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Compressible Flows

Basic concepts of compressibility

Conservation equations

Fanno Flow

Rayleigh flow

Isentropic flows

Normal and oblique shocks

Prandtl Meyer flow

Flow through nozzles and diffusers

Compressible flow

Introduction

Compressibility:

Compressibility is the phenomenon by virtue of which the flow changes its density with
change in speed.

Ques). What is the precise condition under which density changes must be considered?

A quantitative measure of compressibility is bulk modulus of elasticity (E)

𝛥𝑣
𝛥𝑃 = −𝑘 - (1)
𝑣

Where,

change in static pressure

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change in Volume

initial Volume

For ideal gases

For isothermal process

initial static pressure

Pressure equation (2) can be written as

Note:-

𝛥𝑃 ⋅ 𝛥𝑉 is neglected.

Note:

Comparing (3) & (1)

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Pi = k

Hence compressibility may be defined as the volume modulus of the pressure

Limiting condition for compressibility:

Where:

𝑚̇ mass flow rate per unit area

flow velocity

corresponding density of fluid

Substituting into equation (1)

𝑘𝛥𝑣
𝛥𝑝 =
𝑣

Compressibility may also be defined as the density modulus of the pressure.

For incompressible flow

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(Bernoulli’s equation)

Compressibility effect can be neglected if density changes are very small

For gas, speed of propagation of sound wave ‘a’

hence, the condition of incompressibility for gas is

Hence,

: flow can be treated as incompressible

Shorter version:
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Velocity of sound in a fluid:

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cross section area of pipe

Velocity of piston

pressure of fluid in pipe before movement of the piston

density of fluid before the movement of the piston

small interval of time with which piston is moved

Velocity of pressure wave travelling in fluid

Mass of fluid for a length ‘L’ before compression

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Mass of fluid after compression for length

From continuity

Mass of fluid before compression = mass of fluid after compression

𝜌 ∗ 𝐴 ∗ 𝑐 ∗ ⅆ𝑡 = (𝜌 + ⅆ𝜌) ∗ 𝐴 ∗ (𝑐ⅆ𝑡 − 𝑉 ⅆ𝑡)

Net force on fluid element

Combining eq (1) & (2)

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For adiabatic process

Differentiating above equation

For isothermal process:

Differentiate above equation

Hence,

Important points about Sonic velocity:

(1). Sonic velocity depends upon the change in density for a given change in pressure.

(2). It increases with increase in temperature

(3). Sonic velocity is higher in gases with a high value of gas constant
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Mach Number (M):

Defined as square root of the ratio of inertia force of a flowing fluid to the elastic force.

Where,

velocity of fluid

velocity of sound in the fluid

bulk modulus

Mach Angle:
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(a) and (b) the disturbance wave reach a stationary observer before the source of
disturbance could reach him in subsonic flow

Compressible Flow

Basic Equations:

(1). Equation of state

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Where,

= absolute pressure in

= volume occupied by mass of the gas

= mass density in

= Absolute Temperature in Kelvin

= Gas constant

(2). Continuity Equation

(for 1-D steady flow)

Differential form

(3). Momentum Equation (Euler’s Equation)

(4). Energy Equation

Incompressible flow
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Compressible flow

Energy Equation (Bernoulli’s Equation) for incompressible flow

(Euler’s eq)

Integrating above equation

For Compressible flow

(A). for isothermal process

Bernoulli’s Equation for Adiabatic process

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Hence substituting into Euler’s momentum equation

Further

Use
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𝑝1 𝑝2
𝛾 = 𝛾
𝜌1 𝜌2

Above equation can be reduced to

𝑉12 𝑉22
𝐶𝑝 𝑇1 + + 𝑔𝑍1 = 𝐶𝑝 𝑇2 + + 𝑔𝑍2 = 𝑐𝑜𝑛𝑠𝑡
2 2

Steady flow energy equation:

1) No heat exchanges

2) No shaft work

Stagnation point / Stagnation properties:

(at stagnation point)

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Using above equation with at point

(Total specific enthalpy)

𝛾𝑅
𝑐𝑃 =
𝛾−1

From adiabatic relation (or)

Similarly
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Relation between and

(Imp)

Flow of compressible fluid from a reservoir

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At point

Apply Bernoulli’s equation at (Assuming adiabatic process)

Hence
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will be maximum when

is limiting velocity may be achieved by expanding a gas at any given stagnation

condition into vacuum.

For incompressible flow

Hence

Condition for maximum mass flow rate

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For maximum mass flow rate

Value of for maximum mass flow rate

Substituting value of
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Maximum mass flow rate

Velocity at outlet for maximum rate of flow is equal to sonic velocity

Proof :-

Velocity at outlet for maximum mass flow rate is

2
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Imp

Critical value

Critical Mach no

Here
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Isentropic flow of gas through a varying-area channel

Consider the continuity equation-

𝝆𝑨𝑽 = 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕

𝒅𝝆 𝒅𝑨 𝒅𝑽
+ + =𝟎
𝝆 𝑨 𝑽

Consider the differential momentum equation for no viscous, no body forces-

𝒅𝑷 + 𝝆𝑽𝒅𝑽 = 𝟎

Now, we will consider subsonic and supersonic isentropic flow through varying area channel.
Combining the above continuity and momentum equation, we obtain,

𝒅𝝆 𝒅𝑨
𝒅𝑷 + 𝝆𝑽𝟐 [− − ]= 𝟎
𝝆 𝑨

𝝏𝑷
Also, ( ) = 𝒂𝟐
𝝏𝝆 𝒔

𝒅𝑷 𝒅𝑨
Combining above two equations, 𝒅𝑷 + 𝝆𝑽𝟐 (− − )=𝟎
𝝆𝒂𝟐 𝑨

𝒅𝑨
𝒅𝑷(𝟏 − 𝑴𝟐) = 𝝆𝑽𝟐
𝑨

Now divide the above obtained expression by pressure and use the perfect-gas expression for the
𝜸𝑷
speed of sound, 𝒂𝟐 = , we get
𝝆

𝒅𝑷 𝜸𝑴𝟐 𝑷 1
= [ ]
𝒅𝑨 𝟏 − 𝑴𝟐 𝑨

𝒅𝑷
For subsonic flow, M<1, the right-hand side of the expression (1) is positive. Hence, > 𝟎,which
𝒅𝑨
tells us that the pressure and area change in the same direction; which means that increase of area
is accompanied by increase in pressure and vice versa. So, a decrease in area will result in decrease
in pressure (expansion), hat is velocity will increase (flow will accelerate).
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For supersonic flow, M>1, the right-hand side of above equation is negative because 1 − 𝑀2 < 0.
𝒅𝑷
Accordingly, < 𝟎,which indicates that the area and pressure changes happen in opposite
𝒅𝑨
direction. Hence, increase in area is accompanied by decrease in pressure and vice versa. A
decrease in area will result in increase in pressure (compression) and therefore a decrease in
velocity (deceleration).

The above explanations for subsonic and supersonic flows are shown below graphically.

The above figure and equations explain that a subsonic flow cannot be accelerated to a velocity
greater than the velocity of sound in a converging nozzle irrespective of reducing back pressure at
the exit of nozzle. TO accelerate a flow to supersonic velocity, a convergent-divergent duct must
be used with sonic flow at minimum section of the duct (throat).

For M=1 at the throat, from the equation (1), dA=0 for the equation not to be infinite. It can be
found that for this condition 𝒅𝟐 𝑨 > 𝟎. Hence, the location in the convergent-divergent duct where
M=1 occurs is the minimum area section shown below in the figure.
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Area vs Mach No relation:

𝜸+𝟏
𝑨 𝟏 𝟐 𝜸 − 𝟏 𝟐 𝟐(𝜸−𝟏)
= [( )( 𝟏 + 𝑴 )]
𝑨∗ 𝑴 𝜸+ 𝟏 𝟐

1) For each A/A* value, there are

two values of Mach number M

2) The two values for Mach number

M correspond to the subsonic and
supersonic values at that area ratio

3) A/A* =1 for M=1

4) The above equation is valid for

isentropic flow
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Flow in Converging Nozzle

Because, the reservoir has a very large area, we can assume the velocity inside the reservoir to be
very small and close to 0. Hence, static pressure and temperature can be assumed to be same as
the stagnation pressure and temperature. As the flow is accelerated in the nozzle, static pressure
and temperature decrease.
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(1). Curve 1 is for 𝑃𝐵 = 𝑃𝑟 , and there is no flow in the nozzle with no change in pressure
distribution.

(2). As 𝑃𝐵 is reduced below 𝑃𝑟 , flow is induced in the nozzle. Static pressure decreases with x for
this subsonic flow as shown by the curve 2 &3 in figure above.

(3). As 𝑃𝐵 is decreased more, the velocity at the nozzle exit plane increases and eventually it
becomes equal to sonic velocity which is shown by the curve 4.

(4). In a converging nozzle flow can not be accelerated to a velocity greater than the velocity of
sound at the nozzle exit. Once the nozzle exit velocity is sonic, any change/disturbance
downstream of nozzle exit cannot be propagated upstream, because the disturbance is propagated
upstream via wave travelling at a speed of sound. This is explained as follows:

Velocity of fluid= V

Velocity of wave relative to fluid= a

Absolute velocity of wave= a -V

(5). So, when fluid velocity becomes sonic (V=a), the absolute velocity of wave becomes 0 and
hence the disturbance cannot reach upstream of the nozzle exit to the reservoir. Hence, when back
pressure is decreased below that corresponding to curve 4, the reservoir fluid cannot sense this
change in the backpressure and flow is same as that corresponding to that of the curve 4. So, exit
pressure is not equal to back pressure. Instead, the flow will adjust to the back pressure by means
of expansion wave which occurs outside the nozzle.

