The lift coefficient of a thin, jet-flapped wing

Abstract

A solution is given for the inviscid, incompressible flow past a thin, two-dimensional wing at a small incidence α , at the trailing edge of which a thin jet emerges at a small deflexion ז . The flow inside the jet is assumed to be irrotational, and bounded by vortex sheets across which it is prevented from mixing with the main stream. The effect of the jet on the outside flow is the same as that of a vortex sheet of strength proportional to its curvature and to the jet momentum flux, together with a doublet distribution proportional to the jet thickness (and vanishing in the present limiting case). An integral equation is obtained for the slope of the jet, by aligning the trailing vortex sheet in the undisturbed-stream direction. The solution is expressed as the sum of a Fourier series, together with a function possessing the correct (i. e. logarithmic) form of singular behaviour at the trailing edge. The coefficients of a 9-term interpolation to the series have been calculated on an electronic computer, for a number of momentum-flux coefficients C J lying between 0.01 and 10. The convergence is sufficient to justify the truncation of the Fourier series a posteriori . Simple expressions have been obtained for lift, pressure distribution and pitching moment. In particular, the lift coefficient is given by C L = 4 πA 0 τ + 2 π (1+ 2 B 0 ) α , where A 0 and B 0 are the leading Fourier coefficients associated with deflexion and incidence respectively. Close interpolations to their values are provided by the formulae 4 πA 0 = 3.54 C ½ J + 0.325 C J + 0.156 C 3/2 J , 4 πB 0 = 1.152 C ½ J + 1.106 C J + 0.051 C 3/2 J . For small C J these exhibit the dependence on C ½ J which has been predicted by the empirical theories of Stratford (1956) and Woods (1955). Close agreement with the theory is found in tests made at the National Gas Turbine Establishment, Pyestock, by Dimmock (1955) on an 8:1 elliptic cylinder with a narrow deflected jet exit close to its trailing edge, both for τ ≑ 30° and for τ ≑ 60°. The measured lift coefficients lie within 5% of the corresponding calculated curves (an allowance of 12½% having been made for thickness), and the observed centres of lift lie within 2% of the chord of those calculated for a thin aerofoil, over the whole range of C J . The ‘shallow jet’ approxi­mations used in the theory had not been expected to apply for τ as large as 60°, but the experimental agreement still found in this case resembles that obtained at large deflexion angles by the Glauert (1927) small angle theory for hinged flaps, in which similar approxima­tions occur.