The numerical simulation of flows with interfaces and free-surface flows is a vast topic, with applications to domains as varied as environment, geophysics, engineering, and fundamental physics. In engineering, as well as in other disciplines, the study of liquid-gas interfaces is important in combustion problems with liquid and gas reagents. The formation of droplet clouds or sprays that subsequently burn in combustion chambers originates in interfacial instabilities, such as the Kelvin-Helmholtz instability. What can numerical simulations do to improve our understanding of these phenomena? The limitations of numerical techniques make it impossible to consider more than a few droplets or bubbles. They also force us to stay at low Reynolds or Weber numbers, which prevent us from finding a direct solution to the breakup problem. However, these methods are potentially important. First, the continuous improvement of computational power (or, what amounts to the same, the drop in megaflop price) continuously extends the range of affordable problems. Second, and more importantly, the phenomena we consider often happen on scales of space and time where experimental visualization is difficult or impossible. In such cases, numerical simulation may be a useful prod to the intuition of the physicist, the engineer, or the mathematician. A typical example of interfacial flow is the collision between two liquid droplets. Finding the flow involves the study not only of hydrodynamic fields in the air and water phases but also of the air-water interface. This latter part

Direct numerical simulation of free-surface and interfacial flow

Numerical Modeling of Water Waves with the SPH Method

An accurate three-dimensional numerical model, applicable to strongly non-linear waves, is proposed. The model solves fully non-linear potential flow equations with a free surface using a higher-order three-dimensional boundary element method (BEM) and a mixed Eulerian–Lagrangian time updating, based on second-order explicit Taylor series expansions with adaptive time steps. The model is applicable to non-linear wave transformations from deep to shallow water over complex bottom topography up to overturning and breaking. Arbitrary waves can be generated in the model, and reflective or absorbing boundary conditions specified on lateral boundaries. In the BEM, boundary geometry and field variables are represented by 16-node cubic ‘sliding’ quadrilateral elements, providing local inter-element continuity of the first and second derivatives. Accurate and efficient numerical integrations are developed for these elements. Discretized boundary conditions at intersections (corner/edges) between the free surface or the bottom and lateral boundaries are well-posed in all cases of mixed boundary conditions. Higher-order tangential derivatives, required for the time updating, are calculated in a local curvilinear co-ordinate system, using 25-node ‘sliding’ fourth-order quadrilateral elements. Very high accuracy is achieved in the model for mass and energy conservation. No smoothing of the solution is required, but regridding to a higher resolution can be specified at any time over selected areas of the free surface. Applications are presented for the propagation of numerically exact solitary waves. Model properties of accuracy and convergence with a refined spatio-temporal discretization are assessed by propagating such a wave over constant depth. The shoaling of solitary waves up to overturning is then calculated over a 1:15 plane slope, and results show good agreement with a two-dimensional solution proposed earlier. Finally, three-dimensional overturning waves are generated over a 1:15 sloping bottom having a ridge in the middle, thus focusing wave energy. The node regridding method is used to refine the discretization around the overturning wave. Convergence of the solution with grid size is also verified for this case. Copyright © 2001 John Wiley & Sons, Ltd.

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A fully non‐linear model for three‐dimensional overturning waves over an arbitrary bottom

Digital particle image velocimetry (DPIV) measurements of the velocity eld under breaking waves in the laboratory are presented. The region of turbulent fluid directly generated by breaking is too large to be imaged in one video frame and so an ensemble-averaged representation of the flow is built up from a mosaic of image frames. It is found that breaking generates at least one coherent vortex that slowly propagates downstream at a speed consistent with the velocity induced by its image in the free surface. Both the kinetic energy of the flow and the vorticity decay approximately as t 1 . The Reynolds stress of the turbulence also decays as t 1 and is, within the accuracy of the measurements, everywhere negative, consistent with downward transport of streamwise momentum. Estimates of the mometum flux from waves to currents based on the measurements of the Reynolds stress are consistent with earlier estimates. The implications of the measurements for breaking in the eld are discussed. Based on geometrical optics and wave action conservation, we suggest that the presence of the breaking-induced vortex provides an explanation for the suppression of short waves by breaking. Finally, in Appendices, estimates of the majority of the terms in the turbulent kinetic energy budget are presented at an early stage in the evolution of the turbulence, and comparisons with independent acoustical measurements of breaking are presented.

The velocity field under breaking waves: coherent structures and turbulence

https://hal.archives-ouvertes.fr/hal-00294092/document

Three-dimensional Large Eddy Simulation of air entrainment under plunging breaking waves

Numerical simulations describing plunging breakers including the splash-up phenomenon are presented. The motion is governed by the classical, incompressible, two-dimensional Navier–Stokes equation. The numerical modeling of this two-phase flow is based on a piecewise linear version of the volume of fluid method. Capillary effects are taken into account such as a nonisotropic stress tensor concentrated near the interface. Results concerning the time evolution of liquid–gas interface and velocity field are given for short waves, showing how an initial steep wave undergoes breaking and successive splash-up cycles. Breaking processes including overturning, splash-up and gas entrainment, and breaking induced vortex-like motion beneath the surface and energy dissipation, are presented and discussed. It is found that strong vorticities are generated during the breaking process, and that more than 80% of the total pre-breaking wave energy is dissipated within three wave periods. The numerical results are compared with some laboratory measurements, and a favorable agreement is found.

Two-dimensional navier-stokes simulation of breaking waves

Numerical simulations describing plunging breakers including the splash-up phenomenon are presented. The motion is governed by the classical, incompressible, two-dimensional Navier-Stokes equation. The numerical modelling of this two-phase flow is based on a piecewise linear version of the volume of fluid method. Preliminary results concerning the time evolution of the liquid-gas interface and the vorticity field are given for short waves, showing how an initial steep wave undergoes breaking and successive splash-up cycles.

https://arxiv.org/pdf/comp-gas/9605002.pdf

G. Chen

Papers

Two-dimensional navier-stokes simulation of breaking waves

Two-Dimensional Navier-Stokes Simulation of Breaking Waves