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 n 0 modes of motion are present to establish the Rayleigh number at which the ( n 0 +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.

Turbulent convection in liquid metal with and without rotation

“…which is attributed to the marginal stability of the thermal boundary layer (33). Low Pr fluids such as liquid metals are more This article is a PNAS Direct Submission.…”

Laboratory-numerical models of rapidly rotating convection in planetary cores

“…There are two well-known Nu ∼ Ra α scaling regimes of RBC heat transfer that are accessible with our experiments. One classical prediction, first theorized by Malkus (1954), is the α = 1/3 relation. Malkus' arguments apply to systems containing vigorous convective mixing, where the bulk fluid becomes isothermal and the time-averaged temperature gradients are localized to thin thermal boundary layers adjacent to the top and bottom of the fluid layer.…”

“…argue that boundary layer physics controls rotating convective heat transfer in water. By assuming that Malkus' (1954) marginal boundary layer arguments hold in a rapidly rotating system, they develop theoretical arguments predicting that rotating convective heat transfer scales steeply:…”

“…In addition, our systematic approach provides the opportunity to contextualize core fluid dynamics, using the predictions derived in the well-established body of convection physics literature (e.g. Malkus 1954;Kraichnan 1962;Julien et al 1996;Grossmann & Lohse 2000;Sprague et al 2006;Ahlers et al 2009;Grooms & Whitehead 2015).…”

Bounds on dissipation in stress-driven flow

“…Such decompositions can then be used to construct rigorous bounds on flow quantities of interest, consistently with imposed dynamic and kinematic constraints, using conventional techniques of the calculus of variations. Kerswell (1997Kerswell ( , 1998Kerswell ( , 2001) demonstrated that this method produces the complementary variational problem to the earlier approach pioneered by Howard (1963Howard ( , 1972Howard ( , 1990 and Busse (1969aBusse ( , b, 1970Busse ( , 1978, developing ideas originally proposed by Malkus (1954Malkus ( , 1956.…”