Harlow and Welch (Phys. Fluids 8:2182-2189, 1965) introduced a discretization method for the incompressible Navier-Stokes equations conserving the secondary quantities kinetic energy and vorticity, besides the primary quantities mass and momentum. This method was extended to fourth order accuracy by several researchers ,  and . In this paper we propose a new consistent boundary treatment for this method, which is such that continuous integration-by-parts identities (including boundary contributions) are mimicked in a discrete sense. In this way kinetic energy is exactly conserved even in case of non-zero tangential boundary conditions. We show that requiring energy conservation at the boundary conflicts with order of accuracy conditions, and that the global accuracy of the fourth order method is limited to second order in the presence of boundaries. We indicate how non-uniform grids can be employed to obtain full fourth order accuracy.
Sanderse, B., Verstappen, R. W. C. P., & Koren, B. (2014). Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier-Stokes equations. Journal of Computational Physics, 257(Part B), 1472-1505. https://doi.org/10.1016/j.jcp.2013.10.002