L. Van Dommelen

The spontaneous generation of the singularity in a separating laminar boundary layer

Abstract The question whether solutions of the unsteady laminar boundary layer equation may turn singular spontaneously has been in dispute lately. By a case study, here a numerical proof is given that the spontaneous generation of a singularity does indeed occur, thus settling the point. The proof is achieved by abandoning the usual Eulerian boundary layer coordinates in favor of Lagrangian coordinates. The singularity then occurs at a stationary point, the presence of which can be ascertained numerically without ambiguity.

The Genesis of Separation

For the classical boundary-layer problem of the impulsively started circular cylinder [3.40], both the Blasius small-time expansion [3.41] and the Proudman-Johnson solution for the rear stagnation point [3.4, 3.38] imply that the boundary layer remains, to its order of thickness, attached to the wall. However, experimental observations [1.42, 3.42] show “thin vortex layers” which “leave the surface of the body and curl round on themselves.” Very recent experimental results of Bouard and Coutanceau [3.43], which became available after the original presentation of this paper, suggest even more precisely a “vorticity peak” which is “individualised into a ‘vortex’” after roughly 0.75 diameter movement of the cylinder. These inconsistencies with experiment suggest that the Blasius and the Proudman-Johnson results, though locally correct, do not yet qualitatively describe the entire flow development.

Boundary-layer separation control on a thin airfoil using local suction

High-speed incompressible flow past a thin airfoil in a uniform stream is considered. When the angle of attack for a solid airfoil exceeds a certain critical value, the boundary layer in the leading-edge region separates in a process known to lead to dynamic stall. Here suction near the leading edge is studied as a means of controlling separation and thereby inhibiting dynamic stall. First, steady boundary-layer solutions are obtained to determine the nature of suction distributions required to suppress separation on an airfoil at an angle of attack beyond the critical value (for a solid wall). Unsteady boundary-layer solutions are then obtained, using a combination of Eulerian and Lagrangian techniques, for an airfoil at an angle of attack exceeding the critical value; the effects of various parameters associated with the finite-length suction slot, its location and the suction strength are considered. Major modifications of the Lagrangian numerical method are required to account for suction at the wall. It is determined that substantial delays in separation can be achieved even when the suction is weak, provided that the suction is initiated at an early stage.

Interactive Separation from a Fixed Wall

The Kirchhoff free streamline picture has long been believed to be representative of large scale separated flows [410]. The local flow near separation was first described by Sychev [411] in the framework of triple deck theory [391, 412].

Boundary layer separation singularities for an upstream moving wall

The possible singular structure of a steady boundary layer at separation for a regularly prescribed outer flow and an upstream moving wall is examined. By means of a family of similar solutions it is shown that a consistent description can indeed accomodate the flattening of the velocity profile at flow reversal postulated in the so called MRS-model of separation. The conditions in the separated region downstream, being unknown, are reflected in the non-uniqueness of the expansion.

A determinate model of thrust-augmenting ejectors

A theoretical analysis of the compressible flow through a constant-area jet-engine ejector in which a primary jet mixes with ambient fluid from a uniform free stream is pursued. The problem is reduced to a determinate mathematical one by prescribing the ratios of stagnation properties between the primary and secondary flows. For some selections of properties and parameters more than one solution is possible and the meaning of these solutions is discussed by means of asymptotic expansions. Our results further show that while under stationary conditions the thrust-augmentation ratio assumes a value of 2 in the large area-ratio limit, for a free-stream Mach number greater than 0.6 very little thrust augmentation is left. Due to the assumptions made, the analysis provides idealized values for the thrust-augmentation ratio and the mass flux entrainment factor.