The heat transport and spectrum of thermal turbulence

Abstract In this paper a theoretical investigation is made of various properties of the steady-state inhomogeneous turbulent convection of heat in a fluid between horizontal conducting surfaces. An upper limit to the heat transport is found subject to the constraint that some minimum eddy size exists which is effective in this transport. The spectrum of convecting motions, the mean thermal gradients at each point and the eddy conductivity are then determined in terms of the minimum eddy size. The relation between the boundary conditions and eddy size is studied by an extension of the work of Pellew & Southwell using the mean thermal gradients deduced when n0 modes of motion are present to establish the Rayleigh number at which the (n0+1)th mode first becomes unstable. In a final section the spectra and mean-square values of the fluctuating velocity and temperature fields are estimated from the Boussinesq form of the hydrodynamic equations. The previously reported experimental heat transports are within 10% of those predicted. The discrete transitions are within the error limits of the observations. However, further data must be mgathered to justify the use of minimum eddy size as a defining parameter in situations of geophysical scale.