Skip to main content Accessibility help
×
×
Home

Stability transitions and turbulence in horizontal convection

  • Bishakhdatta Gayen (a1), Ross W. Griffiths (a1) and Graham O. Hughes (a1)

Abstract

Recent results have shown that convection forced by a temperature gradient along one horizontal boundary of a rectangular domain at a large Rayleigh number can be turbulent in parts of the flow field. However, the conditions for onset of turbulence, the dependence of flow and heat transport on Rayleigh number, and the roles of large and small scales in the flow, have not been established. We use three-dimensional direct numerical simulation (DNS) and large-eddy simulation (LES) over a wide range of Rayleigh numbers, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}Ra\sim 10^8\mbox{--}10^{15}$ , for Prandtl number $Pr=5$ and a small aspect ratio, and show that a sequence of several stability transitions at $Ra \sim 10^{10}\mbox{--} 10^{11}$ defines a change from laminar to turbulent flow. The Prandtl number dependence too is examined at $Ra = 5.86 \times 10^{11}$ . At the smallest $Ra$ considered the thermal boundary layer is characterized by a balance of viscous stress and buoyancy, whereas inertia and buoyancy dominate in the large- $Ra$ regime. The change in the momentum balance is accompanied by turbulent enhancement of the overall heat transfer, although both laminar and turbulent regimes give $Nu\sim Ra^{1/5}$ . The results support both viscous and inviscid theoretical scaling models from previous work. The mechanical energy budget for an intermediate range of Rayleigh numbers above onset of instability ( $10^{10}<Ra<10^{13}$ ) reveals that the small scales of motion are produced predominantly by thermal convection, whereas at $Ra \ge 10^{14}$ shear instability of the large-scale flow begins to play a dominant role in sustaining the small-scale turbulence. Extrapolation to ocean conditions requires knowledge of the inertial regime identified here, but the simulations show that the corresponding asymptotic balance has not been fully realized by $Ra \sim 10^{15}$ .

