15

D. H. WOOD

(12) tends to zero as X, Y ∞. Using (7) and results

like (10) to (12) as X, Y ∞ shows that

,,

0

xGE xB

FF

(13a)

12

,0 4,

2

12

,0 4,

2

d

d

yGE GE

yB B

Uux

Uv

x

(13b)

Along y = 0, Equation (9) simplifies to

4, 22

4, 22

π

π

GE

B

h

u

h

x

v

xh

Thus

2

,0 4

yGE

Uh

(15)

and

2

,0 4

yB

Uh

(16)

Equation (15) is well known: it is, for example, Equa-

tion (16) of Katz & Plotkin [7] derived from a

lumped-vortex model, and is equivalent to their (6.113)

obtained from the Blasius theorem. (15) can be inter-

preted as the modification of (6a) due to the induced ve-

locity of the image vortex,

–4πh on the “real”

vortex in the direction opposite to U0. This causes a lift

reduction of

24πh

according to (6b).

Equations (4) and (13b) show that the second term in

(15) is due to the non-zero gauge pressure acting on the

ground plane. The total force (per unit length) on the

fluid ρU0Γ is shared between the lifting body and the

surface pressure. Other analyses of ground effect that

include information about the body geometry usually

show an increase in lift as the ground is approached but

only when h is comparable to the chord length c, e.g.

Thwaites [8, p. 527ff] and Katz & Plotkin [7, Section

12.3]. Assuming the usual relation between Γ. and aero-

foil lift coefficient, Cl, gives the ratio of the GE lift to the

aerofoil lift from (15) as

18π

l

cC h (17)

At Cl 1.0 nand 0.5hc, the reduction is only 8%

and can be easily overwhelmed by other effects.

As far as the author can tell, Equation (16) for the lift

on the upper body of a biplane, is new but is easily es-

tablished from (6b). The lower vortex induces a velocity

of

4πh on the upper vortex in the direction of U0

which increases the lift by

24πh

As with the for-

mulation of Crowdy [5], the lift is increased and that on

the lower body reduced by the same amount. Equation

(16) shows the difference in lift increases with Γ, as

found by Crowdy [5], but, in contrast, the difference here

is zero when Γ = 0.

4. Tandem Aerofoils

If two identical lifting bodies are placed at d on the x-

axis then the analysis of the last Section is easily modi-

fied to show that proximity does not alter the lift but

causes a force

24π

x

d

(18)

on the forward aerofoil and an equal and opposite force

on the rear one. The rear vortex induces a vertical veloc-

ity of

4πd on the front vortex which now produces

a thrust (opposite of drag) of

24πd

). The argu-

ment is readily reversed to show that the rear vortex ex-

periences an equal and opposite force.

5. Summary and Conclusions

Momentum balances provide a straightforward proof of

the usual form of the Kutta-Joukowsky Equation, (6a),

for the fluid forces acting on isolated bodies and infinite

cascades of equi-spaced identical bodies. The analysis

implies—but does not assume—that each body is repre-

sented simply as a vortex. These two geometries are

unique in that there is no induced velocity on any of the

bodies. Deriving the forces using momentum balances

shows the close link between cascade flow and that over

a single aerofoil.

For pairs of identical or mirror image bodies the

analysis becomes more complex and it is likely that mo-

mentum balances would become too cumbersome for

larger numbers. The forces acting on aerofoils in ground

effect, biplanes, and tandem aerofoils as determined from

momentum balances are in agreement with the more

general form of the Kutta-Joukowsky Equation, (6b),

which includes the effect of the velocity induced by one

body on the other. This idea is easily generalised, so that,

for example, three aerofoils spaced equally apart by dis-

tance h in the vertical direction will experience the fol-

lowing vertical forces:

2

012πUh

on the up-

per, ρU0Γ on the middle, and

2

0–12πUh

on

the lower.

In practice, the determination of the forces on multiple

bodies can be more complex with differences in circula-

tion for geometrically identical bodies, as opposed to the

assumption of equal circulation made here. For example,

Section 5.5 of Katz & Plotkin [7] shows that a lumped

vortex model of tandem aerofoils requires the upstream

aerofoil to have a greater circulation. Momentum bal-

ances will still give the forces if the circulations differ

Copyright © 2011 SciRes. OJFD