal convection rolls in an inclined layer. J. Fluid Mech. 81, 107127.
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.
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 8521111.
Daniels, K.2002 Pattern formation and dynamics in inclined layer convection. PhD thesis, Cornell University, USA.
Daniels, K. E. & Bodenschatz, E. 2002 Defect turbulence in inclined layer convection. Phys. Rev. Lett. 88, 034501.
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.
Daniels, K. E., Plapp, B. B. & Bodenschatz, E. 2000 Pattern formation in inclined layer convection. Phys. Rev. Lett. 84, 53205323.
Daniels, K. E., Wiener, R. J. & Bodenschatz, E. 2003 Localized transverse bursts in inclined layer convection. Phys. Rev. Lett. 91, 114501.
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.
Egolf, D., Melnikov, I. V., Pesch, W. & Ecke, R. 2000 Extensive spatiotemporal chaos in Rayleigh–Bénard convection. Nature 404, 733736.
Fujimura, K. & Kelly, R. E. 1992 Mixed mode convection in an inclined slot. J. Fluid Mech. 246, 545568.
Gershuni, G. Z. & Zhukhovitzkii, E. M. 1969 Stability of plane-parallel convective motion with respect to spatial perturbations. Prikl. Mat. Mekh. 33, 855860.
Hart, J. E. 1971 Stability of flow in a differentially heated inclined box. J. Fluid Mech. 91, 319335.
Koikari, S. 2009 Planar measurements of differential diffusion in turbulent jets. ACM Trans. Math. Softw. 36, 12,1–20.
Lappa, M. 2009 Thermal Convection, Patterns, Evolution and Stability. Wiley.
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.
Pesch, W. 1996 Complex spatiotemporal convection patterns. Chaos 6, 348357.
Rudakov, R. N. 1967 Spectrum of perturbations and stability of convective motion between vertical planes. Prikl. Mat. Mekh. 31, 349355.
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.
Swinney, H. L. & Gollub, J. P. 1985 Hydrodynamic Instabilities and the Transition to Turbulence, 2nd edn. Springer.
Trainoff, S. P. & Canell, D. S. 2002 Physical optics treatment of the shadowgraph. Phys. Fluids 14, 13401363.
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.
Vest, C. M & Arpaci, V. S. 1969 Stability of natural convection in a vertical slot. J. Fluid Mech. 36, 115.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed