4 Normal Shocks

8/13/2019 4 Normal Shocks

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GasDynamics 1

Normal Shocks





GasDynamics

Formation of Shock WaveA piston in a tube is given a smallconstant velocity increment to the rightmagnitude dV, a sound wave travelahead of the piston.

A second increment of velocity dVcausing a second wave to move into thecompressed gas behind the first wave.

As the second wave move into a gas thatis already moving (into a compressedgas having a slightly elevatedtemperature), the second waves travelswith a greater velocity.

The wave next to the piston tend toovertake those father down the tube. Astime passes, the compression wavesteepens.

3



GasDynamics

T y pe s o f S ho c k W a ve s :

Normal shock wave- easiest to analyze

Oblique shock wave- will be analyzed

based on normalshock relations

Curved shock wave- difficult & will

not be analyzedin this class

- The flow across a shock wave is adiabatic butnot isentropic (because it is irreversible ). So:

0201

0201

PP

T T

!=

!

4



GasDynamics








0201

0201

PP

T T

!=

!



GasDynamics








0201

0201

PP

T T

!=

!

6



GasDynamics

Governing Equations

1

1

1

1

!

T

P

V

2

2

2

2

!

T

P

V Conservation of mass:

Conservation of momentum:

Rearranging:

Combining:

AV AV 2211 !! =

( ) ( )

( )( )122221

121121

1221

V V V PPV V V PP

V V m APP

"=" "=

"

"="

!!

&

2

212122

1

212121

!

!

PPV V V

PPV V V

"="

"="

( ) 21

22

2121

11V V PP "=# #

$ %

&&' (

+"!!

Conservation of energy:

Change of variable:

0

2

22

2

11 22 T cV

T cV

T c p p p =+=+

# # $ % &&'

( "# # $ % &&'

( "

="2

2

1

121

22 1

2!!"

" PPV V

combine

22

2

221

1

1

12

12

V P

V P

+# # $ %

&&' (

"=+# #

$ %

&&' (

" !"

"

!"

"

7



GasDynamics

Continued:

Multiplied by ) 2/p1:

Rearranging:

( ) # # $ %

&&' ( "# #

$ %

&&' (

"=# #

$ %

&&' (

+"2

2

1

1

2121 1

211!!"

"

!!

PPPP

# # $ % &&' ( "# # $ % &&' ( "=# # $ % &&' (

+# # $ % &&' ( "1

2

1

2

1

2

1

2

1211

PP

PP

!!

" "

!!

*+,

-./ "# #

$ %

&&' (

"+

*+

,-.

/ "# # $

% &&'

(

"

+

=

1

2

1

2

1

2

11

11

1

!

!

"

" !

!

"

"

PP

*+,

-./

+# # $ %

&&' (

"+

*+

,-.

/+# #

$

% &&'

(

"

+

=

1

2

1

2

1

2

11

11

1

PP

P

P

"

" "

"

!!or

Governing Equations cont.

8



GasDynamics

*+,

-./

+# # $ %

&&' (

"+

*+,

-./

+# # $ %

&&' (

"+

==

1

2

1

2

2

1

1

2

11

111

PP

PP

V V

"

"

"

"

!

!

2

1

1

2

1

2

!

!

PP

T T

=

*+,

-./

+# # $ %

&&' (

"+

*+,-

./ +# # $ % &&' ( "

+

=

2

1

1

2

1

2

11

11

PP

PP

T T

"

"

" "


From conservation of mass:

From equation of state:

9



GasDynamics


2211 V V !! =

( ) ( )

( ) ( )222

211

2

2222

2111

1221

11 M P M P

Pa

V PV P

V V m APP

" "

!"

!!

+=+

=

+=+

"=" &

( )

( ) ***

+

,

---

.

