Session Q12: Waves: Surface Waves (3:55pm - 4:40pm CST)

Observation of broad-band water waveguiding in shallow water: a revival

We report on the observation and characterization of broad-band waveguiding of surface gravity waves in an open channel, in the shallow water limit. The waveguide is constructed by changing locally the depth of the fluid layer, which creates conditions for surface waves to propagate along the generated guide. We present experimental and numerical results of this shallow water waveguiding, which can be straightforwardly matched to the one-dimensional water wave equation of shallow water waves. Our work revitalizes water waveguiding research as a relevant and controllable experimental setup to study complex phenomena using waveguide geometries.   

A GPU-Accelerated, Ensemble-Based High-Order Spectral Solver for Surface Gravity Waves

Many applications in the physical study and forecasting of nonlinear waves require the efficient computation of ensemble wave fields. In physical studies, e.g. wave turbulence, considerable effort is required to compute statistics from large ensembles of simulations. In marine engineering, ensemble-based forecasting tools are ideally integrated with remote sensing technology, to be deployed on compact devices in offshore environments. For these purposes, we develop an ensemble-based solver of the gravity wave equations for a CPU-GPU heterogeneous architecture that allows for the fast and efficient computation of wave field evolution and statistics via the High-Order Spectral method (HOS). This is achieved by simultaneously computing the evolution of ensemble wave fields, leveraging GPU-acceleration for highly-parallel array operations and fast Fourier transforms. Performance metrics are provided for comparison to serial and MPI-based HOS codes, and the solver's accuracy is validated. We conclude with applications in wave forecasting via the ensemble Kalman filter and a study of the long-term evolution of wave field statistics.   

The Effect of Wind on Shoaling Wave Shape

Wave shape is a key factor in sediment transport, beach morphology, and ship safety. The authors previously showed that wind can affect the skewness and asymmetry of waves in shallow, intermediate, and deep water over a flat bottom. Wave shoaling has long been known to induce wave asymmetry, but it has not yet been shown how wind-generated and shoaling-generated asymmetry interact. In this study, the evolution of waves propagating over a gentle slope $\partial_x h \ll 1$ are examined using a multiple scales analysis. Wind forcing is parametrized through a surface pressure, and the induced changes to skewness and asymmetry are calculated numerically. The relative influences of shoaling and wind are examined by applying both onshore and offshore wind of different strengths. These results will be used to contextualize observations of wind's effect on shoaling wave shape.   

On the mechanism of bound wave generation in irregular wave fields

Bound waves are short wave components on the water surface that are produced by longer waves. In spite of their ubiquity in nature and importance to remote sensing, the generation mechanisms of bound waves are still not well understood. A standard view of bound waves is that they travel at nearly the same speed as the carrier long wave. In this work, we show that this view is incomplete for an irregular wave field, for which a significant portion of the bound waves are dispersive (but not satisfying the linear dispersion relation). We propose a mechanism on the generation of bound waves as the sum and difference interactions between an arbitrary mode and the spectral-peak mode. This mechanism excellently explains the dispersive nature of the bound waves produced in potential flow simulations and two-phase Navier-Stokes simulations. Furthermore, the bound waves generated by wave breaking are studied and their effects on the small-scale wave spectra are elucidated.   

Real-Time Phase-Resolved Ocean Wave Forecast with Data Assimilation

The phase-resolved prediction of ocean waves is crucial for the safety of offshore operations. With the ocean surface obtained from radar measurements as the initial condition, nonlinear wave models such as the high-order spectral (HOS) method can be applied to predict the evolution of the ocean waves. However, due to the error in the initial condition (associated with the radar measurements and reconstruction algorithm) and the chaotic nature of the nonlinear wave equations, the prediction by HOS can deviate quickly from the true surface evolution. To address these issues, the capability to regularly incorporate measured data into the HOS simulation through data assimilation is desirable. In this work, we develop the data assimilation capability for nonlinear wave models, through the coupling of an ensemble Kalman filter (EnKF) with HOS. We also propose a strategy of modifying the Kalman gain to address the problem of the shrinking of the predictable zone. The validity of the developed scheme is benchmarked using both the synthetic data and radar measurements. We show that the EnKF-HOS coupled scheme achieves much higher accuracy in the long-term simulation of nonlinear waves compared to the HOS-only method.   

Faraday-waves contact-line shear gradient induces streaming flow and tracers' self-organization: from rotating rings to spiral galaxy-like patterns

We experimentally demonstrate self-organization of small tracers under the action of longitudinal Faraday waves in a narrow container. We observe a steady current formation dividing the interface in small cells given by the symmetries of the Faraday wave. These streaming currents are rotating in each cell and their circulation increases with wave amplitude. This streaming flow drives the tracers to form patterns, whose shapes depend on the Faraday wave's amplitude: from low to high amplitudes we find dispersed tracers, a narrow rotating ring and a spiral galaxy-like pattern. We first describe the main pattern features, and characterize the wave and tracers' motion. We then show experimentally that the main source of the streaming flow comes from the time and spatial dependent shear at the wall contact line, created by the Faraday wave itself. We end by presenting a 2D model that considers the minimal ingredients present in the Faraday experiment, namely the stationary circulation, the stretching component due to the oscillatory wave and a steady converging field, which combined produce the observed self–organized patterns.   

