Nonlinear Free Surface Research Articles

Some aspects of the pressure field preceding the onset of critical jets in a breaking wave

We consider in this article two-dimensional nonlinear free surface motions in potential theory. There are situations where the onset of critical jet along a free surface can be predicted by analyzing the temporal and spatial variation of the pressure itself and its successive derivatives. In particular, when a local maximum of pressure appears close enough to a free surface where the pressure vanishes, it is expected a great pressure gradient at the free surface and consequently a large Lagrangian acceleration. The present analysis examines this phenomenon. This analysis is facilitated after identifying a line along which the pressure gradient is parallel to one of the eigenvectors of the Hessian matrix of the pressure. In a standard overturning crest, this line connects the tip of the crest to the bottom of the tank in which the flow is simulated. The analysis of its time evolution gives much information into the onset of critical jets. Other types of critical jet can also appear when a region of positive Gaussian curvature of the pressure suddenly grows in the vicinity of the free surface.

Numeričko modeliranje plutajuće kontrukcije primjenom spregnute Euler-Lagrangeove formulacije

Floating structures are complex systems composed of superstructure, floating-mooring components, and anchors. In this study, the behaviours of a pontoon, which is used as a floating structure, and four mooring elements under different wave loading conditions were investigated. A numerical analysis of the coupled motions of the pontoon, mooring lines, and marine environment was performed. While the mooring lines were modelled as wire elements, the pontoon was modelled as a rigid body with six degrees of freedom. Wave loading conditions were represented using two different wave spectra. The first spectrum (Case I) was generated based on a single sinusoidal wave utilising the JONSWAP spectrum, whereas the second one (Case II) was generated based a superimposed multi-sinusoidal wave. Time-varying motions of the pontoon and tensions of the mooring lines were determined based on the numerical analyses for Cases I and II. Critical values were determined using the results of both cases. The numerical solutions were based on bidirectional fluid–structure interaction (FSI) analysis. A fully non-linear free surface simulation was performed using the coupled Eulerian–Lagrangian (CEL) technique. Furthermore, numerical results were compared with the results obtained from analytical solutions of free surface elevations.

An efficient three-dimensional multiphase fluid structure interaction model on GPU for water impact of a moving body with complex geometries

This paper presents an efficient three-dimensional (3D) multiphase fluid structure interaction (FSI) model to simulate water impact of complex moving bodies in ocean engineering. This numerical model adopts an improved ghost cell method (GCM) to enforce the boundary conditions on the moving boundaries and the gradient augmented level set (GALS) method to capture the nonlinear free surfaces. An improved iso-surface representation method is highlighted to track arbitrary moving bodies such as sharp or concave shapes. Also, a GPU-CUDA parallel framework is incorporated to greatly accelerate the computational performance. The accuracy and capability of the present model is validated by simulating 3D water entry of arbitrary free-falling bodies (a sphere, a wedge, a wedge with side plates, and a catamaran). Excellent acceleration performance of the present GPU model is demonstrated. The present results including the slamming coefficient, the acceleration, and the local pressure agree satisfactorily with the experimental data. The complex flow evolutions are well captured such as long cavity wall, the pinch-off of the cavity, and the air entrapment. Also, the air entrapment and 3D effects are confirmed to be nonnegligible for the water impact of complex bodies with enclosed sections, which can be well predicted by the present model.

Efficient Calculation of Hydrodynamic Loads on Offshore Wind Substructures Including Slamming Forces

Abstract Estimation of the hydrodynamic loads based on strip theory using the Morrison equation provides an inexpensive method for load estimation for the offshore industry. The advantage of this approach is that it requires only the undisturbed wave kinematics along with inertia and viscous force coefficients. Over the recent years, the development in numerical wave tank simulations makes it possible to simulate nonlinear 3-h sea states, with computational times in the order of real time. This presents the possibility to calculate loads using wave spectrum input in numerical simulations with reasonable computational time and effort. In the current paper, the open-source fully nonlinear potential flow model REEF3D::FNPF is employed for calculating the nonlinear wave kinematics. Here, the Laplace equation for the velocity potential is solved on a σ-coordinate mesh with the nonlinear free surface boundary conditions to close the system. A technique to calculate the total acceleration on the σ-coordinate grid is introduced which makes it possible to apply strip theory in a moving grid framework. With the combination of strip theory and 3-h wave simulations, a unique possibility to estimate the hydrodynamic loads in real time for all discrete positions in space within the domain of the numerical wave tank is presented in this paper. The numerical results for inline forces on an offshore wind mono-pile substructure are compared with measurements, and the new approach shows good agreement.

