M&AE 305 October 3, 2006

Thin Airfoil Theory

D. A. Caughey

Sibley School of Mechanical & Aerospace Engineering

Cornell University

Ithaca, New York 14853-7501

These notes provide the background needed to implement a simple vortex-lattice numerical methodto determine the properties of thin airfoils. This material is covered in Lecture, but is not inthe textbook [5]. A summary of results from the analytical theory also is provided, as well as acomparison of the thin-airfoil results with those of a complete inviscid theory that accounts forthickness effects.

1 The Vortex Lattice Method

We here describe the implementation of the vortex lattice method for two-dimensional flows pastthin airfoils. The method is even more useful for three-dimensional wings, i.e., for the flow pastwings of finite span, but that problem is not considered here. Instead, the reader is referred tostandard aerodynamics texts, e.g., [2]. In this numerical procedure to solve the thin-airfoil problem,we place a finite number of discrete vortices along the chord line, with the boundary condition thatthe induced vertical velocity

v =dycdx

− α , (1)

be enforced at selected control points to determine the vortex strengths. Equation (1) simply saysthat the net velocity vector, comprised of components due to the free stream, at angle of attack αto the chord line, plus that induced by the point vortices, is tangent to the camber line whose slopeis dyc/dx; the magnitude of the free stream velocity is taken to be unity.

Thus, we discretize the chord line into a finite number N of segments, or panels, as illustrated inFig. 1 (a). On each panel we place a point vortex and a control point, as illustrated in Fig. 1 (b).The most accurate results are obtained by locating the vortex one-quarter of the panel length, andthe control point three-quarters of the panel length, aft of the leading edge of the panel. (Thisstrategy can be shown to reproduce the exact results of analytical thin-airfoil theory for paraboliccamber lines using a single panel , as shown in Section 2.3.1.)

The vertical velocity vi,j induced at the ith control point by the jth point vortex is given by

vi,j =Γj2π

1

xvj− xci

where xvjis the chordwise coordinate of the jth vortex having strength Γj , and xci

is the chordwisecoordinate of the ith control point. The total vertical velocity at the ith control point induced by

1 THE VORTEX LATTICE METHOD 2

y

xx xi i+1

Γi

x xv c

(a) (b)

Figure 1: Sketch of discretization of chord line for implementation of vortex lattice calculation.(a) Chord line subdivided into N panels; (b) Single panel showing location of point vortex andcontrol point.

all the vortices representing the airfoil camber line is thus

vi =

N∑

j=1

Γj2π

1

xvj− xci

=N∑

j=1

ai,jΓj

where

ai,j =1

2π(

xvj− xci

)

is the influence coefficient representing the effect on the induced vertical velocity at the ith controlpoint of a vortex of unit strength located on the jth panel.

If we introduce the vector notation

v = [v1 v2 . . . vN ]T

andΓ = [Γ1 Γ2 . . . ΓN ]

T,

and define the matrix of influence coefficients

A =

a1,1 a1,2 · · · a1,N

a2,1 a2,2 · · · a2,N

· · · · · ·

· · · · · ·

· · · · · ·

aN,1 aN,2 · · · aN,N

,

then the system of equations representing the enforcement of the boundary condition of Eq. (1) ateach of the control points can be written

AΓ = v . (2)

Since the elements of A and v are known, Eq. (2) represents a linear system of equations that canbe solved for the N unknown values Γj .

The net lift on the airfoil is then given by the Kutta-Joukowsky theorem as

` = ρU

N∑

j=1

Γj

1 THE VORTEX LATTICE METHOD 3

whence the lift coefficient is

C` =`

12ρU

2c=ρU

∑Nj=1 Γj

12ρU

2c=2

Uc

N∑

j=1

Γj

or

C` = 2

N∑

j=1

Γj (3)

if we interpret Γ to be normalized by the product Uc (or, equivalently, take U = c = 1).

The pitching moment referenced to the quarter-chord point of the airfoil is similarly given by thesum of the contributions of the individual lifting forces as

mc/4 = −ρU

N∑

j=1

Γj

(

xvj−1

4

)

whence the moment coefficient is

Cmc/4=

mc/4

12ρU

2c2= −

∑Nj=1 Γj

(

xvj−

14

)

12Uc

2= −

2

Uc2

N∑

j=1

Γj

(

xvj−1

4

)

or

Cmc/4= −2

N∑

j=1

Γj

(

xvj−1

4

)

(4)

if we again interpret Γ to be normalized by the product Uc.