(6). Since, reduction in back pressure is not sensed by the reservo ir when nozzle exit velocity is
sonic, more flow cannot be induced in the nozzle. The nozzle is said to be choked under these
circumstances.
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When nozzle is choked, mass flow rate becomes constant and independent of the changes in back
pressure downstream of the nozzle exit. This is shown above in the figure.
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Flow in Converging-Diverging Nozzle

A converging-diverging nozzle is shown above in the figure. C-D nozzle is needed to generate a
supersonic flow. This can be proved from the area-velocity relation and density-velocity relation.

𝒅𝑨 𝒅𝑽 𝒅𝝆 𝒅𝑽
=− (𝟏 − 𝑴𝟐 ) , = −𝑴𝟐
𝑨 𝑽 𝝆 𝑽

It is evident from the above equation that for M>1, with M=1 at the throat, area should be increased
after the throat with increase in velocity. For supersonic flow (M>1), change in density is much
larger than that in subsonic flow with the increase in velocity, and this can be deduced from the
above density-velocity relation. So, when velocity is increased in supersonic flow, change in
density is very large. Since, for a steady flow mass flow remains same; to keep same mass flow,
large and sudden reduction in density due to increase in velocity will requires an increase in area.
Hence, C-D nozzle is required to generate supersonic flow with M=1 at the throat. Choking at
throat is essential for a supersonic flow to exist in the diverging portion of C-D nozzle

A number of flow patterns can be generated in a C-D nozzle by varying the back pressure. This is
discussed below.

When 𝑃𝑒 = 𝑃0 , there is no flow through the nozzle. As exit pressure is reduced below 𝑃0 to a
reduced value of 𝑃𝑒1, then a small favorable pressure gradient across the nozzle will be set up and
will induce flow through the nozzle at low subsonic speeds. The local Mach number will increase
in convergent portion of nozzle and will reach a maximum at the throat. This will lead to decrease
in static pressure in convergent portion of the nozzle with its minimum value at the throat. This
flow condition is shown by curve (a) in the figure below.
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Now, 𝑃𝑒 is reduced further to a value 𝑃𝑒2 creating a stronger favorable pressure gradient and flow
will accelerate more. This will cause large variation in Mach number, pressure and temperature
through the duct. This flow condition is shown by curve (b) in figure above.

If 𝑃𝑒 is kept on reducing continuously, then at some value of 𝑃𝑒3, flow velocity will be sonic at
the throat and then decelerate as subsonic flow shown by curve (c) in the figure above. This
corresponds to exit pressure of 𝑃𝑒3 . For 𝑀𝑡ℎ =1 and 𝑃𝑒 < 𝑃𝑒3, nozzle is choked and no more mass
flow can be induced in the nozzle by lowering the exit pressure. At this choking condition, mass
𝑃
flow rate is constant and maximum. For pressure ratio 𝑒3 < 1, with sonic speed at throat, flow
𝑃𝑡ℎ
will expand further in divergent portion of the nozzle as supersonic flow. For back pressures below
𝑃𝑒3, flow is unchanged in the convergent portion of the duct. But, in the divergent portion, flow
expands as supersonic flow for 𝑃𝑒 < 𝑃𝑒3 .

When 𝑃𝑒 is reduced substantially to a specified value 𝑃𝑒𝑐 , then an isentropic flow in divergent
portion is obtained which results in the shock-free supersonic solution. The variation in pressure
for this flow condition is shown by curve (d).

For the exit pressures between 𝑃𝑒𝑐 and 𝑃𝑒3, there will be a normal shock wave inside the divergent
portion of the nozzle. The normal shock will make the flow subsonic downstream of the shock and
the static pressure will increase to 𝑃𝑒4 at exit. If pressure at exit is below 𝑃𝑒4, the normal shock
moves downstream. The normal shock which moves downstream will stand at the nozzle exit for
𝑃𝑒 =𝑃𝑒5. 𝑃𝑒5 is the static pressure behind the normal shock at the design Mach number of the nozzle.

When value of 𝑃𝑒 is further reduced, such that 𝑃𝑒𝑐 < 𝑃𝑏<𝑃𝑒5, the flow inside the nozzle will
become fully supersonic and isentropic.
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The figure besides

shows the normal
shock at exit of the
nozzle for 𝑃𝑒 =𝑃𝑒5

The flow in the nozzle will be over-expanded for 𝑃𝑒 < 𝑃𝑏<𝑃𝑒5, since the flow has expanded below
the back pressure. So, an oblique shock wave is formed to increase the pressure at exit to match
the back pressure. Conversely, for 𝑃𝑏<𝑃𝑒 , the flow in the nozzle is under-expanded since exit
pressure is more than the back pressure. For under-expansion case, expansion waves form to
reduce the exit pressure to match the back pressure value. These situations are shown below in the
figure.

Over-expanded
nozzle

Under-expanded
nozzle
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The important takeaways from the above discussion are:

𝑷 𝒆𝟏
(1). For <1, the flow is subsonic at throat and divergent portion acts as diffuser. The velocity
𝑷𝟎
increases in convergent portion till the throat and then decreases in the divergent portion of the
nozzle. So, the nozzle acts as venturi.
𝑷 𝒆𝟑
(2). For <1, velocity at throat becomes sonic and 𝑀𝑡ℎ=1. Pressure is minimum at the throat.
𝑷𝟎
𝑷 𝒆𝟑
But >1; therefore, the divergent portion acts as a diffuser and the flow velocity does not
𝑷 𝒕𝒉
accelerate to supersonic speeds.

(3). For the exit pressure equal to 𝑃𝑒𝑐 , there is an isentropic expansion of the flow and a shock-free
supersonic flow in the divergent portion of the nozzle is achieved.
𝑷 𝒆𝒄 𝑷𝒆 𝑷 𝒆𝟑
(4). For < < , there will be supersonic flow and a normal shock in the diverging section of
𝑷𝟎 𝑷𝟎 𝑷𝟎
the nozzle. So, at the exit supersonic flow will not exit due to normal shock in diverging portion
of nozzle. Above a certain back pressure 𝑃𝑒𝑐, supersonic flow will not exist at the exit of nozzle,
and only below 𝑃𝑒𝑐 , there can be shock free supersonic flow till exit.
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Normal Shock wave

A shock wave is on extremely thin region typically on the order of across which the

flow properties such as pressure, temperature, density, Mach number etc. change drastically.

M<1

M>1

Change of property across Normal shock

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Section (1) Section (2)

Flow across shock wave is adiabatic

Assumption of Normal shock wave:-

Thickness of NSW is in the range of

Process is adiabatic and entropy change is non-zero

Flow is 1-D

Conservation equations are valid across the shock

Flow is steady

Governing equations :-
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(1). Continuity equation

(2). Momentum equation

(3). Energy equation

Rankine Hugoniot equation:-

This equation related property across N S

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for

Special case as

Measurement of velocity in supersonic flow :-

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Rayleigh pitot tube formula

Once and is known above equation can be solved for M

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Moving Normal Shock Wave

This topic deals with the traveling waves- which are the pressure waves that move at a finite
velocity relative to a fixed co-ordinate system. This is in contrast to the previous topic in which
the normal shock was stationary and the gas ahead of the wave moves at a finite distance .

The wave properties depend on the velocity of the gas ahead of the wave relative to the wave, so
it doesn’t matter whether the normal shock wave is moving into a stagnant gas or the gas is moving
through a stationary shock. But there is one difference between the moving shock wave and the
stationary shock wave- a traveling wave creates an unsteady flow relative to laboratory fixed
coordinate system, whereas, a flow through a stationary wave is steady with respect to laboratory.

Shock Tube is such an application of moving normal shock in which high pressure and high
temperature gases are produced in order to study the thermodynamic and chemical properties of
gases at temperatures and pressures higher than obtainable in the laboratory.

A Normal Shock moving at constant speed into a motionless gas

V S

b a

(a) Shock moving into a stagnant gas (Stationary coordinate system)

S-V S

2 1

(b) Stationary shock (Moving coordinate system)

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Consider a Normal shock moving at a constant speed into a motion less gas shown above in figure
(a). The shock velocity is S and the velocity of the gas behind the wave is V and both the velocities
are measured with respect to a stationary observer. For a stationary observer, flow is unsteady, as
the conditions at a point are dependent on whether the shock has passed over that point.

We have to analyze the shock and apply the governing equations, and hence, it is necessary to
convert the unsteady problem into a steady problem. This is done by making the observer ride with
the shock, which makes the shock stationary with respect to the observer. Hence, the unsteady
problem is changed to a steady problem.

The observer velocity’s effect on the static and stagnation properties has to be considered. Static
properties are those which are measures by the instrument moving at an absolute flow velocity.
Thus, static properties are independent of the observer’s velocity. From above figure,
𝑷𝟐 𝑷𝒃 𝑻𝟐 𝑻𝒃
= and =
𝑷𝟏 𝑷𝒂 𝑻𝟏 𝑻𝒂

But stagnation properties are measured by bringing the flow to rest. From the figure above, if 𝑇1 =
𝑇𝑎 and 𝑃1 = 𝑃𝑎 , then it can be concluded that, 𝑇𝑜1 > 𝑇𝑜𝑎 and 𝑃𝑜1 > 𝑃𝑜𝑎 . This is because the gas at
state 1 has velocity S, while the gas at state a has zero velocity.

