Cohen I.M., Kundu P.K. Fluid Mechanics

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162 Vorticity Dynamics

2. A tornado can be idealized as a Rankine vortex with a core of diameter 30 m.

The gauge pressure at a radius of 15 m is −2000 N/m

(that is, the absolute pressure

is 2000 N/m

below atmospheric). (a) Show that the circulation around any circuit

surrounding the core is 5485 m

/s. [Hint: Apply the Bernoulli equation between

inﬁnity and the edge of the core.] (b) Such a tornado is moving at a linear speed

of 25 m/s relative to the ground. Find the time required for the gauge pressure to

drop from −500 to −2000 N/m

. Neglect compressibility effects and assume an air

temperature of 25

◦

C. (Note that the tornado causes a sudden decrease of the local

atmospheric pressure. The damage to structures is often caused by the resulting excess

pressure on the inside of the walls, which can cause a house to explode.)

3. The velocity ﬁeld of a ﬂow in cylindrical coordinates (R,ϕ,x)is

= 0 u

= aRx u

= 0

where a is a constant. (a) Show that the vorticity components are

=−aR ω

= 0 ω

= 2ax

(b) Verify that ∇

•

ω = 0. (c) Sketch the streamlines and vortex lines in an Rx-plane.

Show that the vortex lines are given by xR

= constant.

4. Consider the ﬂow in a 90

◦

angle, conﬁned by the walls θ = 0 and θ = 90

◦

Consider a vortex line passing through (x, y), and oriented parallel to the z-axis.

Show that the vortex path is given by

= constant.

[Hint: Convince yourself that we need three image vortices at points (−x,−y),

(−x,y) and (x, −y). What are their senses of rotation? The path lines are given

by dx/dt = u and dy/dt = v, where u and v are the velocity components at the

location of the vortex. Show that dy/dx = v/u =−y

, an integration of which

gives the result.]

5. Start with the equations of motion in the rotating coordinates, and prove

Kelvin’s circulation theorem

(

) = 0

where





(ω + 2)

•

Literature Cited 163

Assume that the ﬂow is inviscid and barotropic and that the body forces are conser-

vative. Explain the result physically.

6. Consider the interaction of two vortex rings of equal strength and similar

sense of rotation. Argue that they go through each other, as described near the end of

Section 8.

7. A constant density irrotational ﬂow in a rectangular torus has a circulation

 and volumetric ﬂow rate Q. The inner radius is r

, the outer radius is r

, and the

height is h. Compute the total kinetic energy of this ﬂow in terms of only ρ, , and Q.

8. Consider a cylindrical tank of radius R ﬁlled with a viscous ﬂuid spinning

steadily about its axis with constant angular velocity . Assume that the ﬂow is in

a steady state. (a) Find



ω · dA where A is a horizontal plane surface through the

ﬂuid normal to the axis of rotation and bounded by the wall of the tank. (b) The tank

then stops spinning. Find again the value of



ω · dA.

9. In Figure 5.11, locate point G.

Literature Cited

Lighthill, M. J. (1986). An Informal Introduction to Theoretical Fluid Mechanics, Oxford, England: Claren-

don Press.

Sommerfeld, A. (1964). Mechanics of Deformable Bodies, New York: Academic Press. (This book contains

a good discussion of the interaction of vortices.)

Supplemental Reading

Batchelor, G. K. (1967). An Introduction to Fluid Dynamics, London: Cambridge University Press.

Pedlosky, J. (1987). Geophysical Fluid Dynamics, New York: Springer-Verlag. (This book discusses the

vorticity dynamics in rotating coordinates, with application to geophysical systems.)

Prandtl, L. and O. G. Tietjens (1934). Fundamentals of Hydro- and Aeromechanics, New York: Dover

Publications. (This book contains a good discussion of the interaction of vortices.)

