Transcript Document
One-dimensional Flow
3.1 Introduction
P1
P2 P1
T1
T2 T1
1
u1
M1 1
2 1
u2 u1
M2 1
Normal shock
In real vehicle geometry, The flow
will be axisymmetric
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 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
u.ds d
t
s
u .ds 0
s
L.H.S of C.V
1u1 A 2 u2 A 0
1u1 2u2
(Continuity eqn for
steady 1-D flow)
The momentum equation
( u )
(
u
.
d
s
)
u
d
f
dv
pd
s
s
v t
v
s
( u .ds )u pds
s
s
P1 1u1 P2 2u2
2
2
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.
( u1 A)u1 u2 Au2 P2 A P1 A
P1 1u1 P2 2u22
2
The energy equation
u2
u2
v qd s pu.ds v ( f .u )dv v t (e 2 )d s (e 2 )u.ds
2
2
u
u
Q ( p1u1A p2u2 A) 1 (e1 1 )u1 A 2 (e2 2 )u2 A
2
2
Q
q
1u1A
u12
u2 2
q h1
h2
2
2
Physical principle of the energy equation is the energy is
the energy is conserved
2
2
u1
u2
Q p1u1 A 1 (e1 )u1 A p2u2 A 2 (e2 )u2 A
2
2
Energy added to the C.V
Energy taken away from the system to the
surrounding
3.3 Speed of sound and Mach number
Mach angle μ
sin
at a 1
vt v M
sin 1
1
M
Wave front called
“ Mach Wave”
Always stays inside the
family of circular sound
waves
Always stays outside the
family of circular sound
waves
1
2
a da
p dp
d
T dT
a
p
T
A sound wave, by definition,
ie: weak wave
( Implies that the irreversible,
dissipative conduction are negligible)
Wave front
Continuity equation
a ( d )(a da) a ad da dda
a
da
d
Momentum equation
p a p dp ( d )(a da)
2
dp a 2 d
2a
a
d
2
dp 2ada a2d
da
a
d
dp a 2 d
da
2a
1 dp a 2
2a d 2a
a2
dp
d
No heat addition + reversible
p
a ( )s
2
p 2 v
v
s
s
p
General equation
a ( )s
s valid for all gas
Isentropic compressibility
For a calorically prefect gas, the isentropic relation becomes
p c
1
p c
p
p
p
r 1
1
c .
s
p
a
RT
For prefect gas, not valid for chemically resting
gases or real gases
Ideal gas equation of state P RT
a aT
Form kinetic theory
C
8RT
8RT
C
8 1.35
a
RT
3
a C
4
a for air at standard sea level = 340.9 m/s = 1117 ft/s
Mach Number
M 1
M 1
M 1
M
V
a
The physical meaning of M
Subsonic flow
Sonic flow
M
2
Kinetic energy
Internal energy
supersonic flow
V2
V2
V2
V2
V2
V2
2
2
2
2
2
2
2
2
2
M 2
a
RT ( 1) RT 1 R T 1 CvT ( 1) e
1
1
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.
