Fluid Mechanics Lecture Notes II

8/11/2019 Fluid Mechanics Lecture Notes II

http://slidepdf.com/reader/full/fluid-mechanics-lecture-notes-ii 1/127

Mechanics of Fluids MECH2008 (2008 – 2009)

Assessment: In-course continuous assessment 10%

• a mid-term test to be announced later

Examination in December 90%

Topics Covered:

1. Flow kinematics with differential vector calculus

2. Differential equations of motion

3. Unidirectional viscous flow and hydrodynamic lubrication

4. Potential flow and stream function

5. Boundary layer and drag

6. Open-channel flow and fluid machines

Prerequisites:-

This is a Level-II Mechanics of Fluids course demanding the knowledge you acquired in thefirst-year fluid and mathematics courses. In particular, the following topics are relevant and

should be reviewed if you have already forgotten the stuffs:-

1) Properties of fluid (density, viscosity);

2) Principles of fluid statics;

3) Fluid dynamics by control volume approach

continuity equation

energy equation (or Bernoulli equation)

momentum equation

head, head loss

4) Differentiation and integration;

5) Vector differential calculus (grad, div, curl, Gauss theorem, etc);

1



Lecture Notes and Worked Examples

(The corresponding section numbers in the textbook or other references are noted wherever

appropriate.)

(I) DIFFERENTIAL ANALYSIS OF FLUID FLOW

A. Description of Fluid Motion (Section 4.1.1)

• Lagrangian description: fluid particles are “tagged” or identified; rate of change of

flow properties as observed by following a fixed particle; variables are functions of

the initial position of particles and time.

• Eulerian description: fluid properties and variables are field variables, which are

functions of position in space (with respect to a fixed frame of reference) and time.

The Eulerian description, which is comparable to the data recorded by a measuring

device fixed in position, is more convenient to use in fluid mechanics.

Eulerian and Lagrangian descriptions

of temperature of a fluid discharging

from a smoke stack ual particles

In the Lagrangian description, one

must keep track of the position and

velocity of individ

•

Rectangular (Cartesian) coordinates:

( )

( )

( )

1 2 3

1 2 3

1 2 3

( , , ) ( , , ) 1, 2,3

( , , ) ( , , ) 1, 2,3

, , , , 1, 2,3

e.g.,

, , ( , , )

i

i

i

i

i

x y z x x x x i

u v w u u u u i

i x y z x x x x

uu v wu v w

x y z x y z x

= = = =

= = = =

⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂= = = =⎜ ⎟⎜ ⎟

∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞ ∂∂ ∂ ∂ ∂ ∂ ∂= + + =⎜ ⎟

∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

x

V

V =i i

z

V y

O

x

• Primitive variables: pressure ( , ) - scalar (0th order tensor)

velocity ( , ) - vector (1st order tensor)

p t

t

x

V x

Deduced variable stre ss ( , ) - 2nd order tensor t τ x

2



In the Eulerian description, one

defines field variables, such as the

pressure field and the velocity field,

at any location and instant in time.

B. Kinematics (Sections 4.2 and 6.1)

• Total (a.k.a. material, substantial) derivative = local rate of change + convective (or

advective) rate of change = the rate of change as observed following a particle of

fixed identity. It is an operator that can be applied to any scalar or vector quantity.

( ) ( )( )( )

( )

( ) ( ) ( )

( )

local rate convective rate of changeof change

e.g., local acceleration =

convective acceleration =

d

dt t

u v wt x y z

t

u v w x y z

∂

= +∂

∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂

∂

∂∂ ∂ ∂

+ +∂ ∂ ∂

i

i

V

V

V V

V V =

V

-

The local rate of change, also called the unsteady term, vanishes identically for

a steady flow. Therefore a flow is steady if and only if / 0t ∂ ∂ ≡ .

-

The quantity ( )iV

is a scalar convective operator that determines the time

rate of change of any property (e.g., velocity, density, concentration,

temperature) of a particle by reason of the fact that the particle moves from a

place where the property has one value to another place where it has a

different value.

The total derivative is defined by

following a fluid particle as it moves

throughout the flow field. In this

illustration, the fluid particle is

accelerating to the right as it moves

up and to the right. A velocity field with respect to a fixedframe of reference ( x, y). A point fixed

in space is occupied by different fluid

particles at different time.

3



•

Translation

+ (rigid body motion)

Rotation

General motion = +

Dilatation (change in volume)

+

Angular deformation (change in shape)

⎧ ⎫⎪⎪⎬⎪⎪⎪ ⎭

⎪⎨⎪⎪⎪⎪⎩

As illustrated by:-

The various modes of deformation can be expressed in terms of the velocity gradients.

• Divergence of velocity is the volumetric strain/dilatation rate (rate of change of

volume per unit volume)

u v w

x y z

∂ ∂ ∂≡ + +

∂ ∂ ∂V i

where u x∂ ∂ , v y∂ ∂ and w z∂ ∂ are the components of the volumetric strain rate due

to elongation of a fluid element in the x-, y-, and z-directions, respectively.

Consider a small element of dimensions x y zδ δ δ × × :

4



Because of the velocity differential uδ over a distance xδ , the element is lengthened

in the x-direction by u t δ δ ⋅ over a small period of time t δ . The corresponding

change in volume is therefore

xV u t y zδ δ δ δ δ = ⋅ ⋅ ⋅ ,

and the volume strain rate (change in volume per volume per time) is

as , 0 xV u t y z u u x t

V t x y z t x x

δ δ δ δ δ δ δ δ

δ δ δ δ δ δ

⋅ ⋅ ⋅ ∂= = =

⋅ ⋅ ⋅ ⋅ ∂ → .

Similarly, for the lengthening of the element in the y- and z-directions

, as , , 0 y z

V V v w y z t

V t y V t z

δ δ δ δ δ

δ δ

∂ ∂= =

⋅ ∂ ⋅ ∂ →

The total volume strain rate is hence given by the divergence of the velocity.

as , , , 0V u v w x y z t V t x y zδ δ δ δ δ

δ

∂ ∂ ∂= + + →⋅ ∂ ∂ ∂

In this incompressible flow, in which the

velocity divergence is identically zero, an

initially square parcel of marked fluid will

deform into a long thin shape (stretch in x-

direction, but shrink in the y-direction) in the

course of movement shown in the figure. Theflow is irrotational is this case.

• Any shear deformation can be decomposed into rigid body rotation and angular

deformation. Consider a small element undergoing shear deformation

5



Because of the velocity differential vδ over a distance xδ , the face OA rotates

counterclockwise by an angle /v t xδα δ δ δ = ⋅ over a small period of time t δ .

Therefore the angular velocity of OA is

as , 0d v

x t

dt x

α α δ δ

∂= = →

∂

Similarly, the face OB rotates clockwise at an angular velocity given by

as , 0d u

y t dt y

β β δ δ

∂= = →

∂

x

y

Oα

β

The deformation can be decomposed into a rigid body rotation at an angular velocity

( )1 1

2 2

v u

x yω α β

⎛ ∂ ∂= − = −⎜ ∂ ∂⎝ ⎠

⎞

⎟ , where counterclockwise rotation is taken to be positive,

x

y

O

( )1

2α β −

( )1

2α β −

and an angular deformation, where the corner angle decreases at a rate given by

v u x y

γ α β ∂ ∂= + = +∂ ∂ , where a positive rate means a decreasing angle,

x

y

O

( )1

2

( )1

2α β +

α β +

γ

6



•

Rate of angular deformation of a 2-D fluid element moving in the x- y plane (angular

deformation is considered to be positive if it is to decrease the original right angle) is

hence defined to be

0lim

xyt

v u

t xδ y

δα δβ γ

δ →

+ ∂ ∂= = +

∂ ∂ .

For a 3-D element in general, the rate of change of the corner angle that is initially a

right angle between the i- j axes

( ) ji

ij

j i

uui j

x xγ

∂∂= + ≠

∂ ∂ ,

which is symmetric, i.e.,ij jiγ γ = .

• Rotation of a fluid element (about an axis which is perpendicular to the plane of the

fluid motion) is the average of the angular velocities of the two mutually

perpendicular sides of the element, where counterclockwise rotation is considered to

be positive:

1rotation about -axis:

2


2


2

z

x

y

v u z

x y

w v x

y z

u w y

z x

ω

ω

ω

⎛ ⎞∂ ∂= −⎜ ⎟

∂ ∂⎝ ⎠

⎛ ⎞∂ ∂= −⎜ ⎟

∂ ∂⎝ ⎠

∂ ∂⎛ ⎞= −⎜ ⎟

∂ ∂⎝ ⎠

Rotation (or angular velocity) vector ( ) x y z , ,ω ω ω ω =

Note that for a 2-D flow in the x- y plane, xω and

yω vanish identically; hence the

rotation vector is always perpendicular to the x- y plane.

• To generalize, we may define

1 1 1shear rate tensor angular deformation rate , and

2 2 2

ji

ij ij

j i

uue

x xγ

⎛ ⎞∂∂= + = =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

1rate of rotation vorticity, where

2

vorticity (curl of velocity)

22 y x

i j k

x y z

u v w

w v u wi j

y z z x

ω ω

=

= ×

∂ ∂ ∂=

∂ ∂ ∂

⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞= − + − +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

ω

ζ V

2 z

v uk

x y

ω

⎛ ⎞∂−⎜ ⎟∂ ∂⎝ ⎠

7



2=

ζ ω

The direction of a vector cross product is

determined by the right-hand rule.

The vorticity vector is equal to twice the

angular velocity vector of a rotating fluid

particle.

The difference between a rotational and irrotational flow: fluid elements in a rotational

region of the flow rotate about their own axis, but those in an irrotational region of the

flow do not.

In this incompressible and rotational flow, an

initially square fluid parcel will not onlyelongate, but also rotate about its axis as it moves

over the time periods shown in the figure.

8



C. The Reynolds Transport Theorem (Section 4.4)

Define:

The Reynolds transport theorem

(RTT) provides a link between

the system approach and thecontrol volume approach.

Two approaches of analyzing a problem.

(a) System approach: follow the fluid as

it moves and deforms; no mass crossesthe boundary. (b) Control volume

approach: consider the changes in a

certain fixed volume; mass crosses the

boundary.

- Material Volume: a volume that contains the same fluid as it moves and deforms

following the motion of the fluid- Material Surface: enclosing surface of a material volume; by definition no fluid

particles can cross it.

-

Control Volume: a volume of fluid in a flow field, usually fixed in space, to be

occupied by different fluid particles at different times.- Control Surface: imaginary or physical enclosing surface of a control volume.

- Flux: amount of property (e.g., mass, momentum, energy) crossing a unit area of asurface per unit time.

We state without proof the Reynolds transport theorem, which provides a basis for

developing differential equations for the various conservation laws:

rate of change of the local rate of change of the property within property within the fixed the material volume control volume that happens

to coincide with the materialvolum

MV CV

d bdV bdV

dt t ρ ρ

∂=

∂∫∫∫ ∫∫∫net out-flux of the

property across theentire control surface

e at that instant

CS b d A ρ +

∫∫ i

V n

where

density of fluid

an intensive property of fluid (property per unit mass)material volume that happens to coincide with at time t

b B MV CV

ρ =

==

9



control volume (fixed in space)

control surface

unit outward normal to

CV

CS

CS

=

=

=n

The integral of over the

control surface gives the net amount

of the property B flowing out of the

control volume (into the control

volume if it is negative) per unit time.