Weakly nonlinear evolution of a cavity on a free-surface

We formulate a second order asymptotic solution in two dimensional Cartesian coordinates to the initial value problem involving a finite-amplitude, localized perturbation resembling a cavity (Gaussian depression) on the free-surface of a horizontally unbounded, infinitely deep liquid. We employ the Hamiltonian theory for inviscid capillary-gravity waves and expand upto the three-wave interaction Hamiltonian. The Zakharov equation [Zakharov (1968), \textit{J. Appl. Mech. Tech. Phys., 9(2)}] is numerically solved for the aforementioned Cauchy data on the canonical coordinates in our reduced Hamilton's equation [Krasitskii (1994), \textit{JFM, 272}]. We compare the time evolution of our weakly nonlinear interfacial profile with the corresponding linear solution to the classical Cauchy-Poisson problem [Poisson (1818) : \textit{Mem. Prés. divers Savants Acad. R. Sci. Inst. 2}; Cauchy (1827) : \textit{Mem. Prés. divers Savants Acad. R. Sci. Inst. 1}] and results obtained from Direct Numerical Simulation (DNS) of the Euler's equation (including both gravity and surface tension) using Basilisk [basilisk.fr].   

Finite amplitude, axisymmetric, capillary waves in a cylindrical container

We obtain the solution to the initial value problem for a surface perturba- tion on a deep pool of liquid contained in a cylindrical container. The solution is formulated as a perturbative expansion upto third order in the wave steep- ness parameter ≡ a0k. The initial surface perturbation is chosen to be an axisymmetric Bessel function i.e. η(r, 0) = a0J0(kr) with k sufficiently large for gravity to be negligible. We solve the nonlinear initial-value problem under the inviscid, irrotational approximation using the Lindstedt-Poincare technique and the Dini series, solving the resultant equations upto O( 3 ), accounting for surface tension. The resultant expression for the time evolution of the inter- face η(r, t) is compared against numerical solutions to the incompressible Euler equation. We compare these results to those obtained recently from a sec- ond order expansion, where both capillary and gravity effects are taken into account (Basak, Farsoiya and Dasgupta, 2020, under review; https: // gfm. aps. org/ meetings/ dfd-2019/ 5d764521199e4c429a9b2bd ). The differences between the finite amplitude capillary wave and the capillary-gravity wave will be highlighted.   

A wide-spacing approximation model for the reflection and transmission of water waves over an array of vertical obstacles

With a view to modelling and optimisation of wave energy farms, a simple recursive formulation is proposed for the reflection and transmission of plane water waves by a number of rows of vertical obstacles, under the wide-spacing approximation. The approach accommodates dissipation along the wave propagation direction. The proposed recursive model is validated by means of experiments in a small-scale wave flume, where waves are reflected and transmitted by one, two and three rows of vertical, flexible blades. For the special case of identical, regularly-spaced rows, well-known analytical formulae are then discussed for the global reflection and transmission coefficients, as a function of the reflection and transmission properties of individual rows. In a `non-dissipative' case, the well-known fact that discrete values of the row-to-row distance $L$ completely cancel reflection is retrieved, as well as the existence of `band-gap' intervals, i.e. intervals for $L$ where reflection is high, with maximum reflection occurring away from the Bragg condition. In contrast, when dissipation takes place, reflection is always nonzero, and, as the number of rows tends to infinity, forms Bragg peaks, reaching unity when $L$ is a multiple of half a wavelength.   

Nonlinear waves through vegetation in coastal regions

The study of fluid flows interacting with vegetative structures in coastal regions presents a significant challenge on account of its multi-scale nature. Hence, much of the previous work has been confined to the small-amplitude regime. In this talk, we show how compact asymptotic reductions allow us to study nonlinear surface waves over vegetative regions. For example, for plants over a horizontal bed, previous work on linear waves predicts a quadratic decay in wave amplitude. Our multiple-scales analysis on cnoidal waves provides new accurate predictions on how amplitude and wavelength modulate along the coast—verified with finite-element simulations. This work is in collaboration with the US Army Corps of Engineers and HR Wallingford.   

