On the forced heat transfer from a hot film embedded in the wall in two-dimensional unsteady flow

Abstract

An incompressible fluid of constant thermal diffusivity D flows with velocity u = Sβ(ωt) y in the x direction, where S is a scaling factor for the velocity gradient at the wall y = 0, and β(ωt) is a positive function of time t, with characteristic frequency ω. The region 0 [les ] x [les ] l of the wall is occupied by a heated film of temperature T₁, the rest of the wall being insulating. Far from the film the fluid temperature is T₀ < T₁. Using boundary-layer theory, we calculate the heat transfer from the film by means of two asymptotic expansions, a regular one for small values of the frequency parameter $\epsilon (x) = \omega(9x)^{\frac{2}{3}} D^{-\frac{1}{3}} S^{-\frac{2}{3}}$ and a singular one (requiring the use of matched asymptotic expansions) for large values of ε. We notice the appearance of eigenfunctions in the large-ε expansion, where they are to be expected on physical grounds in order to take account of upstream conditions. Numerical computations are made for the case of sinusoidal oscillations, where β(ωt) = 1 + α sin ωt, α < 1 (three values of α, = 0·2, 0·5, 0·8, were chosen); there is seen to be no satisfactory overlap between the two expansions–the small-ε expansion is quite accurate for ε < 5·0 (especially for the smaller values of α) and the large-ε expansion is quite accurate for ε > 10·0. Approximate overlap is declared to occur at ε = 8·0.

The theory is used to calculate the response in oscillatory flow of the hot-film anemometer developed by Seed & Wood (1970a, b) to measure blood velocities in large arteries. The velocity gradient over the film (embedded in the surface of a larger probe) is obtained from the theory of the companion paper (Pedley 1972) on the assumption that the probe resembles a semi-infinite flat plate. The deviations observed in unsteady calibration experiments between the unsteady response of the anemometer and its steady response are predicted qualitatively by the theory, but quantitative agreement is in general unsatisfactory. The probable sources of this error, and the possibility of removing them, are discussed. The quasi-steady calibration curve used by Seed & Wood (1971) is suspect at low instantaneous velocities, but is shown to be adequate for the turbulence measurements of Nerem & Seed (1972). The theory is also applied to the experiments of Caro & Nerem (1972) on mass transfer to segments of arterial wall, and it is shown that oscillations characteristic of the cardiovascular system will have a negligible effect on the mean mass transfer.