The transonic flow past a NACA0012 and the von Neumann paradox

The confluence of three shock waves at a common branching point is observed in different shock-interaction patterns, the most prominent example being the irregular reflection of an oblique shock from a solid surface, known as a Mach reflection. During WWII, John von Neumann developed an analytical model of the Mach reflection which fits well the experiments, except when the incident shock is weak. This inconsistency is referred to in the literature as the von Neumann paradox. In this paper we combine numerical simulations performed using a shock-fitting CFD code and shock polar analysis to show that the triple-point that arises in the so-called fishtail shock-structure, which is observed in the transonic flow past airfoils, is not amenable to be described using von Neumann’s model. Our analysis points to the fact that Guderley’s four waves model, which has already been experimentally and numerically validated as a plausible solution to the von Neumann paradox, should be used instead