Nonlinear Free-surface Conditions Research Articles

Numerical Investigation on a High-Speed Transom Stern Ship Advancing in Shallow Water

A high-speed advancing ship will cause significant squats in shallow water, which could increase the risk of grounding. To this end, a program based on the Rankine higher-order boundary element method (HOBEM) is developed to investigate a high-speed displacement ship with a transom stern moving in shallow water. The nonlinear free surface condition is satisfied by adopting an iterative algorithm on the real free surface. The transom condition is considered by implementing a modified transom condition. Computations of wave-making resistance, sinkage and trim in deep water are first performed, and satisfactory agreement is achieved by comparing with the experimental results; the simulations are then extended to the shallow water case. It indicates that the present method can provide a suitable balance of practicability and robustness, which can be considered as an efficient tool for the guidance in ship design stage.

A fully nonlinear BEM-beam coupled solver for fluid–structure interactions of flexible ships in waves

A fully nonlinear hydroelastic numerical method is proposed for the fluid–structure interactions of elastic ships. The boundary value problem is solved by the three-dimensional boundary element method. To simulate the instantaneous evolution on the free surface, an effective mesh optimization solution consisting of the local coordinate system and an elastic mesh technique has been developed. The mixed Eulerian–Lagrangian method is implemented to update the fully nonlinear free surface conditions in the local coordinate system. The hull structure is simplified as a non-uniform Timoshenko beam, and the modal superposition approach is adopted to analyse the symmetric responses in the head sea. Convergence of the model is evaluated for mesh resolutions and the time step size, and then comprehensive investigations of vertical motions as well as vertical bending moment are carried out for an advancing ship in a range of wave frequencies. The numerical solutions are compared with the experimental data and a satisfactory agreement is demonstrated. The distribution of high-order harmonics and whipping responses are predicted by the amplitude spectrum, in which the water entry and water exit are considered. The high-frequency vertical bending moments are investigated, in which the effects of higher harmonics and whipping responses are addressed.

A nonlinear time-domain BEM for the performance of 3D flapping-wing thrusters in directional waves

Recent studies indicate that flapping-foil biomimetic systems can significantly improve ship propulsion in waves. In most cases horizontal wing(s) are used performing vertical oscillatory motion induced by ship heaving and pitching responses in waves, while foil's rotational motion is actively controlled. Aiming to the detailed investigation of the free-surface effects, a GPU-accelerated Boundary Element Method (GPU-BEM) has been developed for the hydrodynamic analysis of 3D lifting bodies in non-linear waves with an angle of incidence and motions. The nonlinear free-surface conditions are satisfied at the exact location of the free-surface boundary. A nonlinear pressure-type Kutta condition is applied at the trailing edge and the dynamic evolution of the free vortex wake is obtained by a time-stepping method. The free-surface boundary conditions and Kutta condition are used to set up the nonlinear and nonlocal evolution equations. For the treatment of the dynamical system, the present BEM is coupled to the dynamical equations and it is integrated in time domain with an iterative scheme. Results are presented for a few subproblems of special interest, such as the wave resistance and the enforced radiation problems of smooth immersed bodies and compared with other methods for verification. Subsequently, the unsteady biomimetic thruster is examined in unbounded domain, beneath the free surface in the absence of incident waves, and in the presence of obliquely incident waves, illustrating the performance of the system. The present method can be exploited for the systematic investigation and the design of flapping-foil thrusters for augmenting ship propulsion in waves.

