## 10

Fig. 3.39 Calculation of pressure coefficient for NACA 0024 aerofoil What are the advantages of the panel method compared to other numerical methods such as finite differences and finite elements Both of the latter are field methods that require that the whole of the flow field be discretized. The panel method, on the other hand, only requires the discretization of the body surface - the boundary of the flow field. The dimensions of the solution are thereby reduced by one compared to the field...

## Eyk

Cl oo (a < *o) - e from Eqn (5.57) r oc cax (a - ao) - e -r a three-dimensional lift slope Fig. 5.35 Three different wing planforms with the same elliptic chord distribution Fig. 5.35 Three different wing planforms with the same elliptic chord distribution for an elliptic chord distribution, so that on substituting in Eqn (5.63) and rearranging This equation gives the lift-curve slope a for a given aspect ratio (AR) in terms of the two-dimensional slope of the aerofoil section used in the...

## 1

As and T vary elliptically so must c, since on the right-hand side C pV2 is a constant along the span. Thus and the general inference emerges that for a spanwise elliptic distribution an untwisted wing will have an elliptic chord distribution, though the planform may not be a true ellipse, e.g. the one-third chord line may be straight, whereas for a true ellipse, the mid-chord line would be straight (see Fig. 5.35). It should be noted that an elliptic spanwise variation can be produced by...

## Lwb

Fig. 5.33 (a) Elliptic distribution gives constant downwash and minimum drag, (b) Non-elliptic distribution gives varying downwash. (c) Equivalent variation for comparison purposes and since J twfi(z) 0 in Eqn (5.53) D (b) oc 5' j Comparing Eqns (5.54) and (5.55) DV(b) Dv it)+-m and since fi(z) is an explicit function of z, since (f,(z))2 is always positive whatever the sign of fi(z). Hence DV(b) is always greater than Dy ). 5.6 Determination of the load distribution on a given wing This is the...

## Info

Fig. 5.32 Loading make-up by selected sine series harmonics are symmetrical. Even harmonics, on the other hand, return to zero again at 7t 2 where in addition there is always a change in sign. For any asymmetry in the loading one or more even harmonics are necessary. With the number and magnitude of harmonics effectively giving all possibilities the general spanwise loading can be expressed as It should be noted that since pVT the spanwise lift distribution can be expressed as The aerodynamic...

## Kof

Fig. 5.29 Circulation superimposed on forward wind velocity and downwash to give lift and vortex drag (induced drag) respectively the end-effects are more dominant. It seems therefore that a wing that is large in the spanwise dimension, i.e. large aspect ratio, is a better wing - nearer the ideal - than a short span wing of the same section. It would thus appear that a wing of large aspect ratio will have better aerodynamic characteristics than one of the same section with a lower aspect ratio....

## JC

The induced drag per unit span (dv), or the induced drag grading, again by the Kutta-Zhukovsky theorem is and by similar integration over the span This expression for Dv shows conclusively that if w is zero all along the span then Dv 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,...

## A

Fig. 5.23 Modelling the lifting effect by a distribution of horseshoe vortex elements 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...

## 4

Where x (z) denotes the leading edge of the wing. In general, Eqn (5.23) is fairly cumbersome and nowadays modern computational techniques like the panel method (see Section 5.8) are used. In the special case of Fig. 5.20 Modelling the displacement effect by a distribution of sources 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...

## B

To find the velocity at P due to the length AB the sum of induced velocities due to all such elements is required. Before integrating, however, all the variables must be quoted in terms of a single variable. A convenient variable is (see Fig. 5.10) and the limits of the integration are since < j> passes through zero when integrating from A to B. sin 0 cos < p, r2 h2 sec2 < p dr d(h tan < j> ) h sec2 < > d < j> The integration of Eqn (5.8) is thus - rCOS diA sin - p + sin - a...

