Wash waves generated by high speed displacement ships

Cliiiia Ocean Eng., Vol. 29, No. 5, pp. 757 - 770

© 2015 Chinese Ocean Engineering .Societ)' and Springer-Verlag Berlin Heidelberg DDI 10.1007/s 13344-015-0053-8, ISSN 0890-5487

Wash Waves Generated by High Speed Displacement Ships"

Z H O U L i - I a n ( M J ^ i J i è ) " ' ' ' ' ' , G A O Gao (M and R. H . M . Huijsmans'^ ° Key Laboratory of High Performance Ship Technolog)' Minisliy of Education, Wuhan University'

of Technology, Wnhan 430063, China

School of Transportation, Wnhan Uinversity of Technolog}', Wuhan 430063, China ' Delft Universit)' of Technolog)', Delft, The Netherlands

(Received 24 August 2013; received revised fonn2I Febniary 2014; accepted 25 May 2014)

A B S T R A C T

It is difficult to compute far-field waves in a relative large area by using one wave generation model when a large calculation domain is needed because of large dimensions of the waterway and long distance of the required computing points. Variation of waterway bathymetr}' and nonlinearity in the far field camiot be included in a ship fixed process either. A coupled method combining a wave generation model and wave propagation model is then used in this paper to simulate the wash waves generated by the passing ship. A NURBS-based higher order panel method is adopted as the stationary wave generation model; a wave spectrum method and Boussinesq-tj'pe equation wave model are used as the wave propagation model for the constant water depth condition and variable water depth condition, respectively. Tlie waves calculated by the NURBS-based higher order panel method in the near field are used as the input for the wave spectmm method and the Boussinesq-type equation wave model to obtain the far-field waves. With tliis approach it is possible to simulate the ship wash waves including the effects of water depth and waterway bathymetry. Parts of the calculated results are validated experimentally, and the agreement is demonstrated. The effects of ship wash waves on the moored ship are discussed by using a diffraction theory method. The results indicate that the prediction of the ship induced waves by coupling models is feasible.

Key words: M'ash waves; wcn'e generation model; wave specti'um method; Boiissinesg t)'pe equation wave model; motions of moored ship

1. I n t r o d u c t i o n

Wake waves generated by high speed ships may cause detrimental impacts on the coastal environment by damaging shorelines and banks. Wash waves i n shallow water may have effects on water plants and animals. I n recent years, complaints f r o m smaller vessels moored along the banks o f inland waterways due to the unexpected motions caused by the wash waves o f passing ships have increased w i t h the increase o f high speed ships. O f t e n such effects occurred due to operation o f fast ferry services, it is significant to predict wash effects in an early stage to reduce/avoid the harmful effects. With the growing awareness o f wake washes to the high-speed seivice, the environment and so on, many naval architects and scholars have conducted the researches on the washes. Macfarlane et al. (2008) analyzed the erosion to the banks by the washes generated by small ships in sheltered waters,

* This work is financially supported by the National Natural Science Foundation of China (Grant Nos. 50879066 and 51409201) and the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 200804970009).

