Free Surface Waves Research Articles

Modeling internal rogue waves in a long wave-short wave resonance framework

Rogue waves in the interior of a stratified fluid are modeled as special breathers (pulsating modes) arising from long-wave--short-wave resonance. Features like the existence condition and waveforms contrast sharply with those of free surface waves governed by the nonlinear Schr\odinger equation.

Numerical Simulations of Linearly Stratified Flow Past Submerged Bodies

Abstract In this study, a methodology was presented to predict density stratified flows in the near-field of submerged bodies. The energy equation in temperature form was solved coupled with momentum and mass conservation equations. Linear stratification was achieved by the definition of the density as a function of temperature. At first, verifications were performed for the stratified flows passing a submerged horizontal circular cylinder, showing excellent agreement with available experimental data. The ability of the method to cope with variable density was demonstrated. Different turbulence models were used for different Re numbers and flow states. Based on the numerical methods proposed in this paper, the stratified flow was studied for the real scale benchmark DAPRA Suboff submarine. The approach used the VOF method for tracing the free surface. Turbulence was implemented with a k − ω based Detached Eddy Simulation (DES) approach. The effects of submarine speed, depth and density gradient on the free surface wave pattern were quantitatively analyzed. It was shown that, with the increasing of the speed of the submarine, the wavelength and wave height of the free surface wave were gradually increasing. The wave height of the free surface wave was gradually reduced as the submarine’s depth increased. Relative to the speed and submarine depth, the changes of the gradient density gradient have negligible effects on the free surface wave field.

Numerical wave basin using incompressible smoothed particle hydrodynamics (ISPH) on a single GPU with vertical cylinder test cases

This paper presents the development of a numerical wave basin for violent hydrodynamics of free-surface waves interacting with vertical cylindrical structures using the incompressible smoothed particle hydrodynamics (ISPH) method accelerated with a graphics processing unit (GPU). ISPH has been implemented on a single GPU based on the open-source DualSPHysics software and using the ViennaCL linear algebra library for solving the pressure Poisson equation. Extensions to the code are presented for a numerical wave basin including development of the solid boundary condition, based on a mirror image particle and moving least squares interpolation approach, for improved accuracy at the free and solid surface interfaces during wave-structure impact. This is applied to focused waves, including breaking for which SPH is well suited, interacting with a cylindrical column to compare with experimental data. Convergence is demonstrated within the limit of 5 million particles for a single GPU. Acceptable agreement with experiment is achieved for force measurements, and variation of free-surface elevation around the column is shown to be significant particularly in larger waves with a correspondingly large force contribution from hydrostatic pressure. While breaking does not have a marked effect on total force, associated high particle velocities amplify local pressures considerably on the column near the water surface. The computer runtime is similar to that for volume-of-fluid and particle-in-cell solvers running on 8 and 80 processors respectively.

A conservative scheme for simulation of free-surface turbulent and wave flows

A numerical scheme with good conservation properties is developed for the simulation of free-surface turbulent and viscous wave flows using a surface-fitted curvilinear grid. The Navier–Stokes equations are written in a strong conservative formulation with respect to the curvilinear coordinates, and are discretized by a pseudo-spectral method in the horizontal directions and a finite-difference method in the vertical direction. Large-eddy simulation (LES) is implemented with the conservative scheme to extend the simulation capability to turbulent flows with higher Reynolds numbers. Fully nonlinear kinematic and dynamic boundary conditions are implemented at the free surface. The numerical scheme is validated using a variety of wave and vortical flow test cases. The results show good agreement with previous theoretical and numerical predictions, whereas the present scheme achieves significant improvement in the conservation of mass and momentum over the non-conservative scheme developed by Yang & Shen [1]. Meanwhile, the present conservative scheme is found to be more stable than the non-conservative scheme for the simulation of sideband waves and broadband waves. The effect of viscous dissipation on the long-term nonlinear wave evolution is also captured by the present scheme. The ability of the present scheme for simulating long-term wave-current-turbulence interaction is demonstrated by the computation of Langmuir circulation, for which the non-conservative scheme produces significant errors in mass and momentum conservation and the simulation fails. Flow features of the Langmuir circulation, such as the counter-rotating vortices and converging-diverging zones, have been successfully captured with our numerical scheme. The turbulence statistics also agree with the characteristics of Langmuir circulation.