A moving system (hatched region)

and a fixed control volume (shaded

region) in a diverging portion of a

flow field at times t and

bV ndA

i ρ

t t + .

D. Conservation of Mass (Section 6.2)

If the property is mass, then b = 1, and

( )L.H.S. mass in 0

(by definition of , which always contains the same fluid)

MV d d dV MV dt dt

MV

ρ = =∫∫∫

( )

by Gauss theorem is stationary

R.H.S.

=

CV CS

CV CV

CV

dV d At

dV dV t

ρ ρ

ρ ρ

∂+

∂∂

+∂

∫∫∫ ∫∫

∫∫∫ ∫∫∫

V n

V

i

i

Equating L.H.S. and R.H.S., and removing the volume integral since CV is arbitrary, we

get the differential form of Continuity Equation

10



( )

( )

( )

0

or, using the identity

0

or in index form 0 , 1,2,3, summation over repeated indexi

i

t

d

dt ud

i jdt x

ρ ρ

ρ ρ ρ

ρ ρ

ρ ρ

∂+ =

∂

=

+ =

∂+ = =

∂

V

V V + V,

V

i

i i i

i

INCOMPRESSIBLE FLOW is defined as one in which the density of a fluid particle

is invariant with time 0d

dt

ρ ⇔ = , which implies

0

(ie, divergence of velocity is zero for incompressible flow)

In Cartesian coordinates, the continuity equation for inc

V =i

ompressible flow reads

0u v w

x y z

∂ ∂ ∂+ + =

∂ ∂ ∂

Note that a flow with constant density is always incompressible, but an incompressible

flow does not necessarily have a constant density (e.g., flow in a stratified sea).

E. Applied Forces

• Body force due to gravity on a small fluid element = dV ρ g

•

Surface stress =s τ ni , where n is the unit outward normal vector to the surface, and

( ) , 1, 2, 3

xx xy xz

yx yy yz ij

zx zy zz

i j

τ τ τ

τ τ τ τ

τ τ τ

⎡ ⎤⎢ ⎥

= = =⎢ ⎥⎢ ⎥⎣ ⎦

τ

are the stress components on an infinitesimal cubic fluid element.

ijτ is a second order tensor, where

the f irst index i denotes the f ace (on which the stress acts) being normal to i x ,

and the second index j denotes the stress component being in the j x direction.

11



In the textbook, the normal stress is denoted byiiσ in order to distinguish it from the

shear stress ( )ij i jτ ≠ .

It can be shown that ijτ is symmetric, ie, ij jiτ τ = . Therefore there are only 6

independent stress components.

F. Conservation of Linear Momentum (Section 6.3)

Apply Newton’s second law of motion to a material volume of fluid:

= +

rate of change surface bodyof momentum stress force

MV MS MV

d dV dA dV

dt ρ ρ ∫∫∫ ∫∫ ∫∫∫V s

g

The L.H.S. can be converted, using the transport theorem and the continuity equation,

into MV MV

d d dV dV

dt dt ρ ρ =∫∫∫ ∫∫∫

V V .

The first term on the R.H.S. is MS MS MV

dA dA dV = =∫∫ ∫∫ ∫∫∫s τ n τ i i on using Gauss theorem.

Plugging these terms back, and removing the volume integral since the volume is

arbitrary, we get the differential form of momentum equation

d

dt

ρ ρ = +V

τ g i

The left hand term is a total derivative, which can be expanded into the Eulerian form:

oriji i

j i

j j

u uu g

t t

τ

x x ρ ρ ρ

⎛ ⎞ ∂∂ ∂∂⎛ ⎞+ = + + = +⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

V V V τ g i i ρ

∂ ∂

By now, there are more unknowns than equations. To close the problem, we need to

introduce CONSTITUTIVE (stress vs strain-rate) relations to relate the stress and the

kinematics.

If the fluid is Newtonian, a linear relationship is followed

1 forwhere , dynamic viscosity coefficient

0 for

jiij ij

j i

ij

uu p

x x

i j

i j

τ δ μ

δ μ

⎛ ⎞∂∂= − + +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

=⎧= =⎨

≠⎩

Finally, on substituting the above relationship, we obtain the Navier-Stokes equations

12



( )

2

2

2

2

2

or in index form,

, 1,2,3, summation over repeated index

1or

(I) (II)

i i i j i

j i j

i i i j i

j i j

pt

u u u pu g i j

t x x x

u u u pu g

t x x x

ρ ρ μ

ρ ρ μ

ν ρ

∂⎛ ⎞+ = − + +⎜ ⎟∂⎝ ⎠

⎛ ⎞∂ ∂ ∂∂+ = − + + =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

∂ ∂ ∂∂+ = − + +

∂ ∂ ∂ ∂

V V V g V i

(III) (IV) (V)

where is the kinematic viscosityν μ ρ = /

Meanings of the five terms:-

(I) – local acceleration;

(II) – convective acceleration (inertia), nonlinear term of the equation;

(III) – pressure gradient;

(IV) – gravity;

(V)

– viscous diffusion of momentum owing to molecular viscosity of the fluid.

Now, we have 4 equations (1 continuity + 3 components of momentum) for the four

variables as functions of space and time, , , and x y zu u u p ( ), , , x y z t . Note that it is the

pressure gradient, rather than the pressure itself that drives the flow.

13



The Equations of Motion for an Incompressible Newtonian Fluid

In Rectangular Coordinates ( x , y, z )

2 2 2

2 2 2

2 2 2

2 2

Continuity: 0

1-component:

1-component:

y x z

x x x x x x x x y z

y y y y y y y

x y z

uu u

x y z

u u u u u u u p x x u u u g

t x y z x x y z

u u u u u u u p y u u u

t x y z y x y z

ν ρ

ν ρ

∂∂ ∂

+ + =∂ ∂ ∂

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂∂+ + + = − + + + +⎜ ⎟

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

∂ ∂ ∂ ∂ ∂ ∂ ∂∂+ + + = − + + +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2

2 2 2

2 2 2

1-component:

y

z z z z z z z x y z

g

u u u u p u u u z u u u

t x y z z x y zν

ρ

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + + +⎜ ⎟

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ zg

( , , ) are the components of the acceleration due to gravity in the , , and directions.

If, say, and are horizontal axes and is positive upward, then 0, and .

Also, the gravity

x y z

x y z

g g g x y z

x y z g g= = = −

( )can be combined implicitly with the pressure term by introducing

* . x y z p p p g x g y g z ρ ρ ≡ − − + + g x =i

g g

In Cylindrical Coordinates (r , , z )

( )

( )

2

2 2

2 2 2 2

1 1Continuity: 0

1-component:

1 1 2

r z

r r r r r z

r r r

ru u u

r r r z

u uu u u u pr u u

t r r r z r

uu uru g

r r r r r z

θ

θ θ

θ

θ

θ ρ

ν θ θ

∂ ∂ ∂+ + =

∂ ∂ ∂

∂ ∂ ∂ ∂ ∂+ + − + = −

∂ ∂ ∂ ∂ ∂

⎡ ⎤∂∂ ∂ ∂ ∂⎛ ⎞+ + − +⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦

+

( )2 2

2 2 2 2

1-component:

1 1 2

-component:

r

r r z

r

z z zr

u u u u u u u pu u

t r r r z r

u uuru g

r r r r r z

uu u u z u u

t r r

θ θ θ θ θ θ

θ θ θ θ

θ

θ θ ρ θ

ν θ θ

θ

∂ ∂ ∂ ∂ ∂+ + + + = −

∂ ∂ ∂ ∂ ∂

⎡ ⎤∂ ∂∂ ∂ ∂⎛ ⎞+ + + +⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦

∂ ∂ ∂+ + +

∂ ∂ ∂

+

2 2

2 2 2

1

1 1

z z

z z z z

u p

z z

u u ur gr r r r z

ρ

ν θ

∂ ∂= −

∂ ∂

⎡ ⎤∂ ∂ ∂ ∂⎛ ⎞+ + + +⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦

14



G. Scaling and Approximation

• Because of the inertia terms (convective acceleration), the Navier-Stokes (NS)

equations are non-linear equations.

•

Except for simple flow geometry, analytical solutions do not exist in general.

• Fortunately, for many practical applications, not all terms in the equations are equally

important, and therefore some subdominant terms can be dropped in favor of a firstapproximation of the problem. The approximate equations can then be solved

(analytically or numerically) with much greater ease than the full-blown ones.

•

It is important to judge, for a particular problem, the relative significance of the

individual terms in the NS equations, which can be reflected from the magnitude of

the corresponding non-dimensional parameters.

For illustration, consider incompressible unsteady flow past a body:

Body

L

U

Characteristic scales:

Length ( L); Time scale of unsteadiness (T ); Velocity (U ); Pressure (P)

Introduce dimensionless variables (distinguished by *):

/ , * / , / , * / , /U t t T L p p P g= = = = =V* V x* x g* g

the normalized NS can be expressed as

2

2 2*

*

L P gL p

UT t U U UL

ν

ρ

∗ ∗⎛ ⎞∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ = − + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

V * V * V*

∗ g* V i

*

The scales have been chosen to be representative of the variables so that all the

dimensionless terms are order unity. Now, the importance of each term (relative to

the inertia) is carried by its bracketed coefficient.

2

2

temporal accelerationStrouhal number (St)

convective accelertion

pressure forceEuler number (E)

inertia

ineritaReynolds number (Re)

viscous force

ineritaFroude number (Fr)gravi

L

UT

P

U

UL

U gL

ρ

ν

= =

= =

= =

= =ty force

15



Possible Cases of Simplification:-

Large Re

Re 1 negligible viscous effect,

1 NS reduces to Euler equations p

t ρ

⇒

∂+ = − +

∂

V V V g

i

Small StSt 1 negligible unsteady effect quasi-steady flow,

The local (temporal) acceleration term can be dropped.

⇒ ⇒

∴

Small Re

Re 1 negligible inertia effect (good news!)

Viscous force is significant, and is to be balanced by pressure gradient.For slow and viscous flow and negligible gravity, the flow is

called

⇒

Creeping Flow

21 0

Nonlinear inertia terms are now gone, analytical solutions are possible

if the flow geometry is simple enough.

p ν ρ

= − + V

Spatial DimensionAlso, it is often the case that the flow varies only in one or two spatial dimensions,

and therefore the problem can be reduced to a one- or two-dimensional problem, for

which only one or two velocity components need to be solved. Some common cases

of one-dimensional flow:

• fully developed pipe or channel flow: axial velocity as a function of radial

distance from center of pipe ( )u u r = , or longitudinal velocity as a function of

distance from the bottom of channel ( )u u y= ;

• axi-symmetrical flow: velocity is symmetrical about an axis (e.g., point

source/sink, vortex).