Since we have lumped the entire contribution of the continuous vorticity distribution γ(x) over eachpanel into a single point vortex, an approximation of the continuous distribution can be determinedfrom

γ(xvj)∆xj = Γj

or

γ(xvj) =

Γj∆xj

. (5)

The jump in chordwise velocity across the vortex sheet is given by

∆u(x) = γ(x) .

To first order, only the chordwise component of velocity contributes to changes in pressure, so fromthe (incompressible) Bernoulli equation

∆p = p− p∞ =1

2ρ[

U2− V 2

]

=1

2ρU2

[

1− (1 + ∆u)2 −∆v2]

= −ρU2∆u

so the change in pressure coefficient across the vortex sheet is

∆Cp =−ρU2∆u

12ρU

2= −2∆u

whence∆Cp(x) = −2γ(x) (6)

That is, the net lifting pressure difference across the camber line (or, equivalently, the vortex sheet)is simply 2γ(x).

2 CLASSICAL THIN-AIRFOIL THEORY 4

2 Classical Thin-Airfoil Theory

In classical thin-airfoil theory, the boundary condition of Eq. (1) is satisfied by a continuous dis-tribution γ(x) of vorticity along the chord line. This distribution of vorticity induces the verticalvelocity

v(x) =1

2π

∫ 1

0

γ(ξ)

ξ − xdξ

at any point on the chord line, so we must solve the integral equation

1

2π

∫ 1

0

γ(ξ)

ξ − xdξ =

dycdx

− α (7)

to determine the vorticity distribution for a given camber line shape at a given angle of attack α.Equation (7) is the continuous analog of Eq. (2).

The solution to Eq. (7) can be obtained by introducing the change of variables

ξ =1− cosφ

2

and

x =1− cos θ

2, (8)

following which Eq. (7) becomes

1

2π

∫ π

0

γ(φ) sinφ

cosφ− cos θdφ = α−

dycdx

. (9)

The vorticity distribution can then be represented by the infinite series

γ(φ) = 2

[

A0 cotφ

2+

∞∑

n=1

An sinnφ

]

. (10)

The Kutta Condition requires that the vorticity strength go to zero at the trailing edge. Since thetrailing edge is located at φ = π, and cotπ/2 = 0 and sinnπ = 0 for all integer values of n, theabove distribution is seen to satisfy the Kutta Condition γ(π) = 0 automatically.

All the terms that need to be integrated once the vorticity distribution of Eq. (10) is substitutedinto Eq. (9) can be evaluated using the Glauert Integral (see, e.g., [4])

∫ π

0

cosnφ

cosφ− cos θdφ = π

sinnθ

sin θ. (11)

Thus, substitution of the vorticity distribution of Eq. (10) into the integral Eq. (9), using the GlauertIntegral results in

A0 −

∞∑

n=1

An cosnθ = α−dycdx

(12)

Integrating this equation from 0 to π gives

A0 = α−1

π

∫ π

0

dycdxdθ, (13)

while multiplying by cosnθ and integrating from 0 to π gives

An =2

π

∫ π

0

dycdxcosnθ dθ, for n ≥ 1. (14)

Note for future reference that the values of An for n ≥ 1 depend only on the shape of the camberline; the only dependence on angle of attack α is that shown explicitly in Eq. (13) for A0.

2 CLASSICAL THIN-AIRFOIL THEORY 5

2.1 The Flat Plate

The camber line for a flat plate airfoil is simply yc = 0, so Eqs. (13) and (14) give

A0 = α

An = 0 for n ≥ 1.(15)

Thus, for the flat plate airfoil

γ(θ) = 2α cotθ

2= 2α

(

1 + cos θ

sin θ

)

or, transforming back to x,

γ(x) = 2α

√

1− x

x(16)

2.2 Lift and Moment Coefficients

The airfoil lift coefficient is defined as

C` =`

12ρU

2c=

112ρU

2c

∫ c

0

ρUγ(x) dx

or, if we interpret γ and x to be normalized by U and c, respectively,

C` = 2

∫ 1

0

γ(x) dx .

Introducing the transformation of Eq. (8) and the vorticity distribution of Eq. (10), we find

C` = 2

∫ π

0

[

A0 cotθ

2+

∞∑

n=1

An sinnθ

]

sin θ dθ

which can easily be evaluated to give

C` = π [2A0 +A1] . (17)

Thus, the lift coefficient is seen to depend only on the first two terms of the infinite series representingthe vorticity distribution. Also, as noted earlier, the only dependence on the angle of attack α isthrough the dependence of A0 shown in Eq. (13). Thus the lift curve slope ∂C`/∂α is given by

∂C`

∂α= 2π . (18)

Thus, the lift curve slope is the same for all thin airfoils, independent of the camber line shape, andis equal to 2π.

