  A new conservative symmetry-preserving second-order time-accurate PISO-based
pressure-velocity coupling for solving the incompressible Navier-Stokes
equations on unstructured collocated grids is presented in this paper. This new
method for implicit time stepping is an extension of the conservative
symmetry-preserving incremental-pressure projection method for explicit time
stepping and unstructured collocated meshes of Trias et al. (2014). In order to
assess and compare both methods, we have implemented them within one unified
solver in the open source code OpenFOAM. We combine both methods with a Butcher
tableau for a family of explicit and diagonally implicit Runge-Kutta temporal
schemes. We assess the energy conservation properties of the implemented
discretisation methods and the temporal consistency of the selected Runge-Kutta
schemes using Taylor-Green vortex and lid-driven cavity flow test cases.
  Although both implemented methods are based on a symmetry-preserving
discretisation, we show that both methods still produce a small amount of
numerical dissipation when the total pressure is directly solved from a Poisson
equation. This numerical dissipation is mainly caused by the corresponding
pressure error which is of $O(\Delta t \Delta h^2)$. When an
incremental-pressure approach is used, where a pressure correction is solved
from a Poisson equation, the pressure error reduces to $O(\Delta t^2 \Delta
h^2)$, yielding better conservation properties: both methods are then
effectively fully-conservative. Furthermore, we conclude that all selected
explicit and implicit higher order temporal schemes suffer from a reduction of
the temporal order to approximately one when the pressure Poisson equation is
based on the total pressure due to the presence of a pressure error of
$O(\Delta t \Delta h^2)$.