16



17



18



19



20



21



22



23



24



25



26



(II) SIMPLE (EXACTLY OR NEARLY ONE-DIMENSIONAL)

VISCOUS FLOW (Section 6.9)

A. Mathematical Formulation for a Fluid Dynamics Problem

Assumptions:

• constant fluid properties (density ρ , viscosity μ )

• Newtonian fluid (linear, isotropic and purely viscous material)

Basic Variables:

( )

( )

Velocity ( , , ) , , , (3 variables)

Pressure , , , (1 variable)

u v w x y z t

p p x y z t

= =

=

V V

Basic Governing Equations:

2

Continuity 0 (1 equation)

1 Navier-Stokes (3 equations) p

t ν

ρ

∂+ = − + +

∂

V =

V V V g V

i

i

Other derived variables:

( )Stress ( , , , ) , isotropic tensor

with stress components (see the definition on page 11):

2 , (no

T

xx

x y z t p

u p

x

μ

τ μ

⎡ ⎤= − ∇ ∇ =⎣ ⎦

∂= − +

∂

τ I + V + V I

rmal stress)

2 , (normal stress)

(shear stress)

yy

xy yx

v p

y

u v

y x

τ μ

τ τ μ

∂= − +

∂

⎛ ⎞∂ ∂= = +⎜ ⎟

∂ ∂⎝ ⎠

etc.

Vorticity w v u w v ui j k y z z x x y

⎛ ⎞ ⎛ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞= × − + − + −⎜ ⎟ ⎜⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ζ V = ⎞⎟

⎠

Boundary Conditions:

• No-slip boundary condition: the velocity of a fluid in contact with a solid

impermeable wall must equal that of the wall

fluid solid along a fluid-solid interface=V V

If in particular the wall is stationary, the fluid adjacent to the wall must have zero

velocity.

27



The development of velocity profiles due to the no-slip condition as a fluid flows past

a blunt nose and a flat plate.

• Interface boundary condition between two fluids: when fluid A and fluid B meets at

an interface, the velocity and stress must match between the two fluids at the interface

A B A B, along a fluid-fluid interface= =V V τ τ

If, say, the interface is flat (along x-direction) and the fluids are moving parallel to the

interface, the continuity of stress implies the continuity of pressure and shear stress at

the interface

A B A B

A B

,du du

p pdy dy

μ μ = =

• Free-surface boundary condition: a degenerate form of the above interface boundary

condition occurs at the free-surface of a liquid, meaning that fluid A is a liquid (say,

water, oil) and fluid B is a gas (usually air). By virtue of the factair liquid

μ μ , the

shear stress at the air-liquid interface is negligibly small, and it is reasonable to

approximate the shear stress to be at the interface, which is hence called a free surface,

liquid atmosphere liquid

liquid

, 0 along the free surfacedu p p

dyμ = =

28



•

Other boundary conditions, such as inlet condition, outlet condition, periodic

condition and symmetry, may also apply to certain types of boundaries, depending on

the problem.

Boundary conditions along a plane of symmetry are defined so as to ensure that the

flow field on one side of the symmetry plane is a mirror image of that on the other

side, as shown above for a horizontal symmetry plane.

Initial Condition If the problem is time dependent (i.e., unsteady), an initial condition alsoneeds to be specified.

*************************************************************************

Let us consider in the following sections a few applications of the Navier-Stokes equations,

in which the flow configuration is simple enough for analytical solutions (exact or

approximate) to be deduced. The assumptions are that the flow is steady ( / 0t ∴ ∂ ∂ = ),

laminar, and incompressible and the fluid is Newtonian.

B. Plane Poiseuille-Couette Flow

Note that this is a unidirectional flow ( ), 0u u y v= = . Therefore there is no dependence

on x for all variables: ./ 0 x∂ ∂ =

The flow is driven by three forcings: (1) motion of the upper plate; (2) pressure gradient

in the x-direction, / a constant p x∂ ∂ = ; (3) gravity, if x is not in a horizontal direction.

Recall the momentum equations:

x

Upper plate moving at a constant speed U

Lower fixed plate

y

u( y)

y = h

y = 0

29



-component:u

xt

∂

∂

uu

x

∂+

∂

uv

y

∂+

∂

2

2

1 p u

x xν

ρ

∂ ∂= − +

∂ ∂

2

2

2

2

1 ....................... (1)

-component:

x

x

ug

y

u pg

y x

v y

t

ν ρ

⎛ ⎞∂+ +⎜ ⎟⎜ ⎟∂⎝ ⎠

∂ ∂⇒ = −

∂ ∂

∂

∂

vu

x

∂+

∂

vv

y

∂+

∂

2

2

1 p v

y xν

ρ

∂ ∂= − +

∂ ∂

2

2

v

y

∂+

∂

1 0 ....................... (2)

y

y

g

pg

y ρ

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠

∂⇒ − + =

∂

Note that the inertia terms are identically zero, which is true for all unidirectional flows

irrespective of the Reynolds number.

Equation (2) simply gives that the pressure ( ) y p p x g y ρ = + .

The R.H.S. of equation (1) is constant, so the equation can be integrated twice with

respect to y, giving2

1 2( )2

x

p yu y g C y C

x ρ

μ

∂⎛ ⎞= − + +⎜ ⎟∂⎝ ⎠

where and are integration constants that can be determined using the boundary

conditions that1

C 2

C

( 0) 0 (no slip at the lower plate), and

( ) (speed of the upper plate).

u y

u y h U

= =

= =

Solving for these constants, we obtain the solution for the velocity profile (see Fig. 6.31

below):

22

( )2

Couette FlowPoiseuille Flow

x

p h y y yu y g U

x h h h ρ

μ

⎡ ⎤∂⎛ ⎞ ⎛ ⎞= − + − +⎢ ⎥⎜ ⎟ ⎜ ⎟∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

Couette flow is caused by the motion of a boundary wall moving in its own plane, while

Poiseuille flow is caused by axial pressure gradient or gravity in the direction of flow.

30



The shear stress in the flow is

constant stresslinear stress distribution due to due to Couette flowPoiseuille flow,zero stress at the centerline

( )

2

xy x

du y p h U g y

dy x h

τ μ ρ μ ∂⎛ ⎞ ⎡ ⎤

= = − + − +⎜ ⎟ ⎢ ⎥∂⎝ ⎠ ⎣ ⎦

The discharge (flow-rate) per unit width of channel is given by

3

0 12 2

h

x

p h hU Q udy g

x ρ

μ

∂⎛ ⎞= = − + +⎜ ⎟

∂⎝ ⎠∫

The volume flow averaged (mean) velocity2

/12 2

x p h U u Q h g x

ρ μ

∂⎛ ⎞= = − + +⎜ ⎟∂⎝ ⎠

It is left as an exercise for you to show the following

Given that / p x−∂ ∂ is a positive constant and 0 xg = , determine the location of the

maximum velocity. It is also the point where the shear stress vanishes (why?).

Hence, find the minimum value of U such that the shear stress will not vanish

throughout the flow.

C.

Circular Poiseuille Flow

We now consider laminar flow through a circular tube:

•

The objective to find the relationship between volumetric flow rate and pressure

change along a pipe of circular section.

• Examples include blood flow in capillaries, air flow in lung alveoli, where the

Reynolds number is not high enough for the flow to become turbulent.

• Navier-Stokes equations in cylindrical coordinates are to be used, where

/ 0θ ∂ ∂ = , since the flow is axially-symmetric (i.e., no dependence on angular

position in a cross-section of the flow).

• We have seen that the gravity can be combined with the pressure gradient in a

trivial manner, so let us ignore gravity in the following analysis.

Again, this is a unidirectional flow: 0, 0r u u uθ

r

( ) zu r

Circular pipe of radius R

z

z= = ≠

is driven by a constant andsteady pressure gradient dp/dz in the axial direction.

31



( )

The continuity equation reduces to

1

r ru

r r

∂

∂

1 u

r

θ

θ

∂+

∂0 must not depend on , ( )

The -component momentum equation is simplified to

z z z

z

uu z u

z

z

u

t

∂+ = ⇒ ∴ =

∂

∂

∂

zu r

zr

uu

r

∂+

∂ zu u

r

θ

θ

∂+

∂ z

z

uu

z

∂+

∂

2

2 2

1

1 1 z z

p

z

u ur

r r r r

ρ

ν θ

∂= −

∂

∂ ∂ ∂⎛ ⎞+ +⎜ ⎟

∂ ∂ ∂⎝ ⎠

2

2

zu

z

∂+

∂ zg

⎡ ⎤+⎢ ⎥

⎢ ⎥⎣ ⎦

2

1 2

, which can be integrated twice with respect to to give

( ) ln4

z

z

d du r dpr r

dr dr dz

r dpu r C r C

dz

μ

μ

⎛ ⎞⇒ =⎜ ⎟

⎝ ⎠

= + +

The two integration constants C and can be determined using the boundary

conditions:1 2C

1

2

2

( 0) is finite 0

( ) 0 (no slip at boundary wall)4

z

z

u r C

R dpu r R C

dzμ

= ⇒ =

= = ⇒ = −

Plugging back, we get the expression for the velocity profile

( )

2 2

214 z

dp R r

u r dz Rμ

⎡ ⎤

= − −⎢ ⎥⎣ ⎦ which is a parabolic distribution with the maximum at the center:

2

max ( 0)4

z

R dpu u r

dzμ = = = −

The flow-rate is4

02

8

R

z z A

R dpQ u dA u rdr

dz= = = −∫ ∫

π π

μ

The mean velocity is half the maximum velocity4

2max

2

8/ ..................... (1)

8 2

R dp

u R dpdzu Q A

R dz

π

μ π μ

−= = = − =

The shear stress at wall is given by

42

zw

r R

du R dp u

dr dz Rτ μ μ

=

= − = − =

For a given length L of the pipe, the pressure drop is ( )/ p dp dzΔ = − L

g

and the head loss

due to friction is h p / f ρ = Δ . Hence we may obtain from equation (1) the Darcy-

Weisbach equation

32





34



35



36



37



38



39



40



41



42





44



45



D. Nearly One-Dimensional Flow by Lubrication Approximation

Lubrication in a Slider Bearing

A slider bearing is designed as a thrust bearing to support very large loads. To carry these

loads, the fluid film between the solid surfaces must develop normal stresses, so we are

interested in predicting the pressure distribution and thus the load-carrying capacity of the

bearing. Typical examples of slider bearings are found in the shafts of screw-propelledships and in the high-speed turbines of electricity-generating stations. For example, the

thrust of the ship’s propellers may be transmitted through a series of pads (see the figure

below) to the hull of the ship. Each pad (slider) may be tilted slightly to account for the

relative effects of pressure, speed and viscosity, and thus maintain the fluid film between

the two surfaces (the slider and its guide) which are in relative motion, and thereby reduce

friction.

In most lubrication problems the relevant Reynolds number is so small that viscous terms

in the Navier-Stokes equation dominate completely. The reason is not necessarily that the

coefficient of viscosity is large; it is more due to the fact that the thickness of the film is

extremely small compared to the lateral dimensions of the bearing. The Reynolds number

may be defined asslider speed film thickness

1 Re ρ

μ

× ×=

Let us now formulate a model of the slider bearing with a planar face, with the further

assumptions that

1.

The lubricant is an incompressible Newtonian viscous fluid with constant viscosity.

2. The bearing has infinite length into the paper, and the bearing guide is flat. The gap

height h( x) between the slider and its guide varies so gently that the flow is nearly

one-dimensional through a section of the bearing.

3.

Gravity can be ignored.4. The flow has settled down and we need consider only the steady problem.