Numerical Analysis of Wave Energy Extraction Performance According to the Body Shape and Scale of the Breakwater-integrated Sloped OWC

Research on the development of marine renewable energy is actively in progress. Various studies are being conducted on the development of wave energy converters. In this study, a numerical analysis of wave-energy extraction performance was performed according to the body shape and scale of the sloped oscillating water column (OWC) wave energy converter (WEC), which can be connected with the breakwater. The sloped OWC WEC was modeled in the time domain using a two-dimensional fully nonlinear numerical wave tank. The nonlinear free surface condition in the chamber was derived to represent the pneumatic pressure owing to the wave column motion and viscous energy loss at the chamber entrance. The free surface elevations in the sloped chamber were calculated at various incident wave periods. For verification, the results were compared with the 1:20 scaled model test. The maximum wave energy extraction was estimated with a pneumatic damping coefficient. To calculate the energy extraction of the actual size WEC, OWC models approximately 20 times larger than the scale model were calculated, and the viscous damping coefficient according to each size was predicted and applied. It was verified that the energy, owing to the airflow in the chamber, increased as the incident wave period increased, and the maximum efficiency of energy extraction was approximately 40% of the incident wave energy. Under the given incident wave conditions, the maximum extractable wave power at a chamber length of 5 m and a skirt draft of 2 m was approximately 4.59 kW/m.

Nonlinear Free Surface Condition Due to Wave Diffraction for a Pair of Cylinders

Nonlinear Free Surface Condition Due to Wave Diffraction for a Pair of Cylinders

Viscous Flow Evolution and Boundary Layer Characteristics during the Head-On Collision of Solitary Waves

Lee, T.-C.; Chang, C.-H., and Tang, C.-J., 2021. Viscous flow evolution and boundary layer characteristics during the head-on collision of solitary waves. Journal of Coastal Research, 37(3), 544–556. Coconut Creek (Florida), ISSN 0749-0208.Viscous shear flow near the bottom during the head-on collision of two solitary waves is investigated using a stream function–vorticity model. Fully nonlinear free-surface conditions are satisfied numerically at every time step. The bottom boundary-layer flow region is locally refined to explore its physical characteristics through detailed vorticity transport in the shear layer beneath the two solitary waves during their collision. Both symmetric and asymmetric head-on collisions are examined and discussed. The wave deformations are compared with the results of other studies for validation, and the fluid mechanisms in the flow field are illustrated with streamline, equivorticity line, path line, and timeline plots to reveal the interaction processes. It is revealed that the wave interaction behaves nonlinearly not only at the free surface but also in the vorticity exchanges near the bottom.

Investigation on responses and wave climbing of a ship in large waves using a fully nonlinear boundary element method

In this study, a fully nonlinear method, in the time domain, is used to simulate ship motion caused by large waves, and the problem of wave climbing along the freeboard in large waves is studied. The method is based on a three-dimensional potential flow theory. Further, the nonlinear free surface condition is adopted, the incident potential and disturbance potential are separated from each other, and the problem of determining the disturbance potential is established. In order to consider the nonlinearity of the body surface, the wetted surface area of the hull is continuously updated with time and extended along the tangential direction of the uppermost grid when the free liquid level exceeds the height of the hull deck. In addition, an artificial damping layer is provided to eliminate the reflection of waves at the outflow boundary. Under the fully nonlinear boundary conditions, the perturbation potential is solved by the Rankine source boundary element method. Introducing auxiliary functions solves the coupling problem between motion and load in the process of solving rigid body motion equations. After the solution is completed, it is updated to the next time step using the Runge-Kutta 4th order method. In this study, the convergence of the computational grid of the numerical model is firstly verified. Then, the motion and load responses of a zero-speed ship in the regular wave of unit wave amplitude using this method are examined and compared with the linear method. The latter verifies the accuracy of this method in the linear field. Afterwards, in order to investigate the superiority of the fully nonlinear method when the nonlinear phenomenon is strong, the motion and load of the ship in the large amplitute wave are simulated and the influence of the nonlinear phenomenon is analyzed. Finally, the phenomenon of wave climbing generated in high waves is analyzed, and the relationship between wave and water heights is evaluated.

