Scaling analysis of the pressure distribution in shock wave/boundary layer interactions on convex surfaces with extended free-interaction theory

Abstract

Predicting the pressure distribution in shock wave/boundary layer interaction (SWBLI) over convex surfaces is critical for hypersonic vehicle design, yet a unified theoretical model remains absent. This work extends the classical free-interaction theory by introducing two dimensionless parameters: normal upper Delta p overbar Subscript r <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 mathvariant="normal">Δ</mml:mi> <mml:msub> <mml:mover> <mml:mi>p</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>r</mml:mi> </mml:msub> </mml:math> $\Delta \overline {p}_r$ , which quantifies the cumulative effect of downstream curvature on reattachment pressure, and normal upper Delta p overbar Subscript s <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 mathvariant="normal">Δ</mml:mi> <mml:msub> <mml:mover> <mml:mi>p</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msub> </mml:math> $\Delta \overline {p}_s$ , which captures the upstream displacement delay due to boundary-layer growth on the curved wall. Based on the relative position of the separation point with respect to the curvature onset, the interaction is classified into two distinct types: type-I, where separation occurs on the flat plate upstream of the curved section ( normal upper Delta p overbar Subscript s Baseline equals 0 <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 mathvariant="normal">Δ</mml:mi> <mml:msub> <mml:mover> <mml:mi>p</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> $\Delta \overline {p}_s = 0$ ), and type-II, where separation occurs on the curved surface itself ( normal upper Delta p overbar Subscript s Baseline less than 0 <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 mathvariant="normal">Δ</mml:mi> <mml:msub> <mml:mover> <mml:mi>p</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:math> $\Delta \overline {p}_s \lt 0$ ). Scaling methods are developed to reconstruct the entire pressure distribution within the separation region from the flat-plate baseline for both interaction types. Comprehensive validation using numerical simulations and experimental measurements demonstrates that the normalised pressure rise normal upper Delta upper F Subscript c Superscript asterisk divided by upper F Subscript p Superscript asterisk <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 mathvariant="normal">Δ</mml:mi> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>c</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:math> $\Delta F^*_c/F^*_p$ collapses linearly onto alpha Subscript r Baseline normal upper Delta p overbar Subscript r plus alpha Subscript s Baseline normal upper Delta p overbar Subscript s <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:msub> <mml:mi>α</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mspace width="thinmathspace"/> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:msub> <mml:mover> <mml:mi>p</mml:mi> <mml:mo accent="false">¯</mml:m