Juan M. Lopez, E.I.T. Research Consultant LeTourneau University Adjunct Lecturer

Fluid Mechanics and Energy TransportFluid Mechanics and Energy TransportBIEN 301BIEN 301

Lecture 2Lecture 2 Introduction to Fluids, Flow Fields, and Dimensional Introduction to Fluids, Flow Fields, and Dimensional

AnalysisAnalysis

Juan M. Lopez, E.I.T.Juan M. Lopez, E.I.T.

Research ConsultantResearch ConsultantLeTourneau UniversityLeTourneau University

Adjunct LecturerAdjunct LecturerLouisiana Tech UniversityLouisiana Tech University

12/05/200612/05/2006 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

History of Fluid MechanicsHistory of Fluid Mechanics

White 1.14 shows us how Fluid Mechanics has evolved in a helical White 1.14 shows us how Fluid Mechanics has evolved in a helical fashion, returning to its roots, with improvements each time.fashion, returning to its roots, with improvements each time.

Pre-historic and early history aqueducts and waterworks – Pre-historic and early history aqueducts and waterworks – Empirically Designed and BuiltEmpirically Designed and Built

Archimedes (200’s B.C.) and Buoyancy / Vector addition – Archimedes (200’s B.C.) and Buoyancy / Vector addition – Theoretical work with Experimental rootsTheoretical work with Experimental roots

200’s B.C. to Renaissance ship and canal building – Empirical 200’s B.C. to Renaissance ship and canal building – Empirical advances, no great amount of experimental workadvances, no great amount of experimental work

Leonardo da Vinci first formulated the one-dimensional Leonardo da Vinci first formulated the one-dimensional conservation of mass equation – Theoretical stemming from conservation of mass equation – Theoretical stemming from empirical observations.empirical observations.


History of Fluid MechanicsHistory of Fluid Mechanics Mariotte (1600’s) built the first wind tunnel – Testing theoretical Mariotte (1600’s) built the first wind tunnel – Testing theoretical

ideas with experimental work.ideas with experimental work. Isaac Newton (1600’s-1700’s) generated the mathematics which Isaac Newton (1600’s-1700’s) generated the mathematics which

allowed fluid momentum to be studied.allowed fluid momentum to be studied. Bernoulli, D’Alembert, Euler, Lagrange, Laplace, all developed Bernoulli, D’Alembert, Euler, Lagrange, Laplace, all developed

their work in frictionless fluids, and showed the need for a their work in frictionless fluids, and showed the need for a formulation that would do away with the paradox of an object with formulation that would do away with the paradox of an object with no drag immersed in a moving stream, a natural result of no drag immersed in a moving stream, a natural result of frictionless fluid assumptions – Theoretical advances mostly.frictionless fluid assumptions – Theoretical advances mostly.

These theoretical results were unsatisfactory to engineers, so as These theoretical results were unsatisfactory to engineers, so as a natural backlash, a natural backlash, hydraulicshydraulics was developed as an almost was developed as an almost purely experimental form by Pitot, Borda, Poiseuille, etc.purely experimental form by Pitot, Borda, Poiseuille, etc.


History of Fluid MechanicsHistory of Fluid Mechanics Late 1800’s, finally there was a trend towards the unification Late 1800’s, finally there was a trend towards the unification

between experimental between experimental hydraulicshydraulics and theoretical and theoretical hydrodynamicshydrodynamics by the likes of Froude, Raylegh, and Reynolds. All of these by the likes of Froude, Raylegh, and Reynolds. All of these gentlemen have dimensionless groups named after them due to gentlemen have dimensionless groups named after them due to the importance of their work.the importance of their work.

Navier and Stokes began to more fully explore viscous flow in Navier and Stokes began to more fully explore viscous flow in the mid to late 1800’s, setting the stage for Prandtl.the mid to late 1800’s, setting the stage for Prandtl.

In the early 1900’s, Prandtl developed boundary layer theory, In the early 1900’s, Prandtl developed boundary layer theory, one of the most important advances in fluid mechanics, identified one of the most important advances in fluid mechanics, identified by White as by White as the single most important tool in modern flow the single most important tool in modern flow analysis.analysis.


History of Fluid MechanicsHistory of Fluid Mechanics

The past tied to the presentThe past tied to the present These past examples of development in fluid mechanics remain These past examples of development in fluid mechanics remain

important due to the individual contributions each advance has important due to the individual contributions each advance has made to our current understanding. made to our current understanding.

In fact, we continue to study many of these individual ideas as In fact, we continue to study many of these individual ideas as simplified examples of fluid behavior.simplified examples of fluid behavior.

