Heat transfer to a quadratic shear profile

Abstract

An operational method is used to obtain an exact solution for the heat transfer from a surface on which the temperature is prescribed to a stream in which the velocity profile is given by \[ u = (\mu_{e}\rho_w)^{-1}\left[\tau_w\eta + \left(\frac{1}{2\rho_w}\frac{dp}{dx}\right)\eta^2\right], \] where \[ \eta = \int_0^y \rho dy \] and other symbols have their usual meanings. The solution is expanded for small and large values of a dimensionless parameter proportional to $\sigma^{\frac{1}{4}}\tau_w/(dp/dx)^{\frac{2}{3}}$: the leading term when this is large is precisely Lighthill's (1950) expression for heat transfer at high Prandtl number, and that when it is small corresponds to Liepmann's (1958) expression for heat transfer at a separation point.

Particular attention is given to the case of heat transfer from a small element of length l maintained at a constant temperature ΔT above that of the surrounding adiabatic wall: this represents the gauge for skin friction measurement described by Bellhouse & Schultz (1966) and Brown (1967 a). It is shown that the Nusselt number for heat transfer from such an element is of the form $\alpha^{\frac{1}{3}}f(\beta/\alpha^{\frac{4}{3}}) $, where \[ \alpha = \sigma\rho_w\tau_w\frac{l^2}{\mu^2_w},\quad\beta = \bigg(\frac{\sigma\rho_w}{\mu^2_w}\bigg)\frac{dp}{dx}l^3, \] and expansions for small and large values of $\beta/\alpha^{\frac{4}{3}}$ are given. Over the whole range both are adequately represented for experimental purposes by the equation \[ c_1\alpha + \frac{c_2\beta}{Nu} = Nu^3, \] which has a form suggested by consideration of the integral approximation of Curle (1962). The experimental application of the results to both laminar and turbulent flows is discussed.