. Integral defect / profile drag relations

As derived in Appendix C, the airfoil’s profile drag is equal to the far-downstream wake momentum defect PTO. Noting that PTE in (4.37) is the sum of the upper and lower surfaces’ P(s) given by (4.36) at the trailing edge, we then have the following result from (4.37). The D’ limit is indicated in the bottom of Figure 4.10.

It is useful to compare the two terms in (4.40) with the friction+pressure drag components, both of which will be further addressed in Chapter 5. Choosing the x-axis to be parallel to V we have

Dfriction — Tw ■ x ds, Dpressure — pw n ■ x ds (4.41)

airfoil airfoil

where n is the airfoil-surface outward unit normal, and Tw is the surface viscous stress vector. Evidently, the first integral in (4.40) can be interpreted as the friction drag, while the second integral must then be the

remaining pressure drag part.

^friction – Tw ds (4.42)

airfoil

^pressure = D’- fiction ~ j~m^dS = I 6*fsdS (4-43)

airfoil+wake airfoil+wake

The friction drag estimate (4.42) is only approximate because the viscous surface force vector Tw is very nearly parallel to the surface, while an exact match with the friction drag definition (4.41) would require Tw to be parallel to the freestream VA along x. However, the angle between Tw and VA is small over most of the surface, especially for thin airfoils, so the friction and pressure drag component estimates (4.42) and (4.43) are still useful conceptually. In particular, the integrals in (4.43) indicate that most of the pressure drag is produced where there is an adverse pressure gradient in the presence of a large mass defect m or displacement thickness 5*. This combination typically occurs over the rear portion of the airfoil at high lift, and can be clearly seen in the top of Figure 4.10 for s > 0.6c.

To relate the kinetic energy defect result (4.38) and (4.39) to profile drag, we first write P and K in terms of the velocity defect Au.

A u =	u — ue		(4.44)
P=	У (ue — u) pu dn =	(—Au pu dn	(4.45)
K=	{u%.— u2) pu d? t =	—Au, (tte + jAw) pu d??.	(4.46)

If Au is very small compared to ue, then K and P become simply related to a good approximation.

K — J-Auu, e pu dn = Pue (if Au ^ ue) (4.47)

This occurs in the far-downstream wake where Au goes to zero as the wake spreads and mixes out. And in the far wake ue also approaches VA, so that the two defects become exactly related far downstream.

(4.48)

Combining this with (4.40) then gives an alternative expression for the overall profile drag in terms of the far-downstream wake kinetic energy defect, and also the dissipation everywhere.

This rather simple result has a power balance interpretation: The drag D’ must be balanced by an external thrust force which moves at speed VA relative to the airmass, and thus exerts a power of D’VA which is all dissipated in the viscous layers. The conclusion is that profile drag is uniquely related to viscous stresses as quantified by the distribution of the dissipation integral D(s), and this dissipation contributes positively to the drag everywhere since D> 0 always. This strictly-positive dependence of drag on viscous forces isn’t immediately obvious from the alternative momentum-based profile drag expression (4.40). Its second pressure term always has some locally negative contributions to the total drag, and its first friction term also has locally negative contributions in separated regions which exhibit reversed flow and hence Tw < 0.

Invoking the friction and pressure drag definitions (4.41), the power-balance relation (4.49) gives an alter­native relation for the friction plus pressure drag.

-^friction + -^pressure = ТГ I V (4.50)

^ airfoil+wake

Since the D and Dfriction terms both depend only on the viscous stresses т(s, n) via their definitions (4.34) and (4.41), the remaining pressure drag ressure term then also depends only on the viscous stresses. In this power balance view, we can then conclude that the pressure field is not the cause of pressure drag, but rather it’s a necessary additional power-transmission mechanism (the surface friction forces alone are insufficient) from the body surface to the flow-field interior where the viscous power dissipation takes place. This has implications for aerodynamic design as will be discussed in Section 4.11.4.