An Introduction to Compressible Flow - geraldcondon.com · compressible flow Understand the ... Thermodynamics laws Isentropic flow Conservation laws. Gas Dynamics ... Speed of sound

Gas Dynamics 1

An Introduction to Compressible Flow

Gas Dynamics

The Course Outcomes

Understand the definition and fundamental aspects of

compressible flow

Understand the definitions and types of shock and

expansion waves: oblique-shock waves, shock-

expansion interaction, and unsteady expansion waves.

Be able to do simple calculations related to

applications of compressible flow on variations in

geometry

Understand the concept of generalization of subsonic

and supersonic flows

Aims

To provide and introductory to the theories of compressible flows as part ofthe curriculum requirement and as a fundamental background to aerospacepropulsion courses

Gas Dynamics

Course Contents• Introduction to compressible flows

– Definitions and equations of compressible flow, Conservation laws, Sonic velocity and Mach number

• Isentropic Flow (IF)– Equations of isentropic flows and stagnation properties, IF in a converging

and converging-diverging nozzle, and application• Normal shock waves

– Stationary and moving normal shock wave, Working equations and shocks in nozzles, and application

• Oblique shock wave– Working equations, Oblique shock analysis

• Prandt-Meyer Flow– Analysis of Prandtl-Meyer flow – Shock-expansion interaction

• Adiabatic flow– Working equations, nozzle operation, and analysis of Fanno flow

• Flow with heat transfer– Analysis of Rayleigh subsonic & supersonic flow.

Gas Dynamics

Recommended Text booksn Fundamentals of Gas Dynamics by Robert D. Zucker : ISBN 0-916460-12-6n Gas Dynamics by James John and Theo Keith : ISBN 0-13-202331-8n Fundamental of Aerodynamics by John D. Anderson, Jr : ISBN 0-07-

001656-9n Dr Zulfa lecture notes – Gas dynamics

Course Assesment

n Midterm 30%n Final Exam 70%

Gas Dynamics 5

De Laval Nozzle• High Speed flows often seem “counter-intuitive” whenCompared with low speed flows

• Example: Convergent-Divergent Nozzle (De Laval)

In 1897 Swedish Engineer Gustav De Laval designedA turbine wheel powered by 4- steam nozzles

De Laval Discovered that if the steam nozzle first narrowed, and then expanded, the efficiency ofthe turbine was increased dramatically

Furthermore, the ratio of the minimum areato the inlet and outlet areas was critical for achievingmaximum efficiency … Counter to the “wisdom” of the day

flow

Convergent / Divergent Nozzle

Credit: NASA GSFC

Gas Dynamics 6

De Laval Nozzle (cont’d)

• Mechanical Engineers of the 19’th century werePrimarily “hydrodynamicists” .. That is they wereFamiliar with fluids that were incompressible … liquidsand Low speed gas flows where fluid density wasEssentially constant

• Primary Principles are Continuity and Bernoulli’s Law

Gas Dynamics 7


AeA I At

pIVIAIρ

peVeAeρ

ptVtAtρ

• When Continuity and Bernoulli are applied to a De Laval Nozzle and density is Assumed constant

At Throat

ContinuityBernoulli

• Pressure Drop• Velocity Increases

“classic” Venturi

High Pressure Inlet

Gas Dynamics 8


AeA I At

pIVIAIρ

peVeAeρ

ptVtAtρ

• When Continuity and Bernoulli are applied to a De Laval Nozzle and density is Assumed constant

At Exit

BernoulliContinuity

• Pressure Increases• Velocity Drops

High Pressure Inlet

Gas Dynamics 9

De Laval Nozzle (concluded)

AeA I At

pIVIAIρ

peVeAeρ

ptVtAtρ

• But De Laval Discovered that when the Nozzle throatArea was adjusted downward until the pressure ratio becamept / pI < 0.5484 -> then the exit Pressure dropped (instead ofRising … compared to the throat pressure)And the exit velocity rose (instead of dropping)… Which is counter to What Bernoulli’s law predicts … he had inadvertently ,,, Generated supersonic flow! …

High Pressure Inlet

• fundamental principle that makes rocket motors possible

Review of prerequisite elements

10

Perfect gasThermodynamics lawsIsentropic flowConservation laws

Gas Dynamics

constant area duct

quasi one-dimensional flow

compressible flow

steady flow

isothermal flow

ideal gas

Diameter (D) 4/2DA π= is a constant

speed (u)

x

u varies only in x-direction

Density (ρ) is NOT a constant

Temperature (T) is a constant

Obeys the Ideal Gas equation

uAm ρ=&Mass flow rate is a constant

Gas Dynamics

Friction factor: (shear stress acting on the wall)