An analytical model for oblique wave interaction with a partially reflective harbor structure

An analytical model of the interaction of oblique incident regular waves with a maritime structure is developed and extended to irregular waves. The structure is composed of a semi-submergible vertical impermeable plate enclosing a chamber. Linear wave theory is considered, taking into account the head loss due to the constriction of the flow. The results are compared to those obtained with computational fluid dynamics models, revealing that the proposed model is capable to efficiently describe the performance of these systems for weakly non-linear incident waves. The influence of fundamental design parameters, such as geometric and wave characteristics, on the hydrodynamic behavior and the loadings on the plate is analyzed.When dealing with irregular waves, the spectra at the seaward region and inside the chamber show a nodal and antinodal structure that varies with the distance to the reflector. This structure, as well as the phase lag between the free surface elevations at both sides of the plate, affect the total loads over the plate. The statistical analysis of the free surface elevation, wave heights, and crests and throughs of the forces is also presented to evaluate the effect of the geometry in both regions and over the plate.The results show that the model allows to efficiently test different configurations to control wave reflection at the structure and the oscillation in the chamber. It can therefore be applied e.g. for the optimization of the design towards (i) harbor tranquility in the seaward region, (ii) energy extraction inside the chamber and (iii) loads on the structure.As an application, a failure mode defined as the excess of a critical predefined load over the plate, is analyzed by means of Monte Carlo simulations using known theoretical distributions fitted to the random variables. The results show the importance of the analysis of the performance of the system regarding not only the effect inside the chamber but also on the structure.

Stabilization of fluid flow rate upstream the verifiable instrument

Results of experimental study of free fluid surface in a fixed long tank (pressure tank) in conditions of mass transfer through the tank are presented. The research has been conducted on an experimental setup that included a model pressure tank to improve the State Primary Standard of volume flow rate GET 64–74 and a simplified system of fluid motion. The effect of free surface wave amplitude in the tank on stability of flow rate maintenance through the metering line of the Standard has been estimated.

Experimental and numerical investigation of supercritical flow in contraction and expansion of open channels

Shock wave formation is very important in studies of supercritical flow. Based on increasing water height more than inlet flow depth, spreading over a wide range of downstream channel and making disturbances in water surface, formation and propagation of these waves are unfavorable in engineering. The present paper is an attempt to study the supercritical flow in contraction and expansion, and wave formation patterns. In this regard, the height and instantaneous velocity at different points of shock waves in contraction and free-surface wave profiles in expansion were measured for four different Froude numbers. In order to 3D simulation, turbulence models of k-e RNG and RSM based on Fluent software were used. For calculating free-surface wave profiles using the k-e RNG and RSM models, the mean relative errors were obtained as 30.3% and 98.2% in contraction: 13% and 2.58% in expansion, respectively. Also, the mean relative errors of the k-e RNG and RSM models for wave velocity computation of contraction were calculated as 5.26% and 2.87%, respectively. The results indicated that RSM turbulence model performs better than k-e RNG in simulating velocity profiles of shock waves.

Study on vibration of dragon wash basin and free surface waves inside

We conducted experimental and numerical studies on the vibrating modes of a dragon wash basin (DWB) and the free surface waves inside the DWB. Both the vibration of the DWB and the sound produced were studied carefully. It was found that the DWB can be excited at different intrinsic modes under different excitation, including striking and rubbing it fast/slowly. However, with gentle rubbing, the DWB will be excited mainly at the first vibrating mode. We showed that the concave side wall of a DWB decreases the intrinsic frequencies (compared with a straight side wall), and the ears of a DWB lead to breaking of the vibrating-axisymmetry and cause the separation of modes I/II and IV/V as well. A theoretical model, in which the water is assumed to be incompressible and inviscid, is applied to the first order vibration and predicts the relation of vibrating frequency versus water depth in the system. The measurement shows that both the radial and azimuthal waves are produced as DWB is working. The frequency of the first component of the surface wave is twice as large as the second one induced by non-linear effects. For both the radial and azimuthal waves, the dispersion relation is presented in the framework of capillary wave theory.