T , a rRT , M
*
A
P
T
M
*
*
* V
a*
For a given M and T at the some
point A
associated with
Its values of T * and a* at the same
point
In the same sprint, image to slow down the fluid elements
isentropically to zero velocity ,
T0 total temperature or stagnation temperature
P0 total pressure or stagnation pressure
Stagnation speed of sound
Total density
a0 RT0
0 P0 / RT0
Note: T0 .0 are sensitive to the reference coordinate system
T . are not sensitive to the reference coordinate
(Static temperature and pressure)
3.5 Alternative Forms of the 1-D energy equation
Q = 0(adiabatic Flow)
2
1
2
calorically
u
u
h1
h2 2
2
2
prefect
2
2
r P1 u1
r P2
u
( )
( ) 2
r 1 1
2 r 1 2
2
B
aB a B
A
aA
aA
*
*
2
2
2
a1
u
a
u
1 2 2
r 1 2 r 1 2
2
a2 u 2
1 *2
a
1 2 2( 1)
If the actual flow field is nonadiabatic
form A to B → aA* aB*
Many practical aerodynamic flows
are reasonably adiabatic
Total conditions - isentropic
u2
CpT
CpT
2
Adiabatic flow
T0
u 1
u 1
1
1
1
1
M
2
T
2 RT
2 a
2
2
2
2
T0
r 1 2
1
M
T
2
isentropic
P0 0 T0
P T
r
r 1
0
1 2 r11
(1
M )
2
P0
1 2 r r1
(1
M )
p
2
Note the flowfiled is not necessary to be isentropic
If not → T0 A T0 B , P0 A P0 B , 0 A 0 B
If isentropic → T0 , P0 , 0
are constant values
2
a2
u2
a0
r 1 2 r 1
a* 2 T *
2
( )
a0
T0 r 1
r 1.4
T*
0.833
T0
P*
0.528
P0
*
0.634
0
r 1 *
a02
a
2(r 1)
r 1
2
( r 21 )
0
*
1
r 1
*
P
P0
2
r
1
r
r 1
a
u
r 1 *
a
r 1 2 2r 1
2
M
2
1
r 1
M
2
a / u
2
2
2
r 1
1
r 1 a
2 2r 1 u
*
2
r 1 1 1 r 1 / M r 1
*
2r 1 M 2
2r 1
2
*2
2
r 1 / M (r 1)
2
*
or
2
M*
M*
( 1) M 2
2 ( 1) M 2
M*
= 1 if M=1
M*
<1
if M < 1
M*
>1
if M > 1
r 1
r 1
If M → ∞
EX. 32
3.6 Normal shock relations
( A discontinuity across which the
flow properties suddenly change)
The shock is a very thin region ,
Shock thickness ~ 0 (a few molecular mean free paths)
~ 105 cm for standard condition)
1u1 2u2
1
Known
2
To be solved
Ideal gas
E.O.S
Continuity
p1 1u1 p2 2u2
2
Momentum
2
adiabatic
Variable : 2 , u2 , p2 , h2 ,T2
2
u
u
h1 1 h2 2
2
2
Energy
5 equations
2
Calorically
perfect
P2 2 RT2 , h2 CpT2
1
M2 *
M1
a u1u2
*2
*
Prandtl relation
Note:
M1 1 M1 1 M 2 1 M 2 1
*
*
1.Mach number behind the normal shock is always subsonic
2.This is a general result , not just limited to a calorically perfect
gas
1M
2 r 1M
2
M
*2
*2
M2
2
1
M 1*
2
1 [(r 1) / 2]M 1
2
rM 1 (r 1) / 2
2
M2
2
Special case 1. M1 1
M2 1
2. M1
M2
Infinitely weak normal shock . ie:
sound wave or a Mach wave
r 1
2r
2
2 u1 u12 u12
(r 1) M 1
*2
*2 M 1
2
1 u2 u1u2 a
2(r 1) M 1
2
2 u1
(r 1) M 1
1 u2 2 (r 1) M 12
u
P2 P1 1u12 2u22 1u1 u1 u2 1u12 1 2
u1
P2
1u12 2 r 1M 12
1
1
r 1M12
P1
p1
P2
2r
2
1
( M1 1)
P1
r 1
T2
p
h
2r
2 (r 1) M 1
2
( 2 )( 1 ) 2 [1
( M 1 1)][
]
2
T1
p1 2
h1
r 1
(r 1) M 1
2
Note : for a calorically perfect gas , with γ=constant
M2,
2 P2 T2
, ,
1 P1 T1
M1 5
M1 5
are functions of M 1 only
Real gas effects
lim M 2
M
1
r 1
0.378
2r
lim
M
2 r 1
6
1 r 1
lim
M
P2
P1
lim
M
T2
T1
1
1
1
Mathematically eqns of
Physically , only
M1 1
2 p2 T2
, , ,M
1 p1 T1 2
M1 1, M1 1
hold for
is possible
The 2nd law of thermodynamics s2 s1 0
s2 s1 Cp ln
T2
P
R ln 2
T1
P1
M1 1
S2 S1 0
M1 1
S2 S1 0
M1 1
S2 S1 0
Why dose entropy increase across a shock wave ?
