  The treatment of complex geometries in Computational Fluid Dynamics
applications is a challenging endeavor, which immersed boundary and cut-cell
techniques can significantly simplify by alleviating the meshing process
required by body-fitted meshes. These methods however introduce new challenges,
as the formulation of accurate and well-posed discrete operators becomes
nontrivial. Here, a conservative cartesian cut cell method is proposed for the
solution of the incompressible Navier--Stokes equation on staggered Cartesian
grids. Emphasis is set on the structure of the discrete operators, designed to
mimic the properties of the continuous ones while retaining a nearest-neighbor
stencil. For convective transport, a divergence is proposed and shown to also
be skew-symmetric as long as the divergence-free condition is satisfied,
ensuring mass, momentum and kinetic energy conservation (the latter in the
inviscid limit). For viscous transport, conservative and symmetric operators
are proposed for Dirichlet boundary conditions. Symmetry ensures the existence
of a sink term (viscous dissipation) in the discrete kinetic energy budget,
which is beneficial for stability. The cut-cell discretization possesses the
much desired summation-by-parts (SBP) properties. In addition, it is fully
conservative, mathematically provably stable and supports arbitrary geometries.
The accuracy and robustness of the method are then demonstrated with flows past
a circular cylinder and an airfoil.
