Three-dimensional Breaking Waves Research Articles

Highly directionally spread, overturning breaking waves modelled with Smoothed Particle Hydrodynamics: A case study involving the Draupner wave

Wave breaking in the ocean affects the height of extreme waves, energy dissipation, and interaction between the atmosphere and upper ocean. Numerical modelling is a critical step in understanding the physics of wave breaking and offers insight that is hard to gain from field data or experiments. High-fidelity numerical modelling of three-dimensional breaking waves is extremely challenging. Conventional grid-based numerical methods struggle to model the steep and double-valued free surfaces that occur during wave breaking. The Smoothed Particle Hydrodynamics (SPH) method does not fall prey to these issues. Herein, we examine the SPH method’s ability to model highly directionally spread overturning breaking waves by numerically reproducing the experiments presented in McAllister et al. (2019). We find that the SPH method reproduces the experimental observations well; when comparing experimental and numerical measurements we achieve coefficient of determination values of 0.92−0.95, with some smaller-scale features less well reproduced owing to finite resolution. We also examine aspects of the simulated wave’s geometry and kinematics and find that existing breaking criteria are difficult to apply in highly directionally spread conditions.

Wake behind a three-dimensional dry transom stern. Part 1. Flow structure and large-scale air entrainment

We present high-resolution implicit large eddy simulation (iLES) of the turbulent air-entraining flow in the wake of three-dimensional rectangular dry transom sterns with varying speeds and half-beam-to-draft ratios $B/D$. We employ two-phase (air/water), time-dependent simulations utilizing conservative volume-of-fluid (cVOF) and boundary data immersion (BDIM) methods to obtain the flow structure and large-scale air entrainment in the wake. We confirm that the convergent-corner-wave region that forms immediately aft of the stern wake is ballistic, thus predictable only by the speed and (rectangular) geometry of the ship. We show that the flow structure in the air–water mixed region contains a shear layer with a streamwise jet and secondary vortex structures due to the presence of the quasi-steady, three-dimensional breaking waves. We apply a Lagrangian cavity identification technique to quantify the air entrainment in the wake and show that the strongest entrainment is where wave breaking occurs. We identify an inverse dependence of the maximum average void fraction and total volume entrained with $B/D$. We determine that the average surface entrainment rate initially peaks at a location that scales with draft Froude number and that the normalized average air cavity density spectrum has a consistent value providing there is active air entrainment. A small parametric study of the rectangular geometry and stern speed establishes and confirms the scaling of the interface characteristics with draft Froude number and geometry. In Part 2 (Hendrikson & Yue, J. Fluid Mech., vol. 875, 2019, pp. 884–913) we examine the incompressible highly variable density turbulence characteristics and turbulence closure modelling.

Numerical simulation of three-dimensional breaking waves and its interaction with a vertical circular cylinder

Wave breaking plays an important role in wave-structure interaction. A novel control volume finite element method with adaptive unstructured meshes is employed here to study 3-D breaking waves. The numerical framework consists of a “volume of fluid” type method for the interface capturing and adaptive unstructured meshes to improve computational efficiency. The numerical model is validated against experimental measurements of breaking wave over a sloping beach and is then used to study the breaking wave impact on a vertical circular cylinder on a slope. Detailed complex interfacial structures during wave impact, such as plunging jet formation and splash-up are captured in the simulation, demonstrating the capability of the present method.

