gas dynamics-shock waves

Shock Wave

GDJP Anna University

Review:Ø It has been observed for many years that a compressible

fluid under certain conditions can experience an abruptchange of state.

ØFamiliar examples are the phenomena associated withdetonation waves, explosions, and the wave system formedat the nose of a projectile moving with a supersonic speed.

ØIn all of those cases the wave front is very steep and thereis a large pressure rise in traversing the wave, which istermed a shock wave.

Here we will study the conditions under which shockwaves develop and how they affect the flow.

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Normal Shock


Introduction:

Ø By definition, a normal shock wave is a shock wave that isperpendicular to the flow.

Ø Because of the large pressure gradient in the shock wave,the gas experiences a large increase in its density anddecrease in its velocity.

Ø The flow is supersonic ahead of the normal shock waveand subsonic after the shock wave.

Ø Since the shock wave is a more or less instantaneouscompression of the gas, it cannot be a reversible process.

Ø Because of the irreversibility of the shock process, thekinetic energy of the gas leaving the shock wave is smallerthan that for an isentropic flow compression between thesame pressure limits.




Cont..


Ø The reduction in the kinetic energy because of the shockwave appears as a heating of the gas to a statictemperature above that corresponding to the isentropiccompression value.

Ø Consequently, in flowing through the shock wave, the gasexperiences a decrease in its available energy and,accordingly, an increase in its entropy.

Ø A shock wave is a very thin region, its thickness is in theorder of m.

Ø The flow is adiabatic across the shock waves.

810−




Development of a Shock wave


Piston A = Constant

Pres

sure

Distance along the duct

q Pressure pulsestransmitted through thegas to the rightwardmovement of the piston.q The waves traveltowards the right withthe acoustic speed.qThe portion of the gaswhich has beentraversed by thepressure waves is set inmotion.qThe pressure wavesin the upstream regiontravel at highervelocities.qThus the upstreamwaves are continuouslyovertaking those in thedownstream region.




Simplifications & Assumptions


The following simplifications to be made without introducing error in the analysis:

1. The area on both sides of the shock may be considered to be the same.2. There is negligible surface in contact with the wall, and thus frictional

effects may be omitted.

Assumptions 1. One-dimensional flow

2. Steady flow

3. No area change

4. Viscous effects and wall friction do not have time to influence flow

5. Heat conduction and wall heat transfer do not have time to influence flow

6. No shaft work

7. Neglect potential




Shock Types


1. Normal Shock (One-dimensional phenomena)

2. Oblique Shock (Two-dimensional phenomena)




Normal Shock – Fundamental Equations





Normal Shock on Fanno & Rayleigh curves





Flow over a slab- Comparison


Subsonic flow

Supersonic flow

Slab

Slab

q In subsonic flow, sound wavescan work their way upstream andforewarn the flow about the presenceof the body.

Therefore, the flow streamlinesbegin to change and the flowproperties begin to compensate for thebody far upstream.

q In contrast, if the flow issupersonic, sound waves can nolonger propagate upstream. Instead,they tend to coalesce a short distanceahead of the body (shock wave)

Ahead of the shock, the flow hasno idea of the presence of the body.Immediately behind the shock, thestreamlines quickly compensate for theobstruction.

Shock wave




Prandtl-Meyer Relation


We know adiabatic energy equation

Applying the above eqn. to the flow before and after the shock wave we get

(1)

First part of this equation gives Similarly the other part is

(2)(3)






From Momentum equation

From Continuity equation

Substitute continuity eqn. in momentum eqn.

Multiply with γ

but

Therefore(4)

Introduction of eqn. (2) and (3) in (4) gives






)1( 2*)1(

2)1( *)1(

2

2

xcycxcycxcycxcyca

xcycxcyca

+=+−=+

=−++

γγγγ

γγγ






relationMeyer -Prandtl the of form useful another is This

1**Therfore

***

*** 1

relationMeyer - Prandtl *

2

2

=×

==

×==

=

yMxM

yaxaa

axc

a

yc

a

xcyc

axcyc




Downstream Mach number


Generally the upstream Mach number (Mx) in a given problem is known and it is desired to determine the Mach number (My) downstream of the shockwave.

For adiabatic flow of a perfect gas gives

(2) & (1) eqn. From

(2) *xC relationMeyer -Prandtl From

(1) 012*

2

2

ayC

RTa

=

−=

γγ

(3)

Substituting these values in eqn. (3)




Cont..





Static pressure ratio across the shock


From Momentum equation

We know that

(1). eqn in 2yM Substitute

(1)




Cont..


121

2

2 1

2

1

2 1

−−

+−+

+=

xM

xM

M

xpyP

γγ

γγ

γγ




Cont..


