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9.6 D’Alembert’s Paradox and the Kutta–Joukowsky Theorem

N

271

S

N

V

C

Fig. 9.12 Flow around an obstacle

Z

Z

T Nd C

.T

@C

v ˝ v/ N d D 0:

†

The first integral is just the opposite of the force F acting on S , so that the previous

relation becomes

Z

v ˝ v/ N d :

(9.55)

F D .T

†

From similar arguments it can be proved that the torque of the force acting on S

with respect to a pole O can be obtained from (5.30), so that

Z

MO D

r

.T

v ˝ v/ N d :

(9.56)

†

At this stage we recall the following theorem of the potential theory, without a proof.

Theorem 9.6. If in the region surrounding a solid C an irrotational flow of an

incompressible fluid satisfies the condition

lim v D V;

r!1

where V is a constant vector (representing the undisturbed motion at infinity) and r

is the distance of any point from an arbitrary origin, then the velocity field assumes

the asymptotic behavior

v D V C O.r

3

/:

(9.57)

In a 2D motion, (9.57) can be replaced by the following relation:

vDViC

. yi C xj/ C O.r

2 r2

2

/;

(9.58)

272

9 Fluid Mechanics

where i and j are the unit vectors of the axes Ox and Oy, i is parallel to V, and the

circulation

I

D v ds

(9.59)

refers to any closed path surrounding the solid body.

Note that in two dimensions the condition r v D 0 does not imply that

D 0, since the region surrounding the obstacle is not simply connected. However,

by applying Stokes’s theorem, it can be proved that assumes a value which is

independent of the path.

By taking into account the asymptotic behavior (9.57) and Bernoulli’s

theorem (9.24), where U D 0 in the absence of body forces, we find that

p D p0 C

1

.V 2

2

v2 / D p0 C O.r

3

/;

where p0 is the pressure at infinity. Substituting T N D Œ p0 C0.r

gives

(9.60)

3

/N into (9.55)

Z

Z

FD

V ˝ V/N d C

.p0 I

†

O.r

3

/d ;

†

so that, assuming that the arbitrary surface † is a sphere of radius R, we have

Z

FD

.p0 I

V ˝ V/

N d C O.R 1 /:

(9.61)

†

Finally, by applying Gauss’s theorem to the right-hand side, we find that

F D 0;

i.e., the irrotational flow of a perfect fluid gives zero drag on any obstacle placed in

a fluid stream (D’Alembert’s paradox).

In a 2D motion, from (9.58), we have

v2 D V 2

V

y C O.r

r2

2

/:

Substituting into Bernoulli’s theorem instead of (9.60), we get

p D p0 C

1

2

yV C O.r

r2

2

/:

(9.62)

9.7 Lift and Airfoils

273

By combining with (9.55), the force acting on the obstacle assumes the expression

Z

Z

FD

p 0 C V 2 i Nx d l p 0 j Ny d l

Vj

2 R2

Vi

2 R2

†

Z

†

.xNx C yNy / d l

Z

†

Z

.yNx

xNy / d l C

†

O.r

2

/ d l;

†

where † is now a circle of radius R surrounding the body C . Since integrals

Z

Z

Nx d l;

Ny d l;

†

†

vanish, the previous expression becomes

Z

Z

FD

V j .xNx C yNy / d l

V i .yNx

2 R2

2 R2

†

†

In addition we have .xNx C yNy / D R N D R and .yNx

Blasius formula

FD

V j

xNy / d l C O.r

2

/;

xNy / D 0, so that the

(9.63)

is obtained.

This formula shows that although a steady flow of an inviscid fluid predicts no

drag on an obstacle in the direction of the relative velocity in the unperturbed region,

it can predict a force normal to this direction. This is a result obtained independently

by W.M. Kutta in 1902 and N.E. Joukowski (sometimes referred as Zoukowskii) in

1906, known as the Kutta–Joukowski theorem. Such a force is called lift, and it is

important for understanding why an airplane can fly.

Before closing this section, we observe that our inability to predict the drag for an

inviscid fluid in the direction of relative velocity does not means we should abandon

the perfect fluid model. Viscosity plays an important role around an obstacle, but,

far from the obstacle, the motion can still be conveniently described according to

the assumption of an inviscid fluid.

9.7 Lift and Airfoils

A lifting wing having the form of an infinite cylinder of appropriate cross section

normal to rulings, is usually called an airfoil.

