Undisturbed Free Surface Research Articles

Numerical simulations of a ship sailing across pack ice area in forward motion under different drafts

Different from the existing CFD-DEM models in which the ship remains stationary, a CFD-DEM model with the ship in forward motion is proposed in this paper to simulate the process of a ship sailing across pack ice area at a forward speed under different drafts. A high-precision method for generating pack ice area on the undisturbed free surface that can be used in conjunction with the proposed model is introduced. Taking an ice-strengthened Panamax bulker as study object, the available model test results are used to verify the reliability of the proposed model, which shows that the model can effectively evaluate the resistance performance and simulate the ship-ice-water interaction. Based on the verified model, the ice resistances on different parts of the hull are first investigated, which reveals that the ice resistance of the bow is most significant, while those of the midship and stern are negligible. Then, the speed dependence of ice resistance under different drafts is studied. It shows a strong nonlinearity under shallow draft, while the nonlinearity gradually weakens as the draft increases. Finally, the proportions of ice resistance and open-water resistance in the total resistance under different drafts and ship speeds are discussed.

Water entry of a sphere moving along a circular path at a constant speed

When a marine propeller rotates in partially submerged conditions, air is entrained from above the undisturbed free-surface, which is called the reference surface, and the ventilated air surrounds the propeller blades, causing thrust loss and excessive vibration, all of which seriously damage the durability of the propeller shaft system of a ship. In the present study, the entry of a propeller blade is simplified by the water entry problem of a sphere moving along a circular path at a constant speed. A high-speed camera was employed to capture the rapidly changing flow structures in detail. Above the reference surface, we focused on the free-surface disturbances, including splash and dome formation. Beneath the reference surface, the development and collapse of ventilated cavities, followed by the line-vortex cavity and cavity undulation, were observed. The ventilated cavity of the present study appears to be more elongated than those of the free-falling sphere's water entry experiments. Two parallel vortical structures appeared after the cavity pinch-off, and bubbles were entrained into these structures to form the line-vortex cavity. The sphere's drag was directly measured via the torque meter attached to the sphere's rotating axis. The relation between the measured drag and the flow around the sphere was identified.

A Maneuvering Model for an Underwater Vehicle Near a Free Surface—Part II: Incorporation of the Free-Surface Memory

Using energy-based modeling techniques, we propose a nonlinear, time-dependent, parametric motion model for an underwater vehicle maneuvering near an otherwise undisturbed free surface. By augmenting the system Lagrangian used to derive Kirchhoff's equations for a rigid body moving through an unbounded fluid, we directly incorporate the free surface into the derivation of the equations of motion. This is done using a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">free-surface Lagrangian</i> , which accounts for the instantaneous energy stored within the free surface due to an impulsive vehicle motion as well as fluid memory effects. The system Lagrangian then enables us to derive the six-degree-of-freedom nonlinear equations of motion using the Euler–Lagrange equations. The model structure is similar to standard maneuvering models for surface ships, although additional complexities are present since the hydrodynamic parameters are shown to depend on the vessel position and orientation relative to the free surface. For the proposed model, the vessel motion is unrestricted. This is in contrast to traditional seakeeping models, which use convolution integrals to incorporate memory effects for a vessel, which experiences only small perturbations from steady, forward motion. The proposed motion model is amenable to real-time simulation, design performance analysis, and nonlinear control design. Other important hydrodynamic effects due to viscous flow, for example, may then be incorporated into a robust, nonlinear, closed-loop control system as lumped parameter effects or model uncertainties.