46



The local film thickness is

2 11

( ) h h

h x h x L

−⎛ ⎞= + ⎜ ⎟

⎝ ⎠ (1)

where so that the film is extremely thin.1 2h h L<

The flow is quasi-one-dimensional, and we may recall the equation for discharge for

combined Poiseuille-Couette flow (on page 31):

3

12 2

dp h hU Q

dx μ = − −

where the gravity has been ignored and U is now in the negative direction. By

conservation of mass, Q must be a constant. If there were Couette flow alone, the

discharge would decrease down the slider as the gap height decreases from to .

Therefore in order to balance the flow, the pressure gradient must not be zero.

Rearranging the terms the above equation gives

2h 1h

3

12

2

dp hU Q

dx h

μ ⎛ = − +⎜

⎝ ⎠

⎞⎟ (2)

We further suppose that both ends of the bearing are exposed to surrounding lubricant,or to the atmosphere. Then we have

0( ) (0) p L p p= =

as the boundary condition for the pressure. It follows that integrating (2) from 0 to L is

equal to zero

0

2 30

( ) (0) 0

02

L

L

dpdx p L p

dx

U Q

dxh h

= − =

⎛ ⎞⇒ − + =

⎜ ⎟⎝ ⎠

∫

∫

47



Now, on substituting (1) for h( x), the above integral can be carried out to give

( )1 2

1 2

h hQ U

h h

−=

+

We may put Q back into (2), which is integrated again, but the upper limit is now a

general position “ x”

0 2 30 0 ( ) 12

2

x xdp U Qdx p x p dx

dx h hμ

⎛ ⎞= − = − +⎜ ⎟

⎝ ⎠∫ ∫

After some algebra, we get

( )(0 12 2 2

2 1

6( )

( )

UL)2 p x p h h h h

h h h

μ = − − −

−

Note that sinc h 0e ,1 ( )h h x≤ ≤2 ( ) p x p≥ or a positive pressure distribution is

established within the bearing fluid to support the normal load. It can be readily shown

that, by setting dp/dx = 0 in (2), the pressure reaches a maximum

( )( ) ( )

2 1 1max 0

1 2 1 2 1 2

3 at

2

UL h h h L p p x

h h h h h h

μ −− = =

+ +

It may be shown that in the left-hand section of the bearing, the pressure gradient is

positive so that it drives fluid in the flow direction, and in the right-hand section the

pressure gradient is negative so that it drives fluid against the flow direction. In this way,

the Poiseuille flow will balance the Couette flow to result in a constant dischargethroughout the pad.

Lubrication performance is found to be favorable, since by using orders of magnitude

we may estimate that

0

2

0

Drag shear stress /1

Bearing load pressure /

L

L

dx U h h

UL h L pdx

τ μ

μ

∫

∫∼ ∼ ∼ ∼

By virtue of the small film thickness, the bearing can support a large load with only

small frictional resistance.

48



(III) INVISCID AND POTENTIAL FLOWS (Sections 6.4-6.6)

Analysis can be considerably simplified if the flow under consideration can be regarded as

INVISCID and IRROTATIONAL.

A.

Inviscid (Nonviscous) Flow (Section 6.4)• Flow of an ideal fluid with zero viscosity ( )0μ = would be inviscid exactly.

• In practice, flow is approximately inviscid when the effects of shear stresses on the

motion are small as compared to other influences. One guiding condition is that the

Reynolds number Re must be very large:

viscous force 11 , where Re

inertia force Re

VL ρ

μ =∼

• Many flows involving water or air, whose viscosity is small, can practically be

considered as inviscid as long as the viscous effects are not dominant (e.g., far from a

wall).• When the viscous force becomes negligible, the Navier-Stokes equations reduce to

Euler’s equations

nonlinear termis still here

1 (viscous term is missing here) p

t ρ

∂+ = − +

∂

V V V g

∇

i

• For incompressible flow, Euler’s equations of motion can be integrated along a

streamline to yield the Bernoulli equation (which you learnt already in Year I; read

Section 6.4.2 for a review)

2

constant along a streamline2

p z

g g ρ + + =V

• It is remarkable that the Bernoulli equation provides an algebraic (rather than vector

differential) relationship between pressure, velocity and position in the earth’s

gravitational field.

B. Irrotational (Potential) Flow (Section 6.4.3)

• Recall that vorticity (curl of velocity) is twice the rotation (angular velocity) of a fluid

element.

• A fluid element will acquire vorticity when acted upon by a couple to cause it to

rotate. One source of rotation is unbalanced shear stresses acting on its periphery.

When shear stresses are absent, it is possible that the flow is irrotational.

• A flow field is irrotational if, at every point, the vorticity vanishes or

0∇ ×V = .

• It can be shown that the flow of an inviscid fluid which is irrotational at a particular

instant of time remains irrotational for all subsequent times. That means, the motion

of an inviscid fluid which is started from rest is always irrotational (provided the flow

lies outside a boundary layer).

•

This result is known as the Persistence of Irrotational Motion of an inviscid fluid. It

is because the setting up of a rotation would require forces tangential to the boundary;

and such forces, which arise through the viscous properties of the fluid, are non-existent in the inviscid fluid model.

49



•

The constant in the Bernoulli equation becomes universal (i.e., not specific to a

streamline) when the flow is irrotational (Section 6.4.4). Therefore, for

incompressible irrotational flow, the Bernoulli equation can be applied between any

two points in the flow field:

2 2

1 1 2 21 2

2 2 p V p V z z

g g g g ρ ρ + + = + +

•

The procedures of finding a solution for an irrotational flow field are typically:

o Firstly, solve for the kinematics (velocity components) from an equation

derived from the condition of zero vorticity, which is the subject matter of the

following sections;

o Secondly, find the pressure from the Bernoulli equation.

You should appreciate that solving irrotational flow equations is usually much simpler

than solving the full Navier-Stokes equations.

• You are cautioned that irrotationality fails to apply to a boundary layer, which is a

thin layer that develops next to a solid wall owing to no-slip of the flow at the wall. No matter how small its viscosity is, a real fluid cannot “slide” past a solid boundary.

The flow in a boundary layer is always viscous and highly rotational (a rapid change

in velocity from zero at wall to the free stream value over a short distance); real fluid

behavior must be accounted for in a boundary layer (Chapter 9).

50



C. The Velocity Potential (Section 6.4.5)

• For any scalar field φ , curl(grad φ ) = 0 is an identity. See a proof below.

•

Alternatively speaking, a velocity field V is irrotational or curl V = 0 if and only if

there exists a scalar field φ such that V = grad φ .

• The scalar function is called velocity potential

Cartesian coordinates: , ,

1Cylindrical coordinates: , ,r z

u v w x y z

u u ur r

θ

φ

φ φ φ

φ φ

θ

≡ ∇

∂ ∂= = =

∂ ∂

∂ ∂= =

∂ ∂

V

∂

∂

z

φ ∂=

∂

Irrotational flow is therefore also called potential flow.

•

The velocity potential satisfies Laplace’s equation on substituting the above relation

into the continuity equation:

2 2 2

2 2 2

2 2

2 2 2

0 0, or 0

Cartesian coordinates: 0

1 1Cylindrical coordinates: 0

x y z

r r r r r z

φ φ

φ φ φ

φ φ φ

θ

= ⇒ = =

∂ ∂ ∂+ + =

∂ ∂ ∂

∂ ∂ ∂ ∂⎛ ⎞+ + =⎜ ⎟

∂ ∂ ∂ ∂⎝ ⎠

i iV

∇ ∇ ∇ ∇

• The immediate upshot is: for irrotational flow, one only needs to solve for a scalar

function (instead of a vector with 2 or 3 components) from one single equation in

order to determine the kinematics (good news!). However, the differential equation

for the scalar function is one order higher than that for the vector function (no free

51



lunch!). Once the potential is found, its spatial gradients will give the velocity

components.

D. Equipotential Lines and Streamlines (Sections 6.2.3 and 6.5)

• A two-dimensional potential flow field can be graphically represented using a flow

net composed of equipotential lines and streamlines.

• Equipotential lines are (contour) lines of constant velocity potential, while streamlines

are lines in the flow field that are everywhere tangent to the velocity. It can be shown

that these two sets of lines are orthogonal (i.e., they intersect each other at right

angles).

• You may recall the following mathematical statement:

It follows that:

equipotential lines streamlines equipotential lines

φ ≡ ⊥⇒ ⊥

V

The grad of a scalar function, say φ , gives the maximum rate of

spatial change of the function, and is in a direction normal to the local

line along which the function is constant..

52



Figure 6.15 shows a flow net for a 90o bend. A

flow net is useful in the visualization of a flow

pattern. To further understand what information a

flow net can provide, we need to knowsomething about stream function.

Stream Function

• For 2-D incompressible flow, another scalar function, viz stream function can be

introduced to identically satisfy the continuity equation.

( ) ( )A stream function , or , is defined such that

, for 2-D flow in Cartesian coordinates,

which satisfie

x y r

u v y x

ψ ψ θ

ψ ψ ∂ ∂= = −

∂ ∂

( )

s 0 identically.

1

, for 2-D flow in Polar coordinates,

which satisfies 0 identical

r

r

u v

x y

u u

r r uru

r

θ

θ

ψ ψ

θ

θ

∂ ∂+ =

∂ ∂

∂ ∂= = −

∂ ∂ ∂∂+ =

∂ ∂ly.

Note that the stream function is introduced based on kinematics consideration only. It

is definable for any two-dimensional incompressible flow fields, irrespective of the

flow being inviscid or not.

•

Physically, ψ is constant along a streamline since

V

ψ is constant

d dx dy vdx udy

x y

ψ ψ ψ

∂ ∂= + = − +

∂ ∂

That means, a line of constant ψ (along which 0d ψ = ) will have its slope in the same

direction of flow: . This is nothing but the defining property for a

streamline.

/ /dy dx v u=

Note that a solid boundary is always a

streamline. At a particular instant of time,

there is no fluid crossing any streamline,

and distinct streamlines cannot cross.

53



•

Given any two points in space whose stream function values are known, then the

volume flow rate across any line joining these two points is equal to the difference in

values of their stream functions.

2

12 1

One can readiy see from the above figure that

dq udy vdx dy dx d y x

q d ψ

ψ

ψ ψ ψ

ψ ψ ψ

∂ ∂= − = + =

∂ ∂

∴ = = −∫

• If the 2-D flow is irrotational, the stream function also satisfies Laplace’s equation,

since

2 2

2 2

0 0

0

0

v u

x y

x x y y

x y

ψ ψ

ψ ψ

∂ ∂× = ⇒ − =

∂ ∂

⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞⇒ − − =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

∂ ∂

⇒ + =∂ ∂

V

Therefore, for two-dimensional irrotational flow, both the velocity potential φ and the

stream function ψ satisfy Laplace’s equation. They are called harmonic functions,

and they are harmonic conjugates of each other. These functions are related, but

their origins are different:

– The stream function is defined by continuity; the Laplace equation for ψ results

from irrotationality.

– The velocity potential is defined by irrotationality; the Laplace equation for φ

results from continuity.

