Effects Of Finite Water Depth Research Articles

Effects of finite water depth and lateral confinement on ships wakes and resistance

Wash waves produced by ships disintegrate river banks and coastal lines. This phenomenon of bank erosion is mainly due to the height of the waves. Various factors govern the formation of these waves and their amplitudes: the geometry of the water channel, the shape and the speed of the boat, etc.. These factors play an important role on the wave generation, in addition on the resistance of the ship and so on its fuel consumption. Whether to study the impact of wash waves on the ship's environment or its resistance, the analysis of the generated wake is essential. Hence a fine characterization of the wave field is necessary. This study proposes a comparison of wakes generated by two generic ships based on a Wigley hull with block coefficients 0.67 and 0.89 respectively representative of maritime and fluvial ships. The wakes generated in deep water and confined water configurations have been measured for different Froude numbers by a non-intrusive optical stereo-correlation method, giving access to a detailed and complete definition of the generated wave fields. The resistance of the ship hulls has been measured in deep and confined water configurations with a hydrodynamic balance. The results permit one to study the influence of both hull and water channel geometries on the ship wake, on the amplitude of the far-field generated waves and on the near-field hydrodynamic response. Moreover, resistance curves are obtained for both configurations and highlight the effect of both hull and water channel geometries on the resistance coefficient of the ship. A comparison of the resistance curves with or without the ship trim is conducted and shows the influence of the trim on the resistance coefficient in the different ship speed regimes.

Finite water depth effect on wave-body problems solved by Rankine source method

Finite water depth effect for wave-body problems are studied by continuous Rankine source method and non- desingularized technique. Free surface and seabed surface profiles are represented by continuous panels rather than a discretization by isolated points. These panels are positioned exactly on the fluid boundary surfaces and therefore no desingularization technique is required. Space increment method is applied for both free surface source and seabed source arrangements to reduce computational cost and improve numerical efficiency. Fourth order Runge-Kutta iteration scheme is adopted on the free surface updating at every time step. The finite water depth effect is studied quantitatively for a series of cylinders with different B/T ratios. The accuracy and efficiency of the proposed model are validated by comparison with published numerical results and experimental data. Numerical results show that hydrodynamic coefficients vary for cylinder bodies with different ratios of B/T. For certain set of B/T ratios the effect of finite water depth increases quickly with the increase of motion frequency and becomes stable when frequency is relatively large. It also shows that water depths have larger hydrodynamic effects on cylinder with larger breadth to draft ratios. Both the heave added mass and damping coefficients increase across the frequency range with the water depths decrease for forced heave motion. The water depths have smaller effects on sway motion response than on heave motion response.

Energy distribution in shallow water ship wakes from a spectral analysis of the wave field

This work presents an experimental study of the effects of finite water depth on the waves generated by a ship in a towing tank. The wakes of two hull forms representative of maritime and river ships are measured for both deep water and shallow water configurations and for several Froude numbers. The free surface deformations are measured with an optical stereo-correlation measurement method to access a full and detailed reconstruction of the wave fields. The spatial resolution of the reconstructed wakes allows us to perform a spectral analysis of the waves generated by the ships and to decompose them into a near-field hydrodynamic response and a far-field undulatory component. First, the spectral analysis method is presented and the effects of finite water depth on a theoretical point of view are studied. The analysis of subcritical, trans-critical, and supercritical ship wakes in both real space and spectral space highlights the effects of the finite water depth, of the ship speed, and of the hull shape on the energy distribution in the ship wakes through these different regimes.

Unsteady hydrodynamic interaction between two cylindroids in shallow water based on high-order panel method

A NURBS-based high-order panel method is developed for predicting the hydrodynamic interaction of two bodies moving along parallel courses in shallow water. Source and dipole are distributed on the body surfaces, while dipole is distributed on the wake surfaces; the singularity strengths are determined by satisfying the boundary condition on the body surfaces and the Kutta condition at the trailing edges. The wave-making effect is neglected under the low speed assumption; the image method is used to deal with the effects of finite water depth and undisturbed free surface. The time-stepping method is used to update the velocity potential and the position of each body. Firstly, calculations are carried out for a cylindroid passing a stationary cylindroid and two cylindroids moving in the same direction with the same speed. Detailed convergence study with respect to panel size, time step and truncated mirrored images is carried out. On the basis of convergence study, numerical results are compared with experimental measurements to validate the numerical method. Then, calculations are conducted for two cylindroids during passing, meeting and overtaking under different lateral distances between cylindroids, different water depths and different cylindroid geometry. Numerical results are presented to demonstrate the effects of these factors.