Fluid mechanics encompasses almost every field of physical Fluid mechanics encompasses almost every field of physical systems, and a basic understanding of the mathematics, systems, and a basic understanding of the mathematics, terminology, and usage will greatly benefit you in any terminology, and usage will greatly benefit you in any engineering field.engineering field.


What is a Fluid?What is a Fluid?

Matter that is unable to resist shear by a Matter that is unable to resist shear by a static deflection. static deflection. (White, 1.2)(White, 1.2)

Fluid will deflect under shear unless opposed by some external force. 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.The rate of strain to stress is dependent on the viscosity of the fluid.



This lack of resistance to shear explains why This lack of resistance to shear explains why fluid take the shape of their containers, or spill fluid take the shape of their containers, or spill when there is no body to contain them.when there is no body to contain them.



Mechanical Description – Mohr’s CircleMechanical Description – Mohr’s Circle



As with everything, we make some assumptions in our As with everything, we make some assumptions in our definition-definition-

Continuum Continuum (White 1.3)(White 1.3)

• Infinitely Divisible – All divisions have same properties in Infinitely Divisible – All divisions have same properties in homogeneous fluidhomogeneous fluid

• For real systems, there are uncertainties brought about by volumes For real systems, there are uncertainties brought about by volumes that are too small or too large.that are too small or too large.

Physical properties are defined and have finite values Physical properties are defined and have finite values throughout the continuumthroughout the continuum

Thermal properties are defined and have finite values throughout Thermal properties are defined and have finite values throughout the continuumthe continuum


Dimensions vs. UnitsDimensions vs. Units

We must inherently have a way to We must inherently have a way to describedescribe the systems the systems we are studying. We describe these systems with we are studying. We describe these systems with Dimensions and quantify these dimensions with Units.Dimensions and quantify these dimensions with Units.

Four Four primary Dimensionsprimary Dimensions in our study of Fluid in our study of Fluid Mechanics:Mechanics:

Mass, {M}Mass, {M} Length, {L}Length, {L} Time, {T}Time, {T} Temperature, {Temperature, {ΘΘ}}


Dimensions vs. UnitsDimensions vs. Units It is imperative that you learn consistency in your dimensional It is imperative that you learn consistency in your dimensional

analysis. Fluid mechanics lends itself to some extremely awkward analysis. Fluid mechanics lends itself to some extremely awkward units, especially in the British system.units, especially in the British system.

For this course, we will primarily stick with the International 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 (SI), but we will refresh our memories from time to time on how to interact with the British Gravitational (BG) units.interact with the British Gravitational (BG) units.

The use of tables is an inherent task in engineering work. Become 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 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.Tables A.1-A.6, and how to properly use them.



Using the tables, perform the following Using the tables, perform the following conversion:conversion:

expression SI equivalent Obtain the

54.1112

:as unitsBG in given isair for constant gas The

Rslug

lbfftRair



) A.4 Table (Matches208

208208208

:obtain wearranging,-Re

207.7565556.0

1*115.429

115.429594.14

1*1684.57

1684.574482.4*378.709

378.7093408.0*54.1112

:book ofcover front and 1.2 Table From

2

2

2

22

Ks

mR

Ks

m

Kkgs

mkgm

Kkg

NmR

Kkg

Nm

K

R

Rkg

Nm

Rkg

Nm

kg

slug

Rslug

Nm

Rslug

Nm

lbf

N

Rslug

lbfm

Rslug

lbfm

ft

m

Rslug

lbfftR

air

air

air


Dimensional ConsistencyDimensional Consistency

Dimensional Homogeneity Dimensional Homogeneity (White 1.4)(White 1.4)

Theoretical Equations – dimensionally Theoretical Equations – dimensionally homogeneoushomogeneous

out) (checks ][][][][

]][[][]][[][][

2

1

221

200

LLLL

TLTTLTLL

gttVSS


Dimensional ConsistencyDimensional Consistency

However, much work in fluid mechanics However, much work in fluid mechanics has been empirical, and this can lead to has been empirical, and this can lead to problematic situations.problematic situations.

units) (awkward

][]/[][

][[?]][2/12/712/12/113

12/12/113

2/1

Cv

Cv

V

MLTLMTL

TLMTL

SG

pCQ


UncertaintyUncertainty

Once we have established a way to describe Once we have established a way to describe these systems, we must also account for the these systems, we must also account for the uncertainty in our experimentation. uncertainty in our experimentation. (White 1.11)(White 1.11)

Instruments and all physical measurements Instruments and all physical measurements have some form of uncertainty. have some form of uncertainty. Accounting for all the measurements is importantAccounting for all the measurements is important Adding them all is simply not realisticAdding them all is simply not realistic A simplified Root Mean Square (RMS) approach is A simplified Root Mean Square (RMS) approach is

recommended.recommended.



