Computation of Total Lift

Basic formulas The local lift coefficient c{(y) of a wing section у is obtained through integration of the pressure distribution over the wing chord in analogy to Eq. (2-54) as

*>•09

Ф )~<^Г)/ ДСЛХ>У)&Х (3-53)

Xf(y)

The total lift coefficient cL=L! Aq00 of the wing is thus obtained with qoc =qXJIo!2 as

cl—JJ Acpdxdy

(A) s

= 2 ФШ dy

Compare also with Eq. (3-13). By using Eq. (3-54#), the total lift is obtained through integration of the pressure distribution over the wing chord. With Eq. (3-15), it may also be obtained from the Kutta-Joukowsky theorem. Here the circulation distribution has to be taken from Eq. (3-39), the distribution of vortex density from Eq. (344). Below, a further expression for the total lift will be derived by applying the momentum law. As in Fig. 3-21, a cylindrical control surface is arranged about the wing. The axis of the cylinder runs in the direction of the incident flow velocity Ux. The two base surfaces I and ЇІ of the cylindrical control surface are assumed to be very far upstream and downstream of the wing, respectively. The diameter of the control cylinder is chosen large enough to make pressure and velocity on the cylindrical surface equal to the values px and of the undisturbed flow on surface I, respectively. In computing the lift from the

Figure 3-21 Computation of lift by means of the momentum law and of the induced drag by the energy law.

momentum law, it can be assumed that the free vortex sheet is parallel to the incident flow direction far downstream of the wing.*

The fluid mass permeating an area element dy dz of surface II per unit time is QUoodydz. It produces, together with the velocity w«, induced by the wing, a momentum component in the z direction of magnitude qU^Woo dy dz. Since the induced velocity on surface I is zero, the integral of the momentum over the surface II represents the force exerted normal to the incident flow direction to the wing, that is, the lift

(3-55)

(3-560 (3-56*)

Now, the identity of Eqs. (3-55) and (3-15) will be shown for the not-rolled-up vortex sheet. The field of the induced velocities very far downstream of the wing can be described by means of a two-dimensional velocity potential Ф(у, z) [see Eq. (3-49b)], where Woo = Ъф/dz. By introducing this expression into Eq. (3-55), integration over z yields

— $

On the boundaries у = ±°° and z = ±°°, the values Ф vanish, whereas in the vortex sheet, at z — ±0, the potential in the z direction from Eq. (3-48b) changes abruptly by the amount ДФ(у, 0)= Фи(у, 0) — Ф^(у, 0)=Г(у). The integration limits y = ±°° may be replaced by у = ±s = ±bj2 because ЛФ(у, 0) = 0 outside of the wing span. Introduction into Eq. (3-56<z) yields Eq. (3-56b), in agreement with Eq. (3-15). The total lift thus depends only on the circulation distribution over the wing span. It is thus immaterial whether the circulation distribution is created by wing

”Kxaemer [79] points out the decisive significance of the inclination of the free vortex sheet for computation of the induced drag by means of the momentum law.

planform (aspect ratio, sweepback, taper), wing twist, or camber of the wing surface.

Certainly, Eq. (3-55) is also valid for the rolled-up vortex sheet as in Fig. 3-8. Now let b0 = 2sо be the distance between the two free vortices of circulation strength Г0, whereby the circulation distribution along the span is symmetric (Fig.

3- 22). The induced velocity w*. at a point of the yz lateral plane very far behind the wing (.x -* °°) becomes, from the Biot-Savart law,

so +У, Sp ~y

(sQ+y)2+z^(so-y?+z2

Introducing this expression into Eq. (3-55) and integrating twice yield

L=QUcorQb0 (3-57)

By taking into account the Kutta-Joukowsky lift theorem, this formula states that the lift of a wing of span b = 2s and of variable circulation distribution Г (у) is equal to the lift of a wing of span b0 and over the span constant circulation distribution Г0. Comparison of Eqs. (3-566) and (3-57) yields the distance between the two free vortices:

s

Ъо=^1Г[у)йу (3’58)