Ship waves on uniform shear current at finite depth: wave resistance and critical velocity

We present a comprehensive theory for linear gravity-driven ship waves in the presence of a shear current with uniform vorticity, including the effects of finite water depth. The wave resistance in the presence of shear current is calculated for the first time, containing in general a non-zero lateral component. While formally apparently a straightforward extension of existing deep water theory, the introduction of finite water depth is physically non-trivial, since the surface waves are now affected by a subtle interplay of the effects of the current and the sea bed. This becomes particularly pronounced when considering the phenomenon of critical velocity, the velocity at which transversely propagating waves become unable to keep up with the moving source. The phenomenon is well known for shallow water, and was recently shown to exist also in deep water in the presence of a shear current (Ellingsen, J. Fluid Mech., vol. 742, 2014, R2). We derive the exact criterion for criticality as a function of an intrinsic shear Froude number $S\sqrt{b/g}$ ($S$ is uniform vorticity, $b$ size of source), the water depth and the angle between the shear current and the ship’s motion. Formulae for both the normal and lateral wave resistance forces are derived, and we analyse their dependence on the source velocity (or Froude number $Fr$) for different amounts of shear and different directions of motion. The effect of the shear current is to increase wave resistance for upstream ship motion and decrease it for downstream motion. Also the value of $Fr$ at which $R$ is maximal is lowered for upstream and increased for downstream directions of ship motion. For oblique angles between ship motion and current there is a lateral wave resistance component which can amount to 10–20 % of the normal wave resistance for side-on shear and $S\sqrt{b/g}$ of order unity. The theory is fully laid out and far-field contributions are carefully separated off by means of Cauchy’s integral theorem, exposing potential pitfalls associated with a slightly different method (Sokhotsky–Plemelj) used in several previous works.

Generation and evolution features of L2-type crescent waves in laboratory conditions

A series of experiments on generation and evolution of L2-type crescent waves are performed. The examined influencing factors includes the effects of different experimental setups, competition of class I and class II instabilities and the effects of finite water depth and wave steepness. The most unstable perturbation waves are introduced as a laboratory experiment technique of crescent wave generation. The measured critical wave steepness for triggering crescent waves is obtained and compared with theoretical results. The evolution features of L2-type crescent wave and Benjamin–Feir instability are discussed.

Experimental Investigation of Trimaran Models in Shallow Water

Experiments were conducted at the U.S. Naval Academy's Hydromechanics Laboratory to determine the effect of finite water depth on the resistance, heave, and trim of two different trimaran models. The models were tested at the same length to water depth ratios over a range of Froude numbers in the displacement speed regime. The models were also towed in deep water for comparison. Additionally, the side hulls were adjusted to two different longitudinal positions to investigate possible differences resulting from position. Near critical speed, a large increase in resistance and sinkage was observed, consistent with observations of conventional displacement hulls. The data from the two models are scaled up to a notional 125-m length to illustrate the effects that would be observed for actual ships similar in size to the U.S. Navy's Independence Class Littoral Combat Ship. Faired plots are developed to allow for rapid estimation of shallow water effect on trimaran resistance and under keel clearance. An example is provided.

Experimental Investigation of Trimaran Models in Shallow Water

Experiments were conducted at the U.S. Naval Academy's Hydromechanics Laboratory to determine the effect of finite water depth on the resistance, heave, and trim of two different trimaran models. The models were tested at the same length to water depth ratios over a range of Froude numbers in the displacement speed regime. The models were also towed in deep water for comparison. Additionally, the side hulls were adjusted to two different longitudinal positions to investigate possible differences resulting from position. Near critical speed, a large increase in resistance and sinkage was observed, consistent with observations of conventional displacement hulls. The data from the two models are scaled up to a notional 125-m length to illustrate the effects that would be observed for actual ships similar in size to the U.S. Navy's Independence Class Littoral Combat Ship. Faired plots are developed to allow for rapid estimation of shallow water effect on trimaran resistance and under keel clearance. An example is provided.

Effects of Finite Water Depth on Natural Frequencies of Suspended Water Tanks

Linear sloshing problems (inviscid irrotational flows) in suspended tanks are revisited, with an intention to address some issues in the previous study based on shallow water wave theory. Time-periodic solutions are considered, which describe the synchronized oscillation of the water and tank, reached after the initial transient dies out. The solutions are developed for arbitrary water depths, and separate explicitly the propagating and evanescent wave components of the fluid motion, illuminating clearly the physics and converging rapidly. At the limit of infinite string length, these solutions describe the sloshing motions in tanks that are free to oscillate horizontally on a frictionless plane. Various effects on the lowest sloshing mode are discussed, emphasizing the physical interpretations and examining the limitations of the shallow water approximations. Comparisons with existing laboratory experiments are made, showing agreements with the analysis.

