Open Journal of Fl ui d Dyn a mi cs, 2011, 1, 12-16
doi:10.4236/ojfd.2011.11002 Published Online December 2011 (http://www.SciRP.org/journal/ojfd)
Deriving the Kutta-Joukowsky Equation and Some of Its
Generalizations Using Momentum Balances
David H. Wood
Department of Mechanical and Manufacturing Engineering, Schulich School of Engineering, University of Calgary,
Calgary, Canada
Received November 18, 2011; revised December 10, 2011; accepted December 19, 2011
Abstract
Momentum balances are used to derive the Kutta-Joukowsky equation for an infinite cascade of aerofoils and
an isolated aerofoil. These derivations are simpler than those based on the Blasius theorem or more complex
unsteady control volumes, and show the close relationship between a single aerofoil and an infinite cascade.
The modification of lift due to the presence of another lifting body is similarly derived for a wing in ground
effect, a biplane, and tandem aerofoils. The results are identical to those derived from the vector form of the
Kutta-Joukowsky equation.
Keywords: Lift, Vorticity, Kutta-Joukowsky Equation, Aerofoils, Cascades, Biplane, Ground Effect,
Tandem Aerofoils
1. Introduction
The Kutta-Joukowsky (KJ) equation can be viewed as
the answer to the question: what is the simplest possible
singularity representation of a lifting body in an inviscid
fluid flow? It is fundamental to aerofoil theory and sub-
sequent developments in turbomachinery, wind turbines,
and propellers. The purpose of this note is to provide a
derivation of the equation using simple techniques of
conservation of momentum and the Reynolds transport
theorem, along with a basic knowledge of singularities
and circulation. The demonstration includes isolated bod-
ies, infinite cascades with application to rotating fluid
machines, and pairs of identical or mirror-image bodies
modeling wings in ground effect and biplanes.
In the educational literature there are three common
developments of the KJ equation for an isolated aerofoil:
1) the demonstration of its validity for a specific body,
often a rotating circular cylinder, followed by an un-
proved statement of its generality, e.g. White [1],
2) the derivation using the Blasius theorem combined
with residue calculus and complex variables, e.g. Panton
[2], and
3) the moving and expanding control volume method
of Batchelor [3], which requires a thorough knowledge
of the unsteady Bernoulli equation and careful consid-
eration of the decay of induced velocities at large dis-
tances.
2. Cascades and Isolated Aerofoils
The following demonstrations are considerably simpler
than 2 or 3. Consider an infinite cascade of identical bod-
ies—usually aerofoils—spaced distance s apart along the
y-axis in Figure 1. Only four bodies are shown. The un-
disturbed velocity of the incompressible fluid is U0. For
simplicity, one body is located at the origin surrounded
by a rectangular control volume (CV) with horizontal
faces at 2ys

. The vertical faces are equidistant
from the y-axis: the actual distance is not important. The
faces are labeled in clockwise order from the upstream
one. Symmetry requires that for faces 2 and 4:
· the pressures are equal at the same x,
· there is no net efflux of x- or y-direction momentum,
and
· the contribution to, the circulation around the con-
tour, will cancel.
Γ is positive in the clockwise direction. Only the flow
through faces 1 and 3 contributes to the momentum bal-
ance. The x-velocity at any point in the flow is U0 + u
where the latter is due to the singularities, as is the verti-
cal velocity, v.
Applying the Reynolds transport theorem to the CV
gives for the vertical force on the body, Fy:

13 D. H. WOOD
 
22
033 011
22
dd
ss
yss
F
Uuvy Uuvy 

 

(1)
where the subscripts on u and v denote the face, or

2
0113
2d
s
ys3
F
Uuvuv 

 
y (2)
Similarly, the x-direction force, Fx, is found from
 

22
2
13 03
22
22
01
2
dd
d
ss
xss
s
s
F
PP yUuy
Uu y




 


 (3)
where P is the pressure which can be removed by as-
suming that the Bernoulli constant, C, is the same for all
streamlines in the flow1:


22
0
1
2
PCU uv   

(4)
Equation (3) is rewritten as


222 22
13013 31
2
12
2
s
xsd
F
uuUuuvv y 


 (5)
The first term in Equation (2) makes it necessary to
represent a lifting body by a vortex of strength Γ. This
representation is now shown to be sufficient as (2) and (5)
are fully satisfied. If all the bodies in the cascade are
replaced by vortices of strength Γ, u is an even function
of y and v is an odd function. Thus uv is odd and the in-
tegral in (2) identically zero. Equi-spacing of the CV
faces 1 and 3 about the y-axis requires



13
uy uy
and so the integrand in (5) is zero for
any y. Thus Equations (3) and (5) reduce to
 
13
–vy vy
0
x
F, 0y
F
U


 (6a)
which is the simplest form of the KJ equation. Note that
the forces are independent of the spacing s. The vector
form is

