Received：2018 - 05 - 06Supported by: National Natural Science Foundation of China (51879159, 51490675, 11432009, 51579145); Chang Jiang Scholars

Program of China (T2014099); Shanghai Universities Orientalist Special Professor Post Tracking Program(2013022); Shanghai Excellent Academic Leadership Program (17XD1402300); Ministry of Industry and Informa⁃tion Technlogy Numerical Tank Innovation Special VIV/VIM Project (2016-23/09)

Author(s): Sun Chenguang, male, born in 1993, master degree candidate. Research interests: computational ship hydrodynamics,ship manoeuvrability in waves. E-mail: sunchenguang1993@qq.comWang Jianhua, male, born in 1988, assistant professor. Research interests: computational ship hydrodynamics, ship ma⁃noeuvrability in waves. E-mail: tjuwjh@126.comWan Decheng, male, born in 1967, Ph.D., Professor, doctoral supervisor. Research interests: computational ship hydro⁃dynamics, meshless particle method, hull form optimization, floating turbine, fluid-structure interaction, vortex-in⁃duced vibration and motion. E-mail: dcwan@sjtu.edu.cn

*Corresponding author：Wan Decheng

CHINESE JOURNAL OF SHIP RESEARCH，VOL.14，NO.2，APR 2019To cite this article：Sun C G, Wang J H, Wan D C. CFD numerical simulations of stopping maneuver of ship model using overset

grid technology[J/OL]. Chinese Journal of Ship Research, 2019, 14(2). http://www.ship-research.com/EN/Y2019/V14/I2/8.

DOI：10.19693/j.issn.1673-3185. 01281

CFD numerical simulations ofstopping maneuver of ship model

using overset grid technology

Sun Chenguang1，2，3，Wang Jianhua1，2，3，Wan Decheng*1，2，3

1 School of Naval Architecture，Ocean and Civil Engineering，Shanghai Jiao Tong University，Shanghai 200240，China

2 State Key Laboratory of Ocean Engineering，Shanghai Jiao Tong University，Shanghai 200240，China3 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration，Shanghai 200240，ChinaAbstract：［Objectives］ With the development of international shipping, ports and waterways are becomingincreasingly crowded. Study on stopping ability of large ships is crucial to their voyage safety.［Methods］ Thenaoe-FOAM-SJTU solver based on the CFD platform OpenFOAM and the ship 6-DOF motion and multi-level objectwith propeller motion solver module and the overset grid technology are used to numerically simulate the emergencystopping maneuver of KVLCC2 model with propeller. First，the rotation speed of propeller is controlled to achieve thesteady self-propulsion state of the ship model；then the propeller is reversed at a moment speed to achieve theemergency stopping maneuver. By generally solving viscous flow field，the motion state and detailed flow fieldinformation of the ship model during self-propulsion and stopping maneuver are presented，the cause of reverse effectis analyzed，and the numerical prediction results are compared with the test data.［Results］The results indicate thatthe discrepancy between numerical prediction results and test data is within 5% and that it is possible to numericallypredict the reverse and stopping maneuver problems by using the naoe-FOAM-SJTU solver.［Conclusions］ Themethod adopted herein can provide reference for preliminary design and maneuver method of a ship in terms of shipstopping issues.Key words：stopping maneuver；naoe-FOAM-SJTU solver；overset grid technologyCLC number: U661.33

0 Introduction

In recent years, in order to reduce shipping costsand improve transportation efficiency, large ship is adevelopment direction. As ports and waterways arebecoming increasingly crowded, marine accidentsdue to collisions occur frequently. In addition, thelarge ship reduces the motion performance of hull,

and both the steering ability and the velocity adjust⁃ment ability have a certain degree of decline, re⁃sulting in inconvenient operation. In order to ensurethe safety of the ship during navigation, it is neces⁃sary to conduct a more in-depth study of the stop⁃ping ability.

For the stopping maneuver of normal large shipsin general situation, the braking mode is still domi⁃

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CHINESE JOURNAL OF SHIP RESEARCH，VOL.14，NO.2，APR 2019nated by reversing the propeller. When the ship usesreversing propeller in a propelling or stationary state,the ship bow will be deflected to the left or to theright under some lateral force of the stern, which isthe reversing effect [1]. A ship with good stopping abil⁃ity should have a minimum stopping distance(stroke), a minimum yaw amount (lateral distance)and a minimum yaw angle [2] corresponding to its shiplength after the stopping maneuver.

