Examination of the von Neumann paradox for a weak shock wave

Abstract

Oblique reflections of a weak shock wave have been investigated both theoretically and experimentally. A modified three-shock theory is introduced to explain the well-known von Neumann paradox for weak Mach reflection. As a physical reality, the triple point is not a mathematical point and the slipstream has a finite thickness. Consequently, the effect of the slipstream divergence behind the triple point and the minute pressure differences on both sides of the slipstream are taken into account, and both these effects are examined numerically. The angle of divergence is given parametrically in order to calculate some characteristics around the triple point, e.g., the angle of reflection. Numerical results are compared with measurements, and characteristics of solutions are examined. It is found that for weak Mach reflection the modified three-shock theory gives physically realistic solutions, even when von Neumann's three-shock theory has no solution. It is also found that the divergence effect of the slipstream is predominant over the pressure difference. All the experimental data are found to exist in the domain bounded by von Neumann's classical theories and the modified three-shock theory proposed here. The experimental evidence of slipstream divergence is presented.