Nonlinear wave interactions on the surface of a conducting fluid under vertical electric fields

In this paper, we are concerned with capillary–gravity waves propagating on a two-dimensional conducting fluid under the effect of an electric field imposed in the direction perpendicular to the undisturbed free surface. This work aims to investigate the impact of the electric fields on the nonlinear wave interactions in this context. The full system is mathematically difficult to solve since it is a nonlinear, two-layered, free boundary problem, and the interface dynamics results from the strong coupling between the Euler equations for the lower fluid layer and an electric contribution from the upper gas layer. To investigate electrohydrodynamic wave interactions, we propose a novel numerical scheme based on a time-dependent conformal map and an interpolation technique to conduct unsteady simulations of the fully nonlinear electrified Euler equations of the two-layered problem. To gain analytical insights, we first derive weakly nonlinear envelope equations based on the method of multiple scales for the resonant triad, long-short wave interaction, and modulation of a single-mode wavetrain (a special case of resonant quartet interaction). Summative remarks are made to illustrate the transition from 3-wave interactions to 4-wave interactions and vice versa, occurring when the coefficients of the associated nonlinear Schrödinger equation become singular. The fully nonlinear results obtained by the prescribed numerical method are compared with the predictions by the weakly nonlinear theories, and good agreement is achieved.

Wave Radiation from Bottom Vibrations in Open-Channel Flow with Uniform Vorticity

The linearized water-wave radiation problem for a 2D oscillating source in an inviscid shear flow with a free surface is investigated analytically. The singular source is located at the bottom, with constant fluid depth. The velocity of the basic flow varies linearly with depth, assuming uniform vorticity. The far-field surface waves radiated out from the 2D bottom source are calculated, based on Euler’s equation of motion with the application of radiation conditions. The present analysis extends the work by Tyvand and Sveen to include a nonzero surface flow. Doppler effects will arise in a system at rest with the undisturbed free surface. Resonance with zero group velocity will occur at a critical Froude number for the surface flow. In the presence of a basic surface flow, there are in total four radiated waves when the surface flow is subcritical. With supercritical flow, there are only two emitted waves, with downstream propagation. The critical Froude number depends on the surface velocity, the shear rate and the oscillation frequency of the bottom source.

Three alternative boundary integral flow representations for wave diffraction-radiation by offshore structures

Three alternatives to the classical boundary integral representation of diffraction-radiation of regular waves by large stationary bodies, such as offshore structures and moored ships, that underlies existing panel methods are defined. These three alternative flow representations are associated with two alternative linear flow models, called ‘free waterplane flow model’ and ‘rigid waterplane flow model’, of potential flow around an offshore structure in regular waves. As was shown previously, these two alternative linear flow models of diffraction-radiation of regular waves by stationary bodies yield identical boundary integral flow representations and are then consistent, although the rigid waterplane flow model precludes irregular frequencies. Moreover, this flow model – and mathematical transformations – yield two other boundary integral flow representations, so that three alternatives to the classical boundary integral flow representation used in existing panel methods are defined. These three alternative flow representations are weakly singular. The first of these flow representations, given previously, involves a surface integral over the waterplane inside the body, while the second flow representation involves a line integral around the waterline of the body, i.e. the intersection curve between the body and the undisturbed free surface. The third flow representation is of particular interest because it involves neither a surface integral over the waterplane nor a line integral around the waterline. Moreover, this boundary integral flow representation does not involve a distribution of dipoles over the body surface; indeed, it expresses the flow potential in terms of the normal and tangential components of the flow velocity at the surface of the body. It is also notable that this boundary integral flow representation is identical to the boundary integral representation previously obtained for flows around ships advancing at a constant speed in calm water or in regular waves, and thus provides a common basis for the analysis of three major classes of flows in ship and offshore hydrodynamics.

The Radiation Problem of a Submerged Oblate Spheroid in Finite Water Depth Using the Method of the Image Singularities System

This study examines the hydrodynamic parameters of a unique geometry that could be used effectively for wave energy extraction applications. In particular, we are concerned with the oblate spheroidal geometry that provides the advantage of a wider impact area on waves, closer to the free surface where the hydrodynamic pressure is higher. In addition, the problem is formulated and solved analytically via a method that is robust and most importantly very fast. In particular, we develop an analytical formulation for the radiation problem of a fully submerged oblate spheroid in a liquid field of finite water depth. The axisymmetric configuration of the spheroid is considered, i.e., the axis of symmetry is perpendicular to the undisturbed free surface. In order to solve the problem, the method of the image singularities system is employed. This method allows for the expansion of the velocity potential in a series of oblate spheroidal harmonics and the derivation of analytical expressions for the hydrodynamic coefficients for the translational degrees of freedom of the body. Numerical simulations and validations are presented taking into account the slenderness ratio of the spheroid, the immersion below the free surface and the water depth. The validations ensure the correctness and the accuracy of the proposed method. Utilizing the same approach, the whole process is implemented for a disc as well, given that a disc is the limiting case of an oblate spheroid since its semi-minor axis approaches zero.