By now, referring back to Figure 6.15, you should understand that in a flow net the

velocity is roughly given by

V n s

φ ψ Δ Δ≈ ≈

Δ Δ

where is the spacing between two adjacent equipotential lines, and is the

spacing between two adjacent streamlines. Therefore, the velocity is higher in a

region where the mesh is finer, and lower where the mesh is coarser.

nΔ sΔ

54



E. Some Simple Plane Potential Flows (Sections 6.5.1-6.5.4)

1) Uniform Flow with constant velocity U

For case (a) where the flow is purely in the x-direction:

( )

( )

Velocity potential cos

equipotential lines are parallel to the -axis

Stream function sin

streamlines are parallel to the -axis

Ux Ur

y

Uy Ur

x

φ θ

ψ θ

= =

∴

= =

∴

Can you write down the corresponding φ and ψ for case (b) where the flow is at an

angle α with the x-axis?

2)

Source and Sink

A 2-D source is a line (from a mathematical perspective)

that runs perpendicular to the plane of flow and injects

fluid equally in all directions. The figure shows the flow

field of a source at the origin, from which fluid particles

emerge and follow radial pathlines. The strength of a

source, denoted by m, is the volume rate of flow

emanating from unit length of the line.

/m V L=

55



By conservation of mass, 2r m ruπ = for any radial distance r from the source located at the

origin. Hence,1

2

mu

r r

1, 0

r ur r r

θ

φ ψ φ ψ

θ

∂ ∂= = = = − =

∂ ∂θ π

∂ ∂=

∂ ∂. On integrating,

( )

( )

Velocity potential ln2

equipotential lines are concentric circles centered on the origin

Stream function2

streamlines are radial lines

mr

m

φ π

ψ θ π

=

∴

=

∴

The radial and tangential velocities are:

02r

mu ur

θ π = =

o when 0m > , the flow is radially outward, the origin is a SOURCE

o when 0m < , the flow is radially inward, the origin is a SINK

o the origin is a singularity where r u → ∞

o conservation of mass is satisfied everywhere except the origin

3) Vortex

In contrast to a source, a vortex has the pathlines

being circles centered on the origin, and fluid

particles move along these circles. The vortex can be used to model the flow round the plughole in a

bathtub. An irrotational vortex is called a free

vortex. The strength of a vortex is measured by the

circulation around a closed curve C

that encloses the center of the vortex. Hence,

C

d Γ = ∫V si

1 1

2r

r r r r r θ

0,u uφ ψ φ ψ Γ

θ θ π

∂ ∂ ∂ ∂= = = = = − =

∂ ∂ ∂ ∂.

On integrating,

( )

( )

Velocity potential2

equipotential lines are radial lines

Stream function ln2

streamlines are concentric circles centered on the origin

r

φ θ π

ψ π

Γ=

∴

Γ= −

∴

The radial and tangential velocities for a free vortex are

02

r u ur

θ π Γ= =

56



o

the flow is not defined at the origin

o the vorticity curl V = 0, except at r = 0, where V is not defined

o free vortex (a) is irrotational flow, tangential velocity decreases radially 1u r θ

−∝

o forced vortex (b) is rotational flow, tangential velocity increases radially u r θ ∝

4)

DoubletConsider a combination of a source and a sink of equal strength m and separated at a

distance 2a (left figure):

If the source and sink are moved indefinitely closer together ( )0a → in such a way

that the product 2am (distance apart × strength) is kept finite and constant, then we

obtain a doublet. The streamline pattern for a doublet is shown in the right figure

above. The line joining the source to the sink is called the axis of the doublet, and is

taken to be positive in the direction from sink to source. The strength of the doubletis /K ma π = .

57





examples, you will see how ideal flows can be described by a combination of basic

solutions. The key thing is to locate the dividing streamline. The general procedures are

as follows. (1) Sketch some streamlines for the combined flow. (2) Find the location of a

stagnation point where the velocity vanishes (you may expect that when flow past a body,

there is a point somewhere on the body surface where the flow velocity is zero). (3)

Evaluate the stream function at the stagnation point stagnationψ . (4) The dividing streamline,

which passes through the stagnation point, can be determined by letting the stream

function be equal to the stagnation stream function ( ),r ψ θ = stagnationψ .

1) Source + Uniform Flow = Flow Past a Half Body

It is more convenient to use polar coordinates ( ),r θ where r is the radial distance

from the source. It is along the negative x-axis where the flow due the uniform flow is

directly opposite to that due the source. At a point x b= − the velocities due to the two

flows cancel each other, and this is identified as the stagnation point.

( )uniform flow velocity radial outward flow due to source

2 2

r U um m

U bb U π π

r b∴ = =⇒ = ⇒ =

( ) uniform flow sourceCombined stream function ,

2

r

mUy

ψ θ ψ ψ

θ π

= +

= +

sin2

mUr θ θ

π = +

The stream function at the stagnation point has the value( )stagnation ,

2

mr b bU ψ ψ θ π π = = = = =

Therefore the dividing streamline that passes through the stagnation point is given by

( )

( )

( )

stagnation ,

or sin2 2

orsin

or

r

m mUr

br

y b

ψ θ ψ

θ θ π

π θ

θ

π θ

=

+ =

−=

= −

59



This streamline, which has the shape shown above, can be considered as a solid

boundary of a half-body that extends from x b= − to x → +∞ . The flow exterior to

this streamline represents the flow past a half-body, whose thickness at large x can be

estimated to be 2 bπ , since

0 as

2

b y

b

π θ

π π

⎧ ⎧→ →⎨ ⎨

−⎩ ⎩

Note that the every fluid particle emanated from the source is completely enclosed

within the dividing streamline. The flow pattern around the half-body is described by

streamlines stagnationψ ψ > . The velocity components and the pressure can then be

determined as described in earlier sections.

2) Source + Sink + Uniform Flow = Flow Past a Rankine Oval

The source and the sink are of the same strength: any mass of fluid injected by the

source is eventually drawn into the sink. The dividing streamline is now a closed

curve. This finite body, called Rankine Oval, has two stagnation points, one at the

front end and the other at the rear end of its boundary.

3) Doublet + Uniform Flow = Flow Past a Circular Cylinder

As the source and the sink combine to become a doublet, the Rankine Oval becomes a

circular cylinder. As the flow past a circular cylinder is of fundamental interest, let us

examine the flow in some detail.

60



The combined stream function is

sinsin

( strength of doublet)

K Ur

r

K

θ ψ θ = −

=

The radial velocity is2

1cosr

K u U

r r

ψ θ

θ

∂ ⎛ ⎞= = −⎜ ⎟∂ ⎝ ⎠

Obviously, the radial velocity vanishes at a circular surface with the radius

( )1/ 2

/r a K U = =

This defines the dividing streamline, which represents the surface of a circular

cylinder of radius a. Substituting a for K , the stream function can be written as2

2

1 sina

Ur r

ψ θ ⎛ ⎞

= −⎜ ⎟⎝ ⎠

from which we obtain the velocity components2

2

2

2

11 cos

1 sin

r

au U

r r

au U

r r θ

ψ θ

θ

ψ θ

⎛ ⎞∂= = −⎜ ⎟

∂ ⎝ ⎠

⎛ ⎞∂= − = − +⎜ ⎟

∂ ⎝ ⎠

On the cylinder surface the tangential velocity is,r a= 2 sinsu U θ θ = − . As expected,

there are 2 stagnation points, at 0,θ π = .

The pressure distribution on the cylinder surface can be found from the Bernoulli

equation

( )

2 2

0

2 2

0

1 1

2 2

1 1 4sin

2

s s

s

p U p u

p p U

θ ρ ρ

ρ θ

+ = +

⇒ − = −

where 0 p is the far upstream pressure. It is

remarkable that the pressure distribution is

symmetrical about the horizontal and thevertical diameters. Therefore there is no net

force arising from the pressure distribution

around the cylinder in both streamwise and

lateral directions. In other words, both drag

and lift forces are exactly zero, as predicted

from the potential flow theory.

This zero drag prediction is contrary to what has been observed in reality. There is

always a significant drag developed on a cylinder when it is placed in a stream of

moving fluid. This discrepancy is called d’Alembert’s Paradox, which was not

explained until the concepts of boundary layer and flow separation were developed.A comparison between the inviscid and the real pressure distributions is shown above.

61

http://en.wikipedia.org/wiki/D'Alembert's_paradox

http://en.wikipedia.org/wiki/D'Alembert's_paradox



4) Free Vortex + Doublet + Uniform Flow = Flow Past a Rotating Circular Cylinder

The effect of adding a vortex is to upset

the symmetry of flow about the

horizontal diameter. Therefore, the

pressure in the upper half of the cylinderis not balanced by the pressure in the

lower half. This results in a net lift

force acting laterally on the cylinder.

5) Sink + Free Vortex = Spiral Flow

6) Two separated sources of equal strength = source flow with a neighboring wall

62



63



64



65





67



68



69



70





72



73



74



(IV) FLOW PAST A BODY AND BOUNDARY LAYER THEORY

(Chapter 9)

A. Introduction (Section 9.1.2)

In 1904, Prandtl developed the concept of the boundary layer, which provides an

important link between ideal-fluid flow (inviscid irrotational flow) and real-fluid flow(viscous rotational flow). It was accepted that for fluids with relatively small viscosity (or

more exactly, flow with a high Reynolds number), the effect of internal friction in the

fluid is appreciable only in a narrow region surrounding the fluid boundaries. Therefore

the flow sufficiently far away from the solid boundaries may be considered as ideal flow

(in which effects of viscosity are neglected). However, flow near the boundaries suffers

retardation by the boundary shear forces and at the boundaries the velocity is zero (no-

slip condition). A steep velocity gradient is therefore resulted in a thin layer adjacent to

the boundaries, which is known as the boundary layer. It is of great significance when

behavior of real fluid is considered. For example, it explains the d’Alembert’s paradox –

the drag force experienced by a cylinder in stream that cannot be predicted with a

potential theory.

Flow of a uniform stream parallel to

a flat plate. The larger the Reynolds

number, the thinner the boundary

layer along the plate at a given x-

location.

75



Flow past a circular cylinder; the boundary layer separates from the surface of the body in

the wake for large Reynolds number.

B. Description of the Boundary Layer (Section 9.2.1)

(1) Development of the Boundary Layer

nominal limit

of boundarylayer

u = 0.99 U y

76



U

• On-coming flow is irrotational and has a uniform velocity U .

• The boundary layer starts out as a laminar boundary layer, in which fluid particles

move in smooth layers and the velocity distribution is approximately parabolic. As the

flow moves on, the continual action of shear stress tends to slow down additional

fluid particles, causing the boundary layer thickness to increase with distance

downstream from the leading edge. See below for a definition of the boundary layer

thickness.

• The flow within the boundary layer is subject to wall shear, and dominated by viscous

forces. The velocity gradient (hence the rotation of fluid particles) is the largest at the

wall, and decreases with distance away from the wall, and tends to zero on matching

with the main stream flow. Roughly speaking, the flow is said to be rotational within

the boundary layer, but is irrotational outside the boundary layer.

•

As the thickness of laminar boundary layer increases, it becomes unstable and some

eddying commences. These changes take place over a short length known as thetransition zone.

• It finally transforms into a turbulent boundary layer, in which particles move in

haphazard paths. Due to the turbulent mixing, the velocity distribution is much more

uniform than that in the laminar boundary layer. The increase of thickness along the

plate continues indefinitely but with a diminishing rate. If the plate is smooth (i.e.,

negligible roughness size), laminar flow persists in a very thin film called the viscous

sub-layer in immediate contact with the plate and it is in this sub-layer that the greater

part of the velocity change occurs.