2
u1
u
Large ( y small)
y
Dissapation can not be neglected
1
u2
6
7
0(10 m ~ 10 m)
entropy
2
2
u
u
CpT1 1 CpT2 2
2
2
CpT01 CpT02
T01 T02 To is constant
across a stationary
normal shock wave
s2 a s1a Cp ln
T2 a
P
R ln 2 a
T1a
P1a
P
s2 s1 Cp ln1 R ln o 2
Po1
Note: 1
To ≠ const for a moving
shock
P02
f ( M 1 )only
P01
s2 s1 R ln
Po2
Po1
Po2
e ( s s ) / R
Po1
2
1
2. s2 s1 P02 P01 The total pressure
Ex.3.4 Ex.3.5 Ex 3.6 Ex 3.7
decreases across a
shock wave
3.7 Hugoniot Equation
u2 u1 (
u1
2
1
)
2
P1 1u1 P2 2u2 P2 2 (
2
P2 P1 2
2 1 1
2
2
u1
u2
h1
h2
2
2
2
u2
2
P2 P1 1
2 1 2
h e
P
1 2
u1 )
2
1
1
1
1
e2 e1 ( P1 P2 )( ) p1 p2 v1 v2 Hugoniot equation
2
1 2
2
It relates only thermodynamic quantities across the shock
General relation holds for a perfect gas , chemically reacting gas, real gas
e
p
v
e
c. f . p
s
Acoustic limit is isentropic flow
e pav v 1st law of thermodynamic with q 0
For a calorically prefect gas
r 1 v1
) 1
P2
r 1 v2
r 1 v1
P1
(
)
r 1 v2
(
In equilibrium thermodynamics , any
state variable can be expressed as a
function of any other two state variable
e e p, v
P2 f P1,v1, v2 p2 f v2
Hugoniot curve the locue of all possible
p-v condition behind normal shocks of
various strength for a given P1 ,v1
h e pv
h e p2v2 p1v1 pv p2v2 p1v1
1
p2 p1 v2 v1 v p
2
h
v
p
h
c. f
p s
dh ds udp
h
v
p s
For a specific u1
u1
2
P2 P1 2 P2 P1
v12
2 1 1 v2 v1
u
P P2 P1
1
v v2 v1
v1
2
Straight line Rayleigh line
12u12
∵supersonic ∴
p
c. f 2 a 2
v s
Note
P
0
v
u1 a
p
p
1u12 2 a 2
v
v s
p p
v v s
Isentropic line down below of Rayleigh line
In acoustic limit (Δs=0) u1→a insentrop & Hugoniot have the same slope
s as function (weak) shock strength for general flow
Shock Hugoniot
h h2 h1 1
(v2 v1 )
p p2 p1 2
v v( p, s)
3
3
1
1 2
1
2
v1 v1 p 2 p 3 p .....
2
2 p s
3! p s
p s
For fluids
h h p, s
h
1 2h
h h2 h1 p 2 p 2 .......
2 p s
p s
Coefficient
For gibbs relation
h
1
v
p s
Tds dh
h
T
l p
dp
dh vdp
2 h v
2
p s p s
3h 2v
3 2
p s p s
2
1 v
1
v
h v1p p 2 2 p 3 T1s ...
2 p s
6 p
1 v
1 2v
1 v
2
3
v1p p 2 p sp
2 p s
4 p s
2 s p
1
s T1
2 s
2
1 1 3
4
p 2 p 0p
4 6 p
p 0
Let
s 0
1 2
Ts 2 p3
12 p h
s 0
s 0
p
For every fluid
“Normal fluid “
2v
p 0 if 2 0
p s
“Compression” shock
2
p 0 if v 0
p 2
s
2v
2 0
p s
“Expansion “shock
p
s=const
u
2v
2 0
p s
s=const
u
3.8 1-D Flow with heat addition
q
e.q 1. friction and thermal conduction
u1
u2
p1
p2
ρ1
ρ2
T1
T2
2. combustion (Fuel + air) turbojet
ramjet engine burners.
