164 Aerodynamics for Engineering Students V+V f- V f- Fig. 4.4 Consider this by the reverse argument. Look again at Fig. 4.3b. By definition the velocity potential of C relative to A (&A) must be equal to the velocity potential of C relative to B (~cB) in a potential flow. The integration continued around ACB gives = ~CA i= ~CB = 0 This is for a potential flow only. Thus, if I' is finite the definition of the velocity potential breaks down and the curve ACB must contain a region of rotational flow. If the flow is not potential then Eqn (ii) in Section 3.2 must give a non-zero value for vorticity. An alternative equation for I' is found by considering the circuit of integration to consist of a large number of rectangular elements of side Sx by (e.g. see Section 2.7.7 and Example 2.2). Applying the integral I' = J (u dx + v dy) round abcd, say, which is the element at P(x, y) where the velocity is u and v, gives (Fig. 4.5). av sx av sx The sum of the circulations of all the areas is clearly the circulation of the circuit as a whole because, as the AI' of each element is added to the AI? of the neighbouring element, the contributions of the common sides disappear. Applying this argument from element to neighbouring element throughout the area, the only sides contributing to the circulation when the AI'S of all areas are summed together are those sides which actually form the circuit itself. This means that for the circuit as a whole over the area round the circuit and av au __-_ -c ax ay This shows explicitly that the circulation is given by the integral of the vorticity contained in the region enclosed by the circuit. Two-dimensional wing theoly 165 Fig. 4.5 If the strength of the circulation remains constant whilst the circuit shrinks to encompass an ever smaller area, i.e. until it shrinks to an area the size of a rectangular element, then: I? = C x SxSy = 5 x area of element Therefore, (4.3) r vorticity = lim area-0 area of circuit Here the (potential) line vortex introduced in Section 3.3.2 will be re-visited and the definition (4.2) of circulation will now be applied to two particular circuits around a point (Fig. 4.6). One of these is a circle, of radius r1, centred at the centre of the vortex. The second circuit is ABCD, composed of two circular arcs of radii r1 and r2 and two radial lines subtending the angle ,6 at the centre of the vortex. For the concentric circuit, the velocity is constant at the value where C is the constant value of qr. 166 Aerodynamics for Engineering Students Fig. 4.6 Two circuits in the flow around a point vortex Since the flow is, by the definition of a vortex, along the circle, a is everywhere zero and therefore cos a = 1. Then, from Eqn (4.2) Now suppose an angle 8 to be measured in the anti-clockwise sense from some arbitrary axis, such as OAB. Then ds = rld8 whence Since C is a constant, it follows that r is also a constant, independent of the radius. It can be shown that, provided the circuit encloses the centre of the vortex, the circulation round it is equal to I?, whatever the shape of the circuit. The circulation I' round a circuit enclosing the centre of a vortex is called the strength of the vortex. The dimensions pf circulation and vortex strength are, from Eqn (4.2), velocity times length, Le. L2T- , the units being m2 s-*. Now r = 2nC, and C was defined as equal to qr; hence I' = 2nqr and r q=- 2nr (4.5) Taking now the second circuit ABCD, the contribution towards the circulation from each part of the circuit is calculated as follows: (i) Rudiul line AB Since the flow around a vortex is in concentrk circles, the velocity vector is everywhere perpendicular to the radial line, i.e. a = 90°, cosa = 0. Thus the tangential velocity component is zero along AB, and there is therefore no contribution to the circulation. (ii) Circular arc BC Here a = 0, cos a = 1. Therefore Two-dimensional wing theory 167 But, by Eqn (4.5), (iii) (iv) Radial line CD As for AB, there is no contribution to the circulation from this part of the circuit. Circular arc DA Here the path of integration is from D to A, while the direction of velocity is from A to D. Therefore a = 180", cosa = -1. Then Therefore the total circulation round the complete circuit ABCD is Thus the total circulation round this circuit, that does not enclose the core of the vortex, is zero. Now any circuit can be split into infinitely short circular arcs joined by infinitely short radial lines. Applying the above process to such a circuit would lead to the result that the circulation round a circuit of any shape that does not enclose the core of a vortex is zero. This is in accordance with the notion that potential flow is irrotational (see Section 3.1). 