Stopping maneuver is an important means of shipmaneuver to avoiding collisions. In 1993, the Interna⁃tional Maritime Organization (IMO) adopted the In⁃terim Standard for Ship Maneuverability, which setsclear requirements for the stopping ability of ship. Atthe end of 2002, IMO completed the revision andsupplement of the Interim Standard for Ship Maneu⁃verability, including the supplementary requirementsfor stopping ability. Since then, the Standard forShip Maneuverability has been officially promulgat⁃ed [3].

Stopping maneuver using reversing propeller hasalways been an important research topic in the fieldof ship maneuver. Many scholars in China andabroad have conducted some study of the stoppingmaneuver using reversing propeller problem of ship.Based on the restrained ship model test, the tradition⁃al numerical study of stopping maneuver [4] mainlyconstructs the mathematical model of stopping ma⁃neuver, separates the relevant hydrodynamic deriva⁃tive, predicts the hull hydrodynamic force throughnumerical simulation, obtains the derivative value ofhydrodynamic force needed in the stopping maneu⁃ver, and calculates and simulates the motion re⁃sponse of ship by the above parameters. Chislett andSmitt [5] determined the existence of the lateral forcecaused by reverse maneuver through restrained shipmodel test. Liu [6] proposed the regression formula ofthe lateral force (moment) due to reverse maneuver,and the prediction result is basically consistent withthe actual ship results. However, he did not considerthe lateral force (moment) due to reverse maneuverwhen the ship has relatively small ratio of draught tolength and relatively small propeller immersion.Zhao and Gu [7] predicted the ability of stopping ma⁃neuver using reversing propeller of ship in shallowwater and discussed the method of stopping maneu⁃ver. Yan [8] studied the maneuverability of large shipsin shallow waters. Zhang et al. [9] studied the spacingcontrol of navigating ships in narrow waterwaysbased on the research of Fujii et al. [10]. Based on theresult of ship maneuverability research in recent

years, Zhang et al. [11] combined the different methodsof solving model parameters and fluid dynamic deriv⁃atives to give the ship maneuver model within portssuitable for the ship maneuver within ports that con⁃siders the reverse characteristics of right-handed sin⁃gle propeller reversing. All the above-mentionedscholars indirectly obtain the relevant parameters ofthe stopping maneuver through formula calculation,which cannot visually reflect the hull stress and flowfield information during the stopping process andcannot deeply study the mechanism of the phenome⁃non. Therefore, it is impossible to provide a strongrecommendation for the preliminary design of shipand the selection of means of stopping maneuver.

At present, the study of stopping maneuver usingreversing propeller mainly adopts the constrainedmodel parameters and maneuverability mathematicalmodel, which cannot study the wake field and the in⁃teraction between ship and propeller during the stop⁃ping maneuver. Therefore, the use of CFD method tonumerically simulate the stopping maneuver using re⁃versing propeller has important research signifi⁃cance. Solving large motion of ships by overset gridtechnology is currently the mainstream method. Saka⁃moto et al. [12] used the ship hydrodynamic softwareCFD Ship-Iwoa Ver. 4 to numerically simulate thestatic and dynamic tests of the Planar Motion Mecha⁃nism (PMM) on the bare hull of the standard shipmodel DTMB 5512 and conducted some verificationaccordingly. Carrica[13] used the overset grid technolo⁃gy to numerically simulate the rotational motion andZ-type maneuver test of the DTMB 5512 ship modelby solving the unsteady RANS equation. Mofidi et al. [14]used overset grid technology to numerically simulatethe Z-type maneuver test on KCS ship model underthe full-coupling condition of ship, propeller andrudder. The coupling calculation for ship, propellerand rudder adopts a multi-level object motion solvermodule. Yoshimura [15] conducted numerical model⁃ing and calculation of ship maneuver in shallow wa⁃ter, and the results are in good agreement with the ex⁃perimental results. Based on open source CFD soft⁃ware OpenFOAM and overset grid technology, the re⁃search group of Professor Wan from Shanghai JiaoTong University developed a hydrodynamic solvernaoe-FOAM-SJTU[16] for ships and offshore struc⁃tures under large motion conditions, and achievedsome achievements in numerically simulating the mo⁃tion of sea floating structures and standard ship mod⁃el in waves and ship maneuverability. With the solv⁃er, Wang et al. [17] realized the numerical simulation

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of self-propulsion of fully appended ONRT shipmodel in waves. Wu et al. [18] successfully predictedthe open water performance of propeller. Yin et al. [19]simulated the navigational resistance of the VLCCship model under real scale, and the results are ingood agreement with the experimental results. Luo etal. [20] simulated the roll motion of KCS ship type andanalyzed the viscous effect.