𝝆𝟐 (𝑺 − 𝑽) = 𝝆𝟏 𝑺

𝑷𝟐 + 𝝆𝟐 (𝑺 − 𝑽) 𝟐 = 𝑷𝟏 + 𝝆𝟏 (𝑺 )𝟐

(𝑺 − 𝑽) 𝟐 (𝑺 ) 𝟐
𝒉𝟐 + = 𝒉𝟏 +
𝟐 𝟐
𝑷𝟐 𝑷𝟏 𝜸𝑹 𝑷
= , 𝑪𝒑 = , h=𝑪𝒑 𝑻, 𝒂𝟐 = 𝜸𝑹𝑻 = 𝜸
𝝆𝟐𝑻𝟐 𝝆𝟏𝑻𝟏 𝜸−𝟏 𝝆

𝝆𝟐 (𝜸 + 𝟏)𝑷𝟐 + (𝜸 − 𝟏) 𝑷𝟏
=
𝝆𝟏 (𝜸 + 𝟏)𝑷𝟏 + (𝜸 − 𝟏) 𝑷𝟐
𝝆𝟏 𝑷 𝟐 𝑽 𝑻𝟐
From all the equations we will get the expressions for , , S, , , which are given below.
𝝆𝟐 𝑷 𝟏 𝑺 𝑻𝟏
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𝝆𝟏 𝑽
=𝟏−
𝝆𝟐 𝑺

𝑷𝟐 𝜸𝑺𝑽
= 𝟏+ 𝟐
𝑷𝟏 𝒂𝟏

𝑷𝟐 𝟐𝜸 𝑺 𝟐 𝜸 − 𝟏
= ( ) −
𝑷𝟏 𝜸+𝟏 𝒂 𝜸 +𝟏

𝑽 𝟐 𝒂 𝟐
= [𝟏 − ( 𝟏 ) ]
𝑺 𝜸+𝟏 𝑺

𝟐
𝜸+𝟏 𝜸+𝟏
𝑺= ( ) 𝑽+ √ [( ) 𝑽 + 𝒂𝟐𝟏
]
𝟒 𝟒

𝜸 − 𝟏 𝑷𝟏 𝜸 + 𝟏 𝑷𝟏 𝑷𝟐
𝑺 = √( )( )+ ( ) ( )( )
𝟐 𝝆𝟏 𝟐 𝝆𝟏 𝑷𝟏

𝑷𝟐 𝜸+𝟏 𝑽 𝟐 𝜸𝑽 𝜸 +𝟏 𝟐 𝑽 𝟐
= 𝟏 + 𝜸( √
)( ) + ( ) 𝟏 + ( ) ( )
𝑷𝟏 𝟒 𝒂𝟏 𝒂𝟏 𝟒 𝒂𝟏

𝑻𝟐 𝒂𝟐𝟐 (𝜸 − 𝟏) 𝑽𝟐
= 𝟐=𝟏+ ( 𝑺𝑽 − )
𝑻𝟏 𝒂𝟏 𝒂𝟐𝟏 𝟐
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Oblique Shock
Oblique shock is on compression wave inclined at an angle to the upstream flow

flow deflection angle/wedge angle

Shock angle

Whenever a supersonic flow is turned onto itself an oblique shock wave will occur.

Oblique shock relations-

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1) Continuity Equation:

∯ 𝝆𝒗 ⋅ 𝒅𝒔 = 𝟎

Evaluate the surface integral over the faces a and d which gives -𝜌1 𝑢1 𝐴1 + 𝜌2 𝑢2 𝐴2 , where A1 and
A2 are areas of faces a and d. The faces b, c, e and f are parallel to the velocity and s0, 𝑣 ⋅ ⅆ𝑠 = 0

-𝝆𝟏 𝒖𝟏𝑨𝟏 + 𝝆𝟐 𝒖𝟐 𝑨𝟐 = 𝟎

𝝆𝟏 𝒖𝟏 = 𝝆𝟐 𝒖𝟐, 𝑢1 and 𝑢2 are velocities normal to the shock wave.

2) Momentum Equation:
(a) For tangential component

∯( 𝝆𝑽 ⋅ 𝒅𝒔)𝒘 = − ∯( 𝑷 𝒅𝒔)𝒕𝒂𝒏𝒈𝒆𝒏𝒕𝒊𝒂𝒍 , P ds vectors on faces b and f are

equal and opposite, so the pressure integral involving two forces cancel out each other
over the faces b and f.

−(𝝆𝟏 𝒖𝟏𝑨𝟏)𝒘𝟏 + (𝝆𝟐𝒖𝟐 𝑨𝟐)𝒘𝟐 = 𝟎

w1 =w2
Tangential component of the flow velocity is constant across the oblique shoch wave

(b) For normal component

∯( 𝝆𝑽 ⋅ 𝒅𝒔)𝒖 = − ∯(𝑷 𝒅𝒔)𝒏𝒐𝒓𝒎𝒂𝒍, Integral is evaluated over faces a & d.

−(𝝆𝟏𝒖𝟏𝑨𝟏 )𝒖𝟏 + (𝝆𝟐𝒖𝟐𝑨𝟐)𝒖𝟐 = −(−𝑷𝟏 𝑨𝟏 + 𝑷𝟐 𝑨𝟐)

𝑷𝟏 + 𝝆𝟏 𝒖𝟐𝟏 = 𝑷𝟐 + 𝝆𝟐 𝒖𝟐𝟐
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3) Energy Equation:
𝑽𝟐
∯ 𝝆 (𝒆 + 𝟐
) 𝑽 ⋅ 𝒅𝒔 = − ∯ 𝒑𝑽 ⋅ 𝒅𝒔, Integral over the faces a and d.

𝑽𝟐𝟏 𝑽𝟐𝟐
−𝝆𝟏 (𝒆𝟏 + ) 𝒖𝟏𝑨𝟏 + −𝝆𝟐 (𝒆𝟐 + ) 𝒖𝟐𝑨𝟐 = −(−𝒑𝟏𝒖𝟏 𝑨𝟏 + 𝒑𝟐 𝒖𝟐𝑨𝟐 )
𝟐 𝟐

On Simplification and combining the terms in the above equation, we get the final energy equation
as follows:

𝑽𝟐𝟏 𝑽𝟐𝟐
𝒉𝟏 + 𝟐
= 𝒉𝟐 + 𝟐

𝑉2
Since, ℎ + = ℎ0, Hence, the total enthalpy is constant across the shock wave
2

For a calorically perfect gas, 𝒉𝟎 = 𝑪𝒑 𝑻𝟎, hence total temperature is constant across the
shock wave

From the figure above, 𝑽𝟐 = 𝒖𝟐 + 𝒘𝟐. But, 𝒘𝟏 = 𝒘𝟐

𝑽𝟐𝟏 − 𝑽𝟐𝟐 = 𝒖𝟐𝟏 − 𝒖𝟐𝟐

𝒖𝟐𝟏 𝒖𝟐𝟐
𝒉𝟏 + = 𝒉𝟐 +
𝟐 𝟐

The governing equations namely, continuity, momentum and energy equations involve normal
components of velocity only. Hence, it is concluded that changes across an oblique shock wave
are governed only by the normal component of velocity
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The angle by which the flow turn towards the shock is called deflection angle.

And shock is inclined at an angle to the upstream supersonic flow is called oblique shock.

𝜸−𝟏
𝟏+ [ ]𝑴𝟐𝒏,𝟏
𝑴𝟐𝒏,𝟐 = 𝟐
𝜸−𝟏
𝜸𝑴𝟐𝒏,𝟏 −
𝟐

:-

𝑴𝟐 𝐬𝐢𝐧𝟐 𝜷−𝟏
𝐭𝐚𝐧 𝜽 = 𝟐 𝐜𝐨𝐭𝜷 (𝑴𝟐(𝐜𝐨𝐬𝟐𝜷+𝜸
𝟏
)+𝟐
)
𝟏

The normal component must be supersonic

This requirement impose the restriction on the wave angle for a given
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(1). For a given 𝑀1 these is maximum deflection angle till which attached oblique shock solution
is obtained. After the maximum deflection angle, attached oblique shock is not p ossible and hence
a detached shock is formed as shown in the figure below. The maximum deflection angle (𝜃𝑚𝑎𝑥 )
increases with an increase in the upstream Mach number.

(2). For deflection angle θ <θmax, there are two solutions for straight oblique shoch waves for a
given upstream Mach number. The smaller value of β corresponds to weak shock solution, and
larger value of β corresponds to strong shock solution. In nature, usually weak shock exists. So,
straight oblique shocks are generally the weak shock solution. The strong shock and weak shock
both are shown below in the figure.
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(2). Effect of increasing upstream Mach: As M1 increases, the wave angle β decreases. Although
this shock is at lower wave angle, still it is a stronger shock. It is so because, across the oblique
shock, changes in flow properties are governed by the normal component of velocity 𝑀𝑛1 .
Although β decreases but due to increase in M1, 𝑀𝑛1 = 𝑀1 sin 𝛽 increases by an amount which
compensates for the decrease in wave angle. The converse of this is also true that, the decrease in
Mach number increases the wave angle and makes the shock weaker.

