1. Fluid Mechanics and Energy Transport
BIEN 301
Lecture 2
Introduction to Fluids, Flow Fields, and Dimensional
Analysis
Juan M. Lopez, E.I.T.
Research Consultant
LeTourneau University
Adjunct Lecturer
Louisiana Tech University
2. 12/05/2006 BIEN 301 – Winter 2006-2007
History of Fluid Mechanics
White 1.14 shows us how Fluid Mechanics has evolved in a helical
fashion, returning to its roots, with improvements each time.
Pre-historic and early history aqueducts and waterworks –
Empirically Designed and Built
Archimedes (200’s B.C.) and Buoyancy / Vector addition –
Theoretical work with Experimental roots
200’s B.C. to Renaissance ship and canal building – Empirical
advances, no great amount of experimental work
Leonardo da Vinci first formulated the one-dimensional
conservation of mass equation – Theoretical stemming from
empirical observations.
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History of Fluid Mechanics
Mariotte (1600’s) built the first wind tunnel – Testing theoretical
ideas with experimental work.
Isaac Newton (1600’s-1700’s) generated the mathematics which
allowed fluid momentum to be studied.
Bernoulli, D’Alembert, Euler, Lagrange, Laplace, all developed
their work in frictionless fluids, and showed the need for a
formulation that would do away with the paradox of an object with
no drag immersed in a moving stream, a natural result of
frictionless fluid assumptions – Theoretical advances mostly.
These theoretical results were unsatisfactory to engineers, so as
a natural backlash, hydraulics was developed as an almost
purely experimental form by Pitot, Borda, Poiseuille, etc.
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History of Fluid Mechanics
Late 1800’s, finally there was a trend towards the unification
between experimental hydraulics and theoretical hydrodynamics
by the likes of Froude, Raylegh, and Reynolds. All of these
gentlemen have dimensionless groups named after them due to
the importance of their work.
Navier and Stokes began to more fully explore viscous flow in
the mid to late 1800’s, setting the stage for Prandtl.
In the early 1900’s, Prandtl developed boundary layer theory,
one of the most important advances in fluid mechanics, identified
by White as the single most important tool in modern flow
analysis.
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History of Fluid Mechanics
The past tied to the present
These past examples of development in fluid mechanics remain
important due to the individual contributions each advance has
made to our current understanding.
In fact, we continue to study many of these individual ideas as
simplified examples of fluid behavior.
Fluid mechanics encompasses almost every field of physical
systems, and a basic understanding of the mathematics,
terminology, and usage will greatly benefit you in any
engineering field.
6. 12/05/2006 BIEN 301 – Winter 2006-2007
What is a Fluid?
Matter that is unable to resist shear by a
static deflection. (White, 1.2)
Fluid will deflect under shear unless opposed by some external force.
The rate of strain to stress is dependent on the viscosity of the fluid.
7. 12/05/2006 BIEN 301 – Winter 2006-2007
What is a Fluid?
This lack of resistance to shear explains why
fluid take the shape of their containers, or spill
when there is no body to contain them.
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What is a Fluid?
Mechanical Description – Mohr’s Circle
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What is a Fluid?
As with everything, we make some assumptions in our
definition-
Continuum (White 1.3)
• Infinitely Divisible – All divisions have same properties in
homogeneous fluid
• For real systems, there are uncertainties brought about by volumes
that are too small or too large.
Physical properties are defined and have finite values throughout
the continuum
Thermal properties are defined and have finite values throughout
the continuum
10. 12/05/2006 BIEN 301 – Winter 2006-2007
Dimensions vs. Units
We must inherently have a way to describe the systems
we are studying. We describe these systems with
Dimensions and quantify these dimensions with Units.
Four primary Dimensions in our study of Fluid
Mechanics:
Mass, {M}
Length, {L}
Time, {T}
Temperature, {Θ}
11. 12/05/2006 BIEN 301 – Winter 2006-2007
Dimensions vs. Units
It is imperative that you learn consistency in your dimensional
analysis. Fluid mechanics lends itself to some extremely awkward
units, especially in the British system.
For this course, we will primarily stick with the International System
(SI), but we will refresh our memories from time to time on how to
interact with the British Gravitational (BG) units.
The use of tables is an inherent task in engineering work. Become
familiar with the tables such as White, Table 1.1, 1.2, and Appendix
Tables A.1-A.6, and how to properly use them.
12. 12/05/2006 BIEN 301 – Winter 2006-2007
Dimensions vs. Units
Using the tables, perform the following
conversion:
expression
SI
equivalent
Obtain the
54
.
