Compressible Flow

Dr. Zaib Ali

 Outline
• Introduction
• Ideal Gas Relationships
• Mach No. and Speed of Sound
™
• Categories of Compressible Flow
™
• Isentropic Flow of an Ideal Gas
™
• Shock Waves
™
Compressible Flow
 Introduction
• Fluid compressibility is a very important consideration in numerous
engineering applications of fluid mechanics. For example,
– The measurement of high-speed flow velocities requires compressible flow theory.
– The flows in gas turbine engine components are generally compressible.
– Many aircraft fly fast enough to involve a compressible flow field.

• The variation of fluid density for compressible flows requires attention to

density and other fluid property relationships.

• The fluid equation of state, often unimportant for incompressible flows,

is vital in the analysis of compressible flows. Also, temperature variations
for compressible flows are usually significant and thus the energy equation
is important.
• Curious phenomena can occur with compressible flows.
 Introduction
• In this study of compressibility effects, we mainly consider the
steady, one-dimensional, constant (including zero) viscosity,
compressible flow of an ideal gas.
 Ideal Gas Relationships

universal gas constant

molecular weight

for the isentropic flow

Both the pressures and temperatures used must be absolute.

 Mach Number and Speed of Sound
• Mach Number: dimensionless measure of compressibility in a
fluid flow.
• It is the ratio of the value of the local flow velocity to the local
speed of sound

• What we perceive as sound generally consists of weak

pressure pulses that move through air with a Mach number of
one. When our ear drums respond to a succession of moving
pressure pulses, we hear sounds.
 Mach Number and Speed of Sound
• Consider the 1D fluid mechanics of a thin, weak pressure
pulse moving at the speed of sound through a fluid at rest.

• Ahead of the pressure pulse,

– fluid velocity is zero
– the fluid pressure is p
– and the fluid density is ρ

• Behind the pressure pulse,

– the fluid velocity is δV
– the pressure is p + δp
– and density is ρ + δρ

(b)
 Mach Number and Speed of Sound
• Continuity

• Momentum

(b)
 Mach Number and Speed of Sound
• If we further assume that the frictionless flow through the control volume
of Fig (b) is adiabatic (no heat transfer), then the flow is isentropic.

• For the isentropic flow of an ideal gas (with constant cv and cp)

• speed of sound for air at 0 °C is 331.4 m/s

 Categories of Compressible Flow
• Compressibility effects are more
important at higher Mach numbers.
• Consider the emission of a weak pressure
pulses from a point source.
• These pressure waves are spherical and
expand radially outward from the point
source at the speed of sound, c.
• For a stationary point source, the
symmetrical wave pattern is involved an
observer anywhere in the pressure field
would hear the same sound frequency (a) Pressure waves at t =3 s,
from the point source. V=0
 Categories of Compressible Flow
• When the point source moves to the left
with a constant velocity, V < c, the wave
pattern is no longer symmetrical.
• the flow is considered subsonic and
compressible
• A stationary observer will hear a different
sound frequency coming from the point
source depending on where the observer is
relative to the source because the wave
pattern is asymmetrical.
• This phenomenon is called the Doppler (b) Pressure waves at t =3 s,
effect. V<c
 Categories of Compressible Flow
• The flow is sonic.
• If you were positioned to the left
of the moving point source, you
would not hear the point source
until it was coincident with your
location.
• The pressure waves are all
tangent to a plane that is
perpendicular to the flow and that
passes through the point source. (c) Pressure waves at t =3 s,
V=c
• This plane is called a Mach
wave.
 Categories of Compressible Flow
• A cone (Mach cone) that is
tangent to the pressure wave can
be constructed to represent the
Mach wave that separates the
zone of silence from the zone of
action.

