The general transport equation in Cartesian co-ordinates reads:
where 
stands for a physical scalar (e.g. temperature, concentration,
turbulence quantities), 
and D are general functions, k is a
symmetric diffusion matrix (d is the space dimension) and 
is a source term. These functions may depend on other unknowns, like 
and .
Translated into general co-ordinates this equation becomes (see
[34], formula 5.4):
with 
. Using (2.20), equation (5.2) can be
written in a form which is suitable for the discretization by the finite volume
method:
with
The transport equation (5.3) is integrated over a pressure cell with center (i,j,k) which yields
whereas the right-hand side is integrated using the midpoint rule:
Next, (5.4) is substituted in (5.5) which completes
the discretization. Since the unknown is only given in the center of a
cell, further approximation is needed. Central differences should be used because
of second order accuracy. The problem here is, however, that such schemes tend
to give rise to oscillations which are found to damage the stability of solutions
of the two-equation models, since negative values of the turbulent quantities
tend
to be enlarged by nonlinearities and strong coupling between the model equations,
which prevents the solutions to converge. This matter will be discussed in more
detail in Section 5.3.
