Mae 3050 Thin Air Foils

M&AE 305

Octob er 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 method to determin determinee the properti properties es of thin airfoi airfoils. ls. This This ma mater terial ial is cove covered red in Lectur Lecture, e, but is not in the textbook textbook [5]. A summar summary y of results results from the analytic analytical al theory theory also also is provide provided, d, as well as a comparison of the thin-airfoil results with those of a complete inviscid theory that accounts for thickness effects.

1

The The Vor Vorte tex x Lat Latti tice ce Meth Method od

We here describe the implementation of the vortex lattice method for two-dimensional flows past thin airfoils. airfoils. The method is even even more useful for three-dimensio three-dimensional nal wings, i.e., for the flow past wings wings of finite span, but that problem problem is not conside considered red here. here. Instea Instead, d, the reader reader is referr referred ed to standard 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 that the induced vertical velocity dyc − α, v  = (1) dx be enforced at selected   control points  to determine determine the vortex strengths. strengths. Equation Equation (1) simply simply says says that the net velocity vector, comprised of components due to the free stream, at angle of attack  α to the chord line, plus that induced by the point vortices, is tangent to the camber line whose slope is 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 N  of segments, or  panels , as illustrated in Fig. 1 (a). On each panel we place place a point vortex vortex and a control control point, as illustrated illustrated in Fig. 1 (b). The most accurate results are obtained by locating the vortex one-quarter of the panel length, and the contro controll point point threethree-qua quarte rters rs of the panel panel length length,, aft of the leading leading edge edge of the panel. panel. (This (This strategy can be shown to reproduce the exact results of analytical thin-airfoil theory for parabolic camber lines  using a single panel , as shown in Section 2.3.1.) The vertical velocity  v i,j  induced at the i the  ith th control point by the  j th point vortex is given by vi,j =

Γj 1 2π xvj − xci

where x where x vj  is the chordwise coordinate of the j the  j th vortex having strength Γ j , and x and  x ci  is the chordwise coordinate of the ith control control point. The total vertical vertical velocity velocity at the i the  ith th control point induced by

1

2

THE VORTEX LATTICE METHOD 

y

Γ 

i

xi

x

x

x i+1

(a)

x

v

c

(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 and control point. all the vortices representing the airfoil camber line is thus N 

vi  =

  j =1

Γj 1 2π xvj − xci

N 

=

ai,j Γj

j =1

where

1

ai,j =



2π xvj

− xci



is the   influence coefficient  representing the effect on the induced vertical velocity at the  ith control point of a vortex of unit strength located on the  j th panel. If we introduce the vector notation v  = [v1

v2   . . .

and Γ  = [Γ1

Γ2

vN ]T  ΓN ]T  ,

...

and define the matrix of influence coefficients

A  =

  

a1,1 a2,1

a1,2 a2,2

··· ···

a1,N  a2,N 

·

·

···

·

·

·

···

·

·

·

···

·

aN, 1

aN,2

···

aN,N 

  

,

then the system of equations representing the enforcement of the boundary condition of Eq. (1) at each 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 can be solved for the N  unknown values Γj . The net lift on the airfoil is then given by the Kutta-Joukowsky theorem as N 

 = ρU 

 j =1

Γj

1

3

THE VORTEX LATTICE METHOD 

whence the lift coefficient is ρU  N   2 j =1 Γj C  = 1 = = 1 2 2 Uc 2 ρU  c 2 ρU  c



or

N 



Γj

j =1

N 

C  = 2



Γj

(3)

j =1

if we interpret Γ to be normalized by the product  U c (or, equivalently, take U  = c = 1). The pitching moment referenced to the quarter-chord point of the airfoil is similarly given by the sum of the contributions of the individual lifting forces as N 

mc/4  =

−ρU 

 

Γj xvj

−

j =1



1 4

whence the moment coefficient is Cmc/

4

mc/4 = 1 2 2 =− 2 ρU  c

N  j =1 Γj

 

1 − 4

xvj

1 2 2Uc



2 =− 2 Uc

or N 

Cmc/ = −2 4

 

Γj xvj

−

j =1

1 4

N 

 

Γj xvj

−

j =1



1 4

 (4)

if we again interpret Γ to be normalized by the product  U c. Since we have lumped the entire contribution of the continuous vorticity distribution  γ (x) over each panel into a single point vortex, an approximation of the continuous distribution can be determined from γ (xvj )∆xj = Γ j or Γj . ∆xj

γ (xvj ) =

(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 from the (incompressible) Bernoulli equation 1 ρ U 2 − V  2 2 1 = ρU 2 1 − (1 + ∆u)2 − ∆v 2  = 2 so the change in pressure coefficient across the vortex sheet is ∆ p = p − p  = ∞





