Free Surface Boundary Research Articles

Fully nonlinear simulation of wave interaction with a cylindrical wave energy converter in a numerical wave tank

Convergence and accuracy of fully nonlinear time-domain simulation of wave interaction with a submerged cylindrical wave energy converter oscillating about an off-center axis which was conducted using a two-dimensional numerical wave tank are examined in this study. A high-order boundary element method is used to compute the fluid flow in the Eulerian frame. Fully nonlinear boundary conditions are applied to achieve the unique solution of the boundary value problem. Hence, kinematics of the fluid particles on the exact free surface boundary is calculated using the material node approach in the Lagrangian frame. Runge-Kutta time integration is employed to update the boundary values and the computational control volume. To implement the body condition, the time derivative of the velocity potential is computed implicitly using the acceleration potential. An artificial wave maker and two artificial sponge layers are provided to retain the open water condition. The power absorption efficiency of the submerged Bristol cylinder is compared with experimental data and numerical solutions for different incident waves to verify the present numerical model and the accuracy of the solution is also investigated. The convergence of the present fully nonlinear model in the simulation of the freely rotating cylinder about the off-centered axis as a wave energy converter is evaluated for different submergence depths, off-center distances, and a robust nonlinear incident wave.

Theoretical and experimental researches of the liquid evaporation during thermal vacuum influences

The mathematical model of the evaporation process of model liquid with the free surface boundary conditions of the "mirror" type under thermal vacuum influence and the numerical estimates of the evaporation process parameters are developed. An experimental stand, comprising a vacuum chamber, an experimental model tank with a heating element is designed; the experimental data are obtained. A comparative analysis of numerical and experimental results showed their close match.

Influence of surge movement in non-uniform water flow on performance of underwater quantum communication

Quantum communication is brand new way of communication in which quantum entanglement is used to transmit information. It is an interdisciplinary subject combining quantum informatics with modern communication theory. Motivated by the communication requirements for underwater sensor networks, submarines, etc., underwater optical communication has been developing rapidly in recent years due to the ideal information security of quantum communication. However, the research on the performance of underwater quantum communication in sea has not yet been fully developed because of a series of factors such as surge, salinity and seaweed and so on. In this paper, the influence of surge in non-uniform water flow on the underwater quantum communication is studied theoretically and experimentally. Firstly, a new Boussinesq equation with a given flow function is derived based on the horizontal and vertical wave velocity of the free surface to represent the free surface boundary conditions. On the other hand, In view of the nonlinear motion of movement, the complexity of change and the randomness of the distribution, the spectrum is used for numerically calculating the surge. The characteristics of wave motion are described by wave height, period and wavelength. Secondly, the influence of surge on the entanglement of underwater quantum channel is analyzed. It is proved that the wave height of surge and the change of the cycle affect quantum communication due to the destruction of the quantum coherence and the reduction in quantum entanglement degree. Thirdly, the influence of surge motion on the quantum channel capacity is studied. The influence of the relation between the wavelength and the transmission cycle on the quantum channel capacity is simulated. The relationship between the physical characteristics of surge wave and the capacity of depolarized channel is established. Fourthly, the influence of surge motion on error rate in quantum key distribution is studied. The simulation results show that when the sea surface wind speed changes in a range of 0-20.5 m/s, the propagation cycle is increased gradually. The channel entanglement is increased from 0.0012 to 0.8426, and the channel capacity is reduced from 0.8736 to 0.1024. In the key distribution process, the quantum bit error rate increases from 0.1651 to 0.4812. Therefore, in underwater quantum communication, the parameters of the system should be adjusted adaptively according to the varying degree of the surge movement.

Simulation of hydraulic characteristics of an inclined overflow gate by the free-surface lattice Boltzmann-immersed boundary coupling scheme

ABSTRACTInclined overflow gates are widely used in open channels to meet different flow rate requirements by promptly adjusting their openings according to varied water levels. The characteristics of such gates are quite complex and need intensive study. To develop an effective and efficient numerical method for studying the hydrodynamic features of similar engineering structures with fluid structure interactions and free-surface flows, we propose a free-surface lattice Boltzmann-immersed boundary (FSLB-IB) coupling model, in which the free-surface flow is modeled by the lattice Boltzmann (LB) method and the moving boundaries are handled by the immersed boundary (IB) method. To solve the incompatibility issue arising from the coupling of the FSLB model and IB method, two treatments, namely the velocity correction in the empty cells for matching the IB speed with the local fluid velocity, and the iterative force-correction for enhancing accuracy, are introduced. The feasibility and accuracy of the proposed coupling scheme are first validated by two benchmark cases. Then the FSLB-IB scheme is applied to analyze the hydraulic characteristics of an inclined overflow gate in different working conditions. The good agreements between the simulated and empirical hydraulic parameters as well as the reasonable flow features show that the proposed coupling scheme is feasible for simulating practical hydraulic problems.

