GAS DYNAMICS
MPE 228
By
Dr. Salem AbdelAziz Anous
Mechanical Power Department
Faculty of Engineering
Zagazig University
Chapter #2
INTRODUCTION TO
COMPRESSIBLE FLOW
Sound Waves
❑ SONIC SPEED
▪ A sound wave is fundamentally a pressure disturbance
that propagates through a medium by particle
interaction.
▪ In other words, sound waves move through a physical
medium by alternately contracting and expanding the
section of the medium in which it propagates.
▪ The rate at which the sound waves propagate through
the medium is known as the speed of sound.
▪ A disturbance at a given point creates a region of
compressed molecules that is passed along to its
neighboring molecules and in so doing creates a
traveling wave.
❑ SONIC SPEED
▪ Waves come in various strengths, which are measured
by the amplitude of the disturbance. The speed at
which this disturbance is propagated through the
medium is called the wave speed.
▪ This speed not only depends on the type of medium and
its thermodynamic state but is also a function of the
strength of the wave. The stronger the wave is, the
faster it moves
▪ If we are dealing with waves of large amplitude, which
involve relatively large changes in pressure and density,
we call these shock waves.
❑ SONIC SPEED
▪ Consider a long constant-area tube filled with fluid and having a
piston at one end, as shown in Figure. The fluid is initially at rest.
At a certain instant the piston is given an incremental velocity dV
to the left.
▪ Assume both negligible losses and no heat transfer as the wave
passes through the fluid, thus, the process is both reversible and
adiabatic, which means that it is isentropic.
❑ SONIC SPEED
In SI Units, 𝒈𝒄 = 𝟏
For an isentropic process (Perfect Gas)
In SI Units, 𝒈𝒄 = 𝟏
▪ For perfect gases, the sonic speed is a function of
the individual gas and temperature only
The Bulk or Volume Modulus of Elasticity Ev
Sonic Speed through any Medium
The reciprocal of the bulk modulus is called the compressibility
Table: Speed of Sound at
25°C in Various Media
In SI Units, 𝒈𝒄 = 𝟏
❑ MACH NUMBER
▪ It is important to realize that both V and a are computed
locally for conditions that actually exist at any given point.
▪ The Mach number is the most important parameter in the
analysis of compressible flows.
0 < M < 0.3 Incompressible
0.3 < M < 1 : Subsonic
M = 1 : Sonic
M  1 : Transonic
M > 1 : Supersonic
M >> 1 : Hypersonic
❑ MACH NUMBER
Example: An aircraft flies at a speed of 400m/s. What is this aircraft’s Mach
number when flying at standard sea-level conditions (T=289K) and at standard
15,200m (T=217K) atmosphere conditions.
Note: Although the aircraft speed did not
change, the Mach number did change
because of the change in the local speed of
sound.
❑ WAVE PROPAGATION
▪ Let us examine a point disturbance that is at rest in a fluid (M=0).
Infinitesimal pressure pulses are continually being emitted and thus
they travel through the medium at sonic speeds in the form of
spherical wave fronts.
▪ To simplify matters we shall keep track of only those pulses that are
emitted every second.
▪ At the end of 3 seconds, the picture will appear as shown in Fig.
below. Note that the wave fronts are concentric.
Fig. Wave fronts from a
stationary disturbance.
▪ Now consider a similar problem in which the disturbance is no
longer stationary. Assume that it is moving at a speed less than
sonic speed, say a/2 (M=0.5).
▪ Figure below shows such a situation at the end of 3 seconds.
▪ Note that the wave fronts are no longer concentric. Furthermore,
the wave that was emitted at t=0 is always in front of the
disturbance itself.
▪ Therefore, any person, object, or fluid particle located upstream
will able to feel the wave fronts pass by and know that the
disturbance is coming.
Fig. Wave fronts from subsonic
disturbance.
▪ Next, let the disturbance move at exactly sonic speed (M=1).
▪ Fig. shows this case and you will note that all wave fronts coalesce
on the left side and move along with the disturbance.
▪ After a long period of time this wave front would approximate a
plane indicated by the dashed line.
▪ In this case, no region upstream is forewarned of the disturbance as
the disturbance arrives at the same time as the wave front.
Fig. Wave fronts from sonic disturbance (a transonic flow condition)
▪The only other case to consider is that of a disturbance moving at
velocities greater than the speed of sound. Fig. below shows a point
disturbance moving at M = 2.0 (twice sonic speed).
▪The wave fronts have coalesced to form a cone with the disturbance
at the apex. This is called a Mach cone. The region inside the cone is
called the zone of action since it feels the presence of the waves.
▪The outer region is called the zone of silence, as this entire region is
unaware of the disturbance. The surface of the Mach cone is
sometimes referred to as a Mach wave; half-angle at the apex is called
the Mach angle and is given the symbol μ. It should be easy to see that
Fig. Wave fronts from supersonic
disturbance.
▪ In this section, we have discovered one of the most significant
differences between subsonic and supersonic flow fields.
▪ In the subsonic case the fluid can “sense” the presence of an object
and smoothly adjust its flow around the object.
▪ In supersonic flow, this is not possible, and thus flow adjustments
occur rather abruptly in the form of shock or expansion waves.
❑ EQUATIONS FOR PERFECT GASES IN TERMS OF MACH NUMBER
➢ CONTINUITY
In SI Units, 𝒈𝒄 = 𝟏
❑ EQUATIONS FOR PERFECT GASES IN TERMS OF MACH NUMBER
In SI Units, 𝒈𝒄 = 𝟏
EXAMPLE: TOTAL TEMPERATURE
Total Temperature
T0
 −1 2
= 1+
M1
T1
2
Static Temperature
Vehicle flight Mach number
• A rocket is flying at Mach M=6 through a portion of
the atmosphere where the static temperature is 200 K
• What temperature does the nose of the rocket ‘feel’?
• T0 = 200(1+ 0.2(36)) = 1,640 K!
In SI Units, 𝒈𝒄 = 𝟏
800 ft/s=243.84 m/s
30 psia= 206.84 kPa
600 R=333.33 K
In SI Units, 𝒈𝒄 = 𝟏
In SI Units, 𝒈𝒄 = 𝟏
2860 ft/s=871.73 m/s
50 psia= 344.74 kPa
600 R=333.33 K
1048 R = 582.2 K
b. ENERGY
CONTINUITY
In SI Units, 𝒈𝒄 = 𝟏
Thank you