Airplane wings are obviously cylinders of finite length, and the effects of

this finite length have an important role in the theory of lift. Nevertheless, even

considering an infinitely long cylinder, essential aspects of lift can be identified.

274

9 Fluid Mechanics

1

S

A

d

2

Fig. 9.13 Flow around an airfoil

First, we must justify a nonvanishing circulation around a wing. To do this,

consider a wing L of section S and the fluid motion around it, described by the

complex potential (9.51) at time t0 . Let be any material curve surrounding the

wing (see Fig. 9.13).

In the region R2 S , the motion is assumed to be irrotational with the exception

of a small portion [ d , near the wing, whose presence is justified by the

following arguments: viscosity acts in a boundary layer just around the wing

and it produces a vortex of area d at the point A. This vortex grows away from

the wing and disappears into the fluid mass, but it is continuously generated. The

material curve is assumed to be large enough to contain S at any instant t . Since

in the region R2 S [ [ d , which is not a simply connected region, the motion is

irrotational, Thomson’s theorem implies that the circulation on is zero at any time.

Furthermore, by applying Stokes’s theorem to the region bounded by the oriented

curves , 1 , and 2 , we see that the circulation on 1 must be equal and opposite to

the circulation on 2 . At this stage, it becomes useful to transform the flow field we

are dealing with into another flow field which is easier to determine. For this reason,

we recall the definition of conformal mapping together with some other properties.

If a complex variable w is an analytic function of z, i.e., w D ˆ.z/, then there is a

connection between the shape of a curve in the z plane and the shape of the curve in

the w plane, as a consequence of the properties of an analytic function. In fact, the

value of the derivative is independent of the path the increments dx and dy follow

in going to zero. Since it can be proved that this transformation preserves angles and

their orientation, it is called conformal mapping.

As already shown in Sect. 9.5, the flow around a given profile can be described

through a convenient complex potential. The simplest wing profile was suggested by

Joukowski. It is obtained from the complex potential F . / of the motion around a

circular cylinder (Example 9.5) in the plane D C iÁ, by means of the conformal

mapping z D x C iy D ˆ. /.

Note that lift can occur if there is an asymmetry due either to the asymmetry of

the body or to a misalignment between the body and the approaching flow. The angle

of misalignment is called the angle of attack. The angle of attack to the cylinder

9.7 Lift and Airfoils

275

y

C

b

T'

T

'

l

U0

2l

,x

C'

Fig. 9.14 Geometry of the airfoil

of the velocity vector is denoted by ', as shown in Fig. 9.14, and the motion is

supposed to be described by the complex potential

"

#

2

a

iÂ

i'

FO . / D U0

C

be e

:

(9.64)

be iÂ e i'

It is obtained from the potential introduced in Sect. 9.5, Example 5, i.e., F . 0 / D

U0 . 0 Ca2 = 0 /, by applying the transformations 0 ! 00 D 0 e i' ! D 00 Cbe iÂ ,

corresponding to a rigid rotation ' of the axes and a rigid translation of the origin

into be iÂ .

Joukowsky’s transformation is defined as

z D ˆ. / D C

l2

;

(9.65)

which is conformal everywhere, with the exception of the origin, which is mapped to

infinity. The inverse transformation cannot be carried out globally, since from (9.65)

it follows that

p

z ˙ z2 4l 2

D

:

2

It can also be proved that the region external to the circle of radius a is in one-toone correspondence with the region external to the curve given by the image of

the circle .

As a consequence, Joukowsky’s conformal mapping allows us to define, in the

region external to ; a new complex potential

F .z/ D FO .ˆ 1 .z//

which describes the motion around the profile . Such a profile, provided that the

parameter l is conveniently selected, is the wing shape in Fig. 9.14. In particular, the

276

9 Fluid Mechanics

point T .l; 0/ in the plane . ; Á/, corresponding to the sharp trailing edge, is mapped

into T 0 .2l; 0/ in the plane .x; y/. This point is singular for the derivative d FO =d

of (9.64), since

Â

dF

dz

Ã

d FO

d

D

zD.2l;0/

d FO

d

D

d FO

d

D

!

!

!

Â

D.l;0/

d

dz

Ã

zD.2l;0/

1

.d z=d /

D.l;0/

D.l;0/

1

Á

1

D.l;0/

:

l2

2

D.l;0/

A nonvanishing circulation around the wing is justified by Joukowsky assuming that

the velocity has a finite value at any point.

