When the co-located arrange is used, all variables belong to the same elemental volume in order to evaluate the properties. However, examining the coefficients of the equation of P’, given by (7.3.20), we notice that we need the parameters at the interfaces. These parameters do not exist at the interfaces, because the motion equations are not solved in these places. The source term of (7.3.21) also requires the velocities at the interfaces. In order to obtain all these terms at the interfaces we create a pseudo-equation to the boundaries from equations of P and from the neighbor volumes (E, W, N, S). Doing this we determine the velocities and parameters at the interface.

The velocity equations for u and v for a volume P are given by:

are the central coefficients for motion equations in the respective volumes I.

We should not forget that for (7.4.1) to (7.4.5), the sums contain all neighbors, including those which do not bound the volume P, i. e.:

Evaluating linear interpolations between (7.4.1a) and each equation from (7.4.2a) to (7.4.5a), and also between (7.4.1b) and each equation from (7.4.2b) to (7.4.5b), we find out the expressions for the at the interfaces:

Velocities at the East face of elemental volume:

As we have mentioned in chapter 4, fictitious volumes do not contain pressure values that would be used to evaluate gradients. Therefore, notice that in the second term on the right side of equations (7.4.9) we use lower orders approximations:


North face volume:

For the gradient of pressure at the West face there are the same approximations given at the East face, when the volume belongs to North or South boundary. We should remark that for the elemental volumes at West face of the domain, the velocities are evaluated using the boundary conditions. The lower order approximations are given by:

For pressure gradients of pressure with lower order approximations the volumes are found at East and West face of the domain. We should not forget for the volumes at North face of the domain, these velocities are obtained by the boundary conditions. The lower order approximations are given by:

As for North face, the pressure gradient of a control volume at the South face should be approximated by lower orders approximation when it is placed at the East or West faces. Remembering again that for the volumes at South face these velocities are obtained by the boundary conditions. The lower order approximations are given by:

Finding, finally, the Cartesian velocities at the faces of the elemental volumes, we can determine the mass balance from the source term of P’ (7.3.19), in which the terms from (7.3.21) are obtained from (3.2) and (3.3), and are given by: