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 $\Mach=1+\lambda\alpha^2$, the inner reflection region is governed by the unsteady transonic small-disturbance (UTSD equation and is controlled, to leading order, by the single parameter $a_0=1/(2\sqrt{\lambda})$, independent of the ratio of specific heats $\gamma$. Thus the known UTSD detachment value $a_d=\sqrt2$ corresponds in this scaling to $\lambda_d=1/8$, with Guderley--Mach reflection for $\lambda>1/8$. The physical trajectory angle is obtained by multiplying the canonical UTSD trajectory function $g(a)$ by the Mach-number strength scale $\delta=\sqrt{2(\Mach^2-1)}$, so that $\chi_{\rm phys}=\delta g(a)+O(\delta^2)=2\sqrt{\lambda}\,\alpha g(a_0)+O(\alpha^3)$. 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 $H(a;\gamma)$, 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.