ZHOU Li-lan,GAO Gao

(a.Key Laboratory of High Performance Ship Technology of Ministry of Education;b.Transportation School,Wuhan University of Technology,Wuhan 430063,China)

1 Introduction

Nowadays,the introduction of fast ferries has caused a lot of complaints and protests from the public about severe wash waves.Small boats and people were surprised by suddenly raised waves in calm weather[1].Ships moored in harbors are subjected to wash waves due to other ships passing nearby[2].In some places,the fast ferry has to slow down to avoid problems.The speed restrictions have reduced the number of complaints,but the problem is not completely solved,and it is not a satisfactory solution either.The speed of a fast ferry is indeed an important feature in the service that is offered.Not only high-speed craft,but also more conventional displacement ships can produce adverse impacts on coastal areas[3].With the growing awareness of wake washes to the high-speed service,the environment,and so on,many naval architects and researchers have begun the researches on the washes,Leer-Anderson et al[4]provided with a good discussion of the challenges facing ship designers and operators under new wake wash regulations.Macfarlane,Bradbury[5]analyzed the erosion to the banks by the washes generated by small ships in sheltered waters,and provided that the single criteria is not enough to evaluate the washes;Mohamad Pauzi Abdul Ghani,Abdul Rahim[6]described the experimental results of a patrol,and also gave some discussion on the numerical calculation results.As we all know,it is difficult and costy to measure the full scale data for each ship,and it is hard to change the wave characteristics of ships after being produced,so predicting the waves correctly in the ship design phase is an attractive choice.But there are many problems when calculating the washes of a ship,the dimension of waterways is always very large,the affected subjects are sometimes far from the passing ship,the width of waterways is always variable,and the topography of the waterways is complex,so it is hard to find a kind of method to get the satisfactory results of waves generated by the passing ships,and using coupled methods seem to be an attractive choice,the calculating area are divided into several parts,models are applied for each part of the area,and each model only needs to include the physics for the specific part.

A stationary wave generation model is used to obtain the waves near the ship,it is based on a 3D incompressible potential flow theory method.A Rankine source is adopted as the Green function source,it does not fulfill any boundary condition except the Laplace equation,so it is convenient to apply to complex conditions(bank,shallow water,etc),but it is difficult to apply to large area calculation because sources are needed to be distributed on each surface in the calculating domain.A wave spectrum method combined with the wave generation model is adopted to get the waves in large area,a wave spectrum method is a method that analyzes the limited wave characteristics,gets the wave amplitude function,and then it is possible to obtain the wave height at any point in the domain from the function,it is easy and can be used for the wave calculation in large areas.

2 Wave simulation(coupling of wave generation model with wave spectrum method)

A NURBS(Non Uniform Rational B Spline)based higher panel method is used as the wave generation model,the specific details are described in references[7-8].NURBS based panel method is that of a 3D incompressible potential flow around the ship.The hull boundary condition requires that there is no flow through the ship hull surface,the ship must be in equilibrium with the hydrostatic and hydrodynamic pressure forces,a kinematic boundary condition(no flow through the wave surface)and a dynamic boundary condition(pressure equal to atmospheric pressure)are imposed on the free surface.In case of shallow water,a condition of zero normal velocity must be satisfied on the bottom of the waterway as well,when considering the effects of waterway walls,the condition of zero normal velocity must also be satisfied on the surface of the walls,the transom stern boundary condition must also be satisfied when the ship is with transom stern,a smooth flow off the transom edge tangentially condition which is proposed and validated by Gao[9-10]is used in this paper.The boundary surfaces(hull surface,free surface,bottom surface,etc.)are divided into several small surface panels which are distributed with source density,the boundary condition equations can be expressed in terms of source strength,they are imposed in a set of collocation points,thus a set of equation system can be got,the wave height can be obtained at any point in the calculated domain solving this equation system.But this method is hard to calculate the waves in a large area because of the source strength distribution on all surfaces.To solve this problem,a wave spectrum method is used in this paper to calculate the waves far from the ships.The wave spectrum method is a method that analyzes wave heights(can be the simulation results by the NURBS based panel method or experimental results)to get the wave amplitude function.The analytical method used in this research is the equivalent singularity distribution method.Green function in finite water depth for a Havelock singularity located at（x′,y′,z′）is given by Wehausen[11],according to Scraggs[12],the equation can be simplified as followings for the far field wave:

Discrete the singularity distribution surface S into N panels,and distribute equivalent strength sources σ on each panel,the source strengths are unknown.The wave elevation at the undisturbed free surface at field point（x,y,z）is:

The far field elevation ζidue to panel i is given by:

The free wave spectrum componentandare as following for a monohull:

where xci′and zci′represent the center point of panel i,Δxi′and Δzi′represent the side lengths in the x and z direction of panel i,respectively.

Change the variable of integration from θ to k,

The value of kcis determined according to the dispersion relationship w2=kgtanh（kh）,details can be found in reference[10].

Suppose we get a set of wave heightsat M points by measurement or calculation,we can define:

Then we get the linear system of equations include N equations:

To determine the unknown source strength,the partial derivative of σishould be zero.

Solving the above equations,we can get the source strength,the wave height at（x,y）can be obtained by Eqs.(1)and(2).