Comparison of laminar andturbulent flat plate

boundary layer profiles

(left: non-dimensionalized

by the boundary layer

thickness; right: in physical

variables).

77



(2) Thicknesses of the Boundary Layer

i) Boundary Layer Thickness δ

The velocity within the boundary layer increases to the velocity of the main stream

asymptotically. It is conventional to define the boundary layer thickness δ as the distance

from the boundary at which the velocity is 99% of the main stream velocity.

There are other ‘thicknesses’, precisely defined by mathematical expressions, which are

measures of the effect of the boundary layer on the flow.

ii) Displacement Thickness δ∗

It is defined by

*

01

udy

U δ

∞ = −

∫

δ∗ is the distance by which the boundary surface would have to be shifted outward if the

fluid were frictionless and carried at the same mass flowrate as the actual viscous flow. It

also represents the outward displacement of the streamlines caused by the viscous effects

on the plate. Conceptually one may ‘add’ this displacement thickness to the actual wall

and treat the flow over the ‘thickened’ body as an inviscid flow.

iii) Momentum Thickness θ It is defined by

01

u udy

U U θ

∞ = −

∫

θ is the thickness of a layer of the main stream whose flux of momentum equals the

deficiency in the boundary layer, equivalent to the loss of momentum flux per unit width

divided by 2U ρ due to the presence of the growing boundary layer. The momentum

thickness is often used when determining the drag on an object.

Note that when evaluating the above integrals for *δ and θ, the upper integration limit

can practically be replaced by δ.

78



C. Laminar Boundary Layer Over a Flat Plate

• Heuristic Analysis

x

δ (x)

U

leading edge

of plate

Consider steady flow past a flat plate at zero incidence. The effect of viscosity is to

diffuse momentum normal to the plate. Consider a fluid element that is close enough

to the wall to be influenced by viscosity. In travelling a distance x, it has been

influenced by viscosity for a time . The influence of viscosity will have

spread laterally to a distance

/t x U ∼

( )

( )

1/ 21/ 2

1/ 21/ 2

or Re x

xt

U

x Ux

ν δ ν

δ ν −

≡

∼ ∼

∼

The above analysis is rather crude, and does not yield a full equation for the growth of

the boundary layer thickness. It however correctly describes one important

relationship for the laminar boundary layer: where( )1/ 2

/ Re x

xδ −

∝ Re / x

Ux ν ≡

x

is

the local Reynolds number in terms of the distance from the leading edge x. This

relationship is found to be valid at a distance far behind the leading edge: / 1.δ

The heuristic analysis can be further carried on to find relations for the wall stress:1/ 2

3

0

or ww

y

u U

y x

τ ν τ ρν ν

ρ δ =

∂=

∂ ∼ ∼

U

The wall shear stress wτ decreases with increase of x until the boundary layer turns

turbulent. The local friction coefficient, which is defined as follows, is given by

( )1/ 2

1 2

2

2 Rew f xC

U

τ

ρ

−≡ ∼

While the numerical factor of 2 is far from the true value, the functional dependence

of C f on Re x is correctly predicted.

79



•

Exact Solution by Blasius (Section 9.2.2)

A more rigorous analysis, using the technique of similarity solution, was developed

by Blasius for the laminar boundary layer over a flat plate. While the details of the

analysis are beyond the scope of this course, it is important to note the following

results derived from Blasius’ solution.

( )

( )

( )

( )

1/ 2

1 12

2

1 12

2

5 boundary layer thickness

Re

0.664friction coefficient

Re

1.328drag coefficient

Re

x

w f

x

D

L

x

x

C U

C U L

δ

τ

ρ

ρ

=

≡ =

≡ =D

/ 2

/ 2

1

whereD

is the skin friction drag force on unit width of a plate of length L:

.0

L

wdxτ = ∫D

D. The Boundary Layer Momentum-Integral Equation (Section 9.2.3)

By virtue of the property that the boundary layer thickness δ is much smaller than the

streamwise length scale (say, L): / Lδ , one may simplify the Navier-Stokes equations

to obtain the boundary-layer approximation:

2

2

continuity 0

1-momentum

1-momentum 0

u v

x y

u u p x u v

u

x y x

p y

y

ν ρ

ρ

∂ ∂

+ = ∂ ∂

y

∂ ∂ ∂+ = − +

∂

∂ ∂ ∂ ∂

= −∂

∂

with the boundary conditions:

( , ) (0,0) at 0

, as

(where , are the velocity and pressure

of the inviscid flow just outside the boundary layer)

u v y

u U p P y

U P

= =

= = → ∞

From the y-momentum equation, it is clear that the pressure in the boundary layer is

constant laterally across the layer and equal to the near-wall pressure of the inviscid flow

outside the boundary layer.

On integrating the x-momentum equation with respect to y from y = 0 to y = δ, and after

some algebra including the use of the continuity equation, one may obtain the Karman

momentum integral equation

80



( )2 *

where wall shear stress

density near-wall velocity of the outer inviscid flow

moment

w

w

d dU U U

dx dx

U

τ ρ θ ρ δ

τ

ρ

θ

= +

=

==

=0

*

0

um thickness 1

displacement thickness 1

boundary layer thickness

u udy

U U

udy

U

δ

δ

δ

δ

= −

= =

=

∫

∫ −

0

This momentum integral equation is applicable to laminar, transitional or turbulent boundary layer.

In particular, in the absence of pressure gradient (e.g., flow over a flat plate), the free stream velocity

U = constant and dU , and therefore the momentum integral equation reduces to/ dx =

2

w

d U

dx

θ τ ρ =

by which the skin friction drag and drag coefficient are simply given by

22 2

1 1 20 0

2 2

2, 2

L L L L

w L DU L

U d dx U dx U C

dx U L L ρ

ρ θ θ θ τ ρ ρ θ

ρ = = = = = =∫ ∫

DD

where Lθ is the momentum thickness at L= .

(1)

Laminar Boundary Layer Over a Flat Plate Revisited – approximate solution by

momentum integral equation

It is remarkable that approximate solutions, which are reasonably close to the exact

ones, can be obtained for the boundary layer thickness and drag coefficients from the

momentum integral equation on adopting an assumed velocity profile

( )

( )

where is the -coordinate normalized with respect to the local boundary

layer thickness.

u f

U

y y

x

η

η δ

=

=

The steps are as follows:-

a) Find the relation between θ and δ

( ) (1

0 0

1 1 , is a numerical constantu u

dy f f d a a

U U a

δ

θ δ η δ

= − = − =

∫ ∫

)

81



b) Find the wall shear stress from Newton’s law of viscosity

( )

0 0

0

/

/

is another numerical constant

w

y y

b

du U u U

dy y

U df U

b bd η

τ ρν ρν δ δ

ρν

ρν δ η δ

= =

=

∂= =

∂

= =

c) Substitute θ and wτ into the momentum integral equation

2

2

1

2

U d b a U

dx

b d d

aU dx dx

ρν δ ρ

δ

ν δ δ δ

=

⇒ = =

Integrating the above equation with respect to x, assuming that δ = 0 at x = 0:

( )

2

1/ 2

2

2 /

Re x

b

aU

b a

x

ν δ

δ

=

⇒ =

Furthermore,

( )

( )

1 12

2

1 12

2

2friction coefficient

Re

2 2drag coefficientRe

w f

x

D

L

abC

U

abC U L

τ

ρ

ρ

≡ =

≡ =D

/ 2

/ 2

)

It turns out that the values of a and b are rather insensitive to the choice of the

approximate velocity profile u U / ( f η = as long as it is a reasonable one

satisfying the boundary conditions.

Some assumed velocity profiles are

( )

2

3

parabolic

cubic2 2

sine2

η η

η

πη

−

2

3

sin

f η η

= −

which satisfy

( )

( )

( )

0 0, (no-slip at 0)

1 1, ( at )

' 1 0 (no stress at )

f y

f u U y

f y

δ

δ

= =

= = =

= =

.

82



(2) Turbulent Boundary Layer Over a Flat Plate (Section 9.2.5)

The one-seven-power law, suggested by Prandtl, is used for the velocity profile in the

turbulent boundary layer with zero pressure gradient:

( )1/ 7

1/ 7 oru y

f U

η η δ

= =

by which the momentum thickness is

( ) ( )1 1

1/ 7 1/ 7

0 0

71 1

72 f f d d θ δ η δ η η η = − = − =∫ ∫ δ

The one-seven-power law fails to describe the velocity profile at , where

. The following empirical formula obtained for pipe flow can be adoptedhere:

0 y =

/u y∂ ∂ → ∞

1/ 4

20.0225w U

U

ν τ ρ

δ

=

Substituting θ and wτ into the momentum integral equation, and integrating with

respect to x:

1/ 4

5/ 44 72 0.0225constant

5 7 x

U

ν δ

× = +

It is assumed that the turbulent boundary layer starts from 0. x = (This is a

contradiction to the fact that the boundary layer starts out as a laminar one, but this

assumption has given good results.) Therefore, the constant = 0. Further simplificationyields

( ) ( ) ( )1/5 1/5 1/52

0.370 0.0288 0.072, ,

Re Re Re

w D

x x

C x U

L

τ δ

ρ = = = .

These results are valid for smooth flat plates with 5 1 .5 7

0 Re 10 L× < <

Note that for the turbulent boundary layer flow the boundary layer thickness increases

with x as4 /5

xδ ∼ and the shear stress decreases as1/ 5

wτ −

∼ . For laminar flow thesedependencies are 1/ 2 and 1/ 2 x− , respectively.

83



E. Effect of Pressure Gradient (Section 9.2.6)

The pressure in the streamwise direction (i.e., along the body surface) will not be constant

if the body is not a flat plate. Consequently, the free stream velocity at the edge of the boundary layer U is also not a constant but a function of x. Whether the free-stream flow

is accelerating or decelerating along the body surface will have dramatically different

effects on the development of the boundary layer.

Let us re-examine flow past a circular cylinder, and find out what causes d’Alembert’s paradox.

You may recall that inviscid flow past a circular cylinder has a symmetrical pressuredistribution around the surface of the cylinder about the vertical axis. This results in a

zero pressure drag, which is however not true in reality for any fluid with a finiteviscosity. Such discrepancy is now referred to as d’Alembert’s paradox.

Despite the discrepancy, the potential theory helps to reveal that the pressure and hence

the free-stream velocity U fs on the cylinder’s surface are not constant. From A to C , the pressure gradient is negative and the flow is accelerating, and from C to F , the opposite istrue.

84



The real fluid flow past a circular cylinder is like this:

Flow Past A-B-C

the streamlines are converging, i.e., flow is accelerating, and the free-stream velocityU reaches a maximum at C .

•

• the pressure is decreasing along the cylinder surface, i.e., / 0 p x∂ ∂ < , net pressure

force is in forward direction, and the pressure gradient is said to be ‘favorable’.

the accelerating flow tends to offset the ‘slowing down’ effect of the boundary on thefluid. Therefore, the rate of boundary layer thickening decreases and flow remains

stable.

