On transonic shocks in two-dimensional variable-area ducts for steady Euler system

This paper concerns transonic shocks in compressible inviscid flow passing a twodimensional variable-area duct for the complete steady Euler system. The flow is supersonic at the entrance of the duct, whose boundaries are slightly curved. The condition of impenetrability is posed on the boundaries. After crossing a nearly flat shock front, which passes through a fixed point on the boundary of the duct, the flow becomes subsonic. We show that to ensure the stability of such shocks, pressure should not be completely given at the exit: it only should be given with freedom one, that is, containing an unknown constant to be determined by the upstream flow and the profile of the duct. Careful analysis shows that this is due to the requirement of conservation of mass in the duct. We used Lagrangian transformation and characteristic decomposition to write the Euler system as a 2 × 2 system, which is valid for general smooth flows. Due to such a simplification, we can employ the theory of boundary value problems for elliptic equations to discuss well-posedness or ill-posedness of transonic shock problems in variable-area duct for various conditions given at the exit.