∆C p  =





−ρU 2 ∆u 1 2 2 ρU 

2

−ρU 

∆u

= −2∆u

whence ∆C p (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

4

CLASSICAL THIN-AIRFOIL THEORY 

2

Classical Thin-Airfoil Theory

In classical thin-airfoil theory, the boundary condition of Eq. (1) is satisfied by a continuous distribution γ (x) of vorticity along the chord line. This distribution of vorticity induces the vertical velocity 1 1 γ (ξ ) v(x) =  dξ  2π 0 ξ  − x at any point on the chord line, so we must solve the integral equation

 

1 2π

1

  0

γ (ξ ) dyc  dξ  = ξ  − x dx

−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 − cos φ 2

x =

1 − cos θ , 2

and

(8)

following which Eq. (7) becomes 1 2π

π

  0

γ (φ)sin φ dyc  dφ = α − . cos φ − cos θ dx

(9)

The vorticity distribution can then be represented by the infinite series





∞

 φ γ (φ) = 2 A0 cot + An sin nφ  . 2 n=1



 

(10)

The Kutta Condition requires that the vorticity strength go to zero at the trailing edge. Since the trailing edge is located at φ = π, and cot π/2 = 0 and sin nπ  = 0 for all integer values of  n, the above 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 substituted into Eq. (9) can be evaluated using the Glauert Integral (see, e.g., [4]) π

  0

cos nφ sin nθ  dφ = π . cos φ − cos θ sin θ

 

(11)

Thus, substitution of the vorticity distribution of Eq. (10) into the integral Eq. (9), using the Glauert Integral results in dyc A0 − An cos nθ = α −   (12) dx n=1 ∞



Integrating this equation from 0 to π  gives

A0  = α −

1 π

π

  0

dyc dθ, dx

 

(13)

while multiplying by cos nθ and integrating from 0 to  π  gives 2 An  = π

π

  0

dyc cos nθ dθ, for n dx

≥

1.

(14)

Note for future reference that the values of  An for n ≥  1 depend only on the shape of the camber line; the only dependence on angle of attack  α  is that shown explicitly in Eq. (13) for A 0 .

2

5

CLASSICAL THIN-AIRFOIL THEORY 

2.1

The Flat Plate

The camber line for a flat plate airfoil is simply  y c  = 0, so Eqs. (13) and (14) give A0  = α An  = 0 for n

 

1.

≥

(15)

Thus, for the flat plate airfoil

  

 θ γ (θ) = 2α cot = 2α 2 or, transforming back to x, γ (x) = 2α

2.2

1 + cos θ sin θ



1−x x

 

(16)

Lift and Moment Coefficients

The airfoil lift coefficient is defined as  C  = 1 = 2c ρU  2

1 1 2 2 ρU  c

c

 

ρU γ (x) dx

0

or, if we interpret γ  and x  to be normalized by  U  and c, respectively, 1

C  = 2

 

γ (x) dx .

0

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

   π

C  = 2

0



∞

 θ A0 cot  + An sin nθ sin θ dθ 2 n=1



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  representing the vorticity distribution. Also, as noted earlier, the only dependence on the angle of attack α is through the dependence of  A 0  shown in Eq. (13). Thus the   lift curve slope  ∂ C /∂α  is given by ∂ C = 2π . ∂α

 

(18)

Thus, the lift curve slope is the same for all thin airfoils, independent of the camber line shape, and is equal to 2π. The moment coefficient about the leading edge of the airfoil is given by Cm

le

mle 1 = 1 2 2 =−1 2 2 ρU  c 2 ρU  c

c

 

ρU γ (x)x dx ,

0

or, if we again interpret  γ  and x  to be normalized by  U  and c, respectively, 1

Cm = 2 le

  0

γ (x)x dx .

2

6

CLASSICAL THIN-AIRFOIL THEORY 

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

   π

Cm = − le

0



∞

 θ A0 cot  + An sin nθ (1 − cos θ)sin θ dθ , 2 n=1



which can easily be evaluated to give Cm

le



π A2 =− A0  + A1 − 2 2



.