CFD verification and validation of green sea loads

An extensive verification and validation for green sea load simulations is presented. The calculations are performed using the Naval Hydro pack, a library based on foam–extend, which is an open source Computational Fluid Dynamics software. The geometric Volume of Fluid method is used for interface advection, while the Ghost Fluid Method is employed to discretise the free surface boundary conditions at the interface. Pressure measured at the deck of a fixed structure is compared to experimental data for nine regular waves. Verification is performed using four refinement levels in order to reliably assess numerical uncertainties. A detailed uncertainty analysis comprises both numerical and experimental data. Comparable uncertainties are exhibited in simulations and experiments, with good agreement of results.

A comparison of the neumann-kelvin and rankine sourcemethods for wave resistance calculations

Calm water wave resistance plays a very important role in ship hull design. Numerical methods are meaningful for this reason. In this study, two prevailing methods, the Neumann-Kelvin and the Rankine source method, were implemented and compared. The Neumann-Kelvin method assumes linearized free surface boundary condition and only needs to mesh the hull surface. The Rankine source method considers nonlinear free surface boundary condition and meshes both the ship hull surface and free surface. Both methods were implemented and the wave resistance of a Wigley III and three Series 60(Cb=0.6, 0.7, 0.8) hulls were analyzed. The results were compared with experimental results and the merits of both numerical techniques were quantified. Based on the results, it is concluded that the Rankine source method is more accurate in the calculation of the wave-making resistance. Using the Neumann-Kelvin method, it is found to be easier to model the hull and can be used for slender ships to solve problems like wave current coupling calculation.

Ship squat prediction using a potential flow Rankine source method

A numerical method has been developed for predicting ship squat, with the goal of providing guidance to ship operators for avoidance of grounding in shallow water. The method assumes potential flow, and computations are sufficiently fast to allow evaluation for large numbers of conditions within practical time scales. Boundary conditions on the hull, free surface, and canal walls are satisfied using planar Rankine sources on solid boundaries and point Rankine sources placed a nominal distance above the calm waterline or free surface. Four different models of the free surface boundary condition have been implemented, ranging from a nominally flat free surface to fully nonlinear. For ships with a transom stern, the influence of flow separation is modelled using a virtual stern, which extends afterward from the transom to a location with zero width. The numerical method is generally robust, except for in the vicinity of critical flow conditions. Validation with model tests has been used to develop an envelope for over-prediction of bottom clearance, which can be considered when developing guidance for ship operators.

Solving the linearized forward-speed radiation problem using a high-order finite difference method on overlapping grids

The linearized potential flow approximation for the forward speed radiation problem is solved in the time domain using a high-order finite difference method. The finite-difference discretization is developed on overlapping, curvilinear body-fitted grids. To ensure numerical stability, the convective derivatives in the free-surface boundary conditions are treated using an upwind-biased stencil. Instead of solving for the radiation impulse response functions, a pseudo-impulsive Gaussian type displacement is employed in order to tailor the frequency-content to the discrete spatial resolution. Frequency-domain results are then obtained from a Fourier transform of the force and motion signals. In order to make a robust Fourier transform, and capture the response around the critical frequency, the tail of the force signal is asymptotically extrapolated assuming a linear decay rate. Fourth-order convergence of the calculations on simple geometries is demonstrated, along with a nearly linear scaling of the solution effort with increasing grid resolution. The code is validated by comparison with analytical and semi-analytical solutions using submerged and floating closed-form geometries. Calculations are also made for a modern bulk carrier, and good agreement is found with experimental measurements.

