230 Aerodynamics for Engineering Students Fig. 5.20 Modelling the displacement effect by a distribution of sources wings having high aspect ratio, intuition would suggest that the flow over most of the wing behaves as if it were two-dimensional. Plainly this will not be a good approxi- mation near the wing-tips where the formation of the trailing vortices leads to highly three-dunensional flow. However, away from the wing-tip region, Eqn (5.23) reduces approximately to Eqn (4.103) and, to a good approximation, the C, distributions obtained for symmetrical aerofoils can be used for the wing sections. For complete- ness this result is demonstrated formally immediately below. However, if this is not of interest go directly to the next section. Change the variables in Eqn (5.23) to % = (x - xI)/c, 21 = z1/c and Z = (z - z1)/c. Now provided that the non-dimensional shape of the wing-section does not change along the span, or, at any rate, only changes very slowly St = d(yt/c)/dZ does not vary with Z and the integral I1 in Eqn (5.23) becomes " 12 To evaluate the integral 12 change variable to x = l/Z so that 1 11 1 1 Finite wing theory 231 For large aspect ratios s >> cy so provided z1 is not close to fs, i.e. near the wing-tips, giving Thus Eqn (5.23) reduces to the two-dimensional result, Eqn (4.103), i.e. (5.24) Lifting effect To understand the fundamental concepts involved in modelling the lifting effect of a vortex sheet, consider first the simple rectangular wing depicted in Fig. 5.21. Here the vortex sheet is constructed from a collection of horseshoe vortices located in the y = 0 plane. From Helmholtz's second theorem (Section 5.2.1) the strength of the circulation round any section of the vortex sheet (or wing) is the sum of the strengths of the vortex filaments CL\ Fig. 5.21 The relation between spanwise load variation and trailing vortex strength 232 Aerodynamics for Engineering Students vortex filaments cut by the section plane. As the section plane is progressively moved outwards from the centre section to the tips, fewer and fewer bound vortex filaments are left for successive sections to cut so that the circulation around the sections diminishes. In this way, the spanwise change in circulation round the wing is related to the spanwise lengths of the bound vortices. Now, as the section plane is moved outwards along the bound bundle of filaments, and as the strength of the bundle decreases, the strength of the vortex filaments so far shed must increase, as the overall strength of the system cannot diminish. Thus the change in circulation from section to section is equal to the strength of the vorticity shed between these sections. Figure 5.21 shows a simple rectangular wing shedding a vortex trail with each pair of trailing vortex filaments completed by a spanwise bound vortex. It will be noticed that a line joining the ends of all the spanwise vortices forms a curve that, assuming each vortex is of equal strength and given a suitable scale, would be a curve of the total strengths of the bound vortices at any section plotted against the span. This curve has been plotted for clarity on a spanwise line through the centre of pressure of the wing and is a plot of (chordwise) circulation (I') measured on a vertical ordinate, against spanwise distance from the centre-line (CL) measured on the horizontal ordinate. Thus at a section z from the centre-line sufficient hypothetical bound vortices are cut to produce a chordwise circulation around that section equal to I'. At a further section z + Sz from the centre-line the circulation has fallen to l? - ST, indicating that between sections z and z + Sz trailing vorticity to the strength of SI' has been shed. If the circulation curve can be described as some function of z,flz) say then the strength of circulation shed (5.25) Now at any section the lift per span is given by the Kutta-Zhukovsky theorem Eqn (4.10) I=pVT and for a given flight speed and air density, I' is thus proportional to 1. But I is the local intensity of lift or lift grading, which is either known or is the required quantity in the analysis. The substitution of the wing by a system of bound vortices has not been rigorously justified at this stage. The idea allows a relation to be built up between the physical load distribution on the wing, which depends, as shall be shown, on the wing geometric and aerodynamic parameters, and the trailing vortex system. (a) It will be noticed from the leading sketch that the trailing filaments are closer together