Numerical Investigation on Higher-Order Harmonic Waves Induced by a Submerged Inclined Plate

In this paper, a two-dimensional time-domain numerical flume has been established to model and investigate nonlinear interactions between nonlinear surface waves and a submerged inclined thin plate. The model solves the Laplace equation and the fully nonlinear free surface boundary conditions within the framework of potential flow theory based on the high-order boundary element method. The mixed Euler–Lagrangian method is applied to update the water surface at each time step, and the fourth-order Runge–Kutta method for time stepping. A so-called four-point method was employed to separate the second-order harmonic free and bounded wave that has the same wave frequency but different wave celerity in front of and behind the submerged plate. It is found that the amplitude of the second-order harmonic free wave increases with the inclination angle of the submerged plate, and the level/amount of the increase is larger for a larger wave steepness. In addition, the amplitudes of both the second-order reflected and transmitted waves are found to increase with the wave steepness, and their empirical relationships are derived for potential use in practical engineering.

Experimental study on the water entry of a 3D bow-flared model with various inclination angles

Oblique water entry of the ship bow has both the complexity of structure shape and the water entry attitude, which involves strong nonlinear free surface effect and thus, complexes the prediction of slamming loads. This paper conducted the drop experimental investigation on the impact hydrodynamics of a 3D bow-flare model with inclination angles to help clarify the impact characteristics. The experimental uncertainty was discussed based on repeatable and oscillation analysis showing good stability. Characteristics of bow-flared model in oblique impact including water entry depth, acceleration, pressure and impact force were analyzed. Differing to simple 3D bodies like cones and cylinders, multiple pressure peaks appear at some positions, which should be paid attention. The influence of inclination angles on pressure was further discussed, showing the pressure at the bottom and leeward side (right side) decreases while it increases on the windward side with inclination angle. These findings offer evidence for a complex interplay between geometric parameters and inclination angles, which could ultimately serve the ship design.

Fully nonlinear dynamics of floating solar platform with twin hull by tubular floaters in ocean waves

Sustainability in harvesting solar power offshore depends on the survival of their platforms under environmental loads such as ocean waves. Therefore, a framework is established in this study to estimate the dynamics of floating solar platforms under ocean waves experimentally and numerically. Hence the platform is taken twin hull with double circular cylinders which exposed to a broad range of waves in the calibrated physical wave tank. On the other hand, the high order boundary element method and mixed Eulerian-Lagrangian approach are used in the numerical model to solve the fully nonlinear fluid field and the acceleration potential method is employed to approximate the response of the platform implicitly. To validate the framework, numerical and experimental results are compared for a single cylinder under the incident waves. Then, the comparison is extended for a twin hull platform. The effect of nonlinear free surface flow amongst the twin cylinders on the dynamics of the platform is studied parametrically. Afterwards, the performance of the present framework is assessed for a near breaking incident wave and an irregular wave. The stream function theory and flux of superposed wave components are used to emulate the high steep wave and the irregular wave respectively.

