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. Considering unit axial length, the inwards force due to the pressures is:

which reduces to 6p(r — ^ 6r)S6. Ignoring i 6r in comparison with r, this becomes r dp 86. The volume of umt length of the element is r 6r 66 and therefore its mass is pr6r86. Its centripetal acceleration is (velocity)2/radius, and the force required to produce this acceleration is:

(velocity)2 q2 , mass--r—— = pr8r66— = per 6r66

radius r

Equating this to the force produced by the pressure gradient leads to r 6p = pq2 8r since 66 ^ 0

Now, since the flow outside the vortex core is assumed to be inviscid, Bernoulli's equation for incompressible flow can be used to give, in this case,

Expanding the term in q + 6q, ignoring terms such as (6q)2 as small, and cancelling, leads to:

Substituting this value for 6p in Eqn (5.13) gives pq2 6r + pqr 6q = 0 which when divided by pq becomes q 6r + r 6q = 0 But the left-hand side of this equation is 6{qr). Thus

qr = constant

This shows that, in the inviscid flow round a vortex core, the velocity is inversely proportional to the radius (see also Section 3.3.2).

When the core is small, or assumed concentrated on a line axis, it is apparent from Eqn (5.15) that when r is small q can be very large. However, within the core the air behaves as though it were a solid cylinder and rotates at a uniform angular velocity. Figure 5.13 shows the variation of velocity with radius for a typical vortex.

The solid line represents the idealized case, but in reality the boundary is not so distinct, and the velocity peak is rounded off, after the style of the dotted lines.

The core

Fig. 5.13 Velocity distribution in a real vortex with a core

The core

Fig. 5.13 Velocity distribution in a real vortex with a core

5.3 The simplified horseshoe vortex

A simplified system may replace the complete vortex system of a wing when considering the influence of the lifting system on distant points in the flow. Many such problems do exist and simple solutions, although not all exact, can be readily obtained using the suggested simplification. This necessitates replacing the wing by a single bound spanwise vortex of constant strength that is turned through 90° at each end to form the trailing vortices that extend effectively to infinity behind the wing. The general vortex system and its simplified equivalent must have two things in common:

(i) each must provide the same total lift

(ii) each must have the same value of circulation about the trailing vortices and hence the same circulation at mid-span.

These equalities provide for the complete definition of the simplified system.

The spanwise distributions created for the general vortex system and its simplified equivalent are shown in Fig. 5.14. Both have the same mid-span circulation To that is now constant along part of the span of the simplified equivalent case. For equivalence in area under the curve, which is proportional to the total lift, the span length of the single vortex must be less than that of the wing.

(a ) Normal loading

( b ) Equivalent simplified loading

(a ) Normal loading

( b ) Equivalent simplified loading

Thus

To2/ = area under general distribution = ^

Hence

V _ total lift s IspVTq

2V is the distance between the trailing vortex core centres. From Eqn (5.47a) (see page 246) it follows that and substituting also

For the general case then:

S1 TV

For the simpler elliptic distribution (see Section 5.5.3 below):

In the absence of other information it is usual to assume that the separation of the trailing vortices is given by the elliptic case.

5.3.1 Formation flying effects

Aircraft flying in close proximity experience mutual interference effects and good estimates of these influences are obtained by replacing each aircraft in the formation by its equivalent simplified horseshoe vortex.

Consider the problem shown in Fig. 5.15 where three identical aircraft are flying in a vee formation at a forward speed V in the same horizontal plane. The total mutual interference is the sum of (i) that of the followers on the leader (1), (ii) that of the leader and follower (2) on (3), and (iii) that of leader and follower (3) on (2). (ii) and (iii) are identical.

(i) The leader is flying in a flow regime that has additional vertical flow components induced by the following vortices. Upward components appear from the bound vortices a2C2, a3C3, trailing vortices C2d2, a3b3 and downward

components from the trailing vortices a2b2 and C3d3- The net result is an upwash on the leader, (ii) These wings have additional influences to their own trails due to the leader and the other follower. Bound vortex aiCi and trailing vortices aibi, a2b2 produce downwashes. Again the net influence is an upwash.

From these simple considerations it appears that each aircraft is flying in a regime in which upward components are induced by the presence of the others. The upwash components reduce the downward velocities induced by the aircraft's own trail and hence its trailing vortex drag. Because of the reduction in drag, less power is required to maintain the forward velocity and the well-known operational fact emerges that each aircraft of a formation has a better performance than when flying singly. In most problems it is usual to assume that the wings have an elliptic distribution, and that the influence calculated for mid-span position is typical of the whole wing span. Also any curvature of the trails is neglected and the special forms of the Biot-Savart law (Section 5.2.2) are used unreservedly.

5.3.2 Influence of the downwash on the tailplane

On most aircraft the tailplane is between the trailing vortices springing from the mainplanes ahead and the flow around it is considerably influenced by these trails. Forces on aerofoils are proportional to the square of the velocity and the angle of incidence. Small velocity changes, therefore, have negligible effect unless they alter the incidence of the aerofoil, when they then have a significant effect on the force on the aerofoil.

Tailplanes work at incidences that are altered appreciably by the tilting of the relative wind due to the large downward induced velocity components. Each particular aircraft configuration will have its own geometry. The solution of a particular problem will be given here to show the method.