A
1u1 2u2
3. laser-heated wind tunnel
4. gasdynamic and chemical
p1 1u1 P2 2u2
2
2
+E.O.S
leaser
2
u12
u
h1 q h2 2
2
2
Assume calorically perfect gas
h CPT
2
u22
u1
q C pT2 C pT1 C p T02 T01
2
2
The effect of heat addition is to directly change the total
temperature of the flow
Heat addition
To
Heat extraction
To
P2 P1 1u1 2u2
2
2
P1M12 P2 M 22
P2
P2
2
2
1 M 1
M2
P1
P1
P 2
u a M M PM 2
2
2
2
P2 1 rM 1
P1 1 rM 2 2
2
T 2 P2 1 P2 u2 P2 M 2 T2
T1 P1 2 P1 u1 P1 M 1 T1
u2 M 2 a2 M 2 T2
u1 M 1 a1 M 1 T1
1
2
1
2
1
2
T2 1 rM1 2 M 2 2
(
)
2) (
T1 1 rM 2
M1
2
T2
P2 M 2
1 rM 1 M 2
(
2)
P1 M 1 1 rM 2 M 1
T1
2
2 u1
M a
M 1 rM 2 M 1
( 1 )( 1 ) ( 1 )(
)
2 )(
1 u2 M 2 a2
M 2 1 rM1 M 2
2
2 1 rM 2 M 1 2
(
)
2 )(
1 1 rM1 M 2
2
P
02
P01
P0
r 1 2
(1
M )
P
2
P P P
02
2
1
P2 P1 P01
r
r 1
r
r 1
r 1 2
1
M 2 1 rM 2
P
1
2
P01 1 r 1 M 2 1 rM 22
1
2
02
T02 T02 T2 T1
( )( )( )
T01
T2 T1 T01
2
T02 1 rM1
2
T01 1 rM 2
2
M2
M1
2
T0
r 1 2
1
M
T
2
r 1 2
M2
1
2
1 r 1 M 2
2
T2
P2
s s2 s1 C p ln R ln
T1
P1
Given: all condition in 1 and q
T02 q C p (T02 T01)
M2
T02
T01
P2 T2 2
, , ....
P1 T1 1
To facilitate the tabulation of these expression , let state 1
be a reference state at which Mach number 1 occurs.
P1 P T1 T 1 P01 P0 T01 T0
*
*
*
*
*
M1 1 M 2 M
P
1
*
P 1 M 2
T
2 1
M
*
2
T
1 M
1 1 M 2
2
*
M 1
P0
1 2 1M 2
*
1 M 2
1
P0
2
r
r 1
T0 1M 2
2
2
1
M
*
2
T0 1 M 2
T
P
s s C p ln * R ln *
T
P
*
Table A.3.
For γ=1.4
Adding heat to a
supersonic flow
M↓
q1 CP (T02 T01)
q1 CP (T01 T0 )
*
*
q2 CP (T02 T0 )
*
*
q2 q1 q1
*
*
To gain a better concept of the effect of heat addition on M→TS diagram
T
P
s s C p ln * R ln *
T
P
*
P
1
P* 1 M 2 1
*
2
P 1 M
P
1
P* 1
M 2
P
T 1 2 P 2
p
1
M * M *
*
2
T 1 M
P M
P
2
P
T
* M
*
P
T
P 1 P* 1
*
P P
2
T
T*
T
P 21
P*
( * ) (1 )( ) 1 *
P
P
T
T
P 2
P*
* ( * ) (1 )( ) 1
T
P
P
P 2
P T
( * ) (1 )( * ) * 0
P
P
T
T
1 4 *
P 1
T
P*
2
2
2
1 12 4 (T )
*
ss
T 1
T
ln *
ln
Cp
T
2
Cp
*
R
rR
1
r 1
Cp
T
T*
At point A
B
A
1.0
Rayleigl line
ds
0
dT
p
2
a
ds 0
ds0
dp udu 0
Momentum eq.
d du
0
u
Continuity eq.