Air entrainment and bubble statistics in breaking waves

We investigate air entrainment and bubble statistics in three-dimensional breaking waves through novel direct numerical simulations of the two-phase air–water flow, resolving the length scales relevant for the bubble formation problem, the capillary length and the Hinze scale. The dissipation due to breaking is found to be in good agreement with previous experimental observations and inertial scaling arguments. The air entrainment properties and bubble size statistics are investigated for various initial characteristic wave slopes. For radii larger than the Hinze scale, the bubble size distribution, can be described by $N(r,t)=B(V_{0}/2{\rm\pi})({\it\varepsilon}(t-{\rm\Delta}{\it\tau})/Wg)r^{-10/3}r_{m}^{-2/3}$ during the active breaking stages, where ${\it\varepsilon}(t-{\rm\Delta}{\it\tau})$ is the time-dependent turbulent dissipation rate, with ${\rm\Delta}{\it\tau}$ the collapse time of the initial air pocket entrained by the breaking wave, $W$ a weighted vertical velocity of the bubble plume, $r_{m}$ the maximum bubble radius, $g$ gravity, $V_{0}$ the initial volume of air entrained, $r$ the bubble radius and $B$ a dimensionless constant. The active breaking time-averaged bubble size distribution is described by $\bar{N}(r)=B(1/2{\rm\pi})({\it\epsilon}_{l}L_{c}/Wg{\it\rho})r^{-10/3}r_{m}^{-2/3}$, where ${\it\epsilon}_{l}$ is the wave dissipation rate per unit length of breaking crest, ${\it\rho}$ the water density and $L_{c}$ the length of breaking crest. Finally, the averaged total volume of entrained air, $\bar{V}$, per breaking event can be simply related to ${\it\epsilon}_{l}$ by $\bar{V}=B({\it\epsilon}_{l}L_{c}/Wg{\it\rho})$, which leads to a relationship for a characteristic slope, $S$, of $\bar{V}\propto S^{5/2}$. We propose a phenomenological turbulent bubble break-up model based on earlier models and the balance between mechanical dissipation and work done against buoyancy forces. The model is consistent with the numerical results and existing experimental results.

A two-phase flow model for three-dimensional breaking waves over complex topography

A two-phase flow model has been developed to study three-dimensional breaking waves over complex topography, including the wave pre-breaking, overturning and post-breaking processes. The large-eddy simulation approach has been adopted in this study, where the model is based on the filtered Navier–Stokes equations with the Smagorinsky sub-grid model being used for the unresolved scales of turbulence. The governing equations have been discretized using the finite volume method, with the PISO algorithm being employed for the pressure–velocity coupling. The air–water interface has been captured using a volume of fluid method and the partial cell treatment has been implemented to deal with complex topography in the Cartesian grid. The model is first validated against available analytical solutions and experimental data for solitary wave propagation over constant water depth and three-dimensional breaking waves over a plane slope, respectively. Furthermore, the model is used to study three-dimensional overturning waves over three different bed topographies, with three-dimensional wave profiles and surface velocities being presented and discussed. The overturning jet, air entrainment and splash-up during wave breaking have been captured by the two-phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems over complex topography.

Numerical simulation of three-dimensional breaking waves on a gravel slope using a two-phase flow Navier–Stokes model

Wave breaking is mainly a three-dimensional flow problem characterized by wave energy dissipation due to turbulence. The understanding of the wave breaking mechanism on a beach is essential in studying coastal processes. The complexity of the wave-induced turbulence flow is also increased by the presence of a two-phase flow, which introduces buoyancy effects. In this work a set of numerical experiments is carried out on wave breaking on a gravel slope. The influence of a one-phase and two-phase flow and the permeability effect of the beach are investigated numerically by means of a Navier–Stokes model known as IH-3VOF, which considers the volume-averaged Reynolds-averaged Navier–Stokes (VARANS) equations (del Jesus, 2011 [3]) to characterize the flow within the porous media. The accuracy of the VARANS equations is demonstrated by means of comparisons with laboratory data. The results are found to be within a 2% error in terms of wave height prior to the broken wave, and up to a 10% error after then, and in the order of 0.20 s in the time domain for the worst case. A further analysis of wave evolution on a permeable beach with alongshore variation of porosity is studied. Three-dimensional wave breaking and post-breaking wave transformations alongshore are analysed according to porosity values.

