2019_CompressibleFlow_Part03_Set0123.ppt

compressible flow

1. CP502
Advanced Fluid Mechanics
Compressible Flow
Part 03_Set 01:
Stationary normal shock in variable area ducts

2. R. Shanthini
21 July 2019
Isentropic
upstream;
s = s
x
Isentropic
downstream;
s = s
y
> s
x
M
x
> 1M
y
< 1
Thin
normal
shock
Flow through a stationary normal shock wave:Flow through a stationary normal shock wave:
Mass, momentum and energy are conserved
through a normal shock.
Entropy increases across a normal shock.

3. R. Shanthini
21 July 2019
RTApMRTMRTpAuAdtdm /)/(/  
eee
TMpTp /12/1
**


4. R. Shanthini
21 July 2019
Isentropic
upstream;
s = s
x
Isentropic
downstream;
s = s
y
> s
x
M
x
> 1M
y
< 1
Control
volume
Flow through a stationary normal shock wave:Flow through a stationary normal shock wave:
Let us write the balances over the control
volume shown, assuming the cross-sectional
areas are nearly the same.

5. Balances across a stationary normal shock waveBalances across a stationary normal shock wave
(Rankine-Hugoniot relations):(Rankine-Hugoniot relations):
ρ
x u
x = ρ
y u
y = constant
= constant
= constant
22
22
y
y
x
x
u
h
u
h 
22
yyyxxx upup  
(3.1)
(3.2)
(3.3)
Assuming steady flow across a normal shock assumed to be adiabatic,
we get
R. Shanthini
09 March 2015
  asrewritten becan (3.2)equation , Using
yxpyx
TTChh 
22
22
y
yp
x
xp
u
TC
u
TC 
(3.4)
R. Shanthini
21 July 2019

6. R. Shanthini
21 July 2019
Unknowns are: Tup and , ,
We require one more equation:
R
T
p
T
p
yy
y
xx
x


(3.5)
Balances across a stationary normal shock waveBalances across a stationary normal shock wave
(Rankine-Hugoniot relations):(Rankine-Hugoniot relations):

7. Pressure change across stationary normal shock wave:Pressure change across stationary normal shock wave:
Combining (3.3) and (3.5):
(3.6)
Using in the above:
2
2
1
1
y
x
x
y
M
M
p
p





22
y
y
y
yx
x
x
x u
RT
p
pu
RT
p
p 
RTMu    
22
1 1
yyxx MpMp  
Rearranging:
R. Shanthini
21 July 2019

8. R. Shanthini
21 July 2019
Temperature change across stationary normal shock wave:Temperature change across stationary normal shock wave:
(3.7)
 
yxp
xxyy
TTC
RTMRTM

2

2

22

Substituting for in (3.4):
x
x
py
y
p
T
RM
CT
RM
C


















2

2

22

Rearranging:
2
2
2
1
1
2
1
1
y
x
x
y
M
M
T
T







Substituting and rearranging:)1/(RC
p
RTMu 

9. (3.8)
y
x
x
y
u
u



From (3.1):
Using in the above:RTMu 
y
x
y
x
yy
xx
x
y
T
T
M
M
RTM
RTM





Using (3.7) in the above:
2
2
2
1
1
2
1
1
x
y
y
x
x
y
M
M
M
M









Density change across stationary normal shock wave:Density change across stationary normal shock wave:
R. Shanthini
21 July 2019

10. R. Shanthini
21 July 2019
Mach number change across stationary normal shock wave:Mach number change across stationary normal shock wave:
Rearranging (3.5):


2
2
22
2
2
12
12
1
1
y
x
y
x
y
x
M
M
M
M
M
M

























x
y
x
y
x
y
T
T
p
p





42
42
422
422
12
12
21
21
yy
xx
yy
xx
MM
MM
MM
MM









Substituting from (3.6), (3.7) and (3.8) in the above:
  
  
42242
42242
21 12
21 12
yyxx
xxyy
MMMM
MMMM





11. R. Shanthini
21 July 2019
  
  
  
  
442244
422222
442424
242222
42244222
42244222
1 121
2 42
1 121
2 42
211 212
211 212
yxyxx
yxyxx
yxyxy
yxyxy
yyxyyx
xxyxxy
MMMMM
MMMMM
MMMMM
MMMMM
MMMMMM
MMMMMM












Mach number change across stationary normal shock wave:Mach number change across stationary normal shock wave:

12. R. Shanthini
21 July 2019
 
    
   0 212
0 212
212 212
222222
22224422
24424242



yxxyxy
xyyxxyxy
yxxxyxyy
MMMMMM
MMMMMMMM
MMMMMMMM



Mach number change across stationary normal shock wave:Mach number change across stationary normal shock wave:

