Numerical wave tank

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Ocean Engineering 31 (2004) 15491565

www.elsevier.com/locate/oceaneng

Numerical reproduction of fully nonlinearmulti-directional waves by a viscous 3D

numerical wave tank

J.C. Park a, , Y. Uno b , T. Sato b , H. Miyata b , H.H. Chun aa Department of Naval Architecture and Ocean Engineering, Pusan National University, 30 Jangjeon-dong,

Geumjeong-gu, Busan, South Koreab Department of Environmental and Ocean Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,

Tokyo, Japan

Received 28 July 2003; accepted 3 December 2003

Abstract

Fully nonlinear multi-directional waves are reproduced by use of a viscous 3D numericalwave tank (NWT) simulation technique. The governing equations, NavierStokes and conti-nuity equations, are discretized by a nite-difference/volume method in the framework of arectangular/body-tted coordinate system, and the boundary values are updated at eachtime step in a time-marching procedure. The fully nonlinear kinematic free-surface conditionis satised by the marker-density function technique. The directed incident waves are gener-ated by multi-segmented wavemaker on the basis of the so-called snake-principle, and theoutgoing waves are numerically dissipated inside an articial damping zone located at theopposite side of the wavemaker.

In this study, the results for the generation of regular waves, including numerical conver-

gence tests, irregular waves, and multi-directional random waves are presented. Further-more, the generation of following waves, which are one kind of directional waves, isexamined using a uid acceleration wavemaker, and the hydrodynamic forces acting on anadvancing ship in such a wave condition are discussed.# 2004 Elsevier Ltd. All rights reserved.

Keywords: Numerical wave tank (NWT); Multi-directional waves; NavierStokes equation; Marker-density function; Directional wave spectrum; Dual-face-serpent-type wavemaker; Fluid accelerationwavemaker; Following waves

Corresponding author. Tel.: +82-51-510-2480; fax: +82-51-512-8836.E-mail address: [email protected] (J.C. Park).

0029-8018/$ - see front matter # 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.oceaneng.2003.12.009


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1. Introduction

Ocean waves are generally recognized to have properties of non-one-dimensional

regularity but irregularity composed of multi-dimensional components, and theirenergy distribution can be expressed as a function of wave frequency and wavepropagation direction. Since marine structures and transportation systems areoperated under the real conditions of the sea, our understanding of multi-direc-tional wave characteristics and their effects on structures should be improved at theinitial design stage.

Multi-directional waves are represented as a superposition of various obliquewaves and can be generated in physical or numerical wave basins by a multi-segmented wavemaker on the basis of the so-called snake principle. In otherwords, directional waves can be generated by the spatially sinusoidal motion of an

innitely long segmented wavemaker system. Salter (1981) has adopted a snake-type wavemaker for the generation of the directional waves on the Huygensprinciple in optics.

During the past few years, due to the continuous and substantial increase incomputer power, a number of fully nonlinear numerical wave tanks (NWTs) havebeen developed to reproduce the main scientic features, nonlinear effects and theirnonlinear interactions with the structures observed in physical wave basins. How-ever, fully nonlinear free-surface simulations are still computationally very inten-sive and therefore, a majority of fully nonlinear NWTs are limited to twodimensions (e.g. Clement, 1996; Tanizawa and Naito, 1998 ). To simulate multi-directional steep waves (e.g. Xu and Yue, 1992 ) and their nonlinear interactionswith 3D bodies, 3D NWTs have to be used. Recently, two fully nonlinear 3DNWTs are developed by the authors based on viscous formulations (e.g. Park et al.,1999; Sato et al., 1999).

To simulate steep 3D directional waves in the viscous NWT, a nite-difference/volume method is used based on the NavierStokes (NS) equation and a modiedmarker and cell (MAC) method. The viscous stresses and surface tension areneglected in the dynamic free-surface condition. The kinematic free-surface con-dition is satised by the marker-density function (MDF) technique devised for twouid layers. The method can simulate wave overturning around a three-dimen-sional body, and the simulation can be continued even after wave breaking. Theincident waves are generated by prescribing exible ap-wavemaker motions at theinow boundary and the outow waves are numerically dissipated inside an arti-cial damping zone located at the end of the tank.

