J. Fluid Nech. zyxwvutsr (1987). zyxwvutsrqp vol. 179, pp. 513-529 Pritttrd zyxwvutsrqp itt Oren/ Rrifnin 513 zy On the breakdown of the steady and unsteady interacting boundary-layer description By R. A. W. M. HENKES Department of Applied Physics, Delft University of Technology, P.O. Box 356, Delft, The Netherlands AND A. E. P. VELDMAN Department of Mathematics and Informatics, Delft University of Technology, P.O. Box 356, Delft, zyxwvu and National Aerospace Laboratory NLR, P.O. Box 90502, Amsterdam, The Netherlands (Received 24 April 1986) zyxw It is known that the classical boundary-layer solution breaks down through the appearance of the Goldstein singularity in a steady solution or Van Dommelen’s singularity in an unsteady solution. Interaction between the inviscid flow and the boundary layer removes the Goldstein singularity, until a new critical parameter is reached, corresponding to a marginal separation in the asymptotic triple-deck description. In earlier studies instabilities were encountered in interacting boundary- layer calculations of steady flow along an indented plate, which might be related to the breakdown of the marginal separation. The present study identifies them as numerical. Further, until now it was unknown whether the unsteady interacting boundary-layer approach would remove Van Dommelen’ssingularity in the classical boundary layer around an impulsively started cylinder. It is shown here that its appearance is at least delayed. The calculations show the experimentally known individualization of a vortex, after which the solution grows without reaching a steady limit; a process that is likely to be related to dynamic stall. 1. Introduction When one relies on the physics to select initial and boundary conditions, the Xavier-Stokes equations are expected to give a complete mathematical description of the flow. For large Reynolds numbers the Navier-Stokes solution can be approximated by an asymptotic description. Critical combinations of parameters (e.g. angle of incidence, geometry size, Reynolds number) can exist where a certain asymptotic description fails to represent the physics and breaks down. This mathematical breakdown can be characterized by a singularity, the non- existence/non-uniqueness of the steady solution, or the growth beyond all bounds at infinitely large time of the unsteady solution. In the worst case the asymptotic solution may diverge from the Naviel-Stokes solution without a special warning. The breakdown indicates the need to introduce new asymptotic equations to continue the asymptotic description of the Navier-Stokes solution. The large effort required to calculate a high-Reynolds-number Navier-Stokes solution, owing to the existence of different time- and lengthscales, motivates the search for these new asymptotic structures. Another reason to continue studying asymptotic descriptions is its great physical relevance. A breakdown might be related to the transition to another