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CFD simulation of 3-dimensional motion of a ship in waves: application to an advancing ship in regular heading waves

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A new computational fluid dynamics simulation method has been developed for the unsteady motion of a ship advancing in waves. The objective is to evaluate the added resistance and predict the performance of a ship in waves. In this study, a finite
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  J Mar Sci Technol (1999) 4:108–116 CFD simulation of 3-dimensional motion of a ship in waves:application to an advancing ship in regular heading waves Yohei Sato, Hideaki Miyata, and Toru Sato Department of Environmental and Ocean Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan properties of importance are motion characteristics inwaves and maneuvering characteristics. The usefulnessof CFD simulation will be increased when these un-steady characteristics can be predicted with sufficientaccuracy by improved techniques.The objective of this research was to develop a newtechnique to simulate ship motion in waves. The un-steady motion of a ship was treated by Akimoto 5  usinga boundary-fitted coordinate system which moves inaccordance with the ship’s motion, and is applied to asailing simulation of an IACC-class racing yacht. Otherstudies simulate maneuvering motion, 6  and the motionof underwater vehicles with controllable wings. 7  How-ever, a realization of the 3D motion of a ship in waveshas not yet been achieved.The ship motion problem has also been treated by thetechnique of theoretical fluid dynamics, with the pos-tulation that a velocity potential exists. Since the mainpart of a ship’s motion in waves is linear, this approachhas provided a lot of useful information for the predic-tion of ship motion and is a commonly used, establishedmethod. However, it cannot be applied to motion whichincludes nonlinear properties, such as large-amplitudemotion, motions with a wave impact load (slamming),and capsizing.There are two ways to approach the simulation of moving bodies.1.By use of the moving grid method, 5,6,8  where themotion of the ship is represented by the deformationof a grid system fitted to the ship’s surface. The gridsystem is also fitted to the free surface. The disadvan-tage of this method is that it cannot cope with large-amplitude motion.2.Within a fixed coordinate system, the motion of theship is treated as body forces introduced into theexternal force term of the Navier–Stokes equation. 7 In this study, the latter technique is employed inorder to cope with large-amplitude motion. The surface Abstract: A new computational fluid dynamics simulationmethod has been developed for the unsteady motion of a shipadvancing in waves. The objective is to evaluate the addedresistance and predict the performance of a ship in waves. Inthis study, a finite volume method, in the framework of aboundary-fitted grid system, is employed. The motion of theship is solved with six degrees of freedom by using the hydro-dynamic forces and moments obtained from the solution of the simulation method. The marker–density–function methodis employed to calculate the nonlinear free surface. Thismethod is applied to the coupled motion problem of heavingand pitching. Key words: computational fluid dynamics, unsteady motion,motion in waves, density function method Introduction Ship hydrodynamics computations based on NS solverswere initiated in the 1980s, and since then a number of useful codes have been developed. These are theTUMMAC-IV code by Miyata et al., 1  the NICE code byKodama, 2  the WISDAM-V code by Zhu et al., 3  andothers. These codes have been used by ship designers.In the last few years, computational fluid dynamics(CFD) techniques have been incorporated into non-linear optimization procedures for hull configuration. 4 In this way, CFD simulation plays an important role inship design.However, these techniques are for a ship on a steadystraight course in a calm sea. The value obtained ismainly used to minimize the required horsepower of themain engine, or to increase the boat’s speed with thesame output from the engine. Other hydrodynamic  Address correspondence to:  H. MiyataReceived for publication on Nov. 15, 1999; accepted on Nov.18, 1999  of the air–water interface is moved according to theship’s motion, and the nonlinear free-surface conditionis implemented by use of the density-function methodwhich can simulate breaking waves. Design by simulation A motion simulation system was developed by combin-ing the NS solver and the motion solution method for anequation of motion with six degrees of freedom.First, an