The Effects of Surface Wind Waves on the Atmospheric Wave and Turbulent Kinetic Energy Budgets

The energy exchanges between the atmosphere and ocean are strongly contingent on small- and large-scale dynamics at the air-sea interface. However, there are relatively few investigations on the airside wave and turbulent kinetic energy budgets. Here, we present laboratory measurements (PIV/LIF) of velocity fields in the turbulent airflow above surface waves for 10-m wind speeds of 2.25 to 16.59 m/s. The wave-induced and turbulent velocity components are then extracted from the instantaneous velocity to examine, in detail, the wave and turbulent kinetic energies (WKE and TKE). The TKE is enhanced downwind of wave crests away from the surface and reduced in a thin layer near the interface. These intense regions of TKE are attributed to the airflow separation events past wave crests. It was further observed that the streamwise turbulent velocity variance carries the bulk of the TKE. Like the TKE, a region of enhanced WKE was observed, but it was located on the upwind side of waves close to the surface. This is consistent with the patterns observed in streamwise wave-induced velocity variances. Finally, the measurements provide direct measurements of the generation of WKE by the mean shear in the airflow. We will discuss the results in the context of the total kinetic energy budget.   

Exploring diffraction with a pilot-wave model

The seminal experiments of Yves Couder and Emmanuel Fort demonstrated that a droplet walking on the surface of a fluid bath may exhibit behavior thought to be peculiar to the microscopic quantum realm. One of their experiments~suggested that single-particle diffraction and interference may be obtained when a~walker~crosses a single- or a double-aperture between submerged barriers (Couder, Y. {\&} Fort, E. Phys. Rev. Lett. 97, 154101, 2006).~Later~experiments with finer control of experimental parameters yielded different results, thus~reopening the question of the extent of the analogy between walkers and quantum particles (Andersen, A. et al. Phys. Rev. E 92, 013006, 2015; Pucci, G. et al. J. Fluid Mech. 835, 1136-1156, 2018; Rode, M. et al. Phys. Rev. Fluids 4, 104801, 2019). Here we use the pilot-wave model developed by Oza et al. (J. Fluid Mech. 737, 552-570, 2013) to explore the diffraction of a two-dimensional, wave-piloted particle by one-dimensional barriers. While our results are generally different from the Fraunhofer diffraction patterns in optics, the statistical distribution of deflection angles generally exhibits~multiple peaks, the number of which depends on the obstacle geometry.   

Modification of Stokes Drift in Coastal Areas due to Surface Wave, Current and Topography Interactions

Surface gravity waves cause floating particles to undergo a slow drift in the direction of wave propagation. This forward drift, commonly known as the Stokes drift, plays a crucial role in transporting various tracer parcels, from sediments to pollutants, in the marine environment. In the classical analysis, the effect of mean current and sea-bed undulations are not factored in while calculating Stokes drift. We find that in the nearshore region, Stokes drift is non-trivially affected by alongshore current and sea-bed undulations. We theoretically show that the time-independent particular solution, arising from mean-current and bottom-ripple interactions, leads to additional terms in the Stokes drift. Next, using High-order Spectral method, we numerically simulate wave-current-topography interactions in a 3D setting and compute the Lagrangian drift. We find that the resulting drift in the presence and absence of sea-bed corrugations have significant differences. Topographic interactions are inevitable in the nearshore region, and our study reveals that sea-bed corrugations can significantly affect cross-shelf exchange of microplastics and other nearshore tracers like pathogens, contaminants, nutrients, larvae, and sediments.   

Not Participating

Surface wave breaking, occurring from the ocean to the coastal zone, is a complex and challenging two-phase flow phenomenon which plays an important role in numerous processes, including air--sea transfer of gas, momentum and energy. Recent modelling attempts are struggling with the lack of physical knowledge of the finest details of the breaking processes. Furthermore, no universal scaling laws for physical variables have been found so far. Hence, parameterizing and characterizing breaking effects becomes very difficult. The pre- and post-breaking events can be quantified, detected and qualified following breaking criteria. These criteria can be directly linked to geometrical quantities (wavelength, amplitude, depth, etc.) in order to predict the breaking type, its localization and the energy dissipated during the breaking process. Numerous accurate numerical simulations were performed to gain further insight on predicted and quantified a breaking wave. We aim at presenting and discussing geometrical characteristics and the energy dissipated for small and intermediate breaking waves.   

A fast high-order boundary element method for the study of nonlinear waves inside wave tanks

The ability to predict the nonlinear dynamics of extreme events inside large wave tanks (dimensions of hundreds of meters) is a critical component in the preparation of modern experimental tests. Currently there is no efficient numerical method capable of solving the inverse problem of extreme waves in large tanks, or effectively providing wavemaker kinematics that generate desired wave conditions for experiments. To overcome this deficiency we develop a fast and robust numerical method for modeling nonlinear waves inside wave tanks based on the Zakharov equation and a perturbation expansion of the velocity potential up to an arbitrary order in wave steepness. The boundary-value problem (BVP) for each term of the expansion is solved through a quadratic boundary element method which is accelerated with a pre-corrected Fast Fourier Transform scheme (pFFT), reducing the computational effort to $O(NlogN)$, where $N$ is the number of collocation points used to discretize the boundaries of the tank. The efficiency and accuracy of the method over current fully-nonlinear potential flow methods is demonstrated through a wave focusing numerical experiment.