# Highlight on Closest Point method

In multiphase flows, surface tension plays an important role, particularly at small scales where it is a dominant phenomenon. Accurate computation of the curvature (that leads the cohesion forces at the interface, called surface tension) is mandatory and is still a challenging task in free surface flows. The Continuum Surface Force (CSF) method is commonly employed in CFD codes.

Curvature computation is based on second derivatives of the surface/interface which, consequently, must be at least 4th order precise. Thus, traditional 2nd order Volume-of-Fluid representation and associated methods (PLIC, Height Function, etc.) diverge for the curvature computation (see [Coquerelle, 2015]).

We have chosen the WENO/Level-Set framework to reach more than 4th order precision (regarding spatial discretization) for the surface transport. Additionally, we have adopted a closest point based approach to extend the curvature from the interface location to the surrounding mesh nodes in the interface thickness. We have proposed a new closest point algorithm which converges at 4th order on the curvature and yields to the reduction of the residual velocity spurious currents (see following figures) at the same precision.

**Figure 1 : Schematic view of the Closest Point algorithms. Errors on gradient vectors have been exaggerated and redundant
notations omitted for the sake of clarity. Green dots stand for intermediate points, blue dots for CP results and the red dot for
the final CP ⊥ result. Normals on the surface are drawn as dotted square ended segments. The top gray dashed paths show an
erroneous closest point due to poor approximations of the direction ∇φ toward the surface.**

**Références**

M. Coquerelle, S. Glockner, A fourth-order accurate curvature computation in a level set framework for two-phase flows subjected to surface tension forces, *Journal of Computational Physics*, 305 (2016), pp838-876 (pdf).

**Figure 2: Spurious currents reduction for the static viscous column equilibrium with variable density test case**.

**Movie 1: Multiple drop impact on a free surface**.