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A Simplified Theory of “Simple Waves“

Published online by Cambridge University Press:  07 June 2016

A. Ghaffari
A. Ghaffari*
Affiliation:
The University, Teheran
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Summary

This note is concerned with a general and exact solution of the two-dimensional equations of motion of a compressible fluid, which is steady, irrotational and isen-. tropic. The solution arrived at is the generalisation of the Prandtl-Meyer celebrated “wedge solution” to include an expansion around any smooth (or abrupt) convex boundary.

We shall obtain the solution of equations for the most general type of motion in which the Cartesian components of velocity are connected by some functional relation throughout the field of flow.

Taking into account the conditions of continuity, zero rotation and Bernoulli's equation, together with the fundamental assumption, which implies that the components of fluid velocity are each a function of fluid speed only, it is found that the motion is necessarily supersonic.

It is shown that the lines of equal speed, pressure (isobars) and density are all straight lines, although they are not necessarily concurrent as in the original Prandtl-Meyer theory, and further that the component of fluid velocity perpendicular to the isobar is equal to the local velocity of sound.

The extended Prandtl-Meyer solution for the flow past an arbitrary fixed boundary is obtained analytically.

Type
Research Article
Information
Aeronautical Quarterly , Volume 1 , Issue 2 , August 1949 , pp. 187 - 194
Copyright
Copyright © Royal Aeronautical Society. 1949

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References

1. Lees, F. (1924). On a Case of Steady Flow of a Gas in Two-Dimensions. Proc. Camb. Phil. Soc, Vol. 22, 1924.Google Scholar
2. Bateman, H. (1930). The Irrotational Motion of a Compressible Inviscid Fluid. Proc. Nat. Acad. Sc, XVI, 1930.Google Scholar
3. Taylor, G. I., and MacColl, W. (1935). Aerodynamic Theory edited by Durand, , Vol. III. Div. H., Chapter IV, 1935.Google Scholar