Nonlinear Potential Flow Theory Research Articles

Numerical Study on Wave Radiation by a Barge with Large Amplitudes and Frequencies

A two-dimensional boundary element method is used to study the hydrodynamics of a single barge with prescribed motions of large amplitudes and high frequencies. The wave radiation problem is solved in the time domain based on the fully nonlinear potential flow theory. For numerical simulations, special treatments like plunging wave cutting and remeshing approaches are presented in detail. The numerical schemes are verified through comparing with analytical results. Both the generated outgoing wave amplitudes and hydrodynamic coefficients can be calculated with sufficient accuracy. Then, we focus on large heave, sway and roll motions to investigate the nonlinear effects on hydrodynamic forces, respectively. In particular, the heave motion with two frequencies is also simulated to study the interactions between results at different frequencies. It is interesting to see the sum and difference frequency components and the envelopes in time histories as a result. For forces caused by forced sway or roll motions, there are only even-order harmonics for vertical forces and only odd-order harmonics for horizontal forces. Finally, a single body with combined sway, heave and roll motion is studied to examine the interactions between motion modes.

Numerical study of effects of gap and incident wave steepness on water resonance between two rectangular barges

A 2D numerical wave tank based on the fully nonlinear potential flow theory is built to simulate the water resonance between two fixed rectangular barges, focusing on the influence of the gap width and incident wave steepness on the liquid motion and hydrodynamic forces on the surrounding bodies. The initial boundary value problem (IBVP) for disturbed velocity potential is solved through boundary element method (BEM) in the time domain. Some special treatments, such as remeshing, smoothing, and jet and plunging wave cutting, are applied to the free surface to suppress the numerical instability. With reference to the gap size, it is found that piston mode is predominant in the case of narrow gap, while higher sloshing modes dominate in the case of wide gap. With the increase in incident wave steepness and hence the associated surface nonlinear effect, the normalised liquid motion in the gap varies slightly when the piston mode dominates, while at higher-order sloshing modes it drops dramatically at or near resonance.

Development of a hybrid model based on mesh and meshfree methods and its application to fluid–elastic structure interaction for free surface waves

In this paper, a hybrid scheme, Fluid–Fluid–Elastic Structure (FFES) model was developed in the time domain to address the wave breaking impact on the structure. The model is developed based on the partitioned approach with different governing equations that describe various regions of the model domain. The fluid–fluid model denotes that two different fluid models were used to describe fluid in the actual physical domain. The method is a physics-based approximation to reduce the computational time, i.e. in the far-field inviscid fluid (fully nonlinear potential flow theory model), and near to the structure, viscous fluid (Navier Stokes model) is used. The coupled model then interacts with the elastic structure (based on Euler–Bernoulli beam theory). The system of equations is strongly coupled both in space and time. The Fluid–Fluid coupling uses an implicit predictor–corrector scheme, and the fluid–structure coupling works based on an iterative scheme. This approach makes the method more robust and for future extension. Three different possibilities for introducing the coupling was identified and implemented. The model was validated against results from the analytical solution and literature. The method proposed is a reliable, robust, and efficient alternative for simulating fluid–structure interaction problems.

Frontal collision of two nonlinear internal solitary waves in a stratified fluid

Head-on collision of two internal solitary waves (ISWs) in a two-layer inviscid fluid with fully nonlinear free-surface (FS) condition is investigated numerically in this paper. The collision process of and nonlinear interaction of internal solitary waves is studied based on nonlinear potential flow theory and a series of numerical simulations are conducted using a multi-domain boundary element method (MDBEM). The numerical model is validated by experiment for the simulation of ISWs propagating. The computation results show that the collision of ISWs is the asymmetric and nonlinear process in which the rundown motion causes the interface displacement penetrates the interface. The collision leaves imprints on small dispersive trailing waves which accounts for the attenuation with each departing ISW tilting slightly backward with respect to the direction of its propagation. In addition, the characteristic of collision process including run up, residence time, surface wave feature is discussed. During the collision, the wave spends more time falling than rising for all cases and the residence time is very long for waves of small amplitude, large depth ratios and density ratios. Considering the free surface of the top layer whether or not makes contributions to the run up and the residence time is larger if the top boundary is free surface.

