The response of laminar skin friction and heat transfer to fluctuations in the stream velocity

Abstract

The laminar boundary layer in two-dimensional flow about a cylindrical body, when the velocity of the oncoming flow relative to the body oscillates in magnitude but not in direction, is analyzed mathematically. It is found that the maxima of skin friction at any point anticipate the maxima of the stream velocity, because the pressure gradient needed to speed up the main stream locally produces a given percentage increase in the slow flow near the wall sooner than it can do so in the main stream itself. For each point on the body surface there is a critical frequency ω 0 , such that for frequencies ω > ω 0 the oscillations are to a close approximation ordinary ‘shear waves’ unaffected by the mean flow; the phase advance in the skin friction is then 45°. For frequencies ω < ω 0 , on the other hand, the oscillations are closely approximated as the sum of parts proportional to the instantaneous velocity and acceleration of the oncoming stream; the phase advance in the skin friction is then tan -1 ( ω / ω 0 ). The part depending on the instantaneous velocity may be called the quasi-steady part of the oscillations. The coefficient of the acceleration of the oncoming stream in the frictional drag of the body may be called the frictional component of the virtual mass. For a flat plate in a stream of speed V , ω 0 = 0∙6 V / x at a distance x from the leading edge. If c is the length of the plate, its transient motion parallel to itself is governed solely by quasi-steady forces and this added virtual mass provided that ωc / V < 0∙6. The frictional component of the virtual mass of a flat plate or any thin obstacle is found to be approximately 0∙5 times the mass of the fluid in the boundary layer’s ‘displacement area’; it is suggested that the coefficient may need to be increased to about 0∙8 for turbulent layers. When the body surface is hot, the maxima in heat transfer from it tend to lag behind those of the stream velocity, as a result of thermal inertia, but this is counteracted to some extent by the effect of convection by the phase-advanced velocities near the wall. For layers with a favourable gradient in the mean flow, one finds that the tendency to lag predominates. For the Blasius layer, however, the two effects appear to cancel out fairly closely; and for layers with adverse pressure gradient in the main stream there seems to be phase advance at the lower frequencies. At frequencies well above ω 0 there is always a phase lag of 90°, but the amplitude of heat-transfer fluctuations is then much reduced, even though that of the skin friction fluctuations is increased. Special attention is paid to the phase lag in the heat transfer from a heated circular wire in a fluctuating stream, in the range of Reynolds number for which a laminar boundary layer exists. Curves for the amplitude and phase of the heat-transfer fluctuations as a function of frequency are given in figure 4, from calculations for the layer of nearly uniform thickness, which covers the front quadrant of the wire, and across which most of the fluctuating part of the heat transfer is believed to occur. For frequencies small compared with ω 0 = 20 V / d (where d is the diameter), the departure of the heat-transfer fluctuations from their quasi-steady form consists essentially of a time lag of the order of 0∙2 d / V .