Effect of surface-related Rayleigh and multiple waves on velocity reconstruction with time-domain elastic FWI

It is critically important to assess the effectiveness of elastic full waveform inversion (FWI) algorithms when FWI is applied to real land seismic data including strong surface and multiple waves related to the air-earth boundary. In this paper, we review the realization of the free surface boundary condition in staggered-grid finite-difference (FD) discretization of elastic wave equation, and analyze the impact of the free surface on FWI results. To reduce inputs/outputs (I/O) operations in gradient calculation, we adopt the boundary value reconstruction method to rebuild the source wavefields during the backward propagation of the residual data. A time-domain multiscale inversion strategy is conducted by using a convolutional objective function, and a multi-GPU parallel programming technique is used to accelerate our elastic FWI further. Forward simulation and elastic FWI examples without and with considering the free surface are shown and analyzed, respectively. Numerical results indicate that no free surface incorporated elastic FWI fails to recover a good inversion result from the Rayleigh wave contaminated observed data. By contrast, when the free surface is incorporated into FWI, the inversion results become better. We also discuss the dependency of the Rayleigh waveform incorporated FWI on the accuracy of initial models, especially the accuracy of the shallow part of the initial models.

Nonlinear and viscous effects on the hydrodynamic performance of a fixed OWC wave energy converter

The hydrodynamic performance of a fixed Oscillating Water Column (OWC) device is experimentally and numerically investigated. Based on the time-domain higher-order boundary element method (HOBEM), by introducing an artificial viscosity term in the dynamic free surface boundary condition, a fully nonlinear numerical wave model is used to simulate the hydrodynamic performance of an OWC device. A set of comprehensive experiments for regular waves is carried out to validate the numerical results as well as to investigate the nonlinear effects on the hydrodynamic performance of OWC. The mechanism of the nonlinear phenomenon is investigated based on the analysis of the experimental and numerical results. The influence of the wave nonlinearity and the viscosity on the hydrodynamic efficiency is quantified by comparing the linear and nonlinear numerical results. It was found that the hydrodynamic efficiency increases with the nonlinearity and viscosity when the incident wave amplitude is small. When the incident wave amplitude is large, the hydrodynamic efficiency is reduced by the weakened transmission of the second-order harmonic wave component due to the strong wave nonlinearity. However, when the wave amplitude is between these two regimes, the wave is weakly nonlinear, the efficiency decreases with the wave amplitude due to the combined effect of the nonlinearity and viscosity.

A framework for efficient irregular wave simulations using Higher Order Spectral method coupled with viscous two phase model

In this paper a framework for efficient irregular wave simulations using Higher Order Spectral method coupled with fully nonlinear viscous, two-phase Computational Fluid Dynamics (CFD) model is presented. CFD model is based on solution decomposition via Spectral Wave Explicit Navier–Stokes Equation method, allowing efficient coupling with arbitrary potential flow solutions. Higher Order Spectrum is a pseudo-spectral, potential flow method for solving nonlinear free surface boundary conditions up to an arbitrary order of nonlinearity. It is capable of efficient long time nonlinear propagation of arbitrary input wave spectra, which can be used to obtain realistic extreme waves. To facilitate the coupling strategy, Higher Order Spectrum method is implemented in foam-extend alongside the CFD model. Validation of the Higher Order Spectrum method is performed on three test cases including monochromatic and irregular wave fields. Additionally, the coupling between Higher Order Spectrum and CFD is validated on three hour irregular wave propagation. Finally, a simulation of a 3D extreme wave encountering a full scale container ship is shown.

Signal transduction in the invadopodia formation using fixed domain method

The degradation of the extracellular matrix is driven by actin-based protrusions known as invadopodia, lead to the cancer cell invasion across the surrounding tissue barriers. Signal transduction through the binding between ligand and membrane associated receptor is important in order to establish the actin polymerization, and consequently push the membrane of migrating cells. In this study, we considered one-dimensional Stefan-like problem of the signal transduction and cell membrane is treated as a free boundary surface to separate any activity that happened on intra- and extra-cellular regions. The velocity concerning the movement of the free boundary is calculated by the decrease of signal gradient on the front. The problem is solved numerically using finite-difference scheme of fixed-domain method. Our results showed that both free boundary positions and signal distributions are increasing as time progresses.

Propulsion of an active flapping foil in heading waves of deep water

The propulsion of a flapping foil in heading waves has been investigated through the velocity potential theory. The trajectory of the flapping foil is prescribed with the combined sinusoidal heave and pitch motions. Boundary element method is introduced to solve the boundary value problem and a time stepping scheme is adopted to simulate the interaction of the foil and Stokes waves of infinite water-depth. The nonlinear free surface boundary conditions are imposed when updating the free surface. The effects of the frequency difference between foil motion and waves have been investigated. When the encountered wave frequency equals the flapping frequency, hydrodynamic performance of the flapping foil can be enhanced significantly.