## 778 Mixed boundary layer flow on a flat plate with zero pressure gradient

Figure 7.26 indicates the symbols employed to denote the various physical dimensions used. At the leading edge, a laminar layer will begin to develop, thickening with distance downstream, until transition to turbulence occurs at some Reynolds number Ret Uxxt v. At transition the thickness increases suddenly from < 5l, in the laminar layer to & rt in the turbulent layer, and the latter then continues to grow as if it had started from some point on the surface distant xjt ahead of...

## 925 Activity factor

The activity factor is a measure of the power-absorbing capacity of the airscrew, which, for optimum performance, must be accurately matched to the power produced by the engine. Consider an airscrew of diameter D rotating at n with zero forward speed, and consider in particular an element of the blade at a radius of r, the chord of the element being c. The airscrew will, in general, produce a thrust and therefore there will be a finite speed of flow through the disc. Let this inflow be ignored,...

## 93 Airscrew pitch

By analogy with screw threads, the pitch of an airscrew is the advance per revolution. This definition, as it stands, is of little use for airscrews. Consider two extreme cases. If the airscrew is turning at, say, 2000 rpm while the aircraft is stationary, the advance per revolution is zero. If, on the other hand, the aircraft is gliding with the engine stopped the advance per revolution is infinite. Thus the pitch of an airscrew can take any value and is therefore useless as a term describing...

## 87 Reduction of induced drag

Aspects of this topic have already been discussed in Chapter 5. There it was shown that, in accordance with the classic wing theory, induced drag falls as the aspect ratio of the wing is increased, ft was also shown that, for a given aspect ratio, elliptic-shaped wings (strictly, wings with elliptic wing loading) have the lowest induced drag. Over the past 25 years the winglet has been developed as a device for reducing induced drag without increasing the aspect ratio. A typical example is...

## 852 Compliant walls artificial dolphin skins

It is widely thought that some dolphin species possess an extraordinary laminar-flow capability. Certainly mankind has long admired the swimming skills of these fleet creatures. Scientific interest in dolphin hydrodynamics dates back at least as far as 1936 when Gray published his analysis of dolphin energetics. It is widely accepted that species like the bottle-nosed dolphin (Tursiops truncatus) can maintain a sustained swimming speed of up to 9 m s. Gray followed the usual practice of marine...

## 851 Laminar flow control by boundarylayer suction

Distributed suction acts in two main ways to suppress laminar-turbulent transition. First, it reduces the boundary-layer thickness. Recall from Section 7.9 that for a fixed pressure gradient a critical Reynolds number based on boundary-layer thickness must be reached before transition is possible. Second, it creates a much fuller velocity profile within the boundary layer, somewhat similar to the effect of a favourable pressure gradient. This makes the boundary layer much more stable with...

## 843 Other methods of separation control

Passive flow control through the generation of streamwise vortices is frequently used on aircraft and other applications. Some of the devices commonly in use are depicted * See the recent review by D. Greenblatt and I. Wygnanski (2000) 'The control of flow separation by periodic excitation', Prog, in Aerospace Sciences, 36, 487-545. Seifert and L.G. Pack (1999) 'Oscillatory control of separation at high Reynolds number', AIAA J., 37(9), 1062-1071. in Fig. 8.32. Figure 8.32a shows a row of...

## 841 Boundarylayer suction

The basic principle was demonstrated experimentally in Prandtl's paper that introduced the boundary-layer concept to the world.* He showed that the suction through a slot could be used to prevent flow separation from the surface of a cylinder. The basic principle is illustrated in Fig. 8.22. The layer of low-energy ('tired') air near the surface approaching the separation point is removed through a suction slot. * A more complete recent account is to be found in M. Gad-el-Hak (2000) Flow...

## 84 Boundary layer control for the prevention of separation

Many of the widely used techniques have already been described in Section 8.3. But there are various other methods of flow-separation control that are used on aircraft and in other engineering applications. These are described here.* Some of the devices used are active, i.e. they require the expenditure of additional power from the propulsion units others are passive and require no additional power. As a general rule, however, the passive devices usually lead to increased drag at cruise when...