(d) Pressure waves at t =3 s,

• Significant pressure, and thus V>c
density variation across the cone
surface.
Categories of Compressible Flow
 Categories of Compressible Flow
• This discussion about pressure wave patterns suggests the
following categories of fluid flow:
▪ Incompressible flow: Unrestricted, nearly symmetrical and
instantaneous pressure communication. Ma < 0.3
▪ Compressible subsonic flow: Unrestricted but noticeably asymmetrical
pressure communication. 0.3 < Ma < 1.0
▪ Compressible supersonic flow: Formation of Mach wave; pressure
communication restricted to zone of action. 1 < Ma
▪ Transonic flows (0.9 < Ma < 1.2)
▪ Hypersonic flows (5 < Ma)
 Example
An aircraft cruising at 1000-m elevation, z, above you moves past
in a flyby. It is moving with a Mach number equal to 1.5 and the
ambient temperature is 20 oC. How many seconds after the plane
passes overhead do you expect to wait before you hear the
aircraft?
 Example
• The angle α in Fig. is related to the elevation of the plane, z,
and the ground distance, x, by

• Also, assuming negligible change of Mach number with

elevation, we can use following Eq. to relate Mach number to
the angle α
Example
 Isentropic Flow of an Ideal Gas
• Steady, one-dimensional, isentropic flow of an ideal gas with
constant specific heat values (cv and cp).
• velocity and fluid property changes in the streamwise direction
only.
• Adiabatic and frictionless (reversible) flow is a form of
isentropic flow.
• An isentropic flow is not achievable with actual fluids because
of friction.
• Helps us to gain an understanding of actual compressible flow
phenomena including choked flow, shock waves, acceleration
from subsonic to supersonic flow, and deceleration from
supersonic to subsonic flow.
 Effect of Variations in Flow Cross-Sectional
Area
• Continuity

• Newton’s second law was applied to the

inviscid (frictionless) and steady flow of
a fluid particle.

can be dropped because of its small size in comparison to the other terms
Effect of Variations in Flow Cross-Sectional
Area
Effect of Variations in Flow Cross-Sectional
Area

• The area associated with is either a minimum or a maximum

amount.
 Converging–Diverging Duct
• For the steady isentropic flow of
an ideal gas, we conclude that
the sonic condition can be
attained in a converging–
diverging duct at the minimum
area location.
• This minimum area location is
often called the throat of the
converging–diverging duct.
• To achieve supersonic flow
from a subsonic state in a duct,
a converging–diverging area
variation is necessary.
 Converging–Diverging Nozzle
• For this reason, we often refer to such a duct as a converging–
diverging nozzle.
• Note that a converging–diverging duct can also decelerate a
supersonic flow to subsonic conditions.
• Thus, a converging–diverging duct can be a nozzle or a
diffuser depending on whether the flow in the converging
portion of the duct is subsonic or supersonic.
 Converging–Diverging Duct Flow
Stagnation State
• The stagnation state is associated with zero flow velocity and
an entropy value that corresponds to the entropy of the flowing
fluid. The subscript 0 is used to designate the stagnation state.
• It can be achieved by decelerating a flow, isentropically, to
zero velocity.
• For an isentropic flow of an ideal gas:

Combining
Converging–Diverging Duct Flow
 Choked Flow
• Choked flow occurs when the Mach number is 1.0 at the
minimum cross-sectional area.
• Information about pressure cannot move upstream when
Ma =1, hence upstream flow not affected.
• The state associated with Ma =1 and the same entropy level as
the flowing fluid is called the critical state, denoted by ( )*

Critical pressure ratio

Critical temperature ratio

 Choked Flow

• The area ratio, is a useful concept for the isentropic flow of an

ideal gas through a converging–diverging duct whether or not
the critical area, is physically present in the flow

Ratio of the local velocity

to the speed of sound at the throat
 Example
A converging duct passes air steadily from standard atmospheric
conditions to a receiver pipe as illustrated in Figure. The throat
(minimum) flow cross-sectional area of the converging ducts
is 1×10-4 m2. Determine the mass flowrate through the duct if the
receiver pressure is (a) 80 kPa (abs), (b) 40 kPa (abs).
 Example

Assuming ideal gas isentropic flow with constant specific heats

To determine the throat Mach number

Example
Example
Example
Example
Example
 Example
A converging duct passes air steadily from standard atmospheric
conditions to a receiver pipe as illustrated in Figure. The throat
(minimum) flow cross-sectional area of the converging ducts
is 1×10-4 m2. Determine the mass flowrate through the duct if the
receiver pressure is (a) 80 kPa (abs), (b) 40 kPa (abs).

