Upper bounds on convective heat transport in a rotating fluid layer of infinite prandtl number: case of intermediate taylor numbers
By means of the Howard-Busse method of the optimum theory of turbulence we obtain upper bounds on the convective heat transport in a heated from below layer of fluid of infinite Prandtl number rotating with a constant angular velocity about the vertical axis. We consider the region of intermediate Taylor numbers: alpha41<<Ta<<alpha61 where alpha(1) is the wave number connected to the 1-alpha-solution of the variational problem. The studied optimum fields possess a three-layer or four-layer structure: in addition to the internal, intermediate, and boundary layers, Ekman layers could arise between the intermediate and boundary ones. For the discussed interval of Taylor numbers the intermediate layers do not expand in the direction of the internal layers. We present an asymptotic theory for the case of the fluid layer with rigid lower boundary and stress-free upper boundary. We use an improved solution of the Euler-Lagrange equations of the variational problem for the intermediate sublayer of the optimum field. This solution leads also to correction of the thicknesses of the boundary layers and to lowering of the upper bounds on the convective heat transport for the cases of fluid layer with stress-free or with rigid boundaries. Thus the known upper bounds for these cases can be treated as upper bounds on the upper bounds on the convective heat transport. For the case of the fluid layer with stress-free boundaries the four-layer optimum fields leads to bounds on the convective heat transport which change from R(1/3) at the lower boundary of their interval of validity to values slightly large than R(2/7) near the upper boundary of the interval of validity. Finally we discuss the area of application of the obtained bounds with respect to the Taylor number Ta and Rayleigh number R.
Vitanov
NK
Max-Planck-Institut fur Physik Komplexer Systeme, Nothnitzer Strasse 38, D-01187 Dresden, Germany.