Numerical upper bounds on convective heat transport in a layer of fluid of finite Prandtl number: Confirmation of Howard’s analytical asymptotic single-wave-number bound

Abstract

By means of the Howard-Busse method of the optimum theory of turbulence we investigate numerically the upper bounds on the Nusselt number in a heated-from-below horizontal layer of fluid of finite Prandtl number for the case of rigid boundaries. The bounds are obtained by the solutions of the Euler-Lagrange equations of a variational problem possessing up to three wave numbers. The obtained results are compared to the numerical results for the case of fluid layer with stress-free boundaries [N. K. Vitanov and F. H. Busse, “Upper bounds on heat transport in a horizontal fluid layer with stress-free boundaries,” ZAMP 48, 310 (1997)] as well as to the numerical and analytical asymptotic results obtained by Howard [“Heat transport by turbulent convection,” J. Fluid Mech. 17, 405 (1963)], Busse [“On Howard’s upper bound for heat transport by turbulent convection,” J. Fluid Mech. 37, 457 (1969)], and Strauss [“On the upper bounding approach to thermal convection at moderate Rayleigh numbers, II. Rigid boundaries,” Dyn. Atm. Oceans 1, 77 (1976)]. We show that for low and intermediate Rayleigh numbers the numerical bounds are positioned below the analytical asymptotic bounds obtained by Howard and Busse. For large Rayleigh numbers the numerical bounds tend to approach the analytical asymptotic bounds. We confirm numerically the bound obtained by Howard for the case of one-wave-number solution of the Euler-Lagrange equations. As the region of validity of the results of the analytical asymptotic theory for solutions of the Euler-Lagrange equations with two and three wave numbers lies in the area of very high Rayleigh numbers the values of the second and third wave numbers are different from their analytical asymptotic values for the values of the Rayleigh number reached by the numerical computation.