An alternative linear flow model and boundary integral flow representation for ship motions in regular waves

The basic issue of formulating a linear flow model for the analysis of potential flow around a ship advancing at a constant speed in regular waves is considered. Specifically, the ‘rigid-waterplane flow model’, previously considered for diffraction–radiation of regular waves by offshore structures, is applied to analyze flows around ships advancing in regular waves. A straightforward application of Green’s fundamental identity to the rigid-waterplane flow model yields a remarkable new boundary integral flow representation. Indeed, this flow representation does not involve a line integral around the mean waterline of the ship hull surface, is weakly singular, does not involve a distribution of dipoles over the ship-hull surface, and is free from irregular frequencies. In the particular case of flow around a ship steadily advancing in calm water, the boundary integral flow representation obtained in the study via an analysis based on the rigid-waterplane flow model is identical to the flow representation associated with the Neumann–Michell theory, which is based on a different linear flow model that only considers the flow region strictly outside the ship but consistently accounts for the linear contribution of the thin band of water between the wave profile along the ship hull surface and the undisturbed free surface. This result strongly suggests that the classical application of Green’s basic identity to analyze linear potential flow around a ship advancing in regular waves or in calm water, which yields a notoriously troublesome line integral around the ship waterline, may be questionable due to possible incompatibilities between the Neumann boundary condition at the ship hull surface and the linear free surface boundary condition associated with flows around ships advancing in calm water or in regular waves. This fundamental difficulty may be alleviated in the rigid-waterplane flow model, which imposes a compatibility condition between the Neumann boundary condition at the rigid lid that artificially closes the open ship hull surface and the linear boundary condition at the free surface above the waterplane-lid. The boundary integral flow representation given in the study also holds for diffraction–radiation of regular waves by offshore structures. Thus, this new flow representation provides a common mathematical basis, which fundamentally differs from the basis steadfastly adopted over the past fifty years, for the analysis of three basic classes of flows in ship and offshore hydrodynamics, and indeed can be applied more generally to other boundary-value problems such as those associated with flexural-gravity waves for ice sheets or very large floating structures.

Hydrodynamic performance of a cone falling into waves in 3DOFs free fall motion

This paper concentrates on the 3DOFs water entry of a cone into an incident wave. Undisturbed free surface is modeled with nonlinear 5th Stokes wave. Based on velocity potential theory, fully nonlinear boundary element method is used to simulate fluid field. Fluid-solid interaction between body's free fall motion and fluid loads is decoupled by auxiliary function method. Stretched coordinate system is constructed to make grids match the dramatically changing wetted surface. Distorted free surface is traced with improved Eulerian method. Water entries with different incident wave heights, initial vertical velocities and cone's densities are simulated respectively. 3DOFs motions including vertical, horizontal and rotational motion of cone with different initial conditions are compared and investigated. Pressure on the wetted surface is given, and discussions of pressure distribution along the cone's generatrices are provided.

Hydrodynamics of a submerged oblate spheroid in finite water depth using the method of ultimate image singularities

The present study provides a semi-analytical formulation for the diffraction problem by an immersed oblate spheroid in a liquid field of finite water depth. The method of image singularities in oblate geometry, allows the expansion of the velocity potential in series of auxiliary potentials and eventually, the derivation of analytical expressions for the hydrodynamic loads on the spheroid subjected to the action of regular surface waves. The axisymmetric placement of the spheroid is considered, namely, having its symmetrical axis perpendicular to the undisturbed free surface. The solution of the problem is based on the method of image singularities. Based on the same approach, a disc is simulated as an oblate spheroid. The whole process that has been performed for the spheroid, is performed for the disc as well. The favorable agreement between the results of the present method and existing data, ensures the accuracy and robustness of the proposed methodology.

