Unsteady three-dimensional flows dominated by free-surface dynamics: numerical insights for fluvial and coastal applications

Abstract

Unsteady, three-dimensional free-surface flows are ubiquitous in coastal and fluvial environments. They are characterised by a dynamic free-surface that evolves and modifies its shape in response to local flow hydrodynamics, and a bulk flow typically in the turbulent regime. The conservation laws that govern the evolution of this system are the incompressible Navier-Stokes equations.

When the free-surface evolution is considered, a set of normal and tangential dynamic boundary conditions, derived from the local stress balance at the free surface, must be incorporated to define a physically-consistent problem.

Disregarding this fundamental aspect of the phenomenon may lead to inaccurate solutions when the problem is addressed numerically, as evidenced by the recurring issues with mass conservation and poorly captured vortical dynamics reported in the literature.

In this work, a fully-unsteady single-phase three-dimensional free-surface viscous numerical model based on an Artificial Compressibility Navier-Stokes solver and the Level-Set method for capturing the free-surface evolution, is presented. Special emphasis is placed on the correct implementation and coupling of physically-consistent free-surface boundary conditions, which is achieved by the combination of the Ghost-Fluid method with a solution procedure based on a modified version of the Weighted Least Squares method.

The evolution of the free surface is captured using the Level-Set method. Its advection equation is advanced in time with an explicit third-order accurate Total Variation Diminishing Runge–Kutta scheme, while the convective term is discretised using a third-order accurate Weighted Non-Oscillatory scheme.

Although these high-order methods ensure an accurate solution procedure, they cannot fully prevent degradation of mass conservation properties over time. This issue is addressed by a mass-preserving geometric redistancing procedure, adapted from that presented by Ausas et al., (2011), and modified to enforce strong mass conservation throughout the entire simulation period.

This combination of methods has proven to be a robust approach for obtaining accurate solutions of unsteady free-surface flows.

The solver is thoroughly tested and validated, and then applied to study the interaction between a solitary wave and two identical in-stream obstacles at varying separations. While the separation had minor effects on the wave run-up and pressure-induced loadings on them, it showed to have a strong influence on lateral friction forces and inter-obstacle bed shear stresses, due to the enhanced streaming observed at smaller spacings. Unlike in-stream structures subjected to shear currents, the wave–obstacle interaction did not lead to the formation of a self-sustained Horseshoe Vortex in front of the obstacle. Instead, higher bed shear stresses were observed at the corners of the structure. The vortical structures that emerged from the interaction between the wave and the obstacles were also studied. They remained confined near the obstacles, not leading to the formation of a self-sustained shear layer. Despite these observations, in order to elucidate the underlying physics of this behaviour and identify conditions that promote stronger vortical activity, extending the study to a wider range of conditions is proposed as future work.