Flow field behind Mach reflection and the Neumann paradox
A.Sakurai
Tokyo Denki University
After so many years with abundance of contributions it has gradually been recognized that the cause of the Neumann paradox is traced to its origin in the flow field behind reflected and Mach stem shock waves. Neumann’s three-shock theory assumes there two parallel uniform flows separated by a slip stream line, and it provides shock angles in excellent agreement with experimental data to strong Mach reflection case, while paradoxically it is in no agreement at all with the weaker case.
In these experiments, we can indeed observe uniform flows with a slip line as such to the stronger case but these become blurred to the weaker case, to which we can see the disturbance from edge corner reaching to the triple point and the entire flow field is non-uniform. This non-uniformity results in differences ⊿p,⊿δ in the pressure p and the velocity direction δ at each of immediately behind the reflected shock and the Mach stem as,
⊿p=α, ⊿δ=β, (1)
where α,β are non-zero for non-uniform flow field , and this should be compared with the original three-shock theory condition:
⊿p=0, ⊿δ=0. (2)
The explicit expressions of α,β should be given by the solution of the flow field equation, so the problem is to find the solution satisfying the shock conditions at bounding reflected and Mach stem shock waves.
Now three-shock configuration appears both in steady and unsteady flows, representatively in supersonic inlet flow to the former and advancing plane shock wave over a sharp wedge surface to the latter and the “Neumann Paradox phenomenon” occurs in both flows to their weak cases, and the governing equation to the flow field is different from every other flow of steady and unsteady, and we should have different approaches of getting solution. In any case, flow fields are almost uniform to their weak case and we can utilize this fact of deriving approximate solutions to each case, by which we can have explicit expressions for α,β above. We use these in eq. (1) to have shock angle values different from the ones given by the original three-shock theory condition of eq.(2), which are consistent with the experimental data values showing no sign of “Paradox”.