Using the developed NWTs, mono-directional and multi-directional randomwaves are generated by prescribing adequate snake-like motions at the inputboundary. Furthermore, the generation of following waves, which are one kind of directional waves is examined using a uid acceleration wavemaker (FAW), andthe hydrodynamic forces acting on an advancing ship in such a wave condition are

to be discussed. It is observed that the present NWT can reliably generate variousnonlinear multi-directional waves and have some possible applications to theirnonlinear interactions with marine structures.

J.C. Park et al. / Ocean Engineering 31 (2004) 154915651550


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2. Numerical method

2.1. Governing equations and nonlinear free-surface conditions

Assuming that the uid is incompressible and homogeneous, the governing equa-tions are given by the following NS equations and continuity equations:

@ u@ t

r pq

a ; 1

r u; 2

where

a u r u m r 2u f : 3

In the above equations, u u; v; w is the velocity vector, p the pressure, t thetime, r the gradient operator, m the kinematic viscosity, f the external force includ-ing the gravitational acceleration, and the density q is assumed to be constant inuid region.

In the time-marching procedure, the solutions of the governing equations in eachregion are obtained separately at each time step. The conguration of the interfaceis determined by applying the fully nonlinear free-surface condition. At the freesurface, the following fully nonlinear kinematic and dynamic conditions can beapplied by neglecting the viscous stress and surface tension:

@ M q@ t

u r M q 0; 4

p 0; 5

where the MDF, M q , takes a value between 0 and 1 all over the computationaldomain. Eq. (4) is calculated at each time step, and the free-surface location isdetermined to be a point where the MDF takes the mean value as

M M q 1

2 : 6

After determining the free-surface location, the two uid regions are treated separ-ately, and the governing Eq. (1) is integrated.

2.2. Algorithm and differencing scheme

The substantial part of the solution algorithm for each layer is similar to the pre-vious NS-MAC NWT method ( Miyata and Park, 1995; Park et al., 1999, 2003 ) inwhich the velocity and pressure points are dened in a staggered manner in a rec-tangular coordinate system. In the time-marching process, the distribution of MDF

is calculated from Eq. (4), and the velocity eld is updated through the NS equa-tion (1). The boundary values of the velocities are then set on a new location of theinterface. The new pressure eld is determined by solving iteratively the following

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Poisson equation in each layer using the Richardson method:

r r P r a un

D t r b; 7P m 1 P m xx r r P r b ; 8a

where P p=q , and xx is the relaxation factor set at a value smaller than unity.The superscript n denotes the time level and m the iteration level. The b term thatincludes convection, diffusion, and other external force terms is called a sourceterm. The NS equation (1) is hyperbolic equations to be solved as an initial-valueproblem, while the Poisson Eq. (7) is an elliptic equation to be solved as a bound-ary-value problem. Eq. (1) is solved by time-marching, and at every time stepEq. (7) is solved by an iterative procedure. The cycle is repeated until the numberof time steps reaches the predetermined value.

The numerical scheme for the convective terms must be carefully chosen since itoften renders decisive inuences on the results. In the time simulation, a ux-splitmethod with variable mesh size, which is like a third-order upwind scheme, isemployed so that a variable mesh system can be used for all three directions. A

second-order-centered scheme is employed for the diffusive terms and the second-order Adams-Bashforth method is used for time integrating.

The dynamic free-surface condition of Eq. (5) is implemented by the so-calledirregular star technique ( Chan and Street, 1970 ) in the solution process of thePoisson equation for pressure. In the free-surface problems, it is very important toextrapolate the physical values onto the free surface. Thus, the pressure on this sur-face is determined by extrapolating from the neighboring uid to the free-surfacelocation. The pressures are extrapolated with zero gradient in approximately a nor-mal direction to the free surface, while the static pressure difference in the verticaldirection due to the gravity is taken into consideration. Similarly, the velocities areextrapolated at the interface with approximately no normal gradient from the uidregion. This treatment accords grossly with the viscous tangential condition at thefree surface.

