Structure of Weak Non‐Hugoniot Shocks

Abstract

As the curvature of shock waves increases, the shock structure becomes two dimensional, and the usual Hugoniot jump conditions no longer hold. An equation has been derived for the structure of such a two‐dimensional non‐Hugoniot shock in the case of weak shocks with Mach numbers close to one. The development of this equation from the Navier‐Stokes equations is based on the assumptions that the vertical velocity is of order (M1* − 1)3∕2 and that the flow within the shock is irrotational. From the derivation it appears that the non‐Hugoniot region behaves as an acoustic wave driven by higher‐order viscous effects. The properties of the above equation, which has been called the viscous‐transonic or V‐T equation have been investigated. The V‐T equation appears to be a combination of Burgers' equation for weak normal shock structure and the transonic equation. It is shown that the structure of oblique shocks is a similarity solution of the V‐T equation. Proper formulation of boundary conditions is considered and a uniqueness proof is given for a particular restricted boundary value problem.