Convective heat transport in a rotating fluid layer of infinite Prandtl number: optimum fields and upper bounds on Nusselt number.

Abstract

By means of the Howard-Busse method of the optimum theory of turbulence we investigate numerically upper bounds on convective heat transport for the case of infinite fluid layer with stress-free vertical boundaries rotating about a vertical axis. We discuss the case of infinite Prandtl number, 1-alpha solution of the obtained variational problem and optimum fields possessing internal, intermediate, and boundary layers. We investigate regions of Rayleigh and Taylor numbers R and Ta, where no analytical bounds can be derived, and compare the analytical and numerical bounds for these regions of R and Ta where such comparison is possible. The increasing rotation has a different influence on the rescaled optimum fields of velocity w(1), temperature theta(1) and the vertical component of the vorticity f(1). The increasing Ta for fixed R leads to vanishing of the boundary layers of w(1) and theta(1). Opposite to this, the increasing Ta leads first to a formation of boundary layers of the field f(1) but further increasing the rotation causes vanishing of these boundary layers. We obtain optimum profiles of the horizontal averaged total temperature field which could be used as hints for construction of the background fields when applying Doering-Constantin method to the problems of rotating convection. The wave number alpha(1) corresponding to the optimum fields follows the asymptotic relationship alpha(1)=(R/5)(1/4) for intermediate Rayleigh numbers. However, when R becomes large with respect to Ta, after a transition region, the power law for alpha(1) becomes close to the power law for the case without rotation. The Nusselt number Nu is close to the nonrotational bound 0.32R(1/3) for the case of large R and small Ta. Nu decreases with increasing Taylor number. Thus, the upper bounds reflect the tendency of inhibiting thermal convection by increasing rotation for a fixed Rayleigh number. For the regions of Rayleigh and Taylor numbers where the numerical and asymptotic bounds on Nu can be compared, the numerical bounds are about 70% lower than the asymptotic bounds.