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MODULE 2

Normal Shock

NOTES- By Dr R. K. HEGDE

Prandtl Meyer equation

Rankine – Hugoniot relation

Normal shock equations

Syllabus Property ratios in terms of upstream
Mach number,
Numericals

Moving Normal Shock wave.

NOTES- By Dr R. K. HEGDE

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About SHOCK WAVE

• A shock wave is a special kind of wave referred to
as a steep finite pressure wave.
• The changes in the flow properties across such a
wave are abrupt. The normal shock wave is
perpendicular to the one-dimensional flow.
• In some situations, shocks are undesirable because
they interfere with the normal flow behaviour.
• For example in turbomachines if the areas and
profiles of flow passages are incorrectly designed
shock may occur on account of supersonic flow
developed due to local accelerations.
• These shocks may be normal or inclined to the
direction of local flow; they may cause boundary
layer separation and deviation of flow from its
designed direction.
NOTES- By Dr R. K. HEGDE

About SHOCK WAVE- Pros and Cons

Cons:
• There is a loss in stagnation pressure and increase in entropy across a shock wave.
• The efficiencies of machines experiencing shock wave are considerably low.
• Other undesirable forms of the shock waves are the sonic boom created by supersonic
aircrafts and the blast waves generated by an explosion.
• These wave have a damaging effect on human life and buildings.
Pros:
• Some useful applications of the shock waves are in the shock tubes and supersonic
compressors.
• A strong moving shock wave is utilized to accelerate the flow to a high Mach number in a
shock tube where flow behaviour at high Mach numbers can be studied.
• On account of abrupt changes of pressure, density, etc: across shock waves,
• they are profitably used in supersonic compressors to obtain considerably high-pressure ratios in
one stage
• in such compressors the pressure ratio developed per stage may be as high as 10.0.

NOTES- By Dr R. K. HEGDE

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DEVELOPMENT OF A SHOCK WAVE

• For some off-design values of the pressure
ratio in a convergent-divergent nozzle, the
variation in the fluid properties is sudden
and the flow changes from supersonic to
subsonic at some sections in the divergent
portion and this is due to the formation of
"finite wave fronts" or shock waves at
these sections.
• The thickness of such waves is of the order
of 10-3 mm which is comparable with the
mean free path of the gas molecules.
• Figure 1 (a) shows a constant area duct
which initially contains a gas at rest.
• Pressure pulses (infinitesimal pressure
waves) are transmitted through the gas to
the right by the rightward movement of the
piston in instantaneous impulses.
NOTES- By Dr R. K. HEGDE

DEVELOPMENT OF A SHOCK WAVE

• The growth of one such wave at time t1, t2… is
shown in Figure 1 (b);
• the waves travel towards the right with the
speed of sound (corresponding to the prevailing
temperature) relative to the gas.
• The portion of the gas which has been traversed
by the pressure waves is set in motion.
• The gas nearer the piston moves at a higher
velocity than the gas in the down stream region.
• Similarly, the pressure of the gas in the regions
closer to the piston is higher than in the regions
downstream.
• On account of this, isentropic expansion of the
higher-pressure gas on the upstream side occurs
continuously towards the lower pressure regions
downstream. NOTES- By Dr R. K. HEGDE

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DEVELOPMENT OF A SHOCK WAVE

• Thus pressure and temperature gradients along the duct
length are established. On account of the higher temperature
the speed of sound is higher in the upstream regions.
• Therefore, the pressure waves in the upstream region travel at
higher velocities on account of higher gas velocity and speed
of sound.
• Thus the upstream waves are continuously overtaking those in
the downstream region.
• On account of the above phenomena the weak pressure wave
generated at time t=t1 grows stronger and steeper as it moves
further towards the right.
• This is due to the other waves overtaking the wave under
consideration and consequently strengthening it.
• If this continuous growth continues, at some stage (as at time
t5) it will overturn; this wave form is not possible on account
of the impossibility of three different values of pressure
occurring simultaneously.
• Therefore, the extreme form of the wave is vertical. This
vertical wave front of finite amplitude is the shock wave across
which the changes in pressure, temperature, density, velocity
and Mach number are abrupt.
NOTES- By Dr R. K. HEGDE

