CHAPTER 3
INTRODUCTION TO
COMPRESSIBLE FLOW
3.1
INTRODUCTION
In incompressible flows, the density is constant.
Hence, the main variables are
(i) velocity
(ii) pressure
These variables may then be solved with the aid of
(i) continuity equation
(ii) momentum equation
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
LECTURE 01 - 2/4
In compressible flows, there are appreciable variations in density
Compressibility is especially important in high speed flows
Large changes in velocity results in large changes in pressure.
These pressure changes are accompanied with significant variations in
(i) density
(ii) temperature
Then two more relations are required for the complete solution of the problem. These are
(i) energy equation
(ii) equation of state
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
LECTURE 01 - 3/4
3.2
WAVE PROPAGATION IN COMPRESSIBLE MEDIA
Stationary
dV
gas
When a sudden push is given to the piston, a layer of gas piles up next to the piston and is compressed
at the first instant, but the remainder of the gas is unaffected
The compression wave created by the piston, moves through the gas and eventually all the gas is able to
feel this movement .
If the created pressure pulse is infinitesimally small, then this wave is called a sound wave and it moves
at the speed of sound.
When the medium is incompressible, there will be no piling up of the fluid and all the fluid feels the
motion instantaneously.
The speed of wave propagation is infinite
(speed of sound)water > (speed of sound)air
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
LECTURE 01 - 4/4
3.3
SPEED OF SOUND
Pressure wave
moving
with velocity a
dV
p + dp
 + d
dV
h + dh
Stationary
pressure wave
p
a  dV

a
V=0
h
Control
volume
p
p + dp
 + d
a  dV
h + dh

a
h
p
p
p + dp
p + dp
p
p
x
x
V
V
dV
x
 (a  dV)
x
a
The transformation is a dynamic transformation, since it only affects dynamic properties (velocity,
stagnation pressure, stagnation temperature) but does not affect the static properties (static pressure,
static temperature)
LECTURE 02 - 5/7
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
a) Continuity Equation
Stationary
pressure wave
a  dV
p + dp
 + d
a  dV
h + dh
Control
volume
p

a
h
For steady flow, mass flow rate is constant
& = constant
m
For one-dimensional flow, properties are uniform (constant) over each cross-section.
& = ρaA= (ρ + dρ)(a - dV )A
m
where A is the cross-sectional area of the tube
0
ρaA= ρaA- ρAdV + aAdρ - AdρdV
dV =
a
dρ
ρ
LECTURE 02 - 6/7
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
b) Momentum Equation
Sum of the external forces acting on the control volume must be balanced by the rate of change of linear
momentum across the control volume
Control
volume
Control
volume
(p + dp)A
pA
 Fx = ( p + dp)A - pA
& (a - dV )
m
&
ma
& + (-m
& ) (a - dV )
 Fx = ma
& + (m
& ) -(a - dV )
 Fx = (p + dp)A - pA  Adp = ma
& - ma
& m
& dV
pA+ Adp - pA = ma
Adp =  AVdV
dV =
1
dp
ρa
LECTURE 02 - 7/7
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
c) Energy Equation
Stationary
pressure wave
p + dp
 + d
a  dV
h + dh
a  dV
Control
volume
p

a
h
Across an infinitesimal pressure wave, the frictional effects can be neglected, since the pressure wave is
very thin
The time is too short for heat transfer so that the process can be considered adiabatic
h+
a2
(a - dV )2
= (h + dh ) +
2
2
0
a2
a2
(dV )2
h+
= h + dh +
 adV 
2
2
2
dh = a dV
LECTURE 02 - 8/7
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
d) Second Law of Thermodynamics
Momentum equation
dV =
1
dp
ρa
 dh =
dp
ρ
dh = a dV
Energy equation
From thermodynamics, Gibbs relation is
Tds = dh -
Hence
dp
ρ
ds = 0
Therefore across an infinitesimal pressure wave, the entropy is constant
Continuity equation
Momentum equation
dV =
dV =
a
dρ
ρ
1
dp
ρa
 a2 =
dp
dρ
To evaluate a thermodynamic property, the property to be held constant during differentiation should be
specified. For the present case, entropy is constant
 p 
a2 = 

