Aerodynamics - Thin Airfoil Theory

Topics: Airfoil, Lift, Fluid dynamics Pages: 19 (3124 words) Published: August 28, 2010
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 method to determine the properties of thin airfoils. This material is covered in Lecture, but is not in the textbook [5]. A summary of results from the analytical theory also is provided, as well as a comparison of the thin-airfoil results with those of a complete inviscid theory that accounts for thickness effects.

1

The Vortex Lattice Method

We here describe the implementation of the vortex lattice method for two-dimensional flows past thin airfoils. The method is even more useful for three-dimensional wings, i.e., for the flow past wings of finite span, but that problem is not considered here. Instead, the reader is referred 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 v= − α, (1) dx be enforced at selected control points to determine the vortex strengths. Equation (1) simply 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 segments, or panels, as illustrated in Fig. 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, and the control point three-quarters of the panel length, aft of the leading edge of the panel. (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 vi,j induced at the ith control point by the jth point vortex is given by vi,j = Γj 1 2π xvj − xci

where xvj is the chordwise coordinate of the jth vortex having strength Γj , and xci is the chordwise coordinate of the ith control point. The total vertical velocity at the ith control point induced by

1

THE VORTEX LATTICE METHOD

2

y

Γ

i

xi

x i+1

x

x

v

x

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

vi =
j=1 N

Γj 1 2π xvj − xci ai,j Γj

=
j=1

where ai,j =

1 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 jth panel. If we introduce the vector notation v = [v1 and Γ = [Γ1 Γ2 ... and define the matrix of influence coefficients  a1,1 a1,2  a2,1 a2,2   · · A=  · ·   · · aN,1 aN,2 v2 ... vN ] T

ΓN ] ,  a1,N a2,N   ·  , ·   ·  aN,N

T

··· ··· ··· ··· ··· ···

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

THE VORTEX LATTICE METHOD

3

whence the lift coefficient is C = or
N 1 2 2 ρU c

=

ρU

N j=1 Γj 1 2 2 ρU c

=

2 Uc

N

Γj
j=1

C =2
j=1

Γj

(3)

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