(3). Effect of increasing the deflection angle: For a fixed M1, increasing the deflection angle θ,
wave angle increases. Since wave angle increases, 𝑀𝑛1 also increases and the shock becomes
stronger.
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Supersonic flow over 2D body (wedge) & 3D body (Cone)

Flow over wedge-

(1). Shock wave from the nose of the wedge is straight and attached.

(2). Flow is uniform downstream of the shock and streamlines are parallel to the surface of the
wedge.

(3). Flow deflection behind the shock is sudden.

(4). Surface pressure is equal to the static pressure (𝑃2 ) behind the oblique shock wave

Flow over cone-

(1). Straight oblique shock generates from the tip.

(2). Flow over a cone is 3 dimensional and hence the flow experiences 3D relieving effect. Flow
over the cone has an extra dimension and space in which it can move and can easily adjust to the
presence of the conical body as compared to the two-dimensional wedge.

(3). The three-dimensional relieving effect makes the shock wave on the cone weaker than that
over the wedge. As a result of the weak shock, shock wave angle is smaller.

(4). Due to the weaker shock, flow is deflected gradually and by smaller angle as compared to that
for the wedge. The flow downstream of the shock over cone is curved rather than being straight.

(5). Du to 3dimensional relieving effect, pressure on the cone surface ( 𝑃𝑐 ) is lesser than the wedge
surface static pressure (𝑃2 ).
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Curved Shock in front of Blunt Bodies

(1) For a blunt body in a

supersonic flow, detached curved
shock wave is formed. This
detached curved shock wave
takes the shape of a bow and
hence, it is called bow shock.

(2) The bow shock stands at some

distance from the nose of the
blunt body. This distance is called
shock detachment distance.

Freestream supersonic flow is coming over the blunt body. A detached (bow) shock is formed at a
distance of 𝜹 from the nose of the body as shown in the figure above. There are some points marked
on the bow shock, namely a, b, c, d, e. At point a, shock wave is normal to the upstream supersonic
flow. So, point a corresponds to the normal shock.

It is observed that as we move away from the point a, the shock wave is gradually getting curved
and weaker and finally evolves in to a Mach wave at large distance from the body as shown in the
figure above by point e.

𝝅
Point a corresponds to the normal shock wave (𝜷 = ) and point e corresponds to the Mach wave
𝟐
𝟏
(𝜷 = 𝐬𝐢𝐧−𝟏 ). So, bow shock contains all possible shock solutions from strongest to weakest
𝑴𝟏
shock.

At point b, little above from the centerline, shock wave is curved and is an oblique shock. But this
oblique shock corresponds to a strong shock solution. The flow gets deflected slightly in the
upward direction after the shock at point b.
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As we move along the shock, shock wave becomes more curved and more oblique. So, the flow
deflection increases until we reach point c. Flow will experience maximum deflection at point c.
After point c all the points on the shock wave till e, all points correspond to the weak shock
solution.

At point c’ flow behind the shock becomes sonic. So, it is observed that from point a to c’, the flow
is subsonic behind the shock wave and from point c’ to e, the flow is supersonic behind the shock
wave. So, from the above discussions, it is concluded that the flow field between curved bow shock
and the blunt body is a mixed region of both supersonic and subsonic flow. The subsonic flow and
supersonic flow field behind the shock wave is divided by a sonic line (shown by dashed line in
the figure).

The shape of shock wave and its

detachment distance 𝛿, and the
complete flow field between the
shock and body depends on the
upstream Mach number (𝑀1 ) and
shape and size of the body.

.
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Prandtl-Mayer Expansion Wave

When a supersonic flow is turned away from itself, an expansion wave is formed. This is sketched
in the figure below. The expansion process in an expansion wave is continuous. The expansion
region can be thought of an infinite number of Mach waves and each Mach wave makes an angle
𝝁 which is called Mach angle.

An expansion wave is bounded by forward Mach wave upstream, which makes an angle 𝝁 𝟏 with
respect to the upstream flow direction, and rearward Mach line which makes an angle 𝝁𝟏 with
respect to the upstream flow direction.

Since the expansion process is continuous as the flow is expanding through a continuous
succession of Mach waves, hence, change in entropy is 0 (dS=0). Hence, the flow expansion is
isentropic.
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Prandtl-Mayer Function :-

𝟐 𝑴 √𝑴𝟐−𝟏 𝒅𝑴
𝜸(𝑴) = ∫𝑴𝟏 ( 𝜸−𝟏 )
𝟏+( )𝑴𝟐 𝑴
𝟐

For isentropic expansion

𝜸(𝑴𝟐 ) = 𝜸(𝑴𝟏 ) + 𝜽

For calculation

(1). For given , obtain from table

(2). Calculate

(3). Obtain from using table

(4). since flow is isentropic, use isentropic relation to obtain other property.

Isentropic flow: 𝑇𝑜2 = 𝑇𝑜1 , 𝑃𝑜2 =𝑃𝑜1 .

𝜸
𝑻𝟐 𝜸−𝟏 𝑷𝟐 𝜸−𝟏
𝑻𝟐 𝑻 𝒐𝟐 𝟏+( )𝑴𝟐𝟏 𝑷𝟐 𝑷𝒐𝟐 𝟏+( )𝑴𝟐𝟏 𝜸−𝟏
𝟐 𝟐
= 𝑻𝟏 = 𝜸−𝟏 , = 𝑷𝟏 = (( 𝜸−𝟏 ) )
𝑻𝟏 𝟏+( )𝑴𝟐𝟐 𝑷𝟏 𝟏+( )𝑴𝟐𝟐
𝑻 𝒐𝟏 𝟐 𝑷𝒐𝟏 𝟐
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Shock-Expansion Theory
This theory is applied to flat plates and supersonic airfoils (diamond shaped). The shock expansion
theory over flat plate and diamond shaped airfoils is used to calculate lift and drag. Consider a flat
plate of length of length c at an angle of attack 𝜶 in a supersonic flow. This is sketched in figure
below.

From the above figure, it is observed that on top surface flow is turned away from its initial
direction, hence an expansion wave is formed at flat plate leading edge. After the expansion wave
on leading edge, pressure on top surface (𝑃2 ) decreases and is less than the freestream pressure
(𝑃2<𝑃1 ). At trailing edge, flow direction should be approximately in the freestream direction,
which requires the flow direction to be turned away and hence oblique shock is formed on top
surface at trailing edge.

On the bottom surface, at leading edge flow is turned in to itself and an oblique shock is formed.
The oblique shock on bottom surface at the leading edge increases the downstream pressure (𝑃3 )
and hence 𝑃3 > 𝑃1 . At trailing edge, the need to turn the flow in the freestream direction creates
an expansion wave. The pressure distribution over top and bottom of the flat plate is uniform and
𝑃3 > 𝑃2. Since 𝑃3 > 𝑃2, there is a pressure difference which generates a net aerodynamic force R
as shown in figure above. Resultant force, lift and drag per unit span are as follows:
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𝑹′ = (𝑷𝟑 − 𝑷𝟐 )𝒄

𝑳′ = (𝑷𝟑 − 𝑷𝟐 )𝒄 𝐜𝐨𝐬 𝜶

𝑫′ = (𝑷𝟑 − 𝑷𝟐 )𝒄𝐬𝐢𝐧 𝜶

In the above equations, 𝑃3 is calculated from the oblique shock properties and 𝑃2 is calculated
from the expansion-wave properties.

Another such application is of diamond shaped airfoil shown in the figure below. Assume the
airfoil is at 0° angle of attack. The incoming supersonic flow gets compressed through oblique
shocks at the leading edge and the flow is deflected by an angle ∈. At the mid-chord, flow gets
expanded through expansion wave and is also deflected away from itself by an angle 2 ∈.

At the trailing edge, flow turns back towards the freestream direction through another set of oblique
shocks. At leading edge after the shock, pressure is increased and after expansion wave at mid -
chord pressure is decreased. The pressure distributions are sh own below in the figure. The
pressures on top and bottom surface are uniform and same as 𝑃2 till the mid-chord. Similarly,
pressures on top and bottom surface after the mid-chord are uniform and equal as 𝑃3 as shown in
the figure below.

In the direction perpendicular to the freestream direction, the pressure distributions on the top
and bottom are equal and cancel each other, which makes the lift 𝐿′ =0. But, in the horizontal
direction, pressure distributions on front face are larger than that in the backward face, which
results in a finite drag.
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𝒕
𝑫′ = 2 (𝑷𝟐 𝒍 𝐬𝐢𝐧 ∈ − 𝑷𝟑 𝒍 𝐬𝐢𝐧 ∈ ) = 2 (𝑷𝟐 − 𝑷𝟑 ) = (𝑷𝟐 − 𝑷𝟑 )𝒕
𝟐

Hence, we observe that at 0 angle of attack, lift is zero but drag is not zero. So , we have a finite
drag at 0 angle of attack and it is in contrast to the D’Alembert paradox. The paradox says that in
an inviscid flow, at 0 angle of attack drag is 0. However, in a supersonic inviscid flow, drag is
finite and the source of this finite drag is shock wave and the pressure difference in freestream
direction associated with the wave. Hence this drag is called wave drag.
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Supersonic Diffusers & Wind Tunnels

Diffusers:

It is a duct which is used to slow down the incoming air velocity to lower velocity t exit of diffuser
with minimum total pressure loss. Ideal diffuser is characterized with an isentropic deceleration to
lower velocity. An ideal isentropic diffuser is shown below in the figure.

