# ENGR2000 Lecture Notes - Lecture 9: Surface Tension, Gravity Bone, John Wiley & Sons

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Prepared by Dr Hongwei Wu Chapter 9 – Page 1 of 13

2nd – Year Fluid Mechanics, Faculty of Engineering and Computing, Curtin University

ENGR2000: FLUID MECHANICS

For Second-Year Chemical, Civil and Mechanical Engineering

FLUID MECHANICS LECTURE NOTES

CHAPTER 9 MODELS, DIMENSIONAL ANALYSIS AND SIMILITUDE

In Chapter 6, we have shown that some of the practical engineering problems can be solved

by theoretical analysis, for example, the calculation of the pressure drop of a laminar pipe

flow. However, there are still a large number (in fact, most) of real fluid problems cannot be

solved by theoretical analysis alone, for example, the pressure drop of a turbulent pipe flow.

Solutions of such problems have to rely on experimental investigations, or at least a

combination of theoretical and experimental investigations. This chapter introduces several

important concepts in fluid mechanics, including model, dimensional analysis and similitude.

These concepts are of critical importance to smart experimental planning and design to obtain,

understand and correlate quality experimental data, producing results that can be widely used.

9.1 Model

9.1.1 What is a model?

A model is an intellectual construct that has two basic characteristics: 1) representing reality;

2) being able to predict the consequences of future actions.

EX9-1: A simple example

Newton’s 2nd motion law:

onAcceleratiMassForce

i.e.

maF

This law is a simple mathematical model of the relation between force,

mass and acceleration. It indeed has been validated so that we have the

confidence to use it.

There are two types of models: computational model and physical model. A computational

model relies on the simulation using mathematical models while a physical model is based on

a physical representation of prototype by scaling-up or scaling-down. In practice, it is very

difficult to have completely reliable computational models. Critical engineering decisions are

very often made on the basis of a combination of computational and physical models.

Therefore, A model must be validated by experimental observations or data before we can

have enough confidence in using this model.

9.1.2 Why do we need a model?

Before a process can be implemented in real scale applications, we need to be sure that it will

work in the way we wanted. Otherwise, there is significant risk to waste large amount of

money or even claim human lives. It is often said that

“Have your failures on a small scale, in private;

Have your successes on a large scale, in public.” - Quoted from reference [1]

Therefore, model scaling-up (or scaling-down) are essential. Typical engineering examples

include models of new aircrafts, chemical reactors, etc.

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Prepared by Dr Hongwei Wu Chapter 9 – Page 2 of 13

9.2 Scaling and the need of dimensionless numbers

9.2.1 Scaling

The proper construction of a model requires the scaling up or down of the prototype.

However, without proper scaling rules, it is NOT guaranteed that the model and the prototype

will have comparable performance. The following is an example from reference [1].

EX9-2: An example – creation of giant human

We assume that the creator has a new task to create giant humans who will

live on the earth. A giant human has not only 10 times as high as a normal

human but also 10 times as wide and 10 times as thick.

A normal human

A giant human

Therefore, a giant human’s weight (W) is about 1000 times the normal

human’s:

normalgiant WW 1000

. However, the cross-section (A) of the

bone of a giant man is only 100 times that of a normal human:

normalgiant AA 100

.

The giant human will NOT be able to stand on the earth!!!

Obviously, holding the shape constant while increasing the size does not guarantee equal

performance by differently sized human, or equipment. To construct a walkable giant human,

the creator would need to do more thinking. In the case of the bones of the giant, one should

keep constant the ratio of the stress and crushing strength, with one of the following choices:

1. As the height of the giant is increased, the crushing strength of the giant’s bones could

be increased.

2. The average density of the giant’s body could be reduced in order to keep the same

bone stress while the height is increased.

3. The giant can live in another planet with a lower acceleration of gravity.

9.2.2 Dimensionless numbers

From the discussion in above section, it is clear that to construct a walkable giant human, the

creator needs to consider a parameter called “Bone Ratio” defined as the following:

bonesofstrengthCrushing

gravityofonAcceleratidensitybodyAverageHeight

RatioBone

If the Bone Ratio is kept as constant, the giant can become any height with the same relative

resistance to bone failure as any normal human has. The bone ratio is dimensionless.

Dimensionless ratios like the Bone Ratio are extremely important in predicting a large piece

of equipment from tests on a small model in engineering practice. Also, such dimensionless

ratios are also valuable in correlating, interpreting, and comparing experimental data. The

following is another simple example.

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Prepared by Dr Hongwei Wu Chapter 9 – Page 3 of 13

EX9-3: An example – investment return

Compare a business venture in which you invest $4000 for a return of $400

per year with one in which you invests $6000 for a return of $720 per year.

Which dimensionless number should we use?

The percentage, the most commonly used dimensionless number.

9.2.3 Importance of dimensionless numbers in fluid mechanics

In Fluid Mechanics, dimensionless numbers play very important roles in investigating

physical phenomena, understanding flow behaviours and solving practical engineering

problems. Let’s visit an example.

We know that key factors influencing pressure gradient along a smooth pipe include the pipe

diameter D, the fluid density

, fluid viscosity

, and the mean velocity V at which the fluid is

flowing through the pipe. The pressure gradient

P/

L (

P is the pressure drop across a pipe

with length

L) can then be expressed as a function as

),,,( VDf

L

P

(E9-1)

One may suggest that the easiest way to work out this function is to design sets of

experiments by varying only one of the four variables at a time while keeping the rest three

variables as constant. At first glance, this sounds indeed a good idea because we can foresee

that Figure 9-1a to Figure9-1d may be obtained through these sets of experiments.

Performing experiments to derive Figure 9-1a is easy, which actually was done by Osborne

Reynolds [3] in 1883 (see Region III of Figure 6-3). Performing experiments to derive Figure

9-1b is also possible, although the range of pipe diameter to be tested will be limited due to

engineering constraints. However, carrying out experiments in Figures 9-1c and 9-1d are very

difficult. The reason is that it is practically extremely difficult, if not impossible, to vary fluid

density while still keep fluid viscosity as constant, or vice versa. Actually, even if we are able

to do all experiments in Figures 9-1a to d, we still do not know the exact correlation.

In fact, life would become much easier if we can do a set of experiments to obtain Figure 9-2.

Rather than exhaust our efforts to obtain Figure 9-1, we can simplify the problem into

performing a set of experiments to derive the correlation between two dimensionless

numbers: friction factor (f) and Reynolds number (Re) as a single curve, i.e.

(Re)f

. This

is what have in the Moody chart, i.e. F6-16 or equation E6-26b.

EX9-4: An example – Experiment Design

We have been requested to investigate the pressure gradient (pressure drop

per unit length) along a smooth pipe. We need to decide the key factors

influencing pressure gradient, how to do the experiments, and how to derive

necessary correlations for practical use.

What should we do?

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