RMS Formulation:RMS Formulation:

2/1222

...

...P

For

m

mm

j

jj

i

ii

nm

nj

ni

x

xn

x

xn

x

xn

P

P

xxx mji



RMS ExampleRMS Example

%92.2%5.03%55.0

%5.03%55.0

%5.0%55P

For

2/122

2/122

32/1

P

PP

P

andwhere


Basic Physical PropertiesBasic Physical Properties Thermodynamics Thermodynamics (White 1.6)(White 1.6)

Principal components of velocity vectorsPrincipal components of velocity vectors• Pressure, Pressure, pp• Density, Density, ρρ• Temperature, Temperature, TT

Principal components of work, heat, and energy balance.Principal components of work, heat, and energy balance.• Internal Energy, Internal Energy, ûû• Enthalpy, Enthalpy, h = û + h = û + ρρ/p/p

Principal transport propertiesPrincipal transport properties• Viscosity, Viscosity, μμ• Thermal Conductivity, Thermal Conductivity, k k

Together, these define the state of the fluid.Together, these define the state of the fluid.


Basic Physical PropertiesBasic Physical Properties Additional Properties Additional Properties (White 1.6)(White 1.6)

Specific Weight, Specific Weight, γγ = = ρρgg Specific GravitySpecific Gravity

• SGSGgasgas = = ρρgas gas / / ρρairair

• SGSGwaterwater = = ρρliquid liquid / / ρρwaterwater

Potential EnergyPotential Energy• --gg●●rr

Kinetic EnergyKinetic Energy• 0.5 0.5 VV22

Total EnergyTotal Energy• e = e = û + û + 0.5 0.5 VV2 2 + + (-(-gg●●rr))


State RelationshipsState Relationships

State Relationships for Gases State Relationships for Gases (White 1.6)(White 1.6)

Thermodynamic properties are related to each Thermodynamic properties are related to each other by state relationships. For gases, there other by state relationships. For gases, there is the is the ideal gas law (perfect-gas law).ideal gas law (perfect-gas law).• p = p = ρρRT where R = cRT where R = cpp – c – cvv (gas constant)(gas constant)

The gas constant is related to the universal The gas constant is related to the universal gas constant, gas constant, ΛΛ by the following equation: by the following equation:• ΛΛ = R = Rgasgas * M * Mgasgas



State Relationships for LiquidsState Relationships for Liquids No direct analog of the ideal gas law exists for No direct analog of the ideal gas law exists for

liquids.liquids. Why? If fluids involves liquids and gases, why Why? If fluids involves liquids and gases, why

can we not get a direct correlation to a liquid can we not get a direct correlation to a liquid form?form?• Compressibility. The ideal gas law assumes Compressibility. The ideal gas law assumes

compressibility, whereas most liquids are mostly compressibility, whereas most liquids are mostly incompressible.incompressible.



State Relationships for LiquidsState Relationships for Liquids As an example of this lack of direct As an example of this lack of direct

relationship, see from White, eq. 1.19:relationship, see from White, eq. 1.19:

BBn

aa

1

Where B and n are dimensionless parameters that Where B and n are dimensionless parameters that vary with temperature.vary with temperature.


Velocity FieldsVelocity Fields

For many of the problems encountered here, the velocity For many of the problems encountered here, the velocity field will be the solution to our given problem, or an field will be the solution to our given problem, or an integral part thereof. integral part thereof. (White 1.5)(White 1.5)

The three-dimensional velocity field can be expressed in The three-dimensional velocity field can be expressed in a variety of ways:a variety of ways:

...

ˆ,,,ˆ,,,ˆ,,,,,,

,,,,,,,,,,,,

etc

ktzyxwjtzyxvitzyxutzyxV

or

ztzyxVytzyxVxtzyxVtzyxV zyx



Simplified problems: in White, example 1.5, we see the Simplified problems: in White, example 1.5, we see the convective result for a 1-Dimensional problem. The convective result for a 1-Dimensional problem. The extended answer for the 3D problem is as follows:extended answer for the 3D problem is as follows:

z

ww

y

ww

x

ww

z

vv

y

vv

x

vv

z

uu

y

uu

x

uu

tzyxa

,,,



Dealing with partial differential equations.Dealing with partial differential equations. Cross out terms ahead of time, simplifies Cross out terms ahead of time, simplifies

calculations.calculations. For the 2D problem, there are no velocity For the 2D problem, there are no velocity

components in the Z direction (no w components in the Z direction (no w magnitude, and no magnitude, and no δδ()() / /δδz.z.



y

uu

x

uu

z

w

y

w

x

w

z

v

y

v

x

v

z

uu

y

uu

x

uu

z

ww

y

ww

x

ww

z

vv

y

vv

x

vv

z

uu

y

uu

x

uu

tzyxa

0

0

0

000

000

,,,


ApplicationApplication

So, what can we do with all of this stuff? Why re-So, what can we do with all of this stuff? Why re-hash over so many of the basics we have seen hash over so many of the basics we have seen in other courses over the years?in other courses over the years? While we may have been exposed to all of these While we may have been exposed to all of these

concepts, they become concepts, they become integralintegral in the study of fluid in the study of fluid mechanics.mechanics.

