Abstract
§14. In this paper, the behaviour of flow at considerable distances from a cylinder of arbitrary cross section placed in a uniform flow of a compressible fluid is considered, primarily with a view to finding general formulae for the lift and moment of the cylinder.
In the first place, based on the method of successive approximation recently proposed by the present writer, the most general expressions are determined for the velocity potential and stream function for flow under consideration (see (5•6) and (5•7)). Thus, it is found that theassumptions concerning the velocity field, which formed the basis of the analyses of GLAUERT and of BATEMAN in extending the well-known KUTTA-JOUKOWSKI theorem to the case of a compressible fluid, are too special in one case and too general in the other to be in accordance with the actual state of affairs.
In the next place, the explicit forms of the asymptotic expressions for the velocity potential and stream function are exactly determined to the order of 1/r. As an example, it is confirmed that the velocity potential for the flow past a circular cylinder with circulation round it, which has recently been obtained by HASIMOTO and SIBAOKA correctly to the order of M4, M being the MACH number of the undisturbed stream, has in fact the asymptotic form predicted by the present analysis.
The asymptotic expressions thus found are then used to obtain the general formulae for the lift and moment, of which the first is naturally just the same as GLAUERT'S extension of the KUTTA-JOUKOWSKI theorem. The formulae for the moment, (12•13a) and (12•13b), express a rather surprising fact that determination of the moment of a cylinder requires only the knowledge of the coefficient of 1/z in the asymptotic expansion, in descending powers of z and z, either of the velocity potential Φ(z, z) or of the stream function Ψ(z, z), where z=x+iy, z=x-iy. Finally, it is shown that the writer's previous formula for the moment, (1•8), can be deduced from the newly found formula as a first approximation, namely, by retaining only the quantities of the order of M2