13. R. Shanthini
21 July 2019
  0 212or
:follows as isSolution
2222

yxxyxy MMMMMM 
2
1

2
1
1
2
2
2








x
x
y
M
M
M
Since is a trivial solution, the shock solution is
(3.9)

xyMM
Mach number change across stationary normal shock wave:Mach number change across stationary normal shock wave:
2
1

2
11
2
2








x
x
x
y
M
M
M
M

14. R. Shanthini
21 July 2019
Summary of relationships across stationary normal shock:Summary of relationships across stationary normal shock:
(3.9)
(3.8)
(3.7)
(3.6)
2
2
1
1
y
x
x
y
M
M
p
p





2
2
2
1
1
2
1
1
y
x
x
y
M
M
T
T







2
2
2
1
1
2
1
1
x
y
y
x
x
y
M
M
M
M









2
1

2
1
1
2
2
2








x
x
y
M
M
M

15. R. Shanthini
21 July 2019
Pressure ratio in terms of Pressure ratio in terms of MM
xx::
(3.10)


   
 
   
 



1
1 2


11
1 2 1

12 1 2
1 2 1

1 2
12
1
1
1
1
2
2
22
22
22
2
2
2
2
2





























x
x
xx
xx
xx
x
x
x
y
x
x
y
M
M
MM
MM
MM
M
M
M
M
M
p
p
Combining (3.6) and (3.9):

16. R. Shanthini
21 July 2019
Temperature ratio in terms of Temperature ratio in terms of MM
xx::
(3.11)


 


 


2
1

1 2
2
1
1



2
1
1 1 2
1 2
2
1
1

1 2
12

2
1
1
2
1
1

2
1
1
2
1
1
2
2
22
2
2
2
22
2
2
2
2
2
x
xx
xx
xx
x
x
x
y
x
x
y
M
MM
MM
MM
M
M
M
M
M
T
T






 









 

























Combining (3.7) and (3.9):

17. Density ratio in terms of Density ratio in terms of MM
xx::
(3.12)

 



2
1
1
2
1

1
1 2

1 2
2
1
1
2
1


2
2
2
22
2
2
x
x
x
xx
x
x
y
y
x
x
y
M
M
M
MM
M
p
p
T
T











 













Combining (3.5), (3.10) and (3.11):
R. Shanthini
09 March 2015
R. Shanthini
21 July 2019

18. R. Shanthini
21 July 2019
Summary of relationships across stationary normal shock:Summary of relationships across stationary normal shock:
(3.9)
(3.8 & 3.12)
(3.7 & 3.11)
(3.6 & 3.10)

1
1 2

1
1
2
2
2










x
y
x
x
y M
M
M
p
p
 
 2
2
22
2
2
2
1

1 2
2
1
1

2
1
1
2
1
1
x
xx
y
x
x
y
M
MM
M
M
T
T






 












2
2
2
2

2
1
1
2
1

2
1
1
2
1
1
x
x
x
y
y
x
x
y
M
M
M
M
M
M















2
1

2
1
1
2
2
2








x
x
y
M
M
M

19. R. Shanthini
21 July 2019
Stagnation temperature change across stationary normal Stagnation temperature change across stationary normal
shock wave:shock wave:
(3.7)
2
2
2
1
1
2
1
1
y
x
x
y
M
M
T
T







Equation (3.7) gives the relationship of the temperature change
across the shock:
(2.6)
20
2
1
1 M
T
T 


Equation (2.6) relates the temperature to stagnation temperature:
Combining the two:
(3.13)yx
TT
00


20. (3.6)

1
1 2
2





x
x
y M
p
p
Equation (3.6) gives the relationship of the pressure change across
the shock:
Stagnation pressure change across stationary normal Stagnation pressure change across stationary normal
shock wave:shock wave:
Equation (2.7) relates the temperature to stagnation temperature:
Combining the two:
120
2
1
1











M
p
p
(2.7)

1
1 2
2
1
1
2
1
1
2
12
0
12
0



























x
xx
yy
M
Mp
Mp
R. Shanthini
21 July 2019

21. Stagnation pressure change across stationary normal Stagnation pressure change across stationary normal
shock wave:shock wave:
(3.14)



1
2
2
1
1
2
12
12
2
0
0

2
1
1
2
1
1
1 2

2
1
1
2
1
1
1
1 2































































x
x
x
x
y
x
x
y
M
M
M
M
M
M
p
p
R. Shanthini
21 July 2019