Some examples calculated by the combined wave generation model and wave spectrum method are shown below.The lines and dimensions of the ship model used for the calculation are shown in Fig.1 and Tab.1,respectively.The coordinate system is fixed on the ship.The velocity of the ship is V.When calculating the near field waves by the NURBS based panel method,we used longitudinal cut and transverse cut method,the longitudinal cut position is always at yc=0.1L,and the transverse cut position is always at the aft of the ship with five cuts,ycis the lateral distance from the center line of the ship.Fig.2 shows the calculated and experimental wave pattern of the monohull,and

Fig.3 shows the calculated and experimental wave pattern of the catamaran with the demihull spacing C0=0.3L,we found that the combined method of the NURBS based panel method and the wave spectrum method is feasible.

Fig.1 Lines of calculated ships

Fig.2 Wave pattern at yc=3.5 m(V=3.03 m/s,H=1.5 m)

Fig.3 Wave pattern at yc=3.1 m(V=3 m/s,H=2 m)

Tab.1 Main dimensions of monohull and the demihull of catamaran

3 Model experiment

Experiments to investigate wake washes of monohull and catamaran were conducted in calm water over a wide range of speeds corresponding to a range of depth Froude numbers Fnhfrom 0.1 to 1.0 in towing tank 118 m×10.8 m×2 m of WHUT,water depth is adjustable,the measurements are conducted with water depth H=2 m,H=1.5 m,H=1 m.The wave heights and ship resistance are recorded during the experiments,wave heights are recorded at five points for each run,Fig.4 is the schematic diagram for the wave probe location,ycis the lateral distance from the center line of the ship,1～5 are the five probes.The locations of the five probes for monohull and catamaran are shown in Tab.2.

Tab.2 Location of probes

Fig.4 Schematic diagram of experimental measurement

4 Effects of some factors on washes of ships

4.1 Water depth

In shallow water,the wave pattern generated by the vessel is influenced by the waves reflecting from the sea bottom.The ratio of the vessel speed to the maximum wave celerity in shallow water is often used to classify the wash produced by a vessel.This ratio,called the depth Froude number is defined as follows,

When the wave is propagating in water with a depth of less than approximately half the length of the wave,the wave induced water particle motion reaches the bottom and the water

depth affects wave characteristics.In shallow water the wave speed of a gravity wave is:C=

In deep water,the vessel length Froude numberis used to classify wash.At a critical value of Fnl～0.4 to 0.55,the wavelength of the transverse waves equals the ship length giving a condition similar to the critical condition in shallow water.For most displacement vessels,in normal speed ranges,there is no strong wave bottom interaction if the ratio of water depth to the draft(d/T)is greater than 3 to 5.

The calculated wave making resistance coefficients are shown in Fig.5,the maximum wave heights obtained experimentally at lateral distance yc=2.6 m are presented in Fig.6,the comparison of the calculated wave making resistance coefficients with the experimental wave making resistance coefficients(subtracted from the experimental total resistance by Lackenby method)is shown in Fig.7.The results show that both of the wave making resistance and the maximum wave height at a specific location increase with the decrease of the water depth with the same velocity.

Fig.5 Wave making resistance coefficient Cw

Fig.6 Maximum wave height at y=2.6 m

Fig.7 Calculated wave making resistance coefficient Cw and the experimental wave making resistance Cw’

4.2 Afloat conditions

Fig.8 is the wave making resistance coefficient of monohull with trim(-1 means down by the stern of the ship,+1 means down by the head)and without trim,Fig.9 is the wave heights at lateral distance yc=L with Fn=0.5.We found that the wave making resistance increased when the ship is down by the head,and the amplitude of the ship waves increased a little;the wave making resistance decreased when the ship is down by the stern,and phase shift of ship waves appeard.

Fig.8 Wave making resistance coefficient of monohull(H=2 m)

Fig.9 Wave heights of monohull(Fr=0.5,yc=L,H=2 m)

4.3 Demihull spacing C0

The aim of this part is to get the effects of the separation on the wave profiles and resistance on the catamaran.Fig.10 shows the wave profile of the simulation results of catamaran with C0=0.3L.Fig.11 shows the experimental total resistance and simulated maximum wave height at lateral distance yc=4L of catamaran with different demihull spacing.The results show that large demihull spacing leads to lower ship resistance and small maximum wave height at the same lateral distance with the same velocity.

Fig.10 Wave profile of catamaran(V=3.25 m/s,H=2 m)

Fig.11 Total resistance and maximum wave heights at y=4L(H=2 m)

5 Conclusions

The coupled NURBS based panel method and spectrum method is feasible for the calculation of waves and washes around the ship.The equivalent singularity distribution method is applicable for the computed or measured wave heights.We found from the results that:

(1)The wave-making resistance increased with the decreasing of the water depth with the same length Froude number;

(2)The maximum wave height increased with the decreasing of the water depth at the same lateral distance with the same Froude number;

(3)The wave making resistance increased when the ship is down by the head,and the amplitude of the ship waves increased a little;

(4)The wave making resistance decreased when the ship is down by the stern,and phase shift of ship waves appeard;

(5)The resistance and the maximum wave height at the same lateral distance decreased with the increasing of the demihull spacing of catamaran ship.

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