Post on 14-Dec-2015
One-dimensional FlowOne-dimensional Flow
3.1 Introduction3.1 Introduction
In real vehicle geometry, The flow will be axisymmetric
12
12
12
12
uu
TT
PP
12 M
1
1
1
1
u
T
P
11 M
Normal shock
One dimensional flow
The variation of area A=A(x) is gradual
Neglect the Y and Z flow variation
3.2 Steady One-dimensional flow equation3.2 Steady One-dimensional flow equation
Assume that the dissipation occurs at the shock and the flow up stream and downstream of the shock are uniform
Translational rotational and vibrational equilibrium
The continuity equation
dt
sdus
. 0. sdu
s
L.H.S of C.V
02211 AuAu 2211 uu
The momentum equation
svvs
spddvfdt
uusdu
)(
).(
ss
spdusdu
).( 2222
2111 uPuP
(Continuity eqn for steady 1-D flow)
Remember the physics of momentum eq is the time
rate of change of momentum of a body equals to the
net force acting on it.
2
222
2
111
122211 )(
uPuP
APAPuAuuAu
sduu
edu
et
dvufsdupdqsvvv s
.)2
()2
().(.22
The energy equationThe energy equation
Auu
eAuu
eAupAupQ 2
2
2221
2
1112211 )
2()
2()(
2 21 2
1 22 2
u uq h h
Au
11
Physical principle of the energy equation is the energy is the energy is conserved
Auu
eAupAuu
eAupQ 2
2
222221
2
11111 )
2()
2(
Energy added to the C.V Energy taken away from the system to the surrounding
3.3 Speed of sound and Mach number3.3 Speed of sound and Mach number
Wave front called “ Mach Wave”
Mach angle μ
Mv
a
vt
at 1sin
M
1sin 1
Always stays inside the family of circular sound waves
Always stays outside the family of circular sound waves
Wave front
A sound wave, by definition,
ie: weak wave
( Implies that the irreversible,
dissipative conduction are negligible)
Continuity equation
daddaadadaada ))((
d
daa
1 2
T
p
a
dTT
d
dpp
daa
Momentum equation
22 ))(( daaddppap
dadaadp 22
a
dadpda
2
2
dadadp
a2
2
a
a
d
dp
a 22
1 2
ddp
a 2
No heat addition + reversible
s
pa )(2
ss
vv
p
2
ss
pa
)(General equation valid for all gas
d
daa
Isentropic compressibility
For a calorically prefect gas, the isentropic relation becomes
cp
1 cp
ppc
p r
s
11 .
RTp
a
For prefect gas, not valid for chemically resting gases or real gases
Ideal gas equation of state RTP
Taa
Form kinetic theory
8RTC
88
1.35
RTC
a RT
Ca4
3
a for air at standard sea level = 340.9 m/s = 1117 ft/s
Mach Number a
VM
1
1
1
M
M
M Subsonic flow
Sonic flow
supersonic flow
The physical meaning of M
2 2 2 2
2 22
2
2 2 2 22 2 2 21 1 1 ( 1)( )1 1
v
V V V VV V
MRa RT RT C T eT
2MKinetic energy
Internal energy
3.4 Some conveniently defined parameters 3.4 Some conveniently defined parameters
Inagine: Take this fluid element and Adiabatically slow it own (if M>1) or speed it up (if M<1) until its Mach number at A is 1.
