Wake curvature and the Kutta condition

Abstract

The potential problem for the flow at high Reynolds numbers R outside the boundary layer and wake of a thin flat plate at small incidence with allowance for displacement thickness is not fully denned unless the position of the wake is known in advance. The Kutta–Joukowski hypothesis does not provide a satisfactory first approximation to this because of the singularity in curvature of the streamline springing from the trailing edge in inviscid flow, which implies that the initial curvature of the wake in the real flow will be large enough to cause a modification to the potential flow. The net vorticity per unit length in a curved wake is approximately U∞ δ_2∞dθ₀/ds, where U∞, δ_2∞ and dθ₀/ds are respectively the undisturbed stream velocity, momentum thickness at infinity and curvature. The outer potential problem is set up with a vortex sheet of this strength to represent the wake, leading to a singular integro-differential equation for θ₀(s). From the particular solution we obtain a proportionate correction – (C_D/4π) (log4/C_D) to the Kutta–Joukowski circulation, where C_D is the drag coefficient. For laminar flow this is of order R^−½ log (1/R). However, the solution also contains an arbitrary constant which cannot be settled without an examination of the near wake. The recent work of Brown & Stewartson (1970) suggests that this may lead to a term of the lower order $R^{-\frac{3}{8}}$, but depends on the assumption, not supported by the present analysis, that the pressure rise across the wake is o(R^−¼).