Calculation of Induced Drag

Referring again to Figure 3.54, the lift vector for a wing section is seen to be tilted rearward through the induced angle of attack, a,. As a result, а component of the lift is produced in the streamwise direction. This com­ponent, integrated over the wingspan, results in the induced drag. For a differential element,

dDi _ dL dy dy .

Defining the induced drag coefficient as

For the special case of an untwisted elliptic wing, a, and Q are constant over the span, so that Equation 4.17 becomes

Сц = a, Cl

The induced angle of attack for this case was given previously by Equation 3.7^. Thus,

(4.19)

This is a well-known and often-used relationship that applies fairly well to other than elliptic platforms. For a given aspect ratio and wing lift coefficient, it can be shown (Ref. 4.1) that Equation 4.19 represents the minimum achievable induced drag for a wing. In other words, the elliptic lift distribution is optimum from the viewpoint of induced drag.

In order to account for departures from the elliptic lift distribution and the dependence of the parasite drag on angle of attack, Equation 4.19 is modified in practice in several different ways. Theoretically one can calculate the downwash and section lift coefficients, either analytically or numerically, according to the methods of the previous chapter. These results can then be substituted into Equation 4.17 to solve for CDi. The final result for an arbitrary planform is usually compared to Equation 4.19 and expressed in the form

(4.20)

8, for a given planform shape, is a constant that is normally small by comparison to unity. It therefore represents, for a given wing, the fractional increase in the induced drag over the optimum elliptic case.

The numerical determination of 8 will now be outlined for the simplified lifting line model and some typical results will be presented. As an exercise, develop your own program; it is not too lengthy if a subroutine is available for solving simultaneous linear algebraic equations.

A lifting line model composed of discrete vortex line elements is pictured in Figure 4.21. For clarity only five trailing vortices are shown on each side. Their strengths and positions are symmetrical about the centerline. These can be equally spaced but, with the loading dropping off faster at the tips, it is better to have a closer spacing in this region.

A particular trailing vortex is located a distance of yv from the centerline. Control points are then chosen midway between the vortex lines. Generally,

Ус, = kyv, + y„(_,) 1 < і < n

Ус, = 0

At a control point, the bound circulation, Г, is equal to the sum of the strengths of the vortex lines shed outboard of the point. Thus, for n trailing vortices,

or, generally,

г,= Іт,

i=j

For computational efficiency, it should be recognized that

Г.+І = Г, – 7,

Thus it is expedient to determine Г, first and then apply this recursion relationship in order to determine Г2, Г3,…, Г„.

The section lift is given by

f-ipWC

It can also be obtained from the Kutta-Joukowski law as

dL т/p

drpVT

Comparing these two relationships for dL/dy, it follows that

Г = 2 cQV

(4.23)

Thus Equations 4.21 to 4.23 are interrelated, since the downwash, w, depends on the Г distribution.

By applying the Biot-Savart law to our discrete vortex line model, w can be determined along the lifting line (neglecting any contribution from the lifting line itself). The downwash at the j’th control point produced by the ith trailing vortex is

In the parentheses, the first term arises from the downwash at a point on the right side of the wing produced by a vortex trailing from the left. The second term is from the vortex of opposite rotation and symmetric position trailing from the right wing. This expression reduces to

(4-24)

or

(4.25)

The total downwash at yCj is then found by summing Equation 4.13 over і from 1 to n.

(4.26)

Without any loss of generality, it is convenient to set the free-stream velocity, V, the density, p, and the wing semispan, b/2, equal to unity. It then follows that

(4.27)

or

This can be rewritten as

where

The preceding is of the form

E Д.7, = Bj j = 1,2,3,…, n ‘ (4.28)

i = l

and represents n simultaneous linear algebraic equations that can be solved for the unknown vortex strengths yt, y2,…. у(,…, yn.

Since b/2 is set equal to unity, the chord lengths, c„ must be expressed as a fraction of b/2. The values of cf are defined at the control point locations. Having determined the – y, values, the wing lift coefficient can be calculated from

L = 2[Г, уг, + Г2(у„2 – yv) + • • • + Г i(y„. – у и,.,) + • • • + Г„(у„„ – ус„_,)] or, in coefficient form,

CL = A jr, yC| + 2Гi(yVi ~ (4.29)

The aspect ratio, A, appears, since the dimensionless reference wing area equals the actual area divided by (b/2)2. w, is calculated from Equation 4.26 so that the induced drag follows from Equation 4.15.

