Improved bounds on horizontal convection

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.