4.1.3 Circulation and lift (Kutta-Zhukovsky theorem) In Eqn (3.52) it was shown that the lift l per unit span and the circulation r of a spinning circular cylinder are simply related by 1=pm where p is the fluid density and Vis the speed of the flow approaching the cylinder. In fact, as demonstrated independently by Kutta* and Zhukovskyt, the Russian physi- cist, at the beginning of the twentieth century, this result applies equally well to a cylinder of any shape and, in particular, applies to aerofoils. This powerful and useful result is accordingly usually known as the KutteZhukovsky Theorem. Its validity is demonstrated below. The lift on any aerofoil moving relative to a bulk of fluid can be derived by direct analysis. Consider the aerofoil in Fig. 4.7 generating a circulation of l-' when in a stream of velocity V, density p, and static pressure PO. The lift produced by the aerofoil must be sustained by any boundary (imaginary or real) surrounding the aerofoil. For a circuit of radius r, that is very large compared to the aerofoil, the lift of the aerofoil upwards must be equal to the sum of the pressure force on the whole periphery of the circuit and the reaction to the rate of change of downward momen- tum of the air through the periphery. At this distance the effects of the aerofoil thickness distribution may be ignored, and the aerofoil represented only by the circulation it generates. * see footnote on page 161. ' N. Zhukovsky 'On the shape of the lifting surfaces of kites' (in German), Z. Flugtech. Motorluftschiffahrt, 1; 281 (1910) and 3, 81 (1912). 168 Aerodynamics for Engineering Students Fig. 4.7 The vertical static pressure force or buoyancy h, on the circular boundary is the sum of the vertical pressure components acting on elements of the periphery. At the element subtending SO at the centre of the aerofoil the static pressure is p and the local velocity is the resultant of V and the velocity v induced by the circulation. By Bernoulli's equation 1 1 PO + - 2 PV' = p + -p[v2 2 + VZ + ~VV sin e] giving p =po - pVvsin8 if v2 may be neglected compared with V2, which is permissible since r is large. The vertical component of pressure force on this element is -pr sin 8 St) and, on substituting for p and integrating, the contribution to lift due to the force acting on the boundary is lb = -l (PO - pVvsine)rsin ode 2iT (4.7) = +pVvrr with po and r constant. Two-dimensional wing theory 169 The mass flow through the elemental area of the boundary is given by pVr cos 8 SO. This mass flow has a vertical velocity increase of v cos 8, and therefore the rate of change of downward momentum through the element is -pVvr cos2 O SO; therefore by integrating round the boundary, the inertial contribution to the lift, li, is 2n li =+I pVvrcos20d0 Jo = pVvr.ir Thus the total lift is: I = 2pVvm From Eqn (4.5): giving, finally, for the lift per unit span, 1: 1 = pvr (4.10) This expression can be obtained without consideration of the behaviour of air in a boundary circuit, by integrating pressures on the surface of the aerofoil directly. It can be shown that this lift force is theoretically independent of the shape of the aerofoil section, the main effect of which is to produce a pitching moment in potential flow, plus a drag in the practical case of motion in a real viscous fluid. 4.2 The development of aerofoil theory The first successful aerofoil theory was developed by Zhukovsky." This was based on a very elegant mathematical concept - the conformal transformation - that exploits the theory of complex variables. Any two-dimensional potential flow can be repre- sented by an analytical function of a complex variable. The basic idea behind Zhukovsky's theory is to take a circle in the complex < = (5 + iv) plane (noting that here ( does not denote vorticity) and map (or transform) it into an aerofoil-shaped contour. This is illustrated in Fig. 4.8. = 4 + i+ where, as previously, 4 and $ are the velocity potential and stream function respect- ively. The same Zhukovsky mapping (or transformation), expressed mathematically as A potential flow can be represented by a complex potential defined by (where C is a parameter), would then map the complex potential flow around the circle in the <-plane to the corresponding flow around the aerofoil in the z-plane. This makes it possible to use the results for the cylinder with circulation (see Section 3.3.10) to calculate the flow around an aerofoil. The magnitude of the circulation is chosen so as to satisfy the Kutta condition in the z-plane. From a practical point of view Zhukovsky's theory suffered an important draw- back. It only applied to a particular family of aerofoil shapes. Moreover, all the * see footnote on page 161. 