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

sdpdfdtvvsdv

)()(

2222221

21111 )( 2

1

AuApApdAuAp xA

A

Automatically balainced

X-dir

Y-dir

The energy equation

s

dvfsdvpdq )(

s

dsvVedVet

)2

()]2

([22

consthuhuh 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)( uAdconstuA

)())(())(( 2

2

dAAduuddAAdpp

pdAAupA

0222 uAdudAudAuAdp

Dropping 2nd order terms

(1)

022 dAuuAdudAu (2)0)( uAd

(1) - (2) = 0 uAduAdp

ududp

constuh 2

2

0)2

(2

uhd

0ududh

Euler’s equation

)()()()(

dudpdudp

5.3 Area-Velocity Relation0)( uAd

uAAudAduudA

0 udud

ddPdP

0AdA

udud

uduM

uaduu

audud 2

2

2

2

∵ adiabatic & inviscid no dissipation mechanism∴

→ isentropic

uduM

AdA )1( 2

Important information1. M→0 incompressible flow Au=const consistent with the familiar continuity eq for

incompressible flow2. 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 flow

an 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

ua

AA *

0

0

*

*

Stagnation density (constant throughout an isentropic flow)

11

22

2* )]

211(

12[1)(

r

r

MrrMA

A

11

20 )2

11( rMr

11

11

*0 )

21()

211(

rr rr

)3.(

211

21

)(2*

2

2

2*

chMMr

Mr

au

(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

*

rrr

pp

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

atmPPP

PPP 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

3M

For

atmPPP

PP

PPP e

e

04.3117.11)

33.101)(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 PP

04.3328.01

02

01 PP

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 )(PP

PP

RTPRT

P

aa

AA

t

t

02

01

1

2

PP

AA

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