This paper will use the naoe-FOAM-SJTU solverbased on the overset grid technology to conduct theCFD numerical simulation of the stopping maneuverusing reversing propeller coordinated by ship andpropeller to predict the parameters such as longitudi⁃nal distance and lateral distance during the shipstopping, and to make detailed analysis of the sur⁃face pressure of the hull and the flow field aroundthe ship, so as to provide reference for the prelimi⁃nary design of ship and the choice of maneuver mode.1 Numerical method

1.1 Flow field calculation

The numerical calculation is solved using thenaoe-FOAM-SJTU solver independently developedby Shanghai Jiao Tong University. The fluid govern⁃ing equation is an unsteady two-phase incompress⁃ible RANS equation:

Ñ ×U = 0 （1）¶ρU¶t

+ Ñ ×[ρ(U - Ug)U ] = -Ñpd - g × xÑρ +

Ñ ×(μeffÑU ) + (ÑU ) × Ñμeff + fσ （2）where x is the position of the grid node; t is thetime; U is the velocity field; Ug is the moving ve⁃locity of the grid; pd is the dynamic pressure, name⁃ly the difference between total pressure and hydro⁃static pressure; ρ is the liquid density; g is thevector of gravity acceleration; μeff is the effective dy⁃namic viscosity coefficient; fσ is the surface tensionterm.

In this paper, the SST k -ω turbulence model isused to achieve the closure of the RANS equation,where k is the turbulent energy of the fluid particleand ω is the characteristic dissipation rate. Such aturbulence model is not affected by the free surface,and it can ensure the accuracy and reliability of thesolution at wall surface. At the same time, the Vol⁃ume of Fluid (VOF) method with artificial compress⁃ible terms is used to treat the free surface [21]. Thetransport equation is defined as

¶α¶t

+ Ñ ×[ρ(U - Ug)α] + Ñ ×[U r (1 - α)α] = 0 （3）

where U r is the velocity field used to compress theinterface; α is the volume fraction of two-phase flu⁃id, which is defined as

ìíî

α = 0 Airα = 1 Water0＜α＜1 Free surface

（4）The RANS equation and the VOF equation in this

paper are all discretized by the finite volume meth⁃od. For the pressure-velocity coupling equation ob⁃tained after the discretization, the Pressure-Implic⁃it-Split-Operator (PISO) algorithm [22] is used for solu⁃tion with loop iteration.1.2 Overset grid technology

The overset grid method means that each compo⁃nent of the object is separately meshed and then em⁃bedded in another set of large grid. After hole cut⁃ting, the grid in non-computational domain is exclud⁃ed from the calculation. The overset region can con⁃vey flow field information by establishing an interpo⁃lation relationship. The overset grid technology al⁃lows for unconstrained relative motion between multi⁃ple independent grids, so it can well handle the rela⁃tive motion of 6-DOF motion.

Based on the open source CFD software Open⁃FOAM platform, the naoe-FOAM-SJTU solver usedin this paper is added with overset grid technologyand multi-level object motion solver module. In theflow field solution, the Suggar++ [23] program is usedto calculate the Domain Connectivity Information(DCI) between the overset grids.2 Calculation model and grid

The calculations in this paper use the KVLCC2ship model with propeller only. The surface model ofthe hull and propeller is shown in Fig. 1. The rele⁃vant parameters of the actual ship and the model areshown in Table 1.