Rearrangement of liquid metal surface waves by a uniform transverse magnetic field

In the fusion engineering, the liquid metal flows are considered to be an alternative solution against the solid surface in designing the plasma facing components. However, it remains unclear about the magneohydrodynamics influences on the liquid metal flows due to the lacking of experimental observations that the limited measurement method for the opaque liquid metal film should be criticized. In the present study, a laser profilometer based on the laser triangulation technique is adapted to measure the thickness of liquid GaInSn film, while a visualization method is introduced to monitor the topography of the free surface, which is formed on a horizontal substrate in the presence of a uniform transverse magnetic field. We will firstly demonstrate the applicability of a laser profilometer by measuring the thickness of GaInSn in a container with a quasi-static or dynamic free surface, to validate that the result is in good agreement with theoretical data. Using the visualization method and the laser profilometer, the evolutions of free surface waves and the time-dependent film thickness across a fixed line are measured by varying the flow rate and the magnetic field continuously. It is found that, under the influences of transverse magnetic field, the free surface waves are strongly suppressed along the spanwise direction, leaving the surface waves to propagate in the streamwise direction. More quantitatively, the measured thickness of the free surface reveals that a moderate magnetic field tends to decrease the averaged value and the root mean square value of the film thickness fluctuations while a strong magnetic field will conversely increase them. Besides, the dominant fluctuation frequencies of surface wave are observed to decrease with the increase of magnetic intensity, indicating that greater suppression is induced on the motion of small waves by the transverse magnetic field.

Free-surface long wave propagation over linear and parabolic transition shelves

Long-period waves pose a threat to coastal communities as they propagate from deep ocean to shallow coastal waters. At the coastline, such waves have a greater height and longer period in comparison with local storm waves, and can cause severe inundation and damage. In this study, we considered linear long waves in a two-dimensional (vertical-horizontal) domain propagating towards a shoreline over a shallowing shelf. New solutions to the linear shallow water equations were found, through the separation of variables, for two forms of transition shelf morphology: deep water and shallow coastal water horizontal shelves connected by linear and parabolic transition, respectively. Expressions for the transmission and reflection coefficients are presented for each case. The analytical solutions were used to test the results from a novel computational scheme, which was then applied to extending the existing results relating to the reflected and transmitted components of an incident wave. The solutions and computational package provide new tools for coastal managers to formulate improved defence and risk-mitigation strategies.

Nonlinear Wave Loads on High-rise Pile Cap Structures in the Donghai Bridge Wind Farm

A new type of bottom-fixed structure, the so-called high-rise pile cap foundation, has been proposed and used to support offshore wind turbines in the Donghai Bridge Wind Farm, China. Engineers are unaware of the wave load mechanisms for this new structure. Using the Navier-Stokes equations and volume of fluid technique, a fully nonlinear numerical wave tank is established to investigate free surface wave loads and moments for the new structure. The interaction between the cap and piles are discussed in detail. In the case of fully nonlinear waves, the maximum horizontal wave load on all the piles with the cap can increase by 30% compared with those without the cap, and the maximum horizontal wave load on a single pile is nearly doubled. The horizontal wave load on the cap with the piles can increase by about 15%, while the vertical wave load decreases slightly. The conventional Morison formula and diffraction theory generally underestimate the wave loads on the piles and the cap as well.

Semi-implicit non-hydrostatic model for 2D nonlinear wave interaction with a floating/suspended structure

A non-hydrostatic model is developed to simulate two-dimensional (2D) nonlinear wave interaction with a fixed floating/suspended structure. The immersed boundary method is incorporated in the model to address arbitrary body surfaces. The model employs a semi-implicit, fractional step algorithm to solve the Reynolds-averaged Navier–Stokes (RANS) equations based on a grid system, which is built from a horizontal rectangular grid by adding dozens of horizontal layers. The horizontal layers are distributed following a general boundary-fitted vertical coordinate system. This grid system may overcome the lack of flexibility of other boundary-fitted coordinate systems (e.g., the σ-coordinate system). The model is validated by using three test cases, namely, a solitary wave interacting with a floating rectangular obstacle, a solitary wave incident on a suspended horizontal plate and a solitary wave incident on a coastal-bridge deck with girders. Numerical results of the free surface elevation and wave force exerted on each structure are compared with the results of other models or experimental data. Generally, good agreements are obtained, demonstrating the proposed model’s capability to resolve nonlinear wave interactions with floating/suspended structures. The velocity fields around the structures are also presented and discussed.

Numerical and Experimental Investigation of Interactions Between Free-Surface Waves and A Floating Breakwater with Cylindrical-Dual/Rectangular-Single Pontoon

This paper investigates the hydrodynamic performance of a cylindrical-dual or rectangular-single pontoon floating breakwater using the numerical method and experimental study. The numerical simulation work is based on the multi-physics computational fluid dynamics (CFD) code and an innovative full-structured dynamic grid method applied to update the three-degree-of-freedom (3-DOF) rigid structure motions. As a time-marching scheme, the trapezoid analogue integral method is used to update the time integration combined with remeshing at each time step. The application of full-structured mesh elements can prevent grids distortion or deformation caused by large-scale movement and improve the stability of calculation. In movable regions, each moving zone is specified with particular motion modes (sway, heave and roll). A series of experimental studies are carried out to validate the performance of the floating body and verify the accuracy of the proposed numerical model. The results are systematically assessed in terms of wave coefficients, mooring line forces, velocity streamlines and the 3-DOF motions of the floating breakwater. When compared with the wave coefficient solutions, excellent agreements are achieved between the computed and experimental data, except in the vicinity of resonant frequency. The velocity streamlines and wave profile movement in the fluid field can also be reproduced using this numerical model.

Breaking of waves on thin films over topographies

In the present study, we performed experiments on the evolution of gravity-driven nonlinear traveling free surface waves over strongly undulated inclines. We focused on the impact of the excitation frequency and amplitude and the substrates’ shape and periodicity on the traveling wave. Thereby, we revealed phenomena concerning the amplitude evolution of convectively unstable waves. We can classify the wave evolution in three categories: (1) A normal exponential growth of the wave until it reaches a saturation amplitude. (2) An exponential growth of the wave and an abrupt collapse of the wave leading to a wave with a saturated amplitude, which is sensibly smaller than the maximal amplitude. (3) An alternating exponential growth and breaking of the wave. By using Fourier analysis, we investigated the waves in more detail. Furthermore, we report (a) a selection of excitation frequencies for the wave breaking, (b) a major impact of the steady state free surface, especially its mean or global curvature, (c) a bubble formation occurring at the wave breaking, (d) an overlap of the wave front, and (e) a formation of a jet during wave breaking.

Odd surface waves in two-dimensional incompressible fluids

We consider free surface dynamics of a two-dimensional incompressible fluid with odd viscosity. The odd viscosity is a peculiar part of the viscosity tensor which does not result in dissipation and is allowed when parity symmetry is broken. For the case of incompressible fluids, the odd viscosity manifests itself through the free surface (no stress) boundary conditions. We first find the free surface wave solutions of hydrodynamics in the linear approximation and study the dispersion of such waves. As expected, the surface waves are chiral and even exist in the absence of gravity and vanishing shear viscosity. In this limit, we derive effective nonlinear Hamiltonian equations for the surface dynamics, generalizing the linear solutions to the weakly nonlinear case. Within the small surface angle approximation, the equation of motion leads to a new class of non-linear chiral dynamics governed by what we dub the chiral Burgers equation. The chiral Burgers equation is identical to the complex Burgers equation with imaginary viscosity and an additional analyticity requirement that enforces chirality. We present several exact solutions of the chiral Burgers equation. For generic multiple pole initial conditions, the system evolves to the formation of singularities in a finite time similar to the case of an ideal fluid without odd viscosity. We also obtain a periodic solution to the chiral Burgers corresponding to the non-linear generalization of small amplitude linear waves.

Investigation on the effects of versatile deformating bed on a water wave diffraction problem

A hydroelastic model is considered to examine the proliferation of water waves over little deformation on a versatile seabed. The versatile base surface is modelled as a thin large plate and depends upon Euler-Bernoulli beam equation. In such circumstances, two different modes of time-harmonic proliferating waves exist rather than one mode of proliferating waves for any particular frequency. The waves with smaller wavenumber proliferate along the free-surface and the other with higher wavenumber spreads along the versatile base surface. The expression for first and second-order potentials and, henceforth, the reflection and transmission coefficients upto second-order for both modes are acquired by the strategy in view of Green's function method. A fix of sinusoidal swells is considered for instance to approve the scientific outcomes. It is seen that when the train of occurrence waves engenders because of the free-surface unsettling influence or the flexural wave movement in the fluid, we generally acquire the reflected and transmitted vitality exchange from the free-surface wave mode to the flexural wave mode. Further, we understand that the practical changes in the flexural unbending nature on the versatile base surface have a remarkable effect on the issue of water wave proliferation over small bottom distortions.

Surface gravity wave interaction with a submerged horizontal flexible porous plate

An analytical study for surface gravity wave interaction with submerged flexible porous plate based on small amplitude water wave theory and structural response is presented. The flexible porous plate is modeled based on the thin elastic plate theory and wave past porous plate is using generalized Darcy's law to incorporate both frictional force and inertia force. The dispersion relation is analyzed to determine the wave motion characteristics of two propagating wave modes in the presence of free surface and flexural waves at the submerged horizontal porous plate in specific cases. Further, the linearized long wave equation under shallow water approximation is derived in a direct manner and the limiting cases are compared. The integral forms of Green’s functions are derived using the fundamental solution associated with the two-dimensional Laplace equation. Using Green’s identity, the generalized expansion formulae for the wave-maker problem associated with the surface wave interaction with the submerged flexible porous plate are obtained in both the cases of finite and infinite water depths. The usefulness of the expansion formula is demonstrated by analyzing a physical problem associated with the surface gravity wave interaction with a moored finite submerged horizontal elastic porous plate in finite water depth. In the numerical results, the accuracy of the numerical computations are checked and the combined effect of mooring stiffness and porous-effect parameter are analyzed on the reflection coefficients, wave energy dissipation coefficients, and vertical forces. It is observed that the wave energy dissipation/absorption depends significantly on the mooring stiffness, porous-effect parameter, and suitable positing of the submerged plate.

An efficient split-step and implicit pure mesh-free method for the 2D/3D nonlinear Gross–Pitaevskii equations

In this paper, a high-efficient, split-step, and implicit corrected parallel smoothed particle hydrodynamics (SS-ICPSPH) method is developed to simulate the dynamic systems of several nonlinear Schrödinger/Gross–Pitaevskii equations (NLSE/GPE). The proposed method is motivated by the split-step for the equation, the corrected symmetric kernel gradient for the traditional SPH and the implicit scheme for time, respectively. Meanwhile, the MPI parallel technique is introduced to enhance the computational efficiency. Firstly, the numerical accuracy and the merits of the proposed method are tested by solving 2D NLSE, and compared with the analytical results. Secondly, the new method is extended to simulate the 2D/3D two-component GPE, compared with high accuracy finite difference results. Thirdly, the proposed method is extended to investigate the sheet-like vortices in rotating Bose–Einstein condensate. Finally, the implicit corrected SPH scheme is tentatively extended to capture the propagation process of free surface wave in a rectangular pool with initial perturbation. All the numerical results show the ability and the reliability of the proposed method.

Optical diagnostic systems based on Shenguang III

Optical diagnostic system plays an important role in inertial confinement fusion (ICF) experiment. Some key physical quantities and parameters in fusion can be diagnosed intuitively. In this review, three typical diagnostic systems on Shengang III facility are introduced, including imaging velocity interferometer system for any reflector (VISAR), fusion-reaction diagnostic system and backscatter light diagnostic system. VISAR can be used to measure velocity of free surface or shock wave traveling history in transparent media, which is an important diagnostic technique for shock wave velocity regulation, material compression characteristics and radiation temperature measurement. The fusion reaction-history diagnostic system can be used to accurately measure the fusion rate of ICF experiment. It’s a key diagnostic technique to study implosion dynamics and understand the mixed effect. Backscatter light diagnostic system is capable to measuring the energetic quotient stimulated by laser and plasma interaction (LPI), which produce important supports for calculating the coupling efficiency of driven laser and target.

Dynamics of Trapped Solitary Waves for the Forced KdV Equation

The forced Korteweg-de Vries equation is considered to investigate the impact of bottom configurations on the free surface waves in a two-dimensional channel flow. In the study of shallow water waves, the bottom topography plays a critical role, which can determine the characteristics of wave motions significantly. The interplay between solitary waves and the bottom topography can exhibit more interesting dynamics of the free surface waves when the bottom configuration is more complex. In the presence of two bumps, there are multiple trapped solitary wave solutions, which remain stable between two bumps up to a finite time when they evolve in time. In this work, various stationary trapped wave solutions of the forced KdV equation are explored as the bump sizes and the distance between two bumps are varied. Moreover, the semi-implicit finite difference method is employed to study their time evolutions in the presence of two-bump configurations. Our numerical results show that the interplay between trapped solitary waves and two bumps is the key determinant which influences the time evolution of those wave solutions. The trapped solitary waves tend to remain between two bumps for a longer time period as the distance between two bumps increases. Interestingly, there exists a nontrivial relationship between the bump size and the time until trapped solitary waves remain stable between two bumps.