An ideal diffuser cannot be achieved in practical. In reality, actual supersonic diffuser is associated
with as series of oblique shockwaves. These shock waves create discontinuity and increases the
entropy, and so, the isentropic nature cannot be achieved in practical. The flow is decelerated by a
series of oblique shock waves in convergent portion and then in the constant area section. The
shocks interact with viscous flow near the wall and hence, the strength of the reflected shock waves
reduce in strength. The series of shock waves ends with a weak normal shock wave at the end of
𝑃
constant-area duct. At the exit of the duct, 𝑃02 <𝑃01 . The aim of efficient diffuser is to have 02
𝑃01
close to 1. The actual supersonic diffuser with series of shocks associated with it is shown in the
figure below.
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Supersonic Wind Tunnels:

The supersonic wind tunnel includes following components and is shown in the figure below.

(a) Converging-Diverging Nozzle

(b) Constant area section (Test-Section)

(c) Oblique shock diffuser

C-D nozzle is used to generate supersonic Mach number required in the test section. The area ratio
of the nozzle is decided by the test section Mach number.

The oblique shock diffuser is used to decelerate the flow with minimum total pressure loss. This
𝑃
gives the lower pressure ratio 𝑜 and hence lower reservoir pressure 𝑃𝑜 is required, which reduces
𝑃𝐵
the power requirement and cost associated with it. The normal shock being the strongest shock
causes maximum total pressure loss, hence, an oblique shock diffuser is used which has a series
of oblique shock waves terminated by a weak normal shock at the end of constant area duct in a
converging-diverging diffuser. The weak normal shock causes less total pressure loss.

Moreover, in reality, normal shock cannot be maintained at a fixed location. Unsteadiness and
disturbances will cause the normal shock to move.

It is observed from the figure above that there exist two throats, namely, nozzle throat (𝐴𝑡1 ) and
diffuser throat (𝐴𝑡2 ). The mass flow rate through the nozzle evaluated at the first throat is 𝑚̇= 𝜌uA.
The mass flow rate through both the throats will be equal under steady state condition.
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𝒎̇ 𝟏=𝒎̇ 𝟐

𝝆∗𝟏 𝒂∗𝟏𝑨𝒕𝟏=𝝆𝟐 𝒖𝟐𝑨𝒕𝟐

𝑨𝒕𝟐 𝝆∗𝟏 𝒂∗𝟏
=
𝑨𝒕𝟏 𝝆∗𝟐 𝒂∗𝟐

The flow is assumed to be adiabatic throughout, so 𝒂∗𝟏=𝒂∗𝟐.

𝑨 𝒕𝟐 𝝆∗𝟏
=
𝑨 𝒕𝟏 𝝆∗𝟐

𝑷∗𝟏
𝝆∗𝟏 𝑹𝑻∗𝟏 𝑷∗
𝝆∗𝟐
= 𝑷∗𝟐
= 𝑷𝟏∗ , Since, 𝑻∗ is constant for an adiabatic flow .
𝟐
𝑹𝑻∗𝟐

𝜸 𝜸
𝟐 𝜸−𝟏 𝟐 𝜸−𝟏
𝑷∗𝟏 = 𝑷𝒐𝟏 (𝜸+𝟏) , 𝑷∗𝟐 = 𝑷𝒐𝟐 (𝜸+𝟏)

𝑨 𝒕𝟐 𝑷𝒐𝟏
=
𝑨 𝒕𝟏 𝑷𝒐𝟐

It is known that total pressure decreases across the shock wave, so, 𝑨𝒕𝟐>𝑨𝒕𝟏. Hence, the second
throat area should be larger than the first throat. So, to pass the same mass flow and swallow the
shock wave, second throat area should be larger than the first throat area. If second throat area is
not larger than the first throat area, then the diffuser will choke. Under this choked condition, we
would no longer have uniform flow at the desired test section Mach number, the supersonic wind
tunnel is said to be un-started.
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Rayleigh Flow
A frictionless flow process in a constant area duct with heat transfer is known as Rayleigh flow.
The assumptions for analysis of Rayleigh flow are as f ollows:

Assumptions:

(1). Perfect gas

(2). Constant area duct

(3). 1-D, Steady frictionless flow

(4). Absence of body forces

The control volume considered for the analysis of Rayleigh flow is shown in the figure above.
Below given are the governing equations:

Continuity Eq-

𝝆𝑽 = (𝝆 + 𝒅𝝆)(𝑽 + 𝒅𝑽)
𝒅𝝆 𝒅𝑽
+ =𝟎 𝝆𝑽 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝝆 𝑽
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Momentum Eq-

𝒅𝑷 + 𝝆𝑽𝒅𝑽 = 𝟎

𝑷 + 𝝆𝑽𝟐 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

Energy Eq-

𝜹𝒒 = 𝒅𝒉 + 𝑽𝒅𝑽

For perfect gas, 𝒅𝒉 = 𝑪𝒑 𝒅𝑻

𝜹𝒒 = 𝑪𝒑 𝒅𝑻 + 𝑽𝒅𝑽

𝑽𝟐 𝑽𝟐
𝒉𝒐 = 𝑪𝒑 𝑻𝒐 = 𝒉 + = 𝑪𝒑 𝑻 +
𝟐 𝟐

𝑪𝒑 𝒅𝑻𝒐 = 𝑪𝒑 𝒅𝑻 + 𝑽𝒅𝑽

𝜹𝒒 = 𝑪𝒑 𝒅𝑻𝒐

Referring to the momentum equation written above,

𝑷 + 𝝆𝑽𝟐 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

𝑽𝟐
𝑷 (𝟏 + ) = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝑹𝑻

𝜸𝑽𝟐
𝑷 (𝟏 + ) = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝜸𝑹𝑻

𝑷(𝟏 + 𝜸𝑴𝟐) = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

𝑷𝟐 𝟏 + 𝜸𝑴𝟐𝟏
=
𝑷𝟏 𝟏 + 𝜸𝑴𝟐𝟐
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𝑃 𝑚̇
𝑇= ,𝜌 = , 𝑉 = 𝑀√𝛾𝑅𝑇,
𝜌𝑅 𝐴𝑉

𝑴𝟐
√𝑻 = (𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕)𝒑𝑴 , 𝑻 = (𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕) 𝟐
(𝟏+𝜸𝑴𝟐)

𝟐
𝑻𝟐 𝑴𝟐𝟐 (𝟏+𝜸𝑴𝟐𝟏)
= 𝟐
𝑻𝟏 𝑴𝟐𝟏 (𝟏+𝜸𝑴𝟐𝟐)

Energy equation,

𝑸 = 𝑪𝒑 (𝑻𝒐𝟐 − 𝑻𝒐𝟏 )

Rayleigh Curve
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𝒅𝑺
The above figure has two critical points which are the maximum entropy point ( = 𝟎) and the
𝒅𝑻
𝒅𝑻
maximum temperature point ( = 𝟎). The Mach number at the point of maximum entropy is 1.
𝒅𝑺
At the maximum temperature point, Mach number is less than 1.

The curve above M=1 belongs to subsonic flow and the curve below M=1 belongs to supersonic
flow. When heat is added to either subsonic or supersonic flow, entropy will increase , and the flow
will move towards sonic velocity (M=1).

For subsonic flow, Mach number increase by addition of heat and decreases by removal of heat.
However, for supersonic flow, Mach number decreases by addition of heating and increases by
removal of heat.

The maximum amount of heat that can be added is dictated by the maximum entropy point. Any
further heat addition will lead to change in the inlet condition and reduced mass flow rate, in which
case, the Rayleigh curve shifts towards right which is characterized by lower mass flow rate. The
maximum entropy point is the choked condition for corresponding amount of heat added. This is
shown in figure in the following page.

Change in property

Heating Heating Cooling Cooling

V
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Note:-

Static Temperature

For

decrease for

For

increase for

Explanation (why):

For subsonic flow static enthalpy or temperature increase up to point beyond that

further heat addition leads to decrease in temperature. This is due to fact that beyond that point
decrease in enthalpy due to pressure drop is greater than the increase by heat addition.

Thermal choking due to heating

Subsonic flow:

Heating subsonic flow cause exit Mach number to increase, if sufficient heat has been added Mach
number will reach 1 at the end of the duct this is known as thermal choking.

Now suppose that we add more heat to the system. The only way that the system can reflect the
required additional entropy change is to move to a new Rayleigh line at a decreased flow rate.
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Supersonic flow:

If heat addition is more than the max value (allowed). The flow will adjust by creating shock wave.
An adiabatic shock somewhere in heating region would not give readjustment since there is no
change in stagnation temperature across the adiabatic shock. Therefore, it should be in the nozzle
which feeds the duct.

A normal shock in the diverging section of the nozzle will act to reduce the Mach no at the duct
inlet, without any reduction in mass flow rate.

Constant-Area Duct Connected to a Reservoir with a Converging Nozzle

Heat is added in a constant area duct which is connected to a reservoir with a converging nozzle
and is shown in the figure below with pressure distribution curve along the distance of the duct
and nozzle.

For back pressure equal to the reservoir pressure, there will be no flow. As we lower the back
pressure, flow starts and subsonic flow exists in nozzle and duct. The static pressure decreases in
both the nozzle and duct. This flow condition is shown in figure below by the curve (a) and (b).

