The theory of the jet-flap for unsteady motion

Abstract

Linearized methods developed in earlier papers (Spence 1956, 1961) are used to discuss the non-stationary flow of a wing with a jet-flap. We consider a thin twodimensional wing at zero incidence in a steady stream of speed U, with a thin jet emerging parallel to the chord at the trailing edge, and study the motion following an instantaneous deflexion of the jet through an angle τ₀. If the momentum-flux coefficient C_J of the jet is [Lt ] 4, the governing equations can be put in a form in which C_J does not appear explicitly, and a similarity solution then gives the shape of the jet at small times t from the start of the motion as a function of $(x -c)\mu ^{-\frac {1}{3}}t^{-\frac {2}{3}}$, where x= c is the trailing edge and $\mu = \frac{1}{2}C_J$. The solution is obtained from a certain third-order integro-differential equation, by constructing the Mellin transform of the non-dimensional shape. When t is large the jet near the wing approaches the shape given by the known results for steady flow, but its shape at distances of the order of Ut downstream changes diffusively under the action of the starting vortex. A similarity solution is also found for the flow in this region in terms of $(x -c)\mu ^{-\frac {1}{3}}t^{-\frac {2}{3}}$, without restriction to small μ. Expressions for the lift coefficient at small and large times are found, and the case of an oscillating deflexion angle is treated by the same methods.