In order to satisfy this condition, Joukowsky introduces the complex potential

due to the circulation (see Example 6 in Sect. 9.5), so that the velocity is zero at T 0 .

The complex potential F .z/, due to the superposition of different contributions, is

then given by

"

F .z/ D U0

be

iÂ

e

i'

#

a2

be iÂ e

C

i'

Ci

ln

2

be iÂ e

a

i'

;

(9.66)

where

and z correspond to each other through (9.65). Figure 9.15 shows

streamlines around the cylinder obtained by considering the real part of the righthand side of (9.66), i.e., the kinetic potential '. ; Á/. Figure 9.16 shows the wing

profile and the related streamlines.

The complex velocity at T 0 .2l; 0/ in the z plane is

Â

dF

dz

Ã

D

zD.2l;0/

d FO

d

!

D.l;0/

d

:

dz

1.5

1

0.5

3

2

1

0.5

1

1.5

1

2

Fig. 9.15 Flow around a cylinder

3

9.7 Lift and Airfoils

277

1.5

1

0.5

3

2

1

2

1

0.5

3

1

1.5

Fig. 9.16 Flow around the airfoil

Since

d

dz

Á

.2l;0/

D 1=.1

l 2= 2/

D 1, to obtain a finite value of

D.l;0/

.dF=d z/.2l;0/ the derivative .dF=d /.l;0/ needs to vanish. This condition is satisfied

by observing that

"

d FO

D U0 e

d

i'

2

be iÂ

e

2i.

Ci

2

be iÂ D

and from Fig. 9.14 it also follows that

D l, from (9.67) we deduce that

U0 1

#

a2 e i'

ˇ/C2i'

Ci

2 a

e

1

;

be iÂ

l C ae i.

i.

ˇ/Ci'

(9.67)

ˇ/

; therefore, when

D 0:

Equating to zero both the real and imaginary parts, we get the value of :

D

4 aU0 sin .

ˇ

'/ D 4 aU0 sin .ˇ C '/ ;

(9.68)

which depends on the velocity of the undisturbed stream, on the apparent attack

angle ', and on ˇ. The last two parameters define the dimension and curvature of

the wing profile, given a and U0 .

Increasing the angle of attack of an airfoil increases the flow asymmetry, resulting

in greater lift. Experimental evidence shows that if the total angle of attack ' C ˇ

reaches a critical value, the airfoil stalls, and the lift drops dramatically.

There are three programs attached to this chapter: Wing, Potential, and

Joukowsky. The program Wing gives the curve in the plane z corresponding to

the circle of unit radius through Joukowsky’s map.

The program Potential provides a representation of the streamlines corresponding to a given complex potential F .z/.

Finally, the program Joukowski gives the streamlines around a wing, allowing

changes of the angle of attack as well as the coordinates of the center of the cylinder.

278

9 Fluid Mechanics

9.8 Newtonian Fluids

The simplest assumption, that the difference between the stress in a moving fluid and

the stress at equilibrium is linearly related to the rate of deformation tensor, is due

to Newton (1687). The 3D case was studied by Navier (1821) for an incompressible

fluid, and later by Poisson (1831) for the general case.

Due to these contributions, we have that the constitutive equations of a linear

compressible or incompressible viscous fluid (see Sect. 7.3), respectively are

T D Œ p. / C . / ID I C 2 . / D;

(9.69)

TD

(9.70)

pI C 2 D:

If (9.69) is introduced into (5.30)1 , the Navier–Stokes equation

vP D b

rp C r. r v/ C r .2 D/

(9.71)

is obtained. Consequently, for a compressible viscous fluid we must find the fields

v.x; t / and .x; t / which satisfy the system

vP D b rp C r. r v/ C r .2 D/;

P C r v D 0;

(9.72)

in the domain C occupied by the fluid, as well as the boundary condition on a

fixed wall

vD0

on @C

(9.73)

and the initial conditions

.x; 0/ D

0 .x/;

v.x; 0/ D v0 .x/

8x 2 C:

For the case of an incompressible fluid, for which

r v D 0, and

2r D D v C r.r v/ D v;

Eq. (9.71) becomes

vP D b

where

1

rp C v;

D = is the coefficient of kinematic viscosity.