***** ,, aV
MrRTaT
A
P
T
M
For a given M and T at the some point A
Its values of and at the same point
*T *a
associated with
Note: are sensitive to the reference coordinate system are not sensitive to the reference coordinate
00.T
.T
In the same sprint, image to slow down the fluid elements isentropically to zero velocity ,
total temperature or stagnation temperature
total pressure or stagnation pressure
0T
0P
Stagnation speed of sound
Total density
00 RTa
000 / RTP
(Static temperature and pressure)
3.5 Alternative Forms of the 1-D energy equation 3.5 Alternative Forms of the 1-D energy equation
= 0(adiabatic Flow)
22
22
2
21
1
uh
uh
22
22
21
21
2121
u
r
au
r
a
2)(
12)(
1
22
2
22
1
1
1 uP
r
ruP
r
r
2*
22
)1(2
1
21a
ua
Q
A
B
Aa*
Aa
Ba*
Ba
If the actual flow field is nonadiabatic form A to B →
Many practical aerodynamic flows are reasonably adiabatic
calorically
prefect
**
BA aa
Total conditions - isentropic
20
2
11 M
r
T
T
120 )2
11( r
r
Mp
P 1
120 )
2
11( rM
CpTu
CpT 2
2
Ma
u
RT
u
T
T 2
2
220
2
11
1
21
1
21
Adiabatic flow
1000
r
r
T
T
P
P
isentropic
Note the flowfiled is not necessary to be isentropic
If not →
If isentropic → are constant values
BABABA PPTT 000000 ,,
000 ,, PT
121
20
22
r
au
r
a
1
2)(
0
*2
0
*
rT
T
a
a1
1
12
0
*
)( r
r 1
0
*
1
2
r
r
rP
P
1)1(2
1 2
0*2
r
aa
r
r
4.1r
634.0
528.0
833.0
0
*
0
*
0
*
P
P
T
T
)1(/1
22
2
*
rMrM
= 1 if M=1
< 1 if M < 1
> 1 if M > 1
*M*M
*M
or 2
2*
)1(2
)1(2
M
MM
2*
22
12
1
21a
r
ru
r
a
2*2
12
1
2
1
1
/
u
a
r
r
r
ua
12
1/1
2
11
12
1
1
1 2*2
*
2
r
rMr
Mr
r
rM
1
1*
r
rM If M → ∞
EX. 32
3.6 Normal shock relations 3.6 Normal shock relations
The shock is a very thin region ,
Shock thickness ~ 0 (a few molecular mean free paths)
~ cm for standard condition)510
1
Known
adiabatic
2
To be solved
2211 uu
2222
2111 upup
22
22
2
21
1
uh
uh
Continuity
Momentum
Energy
( A discontinuity across which the flow properties suddenly change)
22222 , TChRTP p
Ideal gas E.O.S
Calorically perfect
Variable :
22222 ,,,, Thpu 5 equations
21*2
uua *
1
*2
1
MM
Prandtl relation
Note:
1111 2*
2*
11 MMMM
1.Mach number behind the normal shock is always subsonic
2.This is a general result , not just limited to a calorically perfect
gas
2/)1(
]2/)1[(12
1
212
2
rrM
MrM
Special case 1. 2.
11 M 12 M
1Mr
rM
2
12
2
2*
12
12
Mr
MM
2*1
*2
12
MM
21
212*
12*
21
21
21
2
1
1
2
)1(2
)1(
Mr
MrM
a
u
uu
u
u
u
Infinitely weak normal shock . ie: sound wave or a Mach wave
21
21
2
1
1
2
)1(2
)1(
Mr
Mr
u
u
])1(
)1(2)][1(
1
21[))(( 2
1
212
11
2
2
1
1
2
1
2
Mr
MrM
r
r
h
h
p
p
T
T
)1(1
21 2
11
2
Mr
r
P
P
1
22
112111
2
22
2
1112 1u
uuuuuuuPP
2
1
2
1
1
2
11
1
2
1
1211
Mr
Mr
p
u
P
P
Note : for a calorically perfect gas , with γ=constant
1
2
1
2
1
22 ,,,
T
T
P
PM
are functions of only1M
5
5
1
1
M
MReal gas effects
378.02
1lim 2
1
r
rM
M
61
1lim
1
2
1
r
rM
1
2
1
limP
PM
1
2
1
limT
TM
The 2nd law of thermodynamics 012 ss
1
2
1
212 lnln
P
PR
T
TCpss
1
1
1
1
1
1
M
M
M
0
0
0
12
12
12
SS
SS
SS
Why dose entropy increase across a shock wave ?
