The present theory provides an asymptotic-expansion method for inviscid compressible flows with shock in arbitrary two-dimensional slender nozzles. The flow in front of the shock is assumed to be potential, whereas the flow behind the shock is considered to be rotational owing to the presence of the shock. A parameter that measures the slenderness of the nozzle is used as the expansion quantity. It is found that, except for the region immediately behind the shock, the same coordinate scale can be used for the flows both in front of and further downstream behind the shock. The flows for the regions thus obtained show that all the streamlines are approximately affinely similar to the nozzle wall, and the leading term of the transverse pressure gradient is determined by the local wall shape. For the flow region immediately behind the shock, however, the transverse pressure gradient just behind the shock is determined by the shock conditions rather than by the local wall shape, and a solution is found for that region which transforms the transverse pressure gradient from that determined by the shock conditions to that determined by the local wall shape. The well-known flow singularity at the intersection of the wall and the shock is involved in the solution. Meanwhile, a critical shock location at which the flow has no singularity is derived. A numerical example shows also that the inviscid flow may separate from the wall, owing to the different entropy increase across the shock for different streamlines. The predicted separation point, however, is only of qualitative value, since our theory does not account for reverse flows.
A Two-Dimensional Central Non-Oscillatory Scheme for Inviscid Compressible Flows