The moment coefficient about the leading edge of the airfoil is given by

Cmle=

mle12ρU

2c2= −

112ρU

2c

∫ c

0

ρUγ(x)xdx ,

or, if we again interpret γ and x to be normalized by U and c, respectively,

Cmle= 2

∫ 1

0

γ(x)xdx .

2 CLASSICAL THIN-AIRFOIL THEORY 6

Introducing the transformation of Eq. (8) and the vorticity distribution of Eq. (10), we find

Cmle= −

∫ π

0

[

A0 cotθ

2+

∞∑

n=1

An sinnθ

]

(1− cos θ) sin θ dθ ,

which can easily be evaluated to give

Cmle= −

π

2

[

A0 +A1 −A2

2

]

. (19)

Thus, the pitching moment is seen to depend only on the first three terms of the infinite seriesdescribing the camber line.

Now, the pitching moment about any other point on the chord line, say x/c, is related to that aboutthe leading edge by the relation

Cmx/c= Cmle

+C`

(x

c− 0

)

.

Using the results of Eqs. (17) and (19), this relation gives

Cmx/c= 2πA0

(

x

c−1

4

)

+ π

[(

x

c−1

2

)

A1 +1

4A2

]

.

This equation shows that the pitching moment will be independent of A0 – and, therefore, indepen-dent of the angle of attack α – when

x

c=1

4,

that is, when the pitching moment is measured relative to the quarter-chord point. This referencepoint about which the pitching moment is independent of angle of attack is called the aerodynamiccenter ; thus, thin-airfoil theory shows that the aerodynamic center is independent of the shape of thecamber line and located at the quarter chord point . The value of the moment coefficient, referencedto the quarter-chord point, is then given by

Cmc/4= −

π

4(A1 −A2) . (20)

2.3 Circular-arc Camber Line

The simplest non-trivial camber line is a circular arc which, for small amplitudes, can be approxi-mated as the parabola

yc = 4τx (1− x) ,

where the parameter τ expresses the maximum deviation of the camber line from the chord, as afraction of chord length. The camber line slope is thus found to be

dycdx

= 4τ (1− 2x) ,

which, when the transformation of Eq. (8) is used, becomes

yc = 4τ cos θ .

For this camber line, then, Eqs. (13) and (14) give the Fourier coefficients as simply

A0 = α ,

A1 = 4τ ,

An = 0 for n ≥ 2.

(21)

3 COMPARISONS WITH NONLINEAR THEORY 7

The lift and moment coefficients are then seen to be

C` = 2π (α+ 2τ) ,

Cmc/4= −πτ .

(22)

The expression for the lift coefficient can be used to see that the angle for zero lift is

α0 = −2τ . (23)

In these results, it is seen that both the pitching moment about the aerodynamic center and theangle for zero lift are proportional to the amplitude of the camber line. While the result here hasbeen shown only for the case of circular-arc camber, this is a general result (following from thelinearity of the equations we solve in the thin-airfoil approximation). That is, for a camber line ofany shape, given by, say

yc = τf(x)

both the angle for zero lift and the pitching moment about the aerodynamic center will be directlyproportional to the parameter τ .

2.3.1 Connection to Vortex Lattice Method

Note that if we use a vortex lattice approximation to represent the flow past the circular-arc camberline using only one panel , we will find the vortex strength to be given by

Γ = π (α+ 2τ) .

The lift coefficient is therefore given by

C` = 2Γ = 2π (α+ 2τ) , (24)

which agrees exactly with the result given above in Eq. (22). Note that it is the half-chord separationbetween the vortex position and the control point that gives us the correct lift-curve slope of 2π,while the specific location of the control point gives us the correct angle for zero lift. This single-panel approximation for the flow also gives us a constant moment about the quarter-chord point,but gives the incorrect value of zero for this moment. This reproduction of the exact lift coefficientfor the simplest non-trivial camber line, when using a single vortex panel, provides the motivationfor our placement of the vortex and control point in the vortex lattice method.

3 Comparisons with Nonlinear Theory

In this section we compare the results of thin-airfoil theory with full non-linear, but inviscid, theory.The non-linear results are calculated using a numerical procedure [3] to solve for the inviscid, com-pressible flow past complete airfoils. These calculations are, in fact, performed at very low Machnumbers, typicallyM∞ = 0.05, so the results can be interpreted as equivalent to linear, incompress-ible flow computations. So the only difference between these results and those of thin-airfoil theory,are due to the fact that thickness effects are taken into account. Note, in particular, that viscouseffects still are neglected in these computations.

