A fractional step method for unsteady free-surface flow with applications to non-linear wave dynamics

Abstract

A fractional step method is developed for solving the time dependent two‐dimensional Euler equations with full non‐linear free‐surface boundary conditions. The geometry of the free surface is described by a height function, and its evolution is tracked by integrating in time the kinematic boundary conditions based on the free‐surface volume flux. The fluid domain is discretised by adapting a time‐varying curvilinear grid to all boundaries, including the free surface. Mass and momentum equations are discretised by a conservative finite volume formulation, taking into account the time dependency of the grid. A fractional step type method is developed for integrating the fluid motion in time. The method is applied to a non‐linear standing wave in a square container, testing for compliance with mass and energy conservation and comparing computed wave period with other results. Non‐linear travelling waves are simulated in channels with either constant depth or varying depth and non‐linear wave processes involving both triad interactions and quartet interactions are studied. Results are compared with both experimental data and theoretical results and excellent agreement is found. Interaction of waves and currents is studied. The blocking of waves in an opposing current is simulated and found to show good agreement with theoretical results. The method is intended to be a first step towards a full description of wave dynamics interacting with structures and currents. © 1998 John Wiley & Sons, Ltd.

Hydrodynamics of pedestrians' instability in floodwaters

“…This "active sponge" layer is a practical boundary condition, which allows for reducing the number of cells of the computational domain. Mayer et al (1998) and Jacobsen et al (2012) provided a detailed description of the structure of the relaxation function and of the use of relaxation zones as boundary conditions.…”

The impacts of the ALE and hydrostatic-pressure approaches on the energy budget of unsteady free-surface flows

“…(3) shows the straightforward relation between the local rate of change of the free-surface position and the local normal velocity and is used here as a governing equation. Among the reviewed literature, this expression has only been preferred by [5,13]. In addition to the kinematic boundary condition, the free-surface dynamic boundary condition prescribes that forces are in equilibrium across the air-water interface [25,26].…”

“…The third one recalls the integral formulation of the governing equations, as done by [9,10]. In particular, combinations between the ALE approach and finite-volume techniques are reported in [6,8,9,[11][12][13][14][15].…”

Evaluating wave forces on groups of three and nine cylinders using a 3D numerical wave tank

“…The relaxation method (Mayer, Garapon, & Sørensen, 1998) is used for wave generation and absorption in the numerical wave tank. This method uses a wave generation zone and a wave absorption zone to moderate the computational values and the analytical values given by wave theory.…”