The induced velocity at the lifting line is given a special name: the downwash. It is assigned the velocity component symbol w and enters with a negative sign because the induced velocity is in the negative z-coordinate direction.
Recall that the Biot-Savart Law states that for a semi-infinite vortex filament extending from Point Q to infinity, as shown in Fig. 6.4 and Eq. 6.4, the induced velocity is given by VP = Г /4nh. This velocity is normal to the plane containing the filament and Point P. We apply this result to an arbitrary single filament in Fig. 6.8 and assume that the filament trails in the direction of VTC. The induced velocity (i. e., downwash) at a fixed but arbitrary Point y0 on the lifting line, due to a semiinfinite vortex filament of strength dr that originates at the lifting line at a spanwise location y and trails downstream, is given by:
Points Q and 2 (at downstream infinity) in Fig. 6.8 aid in orientation with Fig. 6.4. This induced velocity is in the z direction. We note that because Eq. 6.8 is simply a repeat of Eq. 6.4, an integration over the streamwise length of the trailing – vortex filament already occurred. Eq. 6.8 is written as dw to emphasize the fact that this is the induced velocity due to only one trailing filament of infinitesimal strength.
Substituting the expression for dr from Eq. 6.7,
Finally, we check the sign. Referring to Fig. 6.8, for the spanwise locations chosen, y > y0 and the sense of the trailing-vortex filament is counterclockwise looking upstream. The velocity induced at y0 then should be downward, or dw should be negative. Now, from Fig. 6.8, the distribution over the right half-wing is such that the rate of change with y is negative. Thus, Eq. 6.9 is physically correct as written.
Because Eq. 6.9 represents the downwash contribution at y0 due to one semiinfinite filament, it remains to calculate the total downwash at y0 due to all of the trailing-vortex filaments from the wing that comprise the vortex sheet. This amounts to a summing, or integration, over y; namely:
Notice that as we sum across the span, some filaments induce an upward component and others a downward component at Station y0, depending on the relative magnitudes of y and y0 and also on which half-wing is considered. Equation 6.10 is the sum total of all of these contributions and is directed downward for positive lift on the wing. The singular behavior of the integrand when the integration variable, y, has the specific value y = y0 is addressed in due course.
Figure 6.9 shows a cross section of a wing according to the lifting-line model. The wing is in an oncoming freestream VTC. Because a vortex filament does not induce any velocity (think of the vortex as exhibiting viscous-dominated, solid-body rotation at the center), the only other velocity component at the wing quarter-chord (i. e., at the lifting line) is the downwash due to the trailing-vortex wake, shown in Fig. 6.9. Because the vortex sheet is aligned with VTC, the downwash component w = Vs is perpendicular to VTC. For positive lift, the downwash at the lifting line always is directed downward.
Figure 6.9. Velocity components at the lifting line.