Compressible flow

COMPRESSIBLE FLOW
B.GOVINDHARAJAN
J.PARTHIBAN
ERODE SENGUNTHAR ENGINEERING COLLEGE

2
Objectives
• Develop the general relations for compressible flows encountered when
gases flow at high speeds.
• Introduce the concepts of stagnation state, speed of sound, and Mach
number for a compressible fluid.
• Develop the relationships between the static and stagnation fluid properties
for isentropic flows of ideal gases.
• Derive the relationships between the static and stagnation fluid properties as
functions of specific-heat ratios and Mach number.
• Derive the effects of area changes for one-dimensional isentropic subsonic
and supersonic flows.
• Solve problems of isentropic flow through converging and converging–
diverging nozzles.
• Discuss the shock wave and the variation of flow properties across the
shock wave.
• Develop the concept of duct flow with heat transfer and negligible friction
known as Rayleigh flow.
• Examine the operation of steam nozzles commonly used in steam turbines.

3
STAGNATION PROPERTIES
Aircraft and jet engines involve high
speeds, and thus the kinetic energy
term should always be considered
when analyzing them.
Stagnation (or total) enthalpy
Static enthalpy: the ordinary enthalpy h
Steady flow of a fluid through an
adiabatic duct.
Energy balance (with no heat or work
interaction, no change in potential energy):

4
If the fluid were brought to a complete stop, the energy balance becomes
Stagnation enthalpy: The enthalpy of a
fluid when it is brought to rest adiabatically.
During a stagnation process,
the kinetic energy of a fluid is
converted to enthalpy, which
results in an increase in the
fluid temperature and
pressure.
The properties of a fluid at
the stagnation state are
called stagnation
properties (stagnation
temperature, stagnation
pressure, stagnation density,
etc.).
The stagnation state is
indicated by the subscript 0.
Kinetic energy is converted to enthalpy during a
stagnation process.

5
The actual state, actual stagnation
state, and isentropic stagnation
state of a fluid on an h-s diagram.
Isentropic stagnation state: When the
stagnation process is reversible as well as
adiabatic (i.e., isentropic).
The stagnation processes are often
approximated to be isentropic, and the
isentropic stagnation properties are simply
referred to as stagnation properties.
When the fluid is approximated as an ideal
gas with constant specific heats
T0 is called the stagnation (or total)
temperature, and it represents the
temperature an ideal gas attains when it
is brought to rest adiabatically.
The term V2
/2cp corresponds to the
temperature rise during such a process
and is called the dynamic temperature.

6
The pressure a fluid attains when
brought to rest isentropically is called
the stagnation pressure P0.
Stagnation density ρ0
When stagnation enthalpies are
used, the energy balance for a
single-stream, steady-flow device
When the fluid is approximated as an ideal
gas with constant specific heats
The temperature of an ideal
gas flowing at a velocity V rises
by V2
/2cp when it is brought to a
complete stop.

7
SPEED OF SOUND AND MACH NUMBER
Speed of sound (or the sonic speed):
The speed at which an infinitesimally
small pressure wave travels through a
medium.
To obtain a relation for the speed of
sound in a medium, the systems in the
figures are considered.
Propagation of a small
pressure wave along a duct.
Control volume moving with the small
pressure wave along a duct.

8
The Mach number can be different at different
temperatures even if the velocity is the same.
The speed of sound changes with
temperature and varies with the
fluid.
Ma = 1 Sonic flow
Ma < 1 Subsonic flow
Ma > 1 Supersonic flow
Ma >> 1 Hypersonic flow
Ma ≅ 1 Transonic flow
For an ideal gas
Mach
number
Speed of
sound

9
ONE-DIMENSIONAL ISENTROPIC FLOW
During fluid flow through many devices such as nozzles, diffusers, and turbine
blade passages, flow quantities vary primarily in the flow direction only, and the
flow can be approximated as one-dimensional isentropic flow with good accuracy.
EXAMPLE
Variation of normalized fluid properties and
cross-sectional area along a duct as the
pressure drops from 1400 to 200 kPa.
A converging–diverging nozzle.

