WING THEORY BY THE METHOD. OF VORTEX DISTRIBUTION

3- 2-1 Fundamentals of Prandtl Wing Theory

The creation of lift of a wing is tied to the existence of a lifting (bound) vortex within the wing (Fig. 3-7). This fact has been explained in Sec. 2-2 by means of Fig. 2-4. The position of the bound vortex on the wing planform is described in Sec. 2-3-2 for the inclined flat plate. Accordingly, it is expedient to position the vortex on the one-quarter point of the local wing chord. An unswept wing in symmetric incident flow is therefore represented by a bound vortex line normal to the incident flow direction.

Figure 3-6 Distribution of local lift coefficients for a rectangular wing of aspect ratio л = 5 and profile Go 420. Reynolds number Re — 4.2 • 10s; Mach number Ma = 0.12.

Figure 3-7 Vortex system of a wing of finite span.

Since the pressure differences between lower – and upper-wing surfaces decrease to zero toward the wing tips, producing a circulation around the wing, the flow field of a wing of finite span is three-dimensional. This pressure equalization at the wing tips, shown schematically in Fig. 3-86, causes an inward deflection of the streamlines above the wing and an outward deflection below the wing (Fig. 3-8(2). In this way, streamlines that converge behind the wing have different directions. They form a so-called surface of discontinuity with inward flow on the upper surface, outward flow on the lower surface (Fig. 3-8c). The discontinuity surface tends to roll up farther downstream (Fig. 3-8<f), forming two distinct vortices of opposite

sense of rotation. Their axes coincide approximately with the direction of the incident flow (Fig. 3-8e and f). These two vortices have a circulation strength Г. Thus, behind the wing there are two so-called free vortices that originate at the wing tips (Fig. 3-7). Far downstream, these two vortices are connected by the starting vortex, the evolution of which was explained in Sec, 2-2-2. The bound vortex in the wing, the two free vortices, originating at the wing tips, and the starting vortex together form a closed vortex line in agreement with the Helmholtz vortex theorem. The vortices produce additional velocities in the vicinity of the wing, the so-called induced velocities. They are, as a result of the sense of rotation of the vortices, directed downward behind the wing. They play an important role in the theory of lift.

The starting vortex need not be taken into account in steady flow for treatment of the flow field in the vicinity of the wing. This is understandable when it is realized that the wing has already moved over a long distance from its start from rest. In this case the vortex system consists only of the bound vortex in the wing and the two infinitely long, free vortices. These form again an infinitely long vortex line shaped like a horseshoe, open in the downstream direction. This vortex is called a horseshoe vortex.

The very simplified vortex model of Fig. 3-7, having one bound vortex of constant circulation, is still insufficient for quantitative determination of the aerodynamics of the wing of finite span. A further refinement of the two simple free vortices originating at the wing is necessary. The above-mentioned pressure equalization at the wing tips causes the lift, and consequently the circulation, to be reduced more near the wing tips than in the center section of the wing. At the very wing tips even complete pressure equalization occurs between upper and lower surfaces. The circulation drops to zero. The actual circulation distribution is. similar to that shown in Fig. 3-9; it varies with the span coordinate, Г = Г(у). The variable circulation distribution Г(у) in Fig. 3-9 can be thought of as being replaced by a step distribution. At each step a free vortex of strength А Г is shed in the downstream direction. In the limiting case of refining the steps to a continuous circulation distribution, the free vortices assume an areal distribution (vortex sheet). A strip of this vortex sheet of width dy has the circulation strength dF = (dr[dy)dy. Thus the slope of the circulation distribution F(y) of the bound vortices determines the distribution of the vortex strength in the free vortex sheet.

It was Prandtl [69] who for the first time gave quantitative information on the three-dimensional flow processes about lifting wings based on the above discussed mental picture. Earlier, Lanchester had investigated this problem qualitatively (see von Karman [90]).

