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FUNDAMENTALS OF INVISCID. INCOMPRESSIBLE FLOW 201

see how the superposition of flows involving such vortices leads to cases with finite lift.

Consider a flow where all the streamlines are concentric circles about a given point, as sketched in Fig. 3.26. Moreover, let the velocity along any given circular streamline be constant, but let it vary from one streamline to another inversely with distance from the common center. Such a flow is called a vortex flow. Examine Fig. 3.26; the velocity components in the radial and tangential directions are V, and Va, respectively, where V, = 0 and Ve = constant/ r. It is easily shown (try it yourself) that (1) vortex flow is a physically possible incom-

pressible flow, i.e., V . V = 0 at every point,

and (2) vortex flow is irrotational,

i.e., V x V = 0, at every point except the origin.

 

From the definition of vortex flow, we have

 

const

C

(3.104)

Vo = -- = -

r

r

 

To evaluate the constant C, take the circulation around a given circular streamline of radius r:

or

(3.105)

Comparing Egs. (3.104) and (3.105), we see that

 

r

(3.106)

C= --

21T

Therefore, for vortex flow, Eg. (3.106) demonstrates that the circulation taken

FIGURE 3.26

Vortex flow.

202 FUNDAMENTALS OF AERODYNAMICS

about all streamlines is the same value, namely, f = -21TC. By convention, f is called the strength of the vortex flow, and Eq. (3.105) gives the velocity field for a vortex flow of strength f. Note from Eq. (3.105) that Vo is negative when f is positive; i.e., a vortex of positive strength rotates in the clockwise direction. (This is a consequence of our sign convention on circulation defined in Sec. 2.13, namely, positive circulation is clockwise.)

We stated earlier that vortex flow is irrotational except at the origin. What happens at r = o? What is the value of V x V at r = o? To answer these questions,

recall Eq. (2.128) relating circulation to vorticity:

 

f = - f f (V x V) . dS

(2.128)

s

 

Combining Eqs. (3.106) and (2.128), we obtain

 

21TC = ff(V x V) . dS

(3.107)

s

 

Since we are dealing with two-dimensional flow, the flow sketched in Fig. 3.26 takes place in the plane of the paper. Hence, in Eq. (3.107), both V x V and dS are in the same direction, both perpendicular to the plane of the flow. Thus, Eq. (3.107) can be written as

21TC = ff(V x V) . dS = ffIv x vi dS

(3.108)

 

 

s

s

 

In Eq. (3.1 08), the surface integral is taken over the circular area inside the streamline along which the circulation f = -21TC is evaluated. However, f is the same for all the circular streamlines. In particular, choose a circle as close to the origin as we wish; i.e., let r~ o. The circulation will still remain f = -21TC. However, the area inside this small circle around the origin will become infinitesimally small, and

 

ffIv x vi dS ~Iv x vi dS

(3.109)

 

 

 

s

 

Combining Eqs. (3.108) and (3.109), in the limit as r~O, we have

 

or

21TC

(3.11 0)

IVxvl=-

 

dS

 

FUNDAMENTALS OF INVISCID. INCOMPRESSIBLE FLOW 203

However, as r ~ 0, dS ~ 0. Therefore, in the limit as r ~ 0, from Eq. (3.110), we have

Iv xvl~oo

Conclusion: Vortex flow is irrotational everywhere except at the point r = 0, where the vorticity is infinite. Therefore, the origin, r = 0, is a singular point in the flow field. We see that, along with sources, sinks, and doublets, the vortex flow contains a singularity. Hence, we can interpret the singularity itself, i.e .• point 0 in Fig. 3.26, to be a point vortex which induces about it the circular vortex flow shown in Fig. 3.26.