1u
2u1
2
)10~10(0 76 mm
y
u
Mathematically eqns of hold for 2
1
2
1
2
1
2 ,,, MT
T
p
p
1,1 11 MM
Physically , only is possible 11 M
Large ( small)y
Dissapation can not be neglected
entropy
1
212 ln
Po
PoRss Rsse
Po
Po /)(
1
2 12
22
22
2
21
1
uCpT
uCpT
a
a
a
aaa P
PR
T
TCpss
1
2
1
212 lnln
1
212 ln1ln
o
o
P
PRCpss
0201 CpTCpT
Note: 1 2.onlyMfP
P)( 1
01
02 12 ss 0102 PP
Ex.3.4 Ex.3.5 Ex 3.6 Ex 3.7
0201 TT To is constant across a stationary normal shock wave
To ≠ const for a moving shock
The total pressure decreases across a shock wave
3.7 Hugoniot Equation 3.7 Hugoniot Equation
)(2
112
uu 21
2
122
2222
2111 )( uPuPuP
1
2
12
1221
PP
u
2
1
12
1222
PP
u
22
22
2
21
1
uh
uh
Peh
2121
21
2112 2
1)
11)((
2
1vvppPPee
Hugoniot equation
It relates only thermodynamic quantities across the shock
General relation holds for a perfect gas , chemically reacting gas, real gas
vpe av
pv
e
p
efc
s
.. Acoustic limit is isentropic flow
1st law of thermodynamic with 0q
2
1
2
1
1
2
)1
1(
1)1
1(
v
v
r
rv
v
r
r
P
P
In equilibrium thermodynamics , any state variable can be expressed as a function of any other two state variable
For a calorically prefect gas
2221,12 , vfpvvPfP
Hugoniot curve the locue of all possible p-v condition behind normal shocks of various strength for a given 11,vP
vpee ,
pvvvpp
vpvpvpvpvpeh
pveh
1212
11221122
2
1
vp
h
sp
hfc
.v
p
h
udpdsdh
s
For a specific 1u
21
12
12
1
2
12
1221 v
vv
PPPPu
2
1
1
12
12
v
u
vv
PP
v
PStraight line
21
21 u Note 0
v
P
Rayleigh line
22. av
pfc
s
∵supersonic ∴ au 1
222
11 av
pu
v
p
s
sv
p
v
p
Isentropic line down below of Rayleigh line
In acoustic limit (Δs=0) u1→a insentrop & Hugoniot have the same slope
as function (weak) shock strength for general flow s
Shock Hugoniot
)(2
112
12
12 vvpp
hh
p
h
),( spvv
.....!3
1
2
1
2
1 3
3
32
2
2
11 pp
pp
pp
vvsss
For fluids
.......2
1 2
2
2
12
pp
hp
p
hhhh
ss
sphh ,
Coefficient
vdpdhdp
dhTds
vp
h
s
1 T
l
h
p
43
2
2
1 06
1
4
1
2
1pp
pp
sTs
For gibbs relation
ssp
v
p
h
2
2
ssp
v
p
h
2
2
3
3
...6
1
2
11
3
2
22
1
sTpp
vp
p
vpvh
s
pss
vp
p
vp
p
vpv
pss
2
1
4
1
2
1 3
2
22
1
u
p
s=const
02
2
sp
v
u
p 02
2
sp
v
3
2
2
12
1p
psT
h
0s 0p 02
2
sp
v
0s 0p 02
2
sp
v
For every fluid
“Normal fluid “
“Compression” shock
“Expansion “shock
Let 0
0
s
p
s=const
if
if
3.8 1-D Flow with heat addition3.8 1-D Flow with heat addition
e.q 1. friction and thermal conduction
2. combustion (Fuel + air) turbojet
ramjet engine burners.