The effect of odd-viscosity on Rayleigh–Taylor instability of a liquid film under a heated inclined substrate

We discuss the effect of odd viscosity on Rayleigh–Taylor instability of a thin Newtonian liquid film with broken time-reversal symmetry as it flows down a uniformly heated, inclined substrate. Although considerable experimental and theoretical studies have been performed regarding Rayleigh–Taylor instability, there is still a need to understand the instability mechanism in the presence of odd viscosity, which creates nondissipative effects. Odd viscosity represents broken time reversal and parity symmetries in the two-dimensional active chiral fluid and characterizes deviation of the system from one that contains a passive fluid. Adopting the long-wave approach allows a nonlinear free surface evolution equation of the thin film that considers the influence of odd viscosity to be derived. New, interesting linear stability analysis results illustrate that larger odd viscosity leads to a lower perturbation growth rate ωr and cutoff wave number kc. In other words, odd viscosity has a stabilizing effect on the Rayleigh–Taylor instability. Numerical simulations are conducted using the method of lines to solve the nonlinear evolution equation. The numerical results show that enhancing the odd viscosity effect suppresses the disturbance amplitude and wave frequency. In addition, the numerical results show that the inclination angle and the Weber number have stabilizing effects on the Rayleigh–Taylor instability. However, the Biot number has the opposite effect when the thin liquid film conductivity is poor. Also, the oscillation tends to accumulate downstream of the inclined substrate if the evolution time is sufficiently long.

Numerical and experimental study on free-surface wave generation by a submerged moving body in a towing tank

Free surface waves were generated by a submerged moving body in a towing tank using a numerical and a physical towing tank. Unlike a circulating water channel, a towing tank allows easy to control of acceleration and deacceleration range of the body motion. Numerical and experimental analysis can make the same moving conditions of the submerged body. Using a Numerical Towing Tank (NTT), free surface elevation was calculated according to body velocities, submerged depths and body volumes. The measured experimental data agreed well with the results of a fully nonlinear simulation and the data was also compared with the Russell comprehensive error method. The nonlinearity of the free surface waves increased as the submerged depth of the body became shallower and the body velocity became faster. The increase in wave height according to the increase of the body velocity or the increase in the wave height according to the decrease in the submerged depth of the body generally showed a proportional relationship. Overall, the nonlinear free surface boundary condition needs to be applied when calculating the free surface wave due to the motion of a submerged body. The ratio of wave height difference according to body displacement was similar to the ratio (percentage) of increase in body displacement to NACA shape displacement as body velocity becomes faster.

Numerical investigation on evaluating nonlinear waves due to an air cushion vehicle in steady motion by a higher order desingularized boundary integral equation method

The problem of nonlinear ship waves generated by an air cushion vehicle is studied and modelled in the framework of potential flow theory. A higher order desingularized boundary integral equation method (HODBIEM) using Rankine source is developed. A stationary iterative method is adopted to solve the nonlinear free surface which is discretized into the inner area disturbed by cushion pressure and the outer undisturbed area. Techniques for numerical implementation are described in detail. To save computing time, grid density and domain size are refined.Both straight-ahead and yawed cases are considered for validations. Wave-making resistance versus various speeds in deep and shallow water are calculated for straight-ahead vehicles. The resistance curves obtained by HODBIEM agree well with results of nonlinear CFD simulations. The linear theory predicts higher peak value of resistance than nonlinear methods. The experimental data fit at low Froude number but deviate from numerical predictions when exceeding primary hump speed. Effects of speed and water depth are analyzed by wave patterns and wave profiles. Resistance, side force and yaw moment are calculated at Fn = 0.533 for drift angle from 0° to 90°. Satisfactory results are given by present method. Details about wave patterns and wave angles are discussed.

Multiphase flow simulation of water entry of a structure with complex geometry using a three-dimensional parallel compressible model

Three-dimensional water entry of a structure with complex geometry is investigated by a parallel compressible model. The projection method is adopted for the present high-order difference model. The convection term is discretized by a constrained interpolation profile method. The free surface is captured by a tangent of hyperbola for interface capturing with slope weighting scheme. The compressible multiphase model is realized by supplementing the equation of state. It can automatically handle compressible and incompressible areas, which considers the mixing compressibility effect. The benchmarks are the water entry of a sphere, water entry of a horizontal cylinder, and water entry of the single skeg and twin skeg. Special attention of this study is focused on the parallelism algorithm, compressibility effect, and the application of the asymmetric water entry of the skeg. Furthermore, the displacement, velocity, and acceleration of the structure can be accurately predicted. The highly nonlinear free surface can be simulated with fairly good accuracy. Air pocket and secondary impact can be reproduced well. The accurate numerical prediction of the impact load with the compressibility of asymmetric water entry of the structure with complex geometry is vital for the design of the structure and the safety of the passengers. These are the further development of the present model and help us understand the short and destructive slamming process.