10
Variation of
normalized fluid
properties and cross-
sectional area along a
duct as the pressure
drops from 1400 to
200 kPa.

11
The cross section of a nozzle
at the smallest flow area is
called the throat.
We note from Example that the flow area
decreases with decreasing pressure up to a
critical-pressure value (Ma = 1), and then it
begins to increase with further reductions in
pressure.
The Mach number is unity at the location of
smallest flow area, called the throat.
The velocity of the fluid keeps increasing after
passing the throat although the flow area
increases rapidly in that region.
This increase in velocity past the throat is due
to the rapid decrease in the fluid density.
The flow area of the duct considered in this
example first decreases and then increases.
Such ducts are called converging–diverging
nozzles.
These nozzles are used to accelerate gases
to supersonic speeds and should not be
confused with Venturi nozzles, which are used
strictly for incompressible flow.

12
Variation of Fluid
Velocity with Flow Area
In this section, the relations for the
variation of static-to-stagnation property
ratios with the Mach number for
pressure, temperature, and density are
provided.
Derivation of the differential form
of the energy equation for steady
isentropic flow.
This relation describes the variation of
pressure with flow area.
At subsonic velocities, the pressure
decreases in converging ducts
(subsonic nozzles) and increases in
diverging ducts (subsonic diffusers).
At supersonic velocities, the
pressure decreases in diverging ducts
(supersonic nozzles) and increases in
converging ducts (supersonic
diffusers).

13
We cannot obtain supersonic
velocities by attaching a converging
section to a converging nozzle. Doing
so will only move the sonic cross
section farther downstream and
decrease the mass flow rate.
This equation governs the shape of a nozzle
or a diffuser in subsonic or supersonic
isentropic flow.
The proper shape of a nozzle depends on
the highest velocity desired relative to the
sonic velocity.
To accelerate a fluid, we must use a
converging nozzle at subsonic velocities and
a diverging nozzle at supersonic velocities.
To accelerate a fluid to supersonic
velocities, we must use a converging–
diverging nozzle.

14
Variation of flow properties in subsonic and supersonic
nozzles and diffusers.

15
Property Relations for Isentropic Flow of Ideal Gases
Critical
ratios
(Ma=1)
When Mat = 1, the properties at the nozzle
throat become the critical properties.
The relations between the static
properties and stagnation
properties of an ideal gas with
constant specific heats

17
ISENTROPIC FLOW THROUGH NOZZLES
Converging or converging–diverging
nozzles are found in steam and gas
turbines and aircraft and spacecraft
propulsion systems.
In this section we consider the effects of
back pressure (i.e., the pressure
applied at the nozzle discharge region)
on the exit velocity, the mass flow rate,
and the pressure distribution along the
nozzle.
Converging Nozzles
The effect of back pressure on the pressure
distribution along a converging nozzle.
Mass flow rate through a nozzle
Maximum mass flow rate

18
The effect of back pressure Pb on the
mass flow rate and the exit pressure Pe
of a converging nozzle.
The variation of the mass flow rate through
a nozzle with inlet stagnation properties.

19
Various property
ratios for isentropic
flow through nozzles
and diffusers are
listed in Table A–32
for k = 1.4 for
convenience.
Ma* is the local
velocity
nondimensionalized
with respect to the
sonic velocity at the
throat.
Ma is the local
velocity
nondimensionalized
with respect to the
local sonic velocity.

20
Converging–Diverging Nozzles
The highest velocity in a converging nozzle is limited to the sonic velocity
(Ma = 1), which occurs at the exit plane (throat) of the nozzle.
Accelerating a fluid to supersonic velocities (Ma > 1) can be accomplished
only by attaching a diverging flow section to the subsonic nozzle at the throat
(a converging–diverging nozzle), which is standard equipment in supersonic
aircraft and rocket propulsion.
Converging–
diverging nozzles
are commonly
used in rocket
engines to
provide high
thrust.

