Elimination of spurious velocities generated by curvature dependent surface force in finite element flow simulation with mesh-fitted interface

[[abstract]]It is known that spurious non-physical velocities can occur when one employs the finite element method for simulation of incompressible flows subjected to external forces. In presence of external body forces, the main reason for this is the incompressibility constraint that is satisfied only in a weak sense against test functions from the pressure function space. In case of the two-phase (incompressible) immiscible flow, a surface force, which is a function of the interface curvature, arises and introduces additional problematics to the finite element model. Due to discrete representation of the interface, the question arises on how to approximate the curvature. A particularly natural approach for the finite element method employs the Laplace–Beltrami operator which allows to express the mean curvature in a weak sense. However, once incorporated into the equations governing the fluid flow, Laplace–Beltrami-reconstructed curvature may introduce spurious non-physical forces at the interface if finite element spaces are chosen arbitrarily. The reason for this is that the test space used for curvature calculation is the test space associated with the velocity field. We show that it is necessary for the function space used for the geometry construction to be of the order equal to or higher than the order of the test space involved in curvature evaluation. This leaves two possibilities for practical fluid flow problems: use the same function spaces for the mesh geometry and the velocity field (isoparametric concept) or decouple the curvature calculation from the main problem