2.3. Other boundary conditions

At the inow boundary of the 3D rectangular computational domain, a numericalwavemaker is established by prescribing the inow velocities based on the water par-ticle velocities of the linear wave (or Stokes second-order wave), which is like a ex-ible ap wavemaker. For multi-directional wave generation, a snake-like wavemakermotion is used on the basis of linear wavemaker theory ( Dean and Darlymple,



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1991). The velocities at the wavemaker boundary are given by

uv

XN

n1Anx n

cosh k n z h

sinh k nh

cos hn

sin hn cos k nx x k ny y x nt en

;

8b

w XN

n1Anx n

sinh k n z h sinh k nh

sin k nx x k ny y x nt en ; 8cwhere the wave numbers in xy directions are k x k cos h and k y k sin h. Thesymbols A, h, x and e are amplitude, heading, frequency and phase difference of component waves, respectively; and N is the nite number of wave components. The

unknown variables for velocity components at each panel of wavemaker, which areA, h and x in Eq. (8a), can be decided from the directional wave energy spectrum,while e is given in random between 0 and 1. The pressure and the values of MDF areextrapolated with zero-normal-gradient condition in the horizontal direction.

For the sidewall boundary, the diverse boundary conditions can be imposed ex-ibly. For instances, the free-slip rigid wall and the weakly opened boundary con-ditions were tested in Kim et al. (2001), and one side of a dual-face-snake-typewavemaker with the other side of a numerical beach was employed to increase theeffective test area in Kim et al. (2000). Also the numerical beach on both sidewallsmay be available to equip when the sidewall reection affects the simulation resultsfor wave motions. Inside the numerical beach, an articial damping scheme ( Parket al., 1993, 1999) is employed to dissipate the energy of outgoing waves and themesh size is gradually increased in the horizontal direction to provide additionalnumerical damping.

At the bottom boundary of the computational domain, zero-normal-gradientboundary conditions are given for the velocity, and the hydrostatic pressure isgiven assuming that the vertical distances from the interface are sufficiently large incomparison with the wave height of interest. In case the water depth is not deepenough, the free-slip condition for the velocity and zero-normal-gradient condition

for the pressure are used as bottom boundary conditions, while the pressure at theinow boundary is given by the static pressure.

3. Computational results and discussions

To demonstrate the capability of the developed numerical multi-directional wavetank, we present results here for the generation of regular waves, including numeri-cal convergence tests, irregular waves and multi-directional waves. Furthermore,

the generation of following waves, which are one kind of directional waves, isexamined using an FAW, and the hydrodynamic forces acting on an advancingship in such a wave condition are to be discussed.



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respectively, and N t is the number of sampling times per period. For all cases, themean values of wave height and phase difference for 10 periods were taken, andthey seem to converge as the discretized numbers increase. It is evident that the

chosen mesh size and time step have to be small to achieve desirable accuracy. As aresult, at least 50 cells per wavelength, 20 cells per wave height and 1000 samplingtimes per period are needed to simulate a regular wave accurately. These resultsseem to be applicable for the case of wave steepness H =L < 1=20.

3.2. Numerical simulation of irregular waves

Numerical simulation of irregular waves was made by superposing 64 wave com-ponents selected from the BretschneiderMitsuyasu-type wave energy spectrummodied by Goda (1987) for coastal zones, which is expressed as

S f 0:205H 21=3T 41=3 f 5exp 0:75 T 1=3 f 4; 9

where f is the frequency, H 1/3 the signicant wave height and T 1/3 the signicantwave period. The wave amplitude of each component is calculated byA ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2S f df p , where df is an interval of frequency between neighboring compo-nents. In this study, the signicant wave height and period are set at H 1=3 0:05 mand T 1=3 1:33 s, respectively.

