The dual solution stability gap bounded by sub- and supercritical geometric thresholds in steady shock reflection

TL;DR

The paper analyzes steady shock reflection in a dual-solution domain defined by $(M_{0},\theta_{w})$, proving that the MR lower limit $H_{R,\min }^{(MR)}$ exceeds the RR lower limit $H_{R,\min }^{(RR)}$ and defining subcritical/supercritical geometric thresholds together with a dual-solution stability gap $\Delta g_{\min }^{(MR-RR)}$. It derives lower-limit expressions in terms of $M_{0}$ and $\theta_{w}$, and shows $g_{\min }^{(MR)}>g_{\min }^{(RR)}$ across the domain, implying RR stability and MR instability inside the gap. Since an exact Mach-stem height model is unavailable, the authors propose a linear approximation $H_{T}/H_{A}=A g+B$ with coefficients from high-fidelity data to estimate the gap; they validate the approach at a representative case $(M_{0},\theta_{w})=(4,25^{\circ})$, obtaining $g_{\min }^{(RR)}=0.239$ and $g_{\min }^{(MR)}\approx 0.417$, and a gap $\Delta g_{\min }^{(MR-RR)}\approx 0.178$, corroborated by CFD. The paper then demonstrates dynamic transitions within the gap via DNS under upstream density disturbances, revealing direct RR→MR→unstart and inverted RR→MR→RR pathways and complex MR interactions, such as type IV shock interference. Overall, the work clarifies the stability structure in shock-reflection problems and reveals new possible dynamic transitions driven by geometric thresholds in the dual-solution regime, with practical implications for shock-control strategies and nozzle flow management.

Abstract

In this paper, we examine the significance of the lower geometric limit, defined as the trailing-edge height at which the reflected shock grazes the trailing edge, for both regular reflection (RR) and Mach reflection (MR). We show that this lower limit for MR is greater than that for RR, within the dual-solution domain away from its lower and left boundaries. We thus identify a dual-solution stability gap lying between the subcritical threshold (the lower limit for RR) and the supercritical threshold (the lower limit for MR). Within this gap RR is stable while MR is unstable, implying a new dynamic transition possiblity there: a steady RR configuration (start flow) can undergo a dynamic transition to an unstable MR state (unstart flow) under suitable disturbance of density or other flow parameters. Numerical simulations confirm the existence of this stability gap and illustrate the time history of dynamic transitions, including (1) direct transitions from RR to MR to unstart flow, with complex flow structures such as hybrid MR-type VI shock interference and double MR -- MR reflections, and (2) inverted transitions, in which RR first shifts to MR and then returns back to RR.

Figures (11)

• Figure 1: Illustration of shock reflection. a) RR, b) MR.
• Figure 2: Shock reflection for $H_{R}=H_{R,\min }$ at which the reflecting shock intersects the trailing edge $R$. a) RR, b) MR.
• Figure 3: Schematic display of lower limit setup. a) RR and MR supposed to have the same $H_{R}=H_{R,\min }$. b) MR at $H_{R}=H_{R,\min }$.
• Figure 4: Contourlines of $\Delta g_{\min }=g_{\min }^{(N)}-g_{\min }^{(RR)}$ in the dual soution domain.
• Figure 5: Contourlines of $\Delta g_{\min }=g_{\min }^{MR}-g_{\min }^{RR}$ in the dual soution domain£¬with $\varepsilon =1.515H_{TN}.$a) global view. b) enlarged view.