The anisotropic eddy-viscosity model contains nonlinear elements. Difficulties
for iterative solution methods can be circumvented by writing the stress term
in terms of the implicit Boussinesq formulation plus an
explicit quadratic term. This can be regarded as the well-known defect correction
technique. Hence, we define the extra turbulent stress 
for
the nonlinear model as the difference between the total stress
and the linear model contribution,
which is explicitly calculated at every time step, using the values from the
previous time step. The linear model is treated implicitly, which means that the
laminar viscosity 
is increased by the turbulent viscosity 
in the
deviatoric stress term (4.2). The right-hand side of the
momentum equations is updated with the extra turbulent stress, as follows
Taken 
as an example, the finite volume discretization results in
The discretization is completed by substituting (6.27) in (6.29), followed by central differences and bilinear interpolations. Due to the covariant derivatives of the contravariant velocity components in (6.27) extra Christoffel symbols are introduced in the discretized right-hand side of the momentum equations. At this moment, the above discretization is only carried out in the inner region.