Two cases of numerical simulation are tested with coarser (CASE 1) and ner(CASE 2) grid conditions and the conditions are detailed in Table 1 .

Fig. 2 shows the time-series of wave elevation probed at x 3:8 m. The dimen-sions of the NWT are length 10 m, width 1 m and depth 2:8 m. It is appar-ently seen that higher wave frequencies (short crest waves) are simulated better inCASE 2 than in CASE 1.

To quantitatively compare the simulated wave spectra with the target spectrum,a spectral analysis is performed by an FFT algorithm. The results are depicted inFig. 3, which are averaged through ve times analyses with different phase angles.For CASE 2, overall agreement is excellent except for the high-frequency region( f 4 1:5 Hz) where the numerical dissipation of energy is remarkable. Based on theresults of numerical convergence tests discussed in Section 3.1, it seems that thecomputational condition for CASE 2 is sufficient to reproduce irregular wavesaccurately in the frequency region of f 1:5 Hz. On the other hand, for CASE 1,numerical dissipation seems to contribute seriously to the deviation in comparisonwith the target one in the whole region of frequency because of an insufficientnumerical condition.

Table 1Condition of calculation for 2D irregular wave

Case 1 Case 2

h

x L1/3 /50 L1/3 /150h z H 1/3 /20 H 1/3 /20h t T 1/3 /800 T 1/3 /2000



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3.3. Reproduction of multi-directional wave

In general, a directional spectrum is represented by

S f ; h S f G h; f : 10

Here, S ( f ) indicates frequency spectrum and G (h, f ) directional spreading function.In the present study, Eq. (11) is used for the frequency spectrum and theMitsuyasu-type function expressed in Eq. (13) is used for the directional spreading

Fig. 2. Time-history of simulated wave elevation for irregular waves, which are probed at x 3:8 m,using the coarser grid (upper) and the ner grid (below).



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function

G h; f G 0cos2s h hp

2

; 11

G 0 hmax

hmincos2s h=2 dh

1

; 12

s S max f = f p

5 : f f p ;S max f = f p

2:5 : f > f p ;( 13where h is the azimuth measured counterclockwise from the principle wave direc-tion of hp . Also, f p is the peak frequency ( f p T 1=3=1:13), G 0 a constant to normal-ize the directional function and S is the directional wave energy spreadingdetermined by angular spreading parameter S max (Goda and Suzuki, 1975 ).

In order to generate the multi-directional random sea characterized by the abovewave directional spectrum, the dual-faced snake-type wavemaker is equipped onboth its inow boundary (second face) and on one side of its sidewall boundary(rst face), and the numerical beach is installed on their opposite sides.

The computational condition is summarized in Table 2 . The 16 16 wave com-ponents are used for the components of frequency and wave angle. The length,width and depth of the main NWT were 30, 20 and 2.8 m, respectively. The longi-

tudinal and lateral cell sizes were not sufficient compare to the case of irregularwaves described in Section 3.2. The total number of cells used was about one andhalf million cells, including the numerical beach, and about 400 Mb memory was

Fig. 3. Comparison of target and simulated wave spectra for 2D irregular wave.



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occupied. The CPU time for this simulation by t 50 s took around 5.5 days onan PC machine with 1.2 Gb Pentium-III processor and 512 Mb memory.

A 3D snapshot of a numerically reproduced wave conguration at t 50 s isshown in Fig. 4. The real sea random waves consisting of highly short-crest wavesseem to be well simulated and reproduced numerically. However, the waves in thefar eld from the wavemaker are generated inadequately compared to those in thenear eld from the wavemaker, which are due to the numerical dissipation.

To analyze the directional spectrum obtained from the numerical simulation, aset of wave gauges, which are composed of four probes, is arranged at the location

Table 2Condition of calculation for 3D random waves

Cell size h x L1/3 /20h y L1/3 /20h z H 1/3 /20

Time increment h t T 1/3 /2000Wave spectrum H 1/3 0.05 m

T 1/3 1.33 sS max 5hp 0

v

Wave gauge Probe 1 ( x 10:0 m, y 10:0 m)Probe 2 (x 10:0 m, y 9:5 m)Probe 3 (x 10:5 m, y 10:5 m)Probe 4 (x 9:5 m, y 9:5 m)

Fig. 4. 3D snap-shot of wave conguration for multi-directional waves at t 50 s, which are generatedby NWT.



