10. fm dimensional analysis adam

Chapter 7
DIMENSIONAL ANALYSIS
AND MODELING
Lecture slides by
Adam, KV Sharma

DIMENSIONS AND UNITS
Dimension: A measure of a physical quantity (without numerical
values).
Unit: A way to assign a number to that dimension.
There are seven primary dimensions :
1. Mass m (kg)
2. Length L (m)
3. Time t (sec)
4. Temperature T (K)
5. Current I (A)
6. Amount of Light C (cd)
7. Amount of matter N (mol)

All non-primary dimensions can be formed by some combination
of the seven primary dimensions.

{Velocity} = {Length/Time} = {L/t}
{Force} = {Mass Length/Time} = {mL/t2}
2

The water strider
is an insect that
can walk on water
due to surface 3
tension.

7–2 ■ DIMENSIONAL HOMOGENEITY
The law of dimensional homogeneity: Every additive
term in an equation must have the same dimensions.

You can’t add apples and oranges! 4

7–2 ■ DIMENSIONAL HOMOGENEITY
The law of dimensional homogeneity: Every additive
term in an equation must have the same dimensions.
Bernoulli equation

5

Nondimensionalization of Equations
Nondimensional equation: If we divide each term in the equation
by a collection of variables and constants whose product has those
same dimensions, the equation is rendered nondimensional.

Most of which are named after a notable scientist or engineer (e.g.,
the Reynolds number and the Froude number).

A nondimensionalized form of the
Bernoulli equation is formed by
dividing each additive term by a
pressure (here we use P∞). Each
resulting term is dimensionless
(dimensions of {1}).

6

In a general unsteady fluid flow problem with a free surface, the scaling
parameters include a characteristic length L, a characteristic velocity V, a
characteristic frequency f, and a reference pressure difference P0 − P∞.
Nondimensionalization of the differential equations of fluid flow produces
four dimensionless parameters: the Reynolds number, Froude number, 7
Strouhal number, and Euler number.

In Fluid Mechanics,
•the Reynolds number (Re) is a dimensionless number that gives a
measure of the ratio of inertial forces to viscous forces.
•The Froude number (Fr) is a dimensionless number defined as the
ratio of a body's inertia to gravitational forces. In fluid mechanics, the
Froude number is used to determine the resistance of a partially
submerged object moving through water, and permits the comparison
of objects of different sizes.
•The Strouhal number (St) is a dimensionless number describing
oscillating flow mechanisms.
•The Euler number (Eu) is a dimensionless number used in fluid flow
calculations. It expresses the relationship between a local pressure
drop over a restriction and the kinetic energy per volume, and is used
to characterize losses in the flow, where a perfect frictionless flow
corresponds to an Euler number of 1.

8

DIMENSIONAL ANALYSIS AND SIMILARITY

In most experiments, to save time and money, tests are
performed on a geometrically scaled model, not on the
full-scale prototype.
In such cases, care must be taken to properly scale the
results. Thus, powerful technique called dimensional
analysis is needed.

The three primary purposes of dimensional analysis are
• To generate non-dimensional parameters that help in the design
of experiments and in the reporting of experimental results
• To obtain scaling laws so that prototype performance can be
predicted from model performance
• To predict trends in the relationship between parameters
9

Greek letter Pi (Π) denote a non-dimensional parameter.
In a general dimensional analysis problem, there is one Π that we
call the dependent Π, giving it the notation Π1.
The parameter Π1 is in general a function of several other Π’s, which
we call independent Π’s.

10

The principle of similarity
Three necessary conditions for complete similarity between a model and a
prototype.
(1) Geometric similarity—the model must be the same shape as the
prototype, but may be scaled by some constant scale factor.
(2) Kinematic similarity—the velocity at any point in the model flow must be
proportional (by a constant scale factor) to the velocity at the corresponding
point in the prototype flow.
(3) dynamic similarity—When all forces in the model flow scale by a constant
factor to corresponding forces in the prototype flow (force-scale
equivalence).

To achieve similarity

11

To ensure complete similarity, the model and prototype must be
geometrically similar, and all independent groups must match between
model and prototype.

Kinematic similarity is
achieved when, at all
locations, the speed in the
model flow is proportional to
that at corresponding
locations in the prototype
flow, and points in the same
direction.

In a general flow field, complete similarity between a model and
prototype is achieved only when there is geometric, kinematic, and
dynamic similarity. 12

A 1 : 46.6 scale
model of an Arleigh
Burke class U.S.
Navy fleet destroyer
being tested in the
100-m long towing
tank at the University
of Iowa. The model is
3.048 m long. In tests
like this, the Froude
number is the most
important
nondimensional
parameter. 13

Geometric similarity between a
prototype car of length Lp and a model
car of length Lm. In the case of
aerodynamic drag on the automobile,
there are only two Π’s in the problem.