Three-dimensional hydrodynamic analysis and efficiency optimization of submerged multi-cylindrical oscillating wave surge converter

The intense surge motion, the low installation costs and the less environmental threat, promote the development of the nearshore wave energy technologies which are often classified as Oscillating wave surge converters (OWSCs). This paper investigates the hydrodynamic performance of a multi-cylindrical fully submerged OWSC by developing a 3-D numerical model based on fully nonlinear potential flow theory and time-domain higher-order element method (HOBEM). A local change in the geometric configuration is verified as an effective approach to enhance wave energy extraction in long waves. Then, it is found that a larger incident wave height not only reduces the peak value of the capture efficiency, but it also leads to a change in the real resonant frequency. The geometrical parameters have a significant influence on the energy extraction, with the thicker, wider and shallower device demonstrating both higher average generated power and capture efficiency in low-frequency region. Meanwhile, the maximum efficiency is mainly related to the optimal PTO parameters. A two-value optimization strategy on PTO damping is applied to industriously maximize the absorbed energy over whole wave frequency region, and is easier to be implemented in practical application compared with ideal optimal scheme at each frequency component of incident wave spectrum.

Investigation of interaction between extreme waves and a moored FPSO using FNPT and CFD solvers

To assess the survivability of marine structures, numerical tools that can predict the interaction between extreme waves and structures are needed. Considering the significant nonlinearity associated with the problem, fully nonlinear models, including the fully nonlinear potential theory (FNPT) and general viscous flow theory based on the Navier-Stokes equation (NS) and Continuity equation, are necessary for a reliable prediction. Both methods have relatively higher computational cost compared to the linear or second order wave theories, which are popular in routine design practices. Although the FNPT model generally requires less computational efforts compared to the NS model, its theoretical assumption, i.e. the flow is incompressible, irrotational and inviscid, invalidates its applications to those problems with significant viscous effects and/or breaking waves. It is, therefore, necessary to conduct a comparative study on the accuracy of the FNPT in various problems to quantify its range of application. In this paper, both the Quasi Arbitrary Lagrangian Eulerian Finite Element (QALE-FEM) method based on the FNPT model and the open source Reynolds Average Navier-Stoke (RANS) based code, OpenFOAM, are used to predict the interaction between extreme waves and a moored Floating Production Storage and Offloading (FPSO) model. The extreme waves are generated using the NewWave theory and different wave steepnesses are used. The results, including the wave runup, pressure and force on the FPSO, are compared with the corresponding experimental data obtained from the ocean basin at the COAST Laboratory, University of Plymouth. Satisfactory agreement between the numerical predictions and the experimental measurements are observed. It is also concluded that the differences between the QALE-FEM results and the OpenFOAM results are mainly caused by the effectiveness of the wave generation in the corresponding simulations; the viscous effects may be considerable in the rotational motion of the FPSO when subjected to extreme waves.

A hybrid numerical model based on FNPT-NS for the estimation of long wave run-up

In the present paper, the hybrid numerical model (Sriram et al. (2014)) is used for the estimation of long wave run-up. The model is based on the strong coupling between the fully nonlinear potential flow theory (FNPT) at the far-field and Navier-Stokes (NS) equations in the nearshore. The simulations are carried out for the propagation and evolution of the tsunami-like waves, i.e., elongated single pulses having a realistic timescale. The model is validated from large scale experiments for the wave propagation as well as for run-up (Sriram et al., 2016). The numerical simulations are found to agree well with experiments. The model capability is shown for two different scenarios over a slope: (a) a rapidly rising tide including surging and spilling breaking and (b) undular bore formation and its plunging breaking on a beach. For the first case, the numerical model is also compared with the analytical estimates. The second case, undular bore breaking over a slope, shows the capability of the meshfree method and the requirements of the hybrid model. In many of the existing particle-based methods, dense distribution of particles is required to simulate the rapidly rising tide, which is not needed in the present model.

Fully Nonlinear Potential Flow Simulations of Wave Shoaling Over Slopes: Spilling Breaker Model and Integral Wave Properties

A spilling breaker model (SBM) is implemented in an existing two-dimensional (2D) numerical wave tank (NWT), based on fully nonlinear potential flow (FNPF) theory and the boundary element method (BEM), in which an absorbing surface pressure is specified over the crest of impending breaking waves (detected based on a maximum front slope criterion) to simulate the power dissipated during breaking. The latter is calibrated to match that of a hydraulic jump of parameters identical to local wave properties (height, celerity, $$\ldots $$ ). Although this model is not aimed at being fully physical in the surfzone, it allows more accurately simulating properties of fully nonlinear shoaling waves than standard empirical absorbing beaches (AB). After assessing the convergence with the discretization of 2D-NWT results for strongly nonlinear shoaling waves, simulations are first validated based on laboratory experiments for non-breaking and breaking waves, shoaling up a mild slope; a good agreement is found between both. The NWT is then used to compute fully nonlinear local (height, celerity, asymmetry) and integral (mean-water-level, radiation stress) properties of periodic waves shoaling over mild slopes. Discrepancies with standard results of wave theories are discussed in light of fully nonlinear effects modeled in the NWT. While only periodic waves are considered here, the SBM could be applied to shoaling irregular waves, for which the breaking point location will constantly vary, which would be advantageous compared to an AB fixed in space. For such cases, besides its more physical energy absorption, the SBM would allow better preventing wave overturning that may occur far from shore due to nonlinear wave–wave interactions and otherwise interrupt the FNPF–NWT simulations.

