On laminar separation at a corner point in transonic flow (Q2702874)

Based on the asymptotic analysis of Navier-Stokes equations at large Reynolds numbers, the authors study the separation of the laminar boundary layer from a convex corner on a rigid body contour in transonic flow. It is shown that outside the interaction region, the von Kármán-Guderley equation describing transonic inviscid flow admits a self-similar solution. This solution predicts that the pressure on the body surface is proportional to the cubic root of the distance \((-x)\) from the separation point, and the pressure gradient \(dp/dx\) is proportional to \((-x)^{-2/3}\). The analysis of the boundary layer driven by this gradient reveals that, as the interaction region is approached, the boundary layer splits into two parts: the near-wall viscous sublayer whose thickness may be estimated as \(y\sim \operatorname {Re}^{-1/2}(-x)^{5/12}\), and the main body of the boundary layer where the flow is locally inviscid. It is remarkable that, contrary to what happens in subsonic and supersonic flows, the displacement effect of the boundary layer is primarily due to the inviscid part. The contribution of the viscous sublayer proves to be negligible compared to the leading order. Consequently, the flow in the interaction region is governed by the inviscid-inviscid interaction. The longitudinal extent of this region is estimated as \(\Delta x\sim O(\operatorname {Re}^{-3/7})\). To describe the flow in the interaction region, one needs to solve the von Kármán-Guderley equation for the potential flow outside the boundary layer. The solution of this equation is found in an analytical form, thanks to which the interaction between the boundary layer and external flow can be expressed via the corresponding boundary condition. Formulation of the interaction problem involves one similarity parameter, which in essence is the von Kármán-Guderley parameter suitably modified for the flow at hand. The solution of the interaction problem is constructed numerically.