When back pressure is decreased more to that corresponding to the curve (c), sonic flow will occur
at the duct exit. Now if back pressure is reduced further below that corresponding to curve (c),
then this change will not be sensed by the reservoir and hence for all back pressures below that of
curve (c), mass flow rate and pressure distributions will remain same as th at for curve (c).
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For back pressures corresponding to curve (d) and (e), the flow adjusts to the back pressures by
the expansion waves outside the duct. Hence, the duct is chocked for back pressures equal to and
less than that for the curve (c).
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Fanno Flow
The steady flow of ideal gas in a constant area adiabatic duct, with friction, in absence of work
and no heat transfer, is called fanno flow.

Assumption

(1). Perfect gas with constant specific heats

(2). Constant area duct (dA=0)

(3). 1-D, steady, frictional flow

(4). Absence of heat and work transfer

Governing Equation

(1). Continuity

(2). Momentum equation

(3). Energy Equation

𝑉2
ℎ 01 = ℎ 02 = ℎ +
2
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The figure above shows the control volume for the analysis of a flow in constant-area duct with
friction and no heat transfer. The forces acting on the control volume are shown in the figure below
and are listed below.

𝝉𝒇 = 𝒔𝒉𝒆𝒂𝒓 𝒔𝒕𝒓𝒆𝒔𝒔 𝒅𝒖𝒆 𝒕𝒐 𝒘𝒂𝒍𝒍 𝒇𝒓𝒊𝒄𝒕𝒊𝒐𝒏

𝑨𝒔 = 𝒔𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒗𝒆𝒓 𝒘𝒉𝒊𝒄𝒉 𝒕𝒉𝒆 𝒇𝒓𝒊𝒄𝒕𝒊𝒐𝒏 𝒂𝒄𝒕𝒔

𝑨 = 𝑪𝒓𝒐𝒔𝒔 − 𝒔𝒆𝒄𝒕𝒊𝒐𝒏𝒂𝒍 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒕𝒉𝒆 𝒅𝒖𝒄𝒕

Now, we apply momentum equation for steady flow shown below.

∑ 𝑭𝒙 = ∬ 𝑽𝒙 (𝝆𝑽 ∙ 𝒅𝑨)

𝒑𝑨 − (𝒑 + 𝒅𝒑) 𝑨 − 𝝉𝒇 𝑨𝒔 = (𝝆𝑨𝑽)(𝑽 + 𝒅𝑽) − (𝝆𝑨𝑽)𝑽

−𝑨𝒅𝒑 − 𝝉𝒇 𝑨𝒔 = 𝝆𝑨𝑽𝒅𝑽
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𝑪𝒓𝒐𝒔𝒔−𝑺𝒆𝒄𝒕𝒊𝒐𝒏𝒂𝒍 𝑨𝒓𝒆𝒂 𝟒𝑨
Now, we define hydraulic diameter as, 𝑫𝒉 = 𝟒 =
𝑾𝒆𝒕𝒕𝒆𝒅 𝑷𝒆𝒓𝒊𝒎𝒆𝒕𝒆𝒓 𝑷

Since the area over which the friction force is acting is 𝐴𝑠 and is equal to the perimeter of the
duct times the incremental length dx, it can be replaced by hydraulic diameter to obtain the
equation in following form.

𝟒𝑨
−𝑨𝒅𝒑 − 𝝉𝒇 (𝒅𝒙) = 𝝆𝑨𝑽𝒅𝑽
𝑫𝒉
𝟒𝝉 𝒇
Now we define a friction coefficient, 𝒇 = 𝝆𝑽𝟐
, and the above equation is modified and presented
𝟐
below.

𝝆𝑽𝟐 𝒅𝒙
−𝒅𝒑 − 𝒇 = 𝝆𝑨𝑽𝒅𝑽 , Now this equation is divided by p
𝟐 𝑫

𝒅𝒑 𝜸𝑴𝟐 𝒇 𝒅𝒙 𝒅𝑽
+ + 𝜸𝑴𝟐 =𝟎
𝒑 𝟐 𝑫 𝑽
𝑑𝑉 𝑑𝑝
An expression for M in terms of x can be calculated by replacing and from above equation
𝑉 𝑝
using the continuity equation, perfect gas-law and the definition of Mach number.

The fanno flow curve is shown below in the figure and its features are discussed.

P
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The portion of the curve above the point P, represents the subsonic flow, and the portion below
the point P represents the supersonic flow.

Consider the subsonic portion of the fanno curve, due to the presence of friction the flow is
irreversible and hence ds>0. Hence the entropy increases in the flow direction. So, for a given
mass flow rate, state of the flow towards right on the curve, corresponding to entropy rise. It is
concluded that for the subsonic flow with friction, Mach number increases till unity. Similarly, for
supersonic flow, entropy must increase which leads to the decrease in supersonic Mach number
till unity (movement along the curve towards right).

But it is to be understood that how friction is accelerating the subsonic flow and decelerating the
supersonic flow in a constant area duct. The cause of the friction is the shear stress acting on the
wall. Due to friction, the wall layer grows thicker in the downstream direction, and hence the
growing thickness of wall layer results in the reduction in effective passage area to pass a given
𝑑𝐴
amount of mass flow rate. This reduction in effective flow passage area ( < 0) makes the flow
𝑑𝑥
area converging. It is known that the converging area accelerates the subsonic flow and decelerates
the supersonic flow. Hence, the effect of friction on accelerating the subsonic flow and decelerating
the supersonic flow is due to the effective flow area reduction.

If the length of the duct is sufficiently long for a flow which is initially subsonic to reach M=1,
and then an additional length is added. So, entropy will be ∆𝑠 = ∆𝑠1−𝑠𝑜𝑛𝑖𝑐 + ∆𝑠𝑒𝑥𝑡𝑟𝑎. The flow
Mach number will exceed unity without decreasing the entropy (∆𝑠𝑒𝑥𝑡𝑟𝑎 < 0). But, from second
law of thermodynamics, it is known that this is not possible for an adiabatic flow. Hence the
additional length will cause the mass flow rate to reduce and the flow will jump on to the next
fanno flow curve corresponding to the reduced mass flow rate.

The duct is choked due to the friction for the case explained above. Hence, for a given subsonic
inlet Mach number, there is a certain maximum duct length ( 𝑳𝒎𝒂𝒙 ) beyond which flow rate
reduction will occur.

If the inlet Mach number is supersonic, and the duct length is made larger than 𝑳𝒎𝒂𝒙 to reach Mach
number 1, then the supersonic flow will not be able to sense the changes in duct length occurring
ahead of it and the flow will adjust to the changes in duct length via normal shock rather than the
flow reduction. The location of the normal shock in the duct is decided by the changes in back
pressure imposed on the duct exit.

All the above points discussed are shown below in the figure.
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Properties variation for Fanno flow

Property Subsonic Supersonic

Velocity Increase Decrease

Mach No Increase Decrease

Static enthalpy (T) Decrease Increase

Stagnation enthalpy Constant Constant

Pressure Decrease Increase

Density Decrease Increase

Stagnation Pressure Decrease Decrease

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Relation for fanno flow

At the given Mach no an either subsonic or supersonic branch of the fanno flow, there is a

specific value of the duct length 𝐿𝑚𝑎𝑥 downstream of this section, which will change this Mach
no to unity. For a given duct, this length is an explicit function of the Mach no.
𝜸+𝟏
𝒇𝑳𝒎𝒂𝒙 𝜸+𝟏 𝟏 𝟏 𝜸+𝟏 𝟏
=( ) 𝐥𝐧 ( 𝟐
𝜸−𝟏 ) − (𝟏 −
𝑴𝟐
)− ( ) 𝐥𝐧 (
𝑴𝟐
)
𝑫 𝟐 𝟏+ 𝑴𝟐 𝜸 𝟐𝜸
𝟐

f = Friction factor

D = hydraulic diameter

Flow through a Nozzle and Constant-Area Duct

In real life, such situations are encountered wherein a constant-area duct is fed by the nozzle. In
such cases, duct back pressure and nozzle stagnation pressure are the known quantities. Consider
such case discussed below.

The flow is provided by the reservoir which goes in to the converging nozzle and then feeds the
constant-area duct shown below in the figure. The pressure distribution for isentropic flow in
nozzle and a fanno flow in the duct, versus distance x is shown below for different back pressures.

The reservoir pressure 𝑃𝑟 is maintained constant.

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As 𝑃𝐵 is lowered below 𝑃𝑟 , the flow starts and the pressure decreases in both nozzle and duct, and
the flow is subsonic. This condition is shown in the figure above by curve (a) and (b). As 𝑃𝐵 is
further lowered and is equal to that corresponding to curve (c) shown in the figure above, the flow
in the duct exit becomes sonic. At this flow condition, Mach number at the exit of the duct is unity.

When Mach number at the duct exit becomes 1, then further reduction in the back pressure will
not be sensed by the reservoir and for all back pressures below that corresponding to curve (c) the
flow is under expanded. The under expanded flow condition corresponds to the back pressure
corresponding to the curve (d) shown in the figure above. The mass flow rate for all back pressure
equal to and less than that of curve (c) remains constant and the system is choked by the duct and
not by the converging nozzle. The maximum mass flow rate that can be passed by the system
shown above is less for the same reservoir pressure than that for a converging nozzle with no duct.
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Crocco’s Equation
This equation is a connection between flow kinematics and flow thermodynamics. This is
demonstrated below by starting from Gibb’s identity.