1112
:
as
units
BG
in
given
is
air
for
constant
gas
The
R
slug
lbf
ft
Rair
13. 12/05/2006 BIEN 301 – Winter 2006-2007
Dimensions vs. Units
)
A.4
Table
(Matches
208
208
208
208
:
obtain
we
arranging,
-
Re
207.756
5556
.
0
1
*
115.429
115.429
594
.
14
1
*
1684.57
1684.57
4482
.
4
*
378.709
378.709
3408
.
0
*
54
.
1112
:
book
of
cover
front
and
1.2
Table
From
2
2
2
2
2
K
s
m
R
K
s
m
K
kg
s
m
kg
m
K
kg
N
m
R
K
kg
N
m
K
R
R
kg
N
m
R
kg
N
m
kg
slug
R
slug
N
m
R
slug
N
m
lbf
N
R
slug
lbf
m
R
slug
lbf
m
ft
m
R
slug
lbf
ft
R
air
air
air
14. 12/05/2006 BIEN 301 – Winter 2006-2007
Dimensional Consistency
Dimensional Homogeneity (White 1.4)
Theoretical Equations – dimensionally
homogeneous
out)
(checks
]
[
]
[
]
[
]
[
]
][
[][
]
][
[
]
[
]
[
2
1
2
2
1
2
0
0
L
L
L
L
T
LT
T
LT
L
L
gt
t
V
S
S
15. 12/05/2006 BIEN 301 – Winter 2006-2007
Dimensional Consistency
However, much work in fluid mechanics
has been empirical, and this can lead to
problematic situations.
units)
(awkward
]
[
]
/[
]
[
]
[
[?]
]
[
2
/
1
2
/
7
1
2
/
1
2
/
1
1
3
1
2
/
1
2
/
1
1
3
2
/
1
Cv
Cv
V
M
L
T
L
M
T
L
T
L
M
T
L
SG
p
C
Q
16. 12/05/2006 BIEN 301 – Winter 2006-2007
Uncertainty
Once we have established a way to describe
these systems, we must also account for the
uncertainty in our experimentation. (White 1.11)
Instruments and all physical measurements
have some form of uncertainty.
Accounting for all the measurements is important
Adding them all is simply not realistic
A simplified Root Mean Square (RMS) approach is
recommended.
17. 12/05/2006 BIEN 301 – Winter 2006-2007
Uncertainty
RMS Formulation:
2
/
1
2
2
2
...
...
P
For
m
m
m
j
j
j
i
i
i
n
m
n
j
n
i
x
x
n
x
x
n
x
x
n
P
P
x
x
x m
j
i
18. 12/05/2006 BIEN 301 – Winter 2006-2007
Uncertainty
RMS Example
%
92
.
2
%
5
.
0
3
%
5
5
.
0
%
5
.
0
3
%
5
5
.
0
%
5
.
0
%
5
5
P
For
2
/
1
2
2
2
/
1
2
2
3
2
/
1
P
P
P
P
and
where
19. 12/05/2006 BIEN 301 – Winter 2006-2007
Basic Physical Properties
Thermodynamics (White 1.6)
Principal components of velocity vectors
• Pressure, p
• Density, ρ
• Temperature, T
Principal components of work, heat, and energy balance.
• Internal Energy, û
• Enthalpy, h = û + ρ/p
Principal transport properties
• Viscosity, μ
• Thermal Conductivity, k
Together, these define the state of the fluid.
20. 12/05/2006 BIEN 301 – Winter 2006-2007
Basic Physical Properties
Additional Properties (White 1.6)
Specific Weight, γ = ρg
Specific Gravity
• SGgas = ρgas / ρair
• SGwater = ρliquid / ρwater
Potential Energy
• -g●r
Kinetic Energy
• 0.5 V2
Total Energy
• e = û + 0.5 V2 + (-g●r)
21. 12/05/2006 BIEN 301 – Winter 2006-2007
State Relationships
State Relationships for Gases (White 1.6)
Thermodynamic properties are related to
each other by state relationships. For gases,
there is the ideal gas law (perfect-gas law).
• p = ρRT where R = cp – cv (gas constant)
The gas constant is related to the universal
gas constant, Λ by the following equation:
• Λ = Rgas * Mgas
22. 12/05/2006 BIEN 301 – Winter 2006-2007
State Relationships
State Relationships for Liquids
No direct analog of the ideal gas law exists for
liquids.
Why? If fluids involves liquids and gases, why
can we not get a direct correlation to a liquid
form?
• Compressibility. The ideal gas law assumes
compressibility, whereas most liquids are mostly
incompressible.