/

"+

"+

=# # $ %

&&' (

+=+

22

21

1

2

22

2

21

1

21

1

21

1

22

M

M

T T

V h

V h

"

"

C

O

M

B

INE

Conservation of mass

Conservation of momentum

Conservation of energy ( ) ( )

( )( ) ( )( ) 02

21

1

)2

11(

1

)2

11(

21

112

11

1

21

22

21

22

21

22

41

42

222

22

22

221

21

21

222

2

2212

1

1

222

211

1

1

2211

="+

""""

+

"+

=+

"+

"+

+=

"+

+

=

=

M M

M M M M M M

M

M M

M

M M

M M

M M

M M

RT M RT

P RT M

RT P

V V

" "

"

"

"

"

"

"

"

"

" "

!!

Expanding theequations

10



GasDynamics


( )

( )12

212

1

21

2

""

+"±=

" "

"

M

M M

Solution:

Mach number cannot be negative. So, only the positive value isrealistic .

11



GasDynamics


( )

( )

( )

( )( )

( )( )

11

12

11

121

11

22

11

21

1

21

1

21

1

2

22

21

1

2

21

2

21

21

1

2

22

21

1

2

+

""+

=

+

+=

# # $ %

&&' (

"+

# # $ % &&'

( ""# $ % &

' ( "+

=# # $ %

&&' (

***

+

,

---

.

/

"+

"+

=# # $ %

&&' (

"

"

"

"

"

"

"

"

"

" "

"

"

M PP

M M

PP

M

M M

T

T

M

M

T T

( )( )

( )

( )( )

2)1()1(

121

11

22

11

1221

21

21

1

2

21

2

21

21

21

21

1

1

2

2

1

2

1

2

1

1

2

+"+

=

# # $ %

&&' (

"+

# # $ % &&'

( ""# $ % &'

( "+

""+"

=

==

M M

M

M M

M M

M

T T

M M

V V

"

"

!

!

"

"

" " "

" "

" !

!

!

!

Temp. ratio

Pres. ratio

Dens. ratio

Simplifying:

1

23

12



GasDynamics

Stagnation pressures:

Other relations:

( )012345 + ""6061

2

6364

5

"+

"+

=

=

"

1 12

21

12

11 2

1

1

21

22

01

02

1

2

01

1

2

02

01

02

" " "

"

" "

"

M M

M

PP

PP

PP

PP

PP

2

02

02

01

2

01

1

01

01

02

1

02

PP

PP

PP

P

P

P

P

P

P

=

=


13



GasDynamics

Entropy change:

But, S02=S 2 and S 01=S 1 because the flow isall isentropic before and after shockwave.

So, when applied to stagnation points:

But, flow across the shock wave is adiabatic & non-isentropic:

And the stagnation entropy is equal to the static entropy:

So:

Shock wave

1 2

*+,

-./"*+

,-./

="1

2

1

212 lnln

PP

RT T

css p

*+,

-./"*+

,-./

="01

02

01

020102 lnln

PP

RT T

css p

0201 T T =

1ln 1201

020102

>"=*+,

-./"=" ss

PP

Rss

( )1exp 12

01

02 <""

= R

ssPP Total pressure decreases across shock wave !


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GasDynamics 15



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GasDynamics 18



GasDynamics

Home Works

1. Consider a normal shock wave in air where the upstream flow propertiesare u 1=680m/s, T 1=288K, and p 1=1 atm. Calculate the velocity,temperature, and pressure downstream of the shock.

2. A stream of air travelling at 500 m/s with a static pressure of 75 kPa anda static temperature of 15 0C undergoes a normal shock wave. Determinethe static temperature, pressure and the stagnation pressure, temperatureand the air velocity after the shock wave.

3. Air has a temperature and pressure of 300 0K and 2 bars absoluterespectively. It is flowing with a velocity of 868m/s and enters a normalshock. Determine the density before and after the shock.

19



GasDynamics 0=

s M

11 > M 12

< M

01

01

1

1

1

T

P

T

P

!

0102

0102

12

12

12

T T

PP

T T

PP

=

<

>

>

>

!!

1 M 2 M 1

2

PP

1

2

T T

1

2

!

!

1

2

aa

01

02

PP

1

02

PP

S t a t i ona r y No r m a l S ho c k W a ve Ta b le – A ppe n d i x C :

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GasDynamics

No rm

al S ho

c k W a

ve Ta b

le

21

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4 Normal Shocks