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 objects 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.
When an object 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.

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 number M and the
ratio of specific heats gam of the gas.

nu = {sqrt[(gam+1)/(gam-1)]} * atan{sqrt[(gam-1)*(M^2 - 1)/(gam+1)]} - atan{sqrt[M^2 -1]}

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

As mentioned above, the Mach number of a supersonic flow increases
through an expansion fan. The amount of the increase depends on the
incoming Mach number and the angle of the expansion.
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. The value of the Prandtl-Meyer function is therefore
called the Prandtl-Meyer angle.

Here's a JavaScript program which solves for the Prandtl-Meyer angle .

Isentropic Flow Calculator

Input

Output

Mach

Mach Angle

P-M Ang

p/pt

T/Tt

rho/rhot

q/p

A/A*

Wcor/A

You select an input variable by using the choice button labeled Input
Variable. Next to the selection, you then type in the value
of the selected variable. When you hit the red COMPUTE button,
the output values change.
The default input variable is the Mach number, and by varying Mach number
you can see the effect on Prandtl-Meyer angle. You can also select
Prandtl-Meyer angle as an input, and see its effect on the flow variables
downstream of the turning.

If you are an experienced user of this calculator, 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: