A note on fluctuating heat transfer at small Péclet numbers

Abstract

The hot-wire anemometer, used for recording speed variations in turbulent flow, involves in its working principle the unsteady heat transfer from a hot fixed surface to a fluctuating air stream moving past the surface. If the wire is maintained at a constant (high) temperature, the rate of loss of heat from the wire changes with the velocity of the incident stream, and the compensating rate of gain of heat, produced by the Joule heating effect of the electric current, changes, correspondingly. The accompanying change of current can be measured, and used to calculate the varying velocity of the air stream. The hot wire may have a diameter as low as 10⁻⁴ in. and the Reynolds number of the flow is then of the order of 0.05 for each ft. per sec of velocity. With low velocities, of the order of 10 or 20 ft./sec, the flow past the wire is in the range of small Reynolds number, and the exact equations of flow may be approximated by simpler equations in the manner of Oseen's theory (Lamb 1932). The approximate equations are not easy to solve when the flow is compressible, as it will be in the presence of the large temperature differences imposed by the heat of the wire. If, however, the temperature differences are assumed to be small, the approximate energy equation is no longer linked with the equations of continuity and momentum, and it may be solved without knowledge of the velocity field. The purpose of this note is to give the solution for the temperature field when a warm circular cylinder or a warm sphere is held at rest in a fluctuating stream.