Copyright

Corresponding author

Email address for correspondence: bishakhdatta.gayen@anu.edu.au

References

Hide All
Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503537.
Armenio, V. & Sarkar, S. 2002 An investigation of stably stratified turbulent channel flow using large-eddy simulation. J. Fluid Mech. 459, 142.
Barkan, R., Winters, K. B. & Llewellyn Smith, S. 2013 Rotating horizontal convection. J. Fluid Mech. 723, 556586.
Chiu-Webster, S., Hinch, E. J. & Lister, J. R. 2008 Very viscous horizontal convection. J. Fluid Mech. 611, 395426.
Gayen, B., Griffiths, R. W., Hughes, G. O. & Saenz, J. A. 2013a Energetics of horizontal convection. J. Fluid Mech. 716, R10.
Gayen, B., Hughes, G. O. & Griffiths, R. W. 2013b Completing the mechanical energy pathways in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 111, 124301.
Gayen, B. & Sarkar, S. 2010 Turbulence during the generation of internal tide on a critical slope. Phys. Rev. Lett. 104, 218502.
Gayen, B. & Sarkar, S. 2011 Direct and large eddy simulations of internal tide generation at a near critical slope. J. Fluid Mech. 681, 4879.
Gayen, B., Sarkar, S. & Taylor, J. R. 2010 Large eddy simulation of a stratified boundary layer under an oscillatory current. J. Fluid Mech. 643, 233266.
Germano, M., Piomelli, U., Moin, P. & Cabot, W. H. 1991 A dynamic subgrid-scale eddy viscosity model. Phys. Fluids 3 (7), 17601765.
Griffiths, R. W., Hughes, G. O. & Gayen, B. 2013 Horizontal convection dynamics: insights from transient adjustment. J. Fluid Mech. 726, 559595.
Holford, J. M. & Linden, P. F. 1999 Turbulent mixing in a stratified fluid. Dyn. Atmos. Oceans 30, 173198.
Hughes, G. O., Gayen, B. & Griffiths, R. W. 2013 Available potential energy in Rayleigh–Bénard convection. J. Fluid Mech. 729, 110; R3.
Hughes, G. O. & Griffiths, R. W. 2006 A simple convective model of the global overturning circulation, including effects of entrainment into sinking regions. Ocean Model. 12, 4679.
Hughes, G. O. & Griffiths, R. W. 2008 Horizontal convection. Annu. Rev. Fluid Mech. 40, 185208.
Hughes, G. O., Griffiths, R. W., Mullarney, J. C. & Peterson, W. H. 2007 A theoretical model for horizontal convection at high Rayleigh number. J. Fluid Mech. 581, 251276.
Hughes, G. O., Hogg, A. McC. & Griffiths, R. W. 2009 Available potential energy and irreversible mixing in the meridional overturning circulation. J. Phys. Oceanogr. 39, 31303146.
Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 323.
Lund, T. S.1997 On the use of discrete filters for large eddy simulation. In Annual Research Briefs, pp. 83–95. Centre for Turbulence Research, NASA Ames–Stanford University.
Mullarney, J. C., Griffiths, R. W. & Hughes, G. O. 2004 Convection driven by differential heating at a horizontal boundary. J. Fluid Mech. 516, 181209.
Ni, R., Huang, S.-D. & Xia, K.-Q. 2011 Local energy dissipation rate balances local heat flux in the centre of turbulent thermal convection. Phys. Rev. Lett. 107, 174503.
Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. J. 2001 The wind in confined thermal convection. J. Fluid Mech. 449, 169178.
Paparella, F. & Young, W. R. 2002 Horizontal convection is non-turbulent. J. Fluid Mech. 466, 205214.
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135167.
Prastowo, T., Griffiths, R. W., Hughes, G. O. & Hogg, A. McC. 2008 Mixing efficiency in controlled exchange flows. J. Fluid Mech. 600, 235244.
Prastowo, T., Griffiths, R. W., Hughes, G. O. & Hogg, A. M. C. 2009 Effects of topography on the cumulative mixing efficiency in exchange flows. J. Geophys. Res. 114 (C8), C08008.
Rossby, H. T. 1965 On thermal convection driven by non-uniform heating from below: an experimental study. Deep-Sea Res. 12, 916.
Rossby, H. T. 1998 Numerical experiments with a fluid heated non-uniformly from below. Tellus A 50, 242257.
Scotti, A. & White, B. 2011 Is horizontal convection really non-turbulent? Geophys. Res. Lett. 38, L21609.
Shishkina, O., Stevens, R. J. A. M., Grossmann, S. & Lohse, D. 2010 Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution. New J. Phys. 12, 075022.
Stevens, R. J. A. M., Verzicco, R. & Lohse, D. 2010 Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection. J. Fluid Mech. 643, 495507.
Stewart, K. D., Hughes, G. O. & Griffiths, R. W. 2011 When do marginal seas and topographic sills modify the ocean density structure? J. Geophys. Res. 116, C08021.
Stommel, H. 1962 On the smallness of sinking regions in the ocean. Proc. Natl Acad. Sci. USA 48, 766772.
Tailleux, R. 2009 On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models and the ocean heat engine controversy. J. Fluid Mech. 638, 339382.
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.
Vreman, B., Geurts, B. & Kuerten, H. 1997 Large-eddy simulation of the turbulent mixing layer. J. Fluid Mech. 339, 357390.
Winters, K. B. & D’Asaro, E. A. 1994 Three-dimensional wave instability near a critical level. J. Fluid Mech. 272, 255284.
Winters, K. B., Lombard, P. N., Riley, J. J. & D’Asaro, E. A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115128.
Winters, K. B. & Young, W. R. 2009 Available potential energy and buoyancy variance in horizontal convection. J. Fluid Mech. 629, 221230.
Zang, Y., Street, R. L. & Koseff, J. R. 1993 A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Phys. Fluids A 5 (12), 31863196.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

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