when they are shed from a rapidly diminishing or changing distribution curve. Where the filaments are closer the strength of the vorticity is greater. Near the tips, therefore, the shed vorticity is the most strong, and at the centre where the distribution curve is flattened out the shed vorticity is weak to infinitesimal. (b) A wing infinitely long in the spanwise direction, or in two-dimensional flow, will have constant spanwise loading. The bundle will have filaments all of equal length and none will be turned back to form trailing vortices. Thus there is no trailing vorticity associated with two-dimensional wings. This is capable of deduction by a more direct process, i.e. as the wing is infinitely long in the spanwise direction the lower surface @ugh) and upper surface (low) pressures Figure 5.21 illustrates two further points: Finite wing theory 233 cannot tend to equalize by spanwise components of velocity so that the streams of air meeting at the trailing edge after sweeping under and over the wing have no opposite spanwise motions but join up in symmetrical flow in the direction of motion. Again no trailing vorticity is formed. A more rigorous treatment of the vortex-sheet modelling is now considered. In Section 4.3 it was shown that, without loss of accuracy, for thin aerofoils the vortices could be considered as being distributed along the chord-line, i.e. the x axis, rather than the camber line. Similarly, in the present case, the vortex sheet can be located on the (x, z) plane, rather than occupying the cambered and possibly twisted mid-surface of the wing. This procedure greatly simplifies the details of the theoretical modelling. One of the infinitely many ways of constructing a suitable vortex-sheet model is suggested by Fig. 5.21. This method is certainly suitable for wings with a simple planform shape, e.g. a rectangular wing. Some wing shapes for which it is not at all suitable are shown in Fig. 5.22. Thus for the general case an alternative model is required. In general, it is preferable to assign an individual horseshoe vortex of strength k (x, z) per unit chord to each element of wing surface (Fig. 5.23). This method of constructing the vortex sheet leads to certain mathematical difficulties (a 1 Delta wing ( b ) Swept - back wing Fig. 5.22 Fig. 5.23 Modelling the lifting effect by a distribution of horseshoe vortex elements 234 Aerodynamics for Engineering Students Strength, ksxl /, Strength, (kt all ak 6z,)8xl ,Strength, kSx, ___ I I I 1 kZngth, - 81, sx, 1 (a ) Horseshoe vortices (b) L-shaped vortices Fig. 5.24 Equivalence between distributions of (a) horseshoe and (b) L-shaped vortices when calculating the induced velocity. These problems can be overcome by recom- bining the elements in the way depicted in Fig. 5.24. Here it is recognized that partial cancellation occurs for two elemental horseshoe vortices occupying adjacent span- wise positions, z and z + 6z. Accordingly, the horseshoe-vortex element can be replaced by the L-shaped vortex element shown in Fig. 5.24. Note that although this arrangement appears to violate Helmholtz’s second theorem, it is merely a math- ematically convenient way of expressing the model depicted in Fig. 5.23 which fully satisfies this theorem. 5.5 Relationship between spanwise loading and trailing vorticity It is shown below in Section 5.5.1 how to calculate the velocity induced by the elements of the vortex sheet that notionally replace the wing. This is an essential step in the development of a general wing theory. Initially, the general case is considered. Then it is shown how the general case can be very considerably simplified in the special case of wings of high aspect ratio. The general case is then dropped, to be taken up again in Section 5.8, and the assumption of large aspect ratio is made for Section 5.6 and the remainder of the present section. Accordingly, some readers may wish to pass over the material immediately below and go directly to the alternative derivation of Eqn (5.32) given at the end of the present section. 