Lift and induced drag From the Kutta-Ioukowsky theorem [see Eq. (2-15)], the lift dL of a wing section of width dy and its circulation Г (у) are related by

dL = 5 УГ[у) dy

Figure 3-9 Wing with variable circulation dis­tribution over the span.

The total lift is obtained by integration as

L — qV f Г (у) dy (3-15)

•f-s

As the most important consequence of the formation of free vortices, the airfoil of finite span undergoes a drag even in frictionless flow (induced drag), contrary to the airfoil of infinite span. Physically, the induced drag can be explained by the roll-up of the discontinuity sheet into the two free vortices: During every time increment a portion of the two free vortices has to be newly formed. To this end, work must be done continually; this work appears as the kinetic energy of the vortex plaits. The equivalent of this work is expended in overcoming the drag during forward motion of the wing.

On the other hand, the formation of induced drag may also be understood by means of the Kutta-Joukowsky theorem as follows: The downstream-drifting free vortices produce a downwash velocity w£ behind and at the wing, after Biot-Savart. At the wing the incident flow velocity of the wing profile is therefore the resultant of the incident flow velocity V and this induced downwash velocity w£. Accordingly, the resultant incident flow direction at the wing is inclined downward by the angle а£ against the undisturbed incident flow direction, with

«< = У (3-16)

In general, w£ < V and hence a£ sin щ ~ tan a£.

From the Kutta-Joukowsky theorem (Sec. 2-2-1), the resultant dR of the aerodynamic forces at the wing cross section у (Fig. 3-10) stands normal to the resultant incident flow direction. Hence, normal to the undisturbed flow direction there is a lift component dL = dR cos « dR and parallel to the undisturbed flow direction a drag component dDi — dR sin щ «=» dRat. The latter is the induced drag of the wing cross section y, which, with Eq. (3-16), becomes

Wi

dDt = aidL = dL —

Hence, the total induced drag is obtained through integration over the wing span from у = — s toy = +s, and by noting Eq. (3-14), as

(3-18)

where W/(y) is the distribution of the induced downwash velocity that is variable in

the general case.

The distribution of the induced downwash velocity along the span is obtained by applying the Biot-Savart law to the semi-infinitely long free vortex behind the wing. The contribution of the vortex strip dy at station у to the downwash velocity at the location of the lifting line у (Fig. 3-9) is with iy—y) being the distance of the point under consideration (control point) у from the location у’ of the free vortex line. From this, the induced velocity at the wing is found by integration over the area of the free vortices as*

(3-19)

From this equation, the induced downwash velocity W/ at the location of the lifting line can be computed when the circulation distribution Г(у) is known. Finally, the induced drag can be determined from Eq. (3-18).

It should be mentioned here that the induced downwash velocity very far behind the wing has twice the value of the downwash velocity wt at the wing from

*At station y’ =y, the integrand has a singularity. The analysis shows that the integral has to be evaluated through the Cauchy principal value. Hence, the range у—є<у'<у + є must be excluded during integration and the limit operation

Ihu і f ■ ■ ■ dy’ + f. .. dy’

> О 1 —f. .*»_!_* must be conducted.

Eq. (3-19). This is obvious from the fact that far downstream the free vortices can be taken as being infinitely long vortex lines, leading to

Woo(y) — — 2и>((у) (3-20)

The velocity w0о is taken as positive in the direction of the positive z axis (see Fig. 3-21).

Prandtl’s integral equation of the circulation distribution The above considerations will now be applied to the derivation of an equation for the determination of the spanwise circulation distribution for a given wing of finite span.

The change of the incident flow direction that results from the downwash velocity induced by the free vortices was explained in Fig. 3-10. This change of flow incidence, at equal geometric angles of attack a, is responsible for the reduced lift at the cross section у of a finite-span wing in comparison with the lift at the same cross section of an infinitely long wing.