The ALTERNATE FORM OF ENERGY EQUATION

NOTES- By Dr R. K. HEGDE

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The ALTERNATE FORM OF ENERGY EQUATION

NOTES- By Dr R. K. HEGDE

The ALTERNATE FORM OF ENERGY EQUATION-TOTAL

CONDITIONS

NOTES- By Dr R. K. HEGDE

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The ALTERNATE FORM OF ENERGY EQUATION

NOTES- By Dr R. K. HEGDE

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NOTES- By Dr R. K. HEGDE

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THE GOVERNING EQUATIONS

• Figure shows a normal shock wave in a constant area frictionless
duct; the shock wave is contained in a control volume.
• Continuity, momentum, energy and the state equations govern the
flow through the shock.

NOTES- By Dr R. K. HEGDE

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THE GOVERNING EQUATIONS

NOTES- By Dr R. K. HEGDE

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THE GOVERNING EQUATIONS

𝑚̇
𝑝𝑥 − 𝑝𝑦 = 𝑐𝑦 − 𝑐𝑥 (3)
𝐴
Equations (1) and (3) gives.
(𝑝𝑥 + 𝜌𝑥 𝑐𝑥2 ) = 𝑝𝑦 + 𝜌𝑦𝑥 𝑐𝑦2 (4)

Definition of impulse function in Equation (4) gives

Fx = Fy = constant (5)
The equation of state gives.
h = f (s, ρ)
s = f (p, ρ) (6)
These equations will now be used to define two important curves known as Fanno and Rayleigh lines.

Figure shows the conditions of the flow upstream and

downstream of a shock wave on an enthalpy-entropy
diagram.
It must be noted that the state of the gas before and
after the shock lies on both the curves which satisfy
the aforementioned four equations.

NOTES- By Dr R. K. HEGDE

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The Fanno Line

• Fanno line describes an adiabatic flow process in a constant area duct
with friction.
• The stagnation enthalpy and the flow rate per unit area remain constant.
• On account of friction the process is irreversible.
• The process is governed by the Equations of continuity, energy, and state.
Equations (1), (2) and (6) are rewritten in the following form:

𝜌 = (7)

ℎ =ℎ − 𝑐 (8)
sy =f (py, ρy) (9)
Equations (7), (8) and (9) for different values of cy, define the Fanno line
on the enthalpy-entropy diagram shown in Figure .
The constant pressure lines have also been shown for studying changes in
pressure in each Fanno process.
NOTES- By Dr R. K. HEGDE

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The Fanno Line

Now an infinitesimal Fanno process is considered in the vicinity of the maximum point F. The
changes entropy being small the process may be considered as reversible (Δs = 0) near point F. For
this process, we have
1
ℎ + 𝑐 2 = constant
2
dh =-c dc (energy equation) (10)
Equation (1) gives
ρc = constant
ρ .dc +c. dρ= 0
𝑐
dc = - (continuity equation) (11)
𝜌
For an isentropic process,
1
dh = dp (12)
𝜌
Substituting Equations (11) and (12) in (10) gives
𝑑𝑝 ∂𝑝
𝑐 2 = 𝑎2 = = (13)
𝑑𝜌 ∂𝜌 𝑠

Equation (13) demonstrates that the gas velocity at the maximum entropy point (F) on the Fanno line
is sonic (M = 1). It can also be shown that the upper branch of this curve represents subsonic flow and
the lower branch supersonic flow. NOTES- By Dr R. K. HEGDE

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The Rayleigh Line

Rayleigh line describes a frictionless flow process in a constant
area duct with heat transfer.
• The flow rate per unit area remains constant.
• The process is governed by the equations of continuity,
momentum and state. Equations (1), (4), (6) for this process are,
̇
𝜌 = (7)

𝑝 = 𝑝 +𝜌 𝑐 −𝜌 𝑐 (14)
sy =f (py, ρy) (15)
• Equations (7), (14) and (15) for different values of cy, define the
Rayleigh line on the enthalpy-entropy diagram shown in Figure
4.
• The maximum entropy point on this curve is R.
• An infinitesimal process is now considered in the vicinity of this
point. NOTES- By Dr R. K. HEGDE

