An efficient quasi‐Newton method for two‐dimensional steady free surface flow

Abstract

Steady free surface flows are of interest in the fields of marine and hydraulic engineering. Fitting methods are generally used to represent the free surface position with a deforming grid. Existing fitting methods tend to use time-stepping schemes, which is inefficient for steady flows. There also exists a steady iterative method, but that one needs to be implemented with a dedicated solver.Therefore a new method is proposed to efficiently simulate two-dimensional (2D) steady free surface flows, suitable for use in conjunction with black-box flow solvers. The free surface position is calculated with a quasi-Newton method, where the approximate Jacobian is constructed in a novel way by combining data from past iterations with an analytical model based on a perturbation analysis of a potential flow. The method is tested on two 2D cases: the flow over a bottom topography and the flow over a hydrofoil. For all simulations the new method converges exponentially and in few iterations. Furthermore, convergence is independent of the free surface mesh size for all tests. K E Y W O R D Sfitting method, free surface flow, perturbation analysis, quasi-Newton INTRODUCTIONThis paper considers the numerical simulation of steady free surface flow of incompressible, immiscible fluids with a large density difference, typically water and air. This type of flow is often encountered in the fields of marine and hydraulic engineering, for example to calculate ship hull resistance, analyze ship-wall interactions in narrow straight canals and study flow behavior in river confluences. These flows are governed by the incompressible Navier-Stokes equations, which are typically solved with computational fluid dynamics (CFD). The free surface causes additional difficulties for the flow computations: its position is unknown a priori, and needs to be determined as a component of the solution during the computation. Two classes of methods exist to represent the free surface, namely, surface capturing and surface fitting methods. * In surface capturing approaches, the computational mesh is not aligned with the interface, and the interface can intersect the mesh in an in principle arbitrary manner. Capturing methods are versatile in that they can handle complex phenomena such as wave breaking. Examples are the marker-and-cell method, 2 the volume-of-fluid method, 3 and the level-set method. 4 *Classification and terminology of these methods tends to be inconsistent in the literature; terminology is adopted from Wackers et al. 1Int J Numer Meth Fluids. 2020;92:785-801.wileyonlinelibrary.com/journal/fld

On the effect of nonlinearity and Jacobian initialization on the convergence of the generalized Broyden quasi‐Newton method

“…In practice, this is dealt with by adapting  slightly: in each iteration a constant value is added to the input height to satisfy the inlet condition, and the average is subtracted from the output pressure. See this paper 27 for a more detailed explanation. § Note that the terms "good" and "bad" are no reference whatsoever to Broyden's first and second method; we use these adjectives in the literal sense, to distinguish between a very accurate and a rather poor surrogate model.…”

“…One way to do this is to solve a nonlinear system  (x) = 0, where x contains the (discretized) vertical position of the free-surface and  is a flow solver which returns the free-surface pressure  (x). 27 This corresponds to finding the free-surface shape that yields zero (atmospheric) pressure at the free-surface. ‡ Especially at higher flow velocities, it is important to take viscous and turbulent effects into account to obtain the correct shape.…”

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

“…This results in the Free surface Quasi-Newton method with Least-Squares and Surrogate (FreQ-LeSS), which we recently developed for 2D steady free surface flow. 1 This section describes a version which is adapted to deal with the added complexities of 3D problems.…”

“…We recently proposed a fast method to solve two-dimensional (2D) steady free surface flow problems. 1 That method is extended to three-dimensional (3D) problems in this article. To position the new method in the literature and to explain the main design choices, we first give an overview of existing computational methods for steady free surface flow.…”

“…Following this approach, we successfully developed such a method for 2D flows: the Free surface Quasi-Newton method with Least-Squares and Surrogate (FreQ-LeSS; phonetically freckles). 1 The FreQ-LeSS algorithm considers the normal DBC as a root-finding problem and solves it with an efficient quasi-Newton scheme, yielding a new estimate for the free surface position in each iteration. The flow solver is treated as a black-box solver during the quasi-Newton iterations, ensuring compatibility of FreQ-LeSS with existing CFD packages.…”