Note on the circulation in circuits which cut the streamlines in the wake of an aerofoil at right-angles

Abstract

G. I. Taylor in an Appendix to R. & M. 989 (1924) suggested that, in the two-dimensional flow of a real fluid, the circulations in all circuits enclosing the aerofoil and cutting the streamlines in the wake at right-angles would be very nearly the same. The present writer in R. & M. 1996 (1943) gave a 'proof' that the circulations in such circuits were alI equal, and Temple (1943) gave a more rigorous proof of the same theorem. This theorem is of fundamental importance in the calculations of the lift of aerofoils allowing for the boundary layer (see Preston R. & M. 2725, 1949, and Spence, 1954) and it is re-examined in this note. The theory is developed for convenience and simplicity, for an aerofoil with a jet issuing from the trailing edge and the effect of the wake is deduced from this. Elementary considerations, which are set out below, suggest that, in the case of an aerofoil with jet, the above theorem is not true. It would also appear that it is not quite true for an aerofoil with wake, since the same arguments can be applied. However, in this case the departure of the circulation from a constant value for circuits of the type under consideration may be expected to be small, and the effect of this on the prediction of the lift should be negligible for incidences below the stall. In the case of the aerofoil with a strong jet, the existence of circulation in circuits not enclosing the aerofoil but cutting the jet twice at right-angles may have important effects on the lift.