Heat transfer across rough surfaces

Abstract

It is argued that the heat transfer between a roughened surface and a stream of incompressible fluid flowing over it is dependent on both the viscosity and thermal conductivity of the fluid even when the roughness is large enough for viscosity to have ceased to affect the skin friction.Concentrating on closely spaced roughness, sufficiently large for the skin friction to be independent of Reynolds number, a simple model is constructed of the flow near the surface. It consists of horseshoe eddies which wrap themselves round the individual excrescences and trail unsteadily downstream; the eddies are imagined to scour the surface and thereby to transport heat between the surface and the more vigorous flow in the neighbourhood of the roughness crests. Taken in conjunction with Reynolds analogy between temperature and velocity distributions in the fluid away from the surface, the model leads to an expression for the rate of heat transfer which contains a function of the roughness Reynolds number and the Prandtl number of the fluid whose detailed form is found by appeal to the limited experimental data available. An order-of-magnitude argument suggests that the functional form established empirically is consistent with the assumed model of the flow close to the surface.The object of the work is to establish a basis for the analysis of experimental data and for their extrapolation with respect to Reynolds number and Prandtl number.

On the Relevance of ln(z0/z0T) = kB-1

“…Data obtained in studies of bluff objects in controlled conditions seemed to agree with Owen and Thomson (1963),…”

“…The discussion that follows centres on the turbulent Stanton number, St * = w T /(u * (T 0 − T a )), as initially discussed by Owen and Thomson (1963). This quantity differs from the conventional Stanton number of fluid dynamics by substituting the fluid velocity u by the scale velocity appropriate for turbulent exchange, u * .…”

“…The issue arises in examination of proposed methods to use the thermal roughness length (z 0T ) derived from consideration of the sensible heat flux and the bulk temperature difference between the surface and the air as a basis for atmospheric surface-layer and planetary boundary-layer (PBL) computations. These methods derive from experimental findings that z 0T differs from z 0 , the conventional aerodynamic roughness length associated with the wind profile (Owen and Thomson 1963;Plate 1971;Brutsaert 1982;Garratt 1992). The issue also arises in air pollution modelling.…”

“…Relating either roughness length to actual physical features of the surface is also a continuing challenge. The relationship between z 0 and z 0T was explored in wind-tunnel studies by Owen and Thomson (1963) and in various analyses by, e.g., Chamberlain (1968Chamberlain ( , 1974. Garratt and Hicks (1973, hereafter G&H) extended the resulting formulations to the real world, by summarizing data from micrometeorological field studies as well as from physical modelling.…”

“…1 The variation with surface roughness Reynolds number of the boundary-layer resistance factor kB −1 = ln(z 0 /z 0T ), as determined for the case of heat transfer and as reported by G&H. The solid circles (•) represent the average results from datasets originally available to G&H. Two additional averages are plotted, both discussed below: black up-pointing triangle daytime results obtained over natural fallow in Zimbabwe, and black down-pointing triangle the average of daytime measurements made over grazed pasture in Alabama. The curves drawn are due to, (1) Zhang et al (2001), (2) Owen and Thomson (1963), (3) Brutsaert (1975), and (4) Kanda et al (2007). Curve (5) has been created using random numbers kB −1 decreases uniformly from a maximum value of about 2 at Re * = 10 2 to about zero at Re * = 10 5 , much as might be concluded from Fig.…”