Simulating the evolution of focused waves by a two-layer Boussinesq-type model

Accurate simulation of the evolution of freak waves by the wave phase focusing method requires accurate linear and nonlinear properties,especially in deep-water conditions.In this paper,we analyze the ability to simulate deep-water focused waves of a two-layer Boussinesq-type(BT)model,which has been shown to have excellent linear and nonlinear performance.To further improve the numerical accuracy and stability,the internal wavegenerated method is introduced into the two-layer Boussinesq-type model.Firstly,the sensitivity of the numerical results to the grid resolution is analyzed to verify the convergence of the model;secondly,the focused wave propagating in two opposite directions is simulated to prove the symmetry of the numerical results and the feasibility of the internal wave-generated method;thirdly,the limiting focused wave condition is simulated to compare and analyze the wave surface and the horizontal velocity of the profile at the focusing position,which is in good agreement with the measured values.Meanwhile the simulation of focused waves in very deep waters agrees well with the measured values,which further demonstrates the capability of the two-layer BT model in simulating focused waves in deep waters.

Influences of Floater Motion on Gap Resonance Triggered by Focused Wave Groups

The current study investigates the hydrodynamic characteristics of gap resonance within a narrow gap formed by two adjacent boxes subjected to incident focused transient wave groups.A two-dimensional(2D)numerical wave tank based on the OpenFOAM package is utilized for this purpose.The weather-side box is fixed while the lee-side box is allowed to heave freely under wave actions.The effects of the focused wave amplitude and spectral peak period on the wave amplification within the gap,motion of the lee-side box,and wave forces(including horizontal and vertical wave forces)acting on each box are systematically examined.For comparison,another structural layout consisting of two fixed boxes is also considered.The results reveal that the release of the heave degree of freedom(DoF)of the lee-side box results in remarkably distinct resonance features.In the heave-box system,both its fluid resonant period and the period corresponding to the maximum heave displacement of the lee-side box are significantly larger(i.e.,1.6-1.7 times)than the fluid resonant period of the fixed-box system.However,the wave amplification factor inside the gap in the heave-box system is significantly lower than that in the fixed-box one.Both the variations of the maximum horizontal and vertical wave forces with the spectral peak period and their magnitudes are also significantly different between the two structural systems.

Simulation of Horizontal-Two-Dimension Focused Waves Using A Two-Layer Boussinesq-Type Model

Accurate simulation of the horizontal-two-dimension(H2D)focused wave group in deep water requires high accuracy of a numerical model.The two-layer Boussinesq-type model(Liu and Fang,2016;Liu et al.,2018)with the highest spatial derivative of 2 has high accuracy in both linear and nonlinear properties.Based on the further development of the velocity equations(Liu et al.,2023),the H2D numerical model for water waves is established with the prediction-correction-iteration model in the finite difference method,and a composite fourth-order Adams-Bashforth-Moulton scheme is used for time integration.The wave generation method proposed by Hsiao et al.(2005)is applied and calibrated in this H2D model.The numerical calculations lead to the following three main conclusions:First,compared with the analytical solution of Stokes linear waves,the calculated velocity profiles show higher accuracy by using the improved velocity formulas.Second,the simulations of the focused multidirectional wave group are carried out,and good agreements are found,demonstrating that the present H2D numerical model shows high accuracy in simulating focused multidirectional wave groups,and the effectiveness of the improved velocity formulas is also validated.Furthermore,the velocity profiles throughout the computational domain at the time of maximum wave crest are given.Finally,the FFT method is used to obtain the amplitude with different frequencies for several locations,and the changes of the wavelet energy spectrum at different locations are presented for several cases.

Focused Wave Properties Based on A High Order Spectral Method with A Non-Periodic Boundary

In this paper, a numerical model is developed based on the High Order Spectral （HOS） method with a non-periodic boundary. A wave maker boundary condition is introduced to simulate wave generation at the incident boundary in the HOS method. Based on the numerical model, the effects of wave parameters, such as the assumed focused amplitude, the central frequency, the frequency bandwidth, the wave amplitude distribution and the directional spreading on the surface elevation of the focused wave, the maximum generated wave crest, and the shifting of the focusing point, are numerically investigated. Especially, the effects of the wave directionality on the focused wave properties are emphasized. The numerical results show that the shifting of the focusing point and the maximum crest of the wave group are dependent on the amplitude of the focused wave, the central frequency, and the wave amplitude distribution type. The wave directionality has a definite effect on multidirectional focused waves. Generally, it can even out the difference between the simulated wave amplitude and the amplitude expected from theory and reduce the shifting of the focusing points, implying that the higher order interaction has an influence on wave focusing, especially for 2D wave. In 3D wave groups, a broader directional spreading weakens the higher nonlinear interactions.