Hydrodynamic instabilities of a viscous liquid film flowing down an inclined or vertical plane

In this paper, we have investigated theoretically linear as well as weakly nonlinear stability of a viscous liquid film flowing down an inclined or vertical plane under the action of gravity. The classical momentum-integral method, which is applicable for small as well as large values of Reynolds number Re, has been used to formulate the single nonlinear free surface equation in terms of the dimensionless perturbed film thickness η(x,t). Using sinusoidal perturbation in the linearized part of the surface evolution equation, we obtain the stability criterion and the critical value of the wave number kc which conceives the physical parameters Re, inclination angle θ and Weber number We. However, the linear stability analysis reveals the stabilizing influence of We as well as the destabilizing influence of Re and θ on this flow dynamics. The multiple-scale analysis has been used to derive the complex Ginzburg–Landau type nonlinear equation for investigating the weakly nonlinear stability analysis. We demarcate all the four states of the flow in the Re-k (or θ-k)-plane which are found after the critical value of Rec (or θc) depending upon the values of the other parameters. A novel result which emerges from the nonlinear stability analysis is a simple relationship among the parameters k, Re, We, and θ. This relationship essentially gives us the conditions needed for the existence of an explosive unstable zone when (3Re−3cot θ−4ReWek2)= 0; otherwise, the flow system will be free from this zone. Indeed, this zone decreases with the increase in We, whereas it increases with the increase in Re and θ confirming the stabilizing role of We and destabilizing role of Re and θ as found in linear stability analysis.

Adaptive fully nonlinear potential model for the free surface under compressible air pressure of oscillating water column devices

A two dimensional fully nonlinear numerical wave flume is extended with a pneumatic model to compute the free surface flow inside the chamber of land-based Oscillating Water Column (OWC) devices. Potential theory and the Mixed Eulerian-Lagrangian method are employed to compute the fluid flow using the direct Boundary Element Method (BEM) enhanced by a non-uniform rational B-Spline function. Meanwhile, a pneumatic model is manipulated to obtain the instantaneous air pressure on the chamber's free surface. Spring-like behavior of a perfect gas through an adiabatic process and a linear power take-off mechanism are used in the pneumatic model that makes time derivative of the air pressure turns to a function of reciprocating air and fluid flows in the chamber. Therefore, temporary air pressure is taken into account on the fully nonlinear free surface boundary conditions implicitly along with the time marching algorithm. The hydrodynamic efficiency of an OWC interacting with second-order Stokes waves is compared with former studies to verify the present model. To assess the present model for simulating the air compressibility in a large-scale case study, the present solution is compared with experimental and mathematical results under an irregular wave. Afterwards, an OWC device is exposed to fifth-order Stokes waves in the flume to examine the numerical stability of computing the free surface flow outside and inside of the chamber.

Slow Rhythmical Brain Activity During Coordination of Cognitive Processes

A two dimensional fully nonlinear numerical wave flume is extended with a pneumatic model to compute the free surface flow inside the chamber of land-based Oscillating Water Column (OWC) devices. Potential theory and the Mixed Eulerian-Lagrangian method are employed to compute the fluid flow using the direct Boundary Element Method (BEM) enhanced by a non-uniform rational B-Spline function. Meanwhile, a pneumatic model is manipulated to obtain the instantaneous air pressure on the chamber's free surface. Spring-like behavior of a perfect gas through an adiabatic process and a linear power take-off mechanism are used in the pneumatic model that makes time derivative of the air pressure turns to a function of reciprocating air and fluid flows in the chamber. Therefore, temporary air pressure is taken into account on the fully nonlinear free surface boundary conditions implicitly along with the time marching algorithm. The hydrodynamic efficiency of an OWC interacting with second-order Stokes waves is compared with former studies to verify the present model. To assess the present model for simulating the air compressibility in a large-scale case study, the present solution is compared with experimental and mathematical results under an irregular wave. Afterwards, an OWC device is exposed to fifth-order Stokes waves in the flume to examine the numerical stability of computing the free surface flow outside and inside of the chamber.

