A multigrid ghost-point level-set method for incompressible Navier-Stokes equations on moving domains with curved boundaries

Abstract

In this paper we present a numerical approach to solve the Navier-Stokes equations on moving domains with second-order accuracy. The space discretization is based on the ghost-point method, which falls under the category of unfitted boundary methods, since the mesh does not adapt to the moving boundary. The equations are advanced in time by using Crank-Nicholson. The momentum and continuity equations are solved simultaneously for the velocity and the pressure by adopting a proper multigrid approach. To avoid the checkerboard instability for the pressure, a staggered grid is adopted, where velocities are defined at the sides of the cell and the pressure is defined at the centre. The lack of uniqueness for the pressure is circumvented by the inclusion of an additional scalar unknown, representing the average divergence of the velocity, and an additional equation to set the average pressure to zero. Several tests are performed to simulate the motion of an incompressible fluid around a moving object, as well as the lid-driven cavity tests around steady objects. The object is implicitly defined by a level-set approach, that allows a natural computation of geometrical properties such as distance from the boundary, normal directions and curvature. Different shapes are tested: circle, ellipse and flower. Numerical results show the second order accuracy for the velocity and the divergence (that decays to zero with second order) and the efficiency of the multigrid, that is comparable with the tests available in literature for rectangular domains without objects, showing that the presence of a complex-shaped object does not degrade the performance.