Imposing the free-slip condition with a continuous forcing immersed boundary method

The numerical simulation of spherical and ellipsoidal bubbles in purified fluids requires the imposition of the free-slip boundary condition at the bubble surface. This paper describes a numerical method for the implementation of free-slip boundary conditions in the context of immersed boundary methods. In contrast to other numerical approaches for multiphase flows, the realization is not straightforward. The reason is that the immersed boundary method treats the liquid as well as the gas phase as a field of constant density and viscosity with a fictitious fluid inside the bubble. The motion of the disperse phase is computed explicitly by solving the momentum balance for each of its elements and is coupled to the continuous phase via additional source terms in the Navier-Stokes equations. The paper starts with illustrating that an ad hoc method is unsuccessful. On this basis, a new method is proposed employing appropriate direct forcing at the bubble surface. A central finding is that with common ratios between the step size of the grid and the bubble diameter, curvature terms need to be accounted for to obtain satisfactory results. The new method is first developed for spherical objects and then extended to generally curved interfaces. This is done by introducing a local coordinate system which approximates the surface in the vicinity of a Lagrangian marker with the help of the two principal curvatures of the surface at this point. The numerical scheme is then validated for spherical and ellipsoidal objects with or without prescribed constant angular velocity. It is shown that the proposed method achieves similar convergence behavior as the method for no-slip boundaries. The results are compared to analytical solutions for creeping flow around a sphere and to numerical reference data obtained on a body-fitted grid. The numerical tests confirm the excellent performance of the proposed method.