Solve Example 11.5 using Fig. D.1 of Appendix D.

Example
Example
Example
Example
Example
Example
 Converging–Diverging Nozzle
• 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) requires a
diverging flow section
– Converging-diverging (C-D) nozzle
– Standard equipment in supersonic aircraft and rocket propulsion
• Forcing fluid through a C-D nozzle does not guarantee supersonic
velocity if the back pressure is not in the right range.
• The state of the nozzle flow is determined by the overall pressure
ratio Pb/P0. Therefore, for given inlet conditions, the flow through a
converging–diverging nozzle is governed by the back pressure Pb
Converging–Diverging Nozzle
Converging–Diverging Nozzle
Converging–Diverging Nozzle

• When Pb = P0 (case A), there is

no flow through the nozzle.
• P0 > Pb > PC
– Flow remains subsonic, and mass
flow is less than for choked flow.
Diverging section acts as diffuser
• Pb = PC
– Sonic flow achieved at throat.
Diverging section acts as diffuser.
Subsonic flow at exit. Further
decrease in Pb has no effect on flow
in converging portion of nozzle
Converging–Diverging Nozzle

• PC > Pb > PE
– Fluid is accelerated to supersonic
velocities in the diverging section
as the pressure decreases. However,
acceleration stops at location of
normal shock. Fluid decelerates
and is subsonic at outlet. As Pb is
decreased, shock approaches
nozzle exit.
4. PE > Pb > 0
– Flow in diverging section is
supersonic with no shock forming
in the nozzle. Without shock, flow
in nozzle can be treated as
isentropic.
Converging–Diverging Nozzle
Converging–Diverging Nozzle
Example
Example
Example
Example
Example
 Example
Air flows through a converging–diverging nozzle, with a throat
diameter of 10 cm and an exit diameter of 20 cm, from a
reservoir maintained at 20°C and 300 kPa absolute. Estimate the
two receiver pressures that provide an isentropic flow throughout
the nozzle. Also, determine the associated exit Mach numbers.
Example
 Shock Waves
• Sound waves are created by small pressure disturbances and travel
at the speed of sound
• As discussed, for some back pressures, abrupt changes in fluid
properties occur in C-D nozzles, creating a shock wave
• The flow process through the shock wave is highly irreversible and
cannot be approximated as being isentropic.
• Shock waves that occur in a plane normal to the direction of flow,
called normal shock waves.
• Flow through the shock is adiabatic but irreversible.

• However, that the stagnation pressure decreases across the shock

because of the irreversibilities.
Shock Waves
 Normal Shocks
• Normal shock waves involve
– deceleration from a supersonic to a subsonic
flow,
– a rise in pressure, temperature and density, and
– an increase of entropy.
 Normal Shocks

Applying conservation of mass, momentum and

energy across infinitesimally thin CV

Normal Shock Table

Normal Shocks
Normal Shocks
 Example
If the air flowing through the converging–diverging nozzle experiences a normal shock
wave at the nozzle exit plane, determine the following after the shock: (a) the
stagnation pressure, static pressure, static temperature, and static density; (b) the
entropy change across the shock; (c) the exit velocity; and (d) the mass flow rate
through the nozzle. Approximate the flow as steady, one-dimensional, and isentropic
with k = 1.4 from the nozzle inlet to the shock location.
 Example