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Chapter 6

Irrotational Flow

1. Relevance of Irrotational Flow

Theory ......................... 165

2. Velocity Potential: Laplace

Equation ....................... 167

3. Application of Complex

Variables ....................... 169

4. Flow at a Wall Angle ........... 171

5. Sources and Sinks .............. 173

6. Irrotational Vortex.............. 174

7. Doublet ........................ 174

8. Flow past a Half-Body ......... 175

9. Flow past a Circular Cylinder

without Circulation ............. 178

10. Flow past a Circular Cylinder

with Circulation ................ 180

11. Forces on a Two-Dimensional

Body ........................... 184

Blasius Theorem................ 184

Kutta–Zhukhovsky

Lift Theorem................. 185

Unsteady Flow ................. 188

12. Source near a Wall: Method of

Images ......................... 189

13. Conformal Mapping ............ 190

14. Flow around an Elliptic Cylinder

with Circulation ................ 192

15. Uniqueness of Irrotational Flows. 194

16. Numerical Solution of Plane

Irrotational Flow ............... 195

Finite Difference Form of the

Laplace Equation ............ 196

Simple Iteration Technique ..... 198

Example 6.1 ................... 199

17. Axisymmetric Irrotational

Flow ........................... 201

18. Streamfunction and Velocity

Potential for Axisymmetric

Flow ......................... 203

19. Simple Examples of Axisymmetric

Flows .......................... 205

Uniform Flow .................. 205

Point Source ................... 205

Doublet ........................ 205

Flow around a Sphere .......... 206

20. Flow around a Streamlined

Body of Revolution.............. 206

21. Flow around an Arbitrary Body

of Revolution ................... 208

22. Concluding Remarks ............ 209

Exercises ....................... 209

Literature Cited ................ 212

Supplemental Reading .......... 212

1. Relevance of Irrotational Flow Theory

The vorticity equation given in the preceding chapter implies that the irrotational ﬂow

(such as the one starting from rest) of a barotropic ﬂuid observed in a nonrotating

frame remains irrotational if the ﬂuid viscosity is identically zero and any body forces

165

DOI: 10.1016/B978-0-12-381399-2.50006-X

166 Irrotational Flow

are conservative. Such an ideal ﬂow has a nonzero tangential velocity at a solid surface

(Figure 6.1a). In contrast, a real ﬂuid with a nonzero ν must satisfy a no-slip boundary

condition. It can be expected that viscous effects in a real ﬂow will be conﬁned to

thin layers close to solid surfaces if the ﬂuid viscosity is small. We shall see later

that the viscous layers are thin not just when the viscosity is small, but when a

non-dimensional quantity Re = UL/ν, called the Reynolds number, is much larger

than 1. (Here, U is a scale of variation of velocity in a length scale L.) The thickness

of such boundary layers, within which viscous diffusion of vorticity is important,

approaches zero as Re →∞(Figure 6.1b). In such a case, the vorticity equation

implies that ﬂuid elements starting from rest, or from any other irrotational region,

remain irrotational unless they move into these boundary layers. The ﬂow ﬁeld can

therefore be divided into an “outer region” where the ﬂow is inviscid and irrotational

and an “inner region” where viscous diffusion of vorticity is important. The outer ﬂow

can be approximately predicted by ignoring the existence of the thin boundary layer

and applying irrotational ﬂow theory around the solid object. Once the outer problem

is determined, viscous ﬂow equations within the boundary layer can be solved and

matched to the outer solution.

An important exception in which this method would not work is where the solid

object has such a shape that the boundary layer separates from the surface, giving rise

to eddies in the wake (Figure 6.2). In this case viscous effects are not conﬁned to thin

layers around solid surfaces, and the real ﬂow in the limit Re →∞is quite different

Figure 6.1 Comparison of a completely irrotational ﬂow and a high Reynolds number ﬂow: (a) ideal

ﬂow with ν = 0; (b) ﬂow at high Re.

Figure 6.2 Examples of ﬂow separation. Upstream of the point of separation, irrotational ﬂow theory is

a good approximation of the real ﬂow.

2. Velocity Potential: Laplace Equation 167

from the ideal ﬂow (ν = 0). Ahead of the point of separation, however, irrotational

ﬂow theory is still a good approximation of the real ﬂow (Figure 6.2).

Irrotational ﬂow patterns around bodies of various shapes is the subject of this

chapter. Motion will be assumed inviscid and incompressible. Most of the examples

given are from two-dimensional plane ﬂows, although some examples of axisymmet-

ric ﬂows are also given later in the chapter. Both Cartesian (x, y) and polar (r, θ )

coordinates are used for plane ﬂows.

2. Velocity Potential: Laplace Equation

The two-dimensional incompressible continuity equation

∂u

∂x

∂v

∂y

= 0, (6.1)

guarantees the existence of a stream function ψ, from which the velocity components

can be derived as

u ≡

∂ψ

∂y

v ≡−

∂ψ

∂x

. (6.2)

Likewise, the condition of irrotationality

∂v

∂x

−

∂u

∂y

= 0, (6.3)

guarantees the existence of another scalar function φ, called the velocity potential,

which is related to the velocity components by

u ≡

∂φ

∂x

and v ≡

∂φ

∂y

. (6.4)