758 ZHOULi-km et al I China Ocean Eng., 29(5), 2015, 757 - 770

and provided that the single criteria is not enough to evaluate the washes; Ghani el al. (2008) described the experimental results o f a patrol, and also gave some discussion on the numerical calculation results; Macfarlane (2009) conducted an experimental investigation into the correlation o f model scale wave wake measurements against f u l l scale trials results f o r a catamaran operating at low length Froude numbers; Benassai (2010) discussed the wake wash waves generated by high speed crafts observed at some distance away (typically one or multiple ship lengths) f r o m the line o f travel o f t h e vessel both nuinerically and experimentally; Yaakob et al. (2012) presented results o f a parametric study o f catatnaran hull f o r m to obtain low wake wash hull f o r m configurations or low speed inland waterway boats by CFD simulation and model experiments; Didenkulova (2013) investigated the potential benefits o f wake analysis by means o f a tirne-frequency technique (windowed Fourier transform), and analyzed the characteristic properties o f different vessel wake signals. Pinkster (2004, 2009), van der Hout et al. (2011) calculated the effects o f a passing ship on moored ships; Bunnik et al. (2009) used R A N S method to analyze the responses o f moored ships i n waves due to the passing ship considering the effects o f viscosity, and discussed the effects o f the navigation angle o f the passing ship; L u t h et al. (2011) described the fitU scale test carried out to develop the missing relation between the wash induced motions o f a moored vessel and the hindrance. As known to all, it is difficult and costly to measure the f u l l scale data for each ship, and it is hard to change the wave characteristics o f ships after being produced, so predicting the waves and the effects correctly in the ship design phase is an attractive choice. But there are many problems in calculating the washes o f a ship, the dimensions o f watei-ways are always very large, the affected subjects are sometimes far from the passing ship, the width o f waterways is always variable, and the topography o f the waterways is complex, thus it is hard to find a kind o f method to obtain the satisfactory results o f waves and the effects o f the passing ships. Using coupled methods seems to be an attractive choice, the calculating area is divided into several parts, models are applied for each patt o f the area, and each model only needs to include the physics for the specific part.

A stationary wave generation model combined w i t h a spectruin method is used to calculate the wake waves o f passing ships in deep water with constant water depth (in other words, the effects o f variation o f waterway bathymetty can be ignored in this condition). Wave spectrum method is a kind o f method that calculates the wave spectrum by analyzing the known wave heights, and then the wave height at any point o f the field can be obtained by reconstniction o f the wave spectrum. Calculated and measured wave data both can be used as the input for this method, which makes it feasible to obtain the washes o f ships by doing experiments or using wave generation model in relative small regions. But often, variations o f waterway width or depth may cause changes to the far field wave pattern, which need to be taken into account and resolved. When the depth, w i d t h or cross section o f the waterway varies along the path o f the ship, the fiow field is unsteady principally, which cannot be considered i n above-mentioned coupled method. Therefore, a stationary wave generation model combined w i t h a separate wave propagation model (Boussinesq type equation wave model) is adopted. The Boussinesq type equation wave model used in this paper is a 2D horizontal model f o r the computation o f wave propagation in time domain. The geometry (bathymetry, harbour walls), nonlinear wave-wave interaction, wave breaking, wave run up, wave shoaling, refraction, and diffraction

ZHOU Li-hii et al. I China Ocean Eng., 29(5), 2015, 757 - 770 759

are included in the model.

Vessels moored in ports experience hydrodynamic forces due to waves generated by passing ships. Calculation o f the behavior o f a moored ship in waves generated by passing ships is significant to analyze the effects o f wash waves o f passing ships. I n this paper, calculation o f moored ship motions is based on the wave results obtained by the above-mentioned coupled methods. A diffraction theory method is used to calculate the motions o f moored ship in frequency domain i n this paper. For each input frequency component, the potential is periodic in the form ^ ( . T , j ' , z , / ) = ^(A-,;',z)e'"'and contains eight components: 1) incoming wave potential; 2) six radiation potentials; 3) diffraction potential. The waves obtained at the location o f the moored ship are in time domain, so the wave records have to be transformed into the frequency domain before implemented in the diffraction code. Finally, all results obtained by the diffraction theoiy method are inversely transformed to the yield time domain record o f passing ship effects.

2. W a v e G e n e r a t i o n M o d e l

The prediction o f waves generated by a ship moving in still water has made great progress in the last f e w years. The introduction o f Dawson's method started a new tendency in the prediction o f waves since last centuiy. The Ranltine source is adopted in Dowson's method as the basic source f o r the Green function, which is simple and does not f u l f i l l any boundaiy condition except Laplace equation, so i t is easy to be applied to solve many kinds o f problems (steady and unsteady problems, linear and nonlinear problems). The only problem is that sources need to be distributed on each boundaiy surface, especially the free surface, which causes d i f f i c u l t y o f large calculation domain. I n this model, the coordinate system o-xyz fixed on the ship as shown in F i g . 1 is used, making the wave generation be a steady problem.