21
The effects of back
pressure on the flow
through a converging–
diverging nozzle.
When Pb = P0 (case A), there will be no
flow through the nozzle.
1. When P0 > Pb > PC, the flow remains
subsonic throughout the nozzle, and the
mass flow is less than that for choked
flow. The fluid velocity increases in the
first (converging) section and reaches a
maximum at the throat (but Ma < 1).
However, most of the gain in velocity is
lost in the second (diverging) section of
the nozzle, which acts as a diffuser. The
pressure decreases in the converging
section, reaches a minimum at the
throat, and increases at the expense of
velocity in the diverging section.

22
2. When Pb = PC, the throat pressure
becomes P* and the fluid achieves sonic
velocity at the throat. But the diverging
section of the nozzle still acts as a
diffuser, slowing the fluid to subsonic
velocities. The mass flow rate that was
increasing with decreasing Pb also
reaches its maximum value.
3. When PC > Pb > PE, the fluid that
achieved a sonic velocity at the throat
continues accelerating to supersonic
velocities in the diverging section as the
pressure decreases. This acceleration
comes to a sudden stop, however, as a
normal shock develops at a section
between the throat and the exit plane,
which causes a sudden drop in velocity to
subsonic levels and a sudden increase in
pressure. The fluid then continues to
decelerate further in the remaining part of
the converging–diverging nozzle.

23
4. When PE > Pb > 0, the flow in the
diverging section is supersonic, and the
fluid expands to PF at the nozzle exit
with no normal shock forming within the
nozzle. Thus, the flow through the
nozzle can be approximated as
isentropic.
When Pb = PF, no shocks occur within
or outside the nozzle.
When Pb < PF, irreversible mixing and
expansion waves occur downstream of
the exit plane of the nozzle.
When Pb > PF, however, the pressure
of the fluid increases from PF to Pb
irreversibly in the wake of the nozzle
exit, creating what are called oblique
shocks.

24
SHOCK WAVES AND EXPANSION WAVES
For some back pressure values, abrupt changes in fluid properties occur in a very
thin section of a converging–diverging nozzle under supersonic flow conditions,
creating a shock wave.
We study the conditions under which shock waves develop and how they affect
the flow.
Normal Shocks
Normal shock waves: The shock
waves that occur in a plane normal
to the direction of flow. The flow
process through the shock wave is
highly irreversible.
Schlieren image of a normal shock in a
Laval nozzle. The Mach number in the
nozzle just upstream (to the left) of the
shock wave is about 1.3. Boundary layers
distort the shape of the normal shock near
the walls and lead to flow separation
beneath the shock.

25
Conservation of mass
Conservation of energy
Conservation of momentum
Increase of entropy
Fanno line: Combining the conservation of
mass and energy relations into a single
equation and plotting it on an h-s diagram
yield a curve. It is the locus of states that
have the same value of stagnation
enthalpy and mass flux.
Rayleigh line: Combining the conservation
of mass and momentum equations into a
single equation and plotting it on the h-s
diagram yield a curve.
The h-s diagram for flow
across a normal shock.
Control volume for flow across a
normal shock wave.

26
Variation of flow
properties
across a normal
shock.
The relations between various properties before and after the
shock for an ideal gas with constant specific heats.
Various flow property ratios across
the shock are listed in Table A–33.
This represents the intersections
of the Fanno and Rayleigh lines.

28
Since the flow across the shock is
adiabatic and irreversible, the
second law requires that the
entropy increase across the
shock wave.
Thus, a shock wave cannot exist
for values of Ma1 less than unity
where the entropy change would
be negative.
For adiabatic flows, shock waves
can exist only for supersonic
flows, Ma1 > 1.

30
Oblique Shocks
When the space shuttle travels at supersonic speeds through the atmosphere, it
produces a complicated shock pattern consisting of inclined shock waves called
oblique shocks.
Some portions of an oblique shock are curved, while other portions are straight.
Schlieren image of a
small model of the
space shuttle Orbiter
being tested at Mach 3
in the supersonic wind
tunnel of the Penn State
Gas Dynamics Lab.
Several oblique shocks
are seen in the air
surrounding the
spacecraft.

