The spatial and temporal discretizations yield a set of coupled algebraic equations for the velocity, pressure and scalar quantities. One should have some idea about how this coupled set of equations is going to be solved. The choice affects the coupling of the various unknowns. Basically, the velocity and pressure fields are coupled through the continuity equation. For turbulent flows, the momentum equations are coupled to turbulence transport equations through the eddy-viscosity. Also a strong coupling exists between the turbulence equations.
In principle, there are two approaches for the solution of the coupled set of discretized equations. The first one is to solve all equations simultaneously at each grid point. By contrast, an uncoupled solution technique proceeds sequentially through the equations by treating the other variables as known until the converged solution of the coupled set of equations is obtained. The coupled solution method requires a very large computer memory, but may have better rate of convergence and numerical stability than the uncoupled one. Nevertheless, solving all equations simultaneously may be so complicated that coupled solution procedures are difficult to use. It may then be preferable to employ uncoupled methods or a blended form of both strategies.
Here, the following overall solution algorithm will be used. For each time step,
the process start
by guessing the variables 
, p and 
,
either
initially or from the previous time level. Note that the guessed velocity must
satisfy the incompressibility constraint. Then the continuity equation and
the coupled momentum equations are solved using the non-updated eddy-viscosity, if
applicable.
To ensure a divergence-free velocity field the pressure correction scheme as
will be outlined in Section 9 is used. Note that the linear momentum
and pressure correction equations are solved sequentially. Because of the
nonlinearities this loop ( 
) may be repeated until a
converged nonlinear result is obtained, but one Newton iteration in each
time step is sufficient. Finally, the transport equations and then
turbulence equations are solved in a decoupled way using the updated mean
flow quantities and non-updated eddy-viscosity, if applicable. The transport
equations are solved in the sequence given by their index number, whereas the
equation for 
or 
is solved after k. It may be necessary to
repeat this
loop ( 
) in each time step until convergence
is reached. In addition, the outer loop
( 
),
which contains three inner loops ( ),
( 
) and
( )
coupled via turbulent viscosity, may be repeated until all variables at time level
n+1 converge. However, at this moment one inner
and one outer iteration cycle per time step suffice.