Nonlinear Dynamic Responses of Sliding Isolation Concrete Liquid Storage Tank with Limiting-Devices

Based on potential flow theory (PFT), the lower limit for large amplitude liquid sloshing of rectangular liquid storage tank (RLST) is obtained, and a 3-D numerical model of sliding isolation concrete rectangular liquid storage tank (CRLST) with displacement limiting-devices is established. Results calculated by two-way coupling method are compared with those obtained by finite element method (FEM) based on nonlinear PFT, dynamic responses of isolation and no-isolation CRLSTs are compared, and parameter analysis is conducted. Results show that sliding isolation has a good damping effect on the dynamic responses. The more the dimension of seismic action is, the greater the dynamic responses will be; Dynamic responses increase with the increase of liquid level height; The nonlinear liquid sloshing becomes more pronounced as the PGA increases, but the effect of PGA on tensile stress of the structure is not significant after sliding isolation being taken; the resonance problem caused by excitation frequency should be paid enough attention to.

Fully nonlinear analysis of second-order gap resonance between two floating barges

A two-dimensional boundary element method (BEM) based on the fully nonlinear potential flow theory is adopted to study second-order wave resonance in a gap formed by twin barges. Second-order resonance is related to second-order effects of wave elevations and hydrodynamic forces, which have components of the sum or difference frequency. Firstly, forced body motions (heave, sway, and roll mode) excited second-order resonance is studied with regard to different gap width and resonant mode. Then incident wave induced second-order resonance in the gap is simulated. It is found that second-order resonance is more easily provoked and significant when the gap width over draught ratio is large.

Numerical research on time domain ship motions coupled with sloshing at different liquid levels and forward speeds

In the present study, an reasonable numerical method based on the potential flow theory and boundary element method (BEM) for solving time domain ship motion coupling with sloshing is proposed. Ship motion is solved based on the three dimensional (3D) time domain potential flow theory with forward speed using the BEM. The 3D fully nonlinear time domain potential flow theory and BEM are also adopted to simulate the nonlinear sloshing phenomenon and then numerically coupled with ship motion in a same time step. A corresponding parallel numerical solver is also developed to investigate the coupling effects between 6 degrees of freedom (DOF) ship motion and internal sloshing. To validate the numerical solver and better understand these coupling effects, numerical research on liquid level and forward speed effects is conducted. The developed numerical solver is also validated from previous model experimental results. Computational and experimental results of ship motion response amplitude operators (RAOs) are in reasonable agreement for both 15000 GT and S175 models with liquid tank. The natural frequency of sloshing varies with changes in liquid levels, and coupling effects become more obvious when the natural frequency of sloshing is close to excited frequency of a ship. Our accurate numerical results on the liquid level effects of roll motions indicate that the numerical method proposed can solve the engineering problem related to liquefied natural gas (LNG) and liquefied petroleum gas (LPG) ships traveling through ocean waves. The numerical behavior of a coupling system with forward speed shows that a liquid tank can reduces the resonance frequency of roll motion and has a limited effect on pitch motion. However, as forward speeds increase, such effects follow a gradually increasing trend.

A NONLINEAR AND DISPERSIVE 3D MODEL FOR COASTAL WAVES USING RADIAL BASIS FUNCTIONS

Accurate wave propagation models are required for the design of coastal structures and the evaluation of coastal risks. Nonlinear and dispersive effects are particularly important in the nearshore environment. Two-dimensional cross-shore (2DV) wave models can be used as a preliminary step in coastal studies, but 3D models are needed to capture fully the effects of alongshore bathymetric variations, variable wave incidence, the presence of coastal or harbor structures, etc. Yates and Benoit (2015) developed a numerical model based on fully nonlinear potential flow theory. By assuming non-overturning waves, the kinematic and dynamic free surface boundary conditions are expressed as evolution equations of the free surface elevation and velocity potential, following Zakharov (1968). At each time step, the free surface vertical velocity is estimated by solving the Laplace equation for the velocity potential in the domain. Following Tian and Sato (2008), a spectral approach is used to expand the velocity potential in the vertical as a linear combination of Chebyshev polynomials. The accuracy of the 2DV model was validated with several non-breaking experimental test cases (Benoit et al., 2014; Raoult et al., 2016). Here the model is extended to 3D using scattered nodes (for flexibility) to discretize the horizontal domain. Spatial derivatives are estimated at each node using a linear combination of the function values at neighboring points using Radial Basis Functions (RBF) (Wright and Fornberg, 2006). The accuracy of the method depends on the number of neighboring nodes (Nsten) and the chosen RBF type (e.g. multiquadric, Gaussian, polyharmonic spline (PHS), thin plate spline, etc.), with associated shape factor C for some of them.