170 Aerodynamics for Engineering Students iy z plane 0 U Fig. 4.8 Zhukovsky transformation, of the flow around a circular cylinder with circulation, to that around an aerofoil generating lift members of this family of shapes have a cusped trailing edge whereas the aerofoils used in practical aerodynamics have trailing edges with finite angles. Kkrmkn and Trefftz* later devised a more general conformal transformation that gave a family of aerofoils with trailing edges of finite angle. Aerofoil theory based on conformal transformation became a practical tool for aerodynamic design in 1931 when the American engineer Theodorsen' developed a method for aerofoils of arbitrary shape. The method has continued to be developed well into the second half of the twentieth century. Advanced versions of the method exploited modern computing techniques like the Fast Fourier Transform.** If aerodynamic design were to involve only two-dimensional flows at low speeds, design methods based on conformal transformation would be a good choice. How- ever, the technique cannot be extended to three-dimensional or high-speed flows. For this reason it is no longer widely used in aerodynamic design. Methods based on conformal transformation are not discussed further here. Instead two approaches, namely thin aerofoil theory and computational boundary element (or panel) methods, which can be extended to three-dimensional flows will be described. The Zhukovsky theory was of little or no direct use in practical aerofoil design. Nevertheless it introduced some features that are basic to any aerofoil theory. Firstly, the overall lift is proportional to the circulation generated, and secondly, the magni- tude of the circulation must be such as to keep the velocity finite at the trailing edge, in accordance with the Kutta condition. It is not necessary to suppose the vorticity that gives rise to the circulation be due to a single vortex. Instead the vorticity can be distributed throughout the region enclosed by the aerofoil profile or even on the aerofoil surface. But the magnitude of circulation generated by all this vorticity must still be such as to satisfy the Kutta condition. A simple version of this concept is to concentrate the vortex distribution on the camber line as suggested by Fig. 4.9. In this form, it becomes the basis of the classic thin aerofoil theory developed by Munk' and G1auert.O Glauert's version of the theory was based on a sort of reverse Zhukovsky trans- formation that exploited the not unreasonable assumption that practical aerofoils are * 2. Fhgtech. Motorluftschiffahrt, 9, 11 1 (1918). ** N.D. Halsey (1979) Potential flow analysis of multi-element airfoils using conformal mapping, AZAA J., 12, 1281. NACA Report, No. 411 (1931). NACA Report, No. 142 (1922). Aeronautical Research Council, Reports and Memoranda No. 910 (1924). Two-dimensional wing theory 171 Fig. 4.9 thin. He was thereby able to determine the aerofoil shape required for specified aerofoil characteristics. This made the theory a practical tool for aerodynamic design. However, as remarked above, the use of conformal transformation is restricted to two dimensions. Fortunately, it is not necessary to use Glauert’s approach to obtain his final results. In Section 4.3, later developments are followed using a method that does not depend on conformal transformation in any way and, accordingly, in principle at least, can be extended to three dimensions. Thin aerofoil theory and its applications are described in Sections 4.3 to 4.9. As the name suggests the method is restricted to thin aerofoils with small camber at small angles of attack. This is not a major drawback since most practical wings are fairly thin. A modern computational method that is not restricted to thin aerofoils is described in Section 4.10. This is based on the extension of the panel method of Section 3.5 to lifting flows. It was developed in the late 1950s and early 1960s by Hess and Smith at Douglas Aircraft Company. v. * 4.3 <The general thin