Fig. 1. Schematic coordinate used in wave generation model.

Non-uniform rational B spline (NURBS) method is the most prevailing technique to model the surface nowadays. The application o f N U R B S on the calculated f l u i d mechanics makes the geometrical composition o f panels omit, a k i n d o f continuity is guaranteed by using the N U R B S to express the geometiy o f the hull and the distribution o f source strength, and this also simplifies the problem and become the bridge between CFD and C A D . The N U R B S method is adopted i n the wave generation model in this paper.

The 3D incompressible potential f l o w model is used to calculate ship wave problems. The disturbance velocity potential (zl must f u l f i l l the Laplace equation and the boundary conditions as

760 ZHOUU-lan etal./China Ocean Eng., 29{5), 2015, 757- 770

follows:

(1) H u l l surface boundary condition:

^•it = U-ii; (1)

(2) Free surface kinetic boundary condition:

<f>J}. + ?'.-'7z -<f>y=0, z = rj{x,y); (2)

(3) Free surface dynamic boundaiy condition:

: ^ + i v ^ . V ( ^ + g;7 = 0 , ,j(x,y,t) = z; (3) dl 2

(4) I n f i n i t y boundaiy condition:

lim Vfl) = 0 . (4) (5) Transom stern boundary condition: the common used transom stern boundaiy condition is shown in

reference (Tulin and Hsu, 1986; Nakos and Sclavounos, 1994), but there are drawbacks i n their method. Gao (2006a, 2006b) used a simple and effective method that distributed a column collocation point between the transom stern and its neighbor grid lines o f the free surface, and they f u l f i l l the following condition:

-2<pJAx = -gU[2z(x„y)IAx + zx(x„y)]. (5)

where ii is the outward unit normal vector o f the panel, ij{x,y) is the elevation o f free surface at

{x,y), g is the gravitational acceleration, U is the velocity o f t h e ship, Xis draft o f transom

stern, and;' are the longitudinal coordinate and transversal coordinate o f transom stem respectively, and A,v can be any small distance behind the transom and does not have to be panel length.

Distribute sources on the hull surface and free surface, suppose the source strength is cr(M„) at the point M„ = ^ ( ! ( „ , v „ ) , and S is the surface distributed w i t h sources, then the induced velocity potential at the field point M = p{ii,v) is:

' 4 7 t r ( M , M „ )

Suppose p{ii,v) = {x{u,v),y(ii,v),z{ii,v)) is the coordinates o f point (ii,v) on the three dimensional curved surface, <jiii,v) is the source strength at point {ii,v), and

^ ( M ) = f f _ ^ ( ^ d S (6)

p { u , v ) = ^ , <5{n,v) = ^ . (7)

X E W^N,^, iu)N^^ (V) X I ^^'sA. ( " ) A^.v, (V)

where is the control point f o r panels, JF. is the weight o f control point o f surfaces; 5. is the control point for source strength distribution, IF. is the weight o f control point f o r source strength distribution, a, b, c, and are the degrees o f B spline basic function.

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The difference o f the N U R B S based potential theoiy method with other boundary element methods is that the direct variables in the equations are not the source density, but the control points o f surfaces distributed with sources, so the number o f t h e selected collocation points should be the same as the number o f control points for source strength distribution.

How to f u l f i l l the radiation condition is one o f the difficulties in using the Rankine source as singularity. Raised panels above the fl-ee surface combined w i t h collocation-point shifting up-stream are used conveniently to satisfy the radiation condition numerically. Gao (2006b) analyzed the relevant numerical errors, the specific method and parameters can be found in the reference. Substitute the expression o f velocity potential into the boundaiy condition, free-surface condition, and then we can obtain the algebraic equations with unknown control points, as the Gauss integration method being employed in the calculation. Eq. (6) is used to calculate the velocity potential, and finally the wave height and wave-making resistance can be obtained.