A 3D parallel boundary element method on unstructured triangular grids for fully nonlinear wave-body interactions

This paper presents the development and validation of a three-dimensional numerical wave tank devoted to studying wave-structure interaction problems. It is based on the fully nonlinear potential flow theory, here solved by a boundary element approach and using unstructured triangular meshes of the domain's boundaries. Time updating is based on a second-order explicit Taylor series expansion. The method is parallelized using the Message Passing Interface (MPI) in order to take advantage of multi-processor systems. For radiation problems, with cylindrical bodies moving in prescribed motion, the free-surface is updated with a fully Lagrangian scheme, and is able to reproduce reference results for nonlinear forces exerted on the moving body. For diffraction problems, semi-Lagrangian time-updating is used, and reproduces nonlinear effects for diffraction on monopiles. Finally, we study the nonlinear wave loads on a fixed semi-submersible structure, thereby illustrating the possibility to apply the proposed numerical model for the design of offshore structures and floaters.

THE RESEARCH OF ADDED RESISTANCE IN WAVES BASED ON NONLINEAR TIME-DOMAIN POTENTIAL FLOW THEORY

In order to accurately predict the motion performance of the ship in waves, the numerical calculation of heave, pitch and added resistance of a container ship in a regular wave is studied based on the three-dimensional fully nonlinear time-domain potential flow theory. The boundary element method is used to deal with the quadrilateral elements, and the governing equations are solved by the first-order flat-plate theory and the fourth-order Runge- Kutta time integration method. The ship hull is divided by the fixed mesh. Considering the non-linear superposition of the forward velocity, the stationary ship wave, diffracted wave, diffraction wave and incident wave field, the free surface is generated by the hybrid Euler-Lagrangian method, and the damping area is set manually at the edge. Take the KCS container ship for instance, the calculation results of heave, pitch and added resistance in waves are compared with the experimental values. The results show that the biggest advantage of this method is to get a more accurate prediction of ship motion in a short time, and this method has a broad application prospect in the analysis and motion prediction of ship hydrodynamic performance in waves.

Real-time solution of nonlinear potential flow equations for lifting rotors

Analysis of rotorcraft dynamics requires solution of the rotor induced flow field. Often, the appropriate model to be used for induced flow is nonlinear potential flow theory (which is the basis of vortex-lattice methods). These nonlinear potential flow equations sometimes must be solved in real time––such as for real-time flight simulation, when observers are needed for controllers, or in preliminary design computations. In this paper, the major effects of nonlinearities on induced flow are studied for lifting rotors in low-speed flight and hover. The approach is to use a nonlinear state-space model of the induced flow based on a Galerkin treatment of the potential flow equations.

Proof of the equivalence of Tanizawa–Berkvens’ and Cointe–van Daalen's formulations for the time derivative of the velocity potential for non-linear potential flow solvers

Dealing with freely-floating bodies in the framework of non-linear potential flow theory may require solving Laplace's equation for the time derivative of the velocity potential. At present, there are two competing formulations for the body boundary condition. The first one was derived by Cointe [1] in 2D. It was later extended to 3D by van Daalen [2]. The second formulation was derived by Tanizawa [3] in 2D. It was extended to 3D by Berkvens [4]. In this paper, a proof is given that the Cointe–van Daalen's and the Tanizawa–Berkvens’ formulations are equivalent. It leads to a simplified version of Cointe–van Daalen's formulation. The formulation is validated against the analytical solution for a moving sphere in an unbounded water domain.

Study of water wave diffraction around cylinders using a finite-element model of fully nonlinear potential flow theory

Numerical models employing the fully nonlinear potential flow theory have been studied for over three decades. Adopting the mixed Eulerian–Lagrangian formulation, solution over the fluid domain has been obtained using both the boundary integral/element and finite-element (FE) methods, the latter being the main focus of the present paper. The FE model is derived employing a variational formulation. The main highlight of the FE model developed pertains to the calculation of fluid particle velocities using a C0-type FE solution. The velocity calculation procedure is analogous to the traditional stress calculation approach used in FE analysis of solid mechanics problems and is applicable to unstructured isoparametric hexahedral meshes. The numerical model has been applied to evaluate nonlinear diffraction forces on single cylinder problems and Fourier decomposition has been applied to the pressure force time history obtained. The first-order diffraction forces and moments derived from the present nonlinear model compare very well with literature results. The second-order oscillating and mean forces as well as the second-order mean moments compare fairly well with literature results. However, significant deviations are seen in the second-order oscillating moments. It is conjectured that side wall reflections of higher-order wave components in the numerical wave tank might be responsible for this. This aspect merits further investigation.