Three-dimensional steep wave impact on a vertical plate with an open rectangular section

The present study treats the three-dimensional hydrodynamic slamming problem on a vertical plate subjected to the impact of a steep wave moving towards the plate with a constant velocity. The problem is complicated significantly by assuming that there is a rectangular opening on the plate which allows a discharge of the liquid. The analysis is conducted analytically assuming linear potential theory. The examined configuration determines two boundary value problems with mixed conditions which fully are taken into account. The mathematical process assimilates the plate with a degenerate elliptical cylinder allowing the employment of elliptical harmonics that ensure the satisfaction of the free-surface boundary condition of the front face of the steep wave, away from the plate. This assumption leads to an additional boundary value problem with mixed conditions in the vertical direction. The associated problem involves triple trigonometrical series and it is solved through a transformation into integral equations. To tackle the boundary value problem in the vertical direction a perturbation technique is employed. Extensive numerical calculations are presented as regards the variation of the velocity potential on the plate at the instant of the impact which reveals the influence of the opening. The theory is extended to the computation of the total impulse exerted on the plate using pressure-impulse theory.

A three-dimensional model of coupling dynamics of keyhole and weld pool during electron beam welding

In electron beam welding (EBW), the coupling dynamics between the keyhole and the weld pool play significant roles in joint quality, yet the mechanisms of these dynamics have not been well understood, since the rise of EBW in the 1960s. The present study reported a novel three-dimensional mathematical model of EBW and used it to theoretically understand the coupling behaviors between the keyhole and the weld pool during EBW of a Ti-6-Al-4V alloy. A new phenomenological heat source model, dependent on the dynamic keyhole profile and temperature, was developed. The physical effects, including the beam reflection and absorption of the keyhole wall, scattering of secondary electrons, keyhole evaporation heat loss and beam defocus, were rigorously treated in the heat source model. The transient free surface keyhole dynamics were tracked with a high-resolution level set method (LSM). On the free surface boundary, the majority of the key physical effects, such as the recoil pressure, Marangoni shear stress, the surface tension and hydrodynamic pressure of the molten liquid, were incorporated using a recently developed sharp interface model. The coupling behaviors of the three-dimensional keyhole evolutions, heat transfer, and fluid flow in the weld pool asa function of the process parameters in the EBW of Ti-6-Al-4-V alloys were simulated, of which the results reasonably agreed with independent experimental data. The dynamical keyhole violently exhibited fluctuations within the sub-millisecond characteristic time in the radial direction. The oscillations of the keyhole, which were characterized by its surface area and volume, were theoretically studied asa function of the welding speed and surface tension coefficient. The welding speed has stabilization effects on the keyhole and fully stabilized the keyhole under certain high welding speed conditions, an effect similar to that observed in keyhole mode laser welding. The surface tension governed the keyhole oscillation periods and was proposed asa major source of oscillations. In addition, the complex and violent flow patterns of weld pool were predicated and consistent with the well-accepted X-ray imaging experimental results. The EBW transient keyhole and weld pool dynamics were self-consistent. The present study is to present the successful development of EBW, thereby allowing the establishment of the self-consistent keyhole and weld pool dynamics asa function of the process parameters and material properties, to provide a promising avenue for process optimizations in industrial applications.

Frequency domain finite-element and spectral-element acoustic wave modeling using absorbing boundaries and perfectly matched layer

Wave propagation modeling as a vital tool in seismology can be done via several different numerical methods among them are finite-difference, finite-element, and spectral-element methods (FDM, FEM and SEM). Some advanced applications in seismic exploration benefit the frequency domain modeling. Regarding flexibility in complex geological models and dealing with the free surface boundary condition, we studied the frequency domain acoustic wave equation using FEM and SEM. The results demonstrated that the frequency domain FEM and SEM have a good accuracy and numerical efficiency with the second order interpolation polynomials. Furthermore, we developed the second order Clayton and Engquist absorbing boundary condition (CE-ABC2) and compared it with the perfectly matched layer (PML) for the frequency domain FEM and SEM. In spite of PML method, CE-ABC2 does not add any additional computational cost to the modeling except assembling boundary matrices. As a result, considering CE-ABC2 is more efficient than PML for the frequency domain acoustic wave propagation modeling especially when computational cost is high and high-level absorbing performance is unnecessary.