O–H-type boundary-fitted coordinate sys-tem was generated, and then the NS solver was movedby forces and motions. With these hydrodynamic forcesand moments, the equations of motion are solvedand the translational and rotational accelerations areobtained. The new attitude of the ship is derived, and isthen fed back to the NS solver. This cycle is repeated,and unsteady motion is simulated (Fig. 1).The GMESH code from the Ship Research Institutewas employed for the grid generation. Simulation method Grid system and motion treatment  A boundary-fitted grid system was employed for therepresentation of the hull geometry. An O–H-typegrid system surrounds the hull, as shown in Fig. 2, andthis is not deformed or regenerated by the ship’smotion.The motion of the ship is treated in a fixed coordinatesystem, which means that the grid system also movesaccording to the ship’s motion, as shown in Fig. 3. Thesrcin of the coordinate system is at the center of theship, and the trajectory and attitude of the ship aredetermined in the space-fixed coordinates. Solution method This solution method is for the combined problem of incompressible fluid motion and ship motion. The solu-tion method used for the fluid flow is the finite-volumemethod based on the previously developed WISDAM-V, and the density-function method is used for theimplementation of the free-surface condition.The governing equations for the fluid flow are theNavier–Stokes equation and the equation of continuity.The grid system is fixed to the ship, and then the motionof the ship is expressed by the body force term F  n  w   as   DuDt  puK  w ww  =-—+—+ 1 2 Re (1) Fig. 1. Flow chart of the motion simulation system Fig. 2. Overall view of the O–H grid system Fig. 3. Schematic sketch of the method used. The governingequation is solved in a body-fixed coordinate systemY. Sato et al.: CFD simulation of ship motion109   KF  www  n  = +W (2)   Fur ddt r dV dt   g nwww w  wwwwww w w w  =-¥-¥¥ ( ) -¥- 2 (3)where u w   is the velocity vector of the fluid,  p  is thepressure divided by the density, Re is the Reynoldsnumber, K  w   ¯   is the body force, W w   ¯   is the gravitationvector, w  w   ¯   is the angular velocity vector, r  w   is theposition vector, and V   g w   ¯   is the velocity of the srcin of the body-fixed coordinates.These equations are solved in the MAC-typetime-marching solution procedure, and the results areobtained continuously at each time-level.All vector variables are defined in Cartesian coordi-nates, and the components of velocity and pressure aredefined in the staggered arrangement. Third-orderupstream differencing is used for convective terms,and second-order centered differencing for the otherdiscretization in space. The SOR method is used for thesolution of the pressure. The first-order Euler explicitmethod is used for the time-integration for simplicityand efficiency.The ship’s motion is given by the solution of theequation of motion, with the external forces and mo-ments derived from integration of the pressure and thefrictional force on the hull surface. The time-integrationis also made by the Euler explicit method. Density function method The density function, which is a scalar value of theporosity, is determined at the center of a cell, and isgoverned by the equation   DDtt u r∂r ∂ r  w  =+◊— ( ) = 0 (4)The equation is discretized by the third-orderupstream scheme for space differencing and theAdams–Bashforth scheme for time differencing.The density function takes the value 0.0 in air and 1.0in water, and the free-surface location is determined atthe point where the density function takes the value 0.5:   r rr   freesurfaceairwater  - =+= 205. (5) Boundary conditions and numerical wave-making The boundary conditions for velocity, pressure, anddensity functions are listed in Table 1. The incidentwaves are assumed to be sinusoidal in infinitely deepwater. The wave height z   and the three velocity compo-nents at the space-fixed position (  x ,  y , z ) are given by   xxw  =- ( ) a kxt  cos (6)   Uckekxt V Wckekxt  akzakz =- ( ) ==- ( ) zw zw  cossin0 (7)where c 2   =    g/k , z  a  is the wave amplitude, k  is the wavenumber, w   is the angular velocity, and  g  is the gravita-tional acceleration. The incident waves generated in thespace-fixed coordinates are introduced into the compu-tational domain of the coordinates fixed to the ship’ssurface.In order to avoid unfavorable reflections of the waveson the outer boundaries, the spacing is stretched in theirvicinity. Accuracy evaluation The accuracy of this method is examined by a simula-tion of a ship advancing steadily on a straight course.The wave contours in Wigley’s mathematical modeladvancing at