Consider a fluid element moving through a flow field. The motion of fluid involves both
translational and rotational motion. The translation motion is characterized by velocity V and
rotational motion is characterized by the angular velocity 𝝎. It is known that the curl of velocity
which represents the vorticity of the fluid, is given by 𝜵 × 𝑽 = 𝟐𝝎.

Now, consider the Euler’s equation (without body force and viscous forces) shown below.

𝑫𝑽
𝝆 = −𝜵𝒑
𝑫𝒕
𝝏𝑽
After expanding the substantial derivative in the equation above, 𝝆 + 𝝆(𝛁 ∙ 𝑽)𝑽 = −𝛁𝒑
𝝏𝒕

𝛁𝒑
Now consider the Gibb’s equation from thermodynamics, 𝑻𝛁𝒔 = 𝛁𝒉 −
𝝆

𝟏 𝝏𝑽
Combining above two expressions, we get, 𝑻𝛁𝒔 = 𝛁𝒉 − [−𝝆 − 𝝆(𝛁 ∙ 𝑽)𝑽]
𝝆 𝝏𝒕

𝝏𝑽
𝑻𝛁𝒔 = 𝛁𝒉 + + (𝛁 ∙ 𝑽)𝑽 (1)
𝝏𝒕

𝑽𝟐
Also, 𝒉𝒐 = 𝒉 +
𝟐

𝜵( 𝑽∙𝑽)
𝜵𝒉𝒐 = 𝛁𝒉 + (2)
𝟐

Substitute eq (2) in eq (1)

𝝏𝑽
𝑻𝛁𝒔 = 𝛁𝒉𝒐 − 𝛁 × (𝛁 × 𝑽) +
𝝏𝒕

Crocco’s equation for steady flow- 𝑻𝛁𝒔 = 𝛁𝒉𝒐 − 𝛁 × (𝛁 × 𝑽)

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The above Crocco’s equation can also be written and rearranged as:

∇ × (∇ × 𝑉) = ∇ℎ 𝑜 − 𝑇∇𝑠

Vorticity Total Enthalpy Gradient of

Gradient Entropy

The above equation physically signifies that when a steady -flow has gradients of total enthalpy
and/or total entropy, then the flow is rotational. The practical experience is in the flow field behind
the curved shock wave. In the region ahead of the curved shock, all streamlines in the uniform
freestream have the same total enthalpy. Across the stationary shock wave total enthalpy does not
change so all the streamlines behind the curved shock wave have same total enthalpy, 𝜵𝒉𝒐 = 𝟎.
In the region of the curved shock wave where flow is normal to the shock wave and also in the
region where curved portion is strong shock wave, flow experiences very high gradients of entropy
and so, 𝜵𝒔 ≠ 𝟎. As a result, it is concluded from the above Crocco’s equation, that 𝑽 × (𝜵 × 𝑽) ≠
𝟎, and the flow is rotational.
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Flow Measurements in Compressible Flow

Pressure Measurements

Two types of pressure are known to us namely: static pressure and total pressure

Static pressure is the pressure which is measured by the measuring device which is moving at the
flow velocity so that no disturbance/velocity change is offered to the flow by the device, and the
device sees the freestream flow. Generally, static pressure along a wall is measured by drilling a
small hole normal to the wall surface and this hole is connected to a manometer, or pressure gauge
or similar measuring device. This is shown in the figure below.

The size of the hole must be small with respect to the boundary layer thickness (around 20 percent
of the local laminar boundary layer). The hole should also be free from roughness and burrs, that
can disturb the flow and result in in accurate freestream pressure measurement. It is known that
the change in pressure in the direction normal to the boundary layer is 0, so the pressure measured
in the wall is a true measure of the freestream pressure.

If no wall is present in the flow, static pressure can be measured in the flow by introducing a probe
which will act as a wall. Such a probe should be very thin and aligned with the flow direction in
order to have minimum disturbance to the flow.

The nose of the probe can be round, wedge shaped or conical shaped and the body of the probe is
generally cylindrical. Since, the pressure varies along the surface of the probe, the holes (pressure
tap) are located at a distance of 5-10 times the probe diameter for subsonic flow and 10-15 times
the probe dimeter for supersonic flow. This location is fixed considering that the surface pressure
recovers to the freestream static pressure.

The figure below shows the static pressure probe for subsonic and supersonic flow respectively.
A shock will be formed at the nose and expansion wave at the beginning of the cylindrical part.
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However, the static pressure probe is very sensitive to the flow alignment; and the more the probe
is misaligned, the greater will be the disturbance to the flow by the probe. This sensitivity can be
reduced by drilling several holes (generally, eight equally spaced holes are used) around the
circumference of the probe, so that the resultant pressure is an average pressure.

Stagnation Pressure is the pressure measured by the instrument which brings the flow to rest via
isentropic process. The instrument is called the Pitot Tube, which is a tube with its open end facing
directly into the flow. This instrument was invented by Henri Pitot and is shown in the figure
below.

The Pitot tube consists of three components: (1) instrument tip; (2) the body, which includes the
stem and the hardware to fasten the instrument to wall as well as the lines that connect the tip to a
measuring device; (3) the measuring device. The accuracy of the measurement of the total pressure
is dependent on the size of the tube. The stagnation pressure probe is not that sensitive to the
misalignment of the probe to the flow.
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The stagnation pressure probe and static pressure probe can be joined to form a Pitot-Static probe.
In this tube, the instrument contains a tube within a tube. The inner tube measures the stagnation
pressure, while the outer tube measures the static pressure. The figure below shows the pitot-static
probe.

To reduce the sensitivity to the flow angularity (yaw and pitch angles), a shield is placed round the
total-pressure tube. This type of probe is called Kiel tube. This tube can tolerate flow angle
variations as large as ±45 degrees. This tube is used to measure stagnation pressures in areas with
high total-pressure gradients, such as in boundary layer or turbomachinery flows. Moreover, a
probe can be of traversing mechanism to have a full survey of various regions of the flow.

Kiel Total-
Pressure
Probe
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For supersonic flows, we have a bow shock ahead of the blunt shaped nose of the total pressure
tube. So, the stagnation pressure measured is the stagnation pressure downstream of the portion of
the shock that is normal to the flow direction.

It is generally assumed that the total pressure downstream of shock is same as the pressure on the
probe tip. If the Reynolds number based on the probe tip is too low then this assumption is not
valid, hence it is assumed that Reynolds number is large enough.

𝑷𝒐𝟐 𝑷𝟐
𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 = 𝑷𝒐𝟐 = ( )( )𝑷
𝑷𝟐 𝑷𝟏 𝟏

Where 𝑷𝟏 is the freestream static pressure 𝑷∞ .

𝜸
𝜸 𝟐
𝑷 𝜸 −𝟏 𝟐 𝜸−𝟏 𝜸 −𝟏 + 𝑴𝟐𝟏 𝜸−𝟏
𝜸−𝟏
( 𝟎𝟐 ) = (𝟏 + 𝑴 ) = [𝟏 + ( ) ( 𝟐𝜸 )]
𝑷𝟐 𝟐 𝟐 𝑴𝟐𝟏 − 𝟏
𝜸−𝟏

𝑷𝟐 𝟐𝜸
= 𝟏+ (𝑴𝟐𝟏 − 𝟏)
𝑷𝟏 𝜸+𝟏
𝜸
𝑷𝒐𝟐 𝜸+ 𝟏𝑴𝟐𝟏 𝜸−𝟏 𝟏 − 𝜸 + 𝟐𝜸𝑴𝟐𝟏 Rayleigh Pitot-
=( ) ( )
𝑷𝟏 𝟒𝜸𝑴𝟐𝟏 − 𝟐(𝜸 − 𝟏) 𝜸+𝟏 Tube formula
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Temperature Measurement

The static temperature of a gas flow cannot be measured directly, because the direct measurement
would have to be done by using a device which will not disturb the flow. To make this happen, the
instrument has to travel at a velocity of the flow, which is impractical.

The temperature across the boundary layer varies due to viscous effects so static temperature can
not be measured by locating a thermocouple and thermometric device at the wall surface. Hence a
thermometer when inserted in to the gas flow will read a temperature above freestream temperature
T∞.

In the absence of a wall, stagnation temperature 𝑻𝒐∞ is measured by stagnation temperature probe.
For supersonic flow, measurement of stagnation temperature is unaffected by the presence of
detached shock in front of the probe, since the flow across shock is adiabatic. The stagnation
temperature probe in a supersonic flow is shown below in the figure.

The flow comes to rest within the tube. Vents are provided in the sides of the tube to permit proper
ventilation of the space inside the probe. It is important that the gas is brought to rest at the
thermocouple without any gain or loss of heat. Shields are provided to prevent the loss of heat by
radiation from the thermocouple.

Velocity Measurement

The velocity of a moving gas can be measured by both mechanical or laser/optical probes. Some
are listed below

(a) Pitot-Static tube (c) Laser Doppler Anemometers (LDA)

(b) Hot-Wire Anemometer (d) Particle Image Velocimetry (PIV)

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Pitot-static tube has been discussed earlier. SO, discussions on HWA, LDA and PIV will be taken
up.

