An efficient ψ‐v scheme for two‐dimensional laminar flow past bluff bodies on compact nonuniform grids

Abstract

Summary We recently proposed a second‐order accurate ψ‐v formulation of the steady‐state Navier‐Stokes (N‐S) equations on compact Cartesian nonuniform grids. In the current work, we extend the ideas of the aforesaid formulation and propose a second‐order spatially compact, implicit, stable ψ‐v formulation for the unsteady incompressible N‐S equations. Contrary to the existing ψ‐v finite difference formulations which use grid transformation, the proposed scheme is developed for nonuniform Cartesian grids without transformation specifically designed for two‐dimensional laminar flow past bluff bodies. It has been implemented on problems of internal flows inside curved regions as well as those involving fluid‐embedded body interaction. However, the robustness of the scheme is highlighted by the accurate resolution of a host of complex flows past bluff bodies with different physical set‐ups and boundary conditions. It was seen to handle problems involving both uniform and accelerated flows across a wide range of structures of varied shape, namely, a flat plate, a circular cylinder, inclined square cylinder, and a wedge in channel hinged to the wall. Apart from elegantly capturing all the details of the shedded vortex structures under different circumstances, the scheme was also able to handle both Dirichlet and Neumann boundary with equal ease. In all the cases, our results are found to be extremely close to the available numerical and experimental results.

ψ‐v Computation of steady‐state conjugate heat transfer in backward‐facing step flow

“…Over the years the application of higher‐order compact (HOC) schemes to computing incompressible viscous flows has gained popularity because of their simplicity, accuracy, and advantages associated with compact stencils, both uniform 41–48 and nonuniform 49–55 . For the simulation of 2D incompressible viscous flows, the ‐ form of the N–S equations has a relative advantage over the primitive‐variable form due to the pressure term being present.…”