For a span element dy of a finite-span wing, Eq. (3-12) yields for the lift:

dL = cfy) j V2c(y) dy (3-2 la)

= c’i„ae{y) V2c(y) dy (3-216)

Here c(y) is the wing chord at station у (Fig. 3-9) and q(y) = c’laoote(y) is the local lift coefficient of the area element dA = c(y) dy; ae(y) is termed the effective angle of attack (Fig. 3-10) and = (dci(da)« is called the lift slope for the airfoil of infinite span. The latter value is close to 2тг, from the theory of thin profiles (see Chap. 2). For the inclined flat plate, cj« is exactly equal to 2rr. Equation (3-21) is based on the concept that a profile cross section of a wing of finite span behaves like that of a wing of infinite span (plane flow) at an angle of incidence ae.

The geometric angle of attack о(у), measured from the zero-lift position, the effective angle ae(y), and the induced angle az-(y) [Eq. (3-16)] are related by

qO) = ae0) + af(y)

as shown in Fig. 3-10. The effective angle of attack ae is obtained from Eq. (3-216) with the help of Eq. (3-14), and the induced angle of attack from Eq. (3-19) with ai= wil V as

2rQ)

Vc{y)ct

Introducing Eq. (3-23) into Eq. (3-22) yields the following basic equation for the determination of the circulation distribution:

2Г(у) f dr dy’

VcfyWio[13] + v J dy’ У —у’

-s

This is Prandtl’s integral equation for the circulation distribution of a wing of finite span as first published by Prandtl in 1918 [69]. It is a linear integral equation for the circulation distribution Г(у), where Г depends linearly on the angle of attack a. The profile coefficient ctaa is known from profile theory (Chap. 2).*

With given wing geometry [chord distribution c(y) and angle-of-attack distribu­tion a(y)], the circulation distribution can be determined from Eq. (3-24). This is the so-called direct problem of wing theory. Conversely, if the circulation distribution Г(у) is known, either the angle-of-attack distribution (twist angle) ct(y) can be computed from Eq. (3-24) when the chord distribution c(y) is given, or the chord distribution c(y) when the angle-of-attack distribution a(y) is given. This is the so-called indirect problem of wing theory. In either case, from the circulation distribution Г(у) the lift is obtained from Eq. (3-15) and the induced drag from Eq. (3-18).

From a mathematical viewpoint, the direct problem is considerably more difficult than the indirect problem, because in the former case an integral equation has to be solved while in the latter case only a quadrature has to be performed.

Elliptic circulation distribution A particularly simple solution of Eq. (3-24) that is of great practical importance is found for the elliptic circulation distribution along the span. In this case the circulation becomes

where Г0 is the circulation at the wing center y— 0 (Fig. 3-11). From Eq. (3-15), the lift becomes

L =^obVrQ

4 u

The induced down wash-velocity is obtained from Eq. (3-19). Execution of the integral yields for points within the span, jyj <b/2,

(3-21 a)

(3-276)

This remarkable result shows that, for elliptic circulation distribution, the induced downwash velocity w,-, and consequently the induced angle of attack a,-, are constant over the span (Fig. 3-11).

By introducing Eqs. (3-25) and (3-21 a) into Eq. (3-18), the induced drag is obtained with T0 from Eq. (3-26) as

(3-2Sa)

(3-286)

Here, q — (o/2)V2 is the dynamic pressure resulting from the velocity V. The induced drag is proportional to the square of the lift and inversely proportional to the dynamic pressure and the square of the span. Comparison of Eqs. (3-28b) and (3-27b) confirms the relationship Dt = cqL, given in Eq. (3-17).

The geometry of the corresponding wing is obtained in a particularly simple way when starting from the wing without twist, a(y) = a = const. Since, from Eq. (3-276), the induced angle of attack щ(у) = const, Eq. (3-22) shows that the effective angle of attack along the span must also be constant: ae(y) = const.

Figure 3-11 Elliptic circulation distribution with the corresponding elliptic wing plan-

From Eqs. (3-23a) and (3-25), it follows that the chord is distributed elliptically over the span:

Ф) = cr yj 1 – (3-29)

The elliptic wing planform is shown in Fig. 3-11.* Thus it has been demonstrated that an elliptic wing without twist has an elliptic circulation distribution. From Eq. (3-21), it also has a constant local lift coefficient ct(y) over the span.