FD is the magnitude of the aerodynamic drag on the car, and so on forming
drag coefficient equation.
The Reynolds number is the most well known and useful dimensionless 14
parameter in all of fluid mechanics.

A drag balance is a device used
in a wind tunnel to measure the
aerodynamic drag of a body.
When testing automobile models,
a moving belt is often added to
the floor of the wind tunnel to
simulate the moving ground (from
the car’s frame of reference). 15

If a water tunnel is used instead of a wind tunnel to test their one-fifth
scale model, the water tunnel speed required to achieve similarity is

One advantage of a water tunnel
is that the required water tunnel
speed is much lower than that
required for a wind tunnel using
the same size model (221 mi/h
for air and 16.1 mi/h for water) .

Similarity can be achieved
even when the model fluid
is different than the
prototype fluid. Here a
submarine model is tested
17
in a wind tunnel.

A drag balance is a device used
in a wind tunnel to measure the
aerodynamic drag of a body.
When testing automobile models,
a moving belt is often added to
the floor of the wind tunnel to
simulate the moving ground (from
the car’s frame of reference).
18

THE METHOD OF REPEATING VARIABLES
AND THE BUCKINGHAM PI THEOREM
How to generate the
nondimensional analysis?
There are several method but the
most popular was introduced by
Edgar Buckingham called the
method of repeating variables.

Step must be taken to generate
the non-dimensional parameters,
i.e., the Π’s?

A concise summary of
the six steps that
comprise the method of
20
repeating variables.

THE METHOD OF REPEATING VARIABLES

Step 1

Setup for dimensional analysis of a ball
falling in a vacuum.

Pretend that we do not know the equation
related but only know the relation of
elevation z is a function of time t, initial
vertical speed w0, initial elevation z0, and
gravitational constant g. (Step 1)

21

Step 2

n=5

the six steps that
22

The primary dimensions are [M], [L] and [t].
The number of primary dimensions in the problem are (L and t).
Step 3

Then the number of Π’s predicted by the Buckingham Pi theorem
is

the six steps that
23

Need to choose two repeating parameters since j=2.
Therefore

Step 4
Caution
1. Never choose dependent variable
2. Do not choose variables that can form dimensionless group
3. If there are three primary dimension available , must choose repeating
variables which include all three primary dimensions.
4. Don’t pick dimensionless variables. For example, radian or degree.
5. Never pick two variables with same dimensions or dimensions that differ
by only an exponent. For example, w0 and g.
6. Pick common variables such as length, velocity, mass or density. Don’t
pick less common like viscosity or surface tension.
7. Always pick simple variables instead of complex variables such as
energy or pressure.

the six steps that
24

Step 5: Construct the k Π’s , and manipulate as necessary

25

Need modification for commonly used nondimensional parameters.

Step 6

26

The pressure inside a
soap bubble is greater
than that surrounding
the soap bubble due to
surface tension in the
soap film.
27

If the method of
repeating
variables indicates
zero Π’s, we have
either made an
error, or we need
to reduce j by one
and start over.

28

Although the Darcy friction
factor for pipe flows is most
common, you should be
aware of an alternative, less
common friction factor called
the Fanning friction factor.
The relationship between the
two is f = 4Cf . 37

DIMENSIONLESS PARAMETER

In dimensional analysis, a dimensionless quantity or quantity
of dimension one is a quantity without an associated
physical dimension.
It is thus a "pure" number, and as such always has a
dimension of 1.

Other examples of dimensionless quantities:
- Weber number (We),
- Mach (M),
- Darcy friction factor (Cf or f),
- Drag coefficient (Cd) etc.

39

RAYLEIGH METHOD

Rayleigh's method of dimensional analysis is a conceptual
tool used in physics, chemistry, and engineering.

This form of dimensional analysis expresses a functional
relationship of some variables in the form of an
exponential equation.
It was named after Lord Rayleigh.

40

The method involves the following steps:
•Gather all the independent variables that are likely to influence the
dependent variable.
•If X is a variable that depends upon independent variables
X1, X2, X3, ..., Xn, then the functional equation can be written as X
= F(X1, X2, X3, ..., Xn).
•Write the above equation in the form where C is a dimensionless
constant and a, b, c, ..., m are arbitrary exponents.
•Express each of the quantities in the equation in some fundamental
units in which the solution is required.
•By using dimensional homogeneity, obtain a set of simultaneous
equations involving the exponents a, b, c, ..., m.
•Solve these equations to obtain the value of exponents a, b, c, ..., m.
•Substitute the values of exponents in the main equation, and form the
non-dimensional parameters by grouping the variables with like
exponents. 41

10. fm dimensional analysis adam

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