Numerical and experimental study on seakeeping performance of ship in finite water depth

Vessels operating in shallow waters require careful observation of the finite-depth effect. In present study, a Rankine source method that includes the shallow water effect and double body steady flow effect is developed in frequency domain. In order to verify present numerical methods, two experiments were carried out respectively to measure the wave loads and free motions for ship advancing with forward speed in head regular waves. Numerical results are systematically compared with experiments and other solutions using the double body basis flow approach, the Neumann-Kelvin approach with simplified m-terms, and linearized free surface boundary conditions with double-body m-terms. Furthermore, the influence of water depths on added mass and damping coefficients, wave excitation forces, motions and unsteady wave patterns are deeply investigated. It is found that finite-depth effect is important and unsteady wave pattern in shallow water is dependent on both of the Brard number τ and depth Froude number Fh.

The “avalanche effect” of an elasto-viscoplastic thixotropic material on an inclined plane

The so-called avalanche effect is one of the fingerprints of thixotropic materials. This self-reinforcing process where the decrease in viscosity, due to a rejuvenation process triggered by a stress field, induces a motion which in turn contributes to decrease the viscosity again, is well exemplified by the inclined plane problem. In this situation, the material in its fully-structured state is placed on an inclined plane with respect to the gravity force which is responsible for the beginning of the breakdown process. These thixotropic systems generally have a yield stress, a strength that must be overcome in order to induce rejuvenation. In addition, they exhibit elastic features, especially in the pre-yield state. In the present work we numerically solve the transient evolution of an elasto-viscoplastic thixotropic material subjected to the action of gravity on an inclined plane. In order to handle with the moving free-surface boundary condition encountered in the avalanche effect, we have used a combination of the Marker-And-Cell (MAC) method with the front-tracking scheme. This formulation was successfully employed for this kind of material in the recent paper of Oishi et al. (2016) [28]. In the present work, we have adapted our finite difference formulation to analyze the effects associated with an extended Herschel–Bulkley model in the simulation of a transient complex free surface flow. Concerning the parameters of the flow curve, it is shown that the dimensionless yield stress (plastic number) is the most significant one. However, for a fixed plastic number, different combinations of dimensionless consistency index and dimensionless Newtonian viscosity plateau can lead to a diversity of responses. The thixotropic equilibrium time had a significant impact on shifting the instant when the flow regime changes from an accelerating (when the front part of the material accelerates) to a retardation one (when this front part decelerates). Higher elasticity, as captured by the Weissenberg number, led to longer distances covered by the material.

Thin film flow of bi-viscosity liquid over an unsteady stretching sheet: An analytical solution

Two-dimensional flow of a thin bi-viscosity liquid film over an impulsively stretching sheet has been studied as an unsteady flow problem. Full Navier-Stokes equations and the equation of continuity are solved analytically along with nonlinear free surface boundary conditions. In other words, analytical solutions are obtained while the guiding equations and the boundary equations are both nonlinear PDE. In solving the flow problem, method of singular perturbation and the method of characteristics are used. The results show that the initial non-uniform film thickness will produce planar film at large time while the initial planar film remains planar. Further, the film thins faster for bi-viscosity liquid in compare with Newtonian liquid. It is also found that increase of the bi-viscosity parameter β imparts more resistance on film thinning.

Simulation of wave overtopping using an improved SPH method

An improved Smoothed Particle Hydrodynamics (SPH) method is used to study wave overtopping for different coastal structures. Simulated wave overtopping is too sensitive to the particle movements near the free surface boundary; however, the calculated flow acceleration by means of common SPH methods is not free of errors at this boundary due to the truncation of kernel function and contribution of fewer particles in solving the governing equations. In this paper, this problem is solved by modifying the viscosity of surface particles based on the concept of surface viscosity originally introduced by Xu (2010). By means of the introduced modification, unrealistic particle fluctuation at free surface boundary can be decreased significantly while keeping the model accuracy. This improvement can be used for both Incompressible and Weakly Compressible SPH methods and its implementation is easy and computationally efficient too. Different cases including dam break, solitary wave breaking and wave overtopping at vertical and sloping seawalls are simulated with the modified model and the new model is validated via comparing the results with several experimental and numerical data. Based on this study, free surface boundary can be simulated more accurately by means of the introduced modification and as a result, the predicted values particularly the calculated wave-overtopping rate become more reliable.