Summary of Chapter 6

In this chapter, inviscid incompressible and compressible flows past large and moder­ate aspect ratio wings are studied. Thin wings that only slightly disturb an otherwise uniform flow are defined. The three-dimensional potential flow equation governs the flow. The tangency condition is a mere extension of the two-dimensional one and the linearized pressure coefficient still reads Cp = —2u/U.

First the linear theories are considered and the admissible jump conditions inves­tigated. It is shown that two types of discontinuities can occur, shock waves across which the pressure has a jump, i. e. < Cp >= 0 hence < u >= 0, and vortex sheets across which the pressure is continuous, i. e. < Cp >= 0 hence < u >= 0, and < w >= 0, but < v >= 0. In contrast with two-dimensional flow and its bound vorticity, three-dimensional flow combines bound vorticity in the finite wing and trailed vorticity in the vortex sheet. Forces and moments can be obtained from a momentum balance applied to a large control volume surrounding the finite wing. The formula for lift and pitching moment are consistent with the Kutta-Joukowski lift theorem applied to the bound vortex, whereas the drag exhibits two contributions, one corresponding to the wave drag, the other to the vortex drag or lift induced drag resulting from the trailed vortices.

For large aspect ratio wings, Prandtl lifting line theory is presented and the forces expressed as integrals along the span or lifting line. The matching between the local two-dimensional flow in a wing cross section and the three-dimensional flow super­imposed by the vortex sheet, which determines the local downwash and induced incidence, results in the celebrated integro-differential equation of Prandtl. The gen­eral solution of the latter can be sought as an infinite Fourier series with unknown coefficients for the Fourier modes. Lift and drag are found to be related to the Fourier coefficients in a remarkable way, the lift depending on the first mode only and the induced drag on a summation with positive weights of the squares of the coefficients in the series. This leads to the ideal wing having minimum drag for a given lift and corresponding to the first mode only. The circulation is elliptical and the downwash constant. The geometry of the ideal wing is not unique. In fact, in inviscid flow, there is an infinite number of wings that can produce the ideal loading. They differ by chord and twist distributions. Examples are shown of an elliptic planform without twist and a rectangular planform with twist that satisfy the requirement. Extension to non-straight lifting lines is also discussed and the design and analysis of winglets presented. For wings of arbitrary planform, the numerical simulation is the most practical approach and illustrated with the two examples above of ideal loading. The simulation includes also viscous effects and nonlinear profile polars, with a penalization technique to capture separated flow regions on the wing.

The vortex lattice method is described for flow past moderate aspect ratio wings. It is a natural extension of the lifting line method that has close connection with the treatment of unsteady flow past wings and wind turbines (see Chap. 10).

Compressible flow over moderate aspect ratio wings based on small disturbance approximations, in both subsonic and supersonic regimes, are treated with linearized boundary conditions. General wings in subsonic flows are treated based on Green’s theorems, while for general wings in supersonic flows, Kirchoff’s formula is used. For wings in transonic flows, full potential equations should be solved numerically. For high aspect ratio wings, transonic lifting line theory is briefly discussed.

6.13 Problems