•

Flow Past C-D

the streamlines are diverging, and the flow is retarding.•

• the pressure is increasing along the cylinder surface, i.e., / 0 p x∂ ∂ > , net pressure

force opposes the flow, and the pressure gradient is said to be ‘adverse’ or

‘unfavorable’.

•

y

it reduces the energy and forward momentum of the fluid particles in proximity to thesurface, causing the thickness to increase sharply and fluid near the surface be brought

to a standstill ( /u∂ ∂ at the surface is zero) at D. See figure (b).

85



Flow Past D-E-F

flow close to the cylinder surface starts to reverse at D (separate point), i.e., fluid nolonger to follow the contour of the surface. The phenomenon is termed separation.

•

•

•

large irregular eddies formed in the reverse flow (the wake), in which much energy islost to heat.

the pressure in the wake remains approximately the same as at the separation point D,and is therefore lower than that predicted by the inviscid theory (see figure c). Thislowering of pressure behind the cylinder resulting from flow separation leads to a net

pressure drag on the cylinder. This explain d’Alembert’s paradox. Note that the wider

the wake, the larger the pressure drag, and vice versa.

Influence of the pressure gradient

86





F. Drag (Section 9.3)

Any object moving through a fluid (or a stationary object immersed in a viscous flow)

will experience a drag, – a net force in the direction of flow due to the pressure andshear forces on the surface of the object.

D

Drag = Pressure Drag + Skin Friction Drag

where

Pressure Drag = resultant force arising from the non-uniform and asymmetrical pressure

distribution around the surface the body. It is also called form drag as itdepends on the form or the shape of the body.

Skin Friction Drag = resultant force due to fluid shear stress on the surface of the object.

cos w p dA sin dAθ τ θ = +∫ ∫D

The drag coefficient C D is given by the ratio of the total drag force to the dynamic force

1 2

2

DC U A ρ

= D

where U = relative velocity of fluid far upstream of the object, A = frontal area – the projected area of the object when viewed from a direction

parallel to the oncoming flow if it is a blunt (or bluff) object (e.g., a cylinder);

or the planform area – the projected area of the object when viewed from

above it if it is a streamlined object (e.g., a flat plate).

drag

88



Typically the drag coefficient depends on(i) the shape of the object,

(ii) orientation of the object with the flow (e.g., a flat plate normal to flow has adifferent C D than a flat plate parallel to flow),

(iii) the Reynolds number Re /UD ν = where D is a characteristic dimension of the

object,(iv) surface roughness if the drag is dominated by skin friction and the boundary

layer is turbulent.

Flow Past a Flat Plate

When a flat plate is held normal to flow, the flow is

separated upon past over the plate. A region of

eddying motion (wake) is formed at the rear of the plate, the pressure there being much reduced.

Therefore the pressure drag is dominant, and the

plate is a bluff body in this position. The drag showslittle dependence on the Reynolds number.

When a flat plate is held parallel to flow, formation of the boundary layer over the plate isappreciable and flow separation is

negligible. Therefore the skin frictiondrag is significant. The plate is a

streamlined body in this position.

The drag coefficient increases when the boundary layer becomes turbulent.

Flow Past a Circular Cylinder/Sphere

89



• Re ≤ 1 – creeping flow

– no flow separation

– DC decreases with increasing Re ( 64/ Re D

C = for a sphere)

(Note that a decrease in the drag coefficient with Re does not necessarily imply acorresponding decrease in drag. The drag force is proportional to the square of the

velocity, and the increase in velocity at higher Re will usually more than offset thedecrease in the drag coefficient.)

• Re = 10 – separation starts occurring on the rear of the body forming a pair of vortex bubbles

there – vortex shedding begins at Re ≅ 90, leading to an oscillating Karman vortex street

wake (see next page)

– region of separation increases with increasing Re

– DC continues to decrease with increasing Re until Re = 10

3, at which pressure

drag dominates

•

3 5

10 Re 10< <

– DC remains relatively constant, which is a characteristic behavior of blunt bodies

– flow in the boundary layer is laminar, but the flow in the separated region is

highly turbulent, thereby a wide turbulent wake

•

5 610 Re 10< <

– a sudden drop in DC somewhere within this range of Re

– this large reduction in DC is due to the flow in the boundary layer becoming

turbulent, which moves the separation point further on the rear of the body,

reducing the size of the wake and hence the magnitude of the pressure drag. Thisis in sharp contrast to streamlined bodies, which experience an increase in the drag

coefficient (mostly due to skin friction drag) when the boundary layer turnsturbulent.

90



Laminar boundary layer separation with

a turbulent wake for flow past a circular

cylinder at Re = 2000.

(a) (b)

low over (a) a smooth sphere at Re = 15,000, and (b) a sphere at Re = 30,000 with a tripire; the delay of boundary layer separation is clearly seen by comparing these two

hotographs. The delay of separation in turbulent flow is caused by the rapid fluctuations

f the fluid in the transverse direction, which enables the turbulent boundary layer toavel farther along the surface before separation occurs, resulting in a narrower wake andsmaller pressure drag. Recall also that turbulent flow has a fuller velocity profile as

ompared to the laminar case, and thus it requires a stronger adverse pressure gradient to

vercome the additional momentum close to the wall.

Fw

p

otr a

c

o

Karman Vortex StreetsT

91

he Karman vortex street is one of the best-known vortex patterns in fluid mechanics.

al to

res.

eriodic flow is that the forces on theecause the flow

ually

The vortex street is just a special type of unsteady separation over bluff bodies such as a

cylinder. The vortex street is highly periodic having a frequency which is proportionU/D, where D is the length of the bluff body measured transverse to the flow and U is the

incoming flow speed. This periodicity is responsible for the "singing" of telephone wi

In fact, vortex streets are almost always involved when the wind generates a fairly puretone as it blows over obstacles.

A practical consequence of the regular,

p body are also periodic. B

is asymmetric fore and aft as well as inthe direction transverse to the flow, the

body will experience both an oscillating

drag and lift. If the frequency of theshedding is close to a structural

frequency, resonance can occur, uswith unpleasant results.



92



93





95



96





98



99



100



101



102



(V) OPEN-CHANNEL FLOW (Chapter 10)

A. Introduction

•

Studies of open channel flow are important for design and planning of river control,

inland navigation, surface drainage, irrigation, water supply and urban sanitation.

•

An open channel is a conduit in which the liquid flows with a free surface subjected toatmospheric pressure. It includes river channels (natural water courses), constructed

channels (e.g., canals, flumes) and enclosed conduits (e.g., sewers, culverts) operating

partially full.

• Open channels normally have a very small slope 0.01S < , and the pressure variation

with depth is nearly hydrostatic. The hydraulic grade lines for all the streamtubes are

the same and coincide with the free surface of the flow. Open channel flow is

essentially caused by slope of the channel and self-weight of the liquid.

B.

Types of Flow

1. Uniform and Non-Uniform FlowUniform flow – mean velocity does not change (in both magnitude and direction) from

one section to another along the channel. With a free surface this implies

a constant cross section and flow depth, which is called the normal

depth. Hence the liquid surface is parallel to the base of the channel.

Uniform flow results from an exact balance between the gravity and

frictional effects. (NB: uniform channel flow however does not require

uniformity of velocity across any one section of the flow.)

Non-uniform (varied) flow – the mean velocity V and the fluid depth y change with

distance x along the channel. Non-uniform channel flow can be classified

into gradually, or rapidly varying flow when the flow depth changesslowly ( )/dy dx1 , or rapidly ( )/dy dx ∼1 with distance along the channel.

2. Steady and Unsteady Flow

The flow is unsteady or steady depending on whether or not the depth at a given location

changes with time. Surface waves propagating on a channel is in fact unsteady, but may

appear steady to an observer who travels at the same speed as the waves.

3. Laminar and Turbulent Flow

The flow is laminar or turbulent depending on the magnitude of the Reynolds number,which for open channels can be defined as Re /hVR ν ≡ , where

h R is the hydraulic radius

103



equal to the ratio of the cross section area to the wetted perimeter. Open-channel flows in

rivers, culverts, surface drainage, etc typically involve water as the fluid (with fairly small

viscosity) and have relatively large dimensions so that the flows are invariably turbulent.

For turbulent channel flow, the velocity profile is rather uniform (except close to the

bottom and lateral boundaries) across a section of the flow, and therefore it is a common

practice to use the section-averaged velocity V = V ( x) as the primary variable in openchannel flow.

4. Tranquil (Sub-critical) and Rapid (Super-critical) Flow

Using the momentum principle, one may show that a small-amplitude surface wave (which

is, say, caused by some small disturbance to the flow) will travel in a shallow pool of

liquid at the speed c g= y , where y is the local fluid depth. An open channel flow may

have sharply different behaviors when its flow velocity V is smaller or larger than this

wave speed. The ratio of these two velocities is known as the Froude number, the

magnitude of which corresponds to the following types of flow:

1 s⎧ ub-critical flow

Fr 1 critical flow

1 super-critical flow

V

gy

<⎪

≡ =⎨⎪>⎩

Briefly speaking, a sub-critical flow is so low in speed (thereby called tranquil) that a

disturbance to the flow may send waves both upstream and downstream of the channel. In

sharp contrast, a super-critical flow is a high speed flow (thereby called rapid) and a wave

cannot be transmitted upstream.

C.

Energy Considerations

For channels under consideration, the bottom slope is very small ( 0.001). Streamlines are

therefore virtually straight and parallel, and pressure variation is hydrostatic. This implies

∼

Free surface = hydraulic grade line

Datum

Streamline

Energy line

Channel bed

Lh

2 / 2V g

/ p g ρ

z

104





D. Uniform Flow: Chezy-Manning Equation

Define:

Normal depth n y = depth of uniform flow

Hydraulic radius h R =

flow sectional area

wetted perimeter

A

P=

There is no change in momentum along the channel when the flow is uniform, and therefore

the net force acting on a control volume, as shown above, must be zero. By uniformity, the

two end forces are equal . Consequently, the downward gravity force is exactly balanced by the bottom friction:

1F F = 2

( ) 0

0

sinw

w h

Pl W gAl S

gR S

τ θ ρ

τ ρ

= =

⇒ =

By analogy with pipe flow, the wall stressw

τ can be expressed as

2

8w

f V τ ρ =

where f is the friction factor, which for complete turbulent flow depends only on the

roughness of channel. Combining these equations, we get the so-called Chezy equation for

uniform flow0h

V C R S =

106





E. Specific Energy: Alternative Depth of Flow

Let us recall the specific energy that has been introduced in Section C. The specific energy,

E , is the sum of the pressure and velocity heads2

2

V

E y g= + .

Specific energy is also the energy head referred to the base of the channel. It is a very

important concept in open-channel flow. While the specific energy is constant along the

channel when the flow is uniform, it may increase or decrease down the channel when the

flow is non-uniform.

For the convenience of discussion, let us consider only rectangular cross section from here

onward. The results can be extended to an arbitrary cross section, but will not be considered

here.

1. Variation of Specific Energy with Flow DepthSince the velocity is related to the unit width discharge by /V q y= , we may write the specific

energy as

2

22

q E y

gy= +

Here the equation consists of three variables: E , y, q. It would be of interest to examine the

cases: i) q is constant, and E varies with y; and ii) E is constant, and q varies with y.

If q is kept constant, E varies with y in the following manner.

This is known as the specific energy diagram, in which the following points are notable.

