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 M = 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>2</m:mn> <m:msqrt> <m:mrow> <m:mi>λ</m:mi> </m:mrow> </m:msqrt> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${a}_{0}=1/\left(2\sqrt{\lambda }\right)$ , independent of the ratio of specific heats γ . Thus the known utsd detachment value <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:msqrt> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msqrt> </m:math> ${a}_{d}=\sqrt{2}$ corresponds in this scaling to λ d = 1/8, with Guderley–Mach reflection for λ &gt; 1/8. The physical trajectory angle is obtained by multiplying the canonical utsd trajectory function g ( a ) by the Mach-number strength scale <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>δ</m:mi> <m:mo>=</m:mo> <m:msqrt> <m:mrow> <m:mn>2</m:mn> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msqrt> </m:math> $\delta =\sqrt{2\left({M}^{2}-1\right)}$ , so that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>χ</m:mi> </m:mrow> <m:mrow> <m:mtext>phys</m:mtext> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mi>δ</m:mi> <m:mi>g</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>O</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>δ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>2</m:mn> <m:msqrt> <m:mrow> <m:mi>λ</m:mi> </m:mrow> </m:msqrt> <m:mspace width="0.17em"/> <m:mi>α</m:mi> <m:mi>g</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>O</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${\chi }_{\text{phys}}=\delta g\left(a\right)+O\left({\delta }^{2}\right)=2\sqrt{\lambda } \alpha g\left({a}_{0}\right)+O\left({\alpha }^{3}\right)$ . 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