∞=∞=

==

>>

xPyP

; xM

1 xPyP

; 1xM

1 xPyP

; 1xM

Shock a For




Temperature ratio across the shock


Upstream and downstream of the shock, we get

22

110 & 2

21

10yM

yTyT

xMxTxT −

+=−

+=γγ

therfore , 0T0yT0xT gasperfect afor eqn. energy adiabatic the ==From

eqn. above the in 2yM substitute




Cont..





Density ratio across the shock


Density ratio across the shock also called as ‘Rankine-Hugoniot equations’

Equation of state for a perfect gas gives

( )

−

−

−+

−

−+

=

−

−

−+

−+

+−

−+

=

=

121

222

11

2121

22

1

xy

121

222

11

21

2121

112

12

xy

ratio etemperatur and pressure thefor ngsubstituti ;

xMxM

xMxM

xMxM

xMxM

yTxT

xpyp

xy

γγγ

γγγ

ρ

ρ

γγγ

γγ

γγ

γγ

ρ

ρ

ρ

ρ

−+

+

=2

211

22

1

xM

xM

γ

γ




Cont..


The equation of continuity for constant flow rate through the shock gives

22

1

22

11

xM

xM

yx

xcyc

+

−+

==γ

γ

ρρ

Another expression for the density ratio across the shock can be derived interms of the pressure ratio alone. This is useful for comparing the densityratios in isentropic process and a shock for given values of the pressure ratio.

We know that the pressure ratio across the shock

( )( )

2

1

2121

121

222

1-1

xTyT

; xTyT

in 2M

21

212

xM 112

12

xM

xMxM

xngSubstituti

xpyp

xMxpyp

−+

−

−

+

=

−+

+=⇒

+−

−+

=

γγ

γγγ

γγ

γγ

γγ

γγ




Cont..


( )( )

( )( )

( )( ) 1

1

111

11

14

31

111

14

31

21

21

12

21

12

1 2

11

2 2

1 2

12

11

+−

+

+−

+

=

+−

+−

+

+−

+−

+

=

−+

+−

+

−

−+

+−

−+

+−+

=

γγ

γγ

γγ

γγγ

γγ

γγγ

γγ

γγ

γγ

γγ

γγ

γγ

γγ

γγγ

xpyp

xpyp

xpyp

xpyp

xpyp

xpyp

xTyT

xpyp

xpyp

xpyp

xTyT

After simplifying and rearranging the numerator and denominator




Cont..


xy

11

1xy

11

ratio pressure of terms in ,11

111

xy

xythat Know We;

111

11

ρ

ρ

γγ

ρ

ρ

γγ

γγ

γγ

ρ

ρ

ρ

ρ

γγ

γγ

−−+

−−+

=

+−+

−+

+=∴

=

+−

+

+−

+=

xpyp

Orxpypxpyp

yTxT

xpyp

xpyp

xpyp

yTxT

xpyp

Rankine-Hugoniot equations




R-H and Isentropic relation- Comparison


x

y

PP

xy ρρ

11

−+

γγ

Rankine-Hugoniot relation

Isentropic relation

γ

ρ

ρ

=

xy

xPyP

q It may be observed that fora given density change thepressure ratio across theshock is greater than itscorresponding isentropicvalue.

q But at lower machnumbers the difference isnegligible and the flowthrough the shock wave canbe considered nearlyisentropic.

01

1




Stagnation pressure ratio across the shock


11

112

12

1

22

11

22

1

00

12

2110

12

2110 ;

112

12

00

00

−−

+−

−+

×

−

−+

+

=

−

−+=

−

−+=

+−

−+

=

=

γ

γγ

γγ

γγ

γ

γ

γγγ

γγγγγ

γγ

xMxM

xM

xPyP

yMyPyP

xMxPxP

xMxPyP

xPxP

xPyP

yPyP

xPyP

(1)

(2)

Substitute (2) in (1), on rearrangement gives




Change in entropy across the shock


Change of entropy across the shock is given by

( )

−=

−−=

−−

=∆

−=∆

−−=−=−=∆

xpyp

Rxpyp

pcxpyp

pcs

xpyp

xTyT

xpyp

xTyTpcs

xpyp

pcxTyT

pcxpyp

RxTyT

pcxsyss

0

0ln0

0ln1

1

0

0ln

andsubstitute ;1 ln

ln1lnlnln

γγ

γγ

γγ

γγ

( )

+−

−+−

+

+−

++−

=∆

112

1 2ln

11

11

21

2 ln1R

s

Finally eqn. above the in 0xp0yp

γγ

γγ

γγγ

γγγ

xMxM

Substitute




gas dynamics-shock waves

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