Hot-Wire Anemometer (HWA): it is used when detailed, high frequency or high-resolution flow-
field measurements are required. It is also used to measure mean and fluctuating velocity
components, mean and fluctuating temperatures and vorticity in a turbulent flow. It contains a thin
wire which is electrically heated and is located in the flow stream. The hot wire probe is shown in
the figure below.

The wire is made of tungsten, platinum, or platinum-iridium and the diameter of the sensing
element is 0.5-5 microns with a length of 1-2 mm. The wire is attached to the tips of two support
needles. The two electric connectors are located at the other end of the body of the probe, opposite
to the hot wire. The configuration is shown in the figure above.

Laser Doppler Velocimetry (LDV): The fundamental principle involved in LDV is the Doppler
Shift of the light which is scattered from small particles moving with the fluid. The seed particles
are of the size varying from 0.1-10 microns. The LDV arrangement is shown in the figure below
in which the particles have velocity 𝑉⃗ and a sensor situated at a distance 𝑟𝑑 from a fixed coordinate
system Oxyz measures the shift. The position of the particles relative to the fixed laser source is
given by vector 𝑟1 .
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For a laser of wavelength 𝜆 and frequency of illuminating light 𝜗𝑜 , the frequency of the scattered
light detected at location 𝑟𝑑 is given by:

⃗ ∙ (𝒊𝟏 − 𝒊𝒅 )
𝑽
𝝑 = 𝝑𝒐 +
𝝀

In the above equation, 𝒊𝟏 and 𝒊𝒅 are the unit vectors of 𝒓⃗ 𝟏 and 𝒓⃗ 𝒅 respectively. The doppler
frequency shift depends on linearly on the velocity component in the direction of ( 𝒊𝟏 − 𝒊𝒅 ) and so
by measuring ∆𝑣 component of velocity along the external bisector of the angle formed between
the illumination and observation directions.

Particle Image Velocimetry (PIV): The fundamental principle of PIV method is the calculation of
velocity from a sequence of photographs of seeded and illuminated flow field. PIVs have been
widely used in wide variety of flows, including wind tunnel flows, supersonic jet expansions, two
phase flows, rotating and turbulent flows etc. The basic setup of PIV system is shown below.

A laser is focused onto an optical device, such as cylindrical lens, which creates laser sheet. The
seeded flow field is then illuminated by the sheet if light and the illumination is then photographed.
The plane of photograph is parallel to light sheet. Photographs are taken by computer driven frame
grabber systems. After a short time-interval ∆𝑡, a second image of the flow field is then recorded.
The images are then recorded and displacements ∆𝑥 of the particles are then computed. The
∆𝒙
velocity is computed from 𝒖 = . This process is shown below in the figure.
∆𝒕
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Optical Methods of Measurements

The optical methods depend on the density variation or its derivatives throughout the flow field.
Two optical methods are famous in aerospace applications- Schlieren and Shadowgraph. These
two methods are used to see phenomena such as shock waves, temperature gradients, and natural
convection patterns in liquids without introducing any object or disturbance in the flow.

If the density variation is uniform in the test-section then an entering light ray will not be deflected;
𝜕𝜌
if the density gradient is uniform across the height in the test section, then all rays passing
𝜕𝑦
𝜕2𝜌
through the section will be deflected through same angle; if the curvature of the density is
𝜕2𝑦
uniform across the height of the test-section then the entering rays can interfere with each other.
The relative light intensity sensed on the screen is proportional to the distance between the rays.
The above explanations are shown in the figure below. Consider light rays passing through a
medium then following quantities can be recorded at a screen:

(1) deflection angles of the light ray, 𝜀𝑥 and 𝜀𝑦, measured by a schlieren system

(2) displacements ∆𝑥 and ∆𝑦 of the deflected light ray with respect to an undisturbed light beam,
as measured by shadowgraph instrument.
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Schlieren System
One can visualize the density variations within a transparent medium. The visualization is done by
sensing the refraction of light rays as they pass through a region with density gradient variation.
The schlieren system has been used in supersonic systems. Its applications are:

(1) This system is effective in visualizing the shocks and expansion waves which do not possess
large density gradients.

(2) In subsonic flows, schlieren is used to visualize natural convection. Here. Changes in density
gradient are due to temperature difference.

A schlieren system with point light source is shown below in the figure and discussed. The first
lens L1 in the figure shown below, whose focal point coincides with the location of source makes
the light rays coming from a point source parallel. These light rays continue being parallel through
the test section. These light rays after passing through the test section, are refocused by lens L2.
After passing through the second lens the light is then projected to the screen and an inverted image
of the test section is provided.

The edge of knife K is placed at the focal plane of L 2. The knife can be lowered or raised or rotated.
In the vertical position, if knife is raised too much then light will be obstructed, and the screen will
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be dark. In this way, the image of the test section of the screen will be darker than it would have
been without the knife.

If a gas is flowing through the test section such that there exists a density disturbance at the center
of the test section. The transverse density gradients cause the light rays to be bent from their
original paths and the second lens will focus the rays from each point in the test section to a
corresponding point on the screen. This is just one version of schlieren system and there are other
versions also. One such version is schlieren with mirrors shown in the figure below.
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Shadowgraph Instrument
The shadowgraph technique is used for the flows which are characterized by large and sudden
changes in density, such as flows across strong shock waves. This technique is the cheapest and
simplest method. The basic system is shown in the figure below.

X scr
The lens collimates the light rays coming from the light source. The light rays are then parallel and
these parallel rays pass through the test-section and illuminate the screen. When there is no flow
through the test-section then the screen is uniformly illuminated. When there is a flow through the
test-section then the light rays will be deflected because of the density gradient, and hence dark
and light spots will appear on the screen. These dark and light spots will create shadow patterns
and therefore the image on the screen is shadowgraph.

∆𝑰 𝝏𝟐 𝝆
= −𝑲𝒙𝒔𝒄𝒓 ∫ 𝟐 𝒅𝒙
𝑰𝒕𝒔 𝝏𝒚

∆𝑰= 𝑰𝒔𝒄𝒓 − 𝑰 𝒕𝒔, 𝑰𝒔𝒄𝒓=Intensity at screen, 𝑰𝒕𝒔=Intensity at disturbance

Hence, the screen illumination is proportional to second derivative of the density. By using a
shadowgraph, a shock wave can be seen on the screen as a dark region followed by bright region.
The distribution of density and first and second order derivative of density across shock wave is
shown below in the figure.
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A shadowgraph with a mirror and light source is also used, which is shown below in the figure
below. A shadowgraph plot of shock wave around body is also shown below.
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Numerical Problems
(1). Air flow through a duct. The pressure and temperature at station (1) are atmosphere

and respectively at a second station, the pressure is .Calculate the

temperature and density at second station.

Sol:-
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(2). A ramjet flies with Mach number of 0.9.in diffuser the air is bought to the stagnation
condition so that it is stationary in combustion chamber combustion takes place at constant
pressure and a temperature increase of . The combustion product are then ejected

through nozzle.

(a). Calculate stagnation pressure and temperature

(b). what will be nozzle exit velocity

Sol:-

Hence,

Temperature after combustion

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(3). The pressure, Temperature, Mach No at the entry of a flow passage are , ,

and 1.4. if exit Mach No is 2.5 . Calculate

(1). T,V of gas at exit

(2). Flow rate per square meter of the inlet cross section

(3). Stagnation temperature

Sol:-
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(4). Air enters a axis symmetric duct at

and leaver of at .

The area of cross section flow calculate

(1). Stagnation temperature

(2). Maximum velocity

(3). Mass flow rate

(4). Area of cross section of exit

Sol:-

(ii).

(iii).
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(5). An aircraft flies at at an altitude of , .

The air is isentropic compressed in an inlet diffuser. If the Mach No at the exit of the diffuser is
determine

(a). Entry Mach Number

(b). Velocity, Pressure, Temperature of the diffuser exit

Sol:-

(b). At
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(6). A converging divergimg nozzle is designed to operate with an exit Mach Number of 1.75.
The nozzle is supplied from an reservoir at

Assuming 1-D flow calculate

(a). Maximum back pressure to choke the nozzle

(b). Range of back pressure over which a normal shock will appear in the nozzle

(c). Back pressure for the nozzle to be perfectly expanded to the design Mach Number

(d). Range of back pressure for supersonic flow at the nozzle exit flow
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Sol:-

Design

From isentropic table at ,

(a). Nozzle is checked with at the throat following by subsonic flow in the diverging

portion of the nozzle.

At looking for subsonic solution

Hence,

Nozzle is checked for all bank pressure above this

(2). For normal shock at nozzle exit plate

From isentropic table

For
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For normal shock

At

For normal shock just downstream of nozzle throat the bank pressure is

This Normal Shock will appear in nozzle for range of

(c). from isentropic table at

(d). the nozzle will pressure supersonic flow for all back pressure below

GATE Qus (2016)

Qus (5). Which is not true, across an oblique shock wave

(a).
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(b).

(c).

(d),

Qus(6). For a completely subsonic flow through nozzle which is true

(a).

(b).

(c).

(d).

Qus (30)

1-D,inviscid flow
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Qus (39)

1-D, steady, inviscid compressible flow through convergent nozzle

Flow 1 > chocked with

Now if to increased , with remain same then what is mass flow rate through nozzle

Sol:-

Ans : (B)

GATE Qus (2013)

Qus (33)
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1-D, isentropic flow

for
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GATE Qus (2014)

Qus (33)

Sol:-

From momentum continuity

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