Coefficients Finally, the most important results for the induced angle of attack [Eq. (3-27&)] and for the induced drag [Eq. (3-28b)] will also be expressed through the dimensionless coefficients of lift and induced drag. They are defined as follows:

L = cLqA (3-30a)

Di = cm<lA (3-30 b)

with A being the wing planform area. Consequently, Eqs. (3-27b) and (3-28b) yield

(3-3 la) (3-3 lb)

Here A — b2/A is the aspect ratio of the wing from Eq. (3-9). The important result for the coefficient of the induced drag of Eq. (3-31 b) is compared in Fig. 3-12 with test results for a wing of aspect ratio A = 5. The theoretical curve for the induced drag agrees quite well over the whole cL range with the polar curve of the measured data. The difference between the two curves is about constant over the whole cL range. It is caused by the effect of friction that has been neglected in the above theory. Figure 3-12 suggests splitting up the drag coefficient into a component that is nearly independent of the lift coefficient and a component that is dependent on the lift coefficient. The former is called the coefficient of profile drag cDp, the latter the coefficient of induced drag cDi. They are related by

C-D cDp cDi	(3-32 a)
cl ~CDp 1 їїЛ	(3-32 b)
For the geometric angle of attack, Eqs. (3-22) and (3-3 Ід) yield
, CL a = ae -1— e TtA	(З-ЗЗд)
cl, cL ~ j „ .1	(3-33b)

*The elliptic wing consists of two ellipse halves, the large axis of which is the cj4 line.

From Eq. (3-21), ae = cL/c’Loo because the constant local lift coefficient сг(у) and the total lift coefficient Ci are equal in this case. The latter equation allows one to determine the lift slope of the wing of finite span as a function of the aspect ratio. From Eq. (3-33b) it follows:

with c’L=dcLldot and = 2тг. Equation (3-34h) expresses the degree of reduction of lift slope and consequently also of lift because of the finite aspect ratio when the angles of attack are equal. In Fig. 3-13 this ratio of lift slopes is presented as a function of the aspect ratio.

As will be shown later in more detail, the formulas for induced drag and lift slope found here. for the elliptic wing are valid for other wing shapes in good approximation. This is true particularly for the rectangular wing, as shown by Betz [5]; see Figs. 3-32 and 3-57.

Prandtl’s transformation formulas The above-derived results on the effect of aspect ratio on lift and drag have been checked experimentally by Betz and Wieselsberger [99]. For comparison of the polar curves of two wings of aspect ratios Аг and Л2 at equal angles of attack, Eq. (3-32b) with cDp2 = CDpl yields

Figure 3-13 Ratio of the lift slope of wings of finite and infinite aspect ratios vs. aspect ratio, c’icо = 27г.

In Fig. 3-14a the measured polar curves are plotted for a number of rectangular wings with aspect ratios Ax = 1, 2, , 7. Figure 3-146 shows the result of the

transformation of these polars to the aspect ratio Л2 = 5 from Eq, (3-35). The transformed curves fall well on one curve, confirming experimentally the validity of Eq. (3-35). In Fig. 3-146 the theoretical polar curve of the reduced drag for A — 5 is also included. On the other hand, comparison of the lift curves cl(ol) of two wings of aspect ratios A and Аг of equal lift coefficient yields, with Eq. (3-336),

(3-36)

For the wings of Fig. 3-14, the lift curves were converted to the aspect ratio Лг = 5. Again, the converted curves fall together, confirming experimentally the validity of Eq. (3-36).

The two equations (3-35) and (3-36) can therefore be used for the transformation of measured drag polars cD{cL) and lift curves cL(a) at aspect ratio Лі to those of a wing with a different aspect ratio Л2 if both wings have the same profile. These equations are called, therefore, transformation formulas of the wing of finite span.