For laminar flow in circular pipes:

where Re is the Reynolds number of the flow defined as follows:

For lamina flow in a square channel:

For the turbulent flow regime:

Re/16=f

µρuD

=ReµD

Am&

=µπD

Dm

24 &

=µπD

m&4=

Re/227.14=f

=

εD

f7.3log0.41

10

Quasi one-dimensional flow is closer to turbulent velocity profile than to laminar velocity profile.

2/2ufw ρτ =

Gas Dynamics

Ideal Gas equation of state:

mRTpV =pressure

volume

mass

specific gas constant(not universal gas constant)

temperature

RTVmp =

Ideal Gas equation of state can be rearranged to give

RTp ρ=

kg/m3 J/(kg.K)

K

Pa = N/m2

Gas Dynamics

Review of prerequisite elements

• Perfect gas:Equation of state

For calorically perfect gas

Tqds δ

=RTP ρ=

v

p

vp

v

p

cc

RccdTcdu

dTcdhRTuh

Tuu

=

+==

=+=

=

γ

)(

Entropy

Entropy changes?

−

=−

+

=−

1

2

1

212

2

1

1

212

lnln

lnln

PPR

TTcss

RTTcss

p

v ρρ

p

v

cR

p

cR

v

PP

css

TT

css

TT

−=

−=

1

212

1

2

1

212

1

2

exp

expρρ

TvdPdh

TPdvduds −

=+

=

Gas Dynamics

Review of prerequisite elements Cont.

Forms of the 1st law

dpdhTdspddeTdsewq

υυ

δδδ

−=+=

=+

Tqds δ

≥

The second law

Gas Dynamics


γ

γγ

γ

ρρ

ρρ

=

=

=

−−

1

2

1

2

1

1

2

1

1

2

1

2

PP

PP

TT

For an isentropic flow

γγ

γ

ρρ

ρρ

1

1

2

1

2

1

2

1

1

2

1

2

1

2

−

−

=

=

=

=

PP

PP

TT

TT

p

v

cR

cR

If ds=o

constant=γρP

Gas Dynamics


Conservation of mass (steady flow):

Rate of massenters controlvolume

Rate of massleaves controlvolume

=

1 2dAAdVVd

+++ ρρ

AVρ

dx

flow

AdA

VdV

AdA

VdVd

VdAAdVVAd

dAAdVVdVA

AVAV

mm

−=

=++

=++

+++=

=

=

0

0

))()((

222111

21

ρρ

ρρρ

ρρρ

ρρ

&&

If ρ is constant (incompressible):

Gas Dynamics


Conservation of momentum (steady flow):

Rate momentumleaves controlvolume

Rate momentumenters controlvolume

-Net force ongas in controlvolume

=

( ) ( )12 VmVmFF p && −=+µ

Euler equation (frictionless flow):

∫ =+ constant2

2

ρdpV

1 2

dAAdVVddpp

++++

ρρ

AV

pρ

dx

flow

Gas Dynamics


++−

+++−= e

eeei

iii

CV gzVumgzVumWQdt

dE22

22

&&&&

heat transfer energy transfer due to mass flowwork transfer

Basic principle:• Change of energy in a CV is related to energy transfer by heat, work,

and energy in the mass flow.

Conservation of energy for a CV (energy balance):

Gas Dynamics


iep

pCV

WWW

WWW&&&

&&&

−=

+=

( )

++−

+++−=

+=

+++−

++++−=

++−

+++−−−=

ee

eeii

iiCVCV

ee

eeeeii

iiiiCVCV

ee

eeii

iiiiieeeCVCV

gzVhmgzVhmWQdt

dE

pvuh

gzVvpumgzVvpumWQdt

dE

gzVumgzVumvpmvpmWQdt

dE

22

22

22

22

22

22

&&&&

&&&&

&&&&&&

pvmWAVvmAVm

VFWpAF

p

ppp

&&&&

&

=

=⇒=

=⇒=

ρ

Most important formof energy balance.

Analyzing more about Rate of Work Transfer:• work can be separated into 2 types:

• work associated with fluid pressure as mass entering or leaving the CV.• other works such as expansion/compression, electrical, shaft, etc.

