Chap 5 Quasi-One- Dimensional Flow. 5.1 Introduction Good approximation for practicing gas dynamicists eq. nozzle flow 、 flow through wind tunnel & rocket.

Chap 5

Quasi-One-Dimensional Flow

5.1 Introduction

Good approximation for practicing gas dynamicists eq. nozzle flow 、 flow through wind tunnel & rocket engines

5.2 Governing Equations

• For a steady,quasi-1D flow

The continuity equation :

222111 AuAu

s

dt

sdv

The momentum equation :

s s

sdpdfdt

vvsdv

)(

)(

2222221

21111 )( 2

1

AuApApdAuAp x

A

A

Automatically balainced

X-dir

Y-dir

The energy equation

s

dvfsdvpdq )(

s

dsvV

edV

et

)

2()]

2([

22

consthu

hu

h 0

22

2

21

1 22

peh

total enthalpy is constant along the flow

Actually, the total enthalpy is constant along a streamline in any adiabatic steady flow

PAuρ

P +dPA +dAu +duρ+dρ

dx

In differential forms

0)( uAd

constuA

)())(())(( 2

2

dAAduuddAAdpp

pdAAupA

0222 uAdudAudAuAdp

Dropping 2nd order terms

(1)

022 dAuuAdudAu (2)0)( uAd

(1) - (2) = 0 uAduAdp

ududp

constu

h 2

2

0)2

(2

u

hd

0ududh

Euler’s equation

)()(

)()(

dudp

dudp

5.3 Area-Velocity Relation0)( uAd

uA

AudAduudA

0 udu

d

d

dPdP

0A

dA

u

dud

u

duM

ua

duu

a

udud 2

2

2

2

∵ adiabatic & inviscid no dissipation mechanism∴

→ isentropic

u

duM

A

dA)1( 2

Important information1. M→0 incompressible flow

Au=const consistent with the familiar continuity eq for incompressible flow

2. 0 M≦ ＜ 1 subsonic flow

an increase in velocity (du ， +) is associated with a decrease in area (dA，－ ) and vice versa.

3. M>1 supersonic flowan increase in velocity is associated with an increase in area , and vice versa

4. M=1 sonic flow →dA/A=0

a minimum or maximum in the area

A subsonic flow is to be accelerated isentropically from subsonic to supersonic

Supersonic flow is to be decelercted isentropically from supersonic to subsonic

Application of area-velocity relation

1.Rocket engines

2.Ideal supersonic wind tunnel

Diffuser is to slow down the flow in the convergent duct to sonic flow at the second throat, and then futher slowed to low subsonic speeds in the divergent duct.(finally being exhausted to the atmosphere for a blow-down wind tunnel)

“chocking” “blocking”(When both nozzle with M=1)

Handout – Film Note by Donald Coles

5.4 Isentropic Flow of a Calorically Perfect Gas through Variable-Area Duct

***** auuAAu

u

a

A

A *0

0

*

*

Stagnation density (constant throughout an isentropic flow)

1

12

22

*)]

2

11(

1

2[

1)(

r

r

Mr

rMA

A

1

120 )

2

11( rMr

1

1

1

1

*0 )

2

1()

2

11(

rr rr

)3.(

21

1

21

)(2*

2

2

2

*chM

Mr

Mr

a

u

(1)

(2)

(3)

Area – Mach Number Relation

There are two values of M which correspond to a given A/A* >1 , a subsonic & a supersonic value

Boundary conditions will determine the solution is subsonic or supersonic

)( *AAfM

1. For a complete shock-free isentropic supersonic flow, the exit pressure ratio Pe /P0 must be precisely equal to Pb /P0

2. Pe /P0 、 Te /T0 & Pe /P0 = f(Ae /A*) and are continuously decreasing.

3. To start the nozzle flow, Pb must be lower than P0

4. For a supersonic wind tunnel, the test section conditions are determined by (Ae /A*) 、 P0 、 T0 gas property & Pb

Pb=P0 at the beginning there is no ∴flow exists in the nozzle

Minutely reduce Pb ， this small pressure difference will cause a small wind to blow through the duct at low subsonic speeds

Futher reduce Pb ， sonic conditions are reached (Pb=Pe3)

Pe /P0 & A/At are the controlling factors for the local flow properties at any given section

528.0)2

11( 1

0

*

r

rr

p

p

for r=1.4

tAtUtm

Should use dash-line to indicate irreversible process

What happens when Pb is further reduced below Pe3 ?

Note: quasi-1D consideration does not tell us much about how to design the contour of a nozzle – essentially for ensuring a shockfree supersonic nozzle

Method of characteristics

Wave reflection from a free boundary

Waves incident on a solid (free) boundary reflect in like (opposite) manner , i.e, a compression wave as a compression (expansion wave ) and an expansion wave reflects as an expansion ( compression ) wave

5.5 diffusers

Assume that we want to design a supersonic wind tunnel with a test section M=3Ae/A*=4.23P0/Pe=36.7

3 alternatives

(a) Exhaust the nozzle directly to the atmosphere

(b) Exhaust the nozzle into a constent area duct which serves as the test section

atmPP

P

P

PP e

e

55.3)P10.33

1(36.7)(

02

00

∴ the resvervair pressure required to drive the wind tunnel has markedly dropped from 36.7 to 3.55 atm

(c) Add a divergent duct behind the normal the normal shock to even slow down the already subsonic flow to a lower velocity

atmPPPPPPPPee 04.3)11.171)(10.331(36.7)(022200

3M

For

328.00102PP

04.3328.010201PP atmPP

P

P

P

P

PP e

e

04.3117.1

1)

33.10

1)(7.36(

02

2

2

00

∴ the reservoir pressure required to drive a supersonic wind tunnel (and hence the power required form the compressors) is considerably reduced by the creation of a normal shock and subsequent isentropic diffusion to M ～ 0 at the tunnel exit

Note:

3M 328.001

02 P

P

04.3328.0

1

02

01 P

P

Diffuser - the mechanism to slow the flow with as small a

loss total pressure as possible

Consider the ideal supersonic wind funnel again

If shock-free →P02/P01=1 no lose in total pressure

→a perpetual motion machine!!← something is wrong(1) in real life , it hard to prevent oblique shock wave from occuring inside the duct(2) even without shocks , friction will cause a lose of P0

the design of a perfect isentropic diffuser is physically impossible∴

Replace the normal shock diffuser with an oblique shock diffuser provide greater pressure recovery

Diffuser efficiency

)(

)PP(

01

02

0

d0

PP

actual

D (mostl common one)

If ηD=1→normal shock diffuser

for low supersonic test section Me, ηD>1

for hypersonic conditions ηD<1 (normal shock recovery is about the best to be expected)

Normal shock at Me

Is very sensitive to

At2>At1(due to the entropy increase in the diffuser) proof: assume the sonic flow exists at both throats

*22

*2

*11

*1 aAaA tt

02

01*

2

*1

*2

*2

*1

*1

*2

*1

*2

*1

*2

*1

1

2 )(P

P

P

P

RTP

RTP

a

a

A

A

t

t

02

01

1

2

P

P

A

A

t

t 0102 PP always 12 tt AA

D2tA

At2ηD=max is slightly larger than (P01/P02)At1

the fix- geometry diffuser will operate at an efficiency less than η∴ D,m to start properly

ηD is low it is because At2 is too large the flow pass though a series of ∴

oblique shock waves id still “very” supersonic a strong normal shock form before ∴exit of the diffuser defeats the purpose of are oblique ∴shock diffuser