This paper first describes an apparatus for measuring the Nusselt number N versus the Rayleigh number R of convecting normal liquid 4He layers. The most important feature of the apparatus is its ability to provide layers of different heights d, and hence different aspect ratios T. The horizontal cross-section of each layer is circular, and r is defined by r = D/2d where D is the diameter of the layer. We report results for 2.4 ≤ Γ ≤ 16 and for Prandtl numbers Pr spanning 0.5 ≲ Pr ≲ 0.9. These results are presented in terms of the slope N1— RcdN/dR evaluated just above the onset of convection at Rc. We find that N1is only a slowly increasing function of r in the range 6 ≤ Γ≤ 16, and that it has a value there which is quite close to 0.72. This value of N1is in good agreement with variational calculations by Ahlers et al. (1981) pertinent to parallel convection rolls in cylindrical geometry. Particularly for jT s≲ 6, we find additional small-scale structure in N1associated with changes in the number of convection rolls with changing Γ. An additional test of the linearized hydrodynamics is given by measurements of Rc. We find good agreement between theory and our data for Rc.
Gao, H., Metcalfe, G., Jung, T., & Behringer, R. P. (1987). Heat-flow experiments in liquid 4He with a variable cylindrical geometry. Journal of Fluid Mechanics, 174, 209-231. https://doi.org/10.1017/S0022112087000107