Nonlinear Dynamic Behaviors of A Floating Structure in Focused Waves

Floating structures are commonly seen in coastal and offshore engineering. They are often subjected to extreme waves and, therefore, their nonlinear dynamic behaviors are of great concern. In this paper, an in-house CFD code is developed to investigate the accurate prediction of nonlinear dynamic behaviors of a two-dimensional（2-D） box-shaped floating structure in focused waves. Computations are performed by an enhanced Constrained Interpolation Profile（CIP）-based Cartesian grid model, in which a more accurate VOF（Volume of Fluid） method, the THINC/SW scheme（THINC： tangent of hyperbola for interface capturing; SW： Slope Weighting）, is used for interface capturing. A focusing wave theory is used for the focused wave generation. The wave component of constant steepness is chosen. Comparisons between predictions and physical measurements show good agreement including body motions and free surface profiles. Although the overall agreement is good, some discrepancies are observed for impact pressure on the superstructure due to water on deck. The effect of grid resolution on the results is checked. With a fine grid, no obvious improvement is seen in the global body motions and impact pressures due to water on deck. It is concluded that highly nonlinear phenomena, such as distorted free surface, large-amplitude body motions, and violent impact flow, have been predicted successfully.

Numerical Comparison for Focused Wave Propagation Between the Fully Nonlinear Potential Flow and the Viscous Fluid Flow Models

Numerical simulations on focused wave propagation are carried out by using three types of numerical models,including the linear potential flow,the nonlinear potential flow and the viscous fluid flow models.The wave-wave interaction of the focused wave group with different frequency bands and input wave amplitudes is examined,by which the influence of free surface nonlinearity and fluid viscosity on the related phenomenon of focused wave is investigated.The significant influence of free surface nonlinearity on the characteristics of focused wave can be observed,including the increased focused wave crest,delayed focused time and downstream shift of focused position with the increase of input amplitude.It can plot the evident difference between the results of the nonlinear potential flow and linear potential flow models.However,only a little discrepancy between the nonlinear potential flow and viscous fluid flow models can be observed,implying the insignificant effect of fluid viscosity on focused wave behavior.Therefore,the nonlinear potential flow model is recommended for simulating the non-breaking focused wave problem in this study.

Numerical Study of the High-Frequency Wave Loads and Ringing Response of A Bottom-Hinged Vertical Cylinder in Focused Waves

This paper presents a numerical study on the high-frequency wave loads and ringing response of offshore wind turbine foundations exposed to moderately steep transient water waves.Input wave groups are generated by the technique of frequency-focusing,and the numerical simulation of focused waves is based on the NewWave model and a Fourier time-stepping procedure.The proposed model is validated by comparison with the published laboratory data.In respect of both the wave elevations and the underlying water particle kinematics,the numerical results are in excellent agreement with the experimental data.Furthermore,the local evolution of power spectra and the transfer of energy into higher frequencies can be clearly identified.Then the generalized FNV theory and Rainey’s model are applied respectively to calculate the nonlinear wave loads on a bottom-hinged vertical cylinder in focused waves.Resonant ringing response excited by the nonlinear high-frequency wave loads is found in the numerical simulation when frequency ratios(natural frequency of the structure to peak frequency of wave spectra)are equal to 3–5.Dynamic amplification factor of ringing response is also investigated for different dynamic properties(natural frequency and damping ratio)of the structure.

Numerical investigations of focused wave interact with a moving cylinder

This paper presents a numerical study on focused wave and current interactions with a cylinder.The cylinder is moving in the opposite direction to the wave propagation.An effective computational decomposition method is adopted to reduce the calculation resources.A potential solver high-order spectral(HOS)method is applied to generate focused wave field,while our in-house computational fluid dynamics(CFD)solver naoe-FOAM-SJTU with overset grid takes the charge of achieving the viscous effect around the moving cylinder.The viscous domain moving with the cylinder thus the size and mesh grids in computational domain is greatly reduced.The pressure on cylinder surface and wave fields around cylinder are compared with experimental data,shows a well agreement.Meanwhile,the scattering wave field and vortex shedding are discussed.With the existence of moving cylinder,the classical scattering wave types are still observed.

NUMERICAL SIMULATION OF PROPAGATION AND BREAKING PROCESSES OF A FOCUSED WAVES GROUP

A two-dimensional numerical wave flume is developed to study the focused waves group propagation and the consequent breaking processes. The numerical model is based on the Reynolds-Averaged Navier-Stokes （PANS） equations, with the standard k - c turbulence model to simulate the turbulence effects. To track the complicated and broken free-surface, the Volume Of Fluid （VOF） method is employed. The numerical model combines the ＂Partial Cell Treatment （PCT）＂ method with the ＂Locally Relative Stationary （LRS）＂ concept to treat the moving wave paddle so that various waves can be generated directly in a fixed Cartesian grid system. The theoretical results of the linear and nonlinear waves are used to validate the numerical wave flume firstly, and then a plunging breaking wave created by a focused waves group is simulated. The numerical results are compared to the experimental data and other simulation results, with very good agreements. The turbulence intensity, the flow field and the energy dissipation in the breaking processes are analyzed based on the numerical results. It is shown that the present numerical model is efficient and accurate for studying the waves group generation, the waves packet propagation, and the wave breaking processes.

