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In a transonic nozzle flow in which the velocity is slightly supersonic in some neighbourhood of the nozzle throat, a shock wave may be present either very close to the throat or else somewhat further downstream. In the latter case, relatively simple series solutions in general provide an asymptotic description of the fluid motion except very close to the shock wave. These outer solutions are reviewed for symmetric two-dimensional flow, and it is shown that the shock-wave jump conditions are not satisfied. A correction is then derived in the form of an inner solution for a small region immediately behind the shock. The resulting solution exhibits the singularities in the pressure gradient, streamline curvature and shock-wave curvature which are expected to occur at the intersection of a normal shock wave and a curved wall. An extension to axisymmetric flow is also given.

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Boundary-layer theory for flows at high Reynolds number fails locally in small regions with large gradients, where special solutions are required, with the pressure initially unknown. Examples include the flow near a discontinuity in surface geometry or near a separation point. During the past 15 years, local-interaction problems have been studied extensively for laminar flows, with particular attention to the description and prediction of separation, and a few examples have been worked out for turbulent flows. The basic ideas of asymptotic local-interaction theory are described, and applications are summarized for a variety of flows.