Gas Dynamics-shock Waves

Shock Wave Review: Ø It has been observed for many years that a compressible fluid under certain conditions can experience an abrupt change of state. ØFamiliar examples are the phenomena associated with detonation waves, explosions, and the wave system formed at the nose of a projectile moving with a supersonic speed. ØIn all of those cases the wave front is very steep and there is a large pressure rise in traversing the wave, which is termed a shock wave. Here we will study the conditions under which shock waves develop and how they affect the flow. GDJP PDF created with pdfFactory trial version www.pdffactory.com

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Normal Shock Introduction: Ø By definition, a normal shock wave is a shock wave that is perpendicular to the flow. Ø Because of the large pressure gradient in the shock wave, the gas experiences a large increase in its density and decrease in its velocity. Ø The flow is supersonic ahead of the normal shock wave and subsonic after the shock wave. Ø Since the shock wave is a more or less instantaneous compression of the gas, it cannot be a reversible process. Ø Because of the irreversibility of the shock process, the kinetic energy of the gas leaving the shock wave is smaller than that for an isentropic flow compression between the same pressure limits. GDJP PDF created with pdfFactory trial version www.pdffactory.com

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Cont.. Ø The reduction in the kinetic energy because of the shock wave appears as a heating of the gas to a static temperature above that corresponding to the isentropic compression value. Ø Consequently, in flowing through the shock wave, the gas experiences 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 the order of 10−8 m. Ø The flow is adiabatic across the shock waves. GDJP PDF created with pdfFactory trial version www.pdffactory.com

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Development of a Shock wave A = Constant

Pressure

Piston

Distance along the duct

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q Pressure pulses transmitted through the gas to the rightward movement of the piston. q The waves travel towards the right with the acoustic speed. qThe portion of the gas which has been traversed by the pressure waves is set in motion. qThe pressure waves in the upstream region travel at higher velocities. qThus the upstream waves are continuously overtaking those in the downstream 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 GDJP

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Shock Types 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 q In subsonic flow, sound waves can work their way upstream and forewarn the flow about the presence of the body. Therefore, the flow streamlines begin to change and the flow properties begin to compensate for the body far upstream.

Subsonic flow

Slab

q In

contrast, if the flow is supersonic, sound waves can no longer propagate upstream. Instead, they tend to coalesce a short distance ahead of the body (shock wave) Ahead of the shock, the flow has no idea of the presence of the body. Immediately behind the shock, the streamlines quickly compensate for the obstruction.

Supersonic flow

Slab Shock wave GDJP

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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

(3) (2) GDJP

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Prandtl-Meyer Relation 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 GDJP

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Prandtl-Meyer Relation

2 * (γ + 1)a + c yc x (γ − 1) = 2 γ c yc x 2 * (γ + 1)a = 2 γ c yc x − γ c yc x + c yc x = (γ + 1)c yc x

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Prandtl-Meyer Relation c yc x 1=

2 * = a Prandtl - Meyer relation

c yc x

cy

cx = × a* a*

2 * a a* = a*x = a*y Therfore

M *x × M *y = 1 This is another useful form of the Prandtl - Meyer relation

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Downstream Mach number Generally the upstream Mach number (M x) in a given problem is known and it is desired to determine the Mach number (M y) downstream of the shockwave. For adiabatic flow of a perfect gas gives 2 2γ * a RT 0 = γ −1

(1)

From Prandtl - Meyer relation

2 * C xC y = a

(2)

From eqn. (1) & (2)

(3) Substituting these values in eqn. (3) GDJP

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Cont..

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Cont..

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Static pressure ratio across the shock From Momentum equation

We know that

(1) Substitute GDJP

M 2y in eqn (1). Anna University

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Cont.. 1 + γ M 2 = 2γ px + γ M 2x γ − 1 1 + 2γ M 2x − 1 γ − 1

Py

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Cont.. For a Shock Py >1 Mx > 1; Px Py =1 Mx = 1; Px Py Mx = ∞ ; =∞ Px

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Temperature ratio across the shock Upstream and downstream of the shock, we get T0 x γ −1 2 = 1+ Mx Tx 2

T0 y γ −1 2 = 1+ My Ty 2

&

From the adiabatic energy eqn. for a perfect gas T0x =T0y = T0 , therfore

substitute M 2 y in the above eqn. GDJP

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Cont..

