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
had
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
ℎ
+
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
:
ⅆ
𝜌
ⅆ
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
steady
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.
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