31
Velocity vectors through an oblique
shock of shock angle β and
deflection angle θ.
Unlike normal shocks, in which the
downstream Mach number is always
subsonic, Ma2 downstream of an oblique
shock can be subsonic, sonic, or
supersonic, depending on the upstream
Mach number Ma1 and the turning angle.
An oblique shock of shock angle
(wave angle) β formed by a
slender, two-dimensional wedge
of half-angle δ. The flow is turned
by deflection angle (turning
angle) θ downstream of the
shock, and the Mach number
decreases.

32
The same velocity vectors of Fig.
17–38, but rotated by angle π/2–β,
so that the oblique shock is vertical.
Normal Mach numbers Ma1,n and
Ma2,n are also defined.
Relationships across an oblique shock for an
ideal gas in terms of the normal component of
upstream Mach number Ma1,n.
All the equations, shock tables, etc., for normal shocks
apply to oblique shocks as well, provided that we use
only the normal components of the Mach number.

33
The dependence of straight oblique shock deflection angle θ on shock angle β
for several values of upstream Mach number Ma1. Calculations are for an ideal
gas with k = 1.4. The dashed black line connects points of maximum deflection
angle (θ = θ max). Weak oblique shocks are to the left of this line, while strong
oblique shocks are to the right of this line. The dashed gray line connects
points where the downstream Mach number is sonic (Ma2 = 1). Supersonic
downstream flow (Ma2 > 1) is to the left of this line, while subsonic downstream
flow (Ma2 < 1) is to the right of this line.

34
A detached oblique shock occurs upstream of a
two-dimensional wedge of half-angle δ when δ
is greater than the maximum possible deflection
angle θ. A shock of this kind is called a bow
wave because of its resemblance to the water
wave that forms at the bow of a ship.
Still frames from schlieren videography
illustrating the detachment of an oblique shock
from a cone with increasing cone half-angle δ
in air at Mach 3. At (a) δ =20° and (b) δ =40°,
the oblique shock remains attached, but by
(c) δ =60°, the oblique shock has detached,
forming a bow wave.
Mach angle

35
Shadowgram of a one-half-in
diameter sphere in free flight
through air at Ma = 1.53. The flow
is subsonic behind the part of the
bow wave that is ahead of the
sphere and over its surface back
to about 45°. At about 90° the
laminar boundary layer separates
through an oblique shock wave
and quickly becomes turbulent.
The fluctuating wake generates a
system of weak disturbances that
merge into the second
“recompression” shock wave.

36
Prandtl–Meyer Expansion Waves
We now address situations where supersonic flow is turned in the opposite direction,
such as in the upper portion of a two-dimensional wedge at an angle of attack greater
than its half-angle δ.
We refer to this type of flow as an expanding flow, whereas a flow that produces an
oblique shock may be called a compressing flow.
As previously, the flow changes direction to conserve mass. However, unlike a
compressing flow, an expanding flow does not result in a shock wave.
Rather, a continuous expanding region called an expansion fan appears, composed of
an infinite number of Mach waves called Prandtl–Meyer expansion waves.
An expansion fan in the upper portion of
the flow formed by a two-dimensional
wedge at the angle of attack in a
supersonic flow. The flow is turned by
angle u, and the Mach number increases
across the expansion fan. Mach angles
upstream and downstream of the
expansion fan are indicated. Only three
expansion waves are shown for
simplicity, but in fact, there are an infinite
number of them. (An oblique shock is
present in the bottom portion of this flow.)

37
A cone-cylinder of 12.5° half-angle in a Mach number 1.84 flow. The
boundary layer becomes turbulent shortly downstream of the nose,
generating Mach waves that are visible in this shadowgraph. Expansion
waves are seen at the corners and at the trailing edge of the cone.
Prandtl–Meyer function

38
The complex interactions between shock waves and expansion
waves in an “overexpanded” supersonic jet. The flow is
visualized by a schlierenlike differential interferogram.