0

This relationship can also be interpreted as a statement that the vortex moment about the longitudinal axis (x axis) remains constant during roll-up. For the right

oc(?]) = oiA’i1) + *An) y(v) = УI (v) + yz(v)

zero. Consequently, the circulation distribution of the twisted wing at given angle of attack a = const is given by

y(v) = + Yoiv) (3-63)

The zero distribution 70(h) is obtained from

To (V) = 7g(v) + <*0 T U(V) (3-64)

Through procedures similar to those applied for the lift, integration over 17 for a known circulation distribution 7(17) produces other simple relationships for the lateral distance of the lift center of a wing-half, for the lift force of a wing-half, and also for the rolling moment about the x axis. They are summarized in Table 3-1.

Introduction of a Fourier series Computation of the integrals for the coefficients of lift and rolling moment turns out to be particularly simple when the circulation distribution is expressed as a Fourier polynomial of the form

M

у (#) = 2 £ aft sin, a# (3-65я)

/і=і

This procedure was first introduced by Trefftz [69] and Glauert [23]. The first term in Eg. (3-65a) represents the elliptic circulation distribution 7 = 2аг sin d – = 2*iVl — V2 as treated previously in Sec. 3-2-1. After execution of the integrations over — 1 <7? < 1 and 0< # <7r, respectively, the coefficients of lift and rolling moment are obtained with ch? = — sin d-dd – as

(3-67*)

obtained with jj. = 1 and /і = 2 for the lift coefficient cL and the rolling-moment coefficient cMx (Table 3-1). In addition,-quadrature formulas are also given for the lateral distance of the lift center of a wing-half riL=yLjs and for the lift coefficient of a wing-half c*. Table 3-2 contains the coefficients for the formulas for practical application of the last column of Table 3-1.

«(У) = *«(У) – r »іІУ)

(3-70*)

the effective or, respectively, the induced angle of attack becomes*

*e(v) = f(v)r(v)

dy dr/’

2 n J dr]’ 1] — 7]’

-l

The formula for аг(7?) can also be written, through integration by parts, as

By introducing Eqs. (3-71<z) and (3-71Z?) into Eq. (3-69), the Prandtl integral equation for the dimensionless circulation distribution 7(17) is obtained in the form

*(n) = f(>i)rM+£f

-1

*See footnote on page 139.

Figure 3-23 Wing section y: a(y), angle of attack against zero-lift direction: a. g(y), geometric angle of attack against wing chord; a0 (y), zero-lift angle,

a = dig — a0.

In abbreviated form, the integral equation of the simple lifting-line theory can be written:

z(r/) = «;(>?) H – /(?;) y(y)

Schmidt et al. [76] deal with the mathematical formulation of the simple lifting-line theory and present comprehensive results.

Solution with Fourier polynomials A convenient method of solving Eq. (3-72) for the circulation distribution consists of expressing the circulation distribution as a Fourier polynomial such as Eq. (3-65) (M = n). By introducing Eq. (3-65a) into Eq. (3-71 6), first the induced angle of attack is obtained[14]:

(3-73)

After introduction of Eqs. (3-65a) and (3-73) into Eq. (3-726), the following equation is obtained, defining the Fourier coefficients an

M

<x($) sin$ — a„ [2/(#) sin# – f w] sinw#

Here the distribution of the angle of attack a(&) and the wing planform f(&) are given beforehand. The coefficients ax, a2,. -. , aM are determined by satisfying Eq. (3-74) at M points d-M along the span. This results in a system of M

linear equations for ax to aM. Lotz [56] simplified this procedure by introducing Fourier polynomials for the functions a($) sin # and f(6) sin &. After the Fourier coefficients an have been determined, the circulation distribution is obtained from Eq. (3-65<z) and the distribution of the local lift coefficients from Eq. (3-596). Weinig [93] suggests that the theory of the lifting line be solved by comparison with the corresponding grid flow.

