Transcript Gas Dynamics ESA 341 Chapter 1

Gas Dynamics
ESA 341
Chapter 3
Dr Kamarul Arifin B. Ahmad
PPK Aeroangkasa
Normal shock waves
Definition of shock wave
 Formation of normal shock wave
 Governing equations
 Shock in the nozzle

Definition of shock wave
Shock wave is a very thin region in a
flow where a supersonic flow is
decelerated to subsonic flow. The process
is adiabatic but non-isentropic.
Shock wave
V
P
T
Formation of Shock Wave
A piston in a tube is given a small
constant velocity increment to the
right magnitude dV, a sound wave
travel ahead of the piston.
A second increment of velocity dV
causing a second wave to move into
the compressed gas behind the first
wave.
As the second wave move into a gas
that is already moving (into a
compressed gas having a slightly
elevated temperature), the second
waves travels with a greater velocity.
The wave next to the piston tend to
overtake those father down the tube.
As time passes, the compression
wave steepens.
Types of Shock Waves:
b
Normal shock wave
- easiest to analyze
Oblique shock wave
- will be analyzed
based on normal
shock relations
Curved shock wave
- difficult & will
not be analyzed
in this class
- The flow across a shock wave is adiabatic but
not isentropic (because it is irreversible). So:
T01  T02
P01  P02
Governing Equations
Conservation of mass:
1V1 A  2V2 A
Conservation of momentum:
P1  P2 A  m V2  V1 
P1  P2  1V1 V2  V1 
P1  P2   2V2 V2  V1 
Rearranging:
P P
V1V2  V12  1 2
1
V  V1V2 
2
2
Combining:
P1  P2
2
1
1 
P1  P2     V22  V12
 1  2 
V1
V2
P1
P2
T1
T2
1
2
Conservation of energy:
V12
V22
c pT1 
 c pT2 
 c pT0
2
2
Change of variable:
 2  P1
 2  P2

  V12  
  V22
   1  1
  1  2
 2  P1 P2 
  
V22  V12  
   1  1  2 
combine
Governing Equations
cont.
Continued:
1
1   2  P1 P2 
P1  P2     
  
 1  2     1  1  2 
Multiplied by 2/p1:
 P2   2   2   2 P2 
1    1  
  
 P1  1     1  1 P1 
Rearranging:
   1   2 
 1


P2    1  1 

P1
   1   2 
  

   1  1 
or
   1  P2 
  1

 2    1  P1 

1    1  P2 
  

   1  P1 
Governing Equations
From conservation of mass:
   1  P2 
  1

 2 V1    1  P1 


1 V2    1  P2 
  

   1  P1 
From equation of state:
T2 P2 1

T1 P1  2
 

T2  

T1  

 
 1  P2 
 
 1  P1 
 1  P1 
  
 1  P2 
cont.
Governing Equations
Conservation of mass
1V1   2V2
Conservation of momentum
P1  P2 A  m V2  V1 
P1  1V12  P2   2V22
a2  

P



P1 1  M 12  P2 1  M 22

Conservation of energy
V12
V22
h1 
 h2 
2
2
   1 2 
 T2  1  2 M 1 
   





1
T
2
 1  1 
M2 
2


cont.
C
1V1   2V2
P1
P
M 1 RT1  2 M 2 RT2
RT1
RT2
O
M1
 1 2
M2
 1 2
1

M

1

M2
1
1  M 12
2
1  M 22
2
M
B
I
N
E
 1 2
 1 2
M 1 ) M 22 (1 
M2 )
2
2

2
1  M 12 
1  M 22 2
M 12 (1 
Expanding the equations
  1M 24  M 14   2M 22 M 12 M 22  M 12 


 2 M 22  M 12  0
Governing Equations
cont.
Solution:
M2  
  1M 12  2
2M 12    1
Mach number cannot be negative. So, only the positive value is realistic.
Governing Equations
Temp. ratio
cont.
Dens. ratio
   1 2 
 T2  1  2 M 1 
   

 T1  1    1 M 22 
2


 2 V1 M 1 T1
 
1 V2 M 2 T2

   1 2  2
M 1 
M 12  1
1 
 T2  
2
   1

  
   12  2
 T1 


 2  1  M 1
1


2

1
M1
  1M 12  2
2M 12    1

   1 2  2
2
1

M
M

1



1 
1
2

   1

2
   1  2


 2  1  M 1


Pres. ratio


P2 1  M 12

P1 1  M 22


P2 2M
 1


P1
 1  1
2
1
Simplifying:
2
2
(  1) M 12

1 (  1) M 12  2
3
Governing Equations
Stagnation pressures:
P02 P02 P1 P2

P01 P2 P01 P1

   1 2   1
1
M2  
2

P02 
2

M
1    1
2

 

P01 1    1 M 2  
 1

1
2


Other relations:
P02 P02 P01

P1 P01 P1
P01 P01 P02

P2 P02 P2
cont.
Governing Equations
cont.
Entropy change:
T 
P 
s2  s1  c p ln  2   R ln  2 
 T1 
 P1 
But, S02=S2 and S01=S1 because the flow is
all isentropic before and after shockwave.
Shock wave
1 2
So, when applied to stagnation points:
T 
P 
s02  s01  c p ln  02   R ln  02 
 T01 
 P01 
But, flow across the shock wave is adiabatic & non-isentropic:
T01  T02
And the stagnation entropy is equal to the static entropy:
P 
s02  s01   R ln  02   s2  s1  1
 P01 
So:
P02
 s2  s1 
 exp
1
P01
R
 Total pressure decreases across shock wave !
Group Exercises 3
1.
2.
3.
Consider a normal shock wave in air where the upstream flow
properties are u1=680m/s, T1=288K, and p1=1 atm. Calculate the
velocity, temperature, and pressure downstream of the shock.
A stream of air travelling at 500 m/s with a static pressure of 75
kPa and a static temperature of 150C undergoes a normal shock
wave. Determine the static temperature, pressure and the
stagnation pressure, temperature and the air velocity after the
shock wave.
Air has a temperature and pressure of 3000K and 2 bars absolute
respectively. It is flowing with a velocity of 868m/s and enters a
normal shock. Determine the density before and after the shock.
Stationary Normal Shock Wave Table – Appendix C:
M1
M2
P2
P1
T2
T1
2
1
a2
a1
P2  P1
P1
 2  1
1
T1
P01
T01
T2  T1
P02  P01
T02  T01
M1  1
M2 1
Ms  0
P02
P01
P02
P1