 ρ s
LECTURE 02 - 9/7
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
In isentropic flow, the density and pressure of a gas is related by
p
 constant
ρk
Taking logaritms
(11.6)
lnp - klnρ = ln(constant)
Differentiating
dp
dρ
-k
=0
ρ
ρ
a= k
p
ρ
e) Equation of State
For a perfect gas
p = ρRT
a = kRT
LECTURE 02 - 10/7
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
1.4
MACH NUMBER
M=
Speed of the fluid
V
=
Local speed of sound a
Speed of the fluid V is a measure of directed motion of gas particles and V2 is a measure of kinetic
energy per unit mass of the directed flow .
Local speed of sound a is proportional to T1/2 which is a measure of random motion of gas particles and
a2 is a measure of kinetic energy per unit mass of the random flow .
M=
V V 2 / 2 Directed kinetic energy per unit mass
=
=
a a 2 / 2 Random kinetic energy per unit mass
M<1
Subsonic flow
M=1
Sonic flow
M>1
Supersonic flow
M < 0.3
Incompressible flow
0.9 < M < 1.1
Transonic flow
M>5
Hypersonic flow
LECTURE 02 - 11/7
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
3.4
PRESSURE DISTURBANCES IN A COMPRESSIBLE FLUID
Consider a point source, which is emitting instantaneous infinitesimal disturbances.
These disturbances propagate in all directions at the speed of sound a.
At any time t, the location of the wve front from the disturbance emitted at time t0, will be represented by
a sphere with radius a(t – t0), whose center is coincident with the location of the disturbance at time t0.
a(3t)
a(2t)
at
S
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
LECTURE 03 - 12/6
3.5.1
Stationary Point Source
a(3t)
a(2t)
Sound wave emited at t = 0
at
Sound wave emited at t = t
S
Sound wave emited at t = 2t
Wave pattern at t = 3t
The given wave fronts are in the form of concentric spheres.
LECTURE 03 - 13/6
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
3.5.2
Subsonic Point Source
a(3t)
a(2t)
Sound wave emited at t = 0
at
S0 S 1 S 2 S 3
Direction
of motion
Sound wave emited at t = t
Sound wave emited at t = 2t
Vt VtVt
Wave pattern at t = 3t
If a body moves in a compressible fluid with a subsonic speed , the fluid ahead of the body feels the
presence of the body.
An observer ahead of the body will hear more peaks per unit time as source approaches him then after
the source passes over him. This is known as Doppler’s effect.
The wave fronts are spherical in shape, but they are no longer concentric.
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
LECTURE 03 - 14/6
3.5.3
Sonic Point Source
Mach
wave
a(3t)
a(2t)
at
S0
S1
S2
S3
Direction
of motion
Sound wave emited at t = 0
Sound wave emited at t = t
Sound wave emited at t = 2t
Zone of
action
Zone of
silence
Vt
Vt
Vt
Wave pattern at t = 3t
An observer in front of the source will not hear any sound.
LECTURE 03 - 15/6
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
3.5.4
Supersonic Point Source
Mach cone
Zone of
silence
Zone of
action
Mach angle
a(3t)

a(2t)
at
S1
S0
Vt
S2
Vt
S3
Sound wave emited at t = 0
Direction
of motion
Sound wave emited at t = t
Sound wave emited at t = 2t
Vt
Wave pattern at t = 3t
LECTURE 03 - 16/6
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW
(11.10)
The effect of pressure changes cannot reach ahead of the body and this is known as rule of forbidden
signals
The locus of leading surfaces of the wave fronts is a cone with the body at its apex which is known as
Mach cone.
All disturbances are confined inside the Mach cone and this region is known as the zone of action.
The region outside the Mach cone is unaware of the disturbances and is known as zone of silence.
The half angle of the Mach cone is known as Mach angle.
sinμ =
a(3Δt ) a(2Δt ) a(Δt )
1
=
=
=
V (3Δt ) V (2Δt ) V (Δt ) M
The disturbances are concentrated at the neighborhood of the Mach cone and this is known as the rule
of concentrated action.
LECTURE 03 - 17/6
ME 411 GAS DYNAMICS
3. INTRODUCTION TO COMPRESSIBLE FLOW