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The Rayleigh Line

• The momentum Equation gives,

𝑝 + 𝜌𝑐 =constant

dp +c2dρ +2ρc. dc = 0 (16)

• Substituting from Equation (11) in (16) and simplifying

𝑑𝑝
𝑐 =
𝑑𝜌
• Since the changes are small during the process in the vicinity of point R
it can be considered as isentropic (Δs = 0). Therefore,

𝑐 =𝑎 = = (17)

• This demonstrates that the gas velocity at the maximum entropy point
on the Rayleigh line is sonic (M = 1).

• The upper branch of this curve represents subsonic flow and the lower
branch supersonic flow.

• Since the shock occurs from supersonic to subsonic flow it is shown

from X to Y on Fanno and Rayleigh lines in Figure .
NOTES- By Dr R. K. HEGDE

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The PRANDTL-MEYER RELATION

• Prandtl-Meyer relation is the basis of other Equations for shock waves.
• It is a fundamental relation between the gas velocities before and after the normal shock and the
critical velocity of sound.
• Applying the adiabatic energy Equation to the flow before and after the shock wave, we get

+ 𝑐 = + 𝑐 = 𝑎∗ (x)

• First part of this equation gives,

𝑎 = 𝑎∗ − 𝑐

• Dividing throughout by cx

= − 𝑐 (18)

NOTES- By Dr R. K. HEGDE

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The PRANDTL-MEYER RELATION

Similarly, the other part of Equation (x) gives,
𝑎 1 1𝛾 + 1 ∗
+ 𝑐 = 𝑎
𝛾−1 2 2𝛾 − 1

𝑎 = 𝑎∗ − 𝑐

Dividing throughout by cy,

= − 𝑐 (19)

Equation (3) gives,

̇
𝑝 −𝑝 = 𝑐 −𝑐

̇
Substituting for A/𝑚̇ from Equation (1) i.e =𝜌 𝑐 =𝜌 𝑐

− = 𝑐 −𝑐
NOTES- By Dr R. K. HEGDE

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The PRANDTL-MEYER RELATION

Multiplying throughout 𝛾,

− =𝛾 𝑐 −𝑐

But,

=𝑎 and =𝑎

Therefore,

− =𝛾 𝑐 −𝑐 (20)

Introduction of Equations (18) and (19) in (20) gives

𝛾 + 1 𝑎∗ 𝛾−1 𝛾+1 𝑎 𝛾−1
− 𝑐 − + 𝑐 =𝛾 𝑐 −𝑐
2 𝑐 2 2 𝑐 2

𝑎∗ − + 𝑐 −𝑐 =𝛾 𝑐 −𝑐

𝛾+1 ∗ 𝑐 −𝑐 𝛾−1
𝑎 + 𝑐 −𝑐 =𝛾 𝑐 −𝑐
2 𝑐 𝑐 2
𝛾+1 ∗ 𝛾−1
𝑎NOTES-
+ By Dr R.𝑐 K.𝑐HEGDE
= 𝛾𝑐 𝑐
2 2

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This expression on simplification and rearrangement yields (Prandtl-

Meyer relation)
𝑎∗ = 𝑐 𝑐 (21)

∗ ⋅ ∗ =1

Now, 𝑇∗ = 𝑇 = 𝑇∗ = 𝑇

𝑎∗ = 𝛾𝑅𝑇 ∗ , 𝑎∗ = 𝛾𝑅𝑇 ∗

Therefore, 𝑎∗ = 𝑎∗ = 𝑎∗

Hence, ∗ ⋅ ∗ =1

Or, M x . My = 1 (22)
This is another useful form of the Prandtl-Meyer relation.
NOTES- By Dr R. K. HEGDE

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𝑇 𝑝
Using Equation 𝑆 −𝑆 = 𝐶 ln − 𝑅ln
𝑇 𝑝

NOTES- By Dr R. K. HEGDE

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