Fig.1 The KVLCC2 ship model

ScaleLength of waterline Lw/mWidth of waterline B/m

Draft d/mPropeller diameter D/mPropeller pitch ratio P/D

Full scale1

320.0058.0020.809.860.721

Model1/110.02.9090.5270.1890.0800.721

Table 1 Main parameters of the ship model

Sun C G, et al. CFD numerical simulations of stopping maneuver of ship model using overset grid technology 3

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CHINESE JOURNAL OF SHIP RESEARCH，VOL.14，NO.2，APR 2019The coupling calculation model of ship and propel⁃

ler are constructed using the overset grid method.The calculation domain is divided by the core ofship, and the size is -1.5L≤X≤3L, -1.5L≤Y≤1.5L and -L≤Z≤0.5L, respectively, where L is theship length and the bow is facing the -X direction.The calculation domain is divided into three parts,including background grid, hull grid and propellergrid. All grids are generated by the mesh tool snappy⁃HexGrid built in OpenFOAM. Due to the long timespan of the stopping maneuver time, a large amountof computing resources are needed, and grids shouldnot be too numerous. The number of backgroundgrids used for calculation is 570 000, the number ofhull grids is 2.68 million, and the number of propel⁃ler grids is 500 000. After the Suggar++ program in⁃terpolates values in several sets of grids, the totalnumber of final calculation grids in total flow field is3.75 million, with the computational grid shown inFig. 2.

3 Calculation results and analysis

3.1 Self-propulsion verification

In order to verify the reliability of the numericalmethod used in this paper and the convergence ofthe used grid, the numerical verification of self-pro⁃pulsion of the ship model is required. In this paper,the naoe-FOAM-SJTU solver is used to adjust thepropeller speed in the calculation process, so thatthe thrust of the propeller can make the ship moveforward at a uniform speed, and the self-propulsionpoint of the ship model is obtained.

According to the test data of experimental tank of

the National Maritime Research Institute (NMRI),the design velocity of the ship is selected to be0.76 m/s (the corresponding actual ship velocity of15.5 kn) for numerical simulation. Under this condi⁃tion, the time history curves of the propeller speed nand the ship velocity V of the model are shown inFig. 3 and Fig. 4, respectively. In the figure, CFDstands for forecast value and EFD stands for experi⁃mental data.

Fig. 3 shows that when the predetermined velocityis reached, the predicted propeller speed is17.02 r/s, only 1.05% smaller than the experimentalvalue (17.2 r/s) of the NMRI tank [24]. It can be seenthat the numerical simulation of the ship modelself-propulsion using the numerical calculationmethod in this paper can accurately predict theself-propulsion point of the ship model, and the nu⁃merical simulation of the stopping maneuver usingthe set of grids is suitable.3.2 Numerical simulation of stopping

maneuver

The numerical simulation process for stopping ma⁃neuver is as follows. Firstly, the PI controller is usedto control the ship model to sail at a velocity of0.490 5 m/s (the corresponding actual ship velocity

（a）Position arrangement of overset grid

Hull grid

Propeller grid

Background grid

（b）Longitudinal profile diagram of gridFig.2 Schematic diagram of grid arrangement

Hull grid

Propeller gridBackground grid

Fig.3 Time history curves of propeller revolutions velocity ofship model

Fig.4 Time history curves of speed of ship model0 1 2 3 4 5 6 7 8 9 10 11 12

t/s

CFDEFD0.800.790.780.770.760.750.740.730.72

V/（m·

s-1 ）

0 1 2 3 4 5 6 7 8 9 10 11 12t/s

2422201816141210

n/（r·s

-1 ）

CFDEFD

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of 10 kn). After reaching the stable self-propulsionstate, the propeller is controlled to begin reversingfor the stopping maneuver, with the reverse speed of10.36 r/s. When the longitudinal velocity of the shipmodel drops to 0, the stopping maneuver is complet⁃ed.3.2.1 Forecast of stopping maneuver parame-

tersThe trajectory and velocity versus time curve of

ship model during the stopping maneuver processare shown in Fig. 5.

The dimensionless parameter is used here, inwhich the dimensionless length is used in the trajec⁃tory diagram of Fig. 5(a), and the dimensionless ve⁃locity and time are used in the velocity variation dia⁃gram of Fig. 5(b). The calculation formulas are re⁃spectively: t' = t/ L g ，V' = V/ gL .