The velocity potential for vortex flow can be obtained as follows:

a4>

 

-=V=o

ar

r

1 a4>

r

-- = Ve = ---

r a(J

27Tr

Integrating Eqs. (3.111 a and b), we find

Equation (3.112) is the velocity potential for vortex flow. The stream function is determined in a similar manner:

(3.111a)

(3.111b)

(3.112)

 

 

 

 

 

 

 

(3.113a)

 

 

 

at/!

r

 

 

(3.113b)

 

 

 

-- = Ve = --

 

 

 

 

 

ar

27Tr

 

 

 

TABLE 3.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Type of flow

Velocity

 

 

 

 

 

 

 

 

 

 

 

 

 

Uniform flow in

 

 

 

 

 

 

 

x direction

u = Voo

 

Voox

 

 

 

Source

 

A

 

A

A

 

 

V= -

 

- Inr

- ()

 

 

 

,

21Tr

 

21T

21T

 

 

Vortex

 

r

 

r

r

 

 

Ve = --

-- ()

- Inr

 

 

21Tr

21T

21T

 

 

Doublet

V

K

cos ()

K cos ()

K

sin ()

= ----

21T

21T

r

 

r

217,2

 

 

K

sin ()

 

 

 

 

 

V ----

 

 

 

 

 

e- 21T

r2

 

 

 

 

 

 

 

 

 

 

 

 

204 FUNDAMENTALS OF AERODYNAMICS

Integrating Eqs. (3.113a and b), we have

I ~~271'r In, I

(3.114)

 

Equation (3.114) is the stream function for vortex flow. Note that since '" = constant is the equation of a streamline, Eq. (3.114) states that the streamlines of vortex flow are given by r = constant; i.e., the streamlines are circles. Thus, Eq. (3.114) is consistent with our definition of vortex flow. Also, note from Eq. (3.112) that equipotential lines are given by () = constant, i.e., straight radial lines from the origin. Once again, we see that equipotential lines and streamlines are mutually perpendicular.

At this stage, we summarize the pertinent results for our four elementary flows in Table 3.1.

3.15LIFTING FLOW OVER A CYLINDER

In Sec. 3.13, we superimposed a uniform flow and a doublet to synthesize the flow over a circular cylinder, as shown in Fig. 3.21. In addition, we proved that both the lift and drag were zero for such a flow. However, the streamline pattern shown at the right of Fig. 3.21 is not the only flow that is theoretically possible around a circular cylinder. It is the only flow that is consistent with zero lift. However, there are other possible flow patterns around a circular cylinderdifferent flow patterns which result in a nonzero lift on the cylinder. Such lifting flows are discussed in this section.

Now you might be hesitant at this moment, perplexed by the question as to how a lift could possibly be exerted on a circular cylinder. Is not the body perfectly symmetric, and would not this geometry always result in a symmetric flow field with a consequent zero lift, as we have already discussed? You might be so perplexed that you run down to the laboratory, place a stationary cylinder in a low-speed tunnel, and measure the lift. To your satisfaction, you measure no lift, and you walk away muttering that the subject of this section is ridiculousthere is no lift on the cylinder. However, go back to the wind tunnel, and this time run a test with the cylinder spinning about its axis at relatively high revolutions per minute. This time you measure a finite lift. Also, by this time you might be thinking about other situations: spin on a baseball causes it to curve, and spin on a golf ball causes it to hook or slice. Clearly, in real life there are nonsymmetric aerodynamic forces acting on these symmetric, spinning bodies. So, maybe the subject matter of this section is not so ridiculous after all. Indeed, as you will soon appreciate, the concept of lifting flow over a cylinder will start us on a journey which leads directly to the theory of the lift generated by airfoils, as discussed in Chap. 4.

Consider the flow synthesized by the addition of the nonlifting flow over a cylinder and a vortex of strength r, as shown in Fig. 3.27. The stream function

FUNDAMENTALS OF INVISCID. INCOMPRESSIBLE FLOW 205

+

Nonlifting flow

Vortex of

 

over a cylinder

strength r

Lifting flow over

 

 

a cylinder

FIGURE 3.27

The synthesis of lifting flow over a circular cylinder.

for nonlifting flow over a circular cylinder of radius R is given by Eq. (3.92):

0/1 = (Voor sin 0) ( 1 _ ~22)

(3.92)

The stream function for a vortex of strength r is given by Eq. (3.114). Recall that the stream function is determined within an arbitrary constant; hence, Eq. (3.114) can be written as

0/2

r

 

 

(3.115)

=-In r+const

 

21T

 

 

 

Since the value of the constant is arbitrary, let

 

 

 

 

r

R

(3.116)

Const = -- In

 

 

21T

 

 

Combining Eqs. (3.115) and (3.116), we obtain

 

 

 

r

r

 

(3.117)

 

0/2 = 21T In R

 

Equation (3.117) is the stream function for a vortex of strength r and is just as valid as Eq. (3.114) obtained earlier; the only difference between these two equations is a constant of the value given by Eq. (3.116).