Peaked sloshing in a wedge container

Finite-amplitude free-surface flow in a wedge container is investigated analytically. We study a motionless standing wave of pure potential-flow acceleration with maximal amplitude where its right-angle surface peak falls from rest. The nonlinear free-surface conditions are satisfied by a family of flows where the chosen initial acceleration field is governed by one single dipole plus its three image dipoles. Streamlines and isobars are plotted, with the free surface as the zero-pressure isobar. The key geometric parameters are tabulated for each case, supplied with force calculations for an upright wedge container. The present approach is assessed against established eigenfunctions for linearized standing waves in a wedge container. The present dipole flows constitute a much richer family of peaked free sloshing shapes than the classical Fourier modes of free oscillation.

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.

Optimized perforated bulkhead for sloshing mitigation and control

The phenomenon of sloshing consists in the violent liquid motion inside partially filled containers and it is a major concern for the design of a number of structures in naval and offshore industry. The use of perforated swash bulkheads is an effective technique to mitigate the sloshing. In the present work, a comprehensive investigation of the sloshing within box-shaped tanks equipped with perforated bulkheads is carried out. The investigation considered a broad range of parameters, which includes several open-area ratios and different filling levels under pitching excitations encompassing the first three sloshing resonant modes of a non-compartmented tank. As a result, a relation of optimized open-area ratio as function of the filling ratio is achieved for overall sloshing mitigation. The effectiveness of the obtained relation is confirmed by numerical simulations using proposed swash bulkhead geometries that follow the optimized open-area distribution. As the perforated bulkheads introduces additional complexity to the already challenging sloshing phenomenon, the numerical simulations of the present work were performed by adopting a fully-Lagrangian particle-based method, which easily deals with phenomena involving nonlinear free surface conditions, such as large free surfaces deformation and fragmentation, and complex geometries.

Nonlinear wave resonance from bottom vibrations in uniform open-channel flow

Abstract It is known from linear theory that bottom oscillations in uniform open-channel flow can produce resonant surface waves with zero group velocity and diverging amplitude (Tyvand and Torheim 2012). This resonance exists for Froude numbers smaller than one, at a critical frequency dependent on the Froude number. This resonance phenomenon is studied numerically in the time domain, with fully nonlinear free-surface conditions. An oscillatory 2D bottom source is started, and the local elevation at resonance grows until it may reach a saturation amplitude. Four waves exist at subcritical Froude numbers, where resonance represents the third and the fourth wave merging. In the zero-frequency limit, the dispersive second and fourth wave merge into a steady wave with finite group velocity and amplitude, and no other periodic waves exist. In the time-dependent nonlinear analysis at zero frequency, a transient undular bore may emerge as the dominating phenomenon.

Numerical Simulation of the Potential Flow around a Submerged Hydrofoil with Fully Nonlinear Free-Surface Conditions

ABSTRACT Hu, J.; Guo, L., and Sun, S., 2018. Numerical simulation of the potential flow around a submerged hydrofoil with fully nonlinear free-surface conditions. Hydrofoils have been utilized worldwide to facilitate coastal vessels for transportation, surveillance, and patrol with high speed. Inevitably, the free surface has a significant effect on the hydrofoils because of the shallow draft. In this paper, a numerical procedure based on a boundary-element method is developed for simulating a uniform incoming flow passing a two-dimensional submerged hydrofoil with free-surface effects. In the proposed technique, the boundaries of the computational domain are discretized by small elements over which the known and unknown variables are linearly distributed. To avoid the numerical instabilities induced by the sudden start of the flow through a sharp-edged foil, it is assumed that a vortex sheet with uniform strength Γ has been placed within the foil at the initial time step. The unknown vortex strength Γ, t...

