Predicting Mach stem height in steady supersonic reflection: sensitivity limits of composite models and a calibrated scaling law

Since the early 1990s, numerous theoretical methods have been proposed to predict Mach stem height in steady supersonic shock reflections by assembling sub-models for local flow structures, including incident/reflected shocks, the triple point, the slipline, and Mach stem curvature. We constructed an updated model and employed it as a benchmark to evaluate the performance of various sub-models corresponding to typical flow regions. The results show that the curved assumption for the free part of the slipline outperforms the straight-line approximation, considering the differences in regions after the reflected shock can improve the predictive accuracy, while using compatibility relations in the interactive part of the slipline is superior to the wave reflection model and better captures the linear slope of Mach stem height with wedge trailing edge height. Nevertheless, prediction errors in the slope and systematic biases in the overall Mach stem height prediction persist. To address these shortcomings, we developed a calibrated scaling law for the coefficient of a linear Mach stem model. Grounded in asymptotic reasoning and high-fidelity numerical simulations, this law yields a compact, easy-to-implement expression that achieves substantially higher accuracy than existing analytical composite models across the full parameter space. It retains well-established limiting cases, clarifies how inadequate sub-modelling degrades prediction accuracy, and provides uncertainty estimates for practical engineering applications.