du ud dp duu u 2d
S S*
Cp
u2
dp
d
∴ At
point A , M=1
T
T*
B(M<1)
T
d *
T 0
dM
jump
Heating
A (M=1)
cooling
M<1
2M 1 r 1 rM
2
heating
cooling
M>1
2
2
2
2 4
1 rM 2rM
S S*
Cp
T
1 r 2
2
M
(
)
*
2
T
1 rM
M 1 r 21 rM 2rM 0
1 rM
2 2
2
ds=(dq/T)rev
→addition of heat
ds>0
T
is maximum
*
T
At point B
lower m
MB
1
1.4
MB subsonic
2
0
M2
1
r
1 1 2
T
)
* (
T max 1 1
(1 r ) 2
4r
q
1
2
Supersonic flow
M1>1
M
(M2<M1)
P
(P2>P1)
T
(T2>T1)
T0
(T02>T01)
P0
u
(P02<P01)
(U2<U1)
subsonic flow
M1<1
(M2>M1)
M1
1
2
M1
1
2
P 1
*
P 1 M 2
T
2 1
2
M
(
)
T*
1 M 2
1 1 M 2
(
)
* M 2 1
P0 1 2 ( 1) M 2 1
[
]
*
2
1
P0 1 M
T0 ( 1) M 2
[2 ( 1) M 2 ]
*
2 2
T0 (1 M )
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 subsonic flow heat addition → more closer to A , M →1
a certain value of q M 1 the flow is choked
→ If q > q * , then a series of pressure waves will propagate
upstream , and nature will adjust the condition is region 1
to a lower subsonic M
→ m
decrease
→ for
E.X 3.8
*
3.9 1-D Flow with friction
Fanno line Flow
- In reality , all fluids
are viscous.
- Analgous to 1-D flow
with heat addition.
Momentum equation
s u .d s u s pd s s w.d s
2
4 L
p2 p1 2u2 u 0 w dx
D
dp udu
2
2
1 1
4
1
w dx w u 2 f
D
2
4f
dP
udu
1 dP du
dx
2 2
1
1
D
u
rM P
u 2 u 2
2
2
dp d dT
du dT
p
T
u
T
L
1u1 A 2u2 A p1 A p2 A D wd
2
dM du 1 dT
u Ma
M
u 2 T
Good reference for f : schlicting , boundary layer theory
0
∵ adiabatic , To = const
r 1M 2 dM
T0
r 1 2
dT
M 0
1
M
r 1 2
T
2
T
1
M
2
1
4 fdx
2
1
dM
2
2
1
M
1
1
M
2
M
D
M 2
M2
2
1
x2 4 fdx
r 1
M
x1 D rM 2 2r ln r 1 2
M
1
2
M1
r 1 2
2
M1
T2 T2 T0
2
(
r
1
)
M
1
2
T1 T0 T1 1 r 1 M 2 2 (r 1) M 2 2
2
2
1
P2 M 1 2 (r 1) M 1
P1 M 2 2 (r 1) M 2 2
2
1
P02 M1 2 (r 1) M 2
P01 M 2 2 (r 1) M12
2 M 1 2 (r 1) M
1 M 2 2 (r 1) M
2
1
2
2
2
( r 1)
2
1
2
[ 2 ( r 1)]
Analogous to 1-D flow with heat addition using sonic reference
condition.
T
r 1
T * 2 (r 1) M 2
P
1
r 1
P * M 2 (r 1) M 2
1 2 (r 1) M
* M
r 1
2
1
2
P0
1 2 (r 1) M
*
P0 M
r 1
2
1
2
r 1
2r 1
IF we define x L* are the station where , M = 1
1
L*
0
2
4 fdx 1
r 1
M
ln
2
r 1 2
D
rM
2
r
1
M
M
2
2
4 fL* 1 M
1 (r 1) M 2
ln
2
2
D
M
2
2
(
r
1
)
M
1 L
* 0 fdx
L
*
F: average friction coefficient
Table A.4
s s1 Cv ln
Fanno line
ds < 0
m 1
P
m 2
T
T
u
R ln Cv ln R ln
T1
1
T1
u1
u 2CpT0 T
u2
h0 h
2
chocked
s s1
T r 1 T0 T
ln
ln
C
T1
2
T0 T1
ln T
ds > 0
At point P
1
r 1
0
T 2(T0 T )
u 2C p (T0 T )
C p (T0 T )
r 1
ln T0 T const
2
2
u
2
1
T
r 1
rR u 2 rRT a 2
2
2
u r 1 u
2.
M 1
2 rR
T high
u low
above P , M < 1
T low
u high
below
P,M>1
1-D adiabatic flow with friction
Supersonic flow
M1>1
M
(M2<M1)
P
(P2>P1)
T
T0
(T2>T1)
P0
u
ρ
unchanged
(P02<P01)
(u2<u1)
Subsonic flow
M1<1
(M2>M1)
unchanged