Computation of Induced Drag

Application of the Kutta-Joukowsky theorem The induced drag of an unswept wing of finite span from the Prandtl lifting-line theory, Eq. (3-18), is

S

Di = g J Г(у) Wiiy) dy (3-126)

— S

Here Г(у) is the circulation distribution and Wi(y) is the distribution of the induced downwash velocity over the span — <5, Eq. (3-19).

Now it shall be shown that Eq. (3-126) is also valid for arbitrary wing
planforms. Following Fig. 3-16, let the wing be replaced by so-called elementary wings of the infinitesimal span dy and wing chord c(y). The vortex system of an elementary wing (see Fig. 3-17) consists of a number of horseshoe vortices of width dy arranged in series, one behind the other. In Fig. 3-56, two horseshoe vortices are drawn that originate at the stations xx, yx and x2, у2 of the wing. Their respective widths and circulation strengths are dyx, dy2 and dT 1, dr2. Horseshoe vortex dTг induces at station x2,y2 the upward velocity d2w2X, whereas horseshoe vortex dT2 induced at station xx, yx the upward velocity d2wx2. In analogy to the Kutta-Joukowsky theorem [see Eq. (3-14)], the lifting-circulation elements dFx and dT2 produce forces normal to the upward flow that are the result of the upward-flow velocities d2wn and d2w2X, respectively. These forces represent contributions to the induced drag. The vortex system of Fig. 3-56 produces the partial induced drag

d*Di = —gdrx d2wn dyx — qdT2 d2w2X dy2

where the sense of rotation of the circulation elements has been taken into account. Because dr=kdx, the induced upward velocities of the exterior induction (уx —Уг) are found from Eq. (341) with Eq. (3-50tz), as long as yx Фу2, as

Figure 3-56 Explanatory sketch for computa­tion of induced drag.

It can be seen that the second terms in the square brackets differ by their signs only. When introducing Eq. (3-128) into Eq. (3-127), these terms do not contribute to the induced drag, leading to:

(3-129)

From these considerations there follows immediately that the location of the lifting-circulation elements in the x direction does not affect the drag. Hence, the relationship found earlier [Eq. (3-126)] for the unswept wing is valid for the total induced drag of a wing of arbitrary planform. Since the induced downwash velocity W/(y) from Eq. (3-19) depends only on the circulation distribution over the span, the total value of the induced drag also depends only on the circulation distribution over the span. It is independent of the arrangement of the elementary horseshoe vortices in the. chord direction (flight direction). This result was realized very early by Munk [63, 64] and is known as the Munk displacement theorem. Thus, it is immaterial for the magnitude of the induced drag whether the circulation distribution is caused by the wing planform (aspect ratio, sweepback, taper), by a wing twist, or by camber of the wing surface.

Application of the energy law Although the total lift of a wing can easily be computed by using the momentum law (Sec. 3-3-2), computation of the induced drag by means of the momentum law is considerably more difficult because the inclination of the vortex sheet has to be considered.[17] However, when using the energy law, the inclination of the free vortex sheet (Fig. 3-21), relative to the incident flow direction, can be disregarded. Since the induced velocities on surface I (Fig. 3-21) are zero, the mass gUoodydz permeating the area element dydz of surface II per unit time undergoes an energy increase dEu = (g/2)Uoo(vL + wlo) dy dz. Here u« and vv„ are the induced velocities in the у and z directions, respectively, and the area integral over dEu is the work done by the induced drag UooDi per unit time. Hence, after division by [/«,,

(3-130)

This relationship is valid for both not-rolled-up and rolled-up vortex sheets behind the wing.

The equivalence of Eqs. (3-130) and (3-126) will now be shown for the not – rolled-up vortex sheet. The induced velocity field very far behind the wing with components Vo0(y, z) and wm(y, z) can be expressed through the two-dimensional velocity potential Ф(у, z) as

Неге Ф(у, z) satisfies the potential equation

Introduction of Eq. (3-131) into Eq. (3-130) and integration by parts, the first integral with respect to y, the second with respect to z, yield, with Eq. (3-132),

where surface II has been extended to infinity. Since the values of Ф, ЭФ/ду, and ЭФ/dz vanish at the boundaries y = ±°° and z = ± whereas the potential according to Eq. (348b) changes abruptly in the z direction for z=±0 by the amount Фи(у, 0) — Фг(у, 0) = Г (у), with Eq. (3-13 lb) the induced drag becomes

(3-133)

The integration limits у = ± 00 can be replaced by y = ±s, because Ф(у, 0) = 0 beyond the wing span. Now, by introducing Eq. (3-20) with w«,(y) = — 2w*(y) into Eq. (3-133), Eq. (3-126) is finally obtained, as was to be proved.