Fig. 5 shows that the longitudinal distance, lateraldistance during the ship stopping and the stoppingtrajectory of the ship are in good agreement with theexperimental values. The stroke during the stoppingis 0.75% more than the experimental value, and thelateral distance during the stopping is 4.29% lessthan the experimental value. The stopping time is al⁃so in good agreement with the test value, while the

velocity change after reversing has a certain errorwith the test value. When the residual velocity ishigh, the velocity of the forecast by numerical meth⁃od is reduced faster. The numerical calculation meth⁃od used in this paper can accurately predict the rele⁃vant parameters of stopping maneuver, thus provid⁃ing a powerful means for stopping ability evaluation.

In order to study the hydrodynamics during thestopping maneuver using reversing propeller, theforce during the hull maneuver process is analyzed.The total longitudinal force F t acting on the hull andthe decomposition of the force are shown in Fig. 6.

Fig. 6 shows that the total longitudinal force F tacting on the hull is fluctuating due to the continu⁃ous rotation of the propeller, and the force tends todecrease as the velocity of ship decreases. The totallongitudinal force is decomposed into the resistanceF s acting on the hull and the propeller force Fp . Itcan be found that the hull resistance curve is moreregular. As the navigation time increases, the hull ve⁃locity gradually decreases, and the resistance actingon hull decreases gradually. When the navigationalvelocity is reduced to 0, the hull resistance is also re⁃duced to 0. The propeller force has no obvious regu⁃larity, and its fluctuation near a certain value is toprovide braking force for the ship model.

（a）The trajectory of stopping maneuver

（b） The variable of speed with timeFig.5 The trajectory and the velocity of the ship model

1.00.90.80.70.60.50.40.30.20.10

Y/L

0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4X/L

0 10 20 30 40 50 60 70 80 90 100 110t ′

CFDEFD

CFDEFD

0.100.090.080.070.060.050.040.030.020.01

0

V′

（a）The total longitudinal force

（b）The components of the longitudinal forceFig.6 The longitudinal force of the ship model

0 10 20 30 40 50 60 70 80 90 100 110t ′

3.02.72.42.11.81.51.20.90.60.30

F t/N

0 10 20 30 40 50 60 70 80 90 100 110t ′

2.01.81.61.41.21.00.80.60.40.20

F/N

FsFp

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CHINESE JOURNAL OF SHIP RESEARCH，VOL.14，NO.2，APR 20193.2.2 Analysis of flow field during stopping

maneuverIn order to analyze the hydrodynamic characteris⁃

tics of the ship movement during the stopping maneu⁃ver, the pressure distribution of the ship stern after10 s of stopping maneuver using reversing propeller(the time for corresponding actual ship is 105 s) isshown in Fig. 7. In the figure, pd is the dynamicpressure, which indicates the difference between thetotal pressure at this point and the undisturbed hy⁃drostatic pressure.

Fig. 7 shows that during the stopping maneuver us⁃ing reversing propeller, a high pressure zone isformed on the right side behind the ship hull, whilethe pressure change on the left side is small, whichis negligible compared to that on the right side. Thepressure difference between the left and right sidesbehind the ship hull leads to a lateral force that push⁃es the stern to the left, which causes the bow to beright-handed. And this is called reverse effect.

In order to study the reason for the high pressurezone on the right side of the stern, the flow fieldaround the stern at this time is studied. The distribu⁃tions of the longitudinal velocity Ux and the verticalvelocity Uz of the fluid around the ship are shown inFigs. 8-10.

Fig. 8 shows that there is a certain size of wake onboth sides of the stern. The longitudinal velocity ofthe fluid near the right high pressure zone is largerthan that of the left side, because the fluid on the leftrear side is discharged to the right front of the propel⁃ler under the action of the reversing propeller, and

the propelled water is blocked by the hull during theforward movement, so that it impacts the right rear ofthe hull to generate pressure.

