As a rocket moves through a gas, the gas molecules are deflected
around the rocket. If the speed of the rocket 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 rocket 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 rocket.
When a rocket moves faster than the
speed of sound,
and there is an abrupt decrease in the flow area,
shock waves
are generated.
If the flow area increases, however, a different flow phenomenon is
observed. If the increase is abrupt, we encounter a centered expansion
fan. The word "expansion" denotes that the area is increasing.

There are some marked differences between shock waves and expansion fans.
Across a shock wave, the Mach number decreases, the static pressure increases,
and there is a loss of total pressure because the process is irreversible.
Through an expansion fan, the Mach number increases, the static pressure decreases
and the total pressure remains constant. Expansion fans are
isentropic.

The calculation of the expansion fan involves the use of the
Prandtl-Meyer function.
This function is derived from conservation of
mass,
momentum,
and
energy
for very small (differential) deflections.
The Prandtl-Meyer function is denoted by the Greek letter nu on the
slide and is a function of the
Mach numberM
and the
ratio of specific heatsgam
of the gas. The physical interpretation of the Prandtl-Meyer function is that
it is the angle through which you must expand a sonic (M=1) flow to obtain
a given Mach number.
To compute an expansion from some other Mach number, we denote the upstream
conditions as zone "0" and we calculate the Prandtl-Meyer angle for that
Mach number.

where atan is the
trigonometric inverse
tangent function.
It is also written as shown
on the slide tan^-1. The meaning of atan can be explained
by these two equations:

atan(a) = b

tan(b) = a

We then add the expansion angle a to get the value of the
Prandtl-Meyer function in zone "1".

nu1 = nu0 + a

Then we must iterate (or look up in a table)
to obtain the Mach number in zone "1" which gives that value of the Prandtl-Meyer
function.

Knowing the Mach number in zone "1" and knowing that the flow is
isentropic, we can relate the value of all the flow variables in zone "1" to the
variables in zone "0" through the ratio to the total conditions given on the
isentropic flow
page.

The centered expansion shown on this slide is a special case of the
distributed expansion. Flow expansions can be generated over a long distance
not the sharp edge noted on the slide. Since the flow is isentropic, it is
reversible. Under very careful conditions, we can create isentropic compressions
which take a high Mach number flow down to low Mach numbers, and increase the
static pressure without the loss of total pressure associated with shock waves.
Isentropic compression inlets have been designed for high speed flows, but
typically remain isentropic only over a narrow operating range. If the free
stream Mach number is changed, the isentropic compression waves often coalesce
into a shock wave with the accompanying loss in total pressure.

Here's a Java program which solves the expansion fan problem.

To study the expansion fan problem, push the "Single Wedge" button
at the upper right.
Input to the program is made
using the sliders, or input boxes at the lower left. To
change the value of an input variable, simply move the slider. Or
click on the input box, select and replace the old value, and
hit Enter to send the new value to the program.
For the expansion fan problem, change "Angle 1" to a negative value by
moving the slider to the left.
Output from the program is displayed
in output boxes at the lower right.
The user selects the Zone to display by using the choice button.
The flow variables are presented as ratios to the previous zone (up)
and to the free stream value (0).
The graphic at the upper left shows the wedges (in yellow)
and the shock wave generated by the wedge as a line. The line is colored
green for an oblique shock and white when the shock is a normal shock.
The expansion fan is shown by two magenta lines which define the borders of
the fan.
The user can move the display by clicking on the graphic, holding down,
and dragging the graphic. You can zoom in or out of the graphic by
using the slider at the top. If you loose the graphic, click on "Find"
to restore the initial display.
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: