Steady shock reflection over the full range of geometric set-ups

In steady supersonic flows over a symmetric wedge, the shock reflection pattern is fundamentally constrained by the relative trailing edge height g <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>g</mml:mi> </mml:math> $g$ , especially by the lower limit (for start flow), the upper limit (for interference-free incident shock) and the geometric transition limit. Using a recent calibrated model for the Mach stem height (Li &amp; Bai, 2026, J. Fluid Mech., vol. 1031, A27), we quantify the lower and upper limits of Mach reflection (MR) across the entire parameter space, which has been impossible due to lack of a suitable Mach stem height model. A maximum point of the dual-solution stability gap – i.e. the gap between the lower limits of MR and regular reflection (RR) – is found at Mach number almost equals 6.2 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo>≈</mml:mo> <mml:mn>6.2</mml:mn> </mml:math> $\approx 6.2$ and wedge angle almost equals 28.8 Superscript ring <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo>≈</mml:mo> <mml:msup> <mml:mn>28.8</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> $\approx 28.8^\circ$ . A schematic along the g <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>g</mml:mi> </mml:math> $g$ -axis displays five limits and eight distinct gaps (the unstart-prone gap, the dual-solution stability or interference gaps, the geometric transition gap, the clean RR or MR gaps, the robust reflection gap, and the always RR gap). Within the robust shock reflection gap, shock reflection is safe and free of interference, whenever it is RR and MR. The existence of a potential critical value – the reflection vanished limit – is discussed. Above this limit, the incident shock is expected to degenerate into a Mach wave before reaching the surface. The present study gives a picture of steady shock reflection along the entire g <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>g</mml:mi> </mml:math> $g$ -axis, and deepens our understanding of shock reflection.