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Density ratio across the shock Density ratio across the shock also called as ‘Rankine-Hugoniot equations’ Equation of state for a perfect gas gives

ρy

p y Tx ; substituting for the pressure and temperature ratio = ρx px Ty 1 (γ + 1)2 2 Mx ρ y  2γ γ −1 2 γ −1 2   = Mx − ρx  γ + 1 γ +1   γ − 1 2  2γ 2 M x  M x − 1  1 + 2   γ − 1   γ + 1  2γ γ +1 2  M x2 − 1  M x2  M x ρy 2  γ −1 2    = = ρx   γ − 1 2  2γ γ −1 2   2   M x  M x − 1 1 + Mx  1 + 2 2   γ − 1    GDJP

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

ρx = = cx ρ y cy

γ −1 2 Mx 2 γ +1 2 Mx 2

1+

Another expression for the density ratio across the shock can be derived in terms of the pressure ratio alone. This is useful for comparing the density ratios in isentropic process and a shock for given values of the pressure ratio. We know that the pressure ratio across the shock py γ −1 γ + 1 py γ −1 2γ 2 2 = ⇒ Mx = + Mx − γ +1 px γ + 1 2γ p x 2γ  γ - 1 2  2γ  2  M x  M x − 1  1 + Ty Ty  2  γ − 1  2 Substituti ng M x in ; = Tx Tx 1 (γ + 1)2 2 Mx 2 (γ − 1) GDJP

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Cont..  γ − 1  γ + 1 p y γ − 1    2γ  γ + 1 p y γ − 1        − 1 1 + + + 2  2γ p x 2γ    γ − 1  2γ p x 2γ   T y     = Tx (γ + 1)2  γ + 1 p y + γ − 1  2(γ − 1)  2γ p x 2γ 

After simplifying and rearranging the numerator and denominator

Ty Tx

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=

(γ + 1)3 p y  1 + γ − 1 p y  4γ (γ − 1) p x  γ + 1 p x  = (γ + 1)3  p y + γ − 1  4γ (γ − 1)  p x γ + 1 

py  py γ 1 − 1 + γ + 1 px p x  py γ − 1 + px γ + 1

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   

Cont..

γ −1 px γ + 1 ; = py    1 + γ − 1   1 γ + p x   γ + 1 py 1+ γ − 1 px = γ + 1 py + γ − 1 px py

p y Tx px Ty

∴

ρy ρx

+

ρy

p y Tx We Know that = ρx px Ty

Or , in terms of pressure ratio γ + 1 ρy −1 py γ − 1 ρx = px γ + 1 ρy − γ − 1 ρx GDJP

Rankine-Hugoniot equations

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R-H and Isentropic relation- Comparison  ρy   = Px  ρ x  Py

Isentropic relation

Py

Rankine-Hugoniot relation

Px γ +1 γ −1

1 0

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1

ρy ρx

q It may be observed that for a given density change the pressure ratio across the shock is greater than its corresponding isentropic value. q But at lower mach numbers the difference is negligible and the flow through the shock wave can be considered nearly isentropic.

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γ

Stagnation pressure ratio across the shock P0 y P0 y P y Px = P0 x Py Px P0 x

(1)

P0 x  γ −1 γ −1 2  2γ 2 Mx − M x  = =  1 + ; Px γ + 1 Px  γ +1 2  γ −1 γ P0 y  γ −1 2  M y  =  1 + Py  2  Py

γ −1 γ

Substitute (2) in (1), on rearrangement gives γ γ −1  γ +1 2  −1 γ −1 Mx  P0 y    γ γ − 2 1  2  M 2x − = ×   γ −1 2  P0 x  γ +1 γ +1 Mx + 1   2 GDJP

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(2)

Change in entropy across the shock Change of entropy across the shock is given by

 γ −1  py  ln ∆s = s y − s x = c p ln − R ln = c p ln − c p  Tx px Tx  γ  px py Ty T y Tx ∆s = c p ln ; substitute and px Tx p y p x γ −1 γ py

Ty

(

 ∆s = c p ln  

Ty

)

p0 y p0 x

   

− γ −1 γ

 γ − 1    ln = − c p   γ  

p0 y p0 x

   = − R ln    

p0 y p0 x

   

Substitute p 0y p 0x in the above eqn. Finally    2γ ∆s γ 2 γ − 1 1 γ −1   2  = ln  + + ln Mx −  R γ − 1  (γ + 1)M 2 γ + 1  γ − 1  γ + 1 γ +1 x   GDJP

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