23. 12/05/2006 BIEN 301 – Winter 2006-2007
State Relationships
State Relationships for Liquids
As an example of this lack of direct
relationship, see from White, eq. 1.19:
B
B
n
a
a
1
Where B and n are dimensionless parameters that
vary with temperature.
24. 12/05/2006 BIEN 301 – Winter 2006-2007
Velocity Fields
For many of the problems encountered here, the velocity
field will be the solution to our given problem, or an
integral part thereof. (White 1.5)
The three-dimensional velocity field can be expressed in
a variety of ways:
...
ˆ
,
,
,
ˆ
,
,
,
ˆ
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
etc
k
t
z
y
x
w
j
t
z
y
x
v
i
t
z
y
x
u
t
z
y
x
V
or
z
t
z
y
x
V
y
t
z
y
x
V
x
t
z
y
x
V
t
z
y
x
V z
y
x
25. 12/05/2006 BIEN 301 – Winter 2006-2007
Velocity Fields
Simplified problems: in White, example 1.5, we see the
convective result for a 1-Dimensional problem. The
extended answer for the 3D problem is as follows:
z
w
w
y
w
w
x
w
w
z
v
v
y
v
v
x
v
v
z
u
u
y
u
u
x
u
u
t
z
y
x
a
,
,
,
26. 12/05/2006 BIEN 301 – Winter 2006-2007
Velocity Fields
Dealing with partial differential equations.
Cross out terms ahead of time, simplifies
calculations.
For the 2D problem, there are no velocity
components in the Z direction (no w
magnitude, and no δ() /δz.
27. 12/05/2006 BIEN 301 – Winter 2006-2007
Velocity Fields
y
u
u
x
u
u
z
w
y
w
x
w
z
v
y
v
x
v
z
u
u
y
u
u
x
u
u
z
w
w
y
w
w
x
w
w
z
v
v
y
v
v
x
v
v
z
u
u
y
u
u
x
u
u
t
z
y
x
a
0
0
0
0
0
0
0
0
0
,
,
,
28. 12/05/2006 BIEN 301 – Winter 2006-2007
Application
So, what can we do with all of this stuff? Why re-
hash over so many of the basics we have seen
in other courses over the years?
While we may have been exposed to all of these
concepts, they become integral in the study of fluid
mechanics.
Familiarity with these ideas is no longer enough, we
must master these concepts and learn to apply them
in new and effective ways.
29. 12/05/2006 BIEN 301 – Winter 2006-2007
Application
With these basics we will be able to:
Fully describe and define the subject of our study:
Fluids.
Perform dimensionally consistent calculations,
increasing the skill set required of a modern
professional engineer.
Be conversant and capable in both the BG and the SI
system, able to convert between the two as the
problem requires.
30. 12/05/2006 BIEN 301 – Winter 2006-2007
Application
With these basics we will be able to:
Understand the basic thermodynamic concepts
required to extend our analysis from pure fluid
mechanics to true energy transport problems.
• Heat Transfer
• Temperature-dependent effects
Accurately and professionally report our findings,
accounting for our experimental error and/or
uncertainty.
31. 12/05/2006 BIEN 301 – Winter 2006-2007
Application
As was mentioned before: these skills,
though ideally common throughout all of
our engineering courses, become absolute
cornerstones of success for a subject as
complex and difficult as fluid mechanics.
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Fundamental Approaches
There are two primary approaches to
problem solving in fluid mechanics:
Lagrangian and Eulerian
• Lagrangian: follows a fluid particle as it moves
through a flow field.
• Eulerian: Observes passing fluid particles from a
stationary position relative to the flow field
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Fundamental Approaches
Examples:
Lagrangian –
• A user observes traffic on the freeway as he sits in his
vehicle, travelling down the freeway along with the traffic.
Traffic jams, velocity changes, etc, are all marked and
observed to attempt to describe the flow of traffic through a
section of freeway.
Eulerian –
• A state trooper monitors freeway traffic from a hidden
location under the bridge, monitoring for changes in traffic
that could indicate potential trouble. Multiple state troopers
and cameras along the road give a “big picture” perspective
to traffic managers.
34. 12/05/2006 BIEN 301 – Winter 2006-2007
Fundamental Approaches
Eulerian will be our fundamental approach for this
course.
Probes at different points in the fluid stream are much
more easy to design and monitor for smaller systems
that we’ll concern ourselves with than large
instrumentation designs that follow the flow.
Can you think of an example of Eulerian monitoring
and/or Lagrangian monitoring in biomedical systems?
What are some potential benefits of each type of system
relative to this application?
35. 12/05/2006 BIEN 301 – Winter 2006-2007
Assignment
HW 2 has been posted on blackboard
Project Proposals due soon!
Individual project sign-ups will be available
by tonight on blackboard.