3. F a r F i e l d W a v e P r e d i c t i o n 3.1 Coupling of Wave Generation Model with A Wave Spectrum Method

There are some methods that can be used to extend the waves to the far field f o r uniform water depth, and they depend on the assumption that the waves in the far field are linear. The wave spectrum method is one o f these methods. It analyzes the wave height data and acquires the wave amplitude function, and then the wave height at any point can be obtained. The hull form and dimensions are not necessaiy to be considered. The specific methods o f wave spectrum method are mainly Fourier transform method, equivalent singularity distribution method, X - Y method and so on. We adopt the equivalent distribution method i n this paper.

This method has been exploited by Scragg (2002) f o r the spectrum analysis o f ship waves i n a finite water depth. In the later work, a method has been developed to calculate the wave in the far field (Scragg, 2002, 2003; Belibassakis, 2003) and the results i n the references were quite good. It can be used not only w i t h free-surface elevation data obtained from the linearised solution, but also w i t h wave data f r o m non-linear ship CFD codes, or experimental data measured in a wave tank (however, no practical proof has been conducted yet).

The general expression o f the Green function in a finite water depth f o r a Havelock singularity located at (.Y',;'',Z') is given by Wehausen and Laitone (1960). According to Scragg (2003), the equation can be simplified as follows f o r the far field wave:

= 4 F{d)cos[k(: + h)]cosh[k(:' + h)]sm[k^.(x- x')]cos[*,,(y -y')]d0 ; (8)

^ o + /tcos-^ sec^Ösech(/t/7)e-"', (9)

l-/f„Asec'6'sech'(W7) ^ ' ' W

where and k^. are the components o f wave nuinber k in the ,T- andj'-direction, respectively; h is the

water depth; k„=g/U', U is the velocity o f the ship, g is the gravitational acceleration, and 0 is the direction o f wave propagation.

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We choose the center plane o f the ship as the singularity distribution surface 5. Discrete S into A' panels, and arrange equivalent strength sources a on each panel, as the source strengths being unknown. The wave elevation at the undisturbed free surface {x,y) is:

u d

g 9x

C{x,y) = tc7,ax,y),

(=1

where ^ is the wave elevation due to panel / , it is given by:

(a-, y) = J[" 4 {0) sin(A .Y) COS(/CJO + A,^ (0) cos(A .v) c o s ( / f j ' ) d Ö

The free wave spectrum component (0) and //^ (0) are as follows for a monohull:

(10) (11) (12) F{0,h) K cosh( kh) . . . , r , / , —^sm(/c,xc') c o s h [ ^ ( / 7 4 k • zc')] sin

j

where ( x c . , zc]) represents the center point o f panel /, Av/ and Az.' represent the side lengths in the .r and z direction o f panel /, respectively.

Change the variable o f integration from 0 to /c, ,

^, y) =[ — 4 . {0) sm{k^x) cos{kj>) + — 4 {0) cos(k^x) cos(A ,;Od/f,,;

d0_ dk.. kcos0

2k/{k„sm'0cosh\kh))

(14) sinh(A'/7) cosh( W 7 ) - kh

Suppose a set o f wave heights Ci^^)') obtained at M points by measurement or calculation, we defme:

C ( A - . , 3 ^ ) - ^ < T , C ( A - , , y , )

To determine the unknown source strength, the partial derivative o f cr, should be zero.

da..

: 0 , k = \,2,...,N

(15)

(16)

Then we obtain the linear system o f N equations:

J]CiXj,y,-K(^;,yj) = ta^z,)r)±a,a^z,y^). (17) By solving the above system equation, we can obtain the source strength cr, and then the wave height

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A basic ciiecic o f tlie coupled system has been made in a computation for a N P L model (affined from the 1/8=9 series, where L and B are the ship length and ship width, respectively) with water depths o f 1.5 m and 2 m. Experiments were done in the towing tank o f Wuhan University o f Technology. Five wave height probes were arranged in one line at different distances fi-om the ship centerline as shown in Fig. 2. x is the distance between the probe and the ship centerline, L is the length o f the ship, ƒ/„ is the wave height, Fr is Froude number, Fr = u/^JgL . A longitudinal wave cut at the lateral distance o f 0.8 m calculated by the wave generation mode is used as the input for the wave spectmm method, the number o f meshes is 80, and the wave height number is 200, which are validated to be sufficient to obtain reliable results (Zhou, 2012). The waves simulated by the wave spectrum method at other locations are compared with the experimental data in Fig. 3 and Fig. 4.