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.

The Peregrine Breather on the Zero-Background Limit as the Two-Soliton Degenerate Solution: An Experimental Study

Solitons are coherent structures that describe the nonlinear evolution of wave localizations in hydrodynamics, optics, plasma and Bose-Einstein condensates. While the Peregrine breather is known to amplify a single localized perturbation of a carrier wave of finite amplitude by a factor of three, there is a counterpart solution on zero background known as the degenerate two-soliton which also leads to high amplitude maxima. In this study, we report several observations of such multi-soliton with doubly-localized peaks in a water wave flume. The data collected in this experiment confirm the distinctive attainment of wave amplification by a factor of two in good agreement with the dynamics of the nonlinear Schrödinger equation solution. Advanced numerical simulations solving the problem of nonlinear free water surface boundary conditions of an ideal fluid quantify the physical limitations of the degenerate two-soliton in hydrodynamics.

Nonlinear hydrodynamics of freely floating symmetric bodies in waves by three-dimensional fully nonlinear potential-flow numerical wave tank

This study estimated the nonlinear hydrodynamic performance of freely floating structures in nonlinear water waves. For this, a three-dimensional, fully nonlinear numerical wave tank (NWT) was developed based on potential theory and the boundary element method (BEM). The fully nonlinear NWT evaluates the hydrodynamic force acting on the instantaneous body position and free-surface elevation based on the mixed Eulerian and Lagrangian (MEL) method and acceleration-potential approach. The instantaneous nonlinear free surface was traced at each time step, and the far-field outgoing wave condition was satisfied by placing an artificial free-surface damping zone. In addition, the least square gradient reconstruction method, thin-plate spline method, and modified inverse distance weighting method were adopted to re-grid the calculation nodes and evaluate the spatial derivatives of the physical quantities robustly. The developed NWT was used to evaluate the first-order and second-order motion RAOs and excitation forces of a freely floating vertical cylinder. The results were compared systematically with previously published results using perturbation theory. Moreover, the heave-to-pitch Mathieu-instability phenomena on a deep-draft floating cylinder were demonstrated by the fully nonlinear NWT simulations. The results agreed well with the corresponding experimental observations.

An iterative time-marching scheme for the investigation of hydrodynamic interaction between multi-ships during overtaking

An iterative time-marching scheme is developed to investigate the hydrodynamic interactions between multiple ships. Such an unsteady interactive effect could be magnified in restricted waterways, e.g., a channel or harbor area. To the author’s knowledge, nearly all the research on the ship-to-ship interaction neglecting the free surface effects. The free surface is usually treated as a rigid wall. This assumption is only reasonable when the speed of the ships is very low in deep water condition, due to the hydrodynamic interaction between the ships is mainly induced by near-field disturbances. However, when the moving speeds are moderately higher, especially with a small lateral separation between ships, the far-field effects arising from the ship waves become important. The main objective of the present paper is to develop an iterative time-matching algorithm to solve the hydrodynamic interaction between high-speed ships taking into account the nonlinear free surface boundary condition in time domain.

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

Properties of the pressure field in highly nonlinear free surface flows with critical jet

In this note we examine fluid violent kinematics in a plunging breaker. The fluid motion is computed in the frame of the potential flow theory. The fluid kinematics is basically generated by forcing the motion of a rectangular tank, starting from rest. When a sufficient level of energy is injected in the fluid, the free surface has highly nonlinear behavior, here a plunging breaker. In the vicinity of the main tip crest, a sharp corner (cusp) appears along the surface of the barrel. The appearance of this critical jet is described and discussed in terms of the spatial and temporal variations of the pressure field. In the present case a local pressure maximum is captured that follows a continuous decreasing pressure gradient in a region of positive Gaussian curvature of the pressure. It is also shown that, at the free surface before the appearance of the critical jet, there is a strong correlation between the change of sign of the Gaussian curvature of the pressure on the one hand and the radius of curvature of the free surface profile on the other hand.