## 831 The slat effect

To appreciate qualitatively the effect of the upstream element (e.g. the slat) on the immediate downstream element (e.g. the main aerofoil) the former can be modelled by a vortex. The effect is illustrated in Fig. 8.11. When one considers the component of the velocity induced by the vortex in the direction of the local tangent to the aerofoil contour in the vicinity of the leading edge (see inset in Fig. 8.11), it can be seen that the slat (vortex) acts to reduce the velocity along the edge of...

## 8

Fig. 8.4 Upper-wing-surface pressure distributions with laminar rooftop Fig. 8.4 Upper-wing-surface pressure distributions with laminar rooftop other hand, only modest maximum suction pressures are permissible before sonic conditions are reached. In this case, therefore, the pressure distribution is very flat. An example of the practical application of these ideas for low flight speeds is illustrated schematically in Fig. 8.5. This shows a Liebeck* aerofoil. This sort of aerofoil was used as a...

## 24 Twodimensional flow

Consider flow in two dimensions only. The flow is the same as that between two planes set parallel and a little distance apart. The fluid can then flow in any direction between and parallel to the planes but not at right angles to them. This means that in the subsequent mathematics there are only two space variables, x and y in Cartesian (or rectangular) coordinates or r and 9 in polar coordinates. For convenience, a unit length of the flow field is assumed in the z direction perpendicular to x...

## 0228

When the component of the free-stream velocity perpendicular to the leading edge is greater than the local speed of sound the wing is said to have a supersonic leading edge. In this case, as illustrated in Fig. 6.56, there is two-dimensional supersonic flow over much of the wing. This flow can be calculated using supersonic aerofoil theory. For the rectangular wing shown in Fig. 6.56 the presence of a wing-tip can only be communicated within the Mach cone apex which is located at the wing-tip....

## 35 Computational panel methods

In Section 3.3.7, it was shown how the two-dimensional potential flow around an oval-shaped contour, the Rankine oval, could be generated by the superposition of a source and sink on the ,v axis and a uniform flow. An analogous three-dimensional flow can also be generated around a Rankine body - see Section 3.4.4 above - by using a point source and sink. Thus it can be demonstrated that the potential flow around certain bodies can be modelled by placing sources and sinks in the interior of the...

## 842 Control by tangential blowing

Since flow separation is due to the complete loss of kinetic energy in the boundary layer immediately adjacent to the wall, another method of preventing it is to re-energize the 'tired' air by blowing a thin, high-speed jet into it. This method is often used with trailing-edge flaps (Fig. 8.25). To obtain reasonable results with this (a) Normal coanda flow (b) Jet break-away Fig. 8.26 The Coanda effect - the flow of a jet around a circular cylinder Source Based on Fig. 1 of P.W. Carpenter and...

## 7133 Shockwaveboundarylayer interaction in supersonic flow

One of the main differences between subsonic and supersonic flows, as far as boundary-layer behaviour is concerned, is that the pressure gradient along the flow is of opposite sign with respect to cross-sectional area change. Thus in a converging supersonic flow the pressure rises and in a diverging flow the pressure falls in the stream direction (see Section 6.2). As a result the pressure gradient at a convex corner is negative and the boundary layer will generally negotiate the corner without...

## A solution of integrals of the type of Glauerts integral

o cos 0 cos In Chapters 4 and 5 much use is made of the integral This may be proved, by contour integration, as follows. In the complex plane, integrate the function with respect to z round the circle of unit radius centred at the origin. On this circle z eltf and therefore z2 - 2z cos 0i + 1 7_lr tm - 2eie cos + 1 which, cancelling eifl from numerator and denominator, putting e'e cos 0 + i sin and using De Moivre's theorem, reduces to * This section may be omitted at a first reading. The...

