Theoretical modelling of unsteady shock reflection induced by vertical upward wedge motion

Combining theoretical analysis and numerical simulations, this paper investigates the unsteady effects on shock reflection as the wedge moves vertically upward from an initial steady state. A theoretical model is developed to predict the configuration of the incident shock and the corresponding downstream flow characteristics, validated by numerical simulations. Accordingly, the unsteady evolution is classified into four stages, delimited by the different unsteady wave systems. For cases initialised with regular reflection, an analytical solution for the reflection point velocity is derived, and its dynamic variation is identified as the key factor dominating shock reflection transitions. Based on the temporal evolution of the incoming Mach number and flow deflection angle at the reflection point in a moving frame, a theoretical framework is established to predict the unsteadiness-induced transition behaviours, with the reflection point velocity and the timing of the first transition from regular reflection to Mach reflection accurately predicted, and the timing of the reverse-transition from Mach reflection to regular reflection addressed by introducing a simplified Mach stem correction. Thus five typical flow regimes are summarised and numerically verified, distinguished by the initial conditions and transition history, with two critical motion velocities governing the boundaries between each regime. Of particular interest is a special unsteady-dominated regime: although the flow starts from regular reflection and remains within the dual-solution domain after unsteady effects vanish, the transient unsteadiness drives the flow to transition into Mach reflection, persisting due to the hysteresis effect. This study provides a new perspective and analytical approach for investigating unsteady shock reflection.