CX. The shock strength in supersonic “conical fields.”

Summary In the linearized theory of supersonic “conical fields,” there is a singularity on the Mach cone from the apex. In this paper, using the exact equations of motion, and a general method of rendering approximate solutions uniform, which is described elsewhere, but is here fully explained as far as is needed in the present application, the true behaviour of the flow near the Mach cone is deduced. Near to it is a shock; the most interesting conclusion is that the strength of the shock can be inferred, to a first approximation, from the linearized solution alone (by formulas derived below from the exact equation of motion). The general flow, with disturbing surfaces outside as well as inside the Mach cone, is treated in § 5. In the simpler special case, treated in § 4, when the disturbance on linear theory is confined within the Mach cone, the results can be stated simply. Let the intersection of the cone with a plane perpendicular to the stream be the circle r=l in polar coordinates in that plane; then the pressure on linearized theory has a singular behaviour as r → 1 of the type (p- p 1)/ρ 1U2∼A(ϑ)√(l—r), and it is shown below that the shock (a conical surface slightly outside the Mach cone)has strength which varies with ϑ like 3/4γ(γ+l)M6(M2-1)−1A2(ϑ), to a first approximation, at points where A(ϑ)>0; but which is at most of order the cube of the disturbance velocities at points where A(ϑ)≤0, when the most noticeable phenomenon is a limited region of rapid expansion, at a finite rate independent of the size of the disturbance. Examples are given in § 6 from a number of problems whose linearized solution is known. In the supersonic flow past any conical surface all of whose generators make a small angle with the main stream, the shock is found to be of a uniform strength to a first approximation, which depends only on the solid angle of the cone; two examples from supersonic wing theory are also given. The work explains in detail why linear theory gives, as all writers on the subject have assumed, a correct first approximation to the flow field within the Mach colxe, provided that the boundary condition of continuous velocity across the cone be applied, even though the linear solution is essentially the first term in a series, which diverges near the Mach cone, and although a shock discontinuity is really present in its neighbourhood.