Numerical model of sloshing in rectangular tank based on Boussinesq-type equations

The highly accurate Boussinesq-type equations in terms of velocity potential are adopted for the simulation of sloshing phenomena in a two-dimensional rectangular tank. The fully nonlinear free surface conditions are used and linearized energy dissipation term is added into the dynamic free surface condition. The total velocity potential is divided into two parts: the particular solution and the rest which is calculated by the Boussinesq-type model. Different filling levels of liquid in the tank are considered for the validation of the numerical model, including shallow water, intermediate water depth and finite water depth cases. The linear theory based on the superposition of eigen-modes corresponding to natural sloshing frequencies is introduced. Compared with results calculated by linear theory and results from literatures, the numerical model based on Boussinesq-type equations presents good performances in predicting the sloshing motions in rectangular tank with small excitation amplitude.

Computations of linear and nonlinear ship waves by higher-order boundary element method

A practical method for computing ship waves accurately and efficiently is favorable for hull form optimization in early stage of ship design. In the present study, Rankine source method incorporated with high-order boundary element (HOBEM) discretization is applied to solve linear ship waves at first. Numerical implementation is described in detail. An incomplete LU factorization preconditioned Generalized Minimal Residual method (GMRES) is employed to solve the resulting boundary integral equation in order to improve efficiency. A corresponding Fortran program is developed for the validation study. It is applied to solve ship waves of different hulls, including slender Wigley hull, KRISO Container Ship (KCS) with relatively full form and a fishery patrol boat. Ship wave drag, sinkage, trim and wave pattern over a wide range of Froude numbers are all well predicted.In order to further investigate nonlinear effects on ship waves, a fully nonlinear potential flow method based on stationary iteration is proposed. The same numerical approach of HOBEM is employed. The combined nonlinear free surface condition is solved in each iteration to evaluate the free surface. Numerical investigation for Wigley and KCS hull shows the present nonlinear method are accurate. Comparison study with linear and partial nonlinear solution are also carried out and nonlinear effects on ship waves are detailedly discussed.

A hydrodynamic optimization design methodology for a ship bulbous bow under multiple operating conditions

ABSTRACTThe main objective of this article is to describe an innovative methodology for the hydrodynamic optimization of a ship bulbous bow which considers multiple operating conditions. The proposed method is more practical and effective than the traditional optimization process, which is only based on contractually specified design condition. Parametric form approaches are adopted by employing an F-spline curve in order to generate variants of the hull bulbous bow forms using form design parameters modified, resulting in an optimization system based on improved genetic algorithms. The Rankine source panel method is used for the hydrodynamic evaluation, wherein non-linear free surface conditions and the trim and sinkage of the ship are taken into consideration. The validity and effectiveness of the proposed methodology for a large container ship is investigated by comparing the computational results with experimental data, which demonstrates that the proposed methodology can engage well in the automation process and improve hydrodynamic performance during actual ship design practices.

Nonlinear time-domain NWT simulations for two types of a backward bent duct buoy (BBDB) compared with 2D wave-tank experiments

A fully nonlinear time-domain NWT-simulation tool has been developed with nonlinear viscous-damping and pneumatic pressure terms. The tool was applied to the performance evaluation of fixed/floating sharp-corner and round-corner BBDB WECs. Shape-induced and motion-induced viscous damping for the BBDB system was considered in the potential-flow-based simulation with quadratic pneumatic pressure and fully nonlinear free-surface conditions. Systematic experiments were also conducted in a two-dimensional wave tank to verify the NWT-simulation results. The numerical results were further verified by checking the total power conservation of the entire system. The nonlinear simulations were also compared with linear ones. Since the round-corner BBDB has less energy loss caused by body shape and motion, the available power of the round-corner BBDB is generally bigger than that of the sharp-corner BBDB. The phenomenon of negative drift motion for only the sharp-corner BBDB for a certain wave-length range was also reproduced by the fully nonlinear BBDB simulations. This numerical simulation tool can be used for the reliable design of prototype systems for a given site and environmental conditions.