Overall performances of a propeller operating near the free surface

The paper describes a numerical study on the effects of the interaction of the free surface with a marine propeller operating close to it, including the effects of the immersion on the cross flow. Extensive simulations using the ISIS-CFD naval hydrodynamics computational fluid dynamics solver are conducted to study the open water characteristics (POW hereafter), transient blade loads, and flow behavior around the DTMB 4119 three-bladed propeller, with comparisons to the fully submerged experimental data at two different immersions in respect to the undisturbed free-surface. Computations and experiments for the deeply submerged case show that the cross flow results in an increase of the energy spent and lower efficiency relative to the uniform inflow. Near the water surface, numerical solutions show that effects on thrust and torque increase significantly as the propeller load increases. Furthermore, the presence of the free surface breaks the symmetry resulting in highest blade force losses when the blade is near top dead center. As the propeller approaches the surface, not only the amplitude of the blade higher harmonics increases, but also the efficiency drops down.

Incompressible SPH simulation of flow past horizontal cylinder between plane wall and free surface

Flow past a circular cylinder located between free-surface and wall boundaries is modelled numerically using non-uniform particle size incompressible smoothed particle hydrodynamics (SPH). Several enhancements including Rhie and Chow interpolation, colour function-based free surface tracking, and modified particle shifting are introduced to increase the accuracy and efficiency of the method. A parameter study at constant Reynolds number, Re=150, examines the influence of Froude number (Fr=0.2−0.6), submergence ratio (H∕D=0.5−1.0, where H is the distance between the apex of the cylinder and undisturbed free surface, and D is the cylinder diameter), and bottom gap ratio (G∕D=1.0−5.0, where G is the distance between the base of the cylinder and the bed) on the ambient free-surface flow pattern and hydrodynamic force on the cylinder. It is found that the vortex shedding pattern depends on all three parameters. As the Froude number increases for fixed submergence ratio and bottom gap ratio, vortex shedding is eventually suppressed. As the submergence and bottom gap ratios increase, the threshold of vortex shedding suppression shifts to higher values of Froude number. The mean drag coefficient depends on the submergence ratio and bottom gap ratio but is independent of Froude number. Meanwhile, the lift coefficient depends on submergence ratio and Froude number but is independent of bottom gap ratio. Spectral analysis of force and free-surface elevation time signals shows that the free-surface deformation and lift force are closely related during the vortex shedding regime.

An analytical approach for the three-dimensional water entry problem of solids creating elliptical contact regions

The purpose of the present study is to approach analytically the water entry problem of solids creating elliptical contact regions with the free surface during impact. The governing boundary value problem has been linearized using asymptotic analysis and considering the early stages of the phenomenon. The fundamental assumption is made that the contact region between the solid and the free surface is always an ellipse. This allows to consider the underlying boundary value problem in elliptical coordinates exploiting the properties of elliptical harmonics. It is shown that for elliptical contact regions the solution sought can be detached from the angular dependence. This finding is extremely beneficial as it allows the derivation of a perturbation sequence of mathematical systems of mixed type, involving Fredholm integral equations, which can be treated using rigorous methods of solution for mixed boundary value problems in potential theory. The theory developed for elliptical contact regions during water entry situations, is accordingly applied for anticipating the details of the slamming phenomenon during the free-surface impact of a nonaxisymmetric prolate spheroid assuming that the contact line between the liquid and the solid is always an ellipse. The solid is assumed to descend steadily, without rotating, towards an initially undisturbed free surface. The perturbation technique requires that the slenderness parameter is small compared to unity, while slenderness affects significantly the hydrodynamic load exerted on the solid during water entry. Numerical results are presented and discussed using both the von Karman and the Wagner models of water impact. Finally, the study was extended to consider the 3D Wagner water impact problem of an elliptic paraboloid and to compare computations against available data reported in the past.

