On the Problem of Shock Wave Structure

Abstract

We consider the problem of internal structure of the triple point appearing in Mach reflection, which is considered to be important for the cause study of the von Neumann paradox as well as the shock reflection itself in rarefied gas. We investigate it in an adequately made finite region near the triple point and use analytical approach rather than numerical to have a solution of 2D Navier-Stokes equations system, by which we can avoid the difficulties such as the need for ever finer mesh size for the region not known in the beginning. We consider first one-dimensional flow in a finite region, which gives a flow with a hump unlike conventional one of monotonous change for the infinite region. Then we seek a solution of the 2D Navier-Stokes equations system in polar coordinates to the flow field between two curved boundaries. Results show the incoming parallel flow bents to the direction of the slip flow and the density distribution along the streamline increases similar to that for one-dimensional shock structure but a small hump as the solution over to the flow for a finite range.

Appendix: Solution of Triple-Point Singularity [7, 8]

This is a r-power expansion solution of Eq. (6) in angler region between the reflected and Mach stems satisfying the shock boundary condition at the two shock lines so that it is singular at the triple point r = 0, which in fact represents the flow in the non-R-H zone caused by shock lines with different shock values meeting there. Some of the results relevant to this study are

$$ {\displaystyle \begin{array}{l}{\tilde{V}}_{\theta }={\tilde{V}}_{\theta}^{(0)}+r{\tilde{V}}_{\theta}^{(1)},{\tilde{V}}_r={\tilde{V}}_r^{(0)}+r{\tilde{V}}_r^{(1)},\tilde{\rho}={\tilde{\rho}}^{(0)}+r{\tilde{\rho}}^{(1)},\tilde{p}={\tilde{p}}^{(0)}+\mathrm{r}{\tilde{\mathrm{p}}}^{(1)}\\ {}{\tilde{V}}_{\theta}^{(0)}{=}\tilde{A}\left(1{+} k\theta \right)\sin \left(\theta {+}{\theta}_0\right),{\tilde{V}}_r^{(0)}=-\tilde{A}\left[\left(1{+} k\theta \right)\cos \left(\theta {+}{\theta}_0\right){+}k/7\sin \left(\theta {+}{\theta}_0\right)\right]\\ {}{\tilde{\rho}}^{(0)}=E{\left(1+ k\theta \right)}^{-6/7},{\tilde{p}}^{(0)}={\tilde{\rho}}^{(0)}\left( F\theta +G\right),\\ {}{\tilde{V}}_{\theta}^{(1)}=-\left(1/2\right)\tilde{A}{}_1\;\sin \left(\theta +{\theta}_0-2\alpha \right)\sin \left(\theta +{\theta}_0\right)+{S}_1\left(\theta \right)\\ {}{\tilde{V}}_r^{(1)}=-\left(1/2\right)\tilde{A}{}_1\;\cos \left(\theta -{\theta}_0-2\alpha \right)\sin \left(\theta +{\theta}_0\right)+{S}_2\left(\theta \right),\\ {}{S}_1\left(-{\theta}_0\right)={S}_1\left(-{\theta}_0\right)={\tilde{\rho}}^{(1)}\left(-{\theta}_0\right)={\tilde{p}}^{(1)}\left(-{\theta}_0\right)=0\end{array}} $$