Fig. 9 shows that under the action of the reversingpropeller, the fluid on the right side of the stern isdischarged upward and is blocked by the hull duringthe upward movement, thereby impacting the lower

Fig.7 The pressure distribution on the aft of the ship

pd /Pa-60 -30 0 30 60 ×103

pd /Pa-60 -30 0 30 60×103 （b）Longitudinal velocity distribution in horizontal section

Fig.8 The speed distribution of X direction

0.500.250-0.25-0.50

Ux /（m·s-1）

pd /Pa-60 -30 0 30 60 ×103

（a）Longitudinal velocity distribution in cross section

pd /Pa-60 -30 0 30 60 ×103

0.500.250-0.25-0.50

Ux /（m·s-1）

（b）Vertical velocity distribution in horizontal sectionFig.9 The speed distribution of Z direction

（a）Vertical velocity distribution in cross section

0.300.150-0.15-0.30

Uz /（m·s-1）

pd /Pa-60 -30 0 30 60 ×103

pd /Pa-60 -30 0 30 60 ×103

0.300.150-0.15-0.30

Uz /（m·s-1）

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right side of the stern to generate a certain pressure.In contrast, the fluid on the left side moves down⁃wards and flows deeper into the water, without signif⁃icantly affecting the pressure distribution on the sur⁃face of the hull. Hence, the pressure difference be⁃tween the left and right sides forms a lateral forcethat pushes the stern to the left.

Fig. 10 clearly reflects the mechanism of the stop⁃ping maneuver using reversing propeller: the revers⁃ing propeller discharges the fluid to the front, andthe reactive force of the fluid acts on the propeller toprovide the braking force for the ship through thepropeller shaft. Looking from the stern to the bow,the reverse propeller rotates counterclockwise,changing the movement of the fluid near the stern.Blocked by the hull when moving forward and up⁃ward, the propelled water on the right interacts withthe hull, generating pressure at the hull of the sternand pushes the stern to the left, causing the bow tobe right-handed and leading to a reverse effect.4 Conclusions

This paper expounds the significance of CFD nu⁃merical simulation of stopping maneuver and intro⁃duces the numerical simulation method of the stop⁃ping maneuver using reversing propeller using thenaoe-FOAM-SJTU solver based on overset grid tech⁃nology. The results show that numerical predictionresults (such as the stopping stroke, lateral distanceduring the ship stopping, stopping time) are in good

agreement with the test results, with the error below5% , indicating that the current numerical method isreliable for numerical prediction of the stopping ma⁃neuver problem.

In addition, in the process from the self-propul⁃sion to the stopping maneuver, detailed flow field in⁃formation, such as the surface pressure distributionof the ship stern and the change of the fluid velocityaround propeller, is given, and the reason for the re⁃verse effect is analyzed. The solver used in this pa⁃per can accurately predict the motion process and hy⁃drodynamic characteristics of the ship during stop⁃ping maneuver using reversing propeller, and canprovide a reliable reference for selecting the evalua⁃tion mode of stopping ability and the digital designand control mode of ship.

In the future, the stopping maneuver in shallowwater will be simulated, and different stopping meth⁃ods will be compared and analyzed. At the sametime, the numerical simulation method of ship ma⁃neuver used in this paper will be more extensivelyverified.References［1］ Hong B G. The principle and technology of ship maneu⁃

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基于重叠网格的船模停船操纵CFD数值模拟

孙晨光 1，2，3，王建华 1，2，3，万德成*1，2，3

1 上海交通大学 船舶海洋与建筑工程学院，上海 2002402 上海交通大学 海洋工程国家重点实验室，上海 2002403 高新船舶与深海开发装备协同创新中心，上海 200240

摘 要：［目的目的］随着国际航运业的迅速发展，港口和航道变得日益拥堵，研究大型船舶的停船性能对于航行安

全至关重要。［方法方法］使用基于开源 CFD软件 OpenFOAM自主开发的 naoe-FOAM-SJTU求解器，以及船舶六自由度运动和带桨多级物体运动求解模块，采用重叠网格技术，对带桨 KVLCC2船模进行紧急停船操纵数值模拟。首先，控制螺旋桨转速，使船模达到稳定自航状态；然后，在某时刻控制螺旋桨倒转，以达到紧急停船操纵

的目的。通过对全粘性流场的整体求解，给出船模自航以及停船操纵过程中的运动状态和细致流场信息，分析

倒车效应产生的原因，并将数值预报结果与相关试验数据进行对比验证。［结果结果］结果显示，数值预报结果与相

关试验数据间误差在 5%以内，证明采用 naoe-FOAM-SJTU求解器对船舶倒车停船操纵问题进行数值预报是可靠的。［结论结论］所采用的方法可针对停船问题为船舶前期设计和操纵方式的选择提供参考。

关键词：停船操纵；naoe-FOAM-SJTU求解器；重叠网格技术

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