1712 AIAA JOURNAL VOL. 4, NO. 10

Second-Order Theory for Airfoils in Uniform Shear FlowC. F. CHEN*

Rutgers-The State University, New Brunswick, N. J.

The first- and second-order problems for a uniform shear flow past an airfoil havebeen formulated and solved. It has been assumed that the stream shear, when properly non-dimensionalized, must be of order 1. Expressions for the first- and second-order pressurecoefficients have been obtained. The lift and moment coefficients are given in integral form.Once the airfoil shape is known, the force coefficients can be obtained by integration. Nu-merical calculations have been carried out to obtain lift coefficients on a symmetric Joukowskyairfoil, a circular arc, and a cambered Joukowsky airfoil for different values of stream vor-ticity. It is found that, in general, if the nondimensional stream vorticity (with respect tothe airfoil chord and a reference velocity) is limited to 1 or less, the second-order theory willgive lift coefficients within 5% of the exact theory up to 30 angle of attack. The uniformlyvalid pressure distributions correct to the second order along a symmetric Joukowsky air-foil at 10 incidence are presented to exhibit the effect of stream shear.

I. Introduction

IN many engineering applications of airfoils, the approach-ing flow is not uniform. When airplanes take off intothe wind, the approaching flow is nonunif orm because of windshear. The approaching flow to that section of the wing,which is situated in the slipstream of the propeller, is non-uniform. The rotor blades of helicopters operate in thewakes of the other blades; the propellers of ships operate inthe wake of the stern. In both cases, the propeller bladesare operating in a nonunif orm stream. It was the first in-stance cited that prompted Tsien1 to look into the problemof shear flow past airfoils. He succeeded in obtaining theexact solution of a uniform shear flow past a symmetricJoukowsky airfoil. More recently, Sowyrda2 extendedTsien's result to include cambered Joukowsky airfoils. Toobtain the effects of nonunif orm shear on airfoils, Jones3-4considered the slightly parabolic and hyperbolic shear flowspast Joukowsky airfoils and elliptic cylinders. Experi-mentally the effect of stream shear on the maximum lift ofan airfoil has been investigated by Vidal.5 Recently,Weissinger6 studied the exponential shear flow past airfoils.

Confining our attention to the case of uniform shear flowpast an airfoil, the disturbance flow is irrotational. By theuse of conformal mapping, such a flow past a camberedJoukowsky airfoil has been obtained.1-2 For arbitrary foilsections, the task is considerably more difficult. Since inpractice the airfoils are thin and of small camber, and theangles of attack are usually small, it seems reasonable todevelop a perturbation method to calculate the pressure dis-tributions and the force coefficients of the foil. The pres-sure distribution along the foil may give some indicationof the stalling characteristics of the foil.5

In this paper, we first systematically expand the com-ponents of the disturbance velocity into a power series in e,which is a small parameter characterizing the size of smalldisturbances. The vorticity of the approaching stream neednot be small, but must be of order 1. When the series ex-pansions are substituted into the boundary conditions alongthe airfoil, first- and the second-order problems presentthemselves which are similar to those for an airfoil in uni-

Received January 4, 1966; revision received May 31, 1966.This research was supported by the U. S. Army Research office(Durham), under Contract DA-31-124-ARO(D)-387. Abstractof this paper was presented at the Fifth U. S. National Congressof Applied Mechanics, University of Minnesota, June 14-17,1966.

* Associate professor of Mechanical Engineering. MemberAIAA.

form flow. Solutions can be obtained by using the methodof Lighthill.7 From the pressure distribution along the foil,calculated according to Kuo,8 the lift and moment may beobtained by integration. Numerical calculations have beencarried out to obtain lift coefficients on a symmetric Jou-kowsky airfoil, a circular arc, and a foil section approximatinga cambered Joukowsky airfoil for different values of shearparameter or nondimensional vorticity. It is found that, higeneral, if the nondimensional vorticity (nondimensionalizedwith respect to the airfoil chord and a reference velocity) isequal to or less than 1 and the angle of attack is up to 30,the second-order theory will give lift coefficients within 5%of the exact solutions.1-2 The pressure distributions alonga symmetric Joukowsky airfoil at 10 angle of attack in anapproaching stream with or without shear are also presented.

