Distinguished Scaling and UTSD Structure in Weak Shock Reflection at Nearly Glancing Incidence

Abstract

We study weak shock reflection from a rigid wall in the joint limit of weak shock strength and nearly glancing incidence. In the distinguished scaling $\Mach=1+λα^2$, the inner reflection region is governed by the unsteady transonic small-disturbance (UTSD equation and is controlled, to leading order, by the single parameter $a_0=1/(2\sqrt{\hat{I}}»)$, independent of the ratio of specific heats ${\hat{I}}^3$. Thus the known UTSD detachment value $a_d=\sqrt{2}$ corresponds in this scaling to $\hat{I}{»}_d=1/8$, with Guderley--Mach reflection for $\hat{I}»>1/8$. The physical trajectory angle is obtained by multiplying the canonical UTSD trajectory function $g(a)$ by the Mach-number strength scale $δ=\sqrt{2(\Mach^2-1)}$, so that $\ddot{I}{‡}_{\mathrm{phys}}=\hat{I}´g(a)+O(\hat{I}{´}^2)=2\sqrt{\hat{I}}»\hat{I}\pm g(a_0)+O(\hat{I}{\pm }^3)$. We rederive the self-similar UTSD reduction, sonic parabola, and shock polar in order to make the convention and the detachment map self-contained. We also record a formal adjoint solvability expression for the first correction $H(a;{\hat{I}}^3)$, while specifying the free-boundary data required to evaluate it. Finally, a time-marching solver for the full leading-order canonical UTSD system is benchmarked at $a_0=0.5$: retaining the transverse compression $u>1$ gives a $u=0.5$ contour location consistent with the Hunter--Tesdall triple-point benchmark. This computation is used only as a leading-order benchmark, not as a substitute for an adaptive self-similar Guderley free-boundary solver.

19 pages, 8 figures

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