Equation (3-130).is valid for the not-rolled-up vortex sheet. Kaufmann [41] showed that the same induced drag is obtained for the rolled-up vortex sheet, where it must be assumed, however, that the cores of the two free vortices have finite velocities.

Practical computation of the induced drag From Eq. (3-126) the formula for the coefficient of induced drag is obtained with 7 = Г1Ы1Ж and щ = w-JU„ from Eq. (3-71 b), and with 7]-y/s as

(3-134)

where /1 is the aspect ratio of the wing from Eq. (34). By expressing the circulation distribution 7 by a Fourier polynomial as in Eq. (3-65<z), the result of the integration becomes, with щ from Eqs. (3-73) and (3-65b),

cDi — zc/L У!

(3-135b)

In the second relationship, the lift coefficient cL was determined from the coefficient d of Eq. (3-66a). In Eq. (3-135b) the first term represents the value for the elliptic circulation distribution [see Eq. (3-3lb)]. Since the second term is always positive, the important theorem follows that the induced-drag coefficient for elliptic circulation distribution is a minimum. This theorem is true for fixed aspect ratio and for fixed lift coefficient cL. It was proved first by Muhk [63].

A summation formula for the coefficient of induced drag can be derived in a way similar to that which led to the summation formulas for the lift-related coefficients of Table 3-1. It has the form

where the values for ain have to be computed from Eq. (3-83).

Equation (3-134) for the induced drag will now be applied to trapezoidal wings with symmetric twist. This example explains the relationship between twist and induced drag. In Sec. 3-3-2 it was shown that the circulation distribution of a symmetrically twisted wing can be put together from that of a wing without twist and a zero distribution. In the same way as the circulation distribution was split up in Eq. (3-63), the induced angle of attack of the twisted wing can be split up:

+ <*io(v) (3-137)

When the wing has no twist, the induced drag is determined just by the term C2. The wing with twist requires, in addition to this term, a term Cj that is proportional to cL, and a term C0 that is independent of <?£. Here, the first term represents a linear twist, the latter a quadratic twist. As can easily be seen by comparison with Eq. (3-3lb), the constant C2 is unity for an elliptic circulation distribution. For the wing without twist the coefficient C2 signifies physically, therefore, the ratio of the induced drag to its minimum value for elliptic circulation distribution.

As an example, the induced drag of a trapezoidal wing with twist is given in

Figure 3-57 Induced drag of symmetrically twisted tapered wing of various aspect ratios л and various tapers Л from Eq. (3-138) (lifting-surface theory), (a) Wing planform with linear twist. (b) Induced drag of wing without twist, from Eq. (3-139a). (c), (d) Twist contribution to the induced drag from Eqs. (3-1396) and (3-139c).

Fig. 3-57. It is based on symmetric linear twist with as(rf) — |tj|Qi. The corresponding circulation distribution has been computed from the lifting surface. Figure 3-51 b indicates that the induced drag of the wing without twist has a minimum for a taper X ~0.45 at all aspect ratios A. The value of this minimum is only a little different from that of the elliptic wing (C2 = 1). For delta wings (X = 0) and rectangular wings (X = 1), cDi is in many instances considerably larger than for elliptic wings. The contribution that is independent of the lift, Fig. 3-57c, is always positive. The sign of the contribution that is linearly dependent on lift, Fig. 3-51 d, depends on the value of the taper. When the taper X ~ 0.45, this contribution is zero for all aspect ratios. The reason is found in the nearly elliptic circulation distribution over wings of this taper without twist. Furthermore, by means of Eq. (3-1356), it can be shown that Cx = 0 for elliptic wings with arbitrary twist and that

M

cDio = О, = я A 2 nal (3-140)

w=2

is the contribution of the induced drag caused by the twist for zero lift.

Investigations related to the establishment of the local drag distribution along the span are compiled in [1].