Numerical Simulation of Focused Wave and Its Uncertainty Analysis

On the basis of the transient water wave(TWW) theory,focused wave is generated in the circulating water channel.Numerical simulation of the focused wave is carried out by solving the Reynolds averaged NavierStokes(RANS) equations.The dynamic grid technique is adopted to simulate the motion of the wave maker,and the volume of fluid(VOF) method is used to capture the free surface of the wave.The simulation results are compared with the measured data,and good agreement is obtained.For quantitative estimation of the numerical simulation error and uncertainty,the uncertainty analysis method recommended by the International Towing Tank Conference(ITTC) procedure is performed for the simulation results of the surface elevations at different positions.Both grid-convergence and time-step-size convergence studies are conducted using three types of grids and time step sizes.The simulation results are all monotonously convergent in the verification procedure,and the validations of the simulated surface elevations with the positions at 3.5,4.0 and 4.5 m are all achieved by comparing with the validation uncertainty.It is found that the numerical simulation errors caused by the grid and time-step-size in the convergence studies have the same order of magnitude.In addition,the numerical errors and uncertainties for the surface elevations at different positions are compared and discussed in detail.This paper presents the first attempt to carry out the uncertainty analysis of the simulation of focused wave,and the effectiveness of the proposed verification and validation procedures in the uncertainty analysis is demonstrated.

The response of a semisubmersible model under focused wave groups: Experimental investigation

In recent years,extreme wave events have occurred more frequently than have been predicted using theoretical methods.It is,therefore,a requirement to investigate the impact of these events on coastal and offshore structures.This paper reports on results of an experimental investigation into the interaction between unidirectional waves and a horizontally moored semisubmersible model.The target crest height was created at a focal point and time using the focused wave technique.Different values of wave steepness were tested in order to ascertain the nonlinear effects on the quality of waves generated by a piston-type wavemaker.The measured crest height was in good agreement with the theoretical one within 4%relative error.The magnitudes of heave and pitch motions of the model were found to increase as the wave steepness increased.Overall,the paper contributes towards establishing the application of focused wave technique to floating offshore platforms.

Influence of mode conversions in the skull on transcranial focused ultrasound and temperature fields utilizing the wave field separation method： A numerical study

Transcranial focused ultrasound is a booming noninvasive therapy for brain stimuli. The Kelvin–Voigt equations are employed to calculate the sound field created by focusing a 256-element planar phased array through a monkey skull with the time-reversal method. Mode conversions between compressional and shear waves exist in the skull. Therefore, the wave field separation method is introduced to calculate the contributions of the two waves to the acoustic intensity and the heat source, respectively. The Pennes equation is used to depict the temperature field induced by ultrasound. Five computational models with the same incident angle of 0？and different distances from the focus for the skull and three computational models at different incident angles and the same distance from the focus for the skull are studied. Numerical results indicate that for all computational models, the acoustic intensity at the focus with mode conversions is 12.05%less than that without mode conversions on average. For the temperature rise, this percentage is 12.02%. Besides, an underestimation of both the acoustic intensity and the temperature rise in the skull tends to occur if mode conversions are ignored. However, if the incident angle exceeds 30？, the rules of the over-and under-estimation may be reversed. Moreover,shear waves contribute 20.54% of the acoustic intensity and 20.74% of the temperature rise in the skull on average for all computational models. The percentage of the temperature rise in the skull from shear waves declines with the increase of the duration of the ultrasound.

DETONATION INITIATION INDUCED BY FLAME IMPLOSION AND SHOCK WAVE FOCUSING

Computational simulations on structurally different detonation generator are carried out to study the phenomena,the mechanism and the gas dynamics characteristics of flame implosion and shock wave focusing.Two-dimensional axisymmetric and unsteady Navier-Stokes equations are numerically solved and detailed chemical reaction kinetics of hydrogen/air mixture is used.The simulation results show that the laminar flame generated by low energy spark in the jet flame burner is accelerated under the narrow channel,the jet flame impinging on the axis strengthens shock wave and the shock wave enhances flame acceleration.Under the function of multiple shock waves and flame,a number of hot spots appear between the wave and the surface.The spots enlarge rapidly,thus forming an over-drive detonation with high pressure,and then declining to stable detonation.Through calculation and analysis,the length of detonation initiation and stable detonation are obtained,thus providing the useful information for further experimental investigations.

