Abstract
For the problem of horizontal convection the Nusselt number based on entropy production is bounded from above by as the horizontal convective Rayleigh number for some constant (Siggers et al., J. Fluid Mech., vol. 517, 2004, pp. 55–70). We re-examine the variational arguments leading to this ‘ultimate regime’ by using the Wentzel–Kramers–Brillouin method to solve the variational problem in the limit and exhibiting solutions that achieve the ultimate scaling. As expected, the optimizing flows have a boundary layer of thickness pressed against the non-uniformly heated surface; but the variational solutions also have rapid oscillatory variation with wavelength along the wall. As a result of the exact solution of the variational problem, the constant is smaller than the previous estimate by a factor of for no-slip and for no-stress boundary conditions. This modest reduction in indicates that the inequalities used by Siggers et al. (J. Fluid Mech., vol. 517, 2004, pp. 55–70) are surprisingly accurate.
Highlights
Horizontal convection (HC) is convection generated in a fluid layer by imposing non-uniform buoyancy along a single horizontal surface (Rossby 1965; Hughes & Griffiths 2008)
Horizontal convection can be considered as a basic problem in fluid mechanics, as a counterpoint to the more widely studied problem of Rayleigh–Bénard convection in which the fluid
These strands of HC research are entwined because buoyancy transport is a prime index of the strength of the HC, and of the strength of ocean circulation
Summary
Horizontal convection (HC) is convection generated in a fluid layer by imposing non-uniform buoyancy along a single horizontal surface (Rossby 1965; Hughes & Griffiths 2008). In that context HC is interesting because there is a restrictive bound on the rate of dissipation of kinetic energy and on net vertical buoyancy flux (Paparella & Young 2002; Scotti & White 2011; Gayen et al 2013). These strands of HC research are entwined because buoyancy (or heat) transport is a prime index of the strength of the HC, and of the strength of ocean circulation. While the exponent is unchanged from 1/3, the variational solution is of interest because it may contain clues as to the structure of the Boussinesq flows that might achieve the ultimate scaling
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