The resulting stream function for the flow shown at the right of Fig. 3.27 is

or

 

 

 

 

(3.118)

 

 

 

 

 

From Eq. (3.118), if r = R, then 0/ = 0 for all values of O. Since 0/ = constant

is the equation of a streamline,

r = R is therefore a streamline of the flow, but

206 FUNDAMENTALS OF AERODYNAMICS

r = R is the equation of a circle of radius R. Hence, Eq. (3.118) is a valid stream function for the inviscid, incompressible flow over a circular cylinder of radius R, as shown at the right of Fig. 3.27. Indeed, our previous result given by Eq. (3.92) is simply a special case of Eq. (3.118) with r = o.

The resulting streamline pattern given by Eq. (3.118) is sketched at the right of Fig. 3.27. Note that the streamlines are no longer symmetrical about the horizontal axis through point 0, and you might suspect (correctly) that the cylinder will experience a resulting finite normal force. However, the streamlines are symmetrical about the vertical axis through 0, and as a result the drag will be zero, as we prove shortly. Note also that because a vortex of strength r has

been added to the flow, the circulation about the cylinder is now finite and equal to r.

The velocity field can be obtained by differentiating Eq. (3.118). An equally direct method of obtaining the velocities is to add the velocity field of a vortex to the velocity field of the nonlifting cylinder. (Recall that because of the linearity of the flow, the velocity components of the superimposed elementary flows add directly.) Hence, from Eqs. (3.93) and (3.94) for nonlifting flow over a cylinder of radius R, and Eqs. (3.111a and b) for vortex flow, we have, for the lifting flow over a cylinder of radius R,

Vr = ( 1 - ~22) Voc cos 8

v. = -

(

R2)

V.

r

1+-

sin 8 --

o

r2

00

27fT

To locate the stagnation points in the flow, set Vr = and (3.120) and solve for the resulting coordinates (r, fJ):

V. = ( 1 - R 2) V. cos fJ = 0

r r 2 00

(3.119)

(3.120)

Vo = 0 in Eqs. (3.119)

(3.121)

(3.122)

From Eq. (3.121), r = R. Substituting this result into Eq. (3.122) and solving for

fJ, we obtain

 

 

8 = arc sin (_

r )

(3.123)

 

47TVooR

 

Since r is a positive number, from Eq. (3.123) 8 must be in the third and fourth quadrants. That is, there can be two stagnation points on the bottom half of the circular cylinder, as shown by points 1 and 2 in Fig. 3.28a. These points are located at (R, 8), where 8 is given by Eq. (3.123). However, this result is valid only when I'/47TVooR < 1. If rj47TVooR > 1, then Eq. (3.123) has no meaning. If rj47TVooR = 1, there is only one stagnation point on the surface of the cylinder, namely, point (R, -7Tj2) labeled as point 3 in Fig. 3.28b. For the case of

J

,~

FUNDAMENTALS OF INVISCID, INCOMPRESSIBLE FLOW 207

(a) r<47TVooR

 

(b) r =47TVooR

 

FIGURE 3.28

(e) r> 47TVooR

 

Stagnation points for the lifting flow over a circular cylinder.

