On the integral constraint of the pressure Poisson equation for incompressible flows

We illustrate, using analytical and numerical proofs, how a conservative discretisation of the pressure Poisson equation arising out of the discretisation of the incompressible NavierStokes equations (on a two-dimensional unstructured non-staggered grid) satisfies the integral constraint on the pressure boundary condition without any additional treatment. When discretised in a non-conservative manner, it is seen that the integral constraint is not exactly satisfied, but only to an order , where is an appropriate velocity scale. When solved using an iterative method, such as the Bi-Conjugate Stabilised method, it is proved that the vanishing sum of residuals on all points inclusive of the boundary is a consequence of this integral constraint. This result can then be used as a tool to identify whether the discrete integral constraint has been satisfied or not, especially when the pressure is solved as a Neumann problem.