Eq.(10) will not describe the flow correctly in time, but as the solution converges to steady state, the time dependent term in Eq.(10) vanishes and the continuity equation, Eq.(1) is satisfied.
The equations to be solved numerically are Eq.(10) and Eq.(4). This is done by the use of a cell centered finite volume method combined with an explicit time integration scheme. In the following a brief description of the method is given, for further details the reader is referred to .
The 2D version of the equations, written in conserved form are,
where where has been substituted for p in the temporal term of the continuity equation. is the preconditioning matrix,
This set of equations are marched forward in pseudo time until steady state is obtained. The time integration is done using a similar Runge Kutta scheme as described in further details in the next section.