39
DUCT FLOW WITH HEAT TRANSFER AND
NEGLIGIBLE FRICTION (RAYLEIGH FLOW)
Many practical compressible flow problems
involve combustion, which may be modeled
as heat gain through the duct wall.
So far we have limited our
consideration mostly to isentropic
flow (no heat transfer and no
irreversibilities such as friction).
Many compressible flow problems
encountered in practice involve
chemical reactions such as
combustion, nuclear reactions,
evaporation, and condensation as
well as heat gain or heat loss
through the duct wall.
Such problems are difficult to
analyze exactly since they may
involve significant changes in
chemical composition during flow,
and the conversion of latent,
chemical, and nuclear energies to
thermal energy.
A simplified model is Rayleigh flow.
Rayleigh flows: Steady one-
dimensional flow of an ideal gas with
constant specific heats through a
constant-area duct with heat transfer,
but with negligible friction.

40
Mass equation
x-Momentum
equation
Energy
equation
Entropy
change
Equation
of state
Control volume for flow in a
constant-area duct with heat
transfer and negligible friction.
Consider a gas with known properties R, k, and cp. For a specified inlet
state 1, the inlet properties P1, T1, ρ1, V1, and s1 are known. The five
exit properties P2, T2, ρ2, V2, and s2 can be determined from the above
equations for any specified value of heat transfer q.

41
T-s diagram for flow in a constant-area
duct with heat transfer and negligible
friction (Rayleigh flow). This line is
called Rayleigh line.
1. All the states that satisfy the
conservation of mass, momentum,
and energy equations as well as
the property relations are on the
Rayleigh line.
2. Entropy increases with heat gain,
and thus we proceed to the right
on the Rayleigh line as heat is
transferred to the fluid.
3. Heating increases the Mach
number for subsonic flow, but
decreases it for supersonic flow.
4. Heating increases the stagnation
temperature T0 for both subsonic
and supersonic flows, and cooling
decreases it.
5. Velocity and static pressure have
opposite trends.
6. Density and velocity are inversely
proportional.
7. The entropy change corresponding
to a specified temperature change
(and thus a given amount of heat
transfer) is larger in supersonic flow.
From the Rayleigh line and the
equations

42
During heating, fluid temperature
always increases if the Rayleigh
flow is supersonic, but the
temperature may actually drop if
the flow is subsonic.
Heating or cooling has opposite effects on
most properties. Also, the stagnation
pressure decreases during heating and
increases during cooling regardless of
whether the flow is subsonic or supersonic.

44
Property Relations for Rayleigh Flow
Representative results are given in
Table A–34.

45
Choked Rayleigh Flow
For a given inlet state, the
maximum possible heat transfer
occurs when sonic conditions are
reached at the exit state.
The fluid at the critical state of Ma =1
cannot be accelerated to supersonic
velocities by heating. Therefore, the
flow is choked.
For a given inlet state, the
corresponding critical state fixes the
maximum possible heat transfer for
steady flow:

46
STEAM NOZZLES
The h-s diagram for the isentropic
expansion of steam in a nozzle.
Water vapor at moderate or high
pressures deviates considerably from
ideal-gas behavior, and thus most of
the relations developed in this
chapter are not applicable to the flow
of steam through the nozzles or
blade passages encountered in
steam turbines.
Supersaturated steam: The steam
that exists in the wet region without
containing any liquid.
Supersaturation states are
nonequilibrium (or metastable)
states.
Wilson line: The locus of points
where condensation takes place
regardless of the initial
temperatureand pressure at the
nozzle entrance.
When steam is
assumed ideal
gas with k = 1.3

47
Summary
• Stagnation properties
• Speed of sound and Mach number
• One-dimensional isentropic flow
 Variation of fluid velocity with flow area
 Property relations for isentropic flow of ideal gases
• Isentropic flow through nozzles
 Converging nozzles
 Converging–diverging nozzles
• Shock waves and expansion waves
 Normal shocks
 Oblique shocks
 Prandtl–Meyer expansion waves
• Duct flow with heat transfer and negligible friction
(Rayleigh flow)
 Property relations for Rayleigh flow
 Choked Rayleigh flow
• Steam nozzles

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