Text Book:

Fluid Mechanics With Engineering Applications (10th Edition)

by E. John Finnemore & Joseph B. Franzini

Hydraulic Machinery by S.S.Rattan

Reference Books:

Applied Fluid Mechanics (6th Edition)

by Robert L. Mott

Fluid Mechanics by A.K Jain

A textbook of Hydraulics, Fluid Mechanics and Hydraulic Machines (19th Edition) by R.S. Khurmi

Sessionals – 30%

Mid Term - 20%

Final Exam - 50%

Dimensional Analysis ◦ Rayleigh‟s Method and Buckingham's Pi-theorem

and their applications.

Fluid flow in pipes

◦ Reynold's number and its significance, Instability of

viscous flow.

◦ Viscous flow through circular pipes. Turbulent flow through circular pipes.

◦ Semi-empirical theories of turbulence. Velocity profile in turbulent flow.

◦ Pipe roughness, Nikuradse's experiments. Moody's diagrams.

◦ Pipe networks

Forces on immersed bodies ◦ Development of boundary layer on immersed

Bodies.

◦ Elementary theory of surface drag and form drag.

◦ Simple lift and drag equations and their applications to simple engineering problems.

◦ Separation of boundary layer.

Forces on vanes and turbomachinery ◦ Impulse momentum equation.

◦ Forces on moving flat and curved vanes.

◦ Impulse turbine, construction features and operations, Specific speed.

◦ Reaction turbine, types, construction features and operation, specific speed, cavitation, draft tube.

Centrifugal pumps. ◦ Classification, construction features and operation,

specific speed, cavitation.

Reciprocating Pumps ◦ Single acting and double acting pumps, acceleration

head, maximum suction lift. Use of air vessels

It is a pure mathematical technique to establish a relationship between physical quantities (variables or parameters) involved in a fluid phenomenon by considering their dimensions.

Reliable problem solving approach

Reduces errors in algebra

Reinforces unit conversion

Simplifies computation

Improves understanding of maths applications Multiple ways to solve the same problem

Identify what you are being asked.

Write down what is given or known.

Look for relationships between knowns and unknowns (use charts, equations).

Rearrange the equation to solve for the unknown.

Do the computations, cancel the units, check for reasonable answers

Dimensional Analysis

Dimensional Analysis refers to the physical

nature of the quantity and the type of unit

(Dimension) used to specify it.

Distance has dimension L.

Area has dimension L2.

Volume has dimension L3.

Time has dimension T.

Speed has dimension L/T

Development of an equation for fluid phenomenon

Conversion of one system of units to another

Reducing the number of variables required in an experimental program so that number of experiments to be performed is reduced

Develop principles of hydraulic similitude for model study

MLT system

FLT system

These two systems are inter-related as follow:

F = ma 2nd Law of motion

F = MLT-2

or M = FL-1T+2

[x], Length – L

[m], Mass – M

[t], Time – T

[v], Velocity – LT-1

[a], Acceleration – LT-2

[F], Force – MLT-2

[r], Mass Density – ML-3

[P], Pressure – ML-1T-2

[E], Energy – ML2T-2

[I], Electric Current – QT-1

[q], Electric Change – Q

[E], Electric Field - MLQT-2

Principle of Dimensional Homogeneity

◦ The fundamental dimensions and their respective

powers should be identical on either side of the sign of equality.

Q = A x V [continuity equation is homogeneous]

Is Manning‟s equation dimensionally homogeneous or not?

Rayleigh‟s Method

Buckingham‟s π-method

Functional relationship between variables is expressed in the form of an exponential relation which must be dimensionally homogeneous

if “y” is a function of independent variables x1, x2, x3, ….. xn, then

),.......,,( 321 nxxxxfy

In exponential form as

]),.......()(,)(,)[( 321

z

n

cba xxxxy

Write fundamental relationship of the given data Write the same equation in exponential form Select suitable system of fundamental dimensions Substitute dimensions of the physical quantities Apply dimensional homogeneity Equate the powers and compute the values of the

exponents Substitute the values of exponents Simplify the expression

For further understanding, lets explore the equation for the velocity (V) of a pressure wave through a fluid.