Unsteady hydrodynamic forces of solid objects vertically entering the water surface

We investigate the unsteady hydrodynamic force of solid objects vertically entering water with an air cavity behind the falling body. Physical models are proposed to represent the force components corresponding to the body acceleration, the gravity, and the velocity of the body and the fluid particles. The theoretical or numerical solutions of the physical models are presented to understand the evolution of the force components. The body-acceleration force component is expressed as the high-frequency added mass times the body acceleration. Near the undisturbed free surface, the added mass grows strongly with increasing the submerged depth. It tends to be steady after the submerged depth is greater than a few characteristic lengths. The gravity force component consists of an upward hydrostatic term and a downward dynamic term. Generally, the hydrostatic term, which is obtained by integrating the gravity term in Bernoulli’s equation over the wetted body surface, is much larger than the gravity force component. For the three-dimensional bodies, the gravity force component is found to vary as a power of the submerged depth, where the exponent is about 0.83. The velocity force component is represented as the drag coefficient defined by the V-squared law, which is characterized by the body geometry. The drag coefficient may experience three successive stages with increasing the submerged depth.

Experimental study of cavity characteristic induced by vertical water entry impact of a projectile with a 90° cone-shaped head at different velocities

Experimental studies of the vertical water entry of projectile with 90° cone-shaped head were conducted by using high speed camera. The pinch-off types and evolutionary process of the cavity were comprehensively discussed at different water entry impact velocities. The variations of cavitation closure time, water depth at the closure point and length of the warhead cavitation with water entry velocities, as well as the cavitation diameter at different water depth positions were analyzed. The jet phenomenon caused by the closure of the water curtain and contraction-rising process of the cavity near undisturbed free surface were studied, as well as the coupled effect between them. The cavity wall fluctuation occurring after the deep seal was discussed. The results show that with the increase of water entry velocity, the quasi-static closure, shallow closure, deep closure and surface closure of the cavitation occurs respectively, and each closure mode corresponds to a velocity range; the critical water entry velocity of forming cavitation is 0.657 m/s. The diameter of the cavitation presents a nonlinear increase along with the water depth. The initial cavity expansion velocity increases, the maximum diameter of the cavity decreases, the expansion section shortens, the contraction section lengthens, and the acceleration of the expansion and contraction of the cavity increases along with the increase of the water depth at the same time. When the water curtain closes, there will be upward and downward jets, and when the downward jet velocity is relatively large, the projectile motion will be affected. The strength of a water jet induced by longitudinal upward contraction of the cavity near free surface is proportional to the volume of cavity and pinch-off depth. A large strength upward water jet is induced by the coupling of the above all water jet. The longitudinal contraction of the cavity around the projectile and the impact effect of downward jet on projectile would cause the force change of the projectile after deep seal. The fluctuation of the projectile's velocity because of its force change brings the velocity change of fluid and then leads to the fluctuation of the cavity wall. The fluctuation of cavity wall follows the principle of independent expansion of cavitation section.

Fully nonlinear capillary–gravity solitary waves under a tangential electric field, Part II: Dynamics

The stability and dynamics of two-dimensional fully nonlinear capillary–gravity solitary waves are studied when a uniform electric field is applied in the direction parallel to the undisturbed free surface of a dielectric fluid. For simplicity, we assume that the permittivity of the gas above the fluid is much smaller, therefore the two-layer problem can be reduced to be a one-layer one. The existence of fully localized solitary waves in deep water has been thoroughly studied in our previous work (Tao and Guo, 2014) via weakly nonlinear normal form analysis and fully nonlinear numerical computation. In the present paper, the stability and dynamics of the obtained solitary waves are investigated based on the time-dependent conformal map technique. Similar to the nonelectric capillary–gravity solitary waves, all depression waves, together with elevation waves featuring two big troughs connected by a small dimple, are found to be stable. Furthermore, head-on and overtaking collisions between stable solitary waves are studied via a numerical time integration.