D const ,

(9.74)

D const ,

9.9 Applications of the Navier–Stokes Equation

279

Then the problem reduces to finding the fields v.x; t / and p.x; t / that satisfy

the system

1

vP D b

rp C

v

r v D 0;

(9.75)

the boundary condition (9.73), and the initial condition

v.x; 0/ D v0 .x/

8x 2 C:

9.9 Applications of the Navier–Stokes Equation

In this section we consider a steady flow of a linear viscous fluid characterized by a

velocity field parallel to the axis Ox in the absence of body forces.

If Oy and Oz are two other axes, which together with Ox form an orthogonal

frame of reference, then

v D v.x; y; z/i;

(9.76)

where i is the unit vector associated with the Ox axis. The introduction of (9.76)

into (9.75) gives

1 @p

@v

vC

D

@x

@x

v;

@p

@p

D

D 0;

@y

@z

@v

D 0:

@x

These equations imply that

p D p.x/;

v D v.y; z/;

so that the pressure and the velocity field depend only on x and on y, z, respectively.

It follows that they are both equal to the same constant A:

Â 2

Ã

@v

@p

@2 v

D A;

(9.77)

C 2 D A:

@x

@y 2

@z

The first equation tells us that p is a linear function of x, so that if p0 and p1 are the

values at x D 0 and x D l, it follows that

pD

p1

p0

l

x C p0:

(9.78)

280

9 Fluid Mechanics

Therefore, (9.77)2 becomes

@2 v

@2 v

p1 p0

:

C 2 D

2

@y

@z

l

(9.79)

We integrate (9.79) when v D v.y/ and the fluid is confined between the two

plates y D 0 and y D h of infinite dimension, defined by y D 0 and y D h.

Moreover, the second plate is supposed to move with uniform velocity V along Ox

and p0 , p1 are supposed to be equal. We have

v D ay C b;

where a and b are constant. If there is no relative slip at y D 0 and y D h, i.e.,

v.0/ D 0;

v.h/ D V;

we get

vD

V

y:

h

(9.80)

Now, a flow in a cylinder with axis Ox is taken into account. In cylindrical

coordinates (r, ', x), defined by

rD

p

y 2 C z2 ;

' D arctan

z

;

y

x D x;

provided that v.y; z/ D v.r/, Eq. (9.79) can be written as

Â

Ã

dv

p1 p 0

r

D

r;

dr

l

(9.81)

p1 p 0 2

r C A log r C B;

4 l

(9.82)

d

dr

and the integration gives

vD

where A and B are arbitrary constants. The condition that v has a finite value at

r D 0 requires A D 0. Furthermore, the adhesion condition v.a/ D 0, where a is

the radius of the cylinder, leads to

BD

p1 p0 2

a ;

4 l

so that (9.82) becomes

vD

p1 p 0 2

.r

4 l

a2 /:

(9.83)

9.10 Dimensional Analysis

281

The solution (9.83) allows us to compute the flux Q across the tube:

Z

a

QD

v2 r dr D

a4

0

p0 p1

:

8 l

(9.84)

Formula (9.84) is used to determine the viscosity , by measuring Q, p1

the mass density of the fluid.

p0 , and

9.10 Dimensional Analysis and the Navier–Stokes Equation

It is well known that dimensional analysis is a very useful technique in modeling a

physical phenomenon. By using such a procedure, it can be proved that some terms

of the Navier–Stokes equation can be neglected with respect to other terms, so that

the complexity of the equation is reduced.

For instance, the term rp= in Eq. (9.75) cannot be compared with 4v,

since their order of magnitude is not known a priori. To make this comparison

possible we must introduce suitable reference quantities. If, for sake of simplicity,

our attention is restricted to liquids, i.e., to Eq. (9.75), a characteristic length L and a

characteristic velocity U are necessary. As an example, for a solid body placed in a

moving liquid, L can be identified with a characteristic dimension of the body2 and

U with the uniform velocity of the liquid particles that are very far from the body.

In any case, both L and U have to be selected in such a way that the dimensionless

quantities

v D

v

;

U

r D

r

;

L

D

t

L=U

(9.85)

have the order of magnitude of unity. It follows that

vP D

1

rp D

@v

U @v

U2

C v rv D

C

v

@t

L=U @

L

r v ;

1

r p;

L

v D v ;

r vD

U

r

L

v ;

where the derivatives appearing in the operators r , r , and are relative to

the variables .x ; y ; z / D r . The substitution of these relations into (9.75), after

dividing by U 2 =L, leads to

2

In the 2D case, L can be a characteristic dimension of the cross section.

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