Fluid Mechanics and Energy Transport BIEN 301 Lecture 14 Introduction to Compressible Flows

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

Lecture 14Lecture 14Introduction to Compressible FlowsIntroduction to Compressible Flows

Juan M. Lopez, E.I.T.Juan M. Lopez, E.I.T.Research ConsultantResearch Consultant

LeTourneau UniversityLeTourneau University

Adjunct LecturerAdjunct LecturerLouisiana Tech UniversityLouisiana Tech University

2/6/20072/6/2007 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

Compressible FlowCompressible Flow When density changes become significant, we When density changes become significant, we

have compressible flow.have compressible flow. Occurs in flows moving at sufficient speedOccurs in flows moving at sufficient speed Extremely high pressure gradients in compressible Extremely high pressure gradients in compressible

fluids (gases, blood)fluids (gases, blood) Most important effects of compressible flows:Most important effects of compressible flows:

ChokingChoking Shock WavesShock Waves


Compressible FlowCompressible Flow Mach Number Mach Number (White 9.1)(White 9.1)

Awesome transitional picture, p. 598 White.Awesome transitional picture, p. 598 White. Crossing Mach 1.0, but what is this number?Crossing Mach 1.0, but what is this number?

• Fractional portion of speed of sound in a mediumFractional portion of speed of sound in a medium• Defined as:Defined as:

mediumainsoundofSpeedVelocity

aVMA


Compressible FlowCompressible Flow Mach RegimesMach Regimes

• Ma < 0.3Ma < 0.3 Incompressible FlowIncompressible Flow• 0.3 < Ma < 0.80.3 < Ma < 0.8 Subsonic FlowSubsonic Flow• 0.8 < Ma < 1.20.8 < Ma < 1.2 Transonic FlowTransonic Flow• 1.2 < Ma < 3.01.2 < Ma < 3.0 Supersonic FlowSupersonic Flow• 3.0 > Ma3.0 > Ma Hypersonic Flow Hypersonic Flow

• These are only guides, the individual flow scenarios These are only guides, the individual flow scenarios affect how shock waves might develop.affect how shock waves might develop.


Compressible FlowCompressible Flow Specific Heat RatioSpecific Heat Ratio

• Usually Tabulated for you alreadyUsually Tabulated for you already• Variations have small effects on compressible flow Variations have small effects on compressible flow

calculations, so we will stick with air, k = 1.40 for most calculations, so we will stick with air, k = 1.40 for most calculations in this chapter.calculations in this chapter.

VolumeConstant ,PressureConstant ,

HeatSpecificHeatSpecific

cckv

p


Compressible FlowCompressible Flow Ideal GasIdeal Gas

• Simple state equation we often use to set up boundary Simple state equation we often use to set up boundary behavior calculations.behavior calculations.

slightly.only but re, temperatu vary withk, R,

constantk

constant

v

p

vp

cc

ccRwhere

RTp


Isentropic ProcessIsentropic Process Deals with change in entropyDeals with change in entropy

• The solved integral for constant cThe solved integral for constant cp:p:

entropyswhere

ppR

TTc

ppR

TTcss vp

1

2

1

2

1

2

1

212 lnlnlnln


Isentropic ProcessIsentropic Process Setting entropy equal (isentropic)Setting entropy equal (isentropic)

• The solved integral for constant cp reduces to the The solved integral for constant cp reduces to the power law relationship:power law relationship:

kk

k

TT

pp

ss

1

21

1

2

1

2

12


Speed of SoundSpeed of Sound Adiabatic ProcessAdiabatic Process

Expanding our control Expanding our control volume analysis to a wave volume analysis to a wave moving through a medium, moving through a medium, we can see how friction we can see how friction and heat transfer effects and heat transfer effects are limited to the interior of are limited to the interior of the wave (White, Fig. 9.1)the wave (White, Fig. 9.1)

Speed of sound relates the Speed of sound relates the propagation of a pressure propagation of a pressure pulse of infinitessimal pulse of infinitessimal length through a still fluid.length through a still fluid.

The speed of sound of a The speed of sound of a fluid is given by:fluid is given by:

liquid).or (gas fluidany for

a

wave) theinsideexcept gradients re temperatu(noionapproximat adiabatic

,

21

21

2

Ts

pkp

pa


Steady FlowSteady Flow Adiabatic and Adiabatic and

Isentropic Steady Isentropic Steady Flow Flow (White 9.3)(White 9.3)

Through these Through these simplifications, we are simplifications, we are able to reduce our able to reduce our steady flow to a few steady flow to a few simple relationshipssimple relationships

21

02

1

0

02

02

22max

21

21

TchVVelocity

TcVc

eTemperatur

consthVh

Enthalpy

EquationsStagnation

p

pp


Rigid Body MotionRigid Body Motion Definition Definition (White 2.9)(White 2.9)

All particles are in combined translation and rotation, All particles are in combined translation and rotation, and there is no relative motion between the particles.and there is no relative motion between the particles.