108



• For 0, , (asymptotic to the axis)q

y V E E y

→ = → ∞ ∴ → ∞ .

• For , 0, (asymptotic to the line )q

y V E h y E = . y

→ ∞ = → ∴ →

• Between these two extremes, E declines to a minimum min E at the critical point:

o

depth and velocity at this point are called the critical depth c y and critical

velocityc

V ;

o critical depth represents the least possible specific energy with which the fixed

discharge q is able to flow in the channel of given slope.

• For each other value of E greater than the minimum, there are two possible values of y,

one greater and one less than c y . The two corresponding depths for a given value

min E E > are known as the alternative depths.

• Flow can be classified into

, flow is tranquil or sub-critical

, flow is critical, flow is rapid or super-critical

c c

c c

c c

y y V V

y y V V y y V V

> <

= =< >

2. Criterion for Minimum Specific Energy

Since the minimum E occurs when d E /d y = 0, we may readily obtain, using the relation for E

given above, the following condition for critical flow

2

3

2 22

3

1 0

Fr 1

dE q

dy gy

q V

gy gy

= − =

⇒ ≡ = =

Being consistent with our earlier discussion in Section B.4, the flow is critical, sub-critical or

super-critical depending on the value of the Froude number. Combining all of the above,

min

min

min

1 , sub-critical flow,

Fr 1 , critical flow,

1 , super-critical flow,

c c

c c

c c

y y V V E E V

y y V V E E gy

y y V V E E

< > < >⎧⎪

≡ = = = =⎨⎪> < > >⎩

Also, the critical flow depthc y is given by

1/ 32 2

2

3Fr 1

c

c

q q y

gy g⎛ ⎞≡ = ⇒ = ⎜ ⎟⎝ ⎠

,

and hence the minimum specific energy is

2

min 2

3

2 2c c

c

q E y

gy= + = y .

It is remarkable that, for a given discharge, the critical flow occurs when the specific energy

is the lowest possible value .min E

109





F. Gradually Varied Flow

1. Introduction

• Uniform flow, in which a uniform flow depth is maintained, requires constancy of all

channel characteristics (i.e., cross-sectional shape, bed slope, roughness) throughout the

flow. In natural streams, this condition is hardly attained and flow is invariably non-uniform in nature. Even for artificial channels with uniform cross section, etc., uniform

flow is only a condition to be approached asymptotically. The surface of a varied (i.e.,

non-uniform) flow is not parallel to the bed and takes the form of a curve.

• Steady varied flow is broadly divided into two kinds:-

o Gradually varied flow, in which changes of depth and velocity take place over a

long distance and degree of non-uniformity is very slight. Boundary friction is

significant and is to be accounted for.

o Rapidly varied flow, in which the sectional area of flow changes abruptly within a

short distance. Turbulent eddying loss is more important than boundary friction in

this case. Hydraulic jump, which is a typical example of rapidly varied flow, will

be examined in the next section.

• Gradually varied flow may result from

o a change in cross-sectional shape, bed slope, boundary roughness of the channel.

o the installation of control structures (e.g., sluice gate, weirs, etc.).

2. General Equation for Gradually Varied Flow

When the flow is non-uniform, the specific energy is no longer a constant, but varies with

distance, x, along the channel. It has been derived in Section C that

( ) ( ) ( )0 f E x x E x S S + Δ − = − Δ x

where and0S f S are respectively the slopes of the channel bed and the energy grade line.

This relation implies that the specific energy gradient is equal to the difference between these

two slopes.

0 f

dE S S

dx= −

On the other hand, by the definition of the specific energy,

( )2 2 2

2

2 31 1 F

2 2

dE d V d q q dy dy y y

dx dx g dx gy gy dx dx

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + = + = − = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠r

Combining the two equations above, one may get the general equation governing the free

surface gradient in a gradually varied flow:

.0

21 Fr

f S S −dy

dx=

−

Note that f S and Fr are functions of x, and the geometrical bed slope S may or may not

change with x depending on construction. The equation above may be integrated numerically

to yield the surface profile for any type of gradually varied flow. This kind of profile

evaluation is normally carried out by a commercial package nowadays and will not be

discussed here.

0

111



3. Profile Classification

The general surface profile equation can be utilized in establishing the various forms of

varied flow profile. The equation may be rearranged into

0

0 2

1 /

1 Fr

f S S dyS

dx

−⎛ ⎞= ⎜ ⎟

−⎝ ⎠

.

Let us consider a very wide rectangular channel. As noted earlier in Section D, when the flow

is uniform,

2

0 5/3 f

n

qnS S

y

⎛ ⎞= = ⎜ ⎟

⎝ ⎠

wheren y is the normal depth.

An assumption is made here that for gradually varied flow the energy gradient S can be

related to the local flow depth y in the same manner as in the above formula for uniform

flow. Therefore, when the flow is gradually varied

( )2 2

10/3

0 05/3 5/3, / / f f

n

qn qnS S S S y

y y

⎛ ⎞ ⎛ ⎞= = ⇒ =⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ n y .

On the other hand, the Froude number can be written as

32 22

3 3 3Fr 1c

c

yq q

gy y gy

⎛ ⎞= = =⎜ ⎟

⎝ ⎠∵ .

Putting these relations back into the surface profile equation

( )

( )

10/3

0 3

1 /

1 /

n

c

y ydyS

dx y y

−=

−.

With the equation above, we may outline the various forms of varied flow profiles, with a

classification based on the following features.

Type of Bed Slope

Mild Slope (M): 0 0 ,c nS S y y< > c

c

c

∞

∞

Steep Slope (S): 0 0 ,c nS S y y> <

Critical Slope (C): 0 0 ,c nS S y y= =Horizontal Slope (H): 0 0, nS y= =

Adverse Slope (A): 0 0, nS y< =

( ) y Relative to Normal Depth ( )n y and Critical Depth ( )c y Flow Depth

Type 1: , andn c y y y y> > 0 Backwater curvedy

dx> ⇒

Type 2: is between andn c y y y 0 Dropdown curvedy

dx< ⇒

Type 3: , andn c y y y y< < 0 Backwater curvedy

dx > ⇒

112



Hence, a flow occurring on a mild slope has an M1 curve when its depth is greater than the

normal depth, and has an M3 curve when its depth is less than the critical depth, and so on.

Further notes:-

The critical and normal depth lines (CDL, NDL) and the channel bed form the boundaries of

3 zones. The curves of gradually varied flow surface profile approach each of these zone boundaries in a specific manner:

a) Upper limits of depth – the curves tend to become asymptotic to a horizontal water line.

b) Normal depth line – approached asymptotically (except for C curves).

c) Critical depth line – intersected at right angles (except for C curves).

d) Bed – intersected at right angles.

There are altogether 12 possible profiles of gradually varied flow, as depicted below.

Channel

slope

Depth

Relations

dy

dx

Type of

ProfileSymbol

Type of

FlowForm of Profile

n c y y y> > + Backwater 1 M Sub-

critical

n c y y y> > - Dropdown 2 M Sub-

criticalMild

n c y y> > y + Backwater 3 M Super-

critical

c n y y y> = + Backwater 1C Sub-critical

c n y y y= = Parallel to

bed2C

Uniform,

CriticalCritical

c n y y y= > + Backwater 3C Super-

critical

c n y y y> > + Backwater 1S Sub-

critical

c n y y y> > - Dropdown 2S Super-

criticalSteep

c n y y y> > + Backwater 3S Super-

critical

c y y> - Dropdown 2 H Sub-

criticalHorizontal

c y y> + Backwater 3 H Super-

critical

c y y> - Dropdown 2 A Sub-

criticalAdverse

c y y> + Backwater 3 A Super-

critical

113



G. Hydraulic Jump

1. Introduction

When a change from rapid (super-critical) to tranquil (sub-critical) flow occurs in open

channel, a hydraulic jump appears, through which the depth increases abruptly in the

direction of flow.

In engineering practice, the hydraulic jump frequently appears downstream from overflow

structures (spillways) or underflow structures (sluice gated) where velocities are high. It may

be used as an effective dissipation of kinetic energy (and thus prevent scour) of channel

bottom) or as a mixing device in water or sewage treatment designs where chemicals are

added to the flow.

2. General Equation of Hydraulic Jump

In spite of the complex appearance of a hydraulic jump withits turbulence and air entrainment, it may be analyzed by

application of the momentum equation.

Consider flow in a rectangular channel, and apply

momentum equation for the control volume of unit width

between sections 1 and 2:

( )1 2 2 1F F q V V ρ − = −

Substituting the following relations

114





If the head loss is , the balance of energy heads before and after the jump gives Lh

( )

( )

2 2

1 21 2

2 2

1 2 1 2

2

1 2 2 2

1 2

2 2 22 21 2 1 2

1 2 1 2 1 22 2

1 2

3

2 1

1 2

2 2

1

2

1 1

2

1 1 2 0

4

4

L

L

V V y y h

g g

h y y V V g

q y y

g y y

y y y y q y y y y y y

y y g

y y

y y

+ = + +

⇒ = − + −

⎛ ⎞= − + −⎜ ⎟

⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞+= − + − + − =⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠⎝ ⎠

−=

∵

which also equals the difference in specific energy across the jump.

To ensure a positive magnitude,2 1 y y> and not vice versa.

Therefore, a hydraulic jump is irreversible.

The power dissipated per width of channel is LP gqh ρ = .

116



H. Some Examples of Composite-Flow Profiles

• Downstream of a sluice gate

• A change of bed slope from steep to mild: the hydraulic jump may be formed either

on the steep slope or on the mild slope depending on whether the downstream

conjugate depth2 y is smaller or greater than the normal depth on the mild slope.

• Other types of change of bed slope

117



•

Free fall at the end of a channel

• Flow over a bump (bottom friction is ignored): the free surface over the bump is

depressed or elevated when the flow is sub-critical or super-critical, respectively.

118



When the flow before the bump is sub-critical (state 1a), the flow depth y2 decreases

s the bump height is increased, point 2 continues shifting to the left on the

(state 2a). Since the decrease in flow depth is always greater than the bump height

(why?), the free surface will be suppressed. But if the flow before the bump is super-

critical (state 1b), the flow depth rises over the bump (state 2b), creating a bigger

bump over the free surface. The situation is reversed if the channel has a depression

in its bed: the flow depth increases if the approach flow is sub-critical and decreases ifit is super-critical.

Ab zΔ

m (specific energy diagra while point 1 remains unaffected), until finally reaching the

critical point at which the specific energy is the minimum, and the flow over the bump

is critical. This critical height of a bump is given by the difference between the

original and the minimum specific energy levels1 minbc z E E Δ = − . Since the specific

energy has already reached the minimum level at flow over the bump

will only remain critical even when the bump height is further increased. To

overcome a bump of height greater than bc z

this state, the

Δ , the approach flow must adapt itself(say, either to increase the upstream energy level

1 E or to reduce the flow rate so that

min E is decreased) so that the bump height1 min E E = − is always satisfied. When this

ens, the flow is said to be choked. happ

he fact that flow over a sufficiently high obstruction in an open channel is alwaysT

critical is the working principle of weirs (broad-crested or sharp-crested), which are

used to measure the volume flow rate in open channels.

119



120



121



122



123



124



125



126



Documents

Fluid Mechanics Lecture Notes II