How shallow breaking waves lose energy

A few weeks ago my paper, Inertial energy dissipation in shallow-water breaking waves, was published in the Journal of Fluid Mechanics. It’s about how solitary waves (that is, individual waves not part of a larger wavetrain) approach a beach with a single, uniform slope, and break on it. Although the paper describes various stages in this process, including how the wave changes shape and steepens on approach to the beach, how it breaks, and how it then runs up the beach, it really focuses on how its energy is dissipated as it breaks.

This is a computational study, where the governing equations are solved down to small scales (a technique called direct numerical simulation), so that we can measure the velocity at all points in the fluid and thus determine its total kinetic energy at all points in time. Since we also know the shape of the wave, and hence the height of the water surface, we can also determine the gravitational potential energy. We can track the sum of these two energies as the wave breaks: it converts some potential energy into kinetic energy, and some of this is then dissipated by viscosity into heat. This study varies the height of the incoming wave and the slope of the beach, and measures the energy dissipation from the resulting breakers.

But this is all just a bunch of data unless we have some way to bind it together in an explanation. This is the core of the study: it presents a theoretical model based on related studies in deep water where one assumes the jet of the breaking wave generates a cloud of turbulence underneath the wave. Using information about the wave when it reaches the breaking point – its height, the local depth of the water, the slope of the beach – it turns out that its energy dissipation rate can be predicted quite well, explaining our data in a simple scaling model.

This is a nice result that, firstly, answers basic (“local”) questions about the breaking process in shallow water waves, and could help provide scaling model information for reduced-order models operating over large length and time scales, such as regional coastal wave models. It also opens up more possibilities in direct simulation of shallow-water waves, including extending to 3D to further characterize the nature of the dissipative structures underneath the breaking wave, or progressive wave systems which are more reflective of real coastal environments and hence allow for a more direct relation of dissipative properties to the wave field.

The paper can be read for free (though with some reduced resolution on the figures) here.


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Turbulent bubbly breakups

Recently I contributed to a new study from the Deike research lab at Princeton with lead author Daniel Ruth, published  in the Proceedings of the National Academy of Sciences USA, about how bubbles break up in turbulent flow, particularly with regard to air bubbles in water. This is important in processes like ocean wave breaking (among many others); when a wave breaks, it captures a lot of air into the water in the form of bubbles, which then get entrained in the water, breaking up into smaller bubbles in the process. We care about this because it tells us about how mass is transferred between atmosphere and ocean, a process which is still poorly modelled in climate modeling.

Many studies that focus on the moment when the bubble breaks up – the pinchoff singularity – have focused on an idealized scenario where the surrounding water is not turbulent and the breakup is highly symmetrical. These are crucial for understanding the basic phenomenon, which involves a self-similar thinning of the neck that joins the two lobes of the breaking bubble just prior to pinchoff, but real bubbly flows are not typically this accommodating.

In this study, the same process was investigated experimentally, with some numerical support, in the presence of a turbulent water bulk with the aim of seeing how the turbulence affects the self-similarity of the approach to pinchoff. The turbulence bends and twists the bubble, deforming it, but it turns out that during the pinching process, the effect of the turbulence decreases until, close to the moment of pinchoff, it becomes irrelevant. In this sense the turbulence sets the initial shape of the bubble before pinchoff, but doesn’t directly affect the pinchoff itself.

But by setting the bubble shape in this way and introducing asymmetries into it, the turbulence can indirectly affect aspects of the pinchoff. The shape of the thinning neck can oscillate as it approaches the pinchoff singularity, and, if the introduced asymmetry is severe enough, the pinchoff can escape self-similarity altogether: the neck forms a kink. The chance of this happening appears to depend on the degree of asymmetry introduced into the initial shape. We also tested this by running a numerical case without turbulence, but with a strong asymmetry in the bubble shape and neck to see if the asymmetry could reproduce the escape from singularity, and found that these numerical examples agreed with the experiments in terms of the used metric.

This study is a fundamental step in the understanding of how the pinchoff singularity behaves in realistic flows that don’t feature the nice symmetry aspects that have been necessary in other fundamental studies. Turbulent bubbly flows are now just a bit less mysterious! The study can be found here in the December 17, 2019 issue of the PNAS.

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Facets and singularities

In recent months, in the conclusion of my research at Caltech, two papers have made it into the world. Although they’re pretty heavy on the maths and don’t feature very many illustrative figures, they both make a simple point: with an approximate model, you can show that a large class of shock waves starting off with smooth wobbles on their surface are guaranteed to develop sharp edges after some time which you can quantify. More precisely, there is a time when the shock geometry develops a singularity. Continue reading

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The choice of magnetic fields in MHD implosions: Maximizing stabilization while retaining symmetry

Recently Physical Review Fluids published our paper Magnetohydrodynamic implosion symmetry and suppression of Richtmyer-Meshkov instability in an octahedrally symmetric field, which looks at our previous result that the RM instability can be suppressed in converging MHD flows (e.g. implosions) by an applied magnetic field, albeit with some caveats on the symmetry of the resulting flow, and takes the logical next step of asking whether maximizing the symmetry properties of the applied field also maximizes the symmetry of the implosion, and in particular whether doing so also retains suppression of the RM in a way comparable to the fields we tested earlier.

The study suggests that both these questions may be answered “yes” in the case of a particular field candidate which is octahedrally symmetric. Such a field could be formed by placing an array of six equally spaced electric current loops around the centre of the targeted implosion – this is similar to, but a more extreme form of, the “saddle” field considered in our earlier papers. It turns out that, for the simulations we performed here, the RM instability is suppressed at least comparably to the uniform-field case, which is the field otherwise best at RM-suppression, while retaining a much higher flow symmetry.

The results of the study is encouraging because it suggests that maximal RM suppression can be retained without having to compromise on the flow symmetry by the correct choice of an applied field, making the use of magnetic fields for RM stabilization in converging flow problems more plausible.

The article appeared in the following,

Mostert W., Pullin D.I., Wheatley V., and Samtaney R. (2017) Magnetohydrodynamic implosion symmetry and suppression of Richtmyer-Meshkov instability in an octahedrally symmetric field. Phys. Rev. Fluids2, 013701.

and may be downloaded here for personal use only. Copyright 2017 by The American Physical Society.

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