Applying the free-slip boundary condition with an adaptive Cartesian cut-cell method for complex geometries

The slip condition at the interface of a multiphase flow can occur in situations including micro-and nano-fluidic flow, flow over hydrophobic surfaces, rising bubbles in quiescent liquid, and polymer extrusion processes. The aim of this work is to implement the free-slip boundary condition with an adaptive Cartesian grid method. The Navier-Stokes (NS) equations are solved by a cell-centered collocated finite volume method with adaptive mesh refinement. The arbitrarily-shaped solids imbedded in the computational domain are treated by the cut-cell method where the geometric properties of cut-cells near the boundary are computed through robust geometric operations. In discretized NS equations, the second-order-accurate center difference method is used to estimate the surface fluxes of the regular Cartesian cells in the bulk region, whereas the least-squares method is used to estimate the fluxes of the cut-cells near the boundary. A local coordinate system aligned with the normal and tangential directions of the solid boundary is defined for each cut-cell in order to properly implement the free-slip condition. The tangential velocities at the curved solid boundary are obtained using the free-slip condition and the principal curvatures of the solid surface. The proposed numerical method is implemented in the open-source code Gerris. Numerical tests have been carried out to validate our method. The tests confirm the excellent performances of the proposed method. Although our work focuses on the free-slip condition, the extension of the proposed method to more general slip conditions is straightforward.