Shock Wave
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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
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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.
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Cont..
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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−
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Development of a Shock wave
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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.
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Simplifications & Assumptions
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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
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Shock Types
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1. Normal Shock (One-dimensional phenomena)
2. Oblique Shock (Two-dimensional phenomena)
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Normal Shock – Fundamental Equations
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Normal Shock on Fanno & Rayleigh curves
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Normal Shock on Fanno & Rayleigh curves
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Flow over a slab- Comparison
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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
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Prandtl-Meyer Relation
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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)
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Prandtl-Meyer Relation
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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
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Prandtl-Meyer Relation
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)1( 2*)1(
2)1( *)1(
2
2
xcycxcycxcycxcyca
xcycxcyca
+=+−=+
=−++
γγγγ
γγγ
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Prandtl-Meyer Relation
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relationMeyer -Prandtl the of form useful another is This
1**Therfore
***
*** 1
relationMeyer - Prandtl *
2
2
=×
==
×==
=
yMxM
yaxaa
axc
a
yc
a
xcyc
axcyc
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Downstream Mach number
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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)
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Cont..
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Cont..
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Static pressure ratio across the shock
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From Momentum equation
We know that
(1). eqn in 2yM Substitute
(1)
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Cont..
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121
2
2 1
2
1
2 1
−−
+−+
+=
xM
xM
M
xpyP
γγ
γγ
γγ
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Cont..
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∞=∞=
==
>>
xPyP
; xM
1 xPyP
; 1xM
1 xPyP
; 1xM
Shock a For
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Temperature ratio across the shock
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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
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Cont..
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Density ratio across the shock
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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
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Cont..
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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
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Cont..
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( )( )
( )( )
( )( ) 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
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Cont..
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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
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R-H and Isentropic relation- Comparison
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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
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Stagnation pressure ratio across the shock
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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
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Change in entropy across the shock
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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
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