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Heat Transfer Through a Compressible Laminar Boundary Layer

Published online by Cambridge University Press:  07 June 2016

N. Curle
N. Curle*
Affiliation:
Department of Aeronautics and Astronautics, University of Southampton
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Summary

A simple method is presented for calculating the heat transfer rate through a compressible laminar boundary layer. The temperature of the wall is assumed to be uniform, the viscosity being proportional to the absolute temperature with a ratio which may vary in an appropriate manner with position along the wall; the Prandtl number is arbitrary but greater than about 0·5.

The method assumes, following Lighthill, that heat transfer rates are determined mainly by the form of the velocity field in regions close to the wall. After applying the Ulingworth-Stewartson transformation the velocity is expanded in powers of the distance Y normal to the wall, three terms being retained. The first term, proportional to the skin friction, is zero near to a position of boundary layer separation. The second term is proportional to the pressure gradient and also to the wall temperature. Accordingly it can become close to zero when the wall is sufficiently cooled. The third term becomes important only when the first two are simultaneously close to zero and is proportional to the heat transfer rate. Since the three terms cannot, in fact, ever be all zero at the same position they form a uniformly valid non-trivial approximation to the velocity close to the wall.

With this velocity profile, and following similarity arguments first given by Liepmann, it is possible to reduce the integrated thermal energy equation to a first-order ordinary differential equation for the heat transfer rate, which is easily solved.

A comparison with an accurate solution of Poots, for a particular pressure gradient and wall temperature, yields agreement to within 1 per cent at one third of the distance from leading edge to separation and 3 per cent at twice this distance, with the error rising to 16 per cent at separation where the heat transfer rate is, of course, very low. The modesty of this error is very encouraging.

Type
Research Article
Information
Aeronautical Quarterly , Volume 13 , Issue 3 , August 1962 , pp. 255 - 270
Copyright
Copyright © Royal Aeronautical Society. 1962

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References

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