II. Systematic ExpansionsConsider an airfoil of unit chord is at an angle of attack

ea with respect to an approaching stream of incompressible,inviscid fluid. Let the origin of the stream coordinate sys-tem (x'} y') be at the midpoint of the chord line of the foil,and x' axis be in the direction of the undisturbed flow. Theapproaching flow is uniformly sheared

Ky') (1)in which K is referred to as the shear parameter and physically(KUo) is the vorticity of the approaching stream. Thestream function ^ of such a flow past an airfoil may be de-composed into that due to the undisturbed flow ^o and thatdue to the disturbance \f/

+ (2)where

*o = Uny'[l + (K/2)y'] (3)Since ^ 0 contains the entire vorticity of the flowfield,

V2^ = KUQ (4)

the disturbance flow must be irrotational

V2t = 0 (5)To complete the specification of the problem, the disturbancevelocity must 1) vanish far away from the airfoil, 2) cancelthe normal velocity induced by the approaching flow on thesurface of the foil, and 3) satisfy the Kutta condition at thetrailing edge.

OCTOBER 1966 SECOND-ORDER THEORY UNIFORM SHEAR FLOW 1713

For the solution of the disturbance flow, we first transformto the body coordinate system (x, y ) , whose origin is at thenose of the foil and the x axis is along the chord (see Fig. 1).In the present coordinate system, the velocity componentsfar away from the foil are

(Um cosea, Ua> sinea)U, = U0{1 + K[y cosea - (x - i) sinea]} (6)

Let the foil section bey = ef(x) = e[fc(x) 0 < x < 1 (7)

in which /. is the camber, and ft the thickness. The velocityfield is the sum of the undisturbed shear flow and the dis-turbance velocity caused by the presence of the airfoil:

j V) = (Um cosea + u, Um sinea + v) (8)We assume that the components of the disturbance velocity(u, v) can be expanded into a power series in e:

u =

as well as the disturbance stream function

(9)

(10)It is noted that the ui, Viy ft, etc., are nondimensional. Theproblem reduces now to finding an analytic complex velocityw(z) = u iv of the complex variable z = x + iy withsingularities admitted at the leading edge and satisfying thefollowing boundary conditions: 1) at the foil surface,

sinea + vcosea + u (U)

in which the prime denotes differentiation with respect tothe argument; 2) w vanishes at infinity; and 3) the Kuttacondition is satisfied, i.e., w(l) is bounded.

We substitute Eqs. (6) and (9) into the boundary condi-tion (11). Since Um involves y and we are evaluating thevelocity components at the surface of the foil y = ef(x),we can expand Um cosea and Um sinea into power series ine. Furthermore, the unknown velocity components u andv away from the x axis may be obtained by Taylor seriesexpansions about the x axis, We obtaine/'(*)(l + e{Ul + K\J(x) - a(x - *)]} + 0(e'))

= (a + ,) + 2{a K[f(x) - a(x - |)] +/(aO@i/&y) + v,} + 0(e) (12)

for 0 < x < 1, y = 0 as the case may be. Equating termsof like powers of e, we obtain the boundary conditions for thefirst- and second-order problems.

0(e):0(e2):

a (13)- a(x - i)]2} (14)

The continuity equation (buifbx) + (

1714 C. F. CHEN AIAA JOURNAL

- S O W Y R D A ( r e f . 2 ) -2nd ORDER THEORY

/K-5

/K.I^K=O

Fig. 2 Lift coeffi-cient of a symmetricJoukowsky airfoil inuniform shear flow(r = 0.17, h = 0).

20 30 40 c,degrees

that determines the behavior of the boundary-layer flowalong the wing; the undetermined constant does not detractfrom the usefulness of the pressure expressions.

The lift coefficient CL may be obtained by integration alongthe foil surf ace:

CL = (f) Cp cos[ea -*jr foil

Since ds = dx[l + 0(e2)] and CP may be expanded into aseries, we may write

CL = e Q &Cpdx + e2 o &CPdx + 0(e3) =eCLi + e*CL2 + 0(e3) (22)

in which A(7P denotes the pressure jump across the foil,

The moment coefficient about the midchord due to theforces acting in the x direction is order e3. The principalcontribution is due to the vertical forces :

CM =

= eC 0(e3)

IV. Solutions

0(e3)

(23)

It is noted that the flow shear has no effect on the first-order boundary condition (13); however, it does have aneffect on the pressure coefficient, Eq. (19). The solution tothe first-order problem is the same as that for a uniform flow.Following Lighthill,7 by considering two fundamental solu-tions, one being a symmetric (with respect to the x axis) andthe other yielding an antisymmetric velocity distribution, weobtain the first-order complex velocity as

in which the branch cut is introduced such that the squareroot in front of the second term is negative at y = 0+ , 0