FUΓ (6b)
for a straight line vortex with no internal structure, e.g.
Section 11.4 of Saffman [4]. It will be shown that results
of the momentum balances can be interpreted in terms of
the general from (6b) by appropriately altering the mag-
nitude of the vector velocity, U, from U0.
Figure 1 for a cascade can be replaced by Figure 2 for
an isolated body. This CV extends to ∞ and it is as-
sumed that no x- or y-direction momentum enters or
leaves the horizontal faces. The contribution to the cir-
culation on faces 1 and 3 induced by all the vortices rep-
Figure 1. Control volume for cascade of equi-spaced iden-
tical bodies.
Figure 2. Control volume for an isolated body.
resenting the bodies in Figure 1 is the same as that in-
duced by the single vortex over the infinite faces in Fig-
ure 2. Equations (2) and (5) are unaltered by the change
in CV except that 2
s

are replaced by ∞ and the
argument leading to Equation (6) is the same. This estab-
lishes the KJ equation for an isolated body.
1This is rarely the case for cascades that model fluid machines; large
flow deflections can result in much larger (or smaller) exiting y-direc-
tion velocity than the entering one. Since the x-direction velocity is
constrained by conservation of mass, the pressure and the Bernoulli
constant will change. It may be useful to distinguish between cascades
of blades with these changes and cascades of aerofoils, where they do
not.
3. Aerofoil in Ground Effect and Biplanes
A single lifting body and an infinite cascade of identical

D. H. WOOD
14
bodies are the simplest arrangements in which to estab-
lish the KJ equation because there is no induced velocity
on any of the bodies. For a finite “stack” of lifting bodies,
the analysis becomes considerably more complex, e.g.
Crowdy [5], and momentum balances quickly lose their
attraction. However, for two lifting bodies, there is bene-
fit in extending the present analysis. The geometry and
control volume are shown in Figure 3 for two cases of
vertical separation: in the first the body at –h is a mirror
image of that at h and so has opposite circulation. This is
common model for a lifting body in ground effect, GE.
In the second, the bodies are identical, modeling a bi-
plane, symbolised as B. This case is treated in Chapter
13 of Glauert [6] who gives the lift in terms of elliptic
functions.
The rectangular control volume shown in Figure 3 is
used for both GE and B. It has height Y, and half width
2
X
and it will be necessary to examine the effect of
letting both X and Y tend to infinity. At a distance from
the bodies large compared to h, the biplane acts as a sin-
gle vortex of strength 2Γ, and the GE bodies as having
no circulation. Thus the interaction between the two
bodies must be only an exchange of lift for the biplane
and a mutual increase or decrease in the magnitude of lift
for GE.
For the CV in Figure 3, Equation (2) becomes

01133
0d
Y
y
F
Uuvuv   
y
Figure 3. Aerofoil in ground effect.
Figure 4. Aerofoils comprising a biplane. Control volume as
in Figure 3.

22222
2424
2
1d
2
X
Xuuvv x



 (7)
and the only immediate simplification is that v4 = 0 for
GE and u4 = 0 for B. Similarly, (5) becomes


 
22 22
13013 13
0
2
04 24422
2
12d
2
d
Y
x
X
X
F
uuUuuvv y
Uv vuv uvx







(8)
where




13
uy uy, and v2 and v4 must be even in x. It
is now shown that the first integral in (7) becomes negli-
gible as X, Y  ∞ and the integrand of second reduces to
2
4
u

for GE and for B as X, Y  ∞. The unchanged
first term on the right of (7) requires the continued use of
vortices to represent the bodies. It is trivially easy to
show that the velocities at any point
2
4
v

,

x
y in the flow
are given by
22
12
22
12
2π
11
2π
yh yh
urr
x
vrr

 





 




(9)
with , . The + sign
is for B and the – for GE. All the integrals in (7) and (8)
can be evaluated exactly. For example,

2
22
1
rx yh

2
22
2
rx yh








22
22 2
2
222
22
22
22
2
22
22
d4π
2d
XX
XX
xx
vx
xYhxYh
xx
xYhxYh


 


 




 

 2
(10)
Obviously the integral becomes negligible as X, Y 
∞ for GE and it is easy to show that it does also for B.
This is because, for example,




 
22
22
22
2
22
22
22
22
22
d
1dd
4
X
X
XX
XX
xx
xYhxYh
xx
x
x
Yh xYhxYh


 







 




(11)
and





22
2
2
2
2
2
2
2
2
d
tan -
2- 2
X
X
X
X
xx
xYh
axYh x
Yh xYh






(12)

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
F
Uux
F
Uv









x
(13b)
Along y = 0, Equation (9) simplifies to


4, 22
4, 22
π
π
GE
B
h
u
x
h
x
v
xh




Thus

2
,0 4 yGE
F
Uh

  (15)
and

2
,0 4 yB
F
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
F
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

D. H. WOOD
16
but they will not fix the magnitudes or the ratios of the
circulations.
6. Acknowledgements
This work is part of a program of research on wind tur-
bines and other forms of renewable energy supported by
the National Science and Engineering Research Council
and the ENMAX Corporation.
7. References
[1] F. M. White, “Fluid Mechanics,” 7th Edition, McGraw-
Hill, New York, 2011.
[2] R. L. Panton, “Incompressible Flow,” 3rd Edition, John
Wiley & Sons, New York, 2005.
[3] G. K. Batchelor, “An Introduction to Fluid Dynamics,”
Cambridge University Press, Cambridge, 1967.
[4] P. G. Saffman, “Vortex Dynamics,” Cambridge Univer-
sity Press, Cambridge, 1992.
[5] D. Crowdy, “Calculating the Lift on a Finite Stack of
Cylindrical Aerofoils,” Proceedings of the Royal Society
A, Vol. 462, 2006, pp. 1387-1407.
doi:10.1098/rspa.2005.1631
[6] H. Glauert, “The Elements of Aerofoil and Airscrew The-
ory,” 2nd Edition, Cambridge University Press, Cam-
bridge, 1947.
[7] J. Katz and A. Plotkin, “Low Speed Aerodynamics,” 2nd
Edition, Cambridge University Press, Cambridge, 2001.
[8] B. Thwaites, “Incompressible Aerodynamics,” Clarendon
Press, Oxford, 1960.