NUMERICAL INVESTIGATION OF TOROIDAL SHOCK WAVES FOCUSING USING DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD

A numerical simulation of the toroidal shock wave focusing in a co-axial cylindrical shock tube is inves- tigated by using discontinuous Galerkin （DG） finite element method to solve the axisymmetric Euler equations. For validating the numerical method, the shock-tube problem with exact solution is computed, and the computed results agree well with the exact cases. Then, several cases with higher incident Mach numbers varying from 2.0 to 5.0 are simulated. Simulation results show that complicated flow-field structures of toroidal shock wave diffraction, reflection, and focusing in a co-axial cylindrical shock tube can be obtained at different incident Mach numbers and the numerical solutions appear steep gradients near the focusing point, which illustrates the DG method has higher accuracy and better resolution near the discontinuous point. Moreover, the focusing peak pres- sure with different grid scales is compared.

An Efficient Model for Transient Surface Waves in Both Finite and Infinite Water Depths

A numerical model is developed to simulate fully nonlinear extreme waves in finite and infinite water-depth wave tanks. A semi-mixed Enlerian-Lagrangian formulation is adopted and a higher-order boundary element method in conjunction with an image Green function is used for the fluid domain. The botmdary values on the free surface are updated at each time step by a fourth-order Runga-Kutta time-marching scheme at each time step. Input wave characteristics are specified at the upstream boundary by an appropriate wave theory. At the downstream boundary, an artificial damping zone is used to prevent wave reflection back into the computational domain. Using the image Green function in the whole fluid domain, the integrations on the two lateral walls and bottom are excluded. The simulation results on extreme wave elevations in finite and infinite water-depths are compared with experimental results and second-order analytical solutions respectively. The wave kinematics is also discussed in the present study.

Harmonic Energy Transfer for Extreme Waves in Current

Ning et al. （2015） developed a 2D fully nonlinear potential model to investigate the interaction between focused waves and uniform currents. The effects of uniform current on focusing wave crest, focal time and focal position were given. As its extension, harmonic energy transfer for focused waves in uniform current is studied using the proposed model by Ning et al. （2015） and Fast Fourier Transformation （FFT） technique in this study. It shows that the strong opposing currents, inducing partial wave blocking and reducing the extreme wave crest, make the nonlinear energy transfer non-reversible in the focusing and defocusing processes. The numerical results also provide an explanation to address the shifts of focal points in consideration of the combination effects of wave nonlinearity and current.

Physical Investigation of Directional Wave Focusing and Breaking Waves in Wave Basin

An experimental scheme for the generation of directional focusing waves in a wave basin is established in this paper. The effects of the directional range, frequency width and center frequency on the wave focusing are studied. The distribution of maximum amplitude and the evolution of time series and spectra during wave packet propagation and the variation of water surface parameters are extensively investigated. The results reveal that the characteristics of focusing waves are significantly influenced by wave directionality and that the breaking criteria for directional waves are distinctly different from those for unidirectional waves.

Efficient Focusing Models for Generation of Freak Waves

Four focusing models for generation of freak waves are presented. An extreme wave focusing model is presented on the basis of the enhanced High-Order Spectral （HOS） method and the importance of the nonlinear wave-wave interaction is evaluated by comparison of the calculated results with experimental and theoretical data. Based on the modification of the Longuet-Higgins model, four wave models for generation of freak waves （a. extreme wave model ＋ random wave model; b. extreme wave model ＋ regular wave model; e. phase interval modulation wave focusing model; d. number modulation wave focusing model with the same phase） are proposed. By use of different energy distribution techniques in the four models, freak wave events are obtained with different Hmax/Hs in finite space and time.

Numerical Study of Two-Dimensional Focusing Waves

Two-dimensional focusing waves are generated and investigated by numerical method. The numerical model is developed by introducing the wave maker boundary on the high-order spectral （HOS） method proposed by Dommermuth and Yue in 1987 and verified by theoretical and experimental data. Some cases of focusing waves considering different parameters such as assumed focusing amplitudes, frequency bandwidth, central frequency and frequency spectrum are generated. Characteristics of the focusing wave including surface elevations, the maximum crest, shift of focusing points and frequency spectra are discussed. The results show that the focusing wave characteristics are strongly affected by focusing amplitudes, frequency bandwidth, central frequency and frequency spectrum.

Generation and Properties of Freak Waves in A Numerical Wave Tank

Freak waves are generated based on the mechanism of wave focusing in a 2D numerical wave tank. To set up the nonlinear numerical wave tank, the Boundary Element Method is used to solve potential flow equations incorporated with fully nonlinear free surface boundary conditions. The nonlinear properties of freak waves, such as high frequency components and wave profile asymmetry, are discussed. The kinematic data, which can be useful for the evaluation of the wave forces exerted on structures to avoid underestimation of linear predictions, are obtained, and discussed, from the simulated results of freak waves.