In this section we describe the space discretization of the continuity and momentum equations in the inner region. Discretizations due to the boundary conditions are treated in Section 7.
The equations describing the mean velocity field in incompressible turbulent flow follow from the momentum equations by the Reynolds decomposition of the instantaneous velocity field into a mean and a fluctuating part. The momentum equations in general co-ordinates read (see [34], formula 5.2)
with 
the deviatoric stress tensor given by
Here, 
is the contravariant mean velocity, 
the density, p the
mean
pressure, 
some external force per unit volume, 
dynamic
viscosity and 
the turbulent stress tensor
( 
denotes contravariant velocity fluctuation),
which has to be specified. This specification is accomplished by a two-equation
eddy-viscosity turbulence model. This will be presented in Section 6.
When calculating laminar flows, the turbulent stresses are set to zero.
In the present version all coefficients may depend on space, time and
previous computed solutions. However, with respect to the density a correct
implementation is only guaranteed for is constant. Furthermore, the
discretization presented below has been carried out as if the flow is assumed
laminar.
The continuity equation reads (see [34], formula 5.1):
As unknowns the fluxes 
are used.
Equations (4.1), (4.2) and (4.3) are discretized by a
finite volume method.
We distinguish between the 2D and the 3D case.