5.5.1 Induced velocity (downwash) Suppose that it is required to calculate the velocity induced at the point Pl(x1, zl) in the y = 0 plane by the L-shaped vortex element associated with the element of wing surface located at point P (x, z) now relabelled A (Fig. 5.25). Finite wing theoly 235 t= 2-11 I A /B x-xj-/4q p1 C Fig. 5.25 Geometric notation for L-shaped vortex element Making use of Eqn (5.9) it can be seen that this induced velocity is perpendicular to the y = 0 plane and can be written as svi(xl,~~) = (svi),, + (6vi)Bc -_ - ksx [cosel -cos (5.26) 4n(x - XI) From the geometry of Fig. 5.25 the various trigonometric expressions in Eqn (5.26) can be written as z - z1 cOsel = cose2 = - &x - Xd2 + (2 - x - x1 J(x - + (z + sz - z1)2 z + sz - 21 COS e2 + - = - sin02 = (2 J(x - + (2 + sz - The binomial expansion, i.e. (a + b)" = d + nd-lb + *. . ; can be used to expand some of the terms, for example where r = d(x -XI)' + (z - ~1)~. In this way, the trigonometric expressions given above can be rewritten as 236 Aerodynamics for Engineering Students (5.27) (5.28) (5.29) Equations (5.27 to 5.29) are now substituted into Eqn (5.26), and terms involving (6~)~ and higher powers are ignored, to give In order to obtain the velocity induced at P1 due to all the horseshoe vortex elements, 6vi is integrated over the entire wing surface projected on to the (x, z) plane. Thus using Eqn (5.30) leads to The induced velocity at the wing itself and in its wake is usually in a downwards direction and accordingly, is often called the downwash, w, so that w = -Vi. It would be a difficult and involved process to develop wing theory based on Eqn (5.31) in its present general form. Nowadays, similar vortex-sheet models are used by the panel methods, described in Section 5.8, to provide computationally based models of the flow around a wing, or an entire aircraft. Accordingly, a discussion of the theoretical difficulties involved in using vortex sheets to model wing flows will be postponed to Section 5.8. The remainder of the present section and Section 5.6 is devoted solely to the special case of unswept wings having high aspect ratio. This is by no means unrealistically restrictive, since aerodynamic considera- tions tend to dictate the use of wings with moderate to high aspect ratio for low-speed applications such as gliders, light aeroplanes and commuter passenger aircraft. In this special case Eqn (5.31) can be very considerably simplified. This simplification is achieved as follows. For the purposes of determining the aerodynamic characteristics of the wing it is only necessary to evaluate the induced velocity at the wing itself. Accordingly the ranges for the variables of integration are given by -s 5 z 5 s and 0 5 x 5 (c) For high aspect ratios S/C> 1 so that Ix - XI I << r over most of the range of integration. Consequently, the contributions of terms (b) and (c) to the integral in Eqn (5.31) are very small compared to that of term (a) and can therefore be neglected. This allows Eqn (5.31) to be simplified to where, as explained in Section 5.4.1 , owing to Helmholtz's second theorem (5.32) (5.33) Finite wing theoly 237 Fig. 5.26 Prandtl's lifting line model is the total circulation due to all the vortex filaments passing through the wing section at z. Physically the approximate theoretical model implicit in Eqn (5.32) and (5.33) corresponds to replacing the wing by a single bound vortex having variable strength I', the so-called Zijting Zine (Fig. 5.26). This model, together with Eqns (5.32) and (5.33), is the basis of Prandtl's general wing theory which is described in Section 5.6. The more involved theories based on the full version of Eqn (5.31) are usually referred to as lifting surface theories. Equation (5.32) can also be deduced directly from the simple, less general, theor- etical model illustrated in Fig. 5.21. Consider now the influence of the trailing vortex filaments of strength ST shed from the wing section at z in Fig. 5.21. At some other point z1 along the span, according to Eqn (5.1 l), an induced velocity equal to will be felt in the downwards direction in the usual case of positive vortex strength. All elements of shed vorticity along the span add their contribution to the induced velocity at z1 so that the total influence of the trailing system at z1 is given by Eqn (5.32). 