Wing of elliptic planform The elliptic wing has been treated in Sec. 3-2-1. There it was shown that an elliptic wing without twist has an elliptic circulation distribution over the span. The elliptic wing with twist may be computed from the above formulas very easily as, among others, Schmidt [69] has shown.

For the elliptic wing c = crs/ —r2 — cr sin $ and A = 46/ясу [Eq. (3-9)], and thus from Eq. (3-70),

2 sin # / (#) = — ^L°°

(3-756)

Hence, Eq. (3-74) for the wing with elliptic planform becomes

M

a(&) sin&= £ (k n) ansmn& (3-76)

n— 1

and the coefficients an can be computed directly through a Fourier analysis as

П

an = —^—— f л (г?) sin?? sinnfi d& (3-77)

к – f – П 71 J о

This solution will now be discussed for a few particularly simple angle-of-attack distributions. Setting

a sinm# rn hq

<*W = rm – (3-78)

sm»

the corresponding circulation distribution is obtained with an = 0 for пФт and with am = rml(k + m) for n = m as

y(&) = 2 ——— sin « (3-79)

‘ к -f – m к – f – m

For a wing with the aspect ratio A = 6, that is, к – 3, the results for m = 1, 2, and 3 are presented in Fig. 3-24. Here m = 1 gives the constant angle-of-attack

Figure 3-24 Lift distribution according to the simple lifting-line theory of an elliptic wing at various twists [Eq. (3-78)]; aspect ratio Л = 6. (a) Wing planform. (b) m — 1: wing without twist. (c) m~ 2: linear angle-of-attack distribution, (d) m = 3: parabolic angle-of-attack distribution.

Figure 3-25 Lift slope of elliptic wings vs. aspect ratio; = 2тг. (1) Simple lifting-line theory, Eq. (3-80&). (2) Ex­tended lifting-line theory, Eq. (3-98). (- о -) Exact solutions according to Kin – ner [44] and Krienes [47].

distribution (wing without twist), m — 2 gives the linear angle-of-attack distribution such as, for instance, is encountered in a rolling motion, and m — 3 gives the parabolic angle-of-attack distribution (symmetric twist, cL — 0).

Circulation distribution and coefficients of lift and rolling moment of the elliptic wing with twist are obtained from Eq. (3-66a) and also from Eq. (3-665) by introducing the corresponding coefficients an according to Eq. (3-77). The lift coefficient thus becomes

ГС CL = ——— Га($) sin2# cZ# fc-M я) (3-8 Од) 0 dcj пЛ dot к + 1 (3-805)

For the wing without twist, a = const, the coefficient of lift slope is obtained in agreement with Eq. (3-34#). It is presented in Fig. 3-25 as a function of the aspect ratio /I. Also shown are the results based on the extended lifting-line theory that will be treated further in Sec. 3-34, and the exact solution for the elliptic wing.*

From Eqs. (3-80s) and (3-805) the zero-lift angle is obtained with Eq. (1-23) as

-f-1

Q! o = — ~ J*«(??) І і — rf ch] (3-81)

-i

For approximate computations, the relationships of the elliptic wing can be applied to other wing shapes.

Quadrature method of Multhopp The simplest and most used method for the computation of the lift distribution of unswept wings according to the simple

"Here, the wing planform is an exact ellipse, which, e. g., becomes a circular disk for Л = 4/ir.

lifting-line theory is that of Multhopp [60]. This method will be briefly sketched now: Starting from the expressions for the circulation distribution [Eq. (3-65)] and for the Fourier coefficients [Eq. (3-67)] in connection with Eq. (3-68), the summation expression, Eq. (3-67b), is introduced into Eq. (3-73). The induced angle of attack at the discrete stations

is then obtained in the form

(3-83)[15]

M

о*і„ b, vyt ^, bvnyn (v 1,2,…, ilf)

n= 1

with the universal coefficients

M+ 1 w 4 sin $v

_!-(-! y-n

ш 2(M +1) (cos &v — cos &n)2

By introducing expression (3-83) for the induced angle of attack into the integral equation for the circulation distribution Eq. (3-72b), the following system of equations is obtained for the values of jv:

