Idealized Far-Field Drag

The exact far-field drag is defined as the streamwise component of the total far-field force (5.13).

D = F ■ X = S [ (Po – p) n ■ x – p(V ■ n)(n – Vo) ] dS (5.26)

outer

Appendix C evaluates the 2D form of this relation by integrating around a circular contour far from the airfoil, using potential 2D far-field approximations for p and V, with a special treatment of the viscous wake velocity defect. For a lifting 3D body, the integrand in (5.26) is negligible everywhere except in the downstream Trefftz-plane part of Souter, where the trailing wake exits. Hence, for 3D cases the drag integral above can be conveniently restricted to only this yz Trefftz plane, which has П = X.

D = [ pTO – p – pu(u – Vco)]dy dz (5.27)

TP

We will now decompose the effectively-exact far-field drag expression (5.27) into profile and induced com­ponents. This requires an idealization of the flow in the Trefftz plane, as sketched in Figure 5.7. Specifically, the vortex sheet thickness is assumed to be small compared to its span, and its net strength y(s) is assumed to be in the x direction. The wake roll-up will also be neglected to simplify the sheet’s yz shape. The roll-up issue in drag calculations is discussed in more detail by Kroo [47].

As shown in Figure 5.7, the total velocity at the Trefftz plane is broken down into the freestream VOX, a potential perturbation velocity Vp associated with the streamwise vorticity 7(s), and a streamwise viscous defect An associated with the transverse vorticity (shown in Figure 5.4) inside the viscous wake.

V = (Vo + Au)X + Vp (5.28)

The pressure is related to the potential part of the velocity (excludes An) by the incompressible Bernoulli equation (1.109). This is valid since Trefftz-plane velocities are typically low even for high-speed vehicles.

P = Pc* + pc^V^ – p^ |KoX + V<p|2

= Pc* – Poo Ко Px – 3P00 {pi + Py + pi) (5.29)

Substituting the velocity (5.28) and pressure (5.29) expressions into the drag integral (5.27) and simplifying the result produces a natural decomposition of the drag into its induced-drag and profile-drag components.

The induced drag expression (5.31) is seen to be the crossflow kinetic energy (per unit distance) deposited by the body. This energy is provided by part of the body’s propelling force working against the induced drag. The remaining part works against the profile drag, considered next.