Small Disturbance Approximations for Flows Over a Cone

Von Karman and Moore [3] showed that the perturbation potential may be represented as an integral over a distribution of sources

where в = УM02 – 1. Following the presentation of Liepmann and Roshko [6], a change of variable is recommended from ф to a

ф = x – вг cosh a, ^ dф = – вг sinh ada = –/(x – ф)2 – в2r2da

(7.138)

The integral and the limit of integration become

Notice, u and v are invariant along lines x|r = const. The solution can represent supersonic flow over a cone, where the vertex angle depends on a.

Applying the boundary condition

u = —a cosh 1 ^- v = a^Jcot2 8 – в2 (7.143)

For slender cones, with 8 ^ 1, and cot 8 > в, then

(cot8 ( 2

he pressure rise on the cone is much less than on the wedge, as shown in Fig. 7.23.

Notice the pressure is uniform over the surface, however the distribution of the pressure coefficient along a line r = const. rises continuously downstream of the shock. Also, in the slender body approximation, there is no pressure jump at the nose.

The accuracy of the slender body approximation can be numerically evaluated by comparison to the numerical results of Maccoll and the tables of Kopal. A typical case is shown in Fig. 7.24 (see Liepmann and Roshko [6]).