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