Tsunami evolution and run-up in a large scale experimental facility

In this paper, we study the propagation and run-up of long tsunami-like waves in the 300m long Large Wave Flume (GWK), Hannover, Germany and analyze the feasibility of experiments on tsunami run-up in large facilities. This paper is the continuation of our previous paper (Schimmels et al., 2015, companion paper). The propagation of long period waves over large distances for different shapes is studied experimentally and numerically. Fully nonlinear potential flow theory has been used to model these cases numerically along with Korteweg–de Vries simulations. The theoretical explanation of the observed effects has been discussed. The run-up characteristics of the studied waves with respect to their shape and beach slope were also investigated. Further, a downscaled real tsunami time series is also reproduced experimentally and studied with regard to its possible run-up.

LONG WAVE PROPAGATION, SHOALING AND RUN-UP IN NEARSHORE AREAS

This paper discusses the possibility to study propagation, shoaling and run-up of these waves over a slope in a 300- meter long large wave flume (GWK), Hannover. For this purpose long bell-shaped solitary waves (elongated solitons) of different amplitude and the same period of 30 s are generated. Experimental data of long wave propagation in the flume are compared with numerical simulations performed within the fully nonlinear potential flow theory and KdV equations. Shoaling and run-up of waves on different mild slopes is studied hypothetically using nonlinear shallow water theory. Conclusions about the feasibility of using large scale experimental facility (GWK) to study tsunami wave propagation and run-up are made.

Simulation of floating structure dynamics in waves by implicit coupling of a fully non-linear potential flow model and a rigid body motion approach

We demonstrate the accuracy and convergence of a new numerical model solving wave-structure interactions based on the fully non-linear potential flow (FNPF) theory coupled to a rigid body motion approach. This work extends an earlier model proposed by Guerber et al. (Eng Anal Bound Elements 36(7):1151–1163, 2012), restricted to fully submerged structures, by allowing to solve for freely floating bodies on the free surface. Although we are currently extending the model to three dimensions (3D), the work reported here only considers two-dimensional (2D) problems. We first introduce the FNPF model, originally developed in 2D by Grilli et al. (Eng Anal Bound Elements 6(2):97–107, 1989), Grilli and Subramanya (Comput Mech 17(6):374–391, 1996), and later extended to 3D by Grilli et al. (Int J Numer Methods Fluids 35(7):829–867, 2001). We present the implementation by Guerber et al. (Eng Anal Bound Elements 36(7):1151–1163, 2012) in the 2D-FNPF model of van Daalen’s implicit method for fluid-structure interactions [see van Daalen (Numerical and theoretical studies of water waves and floating bodies, PhD thesis, Universiteit Twente, The Netherlands 1993) and Tanizawa (Proceedings of 4th Osaka colloquium on seakeeping performance of ships 2000)]. We then detail the numerical scheme used for coupling the FNPF model to the motion of a floating rigid body. Moreover, we propose a new numerical strategy for advancing the free surface front inspired by symplectic integrators, which achieves a much better performance for energy conservation. The developed algorithm is first applied to forced motion cases, for which analytical and experimental results can be found in the literature and used as benchmarks. The accuracy of the numerical solution for the fluid and applied forces is then discussed for cases with small or large amplitude motion. In the latter case, a preliminary investigation of non-linear effects is performed for the classical application of a semi-circular heaving cylinder, by comparing the computed hydrodynamic force to the experimental measurements of Yamashita (J Soc Nav Arch 141:61–70, 1977). In particular, the comparison of the magnitude of the force harmonics, up to the third order, shows the importance of simulating non-linear interactions, which become important as the ratio of the radius of the cylinder over the wavelength increases. In a second set of applications, we assess the model accuracy in dealing with freely floating bodies. As a first test case, we consider the decaying motion of a freely heaving horizontal circular cylinder released from a non-equilibrium position above the free surface. In this more demanding computations, we verify that total energy fluid-plus-body motion is well conserved, which confirms the accuracy of the fluid-structure interaction algorithm. As a second test case, we consider the free motion of a rectangular barge in waves and compute the first-order response amplitude operators.