 

f / 417'Vo;)R > 1, return to Eq. (3.121). We saw earlier that it is satisfied by r = R; however, it is also satisfied by e= 17'/2 or -17'/2. Substituting e= -17'/2 into Eq. (3.122), and solving for r, we have

(3.124)

Hence, for f /417' VooR > 1, there are two stagnation points, one inside and the other outside the cylinder, and both on the vertical axis, as shown by points 4 and 5 in Fig. 3.28c. [How does one stagnation point fall inside the cylinder? Recall that r = R, or '" = 0, is just one of the allowed streamlines of the flow. There is a theoretical flow inside the cylinder-flow that is issuing from the doublet at the origin superimposed with the vortex flow for r < R. The circular streamline r = R is the dividing streamline between this flow and the flow from the freestream. Therefore, as before, we can replace the dividing streamline by a solid body-our circular cylinder-and the external flow will not know the difference. Hence, although one stagnation point falls inside the body (point 5), we are not realistically concerned about it. Instead, from the point of view of flow over a solid cylinder of radius R, point 4 is the only meaningful stagnation point for the case f/417'VocR> 1.]

The results shown in Fig. 3.28 can be visualized as follows. Consider the inviscid incompressible flow of given freestream velocity Vo;) over a cylinder of given radius R. If there is no circulation, i.e., if f = 0, the flow is given by the sketch at the right of Fig. 3.21, with horizontally opposed stagnation points A and B. Now assume that a circulation is imposed on the flow, such that r = 417' Vo;)R. The flow sketched in Fig. 3.28a will result; the two stagnation points will move to the lower surface of the cylinder as shown by points 1 and 2. Assume that f is further increased until f = 417'V",R. The flow sketched in Fig. 3.28b will result,

208 FUNDAMENTALS OF AERODYNAMICS

with only one stagnation point at the bottom of the cylinder, as shown by point 3. When r is increased still further such that r> 47T VooR, the flow sketched in Fig. 3.28c will result. The stagnation point will lift from the cylinder's surface and will appear in the flow directly below the cylinder, as shown by point 4.

From the above discussion, r is clearly a parameter that can be chosen freely. There is no single value of r that "solves" the flow over a circular cylinder; rather, the circulation can be any value. Therefore, for the incompressible flow over a circular cylinder, there are an infinite number of possible potential flow solutions, corresponding to the infinite choice for values of r. This statement is not limited to flow over circular cylinders, but rather, it is a general statement that holds for the incompressible potential flow over all smooth two-dimensional bodies. We return to these ideas in subsequent sections.

From the symmetry, or lack of it, in the flows sketched in Figs. 3.27 and 3.28, we intuitively concluded earlier that a finite normal force (lift) must exist on the body but that the drag is zero; i.e., d'Alembert's paradox still prevails. Let us quantify these statements by calculating expressions for lift and drag, as follows.

The velocity on the surface of the cylinder is given by Eq. (3.120) with r = R.

.

r

(3.125)

V = Ve = - 2 V00 sm () ---

 

27TR

 

In turn, the pressure coefficient is obtained by substituting Eq. (3.125) into Eq. (3.38):

 

C = 1- (~)2= 1- (-2 sin () _

r )2

 

 

p

 

Voo

 

27TRVoo

 

or

Cp

= 1 -

[4 sin2 () +2r sin () +(

2

r V )2J

(3.126)

 

 

 

7TRVoo

TTR 00

 

In Sec. 1.5, we discussed in detail how the aerodynamic force coefficients can be obtained by integrating the pressure coefficient and skin friction coefficient over

the surface. For inviscid flow,

cf = O. Hence, the drag coefficient Cd

is given by

Eq. (1.16) as

 

 

 

 

 

 

 

 

 

1 fTE

(Cp,u -

Cp,l) dy

 

 

Cd = ca = ~

LE

 

or

1 fTE

 

1 fTE

Cp,l dy

(3.127)

Cd =-

 

Cp,u dy --

 

 

C

LE

 

C

LE

 

 

Converting Eq. (3.127) to polar coordinates, we note that

 

 

y = R sin ()

dy = R cos () d(}

(3.128)

Substituting Eq. (3.128) into (3.127), and noting that c = 2R, we have

(3.129)