Familiarity with these ideas is no longer enough, we Familiarity with these ideas is no longer enough, we must must mastermaster these concepts and learn to apply them these concepts and learn to apply them in new and effective ways.in new and effective ways.



With these basics we will be able to:With these basics we will be able to: Fully describe and define the subject of our study: Fully describe and define the subject of our study:

Fluids.Fluids. Perform dimensionally consistent calculations, Perform dimensionally consistent calculations,

increasing the skill set required of a modern increasing the skill set required of a modern professional engineer.professional engineer.

Be conversant and capable in both the BG and the SI Be conversant and capable in both the BG and the SI system, able to convert between the two as the system, able to convert between the two as the problem requires.problem requires.



With these basics we will be able to:With these basics we will be able to: Understand the basic thermodynamic concepts Understand the basic thermodynamic concepts

required to extend our analysis from pure fluid required to extend our analysis from pure fluid mechanics to true energy transport problems.mechanics to true energy transport problems.• Heat TransferHeat Transfer• Temperature-dependent effectsTemperature-dependent effects

Accurately and professionally report our findings, Accurately and professionally report our findings, accounting for our experimental error and/or accounting for our experimental error and/or uncertainty.uncertainty.



As was mentioned before: these skills, As was mentioned before: these skills, though ideally common throughout all of though ideally common throughout all of our engineering courses, become absolute our engineering courses, become absolute cornerstones of success for a subject as cornerstones of success for a subject as complex and difficult as fluid mechanics.complex and difficult as fluid mechanics.


Fundamental ApproachesFundamental Approaches

There are two primary approaches to There are two primary approaches to problem solving in fluid mechanics:problem solving in fluid mechanics: Lagrangian and EulerianLagrangian and Eulerian

• Lagrangian: follows a fluid particle as it moves Lagrangian: follows a fluid particle as it moves through a flow field.through a flow field.

• Eulerian: Observes passing fluid particles from a Eulerian: Observes passing fluid particles from a stationary position relative to the flow fieldstationary position relative to the flow field



Examples:Examples: Lagrangian – Lagrangian –

• A user observes traffic on the freeway as he sits in his A user observes traffic on the freeway as he sits in his vehicle, travelling down the freeway along with the traffic. vehicle, travelling down the freeway along with the traffic. Traffic jams, velocity changes, etc, are all marked and Traffic jams, velocity changes, etc, are all marked and observed to attempt to describe the flow of traffic through a observed to attempt to describe the flow of traffic through a section of freeway.section of freeway.

Eulerian – Eulerian – • A state trooper monitors freeway traffic from a hidden A state trooper monitors freeway traffic from a hidden

location under the bridge, monitoring for changes in traffic location under the bridge, monitoring for changes in traffic that could indicate potential trouble. Multiple state troopers that could indicate potential trouble. Multiple state troopers and cameras along the road give a “big picture” perspective and cameras along the road give a “big picture” perspective to traffic managers.to traffic managers.



Eulerian will be our fundamental approach for this Eulerian will be our fundamental approach for this course.course.

Probes at different points in the fluid stream are much Probes at different points in the fluid stream are much more easy to design and monitor for smaller systems more easy to design and monitor for smaller systems that we’ll concern ourselves with than large that we’ll concern ourselves with than large instrumentation designs that follow the flow.instrumentation designs that follow the flow.

Can you think of an example of Eulerian monitoring Can you think of an example of Eulerian monitoring and/or Lagrangian monitoring in biomedical systems? and/or Lagrangian monitoring in biomedical systems? What are some potential benefits of each type of system What are some potential benefits of each type of system relative to this application?relative to this application?


AssignmentAssignment

HW 2 has been posted on blackboardHW 2 has been posted on blackboard Project Proposals due soon!Project Proposals due soon! Individual project sign-ups will be available Individual project sign-ups will be available

by tonight on blackboard.by tonight on blackboard.


Questions?Questions?

Juan M. Lopez, E.I.T. Research Consultant LeTourneau University Adjunct Lecturer

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