# Testing Theories in Fluid Dynamics

Mechanics

Oct 24, 2013 (4 years and 6 months ago)

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Testing Theories in Fluid Dynamics

Wilka Carvalho (SUNY)

References

Acknowledgements

Introduction

Theory and Methods

Objective

Examples

Many

problems

in

mechanical

engineering,

even

seemingly

simple

problems,

cannot

be

solved

analytically
.

Analytical

solutions

are

only

as

accurate

as

the

theories

they

are

grounded

in

so

because

the

physical

world

is

currently

not

perfectly

understood,

analytical

solutions

can

often

be

faulty
.

This

is

often

the

case

in

fluid

dynamics,

as

it

is

necessary

to

experiment

in

order

to

gain

an

accurate

solution

which

may

improve

the

theory

and

allow

for

a

accurate

future

analytical

solutions
.

We

used

Bernoulli's

Principe

and

the

continuity

equation

to

analyze

the

flow

of

water

and

verify

the

analytical

solutions

for

the

behavior

of

water

under

numerous

conditions
.

We

tested

solutions

regarding

the

flow

of

water

in

regards

to
:

Reynolds

Number,

Boundary

Layer

conditions,

Laminar

and

Turbulent

flow,

etc
.

Our

ultimate

goal

was

to

verify

the

theoretical

solutions

with

numerical

and

analytical

solutions
.

The

experiments

done

focused

mainly

on

water

running

through

a

pipe
.

For

all

the

cases,

because

we

dealt

with

water

that

little

variation,

it

was

assumed

that

the

overall

fluid

was

inviscid,

incompressible,

and

irrotational
.

The

water’s

behavior

was

governed

primarily

by

Bernoulli’s

Principle

and

the

continuity

equation
.

Together

the

two

allowed

us

to

define

the

behavior

of

the

fluid

at

different

points

along

the

pipe,

as

well

as

with

different

influences

along

the

pipe

acting

on

it
.

Bernoulli’s

Principle,

in

essence,

is

a

principle

of

conservation

of

mechanical

energy
.

It

states

that

the

sum

of

the

energy

of

a

system

always

remains

equal,

so

the

sum

of

its

components

must

remain

equal

and

can

be

expressed

as
:

𝑃
+
𝜌𝑔

+
𝜌
𝑣
2
2
=
𝑐 𝑎 
,

where

P

is

pressure,

𝜌

is

density,

g

is

gravity,

h

is

height,

and



is

velocity
.

For

the

purpose

of

our

experiments

it

was

interpreted

as
:

the

mechanical

energy

at

one

section

of

a

pipe

is

equal

to

the

mechanical

energy

at

another

section

of

a

pipe
.

Figure 1: This
diagram
depicts the
application of
Bernoulli’s
Principle and
the Continuity
Equation to a
pipe with
varying cross
section and
height

In

fluid

dynamics,

the

continuity

equation

states

that

flow

through

a

tube

of

varying

cross

section

must

be

equivalent

at

all

points

and

can

be

expressed

as
:

𝜌

𝑡
+
𝛻
𝜌


=

0
,

where



is

the

flow

velocity

vector

field
.

It

is

based

on

the

Law

of

Conservation

of

Mass,

which

states

that

mass

can

neither

be

created

nor

destroyed
.

In

a

state

process

(a

process

where

a

variable

remains

constant)

of

a

fluid

flowing

through

a

pipe,

the

rate

at

which

mass

enters

the

system

is

equal

to

the

rate

at

which

it

exits

the

system
.

We

dealt

only

with

incompressible

fluids,

which

by

definition

of

the

continuity

equation

meant

that

the

local

change

in

volume

of

the

fluid

was

0
.

This

meant

that

at

any

two

sections

along

the

pipe,

the

rate

of

change

of

Volume

was

equival ent,

which

led

to

the

simplification
:

𝐴
1


1
=
𝐴
2


2
.

Figure 2: This
graph depicts
the inverse
relationship
pressure has
with height and
velocity under
Bernoulli’s
Principle when
applied to a pipe
of varying cross
section.

Munson et al. (2009).
Fundamentals of Fluid Mechanics.

Jefferson
City: Don
Fowley
.

Turns, S. R. (2000).
An Introduction to Combustion : Concepts
and Applications.

Singapore: McGraw
-
Hill Higher Education.