FUNDAMENTALS OF INVISCID, INCOMPRESSIBLE FLOW 209

The limits of integration in Eq. (3.129) are explained as follows. In the first integral, we are integrating from the leading edge (the front point of the cylinder), moving over the top surface of the cylinder. Hence, e is equal to 17' at the leading edge and, moving over the top surface, decreases to 0 at the trailing edge. In the second integral, we are integrating from the leading edge to the trailing edge while moving over the bottom surface of the cylinder. Hence, e is equal to 17' at the leading edge and, moving over the bottom surface, increases to 217' at the trailing edge. In Eq. (3.129), both Cp," and Cp,1 are given by the same analytic expression for Cp , namely, Eq. (3.126). Hence, Eq. (3.129) can be written as

 

If1T

Cp

cos

IJ21T

Cp

cos

e de

 

Cd = --

 

e de --

1T

 

2

0

 

 

2

 

 

 

or

1 f21T

 

e de

 

 

 

(3.130)

Cd = --

 

Cp cos

 

 

 

 

2

0

 

 

 

 

 

 

 

Substituting Eq. (3.126) into (3.130), and noting that

 

 

 

 

 

 

21T

 

 

 

 

 

 

 

 

f o

cos e de = 0

 

(3.131a)

 

 

 

21T

sin2

 

 

 

 

 

 

 

f o

e cos e de = 0

 

(3.131b)

 

 

 

21T

 

 

 

 

 

 

 

 

fo

sin e cos e de = 0

 

(3.131c)

 

 

 

 

 

we immediately obtain

(3.132)

Equation (3.132) confirms our intuitive statements made earlier. The drag on a cylinder in an inviscid, incompressible flow is zero, regardless of whether or not the flow has circulation about the cylinder.

The lift on the cylinder can be evaluated in a similar manner as follows. From Eq. (1.15) with cf = 0,

 

 

1 fe

1 fe

Cp," dx

 

(3.133)

CI = Cn =-

Cp,1 dx --

 

 

 

 

C 0

C

0

 

 

 

Converting to polar coordinates, we obtain

 

 

 

 

x = R cos e

dx = -

R sin e de

 

(3.134)

Substituting Eq. (3.134) into (3.133), we have

 

 

 

 

lJ2rr

 

lJO

Cp," sin

e de

(3.135)

CI = -"2

1T

Cp,l sin e de +"2

1T

210 FUNDAMENTALS OF AERODYNAMIC'S

Again, noting that Cp,' and Cp,u are both given by the same analytic expression, namely, Eq. (3.126), Eq. (3.l35) becomes

1 f27T

Cp sin e de

(3.l36)

c, = - -

 

2

0

 

 

Substituting Eq. (3.126) into (3.l36), and noting that

 

27T

 

 

 

fo

sin e de = 0

(3.l37a)

27T

 

 

 

fo

sin3

e de = 0

(3.l37b)

27T

 

 

 

f o

sin2

e de = 7T

(3.l37c)

we immediately obtain

 

 

 

 

 

r

(3.l38)

c ---

 

,- RVoo

 

From the definition of c, (see Sec. 1.5), the lift per unit span L' can be obtained from

(3.l39)

Here, the planform area S = 2R(l). Therefore, combining Eqs. (3.l38) and (3.l39), we have

,

1

 

V

2

r

L

=-p

 

2R --

 

2

co

 

co

RVco

or

 

 

 

 

(3.140)

Equation (3.140) gives the lift per unit span for a circular cylinder with circulation r. It is a remarkably simple result, and it states that the lift per unit span is directly proportional to circulation. Equation (3.140) is a powerful relation in theoretical aerodynamics. It is called the Kutta-loukowski theorem, named after the German mathematician M. Wilheim Kutta (1867-1944) and the Russian physicist Nikolai E. loukowski (1847-1921), who independently obtained it during the first decade of this century. We will have more to say about the Kutta-loukowski theorem in Sec. 3.16.

What are the connections between the above theoretical results and real life? As stated earlier, the prediction of zero drag is totally erroneous-viscous effects cause skin friction and flow separation which always produce a finite drag, as will be discussed in Chaps. 15 to 17. The inviscid flow treated in this chapter

r

I