5.5.2 The consequences of downwash - trailing vortex drag The induced velocity at z1 is, in general, in a downwards direction and is sometimes called downwash. It has two very important consequences that modify the flow about the wing and alter its aerodynamic characteristics. Firstly, the downwash that has been obtained for the particular point z1 is felt to a lesser extent ahead of z1 and to a greater extent behind (see Fig. 5.27), and has the effect of tilting the resultant oncoming flow at the wing (or anywhere else within its influence) through an angle where w is the local downwash. This reduces the effective incidence so that for the same lift as the equivalent infinite wing or two-dimensional wing at incidence ax an incidence a = am + E is required at that section on the finite wing. This is illustrated in Fig. 5.28, which in addition shows how the two-dimensional lift L, is normal to 238 Aerodynamics for Engineering Students I 44 J. J ti444 tJ4J 4 J c J.1 w =zero WCP w=2wcp - I Fig. 5.27 Variation in magnitude of downwash in front of and behind wing the resultant velocity VR and is, therefore, tilted back against the actual direction of motion of the wing V. The two-dimensional lift L, is resolved into the aerodynamic forces L and D, respectively, normal to and against the direction of the forward velocity of the wing. Thus the second important consequence of downwash emerges. This is the generation of a drag force D,. This is so important that the above sequence will be explained in an alternative way. A section of a wing generates a circulation of strength I?. This circulation super- imposed on an apparent oncoming flow velocity V produces a lift force L, = pVF according to the Kutta-Zhukovsky theorem (4.10), which is normal to the apparent oncoming flow direction. The apparent oncoming flow felt by the wing section is the resultant of the forward velocity and the downward induced velocity arising from the trailing vortices. Thus the aerodynamic force L, produced by the combination of I? and Y appears as a lift force L normal to the forward motion and a drag force D, against the normal motion. This drag force is called trailing vortex drug, abbreviated to vortex drag or more commonly induced drug (see Section 1.5.7). Considering for a moment the wing as a whole moving through air at rest at infinity, two-dimensional wing theory suggests that, taking air as being of small to negligible viscosity, the static pressure of the free stream ahead is recovered behind the wing. This means roughly that the kinetic energy induced in the flow is converted back to pressure energy and zero drag results. The existence of a thin boundary layer and narrow wake is ignored but this does not really modify the argument. In addition to this motion of the airstream, a finite wing spins the airflow near the tips into what eventually becomes two trailing vortices of considerable core size. The generation of these vortices requires a quantity of kinetic energy that is not recovered Fig. 5.28 The influence of downwash on wing velocities and forces: w = downwash; V = forward speed of wing; V, = resultant oncoming flow at wing; a = incidence; E = downwash angle = w/V; am = (g E) = equivalent two-dimensional incidence; L, = two-dimensional lift; L = wing lift; D, =trailing vortex drag Finite wing theory 239 by the wing system and that in fact is lost to the wing by being left behind. This constant expenditure of energy appears to the wing as the induced drag. In what follows, a third explanation of this important consequence of downwash will be of use. Figure 5.29 shows the two velocity components of the apparent oncoming flow superimposed on the circulation produced by the wing. The forward flow velocity produces the lift and the downwash produces the vortex drag per unit span. Thus the lift per unit span of a finite wing (I) (or the load grading) is by the Kutta- Zhukovsky theorem: I = pvr the total lift being L = /:pVTdz (5.34) The induced drag per unit span (d,), or the induced drag grading, again by the Kutta-Zhukovsky theorem is d, = pwr (5.35) and by similar integration over the span D, = /:pwrdz (5.36) This expression for D, shows conclusively that if w is zero all along the span then D, is zero also. Clearly, if there is no trailing vorticity then there will be no induced drag. This condition arises when a wing is working under two-dimensional conditions, or if all sections are producing zero lift. As a consequence of the trailing vortex system, which is produced by the basic lifting action of a (finite span) wing, the wing characteristics are considerably modi- fied, almost always adversely, from those of the equivalent two-dimensional wing of the same section. Equally, a wing with flow systems that more nearly approach the two-dimensional case will have better aerodynamic characteristics than one where I =pvr L= f spl/rdz -S d, =pwr Fig. 5.29 Circulation superimposed on forward wind velocity and downwash to give lift and vortex drag (induced drag) respectively [...]... distribution for the wing flying straight and level at 89.4m s-l at low altitude From the data: Wing area S = 3.048 + 1 .52 4 x 12.192 = 27.85m2 2 span’ 12.192’ - 5. 