The development of a Cartesian cut cell method for incompressible viscous flows

This paper describes the extension of the Cartesian cut cell method to applications involving unsteady incompressible viscous fluid flow. The underlying scheme is based on the solution of the full Navier-Stokes equations for a variable density fluid system using the artificial compressibility technique together with a Jameson-type dual time iteration. The computational domain encompasses two fluid regions and the interface between them is treated as a contact discontinuity in the density field, thereby eliminating the need for special free surface tracking procedures. The Cartesian cut cell technique is used for fitting the complex geometry of solid boundaries across a stationary background Cartesian grid which is located inside the computational domain. A time accurate solution is achieved by using an implicit dual-time iteration technique based on a slope-limited, high-order, Godunov-type scheme for the inviscid fluxes, while the viscous fluxes are estimated using central differencing. Validation of the new technique is by modelling the unsteady Couette flow and the Rayleigh-Taylor instability problems. Finally, a test case for wave run-up and overtopping over an impermeable sea dike is performed