Hull-form optimization of a container ship based on bell-shaped modification function

In the present study, a hydrodynamic hull-form optimization algorithm for a container ship was presented in terms of the minimum wave-making resistance. Bell-shaped modification functions were developed to modify the original hull-form and a sequential quadratic programming algorithm was used as an optimizer. The wave-making resistance as an objective function was obtained by the Rankine source panel method in which non-linear free surface conditions and the trim and sinkage of the ship were fully taken into account. Numerical computation was performed to investigate the validity and effectiveness of the proposed hull-form modification algorithm for the container carrier. The computational results were validated by comparing them with the experimental data.

Fully nonlinear wave interaction with an array of truncated barriers in three dimensional numerical wave tank

Wave transition due to coinciding with an array of truncated barrier is simulated by a fully nonlinear three dimensional potential Numerical Wave Tank (NWT). The potential theory is used to describe kinematics of the flow field and the isoparametric Boundary Element Method (BEM) is employed to solve the boundary value problem. The Mixed Eulerian–Lagrangian (MEL) approach and fourth order Runge–Kutta time integration applied for time-marching scheme to model the temporary and fully nonlinear free surface. At each time step, solution of Laplace equation in the Eulerian frame is applied to the fully nonlinear free surface conditions in the Lagrangian manner to achieve the new positions and the boundary value of fluid particles for the next time step. Normal flux of potential wave theory is specified on the inflow boundary to stimulate fluid field and to propagate the nonlinear wave along the tank. To minimize the reflected wave energy into the computational domain, two artificial sponger layers are adopted on the free surface at the both ends of the numerical wave tank. Accuracy and convergence of the present numerical procedure is conducted. Also, interaction between a near trapped mode array of truncated barriers and nonlinear input wave is simulated.

Rankine Source Method for Ship–Ship Interaction Problems

A Rankine source method is extended and applied to ship–ship interaction problems. The method covers the nonlinear steady flow and linear seakeeping in the frequency domain. The nonlinear steady flow solution accounts for the nonlinear free-surface conditions, ship wave, and dynamic trim and sinkage. Periodic flow due to waves is linearized with respect to the wave amplitude, taking into account interactions with the nonlinear steady flow following Hachmann approach, which considers the steady perturbation potential as constant in the body-fixed reference frame. This is advantageous for the prediction of ship motions at moderate to high Froude numbers. In this context, a new formulation of the boundary condition for the multibody case is derived. Two examples are considered, overtaking in calm water and replenishment at sea. For a feeder vessel overtaken by a container ship, horizontal forces and yaw moment are computed and compared with reference data. As an example of replenishment operation, interaction between a frigate and a supply vessel is studied. Ship motions are computed for two relative positions and three forward speeds and compared with model test data for the largest forward speed. The Rankine source method proves as more accurate compared with a zero-speed freesurface Green function method. [DOI: 10.1115/1.4029316]

Wave-induced response of a floating two-dimensional body with a moonpool.

Regular wave-induced behaviour of a floating stationary two-dimensional body with a moonpool is studied. The focus is on resonant piston-mode motion in the moonpool and rigid-body motions. Dedicated two-dimensional experiments have been performed. Two numerical hybrid methods, which have previously been applied to related problems, are further developed. Both numerical methods couple potential and viscous flow. The semi-nonlinear hybrid method uses linear free-surface and body-boundary conditions. The other one uses fully nonlinear free-surface and body-boundary conditions. The harmonic polynomial cell method solves the Laplace equation in the potential flow domain, while the finite volume method solves the Navier-Stokes equations in the viscous flow domain near the body. Results from the two codes are compared with the experimental data. The nonlinear hybrid method compares well with the data, while certain discrepancies are observed for the semi-nonlinear method. In particular, the roll motion is over-predicted by the semi-nonlinear hybrid method. Error sources in the semi-nonlinear hybrid method are discussed. The moonpool strongly affects heave motions in a frequency range around the piston-mode resonance frequency of the moonpool. No resonant water motions occur in the moonpool at the piston-mode resonance frequency. Instead large moonpool motions occur at a heave natural frequency associated with small damping near the piston-mode resonance frequency.