3. laser-heated wind tunnel
4. gasdynamic and chemical
leaser
Assume calorically perfect gas
2211 uu
22
22
2
21
1
uhq
uh
2222
2111 uPup
0102
2
11
2
22 22
TTCu
TCu
TCq ppp
+E.O.S
1
1
1
1
ρ
p
u
T 2
2
2
2
ρ
p
u
T
q
A
TCh P
The effect of heat addition is to directly change the total temperature of the flow
Heat addition To
Heat extraction To
22
21
1
2
1
1
rM
rM
P
P
222
211
222
21112
MPMP
uuPP
22222 PMM
PMau
22
1
221
1
2 1 MP
PM
P
P
2
1
222
2
2
1
1
2 )()1
1(
M
M
rM
rM
T
T
2
2
12
1
2
2
1
2 ))(1
1(
M
M
rM
rM
2
1
1
2
1
2
1
2
1
2
1
2
2
1
1
2
1
2
T
T
M
M
P
P
u
u
P
P
P
P
T
T
2
1
1
2
1
2
1
2
1
2
1
2
T
T
M
M
a
a
M
M
u
u
1
22
2
2
1
1
2
1
22
1
1
2 )1
1(
M
M
rM
rM
M
M
P
P
T
T
))(1
1)(())((
2
12
1
2
2
2
1
2
1
2
1
2
1
1
2
M
M
rM
rM
M
M
a
a
M
M
u
u
2
2
2
1
1
2
1
2
2
01 1
1
21
1
21
102
rM
rM
Mr
Mr
P
Pr
r
2
22
2
1
2
2
22
21
01
02
21
1
21
1
1
1
Mr
Mr
M
M
rM
rM
T
T
1
2
1
212 lnln
P
PR
T
TCsss p
011201
120202
P
P
P
P
P
P
P
P 1
)2
11( 20
r
r
Mr
P
P
))()((01
1
1
2
2
02
01
02
T
T
T
T
T
T
T
T 20
2
11 M
r
T
T
Given: all condition in 1 and q
02T
2M
....,,1
2
1
2
1
2
T
T
P
P
)( 0102 TTCq p
To facilitate the tabulation of these expression , let state 1 be a reference state at which Mach number 1 occurs.
01
02
T
T
*1 PP *
1 TT *1 *
001 PP *001 TT
11 M MM 2
2* 1
1
MP
P
2
22
* 1
1
M
MT
T
1
11 2
2*
M
M
12
2*0
0
1
12
1
1
r
r
M
MP
P
2
22
2
*0
0 121
1M
M
M
T
T
*** lnln
P
PR
T
TCss p
Table A.3.
For γ=1.4
Adding heat to asupersonic flow M ↓
1*
1*
2
*002
*2
*001
*1
01021
)(
)(
)(
qqq
TTCq
TTCq
TTCq
P
P
P
To gain a better concept of the effect of heat addition on M→TS diagram
*** lnln
P
PR
T
TCss p
2*2*
2*
11
1
1
1
1M
P
PM
P
P
MP
P
**
2
2
*
2
2
2*
1
1
1
T
T
MP
pM
P
PM
MT
T
** T
TM
P
P
11 *
*
P
P
P
P
2
41
2
1 *
2
*
TT
P
P
2
)(411ln
1ln
*2
*
*T
T
T
T
C
ss
p
1
rR
Cp
Cpr
R
1
*
*2
*1))(1(
1)(
T
T
P
P
P
P
0))(1()(**
2*
T
T
P
P
P
P
1))(1()(
*2
** P
P
P
P
T
T
*T
T
Cp
SS *
1.0
B
A
At point A
0dT
ds0ds
2
0
ap
ds
0 ududp
0u
dud
ddp
u 2
∴ At point A , M=1
Rayleigl line
Momentum eq.
Continuity eq.