aerofoil theory For the development of this theory it is assumed that the maximum aerofoil thickness is small compared to the chord length. It is also assumed that the camber-line shape only deviates slightly from the chord line. A corollary of the second assumption is that the theory should be restricted to low angles of incidence. Consider a typical cambered aerofoil as shown in Fig. 4.10. The upper and lower curves of the aerofoil profile are denoted by y, and yl respectively. Let the velocities in the x and y directions be denoted by u and v and write them in the form: u= UCOSQ+U’. v= Usincu+v‘ Fig. 4.10 172 Aerodynamics for Engineering Students u’ and v’ represent the departure of the local velocity from the undisturbed free stream, and are commonly known as the disturbance or perturbation velocities. In fact, thin-aerofoil theory is an example of a small perturbation theory. The velocity component perpendicular to the aerofoil profile is zero. This constitutes the boundary condition for the potential flow and can be expressed mathematically as: -usinp+vcosp=O at y=yu and y1 Dividing both sides by cos p, this boundary condition can be rewritten as -(Ucosa+ul)-+Usina+v’=O dY at y=yu and y1 (4.11) By making the thin-aerofoil assumptions mentioned above, Eqn (4.11) may be simplified. Mathematically, these assumptions can be written in the form dx dYu dyl yu and yl e c; a,- and - << 1 dx dx Note that the additional assumption is made that the slope of the aerofoil profile is small. These thin-aerofoil assumptions imply that the disturbance velocities are small compared to the undisturbed free-steam speed, i.e. ut and VI<< U Given the above assumptions Eqn (4.1 1) can be simplified by replacing cos a and sina by 1 and a respectively. Furthermore, products of small quantities can be neglected, thereby allowing the term u‘dyldx to be discarded so that Eqn (4.1 1) becomes (4.12) One further simplification can be made by recognizing that if yu and y1 e c then to a sufficiently good approximation the boundary conditions Eqn (4.12) can be applied at y = 0 rather than at y = y, or y1. Since potential flow with Eqn (4.12) as a boundary condition is a linear system, the flow around a cambered aerofoil at incidence can be regarded as the superposition of two separate flows, one circulatory and the other non-circulatory. This is illustrated in Fig. 4.1 1. The circulatory flow is that around an infinitely thin cambered plate and the non-circulatory flow is that around a symmetric aerofoil at zero incidence. This superposition can be demonstrated formally as follows. Let yu=yc+yt and H=yc-yt y = yc(x) is the function describing the camber line and y = yt = (yu - y1)/2 is known as the thickness function. Now Eqn (4.12) can be rewritten in the form dYc dYt VI= u Ua f u- dx dx Circulatory Non-circulatory where the plus sign applies for the upper surface and the minus sign for the lower surface. Two-dimensional wing theory 173 Cumbered plate at incidence (circulatory flow ) Symmetric aerofoil at zero incidence ( non-circulatory flow) Fig. 4.11 Cambered thin aerofoil at incidence as superposition of a circulatory and non-circulatory flow Thus the non-circulatory flow is given by the solution of potential flow subject to the boundary condition v' = f U dyt/dx which is applied at y = 0 for 0 5 x 5 c. The solution of this problem is discussed in Section 4.9. The lifting characteristics of the aerofoil are determined solely by the circulatory flow. Consequently, it is the solution of this problem that is of primary importance. Turn now to the formulation and solution of the mathematical problem for the circulatory flow. It may be seen from Sections 4.1 and 4.2 that vortices can be used to represent lifting flow. In the present case, the lifting flow generated by an infinitely thin cambered plate at incidence is represented by a string of line vortices, each of infinitesimal strength, along the camber line as shown in Fig. 4.12. Thus the camber line is replaced by a line of variable vorticity so that the total circulation about the chord is the sum of the vortex elements. This can be written as r = L'kds (4.13) Fig. 4.12 Insert shows velocity and pressure above and below 6s [...]... (4. 51): +,( (1 -; ) COS4Il - (1 (4. 54) where ’t Fig 4. 16 Two-dimensional wing theory 1 1 3 = LTsinnOsinOdO = - n-1 sin(n - 2 )4 n-2 14 = In the usual notation CH = b l a From Eqn (4. 54) : bl =- + b277, where LT(+ 1 COS 0) (cos 1 4 - COS 0)d0 giving (4. 