A coupled high-resolution fractional-step artificial compressibility and pressure-projection formulation for solving incompressible multi-species variable density flow problem at low Reynolds numbers

In this paper, a coupled fractional-step (FS) artificial compressibility (AC) and pressure-projection (PP) formulation, thus labeled as FSAC-PP method, have been developed for solving incompressible, multi-species, variable density flow problems. The advantageous features of hyperbolic system of AC formulation and the elliptic-type FS-PP method have been put together in order to a) overcome the numerical stiffness of the classical AC method at low Reynolds numbers; b) combine the accuracy properties of the characteristics-based (CB) scheme with PP method; and c) improve further the accuracy and efficiency of the standard AC solution procedure. The proposed FSAC-PP formulation has been developed in the framework of high-resolution Godunov-type algorithm in conjunction with CB scheme. In the FSAC-PP formulation, similarly to the classical FS-PP method, the pressure gradient is neglected in the momentum equation of the dual-time stepping procedure. In order to approach and enforce the divergence-free constraint, after performing the pseudo-time advancement by using a fourth-order explicit Runge-Kutta integration scheme, a pressure-Poisson equation has been solved in a similar way to the original FS-PP method. After solving the pressure-Poisson equation, the velocity field has been updated. The proposed FSAC-PP method has also been coupled with a non-linear, full-multigrid and full-approximation storage (FMGFAS) acceleration technique to increase the efficiency of the developed method. In this paper, a numerical example has been presented for steady-state, incompressible, multi-species, variable density flow in a three-dimensional Y-junction microfluidic channel. The proposed FSAC-PP algorithm, by employing 3rd-order intercell flux interpolation, shows better convergence properties than the classical AC and FS-PP methods at low Reynolds number. Therefore, the incompressible, multi-species, variable density FSAC-PP formulation may also be used in the wide range of scientific and engineering applications in the field of microfluidics.