duududpuddu 2
At point B *T
Tis maximum
22
2*
)1
1(
rM
rM
T
T
r
r
T
T
4
)1(
)11
1(
1
2
2
max*
0*
dMTT
d
rM
12
1
BM
4.1
A (M=1)
01
212111242
222222
rM
rMrMrMrMrM
021 22 rMrM
MB subsonic
*T
T
Cp
SS *
B(M<1)
M<1
Heating
cooling
heating
M>1
jump
cooling
lower m
ds=(dq/T)rev
→addition of heat ds>0
1 2
q
Supersonic flowSupersonic flow
MM11>1>1
subsonic flowsubsonic flow
MM11<1<1
MM (M(M22<M<M11)) (M(M22>M>M11))
PP (P(P22>P>P11))
TT (T(T22>T>T11))
TT00 (T(T0202>T>T0101))
PP00 (P(P0202<P<P0101))
uu (U(U22<U<U11))
2
1
1
M
2
1
1
M
])1(2[)1(
)1(
]1
)1(2[
1
1
)1
1(
1
)1
1(
1
1
222
2
*0
0
12
2*0
0
2
2*
22
2*
2*
MM
M
T
T
M
MP
P
M
M
MM
T
T
MP
P
For supersonic flow Heat addition → move close to A M → 1
→ for a certain value of q , M=1 the flow is said to be “ choked ”
∵ Any further increase in q is not possible without a drastic revision of
the upstream conditions in region 1
→ for a certain value of the flow is choked
→ If q > , then a series of pressure waves will propagate
upstream , and nature will adjust the condition is region 1
to a lower subsonic M
→ decrease
For subsonic flow
*q
1M*q
m
E.X 3.8
heat addition → more closer to A , M →1
3.9 1-D Flow with friction3.9 1-D Flow with friction
- In reality , all fluids
are viscous.
- Analgous to 1-D flow with heat addition.
Fanno line Flow
Momentum equation
L
wdDApApAuAu021
222
211
s
s s
w sdspdusdu ..
L
wdxDuupp
0
211
22212
4
Good reference for f : schlicting , boundary layer theory
u
du
P
dP
rMu
udu
u
dPdx
D
f2
22
12
21
21
4
fudxD
ududp ww2
2
14
T
dT
u
du
T
dTd
p
dp
MauT
dT
u
du
M
dM
2
1
2
1
2
1 2
2
2
21
1ln
2
114
M
M
x
xM
rM
r
r
rMD
fdx
22
21
22
21
1
0
0
2
1
2
)1(2
)1(2
21
1
21
1
Mr
Mr
Mr
Mr
T
T
T
T
T
T
∵ adiabatic , To = const
0
21
1
1
2
11
2
2
20
Mr
MdM
Mr
T
dTM
r
T
T
M
dMMM
MD
fdx1
22
21
2
111
24
21
22
21
2
1
1
2
)1(2
)1(2
Mr
Mr
M
M
P
P2
1
22
21
2
1
1
2
)1(2
)1(2
Mr
Mr
M
M
)]1(2[)1(
21
22
2
1
01
02
)1(2
)1(2
r
r
Mr
Mr
M
M
P
P
Analogous to 1-D flow with heat addition using sonic reference condition.
2* )1(2
1
Mr
r
T
T
2
1
2* )1(2
11
Mr
r
MP
P
2
12
* 1
)1(21
r
Mr
M
12
12
*0
0
1
)1(21
r
r
r
Mr
MP
P
2
22
2
*
)1(2
)1(ln
2
114
Mr
Mr
M
M
D
Lf
Table A.4
IF we define are the station where , M = 1
*Lx
*
0
1
2
2
2
21
1ln
2
114L
M
MrM
r
r
rMD
fdx
F: average friction coefficient *
0*
1 Lfdx
L
11111 lnlnlnln
u
uR
T
TCR
T
TCss vv
constTTr
T
TT
TTr
T
T
C
ss
0
10
0
1
1
ln2
1ln
ln2
1ln
P
At point P 0)(2
11
0
TT
r
T 2)(
2
0
uTTC p
)(2 0 TTCu p T high u low above P , M < 1
T low u high below P , M > 1
22 12
.2
11
u
rR
rRru
r
T
22 arRTu
TTCpu 022
2
0
uhh
Fanno line
1M
ds > 0
ds < 0
chocked1m
2m
Supersonic flowSupersonic flow
MM11>1>1
Subsonic flowSubsonic flow
MM11<1<1
MM (M(M22<M<M11)) (M2>M1)(M2>M1)
PP (P(P2>2>PP11))
TT (T(T22>T>T11))
TT00 unchangedunchanged unchangedunchanged
PP00 (P(P0202<P<P0101))
uu ((uu22<<uu11))
ρρ
1-D adiabatic flow with friction