A residual-based non-orthogonality correction for force-balanced unstructured volume-of-fluid methods

Abstract The collocated unstructured finite volume method is widely used to simulate flows in geometrically complex flow systems. It supports arbitrary finite volume shapes, significantly simplifying the generation of boundary-fitted domain discretizations (meshes). However, fitting the mesh to the domain boundary introduces the so-called non-orthogonality discretization error, severely impacting the stability and accuracy of two-phase flow simulations where the fluid interface meets the domain boundary. We show that the non-orthogonality error introduces a significant imbalance of forces at fluid interfaces when the one-field formulation of the Navier–Stokes equations is discretized by the collocated unstructured volume-of-fluid method on boundary-fitted non-orthogonal meshes. This force imbalance error leads to large errors when the flow is driven by the surface tension or gravitational force, manifesting as artificial (parasitic) velocities. We deterministically recover the force balance using a residual-based control of the non-orthogonality defect correction. Our approach for applying the non-orthogonality correction offers three key advantages for simulations of wetting processes on geometrically complex surfaces. First, contrary to conventional non-orthogonal correction methods, in our approach, the number of non-orthogonal corrections is no longer a user-defined parameter, to be determined by trial and error. Second, our approach significantly increases accuracy and simultaneously speeds up the solution algorithm for the one-field formulation of the Navier–Stokes equations for two-phase flows. Finally, our approach can be applied with minimal modifications to existing methods. The canonical stationary droplet and stationary water column verification cases and a complex droplet wetting a spherical obstacle verify and validate the proposed method.