hai

has any body attempted or had come across any literature to derive a 4th order accurate pressure Poisson solver on a staggered Cartesian grid in the numerical simulation of incompressible flows. is it really makes difference with a second order accurate PPS which is already well established.

ya I have applied. Not much differnece in solutions. I feel.

Suggest that BC's will be implemented around 1.5 to 2nd order accurate which would pollute the interior solution kernels set to 4th order spatial accuracy. Also, source terms of PPE will likely only be 1st or 2nd order (read advection terms and DIV operators). Also, near boundary spatial DELTA() term will reduce to 2nd order with the available points in the stencil

Thus, I would not expect any gain at all.

The accuracy of the poisson solver will also depend on the accuracy of the momentum equations. If you use algorithms like PISO which are second-order accurate, the increase in accuracy of the poisson solver will not be of much difference. Also you need to keep in mind with high-order accuracy your solution might become unbounded.

"f you use algorithms like PISO which are second-order accurate,.."

Actually PISO's only first order accurate (in time) - it's accuracy is often misquoted in the literature, I've seen comments stating that it's first, second and even third order. PISO algorithm really has very little to do with the spatial accuracy.