• ' f \ ' ly^<\\' I " \ f \ T " •

i-\

QI

^ \ ^ ^ ^ ^ T5

Fig. 2. Sketch of arrangement of wave height probes.

• v " . - ( J . l -* x!l. - I I . I L

Fig. 3. Wave pattern at ;^2.6 m (/)=! .5 m, F;-=0.679). Fig. 4. Wave pattern at j^4.4 ra (/i=2 m, F;=0.474).

3.2 Coupling of Wave Generation Model with A Boussinesq Type Equation Model

A combination o f wave generation model and wave spectrum method worked well i n the calculation o f ship wash waves in uniform water field. But in some cases, especially in shallow water, the effects o f variable bathymetiy cannot be neglected. A Boussinesq type equation model can simulate the pi-opagation o f waves in shallow water w i t h variable water depth and can include the nonlineai-effects. Raven (2000), Doorn el al. (2002) combined a 3D potential flow model w i t h a 2 D Boussinesq type model to calculate the waves o f ships, Jiang and Sharma (2002) combined a slender ship theoiy with a extended Boussinesq type model to calculate the wake washes o f ships taking account o f varying topography, van der Molen and Wenneker (2008), van der f ï o u t et al. (2011) used a time domain Boussinesq type model to calculate the wake washes o f ships, and validated the results by experimental data. Thus, a Boussinesq type equation model is a good choice to simulate the wave propagation considering the effects o f variable bathymetry and nonlinearity. The Boussinesq type

764 ZHOU Li-Ian el al. I China Ocean Eng., 291,5), 2015, 757 - 770

equation model used in this paper is the time domain Boussinesq model proposed by Deltares (van der Hout el al., 2011). This model is a 2 D Boussinesq type equation model denved f r o m 3D potential f l o w equations with free suiface conditions by taking a tinncated polynomial expansion o f the particle velocities over the water depth to the vertical dimension. In this Boussinesq type wave model, // = kh

(k is the typical wave number, h is the typical water depth) is adopted to represent the importance o f the linear dispersion, e = a / / ? ( a i s the typical wave amplitude) is used to indicate the importance o f the nonlinear effects. With higher order o f the Boussinesq type model, the upper limit o f and £ w i l l be higher, and the restriction w i l l be smaller, but it w i l l be more complex and w i l l cost much more calculation time (Borsboom el al., 2000). A restriction o f kh < 4 and s < 0.25 is used in this Boussinesq type wave model. The Boussinesq type equation model employed in this paper conserved both mass and momentum (Borsboom el al., 2000).

(1) Mass equation:

— + £V • ( / / » ) = 0 , (18) dt

where H = h + si^ is the total water depth, t is time, V is the gradient operator (ö / dx,dl dy) ,q\s.

the depth integrated velocity, ü = cjl H is the depth averaged velocity, g is the gravitational

acceleration, and h is the water depth with respect to some reference level. (2) Momentum equation:

dm

where / / is a function o f the total water depth H and bottom slope V/?:

H - aH'H^y-H- Pn'HVh • VH

( f = H- a - - fi'H'-TH- p— /.I'HVh-VH

-^(V/i-Vli)H-^(HV'li)H. (20) By optimizing the parameters a and / ? , the modeling o f linear dispersion and first-order linear

shoaling can be improved.

Various types o f boundaiy conditions can be modeled in this model. A t seaward boundaries, incident waves in the form o f time series or spectra can be prescribed, while at the same time, waves can leave the domain without spurious reflections (Borsboom et al., 2001). Full and partial reflection at arbitrarily shaped inner domain structures (e.g. quay walls) can be modeled as well (Wenneker and Borsboom, 2005). This implies that the model is capable o f accurately treating complex waterway geometries and bottom bathymetiy.