Because a velocity potential must exist in all irrotational ﬂows, such ﬂows are fre-

quently called potential ﬂows. Equations (6.2) and (6.4) imply that the derivative

of ψ gives the velocity component in a direction 90

◦

clockwise from the direction

of differentiation, whereas the derivative of φ gives the velocity component in the

direction of differentiation. Comparing equations (6.2) and (6.4) we obtain

∂φ

∂x

∂ψ

∂y

Cauchy–Riemann conditions

∂φ

∂y

=−

∂ψ

∂x

(6.5)

from which one of the functions can be determined if the other is known. Equipoten-

tial lines (on which φ is constant) and streamlines are orthogonal, as equation (6.5)

implies that

∇φ

•

∇ψ =



∂φ

∂x

+ j

∂φ

∂y



•



∂ψ

∂x

+ j

∂ψ

∂y



∂φ

∂x

∂ψ

∂x

∂φ

∂y

∂ψ

∂y

= 0.

This demonstration fails at stagnation points where the velocity is zero.

168 Irrotational Flow

The streamfunction and velocity potential satisfy the Laplace equations

∇

φ =

∂

∂x

∂

∂y

= 0, (6.6)

∇

ψ =

∂

∂x

∂

∂y

= 0, (6.7)

as can be seen by cross differentiating equation (6.5). Equation (6.7) holds for

two-dimensional ﬂows only, because a single streamfunction is insufﬁcient for

three-dimensional ﬂows. As we showed in Chapter 4, Section 4, two streamfunctions

are required to describe three-dimensional steady ﬂows (or, if density may be regarded

as constant, three-dimensional unsteady ﬂows). However, a velocity potential φ can be

deﬁned in three-dimensional irrotational ﬂows, because u = ∇φ identically satisﬁes

the irrotationality condition ∇ × u = 0. A three-dimensional potential ﬂow satisﬁes

the three-dimensional version of ∇

φ = 0.

A function satisfying the Laplace equation is sometimes called a harmonic func-

tion. The Laplace equation is encountered not only in potential ﬂows, but also in heat

conduction, elasticity, magnetism, and electricity. Therefore, solutions in one ﬁeld of

study can be found from a known analogous solution in another ﬁeld. In this manner,

an extensive collection of solutions of the Laplace equation have become known. The

Laplace equation is of a type that is called elliptic. It can be shown that solutions

of elliptic equations are smooth and do not have discontinuities, except for certain

singular points on the boundary of the region. In contrast, hyperbolic equations such

as the wave equation can have discontinuous “wavefronts” in the middle of a region.

The boundary conditions normally encountered in irrotational ﬂows are of the

following types:

(1) Condition on solid surface—Component of ﬂuid velocity normal to a solid

surface must equal the velocity of the boundary normal to itself, ensuring that

ﬂuid does not penetrate a solid boundary. For a stationary body, the condition is

∂φ

∂n

= 0or

∂ψ

∂s

= 0 (6.8)

where s is direction along the surface, and n is normal to the surface.

(2) Condition at inﬁnity—For the typical case of a body immersed in a uniform

stream ﬂowing in the x direction with speed U , the condition is

∂φ

∂x

= U or

∂ψ

∂y

= U (6.9)

However, solving the Laplace equation subject to boundary conditions of the

type of equations (6.8) and (6.9) is not easy. Historically, irrotational ﬂow theory

was developed by ﬁnding a function that satisﬁes the Laplace equation and then

determining what boundary conditions are satisﬁed by that function. As the Laplace

equation is linear, superposition of known harmonic functions gives another harmonic

function satisfying a new set of boundary conditions. A rich collection of solutions

has thereby emerged. We shall adopt this “inverse” approach of studying irrotational

3. Application of Complex Variables 169

ﬂows in this chapter; numerical methods of ﬁnding a solution under given boundary

conditions are illustrated in Sections 16 and 21.

After a solution of the Laplace equation has been obtained, the velocity com-

ponents are then determined by taking derivatives of φ or ψ. Finally, the pressure

distribution is determined by applying the Bernoulli equation

p +

ρq

= const.,

between any two points in the ﬂow ﬁeld; here q is the magnitude of velocity. Thus,

a solution of the nonlinear equation of motion (the Euler equation) is obtained in

irrotational ﬂows in a much simpler manner.