First it should be visualised what physical factors actually influence the velocity

Compressibility density ρ and kinematic viscosity υ (nu) are the physical factors influencing the motion

E

The dimensions of these quantities will be

T

L

L

M

LT

M

L

FE

T

LV

2

3

22

,

,

r

C is the dimensionless constant. Substituting the dimensions

dbaCEV r

dba

T

L

L

M

LT

M

T

L

2

32

For dimensional homogeneity, the exponents of each dimension must be identical on both sides.

For M

For L

For T

ba0

dba 231

da 21

Solving the above equations, we get

So finally,

0

2/1

2/1

d

b

a

rE

CV

Dimensional analysis was developed in such way by Lord Rayleigh.

Very serviceable method but has been superseded.

Developed by E. Buckingham (1915)-a more generalized method of dimensional analysis.

“If „n‟ is the total number of variables in a dimensionally homogenous equation containing „m‟ fundamental dimensions, then they may be grouped into (n-m) π terms.

if y=f(x1, x2, ……xn) then the functional relationship will be written as

Suitable where

Not applicable if (n-m) = 0

0).......,( 21 mn

4n

List all physical variables and note „n‟ and „m‟

n = total no. of variables

m = fundamental dimensions

Compute number of π-terms

Write the eq. in functional form

Write eq. in general form

Select repeating variables. Must have all of the „m‟ fundamental dimensions and should not form a π among themselves (mass, geometry and kinematics)

Solve each π-term for the unknown exponents by dimensional homogeneity.

0.],.........,,[ 321

Consider the factors affecting the drag force on a sphere.(done experiment in lab)

These include size of the sphere D, velocity of the sphere V, density ρ and viscosity μ. So

Here we see that n=5

0),,,,( rVDFf D

Choose a Dimensionless system (MLT or FLT) and determine the number of fundamental dimensions involved in the system (m).

We will choose MLT system so the corresponding dimension will be

We can see that here m=3

LT

M

L

M

T

LL

T

ML,,,,

32

Determine the number of π-terms needed. In this case they will be n-m=5-3=2.

Select the primary or repeating variables such that they must contain all of the m fundamental dimensions and must not form a π (dimensionless group) among themselves.

Choose ρ, D and V as the repeating variables. The π-terms will then be

Using the principle of dimensional homogeneity, we can solve for the exponents on each side of the equation.

Since π-terms are dimensionless, they can be replaced by M0L0T0.

D

cba

cba

FVD

VD

222

111

2

1

r

r

So our expressions will look like

10:

130:

10:

1

111

1

3

0001

1

1

cT

cbaL

aM

LT

M

T

LL

L

MTLM

c

b

a

Solving,

As

1

111

1

111 1;1;1

r

r

r

DV

DVVD

thus

cba

r /DVR

222

1

1

VD

F

and

R

D

r

0,

0,

22

1

21

VD

FR D

r

This shows that the drag force depends upon the Reynold‟s number which is the ratio of inertial forces to viscous forces.

Dimensional analysis only provides a partial solution to the fluid problems as it depends entirely on the ability of the individual to perceive the factors influencing a fluid phenomenon.

So if an important variable is omitted, then the results could be entirely different.

PROBLEM (1)

Derive an expression for the flow rate q over the spillway shown in the figure per foot of spillway perpendicular to the sketch. Assume that the sheet of water is relatively thick, so that surface-tension effects are negligible. Assume also that gravity effects predominate so strongly over viscosity that we can neglect viscosity.

SOLUTION:

PROBLEM (2

Derive an expression for the shear stress at the pipe wall when an incompressible fluid flows through a pipe under pressure. Use dimensional analysis with the following significant parameters: pipe diameter D, flow velocity V, and viscosity µ and density ρ of the fluid.

SOLUTION:

PROBLEM (3)

The critical depth yc in a V-shaped channel depends upon the discharge Q, acceleration due to gravity g, and the vertex angle θ made by the two side-walls of the channel. Using the method of dimensional analysis deive an equation for the critical depth.

SOLUTION: