A new formulation toward unifying the velocity role in collocated variable arrangement

One main challenge in numerical treatment of incompressible fluid flow problems is to suppress the decoupling of pressure and velocity fields. The challenge has prompted research toward suggesting and implementing various coupling strategies. In this work, a novel strategy which suitably couples pressure and velocity in a collocated grid arrangement is presented. The current strategy develops a unique cell-face velocity expression which provides infinite cell-face velocity magnitudes in the algorithm. A smoothing factor is incorporated in the cell-face velocity expression in order to produce a wide range of velocity magnitudes. The smoothing factor provides a smooth transition from an unreal zigzag pressure field to an acceptable physical distribution. The extended formulation is then examined in a domain with a source and a sink which represent either velocity or pressure discontinuities in the domain. The discontinuities may in turn lead to nonphysical solution if the pressure-velocity coupling is weak or inappropriate. The current investigation shows that the new extended formulation provides a robust approach to cure the decoupling problem.