Evolution mechanism of solitary waves on the inviscid film flow under an electrostatic field

To investigate the nonlinear wave motion on an inviscid film flow under an electrostatic field a modified Korteweg-de Vries (KdV) equation has been derived by employing the long-wave and shallow-water approximations. A uniform or non-uniform electric potential on a charged electrode foil at some distance above the film is applied in the direction normal to the undisturbed free surface. The modified KdV equation is numerically solved with spatially periodic boundary conditions by using Crank-Nicholson scheme. Consequently, the uniform electrostatic field hinders the solitons from moving downward. In the non-uniform case where the electrostatic potential slowly varies with space, the wave energy changing rate has a maximum negative value as a new solitary wave is about to produce from the effective region of electrostatic field, and already-existed external solitons are getting accelerated if they pass through this region.

On the weakly nonlinear seakeeping solution near the critical frequency

We study theoretically and numerically the nonlinear seakeeping problem of a submerged or floating body translating with constant forward speed $U$ parallel to the undisturbed free surface while at the same time undergoing a small oscillatory motion and/or encountering small-amplitude waves at frequency $\unicode[STIX]{x1D714}$. It is known that at the critical frequency corresponding to $\unicode[STIX]{x1D70F}\equiv \unicode[STIX]{x1D714}U/g=1/4$, where $g$ is the gravitational acceleration, the classical linear solution is unbounded for a single point source, and the inclusion of third-order free-surface nonlinearity due to cubic self-interactions of waves is necessary to remove the associated singularity. Although it has been shown that the linear solution is in fact bounded for a body with full geometry rather than a point source, the solution still varies sharply near the critical frequency. In this work, we show theoretically that for a submerged body, the nonlinear correction to the linear solution due to cubic self-interactions of resonant waves in the neighbourhood of $\unicode[STIX]{x1D70F}=1/4$ is of first order in the wave steepness (or body motion amplitude), which is the same order as the linear solution. With the inclusion of nonlinear effects in the dispersion relation, the wavenumbers of resonant waves become complex-valued and the resonant waves become evanescent, with their amplitudes vanishing with the distance away from the body. To assist in understanding the theory, we derive the analytic nonlinear solution for the case of a submerged two-dimensional circular cylinder in the neighbourhood of $\unicode[STIX]{x1D70F}=1/4$. Independent numerical simulations confirm the analytic solution for the submerged circular cylinder. Finally, we also demonstrate by numerical simulations that similar significant nonlinear effects for a surface-piercing body exist in the neighbourhood of $\unicode[STIX]{x1D70F}=1/4$.

Numerical simulation of water exit of an initially fully submerged buoyant spheroid in an axisymmetric flow

The free water exit of an initially fully submerged buoyant spheroid in an axisymmetric flow, which is driven by the difference between the vertical fluid force and gravity, is investigated. The fluid is assumed to be incompressible and inviscid, and the flow to be irrotational. The velocity potential theory is adopted together with fully nonlinear boundary conditions on the free surface. The surface tension is neglected and the pressure is taken as constant on the free surface. The acceleration of the body at each time step is obtained as part of the solution. Its nonlinear mutual dependence on the fluid force is decoupled through the auxiliary function method. The free-surface breakup by body penetration and water detachment from the body are treated through numerical conditions. The slender body theory based on the zero potential assumption on the undisturbed flat free surface is adopted, through which a condition for full water exit of a spheroid is obtained. Comparison is made between the results from the slender body theory and from the fully nonlinear theory through the boundary-element method, and good agreement is found when the spheroid is slender. Extensive case studies are undertaken to investigate the effects of body density, dimensions and the initial submergence.

Experimental assessment of buoyant cylinder impacts through high-speed image acquisition

In this paper, we experimentally study the water entry of a set of buoyant bodies with a circular cylindrical shape, which is relevant for several engineering applications, being related to aircraft fuselages in sea landing, solid rocket boosters, and bulbous bows of large ships. Free fall experiments are conducted for three different specimen weights and four different falling heights. The hydrodynamics of the interaction of the buoyant cylinders with the water free surface upon impact is analyzed through an accelerometer, a high-frequency potentiometer and a high-speed camera. Results are presented in terms of impact velocity, kinematics of the cylinder penetration and free surface dynamics, comprising the pile-up and water jet dynamics. Further, the air cavity development is analyzed after the cylinder is completely immersed below the undisturbed free surface.