M&AE 305 October 3, 2006Thin Airfoil TheoryD. A. CaugheySibley School of Mechanical & Aerospace EngineeringCornell UniversityIthaca, New York 14853-7501These 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 MethodWe 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 velocityv =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,jinduced at the ith control point by the jth point vortex is given byvi,j=Γj2π1xvj− xciwhere xvjis the chordwise coordinate of the jth vortex having strength Γj, and xciis the chordwisecoordinate of the ith control point. The total vertical velocity at the ith control point induced by1 THE VORTEX LATTICE METHOD 2yxx xi i+1Γix 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 thusvi=NXj=1Γj2π1xvj− xci=NXj=1ai,jΓjwhereai,j=12π¡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 notationv = [v1v2. . . vN]TandΓ = [Γ1Γ2. . . ΓN]T,and define the matrix of influence coefficientsA =a1,1a1,2· · · a1,Na2,1a2,2· · · a2,N· · · · · ·· · · · · ·· · · · · ·aN,1aN,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 writtenAΓ = 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` = ρUNXj=1Γj1 THE VORTEX LATTICE METHOD 3whence the lift coefficient isC`=`12ρU2c=ρUPNj=1Γj12ρU2c=2UcNXj=1ΓjorC`= 2NXj=1Γj(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 asmc/4= −ρUNXj=1Γjµxvj−14¶whence the moment coefficient isCmc/4=mc/412ρU2c2= −PNj=1Γj¡xvj−14¢12Uc2= −2Uc2NXj=1Γjµxvj−14¶orCmc/4= −2NXj=1Γjµxvj−14¶(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 determinedfromγ(xvj)∆xj= Γjorγ(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∞=12ρ£U2− V2¤=12ρU2£1 − (1 + ∆u)2− ∆v2¤= −ρU2∆uso the change in pressure coefficient across the vortex sheet is∆Cp=−ρU2∆u12ρU2= −2∆uwhence∆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 42 Classical Thin-Airfoil TheoryIn 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 verticalvelocityv(x) =12πZ10γ(ξ)ξ − xdξat any point on the chord line, so we must solve the integral equation12πZ10γ(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 variablesξ =1 − cos φ2andx =1 − cos θ2, (8)following which Eq. (7) becomes12πZπ0γ(φ) sin φcos φ − cos θdφ = α −dycdx. (9)The vorticity distribution can then be represented by the infinite seriesγ(φ) = 2"A0cotφ2+∞Xn=1Ansin nφ#. (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 cot π/2 = 0 and sin nπ = 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])Zπ0cos nφcos φ − cos θdφ = πsin nθsin θ. (11)Thus, substitution of the vorticity distribution of Eq. (10) into the integral Eq. (9), using the