## Aircraft Downwash

5.2.3 Variation of velocity in vortex flow To confirm how the velocity outside a vortex core varies with distance from the centre consider an element in a thin shell of air (Fig. 5.12). Here, flow conditions depend only on the distance from the centre and are constant all round the vortex at any given radius. The small element, which subtends the angle 86 at the centre, is circulating round the centre in steady motion under the influence of the force due to the radial pressure gradient....

## 933Effect of geometric pitch on airscrew performance

Consider two airscrews differing only in the helix angles of the blades and let the blade sections at, say, 70 radius be as drawn in Fig. 9.5. That of Fig. 9.5a has a fine pitch, whereas that of Fig. 9.5b has a coarse pitch. When the aircraft is at rest, e.g. at the start of the take-off run, the air velocity relative to the blade section is the resultant Vr of the velocity due to rotation, 2irnr, and the inflow velocity, Vm. The blade section of the fine-pitch airscrew is seen to be working at...

## 836 Gurney flaps

As well as being a great racing-car driver, Dan Gurney is also well-known for his technical innovations. His most widely emulated innovation is probably the now-obligatory practice of winning drivers spraying their supporters with champagne from vigorously shaken bottles. But it is for the Gurney flap that he is known in aerodynamics. This is a deceptively simple device consisting merely of a small plate fixed to and perpendicular to the trailing edge of a wing. It can be seen attached to the...

## Twodimensional wing theory

Here the basic fluid mechanics outlined previously is applied to the analysis of the flow about a lifting wing section. It is explained that potential flow theories of themselves offer little further scope for this problem unless modified to simulate certain effects of real flows. The result is a powerful but elementary aerofoil theory capable of wide exploitation. This is derived in the general form and applied to a number of discrete aeronautical situations, including the flapped aerofoil and...

## Finite wing theory

Whatever the operating requirements of an aeroplane may be in terms of speed endurance, pay-load and so on, a critical stage in its eventual operation is in the low-speed flight regime, and this must be accommodated in the overall design process. The fact that low-speed flight was the classic flight regime has meant that over the years a vast array of empirical data has been accumulated from flight and other tests, and a range of theories and hypotheses set up to explain and extend these...

## 82 Maximizing lift for singleelement aerofoils

This section addresses the question of how to choose the pressure distribution, particularly that on the upper wing surface, to maximize the lift. Even when a completely satisfactory answer is found to this rather difficult question, it still remains to determine the appropriate shape the aerofoil should assume in order to produce the specified pressure distribution. This second step in the process is the so called inverse problem of aerofoil design. It is very much more demanding than the...

## 741 Separation bubbles

On many aerofoils with relatively large upper-surface curvatures, high local curvature over the forward part of the chord may initiate a laminar separation when the aerofoil is at quite a moderate angle of incidence Fig. 7.14 . Small disturbances grow much more readily and at low Reynolds numbers in separated, as compared to attached, boundary layers. Consequently, the separated laminar boundary layer may well undergo transition to turbulence with characteristic rapid thickening. This rapid...

## 837 Movable flaps artificial bird feathers

This concept is illustrated in Fig. 8.20. Superficially it appears similar to the Gurney flap. However, the mode of operation is quite different. And, in any case, for positive high lift the Gurney flap would be attached to the trailing edge pointing downwards. The basic idea here is that at high angles of attack when flow separation starts to occur near the trailing edge, the associated reversed flow causes the movable flap to be raised. This then acts as a barrier to the further migration of...

## 835Use of multielement aerofoils on racing cars

In the 1960s and early 1970s several catastrophic accidents occurred in which racing cars became airborne. In some cases aerodynamic interference from nearby competing vehicles was undoubtedly a factor. Nevertheless, these accidents are a grim reminder of what can happen to a racing car if insufficient aerodynamic downforce is generated. Modern Grand Prix cars generate their prodigious aerodynamic downforces from two main sources, namely 'ground effect' and inverted wings. Under current...