As an aircraft moves through the air, the air molecules near the
aircraft are disturbed and move around the aircraft. If the aircraft passes
at a low speed, typically less than 250 mph, it is observed that the
density
of the air remains constant. For higher speeds, some of the
energy of the aircraft goes into compressing the air and locally
changing the density of the air. This compressibility
effect alters the amount of resulting force on the aircraft since the
aerodynamic force
depends on the air density.
The effect becomes more important as speed increases. Near and beyond
the speed of sound, about 330 m/s or 760
mph at sea level, small disturbances in the flow are transmitted
to other locations
isentropically or with constant entropy.
Sharp disturbances generate
shock waves
that affect both the lift and drag of the aircraft, and the flow
conditions downstream of the shock wave.
On this slide, we will investigate the dependence of the density change on the
Mach number
of the flow.
The Mach number is the
ratio
of the speed of the aircraft, or the speed of the gas, to the speed
of sound in the gas.
The speed of sound is equal to the speed of transmission of small, isentropic
disturbances in the flow.
To determine the role of the Mach number on compressibility effects.
we begin with the
conservation of momentum
equation:
rho * V dV =  dp
where rho is the fluid density, V is the velocity,
and p is the
pressure.
dV and dp denote differential changes in the velocity
and pressure. From our
derivation
of the conditions for
isentropic flow,
we know that:
dp/p = gamma * drho/rho
dp = gamma * p / rho * drho
where gamma is the
specific heat ratio.
We can use the ideal
equation of state
to simplify the expression on the right:
p = rho * R * T
dp = gamma * R * T * drho
where R is the specific gas constant and T is the
absolute temperature. We recognize that:
gamma * R * T = a^2
where a is equal to the
speed of sound.
So,
dp = a^2 * drho
Substituting this expression for the change of pressure into the
conservation of momentum equation gives:
rho * V dV =  a^2 drho
 (V^2 / a^2) dV / V = drho / rho
 M^2 dV / V = drho / rho
where M is the Mach number.
What does this expression tell us about the role of the Mach number
in compressible flows?

For low speed, or subsonic
conditions, the Mach number is less than one, M < 1
and the square of the Mach number is very small. Then the left
hand side of the equation is very small, and the change in density
is very small.
For the low subsonic conditions, compressibility can be ignored.

As the speed of the object approaches the speed of sound, the
flight Mach number is nearly equal to one, M = 1,
and the flow is said to be transonic.
If the Mach number is near one, the square of the Mach number
is also nearly equal to one. For transonic flows, the change in
density is nearly equal to the change in velocity, and compressibility
effects can not be ignored.

As the speed increases beyond the speed of sound, the flight
Mach number is greater than one M > 1 and the flow is
said to be supersonic
or
hypersonic.
For supersonic and hypersonic flows, the density changes faster
than the velocity changes by a factor equl to the square of the
Mach number. Compressibility effects become more important with
higher Mach numbers.
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