An efficient quasi-Newton method for three-dimensional steady free surface flow

In this article, we introduce the Free surface Quasi-Newton method with Least-Squares and Surrogate (FreQ-LeSS) to solve steady free surface flows, as encountered in, for example, marine applications. The method has two important advantages over other methods. First of all it is fast thanks to the efficient iterative scheme based on quasi-Newton iterations that make use of an advanced surrogate model. Second, it has good compatibility because it poses no special requirements on the flow solver that is called in the algorithm. The excellent convergence speed of the FreQ-LeSS method is demonstrated for a shallowly submerged submarine: good quality wave patterns are obtained in few iterations. These results match with ISIS-CFD, validating the FreQ-LeSS method. The total cost of solving this steady free surface problem is only a few times that of solving a steady single-phase flow with fixed free surface position. We conclude that FreQ-LeSS promises to be much faster than the two-phase time-stepping methods typically used to solve these problems, and provides a sharper interface at the same time thanks to the fitting approach. In the present work, we restrict ourselves to problems without surface-penetrating objects, to avoid the associated complications related to the contact-line motion.