Estimation of Sway Velocity-Dependent Hydrodynamic Derivatives in Surface Ship Manoeuvring Using Ranse Based CFD

The hydrodynamic derivatives appearing in the manoeuvring equations of motion are the primary parameters in the prediction of the trajectory of a vessel. Determination of these derivatives poses major challenge in ship manoeuvring related problems. This paper deals with one such problem in which an attempt has been made to numerically simulate the conventional straight line test in a towing tank using computational fluid dynamics (CFD). Free-surface effects have been neglected here. The domain size has been fixed as per ITTC guide lines. The grid size has been fixed after a thorough grid independency analysis and an optimum grid size has been chosen in order to ensure the insensitivity of the flow parameters to grid size and also to have reduced computational effort. The model has been oriented to wider range of drift angles to capture the non-linear effects and subsequently the forces and moments acting on the model in each angle have been estimated. The sway velocity dependent derivatives have been obtained through plots and curve-fits. The effect of finite water depth on the derivatives has also been looked into. The results have been compared with the available experimental and empirical values and the method was found to be promising.

On the possibility of a bimodal solar dynamo

AbstractA simple way to couple an interface dynamo model to a fast tachocline model is presented, under the assumption that the dynamo saturation is due to a quadratic process and that the effect of finite shear layer thickness on the dynamo wave frequency is analogous to the effect of finite water depth on surface gravity waves. The model contains one free parameter which is fixed by the requirement that a solution should reproduce the helioseismically determined thickness of the tachocline. In this case it is found that, in addition to this solution, another steady solution exists, characterized by a four times thicker tachocline and 4–5 times weaker magnetic fields. It is tempting to relate the existence of this second solution to the occurrence of grand minima in solar activity. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Unsteady ship waves in shallow water of varying depth based on Green–Naghdi equation

Based on Green–Naghdi equation this work studies unsteady ship waves in shallow water of varying depth. A moving ship is regarded as a moving pressure disturbance on free surface. The moving pressure is incorporated into the Green–Naghdi equation to formulate forcing of ship waves in shallow water. The frequency dispersion term of the Green–Naghdi equation accounts for the effects of finite water depth on ship waves. A wave equation model and the finite element method (WE/FEM) are adopted to solve the Green–Naghdi equation. The numerical examples of a Series 60 ( C B =0.6) ship moving in shallow water are presented. Three-dimensional ship wave profiles and wave resistance are given when the ship moves in shallow water with a bed bump (or a trench). The numerical results indicate that the wave resistance increases first, then decreases, and finally returns to normal value as the ship passes a bed bump. A comparison between the numerical results predicted by the Green–Naghdi equation and the shallow water equations is made. It is found that the wave resistance predicted by the Green–Naghdi equation is larger than that predicted by the shallow water equations in subcritical flow (F r= gh <1 ) , and the Green–Naghdi equation and the shallow water equations predict almost the same wave resistance when F r= gh >1 , the frequency dispersion can be neglected in supercritical flows.

Breaking probabilities for dominant surface waves on water of finite constant depth

This paper extends our previous study of the breaking probability of dominant deep water gravity surface waves into the finite water depth environment. It reports a unified behavior of the mean breaking statistics once the effects of finite water depth are taken into account. The shallow water wave data that form the basis of this study were acquired at a field experiment site at Lake George, New South Wales, Australia. The breaking events were detected through visual observation of videotaped records of the wave field in combination with acoustic signatures of the breaking waves from a collocated hydrophone. Following Banner et al. [2000], we argue that when constant finite depth bottom influence is operative, nonlinear hydrodynamical effects associated with energy and momentum fluxes within deforming wave groups remain the primary determinant of breaking onset. This underpins our proposed finite depth water parameterization for the environmental dependence of dominant wave breaking probability, given by the average number of breakers passing a fixed point per dominant wave period. The additional influence of bottom interaction with the wind drift current shear and wind forcing are also included in our finite constant depth formulation. This is a natural extension of our recently proposed deep water dependence and reduces to it as the significant wave height becomes much smaller than the water depth. In common with the deep water case we propose that there exists a threshold of the significant peak steepness below which negligible dominant wave breaking occurs. The available data show encouraging agreement with our proposed dependence, with a correlation coefficient approaching 0.9.

