The regular reflection→Mach reflection transition in unsteady flow over convex surfaces

Abstract

The non-stationary transition from regular reflection (RR) to Mach reflection (MR) over convex segments has been the focus of many recent studies. Until recently, the problem was thought to be very complicated because it was believed that many parameters such as the radius of curvature, initial angle and geometrical shape of the reflecting surface influenced this process. In this study, experiments and inviscid numerical computations were performed in air ($\unicode[STIX]{x1D6FE}=1.4$) at an incident shock-wave Mach number of 1.3. The incident shock waves were reflected over cylindrical and elliptical convex surfaces. The computations were validated by high-resolution experiments, which enabled the detection of features in the flow having characteristic lengths as small as 0.06 mm. Therefore, the RR →MR transition and Mach stem growth were successfully validated in the early stages of the Mach stem formation and closer to the surface than ever before. The evolution of the RR, the transition to MR and the Mach stem growth were found to depend only on the radius of the reflecting surface. The reflected shock wave adjusts itself to the changing angles of the reflecting surface. This feature, which was demonstrated at Mach numbers 1.3 and 1.5, distinguishes the unsteady case from the self-similar pseudo-steady case and requires the formulation of the conservation equations. A modification of the standard two-shock theory (2ST) is presented to predict the flow properties behind a shock wave that propagates over convex surfaces. Until recently, the determination of the time-dependent flow properties was possible solely by numerical computations. Moreover, this derivation explains the controversial issue on the delay in the transition from the RR to the MR that was observed by many researchers. It turns out that the entire RR evolution and the particular moment of transition to MR, are based on the essential ‘no-penetration’ condition of the flow. Therefore, we proposed a simple geometrical criterion for the RR →MR transition.

Similarity in Mach stem evolution and termination in unsteady shock-wave reflection

“…The numerical scheme has been successfully applied to the complementary reflection over a convex cylindrical surface in a former study (Geva et al . 2018). Owing to the use of a commercial program, rigorous validation and verification were done against the highest temporal and spatial resolution experiments available nowadays (Ram et al .…”

“…Further validation using the same experimental apparatus and numerical schemes appears in Geva et al . (2018). Since the simulations are considered to be in good agreement with the experiments and analytical predictions presented, the following analyses relied exclusively on the computational results.…”

“…The following subsections present concise descriptions of the numerical computations, experimental procedure, validation and post-processing. For further information about the computations and extended validation, the interested reader is referred to Geva et al (2013), Ram et al (2015) and Geva, Ram & Sadot (2018).…”

“…Transient terms in the differential equations were calculated by second-order explicit time-stepping integration based on a four-stage Runge-Kutta method (Jameson, Schmidt & Turkel 1981). The numerical scheme has been successfully applied to the complementary reflection over a convex cylindrical surface in a former study (Geva et al 2018). Owing to the use of a commercial program, rigorous validation and verification were done against the highest temporal and spatial resolution experiments available nowadays (Ram et al 2015).…”

On regular reflection to Mach reflection transition in inviscid flow for shock reflection on a convex or straight wedge

“…Recently, high-resolution experiments and inviscid numerical simulations have been conducted to study the RR → MR transition on convex surfaces (Geva, Ram & Sadot 2018). The authors stated that, different from the shock reflection in pseudo-steady flow, in which the reflected shock wave has only a translation motion along the surface, the reflected shock wave in unsteady shock reflection not only translates but also rotates about the reflection point.…”