We study the flow past the five-digit NACA 230xx camber line at its design angle of attack α = 1.65◦.This camber line has its maximum amplitude at the 15 per cent chord station, has zero curvature aftof a point just behind the maximum camber station, and has a design lift coefficient of C` = 0.30 [1].Figure 2 shows the camber line shape, the camber line slope, and the total lifting pressure distribution

REFERENCES 8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

0

0.5

1

1.5

2

2.5Thin Airfoil Theory

x

Cl = 0.30087

Cm(c/4)

= −0.012838

yc (x 10)

dy/dxDelta C

p

Figure 2: Thin-airfoil solution for NACA 230 camber line at design angle of attack α = 1.65◦.

∆Cp, as functions of chordwise position, computed according to the thin-airfoil approximation.Although this solution was computed numerically using 128 vortex panels, it can be considered tobe essentially an exact solution to the thin-airfoil problem.

The results of the full, inviscid computations are shown in Fig. 3. Figures 3 (a)-(d) show that theactual pressure distributions vary significantly with the added contribution due to thickness, butFigs. 3 (e) and (f) show that the net lifting pressure distributions vary only weakly with thickness.

Figure 4 quantifies the effect of thickness ratio on the lift and quarter-chord moment coefficients.The moment coefficient is seen to be nearly independent of thickness ratio, while the lift coefficientvaries more strongly, but differs from the thin-airfoil value by only about 15% for a thickness ratioof 12 per cent.

References

[1] Ira H. Abbott & Albert E. von Doenhoff, Theory of Wing Sections, Dover, New York, 1959.

[2] John J. Bertin, Aerodynamics for Engineers, Fourth Edition, Prentice-Hall, New York,2002.

[3] A. Jameson & D. A. Caughey, How Many Steps are Required to Solve the Euler Equations of

Steady, Compressible Flow: In Search of a Fast Solution Algorithm, AIAA Paper 2001-2673,AIAA 15th Computational Fluid Dynamics Conference, June 11-14, Anaheim, California.

[4] L. M. Milne-Thompson, Theoretical Aerodynamics, Dover, New York, 1958.

[5] Richard S. Shevell, Fundamentals of Flight, Second Edition, Prentice-Hall, New York, 1989.

REFERENCES 9

(a) NACA 23003 (b) NACA 23012

-2

-1.5

-1

-0.5

0

0.5

1

1.5 0 0.2 0.4 0.6 0.8 1

Pre

ssur

e C

oeffi

cien

t

Chordwise position, x/c

Surface Pressure, Total Enthalpy, and Entropy Change Distributions

Pressure coefficient Total enthalpy (x100)Total pressure (x10)

-2

-1.5

-1

-0.5

0

0.5

1

1.5 0 0.2 0.4 0.6 0.8 1

Pre

ssur

e C

oeffi

cien

t

Chordwise position, x/c

Surface Pressure, Total Enthalpy, and Entropy Change Distributions

Pressure coefficient Total enthalpy (x100)Total pressure (x10)

(c) NACA 23003 (d) NACA 23012

-2

-1.5

-1

-0.5

0

0.5

1

1.5 0 0.2 0.4 0.6 0.8 1

Pre

ssur

e C

oeffi

cien

t

Chordwise position, x/c

Lifting Surface Pressure Distribution

Lifting Pressure Coefficient Lifting Pressure Coefficient

Thin Airfoil Theory

-2

-1.5

-1

-0.5

0

0.5

1

1.5 0 0.2 0.4 0.6 0.8 1

Pre

ssur

e C

oeffi

cien

t

Chordwise position, x/c

Lifting Surface Pressure Distribution

Lifting Pressure Coefficient Lifting Pressure Coefficient

Thin Airfoil Theory

(e) NACA 23003 (f) NACA 23012

Figure 3: Full inviscid solutions for flow past NACA 23003 and 23012 airfoils at design incidenceα = 1.65◦ and M∞ = 0.05. (a), (b) show contours of constant pressure coefficient in ∆Cp = 0.05increments in the vicinity of the airfoil surface; (c), (d) show surface pressure distributions; (e), (f)compare lifting pressure distributions with those of thin-airfoil theory for the same camber line.

REFERENCES 10

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

For

ce/M

omen

t Coe

ffici

ent

Thickness ratio, t/c

NACA 230xx Family of Airfoils

LiftMoment

Figure 4: Lift and quarter-chord moment coefficients for airfoils with NACA 230 camber line atdesign angle of attack α = 1.65◦ as functions of thickness ratio. Finite thickness ratio results arecomputed using full, nonlinear theory; zero thickness ratio result is from thin-airfoil theory.