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where the test model is usually placed in the NWT shown in Fig. 5. The detailedlocations of the probes are included in Table 2 .

Fig. 6 shows the time-history of wave elevation collected at a set of wave gauges,and the numerically obtained contour map of directional spectrum using the simu-lated wave proles from Fig. 6 is shown in Fig. 7 with the target spectrum. Foranalyzing the directional spectrum, the extended maximum entropy principle(EMEP) method by Hashimoto et al. (1994) was employed. In Fig. 7, the simulated

Fig. 5. Arrangement of wave gauge array to analyze directional wave spectrum.

Fig. 6. Time-history of wave elevation at four wave probes.



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directional energy contour by the NWT is similar to the target one, which impliesthe possibility of numerical reproduction for random waves inside a 3D-NWT.However, the peak frequency of the NWT is slightly shifted to a high-frequencyregion, which may result in an insufficient numerical condition and lack datapoints used for the spectral analysis.

Fig. 8 shows the comparison of target and simulated directional function at thepeak frequency. The simulated peak value and angle differ a little from the target,but overall agreement is satisfactory considering the coarse condition of simula-

Fig. 7. Comparison of target and simulated directional spectrum in energy contour map.

Fig. 8. Analyzed directional function at peak frequency.



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tion. Some asymmetric features observed in the NWTs results might be due to theeffect of obliquely reected waves from the other side of the wavemaker, as dis-cussed in Hiraishi et al. (1998) .

3.4. Wave generation technique by a uid acceleration wavemaker

One of the most important issues in NWT simulation is how to establish a well-designed wavemaker that can be simultaneously transparent to the reection and/or diffraction waves propagating toward the wavemaker due to the presence of obstacles. Here, a new concept of numerical wave generation technique by Iida(2000), named the uid acceleration wavemaker (FAW), is introduced and appliedto the investigation of nonlinear ship hydrodynamics around a ship advancing infollowing waves (Fig. 9).

Assume the following waves are generated from the downstream of an advanc-ing ship, then the velocity potential, / , at the location of the wavemaker can berepresented by a superposition of three-component waves generated by the uniformow, hull and wavemaker, thus

/ / uflow / ship / wave : 14

The velocity vector u derived from Eq. (14) is then expressed as

u r / r / uflow r / ship r / wave

u

uflow u

ship u

wave

15

If the numerical wavemaker is xed by imposing the dirichlet condition for velo-city in the zone where the ow disturbances and waves exist together, it may act asa rigid wall and generate more reected waves unnecessarily by interacting withthem. This means that it would no longer be allowed the velocity dirichlet

Fig. 9. A snapshot of wave conguration generated by a uid acceleration wavemaker at t 40 s.



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condition for the wavemaker condition in this case. Therefore, for an applicablewavemaker condition, it can be assumed that the acceleration components derivedfrom the velocity potential instead of the velocity components are imposed. Then

each component of the uid acceleration based on wave orbital motion in 2D caseis written as follows:

ax ddt

@ /@ x Ax kU x cosh k z h sinh kz sin k x Ut x t ; 16a

az ddt

@ /@ z Ax kU x sinh k z h sinh kz cos k x Ut x t ; 16b

where U is the advancing speed of the ship.Consequently, the terms for uid acceleration by wave motion in Eq. (16a) and

(16b) would be added as an external force into the source term, a, of NS equationin Eq. (3). On the other hand, the additional numerical beach should be attachedbehind the wavemaker to make it transparent to the reected and/or diffractedwaves.