ZHOU Li-Ian el al. I China Ocean Eng., 29(5), 2015, 757- 770 765

calculation domain is divided into two parts, one is the inner domain where the wave generation calculated, and the other is the outer domain where the wave propagation calculated. A longitudinal cut is obtained at the interface o f these two domains, the wave signals are transformed into time domain according to the coordinates o f the points and ship velocity, and imposed along the edge o f the wave incoming boundary o f the outer domain. These two models are only just one way coupling, and the boundaiy variations in the outer domain w i l l not affect the waves in the inner domain.

A n example o f this coupled method is given below. The hull form used is the Wigley hull f o r m . The dimensions o f the wave generation calculation domain are 3L x 2L ( L is the ship length), the longitudinal wave cut at lateral distance O.IZ, o f f the ship centerline is imposed on the incoming boundary o f the wave propagation calculation domain. The composite plot in Fig. 5 shows the computed wave pattern, the small part below is calculated by the wave generation model; and the large part above is the extension computed by the wave propagation model. The two solutions match w e l l . Fig. 6 compares a longitudinal wave cut computed by the panel method alone with the results computed by the coupled system. The agreement is fair, but with some overestimation o f the wave amplitude by the coupled system.

10; 0 0.06 0.04 0.02 0 -0.02 0.04 Coupled system

i —Wave generation model •

0 10 - 20 - 30

Fig. 5. Wave pattern calculated by tlie coupled method.

0 10 20 30 40

Fig.6. Waves at lateral distance 0.3i off the ship centerline (solid line is computed by the wave generation model, dashed line is computed by the coupled method).

Figs. 7 and 8 are the results obtained by the wave propagation with the boundaiy input o f experimental data in uneven bottom wave field. The experiment details can be found in reference (Latarie et al., 2009). H a l f o f the cross section o f the experiment used in this case is shown in Fig. 9; the remainder left part is with flat bottom. The comparison indicated that the calculated results and experimental data match well.

Tu U Simulation . ' —E.\pcrimcnl 50 100 1.50 5 I 01 • i , •5 5 I ' 10. - 15. x | 0 ' Simulalion — lixpcriinciit (1 50 /(s) 100 1.50 Fig. 7. Wave pattern at .1^=1.34 m. Fig. 8. Wave pattern at;'=2.24 m.

766 ZHOULi-lan et al. I China Ocean Eng., 29(5), 2015, 757- 770

Fig. 9. View of tlie cross section of tlie taiil: (unit: m).

4. E f f e c t s of W a s h W a v e s on the M o o r e d S h i p

Wash waves generated by passing ships may have effects on the environment, cause bank erosion, threaten small crafts, and affect the operation o f moored ships. H o w to reduce the effects o f wash waves is an urgent problem to be solved by related departments and researchers. Moored vessels experience hydrodynamic forces due to waves generated by the passing ships. Prediction and evaluation o f the motions o f the moored ship is a way to evaluate the effects o f the wash waves o f passing ships. A 3D frequency domain diffraction theory model D E L F R A C (Pinkster, 1995) is adopted in this paper to calculate the motions o f a moored ship in waves. The time record wave signals obtained by the above coupling method are transformed into frequency domain by the Fourier analysis, each wave frequency corresponding to unique wave amplitude, wave direction and phase angle according to the dispersion relationship. Finally, the frequency domain calculations by the diffraction theoiy codes are carried out to yield frequency domain results o f forces, velocities, pressures, motions and so on. These resuhs are transformed into time domain by the inverse Fourier transform method.

In regular waves a linear potential 0, which is a function o f the earth-fixed coordinates and o f time /, can be written as a product o f a space-dependent term and a harmonic time-dependent tenn as follows:

0{x,y,z,t) = ,t>{x,y,z)t-. (21) The potentials have to satisfy the following boundary conditions:

Laplace equation: V'tf) = 0 ; (22) Hull boundary condition: (j>^^ =v-n; (23) Sea bed boundaiy condition: (zi„ = 0 , z = -h,^;

Free-surface condition: -m^(j> + g<j>,=*i, z = 0 ;

Radiation condition: lim ij> = Q .