For quick reference, the important equations in polar coordinates are listed in the

following:

∂

∂r

(ru

) +

∂u

∂θ

= 0 (continuity), (6.10)

∂

∂r

(ru

) −

∂u

∂θ

= 0 (irrotationality), (6.11)

∂φ

∂r

∂ψ

∂θ

, (6.12)

∂φ

∂θ

=−

∂ψ

∂r

, (6.13)

∇

φ =

∂

∂r



∂φ

∂r



∂

∂θ

= 0, (6.14)

∇

ψ =

∂

∂r



∂ψ

∂r



∂

∂θ

= 0, (6.15)

3. Application of Complex Variables

In this chapter z will denote the complex variable

z ≡ x + iy = re

iθ

, (6.16)

where i =

√

−1, (x,y) are the Cartesian coordinates, and (r, θ ) are the polar coordi-

nates. In the Cartesian form the complex number z represents a point in the xy-plane

whose real axis is x and imaginary axis is y (Figure 6.3). In the polar form, z repre-

sents the position vector 0z, whose magnitude is r = (x

+ y

)

1/2

and whose angle

with the x-axis is tan

−1

(y/x). The product of two complex numbers z

and z

= r

i(θ

+θ

)

Therefore, the process of multiplying a complex number z

by another complex

number z

can be regarded as an operation that “stretches” the magnitude from r

and increases the argument from θ

to θ

+ θ

170 Irrotational Flow

Figure 6.3 Complex z-plane.

When x and y are regarded as variables, the complex quantity z = x + iy is

called a complex variable. Suppose we deﬁne another complex variable w whose real

and imaginary parts are φ and ψ:

w ≡ φ + iψ. (6.17)

If φ and ψ are functions of x and y, then so is w. It is shown in the theory of complex

variables that w is a function of the combination x + iy = z, and in particular has

a ﬁnite and “unique derivative” dw/dz when its real and imaginary parts satisfy the

pair of relations, equation (6.5), which are called Cauchy–Riemann conditions. Here

the derivative dw/dz is regarded as unique if the value of δw/δz does not depend on

the orientation of the differential δz as it approaches zero. A single-valued function

w = f(z)is called an analytic function of a complex variable z in a region if a ﬁnite

dw/dz exists everywhere within the region. Points where w or dw/dz is zero or

inﬁnite are called singularities, at which constant φ and constant ψ lines are not

orthogonal. For example, w = ln z and w = 1/z are analytic everywhere except at

the singular point z = 0, where the Cauchy–Riemann conditions are not satisﬁed.

The combination w = φ + iψ is called complex potential for a ﬂow. Because

the velocity potential and stream function satisfy equation (6.5), and the real and

imaginary parts of any function of a complex variable w(z) = φ + iψ also satisfy

equation (6.5), it follows that any analytic function of z represents the complex poten-

tial of some two-dimensional ﬂow. The derivative dw/dz is an important quantity in

the description of irrotational ﬂows. By deﬁnition

= lim

δz→0

δw

δz

As the derivative is independent of the orientation of δz in the xy-plane, we may take

δz parallel to the x-axis, leading to

= lim

δx→0

δw

δx

∂w

∂x

∂

∂x

(φ + iψ),

4. Flow at a Wall Angle 171

which implies

= u − iv.

(6.18)

It is easy to show that taking δz parallel to the y-axis leads to an identical result. The

derivative dw/dz is therefore a complex quantity whose real and imaginary parts give

Cartesian components of the localvelocity;dw/dz istherefore called the complex velo-

city. If the local velocity vector has a magnitude q and an angle α with the x-axis, then

= qe

−iα

. (6.19)

It may be considered remarkable that any twice differentiable function w(z), z = x +iy

is an identical solution to Laplace’s equation in the plane (x, y). A general function of

the two variables (x, y) may be written as f(z,z

∗

) where z

∗

= x − iy is the com-

plex conjugate of z. It is the very special case when f(z,z

∗

) = w(z) alone that we

consider here.

As Laplace’s equation is linear, solutions may be superposed. That is, the sums

of elemental solutions are also solutions. Thus, as we shall see, ﬂows over speciﬁc

shapes may be solved in this way.

4. Flow at a Wall Angle

Consider the complex potential

w = Az

(n 

), (6.20)

where A is a real constant. If r and θ represent the polar coordinates in the z-plane,

then

w = A(re

iθ

)

= Ar

(cos nθ + i sin nθ),

giving

φ = Ar

cos nθ ψ = Ar

sin nθ. (6.21)

Foragivenn, lines of constant ψ can be plotted. Equation (6.21) shows that ψ = 0

for all values of r on lines θ = 0 and θ = π/n. As any streamline, including the

ψ = 0 line, can be regarded as a rigid boundary in the z-plane, it is apparent that

equation (6.20) is the complex potential for ﬂow between two plane boundaries of

included angle α = π/n. Figure 6.4 shows the ﬂow patterns for various values of n.

Flow within a certain sector of the z-plane only is shown; that within other sectors

can be found by symmetry. It is clear that the walls form an angle larger than 180

◦

for n<1 and an angle smaller than 180

◦

for n>1. The complex velocity in terms

of α = π/n is

= nAz

n−1

Aπ

(π−α)/α