Viscous-Inviscid Interaction in a Transonic Flow Caused by a Discontinuity in Wall Curvature

The work addresses an important question of whether a discontinuity inwall curvature can cause boundary layer separation at transonic speeds.Firstly an inviscid transonic flow in the vicinity of a curvature breakis analysed. Depending on the ratio of the curvatures, several physicallydifferent regimes can exist, including a special type of supersonic flowswhich decelerate to subsonic speeds without a shock wave, transonicPrandtl--Meyer flow and supersonic flows with a weak shock. It is shown thatif the flow can be extended beyond the limiting characteristic, itsubsequently develops a shock wave. As a consequence, a fundamental linkbetween the local and the global flow patterns is observed in our problem.From an asymptotic analysis of the Karman--Guderley equation it followsthat the curvature discontinuity leads to singular pressure gradientsupstream and downstream of the break point. To find these gradients, we perform computations and employ both the hodograph method and the phase portrait technique.The focus is then turned to analysing how the given pressure distributionaffects the boundary layer. It is demonstrated that the singular pressuregradient, which appears to be proportional to the inverse cubic root of the distance form the curvature break, corresponds to a special resonant case for the boundary layer upstream of the singularity. Consequently, the boundary layer approaches the interaction region in a pre-separated form. This changes the background on which the viscous-inviscid interactiondevelops, allowing to construct an asymptotic theory of the incipient viscous-inviscid interaction for our particular problem. The analysis of the interaction which takes place near a weak curvature discontinuity leads to a typical three-tier structure. It appears to bepossible to obtain analytical solutions in all the tiers of the tripledeck when the curvature break is small. As a result, the interactionequation may be derived in a closed form. The analytical solution of the interaction equation reveals a local minimum in the skin friction distribution, suggesting that a local recirculation zone can develop near thecurvature break. In fact, the recirculation zone is formed when the ratioof the curvatures is represented as a series based on negative powers of the logarithm of the Reynolds number. This proves that a discontinuity in wall curvature does evoke boundary layerseparation at transonic speeds. The result is fundamentally different fromthe effect of a curvature break at subsonic and supersonic speeds, asno separation takes place in these two regimes (Messiter & Hu 1975).