Dynamics of crescent water wave patterns

The nonlinear dynamics of three-dimensional instabilities of uniform gravity-wave trains evolving to crescent wave patterns is investigated numerically. A new mechanism of generation of oscillating horseshoe patterns is proposed and a detailed discussion on their occurrence in a water wave tank is given. It is suggested that these patterns are more likely to be observed naturally in water of finite depth. A critical wave steepness for the onset of three-dimensional wave breaking due to the nonlinear evolution of quintet resonant interactions corresponding to the phase-locked crescent-shaped structures (class II instability) is provided when the quartet resonant interaction (class I instability) is absent. The nonlinear coupling between quartet resonant interactions (class I instability) and quintet resonant interactions (class II instability) leading to three-dimensional breaking waves, as shown experimentally by Su & Green (1984, 1985), is numerically investigated.

Fully three-dimensional direct numerical simulation of a plunging breaker

The scope of this paper is to show the results obtained for simulating three-dimensional breaking waves by solving the Navier–Stokes equations in air and water. The interface tracking is achieved by a Lax–Wendroff TVD scheme (Total Variation Diminishing), which is able to handle interface reconnections. We first present the equations and the numerical methods used in this work. We then proceed to the study of a three-dimensional plunging breaking wave, using initial conditions corresponding to unstable periodic sinusoidal waves of large amplitudes. We compare the results obtained for two simulations, a longshore depth perturbation has been introduced in the solution of the flow equations in order to see the transition from a two-dimensional velocity field to a fully three-dimensional one after plunging. Breaking processes including overturning, splash-up and breaking induced vortex-like motion beneath the surface are presented and discussed. To cite this article: P. Lubin et al., C. R. Mecanique 331 (2003).

Numerical simulation of three-dimensional breaking waves

Three-dimensional (3D) wave breaking around bodies of complex geometry has been numerically investigated by use of two types of Navier-Stokes solvers, namely the finite-difference and the finite-volume methods employing rectangular and curvilinear coordinate systems, respectively. Both methods employ the density-function technique to capture the free surface location and can cope with complicated free surface configurations such as breaking waves. The accuracy of the density-function method is examined through the comparison with experimental results, and it is confirmed to be satisfactory when the grid spacing and the time increment are sufficiently small.

An experimental investigation of the collective oscillations of bubble plumes entrained by breaking waves

Laboratory measurements of the sound produced by mechanically generated two-dimensional (2-D) and three-dimensional (3-D) breaking waves are presented. In the 2-D breaking experiments it was observed that the mean-square pressure at frequencies below 1 kHz correlated strongly with the fractional energy dissipated by breaking and the volume of air entrained. In addition, the volume of air entrained was found to be proportional to the fractional energy dissipated. These results imply that measurements of the low-frequency sound may be useful for studying the dynamics of breaking waves in the field. It was found that 2-D plunging breakers produced significant increases in spectral levels at frequencies below approximately 500 Hz but that spilling breakers did not. Large-amplitude low-frequency signals were observed to begin up to 1/3 of a wave period after the start of active breaking in both the two- and three-dimensional experiments. It is believed that these signals were due to the collective oscillation of the entrained bubble plumes. A model of a cylindrically shaped plume located immediately below the free surface and the void-fraction measurements of Lamarre and Melville [J. Acoust. Soc. Am. 95, 1317–1328 (1994)] were used to compute the eigenfrequencies of the volume mode of collective oscillation. The computed eigenfrequencies closely matched the frequencies of the observed signals. This agreement between the experimental observations and theory provides considerable support for the hypothesis that the observed bubble plumes were oscillating collectively in their volume mode. This leads to the conclusion that the collective volume oscillations of bubble plumes entrained by breaking waves may be a source of low-frequency sound in the ocean.

A Finite-Difference Simulation Method for Strongly Interacting Two-Layer Flow

A variety of two-layer flows exist that are important in engineering. For numerical simulation, the imposition and implementation of proper boundary conditions at the interface is quite important. When the interface is characterized by strongly interacting motion, however, the simulation techniques that have so far been developed cease to be valid. The authors have devised a new finite-difference method that can deal with such a flow. This paper focuses on demonstrating the applicability of the method to the simulation of three-dimensional breaking waves about a vertical cylinder piercing the air-water interface and to the simulation of the effect of water flow on the motion of a sand layer beneath it.