Assumptions of the Lifting-Line Theory
In classic lifting-line theory, the wing is represented by a single finite-strength vortex or lifting line and the trailing-vortex sheet is assumed to be planar and parallel to the oncoming flow. Normally, the vortex sheet is assumed to be in the plane z = 0. Thus, the wake can induce velocity components in the x-y plane only in the z direction—that is, in a direction perpendicular to the lifting line. In particular, the wake cannot induce any velocity components in the spanwise (y) direction in the plane z = 0 containing the lifting line. Of course, the wake also can induce velocity components above, below, and beyond the span of the lifting line, but none of these are of interest in our flow model.
The basic lifting-line theory, then, neglects the spanwise velocity components induced at the lifting line by the trailing-vortex sheet. Thus, each airfoil section behaves as if the flow were locally two-dimensional. This assumption is at odds with the physical argument used previously that the spanwise velocity mismatch at the wing trailing edge gives rise to the trailing-vortex filaments. However, neglecting the spanwise-velocity components gives satisfactory results because they are small, typically a few percentage points of VTC. The assumption of local two-dimensionality is worst at the wing tips, where the spanwise flow is large, but this area represents only a small fraction of the entire wing. One advantage of this assumption is that it allows a two-dimensional treatment of the downwash at the lifting line. Also, if each airfoil section behaves as if it were in a locally two-dimensional flow, then the chordwise pressure-distribution detail on the wing—which is lost in the lifting-line model—is available from two-dimensional airfoil theory or experimental data as in Chapter 5. The fact that the two-dimensional airfoil section is part of a finite wing is manifested by a modification in the angle of attack at which the section is operating. Finally, experimental data for two-dimensional airfoils may be used to predict the viscous drag of the wing. Thus, the advantages of using this assumption of negligible spanwise flow are many and the theory still exhibits satisfactory accuracy. If the wing is highly swept or of very low AR, then the spanwise-flow component is large, the assumption is invalid, and the simple lifting-line theory breaks down. The simple theory has been extended to treat swept wings, but this is not discussed here. The method becomes complicated and the problem is solved more easily numerically.
Because the lifting line represents the wing, it is positioned at the wing quarter- chord. This is because (1) each airfoil section comprising the wing acts as if the flow were two-dimensional (i. e., the trailing vortices present in the model induce no span – wise-velocity components); and (2) the lift force on an arbitrary two-dimensional airfoil in incompressible flow (see Chapter 5) acts at the aerodynamic center (i. e., at c/4), according to thin-airfoil theory. Recall that the assumed chordwise-vorticity distribution that led to this result (see Fig. 5.17) was chosen so as to satisfy the Kutta condition. Thus, placing the lifting line at the wing quarter-chord results in the Kutta condition being approximately satisfied all along the wing trailing edge.