Theoretical model for the separated flow around an accelerating flat plate using time-dependent self-similarity

Abstract

We present a model appropriate to the initial motion (2-3 chords of travel) of a flat-plate airfoil accelerating in an inviscid fluid. The separated flow structures are represented as vortex sheets in the conventional manner and similarity expansions locally applicable to the leading and trailing edges of the plate are developed. The topological character of vortex sheets is maintained rather than resorting to point vortex discretizations. Beyond this, there are two theoretical novelties to our approach as compared to previous studies. First, an expansion is applied to the attached outer flow rather than the vortex sheet circulations and positions. This allows the asymmetric effect of the sweeping component of the free-stream flow parallel to the plate to be built-in to the same governing equation as the singular-order flow. Second, we develop a time-dependent self similarity procedure that allows the modeling of more complex evolution of the flow structures. This is accomplished through an implicit time variation of the similarity variables. As a collective result, the predicted vortex dynamics and forces on the plate compare favorably to Navier-Stokes simulations. Lastly, the model is utilized to provide some new intuition about the separated flow at the leading edge.