As an object moves through a gas, the gas molecules are deflected
around the object. If the speed of the object is much less than the
speed of sound
of the gas, the density of the gas remains constant and the flow of
gas can be described by conserving momentum and energy.
As the
speed of the object increases towards the speed of sound, we
must consider
compressibility effects
on the gas. The density of the gas varies locally as the gas is
compressed by the object.

For compressible flows with little or small
flow turning, the flow process is reversible and the
entropy
is constant.
The change in flow properties are then given by the
isentropic relations
(isentropic means "constant entropy").
But when an object moves faster than the speed of sound,
and there is an abrupt decrease in the flow area,
the flow process is irreversible and the entropy increases.
Shock waves are generated
which are very small regions in the gas where the
gas properties
change by a large amount.
Across a shock wave, the static
pressure,
temperature,
and gas
density
increases almost instantaneously.
The
Mach number
and speed of the flow decrease across a shock wave.
If the upstream Mach number is in the low supersonic regime, the
specific heat ratio of air
remains a constant value (1.4) and air is said
to be calorically perfect.
But under low hypersonic flow conditions
or high total temperature conditions, the specific heat ratio
changes and air is then said to be
calorically imperfect.
Derived flow variables, like the
speed of sound and the
isentropic flow relations
are slightly different for a calorically imperfect gas
than the conditions predicted for a calorically perfect gas
because some of the energy of the flow excites the vibrational
modes of the diatomic molecules of nitrogen and oxygen in the air.

Because a shock wave does no work, and there is no heat addition, the
total
enthalpy
and the total temperature are constant through the shock.
But because the flow is non-isentropic, the
total pressure downstream of the shock is always less than the total pressure
upstream of the shock; there is a loss of total pressure associated with
a shock wave.
The ratio of the total pressure is shown on the slide.
Because total pressure changes across the shock, we can not use the usual (incompressible) form of
Bernoulli's equation
across the shock.

If the
shock wave is perpendicular to the flow direction it is called a normal
shock. On this web page, we have listed the equations which describe the change
in flow variables for flow across a normal shock.
There are two groups of equations and two slides; one for the calorically
perfect gas, and the other for the calorically imperfect gas.
The equations presented here were derived by considering the conservation of
mass,
momentum,
and
energy.
for a compressible gas while ignoring viscous effects.
The equations have been further specialized for a one-dimensional flow
without heat addition.

The equations can be applied to the
two dimensional flow past a wedge for the following combination of
free stream Mach number M and wedge angle c :

where gam is the
ratio of specific heats.
If the wedge angle is less than this detachment angle, an attached
oblique shock
occurs and the equations are slightly modified.

Beginning with the calorically perfect gas,
across the normal shock wave
the Mach number decreases to a value specified as M1:

The right hand side of all these equations depend only on the free stream
Mach number. So knowing the Mach number,
we can determine all the conditions associated with
the normal shock.
The equations describing normal shocks
were published in a NACA report
(NACA-1135)
in 1951.

Now, turning to the calorically imperfect gas, the procedure for
determining the conditions across a normal shock are much more
complicated than the procedure for the calorically perfect gas.
Mathematical models
based on a simple harmonic vibrator have been developed for the calorically
imperfect gas.
The details of the analysis were given by Eggars in
NACA Report 959.
A synopsis of the report is included in
NACA-1135. A collection of the equations and a description of the
method is shown on this slide:

As with the calorically perfect case, the conservation of mass, momentum,
and energy equations are solved simultaneously to determine the flow
conditions across the normal shock. For the calorically imperfect case,
additional terms are added to the energy equation to account for the
vibration of the molecules at high temperatures. The resulting equation for the
static temperature downstream of the shock is:

(u + (R*T/u))^2 - (u + (R*T/u)) * sqrt[(u + (R*T/u))^2 - 4 * R * T1] - 2 * R * T1 - 2 * u^2
+ (gam / (gam -1)) * 4 * R * (T1 - T) + 4 * R * theta * [(1/(e^{theta/T1} - 1)) - (1/(e{theta/T} - 1))] = 0

where R is the gas constant, u is the upstream velocity,
T is the upstream static temperature, T1 is the downstream static temperature,
theta is a constant equal to 5500 Rankine, and gam is the calorically perfect
ratio of specific heats. There is no known direct solution to this equation, so it must be
solved iteratively for the downstream temperature. Having the downstream static temperature, we
can use the calorically imperfect
isentropic relations
upstream and downstream of the shock to determine all of the other variables. Total
temperature remains a constant across the shock wave. We have listed some of the
isentropic ratios on the slide which must also be solved iteratively.

Here's a Java program that solves the normal shock equations for the calorically
perfect and the calorically imperfect conditions:

To change input values, click on the input box (black on white),
backspace over the input value, type in your new value. Then hit
the red COMPUTE button to
send your new value to the program.
You will see the output boxes (yellow on black)
change value. You can use either Imperial or Metric units.
Just click on the menu button and click on your selection.
For the given altitude input, the program computes the upstream
standard day conditions. The ratios are computed
using the
Mach number
and
specific heat ratio
inputs and the equations given in the figure. The
downstream conditions are computed from the upstream conditions and
the ratios.
If you are an experienced user of this simulator, you can use a
sleek version
of the program which loads faster on your computer and does not include these instructions.
You can also download your own copy of the program to run off-line by clicking on this button:

There is more complete shock simulation program that is also available at this web site. The
ShockModeler
program models the intersection and reflection of multiple shock waves.