(a) The fluid properties at the exit of the nozzle just before the shock (denoted
by subscript 1) are
Example
 Example

in earlier example
 Example
A total pressure probe is inserted into a supersonic air flow. A shock wave
forms just upstream of the impact hole and head as illustrated in Fig. The
probe measures a total pressure of 414 kPa. The stagnation temperature at the
probe head is 555 K. The static pressure upstream of the shock is measured
with a wall tap to be 82 kPa. Determine the Mach number and velocity of the
flow.
 Example

Since the stagnation temperature

remains constant across a normal
shock

= 555K

For the isentropic flow upstream of

the shock, Fig. D.1 can be used.
For
 Example

Tx= (0.59)(555K) = 327 K

= 1.9 [ (287) (327) (1.4)]1/2

Vx = 678 m/s
 Oblique Shocks
• An oblique shock wave, unlike a normal shock, is
inclined with respect to the incident upstream flow
direction.
• It will occur when a supersonic flow encounters a
corner that effectively turns the flow into itself and
compresses.
• The upstream streamlines are uniformly deflected
after the shock wave.
• The most common way to produce an oblique shock
wave is to place a wedge into supersonic,
compressible flow.
• Similar to a normal shock wave, the oblique shock
wave consists of a very thin region across which
nearly discontinuous changes in the thermodynamic
properties of a gas occur.
• While the upstream and downstream flow directions
are unchanged across a normal shock, they are
different for flow across an oblique shock wave.
Oblique Shocks
 Oblique Shocks

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.
 Oblique Shocks
• The tangential component of the flow velocity is preserved across an oblique shock wave.
(can be checked by applying the momentum eq. across the CV)
• The changes across an oblique shock wave are governed by the normal component of the
free-stream velocity.

Ma1,𝑛 = Ma1 sin 𝛽

The same velocity vectors of rotated by angle

• 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.
 Oblique Shocks

The dependence of straight oblique shock deflection angle  on shock angle  for several values of
upstream Mach number Ma1. Calculations are for an ideal gas with k = 1.4. The dashed red line
connects points of maximum deflection angle ( =  max). Weak oblique shocks are to the left of this
line, while strong oblique shocks are to the right of this line. The dashed green 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.
 Oblique Shocks
A detached oblique shock occurs upstream of a two-
dimensional wedge of half-angle  when  is
greater than the maximum possible deflection angle
. 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 
in air at Mach 3. At (a)  =20 and (b)  =40, the oblique
shock remains attached, but by (c)  =60, the oblique
Mach angle shock has detached, forming a bow wave.
Oblique Shocks

Color schlieren image of Mach 3.0

flow from left to right over a sphere.
A curved shock wave called a bow
shock forms in front of the sphere
and curves downstream.
Prandtl–Meyer Expansion Waves
 Prandtl–Meyer Expansion Waves
• Lets 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 .
• This type of flow is referred 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 An expansion fan in the upper portion of
compressing flow, an expanding flow does the flow formed by a two-dimensional
wedge at the angle of attack in a
not result in a shock wave. supersonic flow. The flow is turned by
• Rather, a continuous expanding region angle u, and the Mach number increases
across the expansion fan. Mach angles
called an expansion fan appears, composed upstream and downstream of the expansion
of an infinite number of Mach waves called fan are indicated. Only three expansion
Prandtl–Meyer expansion waves. 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.)
 Prandtl–Meyer Expansion Waves

Prandtl–Meyer function
Prandtl–Meyer Expansion Waves

The complex interactions between

shock waves and expansion waves in
an “overexpanded” supersonic jet.
(a) The flow is visualized by a
schlieren-like differential interferogram.
(b) Color shlieren image. (c) Tiger tail
shock pattern.
 Fanno Flow
• Steady, frictional, and adiabatic flow of an ideal gas with
constant specific heats through a constant-area duct is called
Fanno flow.
 Rayleigh Flow

• Steady one-dimensional flow of an ideal gas with constant

specific heats through a constant-area duct with heat transfer,
but with negligible friction.