Froude number 0.289 are calculated underthe conditions of Case A in Table 2 and compared withthe experimental results in Figs. 4 and 5. The waveprofile on the hull surface is also compared in Fig. 6.The wave contours in Figs. 4 and 5 show that thecomputed wave height is much smaller than the mea-surements taken far away from the ship. This is mainlydue to the coarse grid spacing in the far field. However,the agreement with the wave profiles seems to be satis-factory in Fig. 6, except that some discrepancy is notedat the stern. Table 1. Boundary conditionsVelocityPressureDensityInflowDirichletDirichletDirichletOutflowExtrapolationExtrapolationExtrapolationSide boundaryExtrapolationExtrapolationExtrapolationFree surfaceExtrapolationPressure of air ( ∫ 0)0.5110Y. Sato et al.: CFD simulation of ship motion  The effect of grid spacing is tested with the two gridconditions listed in Table 2, and the results are com-pared with the wave profile on the hull surface in Fig. 6.It is probable that such small differences in spacing haveonly a slight influence. Application to pitching motion in heading waves Conditions of simulation This method was applied to ships advancing in regularheading waves. The pitching and heaving motions arefree, and other motions are not permitted. Two hulls,the modified Wigley model 9  and the Series 60 ( Cb   =  0.6)model were chosen, and the advance speed of eachmodel was set at Froude numbers 0.20 and 0.24, respec-tively. The conditions of computation are given in Table 2. Conditions of computationWigley modelWigley modelCase ACase BGrid points130 ¥  30 ¥  110130 ¥  40 ¥  130( =  429000)( =  676000)Computational domain L   =  2.5, radius =  0.8 L   =  2.5, radius =  0.8Minimum grid space z  1 2.0 ¥  10 - 3 2.0 ¥  10 - 3 Minimum grid space z  2 5.0 ¥  10 - 4 5.0 ¥  10 - 4 Minimum grid space z  3 9.4 ¥  10 - 3 6.3 ¥  10 - 3 Reynolds number1.0 ¥  10 6 1.0 ¥  10 6 Froude number0.2890.289Time of simulation4.04.0Step number97929640Time for acceleration1.01.0Maximum dt  1.0 ¥  10 - 3 1.0 ¥  10 - 3 CFL0.50.5 z  1 , longitudinal direction of ship; z  2 , radial direction; z  3 , girth directionThe units are nondimensionalized by the length of the ship, the velocity of advance, and thedensity of the water Fig. 4. Computed wave contours of the Wigley model at Fn   = 0.289. The contour interval is 0.001 Fig. 5. Measured wave contours of the Wigley model at Fn   = 0.289. The contour interval is 0.001 Fig. 6. Comparison of wave profiles on the hull surface be-tween experiment, coarse grid (case A), and fine grid (case B)(Wigley model, Fn   =  0.289)Y. Sato et al.: CFD simulation of ship motion111  Table 3, and the grid systems of the modified Wigleymodel and the Series 60 model are shown in Figs. 7 and8, respectively. Results of simulation A comparison is made between computation and mea-surement. The experimental and theoretical data of themodified Wigley model are from the results of DelftUniversity, 10  and the experiments were performed on aSeries 60 model at the University of Tokyo. The prin-cipal particulars of the two ships which were used for thecomputations and the experiments are shown in Table 4.The amplitude and phase difference of the pitchand heave are defined below. The phase differenceis taken as the difference in the wave height at thecenter-of-gravity position with the heave and pitchamplitude.   zzt t  aezae =◊+ ( ) =◊+ ( ) coscos wez qqwez  q  (8)where z  is the heave and z a  is its amplitude, q   isthe pitch and q  a  is its amplitude, w  e  is the encounterfrequency, and e  z z   and e  q  z   are the phase differences. Table 3. Conditions of computationModified Wigley modelSeries 60 ( Cb   =  0.6)Grid points115 ¥  30 ¥  110118 ¥  30 ¥  114Computational domain L   =  2.5, radius =  1.0 L   =  2.5, radius =  0.8Minimum grid space z  1 2.0 ¥  10 - 3 2.0 ¥  10 - 3 Minimum grid space z  2 5.0 ¥  10 - 4 5.0 ¥  10 - 4 Minimum grid space z  3 5.0 ¥  10 - 3 5.0 ¥  10 - 3 Reynolds number1.0 ¥  10 6 1.0 ¥  10 6 Froude number0.240.20Time of simulation15.015.0Time for acceleration1.01.0Start of wave making5.05.0Maximum dt  1.0 ¥  10 - 3 1.0 ¥  10 - 3 CFL0.50.5Amplitude of5.00 ¥  10 - 3 5.56 ¥  10 - 3 incident waves z  a 1.11 ¥  10 - 2 z  1 , longitudinal direction of ship; z  2 , radial direction; z  3 , girth directionThe units are nondimensionalized by the length of the ship, the velocity of the advance, and thedensity of the water Fig. 7. Grid system on the horizontal plane and hull surface of the modified Wigley model Fig. 8. Grid system on the vertical plane and hull surface of the Series 60 ( Cb   =  0.6)112Y. Sato et al.: CFD simulation of ship motion

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