Re: pressure correction
You will need to incorporate the variation of density with pressure in the continuity equation, i.e. the pressure correction you solve for will be a function of the velocity perturbations and density fluctuation. This is usually done by appealing to the thermodynamic relation rho = rho(p,h) and assuming the enthalpy is fixed during the outer iterations so that one can approximate drho/dt ~ (drho/dp)*dp/dt. In addition, the enthalpy equation is solved along with the momentum equations to give approximations to (u,v,h). Then the pressure correction equation is solved and the quantities are updated as appropriate. More on this can be found in Peric and Ferziger, pp. 278-287.