Di = 2[r1y„1w, + r2(yt,2-y„1)w2+ • • – + Г;(у„,.-yc.^)Wi + • • • + Г„(у„л – y„„.,)w„] or, in coefficient form,

Cd, = A JViy0|w, + 2 Г,-Wj(yVi – y„, ,)] (4.30)

The accuracy of the foregoing numerical formulation of the lifting line model can be tested by applying it to the elliptical planform. Using 25 vortices (n = 25) trailing from each side of the wing, Equations 4.26 to 4.30 are evaluated for flat, untwisted elliptical planform shapes having aspect ratios of 4, 6, 8, and 10 using a theoretical section lift curve slope of 2rrC//rad. The numerical results are presented in Table 4.1 for CL and CDi where they are

Table 4.1 Comparison of CL and CD, Values for Elliptic Wings as Predicted by Numerical and Analytical Methods (a = 10°, C,„ = 2tt) A Cl (Equation 4.29) Cl (Equation 3.73) cDl (Equation 4.30) cDl (Equation 4.19) 4 0.74 0.73 0.0422 0.0425 6 0.83 0.92 0.0354 0.0359 8 0.89 0.88 0.0301 0.0306 10 0.92 0.91 0.0261 0.0266

compared with corresponding analytical values. The CL values are seen to agree within W of each other, while the CDi values differ by only 2% at the most.

With confidence established in the numerical program, the calculations presented in Figures 4.22, 4.23, and 4.24 can be performed. Here unswept wings with linearly tapered planforms are investigated. For this, family of wings, the span wise chord distribution is defined by

c

or

-= 1-(1-A)X

Co

where A equals the taper ratio, cr/c0, cT and c0 being the tip and root chord, respectively. X is the distance along the span measured out from the centerline as a fraction of the semispan.

Figure 4.22 presents 8 as a function of the taper ratio for aspect ratio values of 4, 6, 8, and 10. For all of the aspect ratios, 8 is seen to be a minimum at a taper ratio of around 0.3, being less than 1% higher than the ideal elliptic case at this value of A. The rectangular wing is represented by a A value of 1.0. For this planform shape, used on many light, single-engine

Figure 4.22 Induced drag factor for unswept linearly tapered wings.

Figure 4.23 Spanwise distribution of bound circulation according to numerical solution of lifting line model (n = 25).

aircraft, the induced drag is seen to be 6% or higher than that for the elliptic wing for aspect ratios of 6 or higher.

*• The results of Figure 4.22 can be explained by reference to Figure 4.23, which presents spanwise distributions of Г for the elliptic, rectangular, and

0. 25 taper ratio wings. Observe that the distribution for A = 0.25 is close to the elliptic distribution. Thus the kinetic energies of the trailing vortex systems shed from these two Г distributions are about the same. On the other hand, the Г distribution for the rectangular planform is nearly constant inboard out to about 70% of the semispan and then drops off more rapidly than the elliptic distribution toward the tip. Thus the kinetic energy per unit length of the trailing vortex system shed from the rectangular ■ wing is approximately 6% higher than the energy left in the wake by the tapered or elliptic wing.

In view of the preceding one might ask why rectangular planforms are used in many general aviation airplanes instead of tapered planforms. Part of the answer lies with the relative cost of manufacture. Obviously, the rec­tangular planform with an untapered spar and constant rib sections is less costly to fabricate. Figure 4.24 discloses a second advantage to the rec­tangular planform. Here, the section lift coefficient is presented as a ratio to the wing lift coefficient for untwisted elliptic, rectangular, and linearly tapered planforms. For the elliptic wing, the section Q is seen to be constant and equal to the wing CL except in the very region of the tip, where numerical errors show an increase in Cj contrary to the analytical solution. The rec­tangular planform shows the section Q to be higher than the wing Cl at the

2У b Figure 4.24 Spanwise section G distribution according to numerical solution of lifting line model (n = 25).

centerline and gradually decreasing to zero at the tip. The tapered planform, however, has a section Q that is lower than the wing Cl at midspan. Its Q then increases out to approximately the 75% station before decreasing rapidly to zero at the tip. Thus, again reiterating the discussions of the previous chapter, the tapered planform, unless twisted, will stall first outboard, resul­ting in a possible loss of lateral control.