Fig. 10 shows the time-series of wave elevation probed at x 1L and 2L fromthe start position of the wavemaking zone. It is observed that the wave trains reacha steady state after t 20 s, and the proles maintain uniform shapes after that.As a result, it seems that the new concept of wave generation technique by the

FAW was introduced successfully in this study, and it can be very useful and appli-cable to the various engineering problems dealing with wave motions.

Fig. 10. Time-series of wave elevation probed at two locations in NWT; x 1L and 2L. The waves aregenerated by an FAW.



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3.5. Nonlinear hydrodynamics around a ship advancing in following sea

Now, the FAW technique is applied to investigate the nonlinear hydrodynamic

features around a ship advancing in following waves. The series 60 ( C b 0:6)model was chosen and conditioned that advances the Froude number of Fr 0:2in following waves of L=L ship 2:0 and H =L ship 0:012 .

One more NWT simulation technique named the WISDAM-V motion method(Sato et al., 1999 ) was used for solving the present problem. The governing equa-tions dened in the body-xed coordinates are discretized by a nite-volumemethod in the framework of a boundary-tted coordinate system. The solution of the NS equation is combined with the solution for the equation of motion in thetime-marching procedure. Actually, it can simulate the ship motion with sixdegrees of freedom, but in this study there is no allowance for any degrees of free-dom. The details for motion simulation are referred in Sato et al. (1999) andMiyata and Gotoda (2000) .

A perspective view of the wave conguration around a Series 60 model advanc-ing in the following waves is shown in Fig. 11.

In Fig. 12, the time histories of numerically obtained hydrodynamic forces actingon a ship and wave prole probed at mid-ship are compared with experimentalresults (Iida, 2000) that were conducted in the towing tank at the University of Tokyo. From the gure, it is seen that the agreement between the two results is

Fig. 11. Perspective view of wave conguration around a Series 60 model ( C b 0:6) advancing atF r 0:2 in following waves of L=L ship 2:0 and H =L ship 0:012 .



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quite good. From the comparison of the numerical and experimental results, itseems that the phase of horizontal force F x is same as that of waves, while thephase of vertical force F z is delayed about a quarter of wave period. This meansthat investigating numerically the nonlinear ship motion responses, such as thebroaching motion of a ship in a following and a quartering sea, will be our nextstep in this research eld.

4. Concluding remarks

A nite-difference/volume simulation using a modied marker and cell (MAC)method was applied to investigate the characteristics of nonlinear mono-direc-tional, bi-directional and multi-directional waves inside a three-dimensionalnumerical wave tank (NWT). The NavierStokes (NS) equation was solved bytime-marching, and at every time step, the Poisson equation was solved by an iter-ative procedure to obtain a pressure eld. The fully nonlinear free-surface con-dition was satised by the MDF technique. The directional incident waves weregenerated by prescribing a snake-like motion along the wavemaker, and the out-ow waves downstream were dissipated by using both articial and numericaldamping.

Using the developed NWT, mono-directional, bi-directional and multi-direc-tional random waves were generated by prescribing adequate snake-like motions atthe input boundary. Furthermore, the generation of following waves, which areone kind of directional waves, was examined using a FAW and the hydrodynamicforces acting on an advancing ship in such a wave condition were discussed, show-

ing a good agreement between the numerical and experimental data. It can beconcluded that the viscous NWT can reliably generate various nonlinear multi-directional waves.

Fig. 12. Time-history of wave elevation and hydrodynamic forces; (a) the NWT simulation and (b) theexperiment in the towing tank at the University of Tokyo.



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Acknowledgements

This work was supported by KSGP program of the Ministry of Maritime and

Fisheries, Korea, and by ASERC of the Korea Science and Engineering Foun-dation.The authors thank to Dr. Hashimoto, who is the chief of Hydraulic Lab. at Port

and Airport Research Institute (PARI), Japan, for providing the EMEP programand the discussion on the results.

References

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Miyata, H., Gotoda, K., 2000. Hull design by CAD-CFD simulation. In: 23rd Symposium on NavalHydrodynamics, Val de Reuil, France, pp. 8392.

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