Study of a staggered fourth‐order compact scheme for unsteady incompressible viscous flows

The objective of this paper is the development and assessment of a fourth-order compact scheme for unsteady incompressible viscous flows. A brief review of the main developments of compact and high-order schemes for incompressible flows is given. A numerical method is then presented for the simulation of unsteady incompressible flows based on fourth-order compact discretization with physical boundary conditions implemented directly into the scheme. The equations are discretized on a staggered Cartesian non-uniform grid and preserve a form of kinetic energy in the inviscid limit when a skew-symmetric form of the convective terms is used. The accuracy and efficiency of the method are demonstrated in several inviscid and viscous flow problems. Results obtained with different combinations of second- and fourth-order spatial discretizations and together with either the skew-symmetric or divergence form of the convective term are compared. The performance of these schemes is further demonstrated by two challenging flow problems, linear instability in plane channel flow and a two-dimensional dipole–wall interaction. Results show that the compact scheme is efficient and that the divergence and skew-symmetric forms of the convective terms produce very similar results. In some but not all cases, a gain in accuracy and computational time is obtained with a high-order discretization of only the convective and diffusive terms. Finally, the benefits of compact schemes with respect to second-order schemes is discussed in the case of the fully developed turbulent channel flow. Copyright © 2008 John Wiley & Sons, Ltd.