A convenient formulation can be obtained by writing:

^ = -ui'2](Z*;?, = - i i i

(24)

(25)

(26)

ZHOU Li-Ian et al. I China Ocean Eng., 29(5), 2015 , 757 - 770 767 . g cosh[/f(z + / ; ) ] - « - ( . v c o s / ï t v s m / ; ) . cosh(/f/?) ( ^ ^ , ( A - , j , z ) = | a / . v ' , y , z ' ) G ( A - , j , z , . v ' , y , z ' ) d 5 , 7 = 1, 2, . . . , 7 ; 9(zi, 1 1 r — i = n = - - c r (A-,3',z) + - — C7 ( A - ' , J ' , Z ' X ? ( A - , 3 ' , z , x ' o / , z ' ) d i . dn 2 471 J'

Solving the source strengths cr(A', j ' ' , z ' ) , we can obtain the fluid pressure f r o m the Bernoulli

equation as follows:

d0

= - \\pn,ds = - p u ' V o e " ' " \\{(j>, + <j>,)n,ds . s s

The oscillating hydrodynamic forces and moments in the /f-th direction:

Finally, the motions gj are determined f r o m the solution o f the f o l l o w i n g coupled equations o f

motions for six degrees o f freedom: 6

Z ( ' " < ; + ' h j ) - + c , , ] - g j = X , , (34)

where, 7;;,^ is inertia matrix o f the body f o r inertia coupling in the k mode for acceleration i n the J

mode; a,., is added mass coefficient.

Panel method is used to solve the problem in this paper. Two container ships are used, one is

moored, and the other is passing by. The location o f the measuring wave heights and the defmition o f the forces and moment near the moored ship are shown in F i g . 10, and the forces on captive model are

measured. Two transverse forces and one longitudinal force are measured to yield surge, sway forces, and yaw moment. A case w i t h the speed V=6.5 kn o f the passing container ship is used to validate the

method used i n this paper, the transverse distance o f the two ships is 100 m i n this case, and the passing ship is proceeding straight. Fig. 11 shows the wave signals in frequency domain at location 1

and the comparison o f the calculated and experimental wave forces and moments, and the overall results shows that the calculated results and experimental results match w e l l , as the method used in this

paper is feasible. Experimental results are obtained in M A R I N for the case o f a container ship o f 12500 T E U passing a Panamax container ship (the moored ship).

Fig. 10. Location of tlie measuring wave heiglits and tlie definition of the forces and moments.

^>

2

768 ZHOULi-km et al. I China Ocean Eng., 29(5), 2015, 757 - 770 H 0.2 r 0.15 Frcqui-'ncy Spctrujii .100 200 S 100 i I 2 3 4 5 6 7 8 9 Fret|Licncy(Hz)

(a) I'icqueiKy cloiiiain signal o l ' « a v c Exp. Cai. 1 0 0 lO^"*» 200 700 ).m J300 J500 /(s) (c) Sway Ibrce 1000 15000 I OOOO 5000 0 5000 10000 (b) Surge force

Ar

5. C o n c l u s i o n

In the present work, coupled methods are presented to calculate the wash waves in a large domain w i t h constant or variable water depth. The examples show that they are feasible in the ship wash wave prediction, and can be useful for comparing ship designs based on minimizing the ship wash waves. The calculation o f ship motions in waves generated by the passing ship indicates that the extension o f the coupled wave calculation system to a diffraction theoiy method is also possible, which can be the basis for the wash minimization. Because o f the experimental limitation, the comparison between the simulation and experimental results are confined to relative near field waves. For veiy far field waves, a further refinement and validation are needed. I n general, the coupling o f wave generation model w i t h wave propagation model peimits to address a variety o f problems.

References

Belibassakis, K. A., 2003. A coupled-mode technique for the transformation of ship-generated waves over variable bathymetry regions, ^pp/. Ocean Res., 25(6): 321-336.

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