55) Similarly from Eqn (4. 54) b2 =- 1 % = - x coefficient of r] in Eqn (4. 54) F2 This somewhat unwieldy expression reduces to* 1 b2 = -{ (1 - cos 2 4 ) -. .. Eqn (4. 87) into Eqn (4. 41) gives m [(2p- l)O, TP2 + sine,] + m -P) [(2p- l)(n - 6,) - sin e,] (4. 88) Similarily from Eqn (4. 42) (4. 89) (4. 90) Example 4. 2 The NACA 44 12 wing section For a NACA 44 12 wing section rn = 0. 04 and p = 0 .4 so that 0, = cos-l(l - 2 x 0 .4) = 78 .46 “ = 1.3694rad making these substitutions into Eqns (4. 88) to (4. 90) gives A I = 0.163 A0 = 0.0090, and A2 = 0.0228 Thus Eqns (4. 43)... comparison wt Eqn (4. 35) by writing C x = - ( 1 -case) 2 or 1 x1 = - ( 1 -case) 2 dYl -= L ( i - 2cose + cosze) + a(b - 1) - a(b - 1) case - ab dxl 4 (4. 81) Comparing Eqn (4. 81) and (4. 35) gives 3a A2 = s 8c Thus to satisfy (iv) above, A1 = A2, i.e -( ;+ub):=ai3s giving b = - - 7 8 (4. 82) The quadratic in Eqn (4. 80) gives for xo on cancelling a, x(j = -2 (b - 1) f d 2 2 ( b - 1)2 + 4 x 3b 6 - (1 - b) &&z +b+... = 0.0090, and A2 = 0.0228 Thus Eqns (4. 43) and (4. 47) give CL = T ( A ~ 2x40) + 2 - ~ ~(0.163 2 x 0.009) 0 ~ T T +2 ~ = 0 .45 6 + 6.28320 a CM,,,= - - ( A I - A z ) = (0.163 - 0.0228) = -0 .110 4 4 (4. 91) (4. 92) 196 Aerodynamics for Engineering Students In Section 4. 10 (Fig 4. 26), the predictions of thin-aerofoil theory, as embodied in Eqns (4. 91) and (4. 92), are compared with accurate numerical solutions... since lT sin ne sin me dB = 0 when n # rn, or (4. 44) In terms of the lift coefficient, CM, becomes C M , = - 5 [ 1 + w A1 - A2 ] 4 Then the centre of pressure coefficient is (4. 45) and again the centre of pressure moves as the lift or incidence is changed Now, from Section 1.5 .4, (4. 46) and comparing Eqns (4. 44) and (4. 45) gives 7r - C M , p = - (A1 - A2) 4 (4. 47) This shows that, theoretically, the pitching... Z, = -pV/ '- a 3 +2 4 giving (4. 70) In a similar fashion the contribution to Mq and mq can be found by differentiating the expression for MCG,with respect to q, i.e from Eqn (4. 68) 2K-alvsc2 +- (4. 71) 32 giving for a rectangular wing (4. 72) For other than rectangular wings the contribution becomes, by strip theory: Mq = - p v / - s ( ; ( l - 2h)2 s +- 27r - 32 ") c3 dy (4. 73) and mq = (4. 74) 190 Aerodynamics. .. U 2 c l T ( l+cosO)dO = 7i-apU2c It therefore follows that for unit span I CL = (4. 31) ($q) =27ra The moment about the leading edge per unit span MLE= -lCp dx Changing the sign Therefore for unit span - 7i- (4. 32) Comparing Eqns (4. 31) and (4. 32) shows that CL CMLB= 4 (4. 33) The centre of pressure coeficient kcp is given for small angles of incidence approximately by (4. 34) and this shows a fixed... Eqn (4. 82), b = - - gives 8 22.55 7 .45 xo = - or 24 24 i.e taking the smaller value since the larger only gives the point of reflexure near the trailing edge: y = 6 when x = 0.31 x chord Substituting xo = 0.31 in the cubic of Eqn (4. 80) gives a= ' 0.121 - 8.28 192 Aerodynamics for Engineering Students The camber-line equation then is (4. 83) This cubic camber-line shape is shown plotted on Fig 4. 20... stagnation point Two-dimensional wing theoly y/s 1 Camberline ordinates for CMk'O 0.2 x/c y/8 x/c 0 0 0 .4 0.6 0.8 1.0 x/c 0.05 01 0.15 0.2 0.25 0.3 0.35 0 .4 0 4 5 0 9 8 0. 943 O.St0, 8 0.3 24 0.577 0.765 0.8 94 0.970 0.999 0.5 0 5 0.6 6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 y/6 0,776 0.666 0. 546 0 .42 4 0.3 04 0.1 94 0 0 90.026 5.019 5.03 9 x/c 0 k/2nU 03 1.0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 .4 0 .45 0,127 0.123 01... ideal aC,/& = 2 r by a reasonable value, Q, that accounts 7 for the aspect ratio change (see Chapter 5) The lift coefficient of a pitching rectangular wing then becomes (4. 64) Similarly the pitching-moment coefficient about the leading edge is found from Eqn (4. 44) : 7r 7rqC =-( A2-A1) -4 8V 1 4cL (4. 65) which for a rectangular wing, on substituting for CL,becomes The moment coefficient of importance in the . per unit span MLE = -lCp dx Changing the sign Therefore for unit span 7i- - - Comparing Eqns (4. 31) and (4. 32) shows that CL CMLB = 4 (4. 31) (4. 32) (4. 33) The centre of. (4. 46) and comparing Eqns (4. 44) and (4. 45) gives (4. 47) 7r -CM,p = - (A1 - A2) 4 This shows that, theoretically, the pitching moment about the quarter chord point for a thin aerofoil is. -5 [1+w] A1 - A2 4 Then the centre of pressure coefficient is (4. 44) (4. 45) and again the centre of pressure moves as the lift or incidence is changed. Now, from Section 1.5 .4, (4. 46)