Finite-Strength Sensitivity and Euler--UTSD Correspondence for Guderley--Mach Reflection

Weak shock reflection at nearly glancing incidence is governed, after the transonic weak-shock scaling, by a self-similar unsteady transonic small-disturbance (UTSD) free-boundary problem. We derive and differentiate the first finite-strength perturbation of the corresponding isentropic potential-flow problem along paths of fixed canonical incidence $a=\alpha \mathrm{/}\delta$, where $\mu ={\delta }^2=2(M^2-1)$. A second-order refluxed adaptive finite-volume method, exact discrete tangents and adjoints, and a Rankine–Hugoniot-constrained fitted principal front give the fixed-$a$ canonical sensitivity $H_{2,a}^{\text{PF}}(0.5;1.4)=-0.217\pm 0.012$; a fully differentiated physical back-map gives the diagnostic fixed-$a$ coefficient $K_{2,a}^{\text{PF}}\simeq -0.266$. We derive the exact chain rule that converts these quantities to the distinguished fixed-$\lambda$ path $\lambda =(M-1)\mathrm{/}{\alpha }^2$, showing explicitly that the conversion requires the independent incidence derivative of the leading UTSD branch and therefore cannot be inferred from the fixed-$a$ calculation alone. A matched-boundary self-similar Euler study with strength-dependent refinement contains 21 qualified nonlinear states and 252 evaluations of a common front-functional family. Coupled extrapolation gives $g_0^{\text{Eul}}=0.510\pm 0.006$ and the physical leading-angle coefficient $G_0^{\text{Eul}}=0.256\pm 0.004$, consistent with the shock-fitted UTSD limits. A separate same-strength phase-bracket audit using 18 Euler states shows that the captured-shock subcell phase is comparable to the desired cubic signal. The resulting finite-resolution Euler secants are compatible with the potential-flow correction, but their $\rho =h_{\eta }\mathrm{/}\sqrt{\mu }\to 0$ extrapolation is not model-stable. Thus leading-order Euler–UTSD correspondence is numerically verified, whereas cubic-order Euler correspondence remains unresolved.