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The behaviour of an ideal compressible gas is described by the Euler equations:
with
 |
(4) |
the internal energy of the gas.
Reducing these to one spatial dimension and using cartesian
coordinates one can write them as
 |
(5) |
with
the so called state vector (
,
,
)
, and
the flux vector (
,
, 
). The
elements of the state vector are also called the conserved
quantities since the Euler equations basically say that mass,
momentum and energy are conserved. The variables
,
, and
are often referred to as the primitive variables.
The Euler equations are not complete without an equation of state. We
choose an ideal gas for which
 |
(6) |
where
for a monatomic gas, and
 |
(7) |
with
the mean molecular weight.
A more general form of the Euler equations is
 |
(8) |
in which
now is an arbitrary spatial coordinate and
is
the source vector.
can be split into two parts:
geometrical source terms which arise in the case of non-cartesian
coordinates, and physical source terms such as radiative heating and
cooling, gravitation, etc.
We will be using the spherical coordinate
. For this the momentum
equation is
 |
(9) |
and similar expressions can be written down for the continuity and
energy equations. To write this in the form of Eq. (8) we take
is still the vector of conserved quantities vector since it
is total mass, momentum and energy which are conserved, not their
densities. In spherical coordinates a volume element is proportional
to
(really
but we are not considering the
coordinate here), so by multiplying the densities by the
position dependent volume factor
, one obtains the conserved
quantities.
One sees that choosing a non-cartesian coordinate leads to the
appearance of a geometrical source term. There are actually several
ways in which to split off the geometrical source terms, but here we choose
the one shown in Eqs. (10)-(12).
Next: Properties of the Euler
Up: Introduction
Previous: Introduction
Garrelt Mellema
2003-01-20