 

(19)

Thus, the pitching moment is seen to depend only on   the first three terms   of the infinite series describing the camber line. Now, the pitching moment about any other point on the chord line, say  x/c, is related to that about the leading edge by the relation Cmx/c =  C m  + C le

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



 1 A1  + A2 . 4

This equation shows that the pitching moment will be independent of  A 0 –   and, therefore, independent of the angle of attack  α  – when x 1 =  , c 4 that is, when the pitching moment is measured relative to the quarter-chord point. This reference point about which the pitching moment is independent of angle of attack is called the  aerodynamic  center ; thus, thin-airfoil theory shows that the aerodynamic center is  independent of the shape of the  camber line and located at the quarter chord point  . The value of the moment coefficient, referenced to the quarter-chord point, is then given by π 4

Cmc/ = −  (A1 − A2 ) . 4

2.3

 

(20)

Circular-arc Camber Line

The simplest non-trivial camber line is a circular arc which, for small amplitudes, can be approximated as the parabola yc  = 4τ x (1 − x) , where the parameter τ  expresses the maximum deviation of the camber line from the chord, as a fraction of chord length. The camber line slope is thus found to be dyc = 4τ  (1 − 2x) , dx 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

(21) ≥

2.

3

7

COMPARISONS WITH NONLINEAR THEORY 

The lift and moment coefficients are then seen to be C  = 2π (α + 2τ ) , Cmc/ = −πτ .

 

(22)

4

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 the angle for zero lift  are proportional to the amplitude of the camber line  . While the result here has been shown only for the case of circular-arc camber, this is a general result (following from the linearity of the equations we solve in the thin-airfoil approximation). That is, for a camber line of  any shape, given by, say yc  = τ f (x) both the angle for zero lift and the pitching moment about the aerodynamic center will be directly proportional 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 camber line   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  separation  between the vortex position and the control point that gives us the correct lift-curve slope of 2 π, while the specific   location   of the control point gives us the correct angle for zero lift. This singlepanel 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 coefficient for the simplest non-trivial camber line, when using a single vortex panel, provides the motivation for 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, compressible flow past complete airfoils. These calculations are, in fact, performed at very low Mach numbers, typically  M  = 0.05, so the results can be interpreted as equivalent to linear, incompressible 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 viscous effects 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 aft of 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 ◦

8

REFERENCES 

Thin Airfoil Theory 2.5 y

 (x 10)

c

C  = 0.30087 l

2

C

m(c/4)

dy/dx Delta C

 = −0.012838

p

1.5

1

0.5

0

−0.5 0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

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

∆C p , 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 to be 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 the actual pressure distributions vary significantly with the added contribution due to thickness, but Figs. 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 coefficient varies more strongly, but differs from the thin-airfoil value by only about 15% for a thickness ratio of 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.

9

REFERENCES 

(a) NACA 23003

(b) NACA 23012

Surface Pressure, Total Enthalpy, and Entropy Change Distributions -2

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

-1.5

   t   n   e    i   c    i    f    f   e   o    C   e   r   u   s   s   e   r    P

Surface Pressure, Total Enthalpy, and Entropy Change Distributions -2

-1.5

-1

   t   n   e    i   c    i    f    f   e   o    C   e   r   u   s   s   e   r    P

-0.5

 0

-1

-0.5

 0

 0.5

 0.5

 1

 1

 1.5

 0

0.2

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

0.4

0.6

0.8

 1.5

1

 0

0.2

Chordwise position, x/c

(c) NACA 23003

   t   n   e    i   c    i    f    f   e   o    C   e   r   u   s   s   e   r    P

   t   n   e    i   c    i    f    f   e   o    C   e   r   u   s   s   e   r    P

-0.5

 0

 0

 1

 1

0.4

0.6

Chordwise position, x/c

(e) NACA 23003

1

-0.5

 0.5

0.2

0.8

-1

 0.5

 0

1

Lifting Pressure Coefficient Lifting Pressure Coefficient Thin Airfoil Theory

-1.5

-1

 1.5

0.8

Lifting Surface Pressure Distribution -2

Lifting Pressure Coefficient Lifting Pressure Coefficient Thin Airfoil Theory

-1.5

0.6

(d) NACA 23012

Lifting Surface Pressure Distribution -2

0.4

Chordwise position, x/c

0.8

1

 1.5

 0

0.2

0.4

0.6

Chordwise position, x/c

(f) NACA 23012

Figure 3: Full inviscid solutions for flow past NACA 23003 and 23012 airfoils at design incidence α = 1.65 and M = 0.05. (a), (b) show contours of constant pressure coefficient in ∆ C p = 0.05 increments 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. ◦

∞

10

REFERENCES 

NACA 230xx Family of Airfoils  0.4

Lift Moment

 0.35  0.3    t   n   e    i   c    i    f    f   e   o    C    t   n   e   m   o    M    /   e   c   r   o    F

 0.25  0.2  0.15  0.1  0.05  0 -0.05  0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Thickness ratio, t/c

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