Example 5.1 Let the tailplane of an aeroplane be at distance x behind the wing centre of pressure and in the plane of the vortex trail (Fig. 5.16). Assuming elliptic distribution, the semi-span of the bound vortex is given by Eqn (5.18) as

Aircraft Downwash
rH*

Ground level

Ground level

5.3.3 Ground effects

In this section, the influence of solid boundaries on aeroplane (or model) performance is estimated and once again the wing is replaced by the equivalent simplified horseshoe vortex.

Since this is a linear problem, the method of superposition may be used in the following way. If (Fig. 5.17b) a point vortex is placed at height h above a horizontal plane, and an equal but opposite vortex is placed at depth h below the plane, the vertical velocity component induced at any point on the plane by one of the vortices is equal and opposite to that due to the other. Thus the net vertical velocity, induced at any point on the plane, is zero. This shows that the superimposition of the image vortex is equivalent in effect to the presence of a solid boundary. In exactly the same way, the effect of a solid boundary on the horseshoe vortex can be modelled by means of an image horseshoe vortex (Fig. 5.17a). In this case, the boundary is the level ground and its influence on an aircraft h above is the same as that of the 'inverted' aircraft flying 'in formation' h below the ground level (Figs 5.17a and 5.18).

Before working out a particular problem, it is clear from the figure that the image system reduces the downwash on the wing and hence the drag and power required, as well as materially changing the downwash angle at the tail and hence the overall pitching equilibrium of the aeroplane.

Example 5.2 An aeroplane of weight W and span Is is flying horizontally near the ground at altitude h and speed V. Estimate the reduction in drag due to ground effect. If W = 22 x 104N, h = 15.2m, s = 13.7m, V = 45m s-1, calculate the reduction in Newtons.

With the notation of Fig. 5.18 the change in downwash at y along the span is Aw f where

On a strip of span by at y from the centre-line, lift I = piT0 6y and change in vortex drag

filaments. In order to satisfy Helmlioltz's second theorem (Section 5.2.1) each filament must either be part of a closed loop or form a horseshoe vortex with trailing vortex filaments running to infinity. Even with this restriction there are still infinitely many ways of arranging such vortex elements for the purposes of modelling the flow field associated with a lifting wing. For illustrative purposes consider the simple arrangement where there is a sheet of vortex filaments passing in the spanwise direction through a given wing section (Fig. 5.19). It should be noted, however, that at two, here unspecified, spanwise locations each of these filaments must be turned back to form trailing vortex filaments.

Consider the flow in the vicinity of a sheet of fluid moving irrotationally in the xy plane, Fig. 5.19. In this stylized figure the 'sheet' is seen to have a section curved in the xy plane and to be of thickness 8n, and the vorticity is represented by a number of vortex filaments normal to the xy plane. The circulation around the element of fluid having sides 8s, 8n is, by definition, Ar = (6s. 8n where ( is the vorticity of the fluid within the area 8s 8n.

Now for a sheet 8n0 and if Ç is so large that the product (6n remains finite, the sheet is termed a vortex sheet of strength k = Ç8n. The circulation around the element can now be written

An alternative way of finding the circulation around the element is to integrate the tangential flow components. Thus

Comparison of Eqns (5.21) and (5.22) shows that the local strength k of the vortex sheet is the tangential velocity jump through the sheet.

Alternatively, a flow situation in which the tangential velocity changes discontinu-ously in the normal direction may be mathematically represented by a vortex sheet of strength proportional to the velocity change.

The vortex sheet concept has important applications in wing theory.

5.4.1 The use of vortex sheets to model the lifting effects of a wing

In Section 4.3, it was shown that the flow around a thin wing could be regarded as a superimposition of a circulatory and a non-circulatory flow. In a similar fashion the same can be established for the flow around a thin wing. For a wing to be classified as thin the following must hold:

• The maximum thickness-to-chord ratio, usually located at mid-span, must be much less than unity.

• The camber lines of all wing sections must only deviate slightly from the corresponding chord-line.

• The wing may be twisted but the angles of incidence of all wing sections must remain small and the rate of change of twist must be gradual.

• The rate of change of wing taper must be gradual.

These conditions would be met for most practical wings. If they are satisfied then the velocities at any point over the wing only differ by a small amount from that of the oncoming flow.

For the thin aerofoil the non-circulatory flow corresponds to that around a symmetrical aerofoil at zero incidence. Similarly for the thin wing it corresponds to that around an untwisted wing, having the same planform shape as the actual wing, but with symmetrical sections at zero angle of incidence. Like its two-dimensional counterpart in aerofoil theory this so-called displacement (or thickness) effect makes no contribution to the lifting characteristics of the wing. The circulatory flow - the so-called lifting effect - corresponds to that around an infinitely thin, cambered and possibly twisted, plate at an angle of attack. The plate takes the same planform shape as the mid-plane of the actual wing. This circulatory part of the flow is modelled by a vortex sheet. The lifting characteristics of the wing are determined solely by this component of the flow field. Consequently, the lifting effect is of much greater practical interest than the displacement effect. Accordingly much of this chapter will be devoted to the former. First, however, the displacement effect is briefly considered.

Displacement effect

In Section 4.9, it was shown how the non-circulatory component of the flow around an aerofoil could be modelled by a distribution of sources and sinks along the chord line. Similarly, in the case of the wing, this component of the flow can be modelled by distributing sources and sinks over the entire mid-plane of the wing (Fig. 5.20). In much the same way as Eqn (4.103) was derived (referring to Fig. 5.20 for the geometric notation) it can be shown that the surface pressure coefficient at point (xi, ji) due to the thickness effect is given by

+1 0

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