333 Aspect ratio (AR) = area 27. 85 I);( At any section z from the centre-line [B from the wing-tip] [ chord c = 3.048 1 - 3.048 - 1 .52 4 3.048 (1 3 [ 5* 55T: 5 (1 3 (2)m=a =5. 5[1 +- a o = 5 5 1 + OSCOSB] = 3.048[1 5 ’ 5 5 ~ ~ ’ 8 = 5. 5[1 - 0. 054 55 cos... zero-lift incidence is the same, then c, = uoo[aoo ao]= u[a- a01 - (5. 56) Taking the first equation with a = Q - E , CL = u,[(a - o ) - €1 (5. 57) But equally from Eqn (4.10) c, = lift per unit span =- I f pV2c =- P fpVc W 4pv2c 27 1 c, =-VC Equating Eqn (5. 57) and (5. 58) and rearranging: 27 1 -= VI( - a01 - 4 cam (5. 58) Finite wing theory 251 and since VE= w = - ' / ' M d z 4 r -3 z-21 7 from Eqn (5. 32)... (5. 60) lead to the following four simultaneous equations in the unknown coefficients + + + 0.004739 = 0.22079 A1 0.89202 A3 1. 251 00 A5 0.66688 A7 0.011637 = 0.663 19 A1 f0.98 957 A3 - 1.3 15 95A5 - 1.64234 A7 0.0216 65 = 1.1 15 73 A1 - 0.679 35 A3 - 0.896 54 A5 2.688 78 A7 0.032998 = 1.343 75 A I - 2.031 25 A3 - 2.718 75 A5 - 3.406 25 A7 + These equations when solved give A1 = 0.020 329, A3 = -0 .000 955 ;... 55 cos B] = 5. 5[1 + 0.363 64 cos e] Finite wing theory 253 Table 5. 1 7~/8 ~ 1 4 3~18 71 52 0.382 68 0.707 11 0.923 88 1.ooo 00 0.923 88 0.707 11 -0 .382 68 - 1.ooo 00 0.923 88 -0 .707 11 -0 .38268 1.ooo 00 0.382 68 -0 .707 11 0.923 88 - 1.ooo 00 0.923 88 0.707 11 0.38268 0.000 00 This gives at any section: and par = 0.0329 95( i+o.5cOse)(i - o.o5 455 ~0se)(i +0.36364cosq where a! is now in radians For convenience... the lift-versus-incidence curve for an aerofoil section of 250 Aerodynamics for Engineering Students - - Incidence m 0 c c 0 c - e Lc P Fig 5. 34 Lift-versus-incidence curve for an aerofoil section of a certain profile, working two-dimensionally and working in a flow regime influenced by end effects, i.e working at some point along the span of a finite lifting wing a certain profile working two-dimensionally... aerofoil lift coefficient and geometry Downwash for elliptic distribution Here Substituting this in Eqn (5. 32) wz, = z dz d G ( Z - z1) 242 Aerodynamics for Engineering Students + Writing the numerator as (z - z) z1: l =$[I -s&E-7 s d z +zl 47rs dz -sd?Zf(z-z1) Js 1 Evaluating the first integral which is standard and writing I for the second wz, T O (5. 40) =-[ 7r+z1l] 47rs Now as this is a symmetric flight... drag (b) Non-elliptic distribution 3 gives varying downwash (c) Equivalent variation for comparison purposes Finite wing theory 249 For (b): and since S”_,ritfl(z)= 0 in Eqn (5. 53) (5. 55) Comparing Eqns (5. 54) and (5. 55) and since fl(z) is an explicit function of z, J_:(fl(Z))2dZ 0 > since (f1(z))2 is always positive whatever the sign of fl(z) Hence greater than D v ( ~ ) DV(b) is always 5. 6 Determination... solved give A1 = 0.020 329, A3 = -0 .000 955 ; A5 = 0.001 029; A7 = -0 .000 2766 Thus r = 4sY{0.020 329 sin 8 - 0.000 955 sin 38 + 0.001 029 sin 50 - 0.000 2766 sin 78) and substituting the values of 8 taken above, the circulation takes the values of: 4s rm2s-I Firo 1 0 0 0.924 16. 85 0.343 0.707 28.7 0.383 0.383 40.2 0.82 0 49.2 1.o 254 Aerodynamics for Engineering Students As a comparison, the equivalent... v', w') (5. 70) streamlines in / transverse plane Fig 5. 39 Approximate flow in the transverse plane of a slender delta wing from two-dimensional potential flow theory 262 Aerodynamics for Engineering Students Let the velocity potential associated with the perturbation velocities be denoted by 9' For slender-wing theory cp' corresponds to the two-dimensional potential flow around the spanwise wing-section,... momentum in the flow is the same for both Thus for (a) L 0; 1:mwodz (5. 51) and for (b) (5. 52) where riz is a representative mass flow meeting unit span Since L is the same on each wing l ) l f l ( z ) d z= 0 (5. 53) Now the energy transfer or rate of change of the kinetic energy of the representative mass flows is the induced drag (or vortex drag) For (a): (5. 54) Fig 5 3 (a) Elliptic distribution gives . Eqn (5. 32) wz, = z dz dG(Z - z1) 242 Aerodynamics for Engineering Students Writing the numerator as (z - zl) + z1: 1 =$[I sdz +zl Js dz 47rs -s&E-7 -sd?Zf(z-z1). 5A: sin2 58 + .)dB 7T 7r =-[ A;+3A:+5A:+ ] =2cnAi 2 This gives 1 2 2 DV = 4pV2?ZcnAi = C,-pV2S whence From Eqn (5. 47) CDv = .rr(AR) (5. 49) (5. 50) 248 Aerodynamics for. for comparison purposes Finite wing theory 249 For (b): and since S”_,ritfl(z) = 0 in Eqn (5. 53) (5. 55) Comparing Eqns (5. 54) and (5. 55) and since fl(z) is an explicit function of z,