On the Flow with Circulation of a Compressible Fluid past a Circular Cylinder

Abstract

§9. In the present paper, the two-dimensional irrotational subsonic flow involving circulation of a compressible fluid past a circular cylinder has been investigated by employing the so-called JANZEN-RAYLEIGH'S method of successive approximations, with a special intention of studying the manner in which the values of the ratio L_c/L_i vary as the degree of approximation increases, where L_c denotes the lift acting on the circular cylinder placed in the compressible fluid flow, while L_i is the lift which the same cylinder would experience when placed in the incompressible fluid flow.
In the first place, the expression correct to the order of M⁴ has been obtained for the velocity potential for the flow, where M is the MACH number formed with the velocity U of the undisturbed stream at infinity and the speed of sound c₀ associated with the undisturbed flow so that M=U/c₀. Then, by assuming that the forward and the rear stagnation points occur at certain definite points on the surface of the cylinder, irrespective of whether the fluid is compressible or not, we have determined the value of the circulation κ round the body and have obtained the approximate formula for κ correctly to the order of M⁴. Also, using the value of κ thus determined, we have obtained the approximate expression for the ratio L_c/L_i, which is also correct to the order of M⁴.
By performing various numerical calculations, the values of the critical MACH number, M_crit, as well as of the ratio L_c/L_i have been calculated.
It has thus been found that the second approximate values of M_crit calculated by using the second approximate expressions correct to the order of M⁴ for various quantities, are smaller, by about 3 per cent at most, than the corresponding first approximate values which have been calculated with the aid of the first approximate expressions correct to the order of M². Also, it has been found that the second approximate values of L_c/L_i are, in general, somewhat greater than the corresponding first approximate values and that the difference between the first and the second approximate values is inappreciable, however, when the value of the MACH number M is well below the critical MACH number