Work due to fluid pressure:• fluid pressure acting on the CV boundary creates force.

Gas Dynamics


( ) ( )2

22

22

22

ieie

iii

eee

VVhhdwdq

VhmVhmWQ

−+−=−

+−

+=− &&&&

1 2dVVdhhdTT

+++

VhT

dx

flowTch p=

For adiabatic flow (no heat transfer)and no work:

For calorically perfect gas (dcp=dcv=0):

0=+VdVdTcp

Gas Dynamics

Conservation of mass(compressible flow):

Conservation of momentum(frictionless flow):

Conservation of energy(adiabatic):

021 =++→=A

dAVdVdmm

ρρ&&

( ) ( ) 012 =+→−=+ VdVdPVmVmFF p ρµ &&

( ) ( ) 02

22

=+→−

+−=− VdVdTcVVhhdwdq pie

ie


Conservation laws

Gas Dynamics

Exercises 11. Given that standard atmospheric conditions for air at 150C are a

pressure of 1.013 bar and a density of 1.225kg, calculate the gas constant for air. Ans: R=287.13J/kgK

2. The value of Cv for air is 717J/kgK. The value of R=287 J/kgK. Calculate the specific enthalpy of air at 200C. Derive a relation connecting Cp, Cv, R. Use this relation to calculate Cp for air using the information above. Ans: h=294.2kJ/kgK,Cp=1.004kJ/kgK

3. Air is stored in a cylinder at a pressure of 10 bar, and at a room temperature of 250C. How much volume will 1kg of air occupy inside the cylinder? The cylinder is rated for a maximum pressure of 15 bar. At what temperature would this pressure be reached? Ans: V=0.086m2, T=1740C.

Gas Dynamics

Sound Speed and Mach Number

24

Gas Dynamics

Speed of sound

0=VT

Pρ

dVVdTTddPP

=+++

ρρ

Sound wave

Sounds are the small pressure disturbances in the gas around us, analogous to the surface ripples produced when still water is disturbed

aVT

P

=

ρ

dVaVdTTddPP

−=+++

ρρ

Sound wave

Sound wave movingthrough stationary gas

Gas moving throughstationary sound wave

Gas Dynamics

Derivation of speed of sound

Speed of sound cont.

( )( )

adVd

AdVadaAm

=

−+==

ρρ

ρρρ&

( ) ( )adVdP

amdVamAdPPPAρ=

−−=+− &&

ρddpa =

constant

1

2

1

2

=

=

γ

γ

ρ

ρρ

PPP

RTPa

PddP

γρ

γ

ργ

ρ

==

=

Conservation of mass

Conservation of momentum

Combination of mass and momentum

For

isentropic flow

Finally

Gas Dynamics

What is a Mach number?

Definition of Mach number (M):

M ≡Speed of the flow (u)

Speed of sound (c) in the fluid at the flow temperature

Incompressible flow assumption is not valid if Mach number > 0.3

RTc γ=For an ideal gas,

specific heat ratiospecific gas constant (in J/kg.K)absolute temperature of the flow at the point concerned (in K)

Gas Dynamics

For an ideal gas,

Unit of c = [(J/kg.K)(K)]0.5

= [m2/s2]0.5 = m/s

= [kg.(m/s2).m/kg]0.5

M =u u

RTγc=

Unit of u = m/s

= [J/kg]0.5 = (N.m/kg)0.5

Gas Dynamics

Mach Number

M=V/a

Source of disturbance

Distance traveled = speed x time = 4at

Zone of silence

Region of influence

If M=0

M<1 Subsonic

M=1 Sonic

M>1 Supersonic

M>5 Hypersonic Distance traveled = at

Gas Dynamics

Mach Number cont.


If M=0.5

Original location of source of

disturbance

Gas Dynamics 31

• As the object approaches the speed of sound, it begins to catch up withthe pressure waves and creates an infinitesimally weak flow discontinuityjust ahead of the aircraft

Mach Number cont.

Gas Dynamics

Mach Number cont.


If M=2

Original location of source of

disturbance

utut

utut

Mutat 1sin ==α

Mach wave:

Directionof motion

An Introduction to Compressible Flow - geraldcondon.com · compressible flow Understand the ... Thermodynamics laws Isentropic flow Conservation laws. Gas Dynamics ... Speed of sound

Documents