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We introduce a mimetic Cartesian cut-cell method for incompressible viscous flow that conserves mass, momentum, and kinetic energy in the inviscid limit, and determines the vorticity such that the global vorticity is consistent with the boundary conditions. In particular we discuss how the no-slip boundary conditions should be applied in a conservative way on objects immersed in the Cartesian mesh. We use the method to compute the flow around a cylinder. We find a good comparison between our results and benchmark results for both a steady and an unsteady test case. To compute fluid flow in complicated geometries often curvilinear or unstructured meshes are used. The generation of these meshes is difficult and time-consuming. When the geometry depends on time, the mesh has to be updated after every time step and the cost of mesh generation will take a significant part of the total computing time. Immersed boundary methods form an increasingly popular alternative. Immersed boundary methods are methods in which one Cartesian mesh is used for the complete flow domain, with the boundaries of objects immersed in this Cartesian mesh. Near the immersed boundary the Cartesian method is adapted for the no-slip boundary condition. Within the class of immersed boundary methods essentially two approaches for modeling the boundary conditions exist. In the first approach the influence of the boundary on the fluid is modeled by an extra force term in the Navier-Stokes equations. In the second approach the sharp interface of the boundary is maintained and the boundary condition is taken into account by adjusting the discretized Navier-Stokes equations.

A staggered discretization of the incompressible Navier-Stokes equations is presented for polyhedral non orthogonal nonsmooth meshes admitting a barycentric dual mesh. The discretization is constructed by using concepts of discrete exterior calculus. The method strictly conserves mass, momentum and energy in the absence of viscosity.