A B-spline Galerkin scheme for calculating the hydroelastic response of a very large floating structure in waves

This paper presents an effective scheme for computing the wave-induced hydroelastic response of a very large floating structure, and a validation of its usefulness. The calculation scheme developed is based on the pressure-distribution method of expressing the disturbance caused by a structure, and on the mode-expansion method for hydroelastic deflection with the superposition of orthogonal mode functions. The scheme uses bi-cubic B-spline functions to represent unknown pressures, and the Galerkin method to satisfy the body boundary conditions. Various numerical checks confirm that the computed results are extremely accurate, require relatively little computational time, and contain few unknowns, even in the region of very short wavelengths. Measurements of the vertical deflections in both head and oblique waves of relatively long wavelength are in good agreement with the computed results. Numerical examples using shorter wavelengths reveal that the hydroelastic deflection does not necessarily become negligible as the wavelength of incident waves decreases. The effects of finite water depth and incident wave angle are also discussed.

“Esso Osaka” tanker manoeuvrability investigations in deep and shallow water, using PMM

Results of the experimental investigation of the effects of finite water depth on the hydrodynamic forces and moments acting on ship model moving in the horizontal plane are presented. The structure of the mathematical model used is verified after th

Weakly-Nonlinear, Long Internal Gravity Waves in Stratified Fluids of Finite Depth

This paper presents an analytical investigation of the propagation of a weakly-nonlinear, long internal gravity wave in a stratified medium of finite total depth. The governing equation is derived and shown to reduce to the KdV equation in the shallow-water limit and to the Benjamin/Ono equation in the deep-water limit. The equation is also shown to possess four conserved quantities, just as was the case for the deep-water waves considered by Ono. The equation suggests the existence of a steady-state waveform described by two parameters which degenerates into the one-parameter, steady-state waveforms discussed by Benjamin. A numerical approach using Fornberg's pseudospectral method is used to examine the solution. The results demonstrate the existence of a solitary wavelike steady-state solution with a solitonlike behavior. The effect of finite water depth on the waveform and the wave speed of the steady-state solitary waves is discussed.

On the Three-dimensional Correction Factors for the Added Mass and the Added Mass Moment of Inertia Related to Manoeuvrability in Shallow Water

When we want to estimate the added mass and the added mass moment of inertia of a ship, the strip method which has proved to be very useful for the calculation of ship motion in waves, seems to be suitable, because it is possible to take the hull form of a ship into consideration in detail. However, since the strip method is based on two-dimensional consideration of the flow field around a three-dimensional body, both the added mass and the added mass moment of inertia obtained by the strip method are always excessively large compared with their exact values.Therefore in this paper, the correction factors are introduced separately for the added mass as to the swaying motion, and for the added mass moment of inertia as to the yawing motion in an analogous way to the correction factor for the added mass of the vertical ship hull vibration.Especially, the effect of finite water depth on these correction factors is investigated from the viewpoint of ship's manoeuvrability in restricted waters. As a result, it is found that the correction factors decrease with decrement of the water depth, and that the correction factor for the added mass moment of inertia is smaller than that for the added mass.

Sway Added-Mass Coefficients for Rectangular Profiles in Shallow Water

This paper presents analytic and numerical results for the added-mass coefficients of two-dimensional rectangular cylinders, accelerating laterally in shallow water. The free surface is assumed rigid, corresponding to low-frequency oscillations in waves or low Froude number maneuvers in calm water. The effects of finite water depth are most significant for rectangles of large beam-draft ratio.

Studies on Manoeuvrability of Ships in Restricted Waters

The main aim of this report is to make clear the effects of finite water depth and finite water channel width on ship manoeuvrability both qualitatively and quantitatively.For this object, the forced yawing technique, by which the author has measured the coefficients of the equations of motion in last four years with enough accuracy, was used to measure the effects of the restricted waters on each coefficient, to say, acceleration, static and rotary derivatives. Besides these coefficients, the two more coefficients which are representative of the effectiveness of the presence of the channel walls, are measured by the straight tow test with variation of the position of the ship in the water channel. In the following, they will be called as the asymmetric hydrodynamic iorce coefficients.From these results of experiments, the manoeuvrability of ships, specially the course keeping quality, are discussed in detail. According to them, it can be said that in certain circumstances even a ship which is stable in deep water, becomes unstable in a specific range of water depth, but in more shallow water, the ship returns to be more stable rather than in deep water.The asymmetric hydrodynamic force coefficients play a significant role on determination of the course keeping quality in narrow water channels. In the most wide and deep channel that was examined in this investigation, that is W/B=6 and H/d=1.9, the ships are unstable in the course keeping quality. But it can be concluded that the unstable quality can be improved by simple automatic steering-by directional control system.