One-dimensional Flow
3.1 Introduction
P1
P2  P1
T1
T2  T1
1
u1
 2  1
u 2  u1
M1  1
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 Au2  P2 A  P1 A
 P1  1u1  P2   2u22
2
 The energy equation


u2 
u2  
 
v qd  s pu.ds  v  ( f .u )dv  v t   (e  2 )d  s  (e  2 )u.ds


2
2
u1
u2

Q  ( p1u1A  p2u2 A)   1 (e1  )u1 A   2 (e2 
)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
a  da
A sound wave, by definition,
p
p  dp
ie: weak wave
  d
( Implies that the irreversible,

T
T  dT
dissipative conduction are negligible)
Wave front
Continuity equation
a  (   d )(a  da)  a  ad  da  dda
a  
da
d
Momentum equation
p  a  p  dp  (   d )( a  da)
2
dp  a 2 d
 2a
a  
d
2
dp  2ada  a 2d
da
a  
d
dp  a 2 d
da 
 2a
1 dp a 2

2a d 2a
dp
a 
d
2
No heat addition + reversible
p
a  ( )s

2
 p  2 v
  v 
s
   s
p

a  ( )s 

s
Isentropic compressibility
General equation
valid for all gas
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  aT 
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
V
a
The physical meaning of M
Subsonic flow
M 1
Sonic flow
M 1
supersonic flow
M
2
Kinetic energy
Internal energy
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
u
r P2
u
( ) 1 
( ) 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 r11
 (1 
M )

2
P0
  1 2 r r1
 (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
a

 0
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 21 )
0
*
1
r 1
*
P
P0
 2 


r

1


r
r 1
a / u 
a
u
r 1 *
 
a
r  1 2 2r  1
2
2
2
2
 M 
2
1
r 1
M 
2
r 1

1
r 1  a 

 
2 2r  1  u 
*
2
r  1  1  1 r  1 / M  r  1
 *  
2r  1  M  2
2r  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)
~ 10 5 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  C pT2
a  u1u2
*2
M2
*
1
 *
M1
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
  1M

2  r  1M
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) M1
*2
 
 *2  M1 
2
1 u2 u1u2 a
2(r  1) M1
2
 2 u1
(r  1) M1
 
1 u2 2  (r  1) M12
 u 
P2  P1  1u12   2u22  1u1 u1  u2   1u12 1  2 
 u1 
P2
1u12  2  r  1M 12 
1
1

r  1M 12 
P1
p1 
P2
2r
2
 1
( M 1  1)
P1
r 1
T2
p 
h
2r
2  (r  1) M1
2
 ( 2 )( 1 )  2  [1 
( M1  1)][
]
2
T1
p1  2
h1
r 1
(r  1) M1
2
Note : for a calorically perfect gas , with γ=constant
M2,
 2 P2 T2
, ,
1 P1 T1
M1  5
M1  5
are functions of M1 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
M 1  1, M 1  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 ln 1  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
u 2  u1 (
u1 
2
1
)
2
P1  1u1  P2   2u2  P2   2 (
2
P2  P1   2 
 
 2  1  1 
2
2
u
u
h1  1  h2  2
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   pav v
 e 
c. f .    p
   s
Acoustic limit is isentropic flow
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   pv  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

  


1
1   2 
1


2
3
  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  v1p    p 2   2 p 3  T1s  ...
2  p  s
6  p 
1  v 
1   2v 
1  v 
2
3
 v1p    p   2  p    sp
2  p  s
4  p  s
2  s  p
1      1 1   2  3

s   T1   p      2 p  0p 4 
2  s    4 6  p 

p  0
Let
s  0
1   2 
Ts   2  p 3 
12  p h
For every fluid
s  0
  2v 
p  0 if  2   0
 p  s
s  0
2


p  0 if  v   0
 p 2 

s
p
  2v 
 2   0
 p  s
“Normal fluid “
“Compression” shock
“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
 P1M 12  P2 M 22
P2
P
2
 1  M 12   2 M 2
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  rM 1 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
M1 a1
M1 1  rM 2 M1
  ( )( )  ( )(
)
2 )(
1 u2 M 2 a2
M 2 1  rM 1 M 2
2
2 1  rM 2 M1 2
(
)
2 )(
1 1  rM 1 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  rM 1 

 
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    1M 2 



*
1  M 2 
 1
P0

2
r
r 1
T0   1M 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  
 P*  1
  
 P  
2
T
T*
 T
P 21
P*
( * ) (1   )( )  1  *
P 
P
 T

T
P 2
P*
 *  ( * ) (1   )( )  1
T
P 
P

1   
2
P 1 


*
P
2
P 2
P T
( * )  (1   )( * )  *  0
P
P
T
T 
 4  * 
T 
2
   1    12  4 (T ) 
*
ss
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 

ds0
dp  udu  0
Momentum eq.
d du
 0
 u
Continuity eq.
du  ud  dp  duu  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   21  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  wdx
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
  u A  2u2 A  p1 A  p2 A   D wd
2
1 1
dM du 1 dT


 u  Ma
M
u 2 T
Good reference for f : schlicting , boundary layer theory
0
∵ adiabatic , To = const

r  1M 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

x2
x1



2


4 fdx  1
r 1
M

 

ln 
2
D
2r  1  r  1 M 2 
 rM



2

 M 1
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 M 1  2  (r  1) M 2 



P01 M 2  2  (r  1) M 12 
 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
2  r 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  2CpT0  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

u 2 r  1 u 2
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