M

(bw +fv)7v = ‘ «V + 2’ bvn yn (v = 1, 2, . . . , M) (3-85)

П= 1

This is a system of M linear equations for the M circulation values jv = 7(17*,) with v = 1, 2,. .., M. In Eq. (3-85), the following relationships apply:

&v = a(r? v) fv = ~T~ with cv = c(rjv) (3-86)

C l<x> Cp

For M~1 and M— 15, the universal coefficients are compiled in Tables 3-3 and 34. The values bm for ^—«1=2, 4,.. . are equal to zero. For the numerical solution of the system of equations, it is significant that the system of M equations can be split up into two systems of (M + l)/2 or (M —1)/2 equations, respectively, which can be solved conveniently by iteration. By splitting an arbitrary angle-of – attack distribution into its symmetric and antisymmetric contributions, the procedure of the numerical solution can be further simplified. For a continuous behavior of wing chord c{r) and angle of attack a(rf), usually M = 15 points along the span are sufficient for all practical purposes. For discontinuous angle-of-attack distributions, as found for flap deflections, Multhopp recommends that one split off

Table 3-3 Universal coefficients and bvn for the computation of circulation distributions, forM= 7, according to Eq. (3-84)a V 1 (7) 3 (5) 2 (6) 4 Vv 0.9239 0.3827 0.7071 0.0000 bvv 5.2262 2.1648 2.8284 2.0000 n n 2(6) 1.8810 0.8398 1(7) 1.0180 0.0560 2. 4(4) 0.1464 0.8536 3(5) 1.0972 0.7887 6(2) 0.0332 0.0744 5(3) 0.0973 0.7887 — — — 7(1) 0.0180 0.0560 aAfter Multhopp [60].

the discontinuity stations before applying the above computational procedure; see Chap. 8.

Equations (3-85) are valid for unswept but otherwise arbitrary wings of sufficiently large aspect ratio (/1 > 3) and also for arbitrary angle-of-attack distributions.

Further results of the simple lifting-line theory In Fig. 3-25 the dependence of the lift slope on the aspect ratio is shown for a wing of elliptic planform. This result is approximately valid also for wings of different—for instance, trapezoidal—planforms.

To demonstrate the effect of the aspect ratio on the lift distribution, the circulation distributions over the span were computed for three rectangular wings with c = const and aspect ratios A — 6, 9, and 12. When A increases, the circulation distribution approaches more and more a rectangular distribution. Figure 3-26 demonstrates this fact. Illustrated are the local lift coefficients ct with reference to the total lift coefficient cL along the span. For A 00 (plane problem), Ci/cL = 1, and for very small aspect ratios {A -*0) the lift distribution is elliptic. This can easily be seen from Eq. (3-12b), which for /l-»0 goes to a(rj) = a2-(v) because f(rj) = 0. Hence, for a= const, cq = const, meaning, from Sec. 3-2-1, that the circulation distribution is elliptic.

To show the effect of wing taper on the lift distribution, Fig. 3-27 illustrates the circulation distribution for four trapezoidal wings without twist of aspect ratio A=6 and tapers X=ctfcr = 0, and 1. The taper has a strong effect on the distribution of the local lift coefficients along the span. This can be seen in Fig.

3- 28 in which the curves Ci/cL are shown. The strongly tapered wings have, near the wing tip, local lift coefficients that are considerably larger than the total lift coefficient cL. This fact is significant for the flow-separation characteristics of such

Table 3-4 Universal coefficients bw and bvn for the computation of circulation distribution, for M = 15, according to Eq. (3-84)a

я After Multhopp [60].

Figure 3-26 Lift distribution ciJci of rectangular wings without twist of aspect ratios A — 6, 9, 12; also limiting curves for Л -* 0 and Л = 2rr.

wing shapes at high lift coefficients. With increasing angle of attack, separation begins approximately at the station of maximum local lift coefficient, hence on strongly tapered wings close to the wing tips, but on rectangular wings in the middle of the wing.