Recalling our last lecture, what terms drop out?Recalling our last lecture, what terms drop out?

Vgpa 2

ag

agp

gpa


Steady FlowSteady Flow

Mach Number RelationshipsMach Number Relationships

11

21

100

12100

20

1211

1211

211

kk

kk

kk

MakTT

MakTT

pp

aVMaMak

TT


Bernoulli’sBernoulli’s The isentropic, adiabatic process is actually a process The isentropic, adiabatic process is actually a process

that can be described via Bernoulli’s.that can be described via Bernoulli’s.

)'(0

:,

0

sBernoulliVdVdpgetwecombining

dpdhisentropic

dvVdhadiabatic


Examples, HomeworkExamples, Homework Examples for this chapterExamples for this chapter

P9.10, P9.33P9.10, P9.33 No homework from this section for next No homework from this section for next

time.time. Homework from Ch. 5 will be due next Homework from Ch. 5 will be due next

Thursday (will discuss in a minute)Thursday (will discuss in a minute)


Homework, ReportsHomework, Reports Due Date:Due Date:

Remember, last two homework assignments (Ch. 7, Ch. 8), as Remember, last two homework assignments (Ch. 7, Ch. 8), as well as the Individual and Group Projects are due next week: well as the Individual and Group Projects are due next week: Tuesday, Feb. 13.Tuesday, Feb. 13.

Any questions thus far?Any questions thus far? I’ll be posting the numerical analysis method file soonI’ll be posting the numerical analysis method file soon


Chapter 5 AssignmentChapter 5 Assignment Because there is not class on Thursday, we are going to be reading Chapter 5, and Because there is not class on Thursday, we are going to be reading Chapter 5, and

answering a few questions (DUE THURSDAY, FEB. 15).answering a few questions (DUE THURSDAY, FEB. 15). Define “Dimensional Analysis”, and describe how this is important for our work in Define “Dimensional Analysis”, and describe how this is important for our work in

fluid mechanics.fluid mechanics. Describe the Principle of Dimensional Homogeneity (PDH)Describe the Principle of Dimensional Homogeneity (PDH) Explain the Pi theoremExplain the Pi theorem Explain why nondimensionalization is important for some types of analyses.Explain why nondimensionalization is important for some types of analyses. Explain the following terms:Explain the following terms:

• Geometric SimilarityGeometric Similarity• Kinematic SimilarityKinematic Similarity• Dynamic SimilarityDynamic Similarity

In as much detail as possible, describe the steps you would consider in In as much detail as possible, describe the steps you would consider in designing a test stand to study a scaled-up version of a blood flow obstruction to designing a test stand to study a scaled-up version of a blood flow obstruction to determine wall shear stresses and stagnation points at the obstruction. determine wall shear stresses and stagnation points at the obstruction.

• Which of the items above would be most important? Which of the items above would be most important? • How would you go about ensuring the validity of your scaled-up experiment? How would you go about ensuring the validity of your scaled-up experiment? • Why would doing this scaled-up test be useful in the first place?Why would doing this scaled-up test be useful in the first place?


Exam 2 and HomeworkExam 2 and Homework No curve on examNo curve on exam

Definitely none needed.Definitely none needed. Exam worked as planned:Exam worked as planned:

• If you have been faithfully doing ALL of your homework and If you have been faithfully doing ALL of your homework and your OWN work without taking ANY shortcuts or giving up on your OWN work without taking ANY shortcuts or giving up on problems in homework, you did very well.problems in homework, you did very well.

• If you did not do well, consider how much of your homework If you did not do well, consider how much of your homework habits may have hindered you.habits may have hindered you.

• Rewarded those who have been working extremely faithfully Rewarded those who have been working extremely faithfully at getting their homework done RIGHT. Offset the cheating at getting their homework done RIGHT. Offset the cheating that was going on.that was going on.


RememberRemember

No study session tomorrow night.No study session tomorrow night. I’ll be leaving for Houston this afternoon. I’ll be leaving for Houston this afternoon.

Will not be available for office hours. You Will not be available for office hours. You may reach me via e-mail.may reach me via e-mail.


Flow VisualizationsFlow Visualizations Came across some amazing flow visualizations Came across some amazing flow visualizations

(computational and actual video). I’ll post the link (computational and actual video). I’ll post the link on Blackboard.on Blackboard. I’ll play them now while I hand back the exams and I’ll play them now while I hand back the exams and

homework, and the presenter for today sets up.homework, and the presenter for today sets up. After the presentation, I’ll review the exam for anyone After the presentation, I’ll review the exam for anyone

that would like to stay behind.that would like to stay behind. Your grades are recorded in my notebook, I’ll post Your grades are recorded in my notebook, I’ll post

them to Blackboard later on today.them to Blackboard later on today.


Questions?Questions?

Fluid Mechanics and Energy Transport BIEN 301 Lecture 14 Introduction to Compressible Flows

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