Numerical Study of Transition Boundary between Regular and Mach Reflection for a Planar Shock Striking a Wedge James J. Gottlieb, Jee-Whan Nam and Clinton P.T. Groth Institute for Aerospace Studies, University of Toronto, Toronto, ON, Canada M3H 5T6 Email: gottlieb@utias.utoronto.ca ABSTRACT The interaction of a planar shock wave with a rigid wedge in an air-filled shock tube is studied numerically by computing the unsteady flow field behind the inci- dent shock for the cases of two-shock regular reflec- tion, when the wedge angle is large, and three-shock Mach reflection, when the wedge angle is small. The transition boundary between regular and Mach reflec- tion is determined numerically and compared to the transition boundaries that are predicted analytically by various two- and three-shock theories (sonic, detach- ment and mechanical-equilibrium criteria) and that ob- tained from shock-tube experiments. The objective of this numerical study is to help resolve the von Neu- mann paradox of why experimental regular shock re- flections persist across all predicted transition bound- aries into the Mach reflection region. 1 I NTRODUCTION The interaction of a planar shock wave moving initially in a constant-area shock tube and then striking an in- clined flat surface with a wedge angle θ w results in a single reflected shock when the wedge angle is large, as shown in Fig. 1(a), and results in a reflected shock with a Mach stem to the wedge surface and a trailing slip surface (dashed) when the wedge angle is small, as shown in Fig. 1(b). Analytical predictions using two-and three-shock theories of the transition bound- ary between regular and Mach reflection depend on various assumptions (see Henderson [1], Ben-Dor [2] wedge S i S r θ w (a) S i S r θ w (b) Fig. 1: Two-shock regular reflection (a) and three- shock Mach reflection (b) from a wedge for shock Mach numbers 1 < M s < 2. and Glass and Sislean [3]), and three predictions using the detachment and sonic criteria (nearly the same re- sults) and using the mechanical-equilibrium criterion are shown in Fig. 2, in which the wedge angle for tran- sition boundary is plotted versus the incident shock strength (shock Mach number M s ). Experimental data for planar shock waves interacting with wedges in a shock tube are also shown to illustrate that the experi- mental wedge angles are below the predicted transition boundaries. Hence, regular reflection persists beyond all predicted transition boundaries into the Mach re- flection region. This persistence of regular reflection in the Mach reflection region is called the von Neu- mann paradox (originally by Birkoff [4] in 1950), and it has been unresolved since von Neumann’s original work in 1943 [5]. The goal of this study is to solve shock and wedge interaction problems numerically to provide new in- formation to help resolve the von Neumann’s paradox. The intent is to solve shock on wedge reflection prob- lems using both the Euler and Navier-Stokes equa- tions, so that viscous and heat-conduction effects in 1 2 3 20 30 40 60 M s θ w (deg.) regular reflection region Mach reflection region dual region mechanical equi- librium criterion sonic criterion detachment criterion experimental data for the transition Fig. 2: Predicted and experimental transition boundaries between the regular and Mach reflection regions.