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

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:

, which quantifies the cumulative effect of downstream curvature on reattachment pressure, and

normal upper Delta p overbar Subscript s Δ p ¯ s $\Delta \overline {p}_s$

, 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 Δ p ¯ s = 0 $\Delta \overline {p}_s = 0$

), and type-II, where separation occurs on the curved surface itself (

normal upper Delta p overbar Subscript s Baseline less than 0 Δ p ¯ s < 0 $\Delta \overline {p}_s \lt 0$

). 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 Δ F c ∗ / F p ∗ $\Delta F^*_c/F^*_p$

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 α r Δ p ¯ r + α s Δ p ¯ s $\alpha _r\,\Delta \overline {p}_r + \alpha _s\,\Delta \overline {p}_s$

with

upper R squared greater than 0.98 R 2 > 0.98 $R^2 \gt 0.98$

, and the predicted overall pressure profiles agree excellently with numerical data. Empirical correlations for the model coefficients and a critical curvature radius are provided to define the applicability range. This work elucidates the physical mechanisms governing pressure distributions in curved-wall SWBLI, and establishes a systematic theoretical framework for pressure prediction in such configurations.