Deriving the Kutta-Joukowsky Equation and Some of Its Generalizations Using Momentum Balances

doi:10.4236/ojfd.2011.11002

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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

E-mail: [email protected]

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:

D. H. WOOD

 

033 011

yss

Uuvy Uuvy





 



(1)

where the subscripts on u and v denote the face, or



0113

ys3

Uuvuv





 

y (2)

Similarly, the x-direction force, Fx, is found from

 



13 03

xss

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:





PCU uv



 



(4)

Equation (3) is rewritten as





222 22

13013 31

xsd

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







uy uy

and so the integrand in (5) is zero for

any y. Thus Equations (3) and (5) reduce to

 

–vy vy

F, 0y





 (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



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

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

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

Uuvuv



 

y

Figure 3. Aerofoil in ground effect.

Figure 4. Aerofoils comprising a biplane. Control volume as

in Figure 3.



22222

2424

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

04 24422

12d

uuUuuvv y

Uv vuv uvx











(8)

where









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



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





y in the flow

are given by

2π

yh yh

urr

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,



rx yh



rx yh

















22 2

222

d4π

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,









 

1dd

xYhxYh

Yh xYhxYh





 















 









(11)

and









tan -

2- 2

xYh

axYh x

Yh xYh











(12)

D. H. WOOD

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

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

xGE xB

FF

(13a)

,0 4,

yGE GE

yB B

Uux

















x

(13b)

Along y = 0, Equation (9) simplifies to



4, 22









Thus



,0 4

yGE





 (15)

and



,0 4





 (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

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



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



  (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:



012πUh



  on the up-

per, ρU0Γ on the middle, and





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

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.