Numerical Study on the Hydrodynamic Interaction of Ship-ship Models in Calm Water

ZHOU Guang-li,DONG Wen-cai,XIAO Wen-bin

(Department of Naval Architecture,Naval University of Engineering,Wuhan 430033,China)

0 Introduction

Along with the development of shipping business and increasing of the ship tonnage in recent years,some ship maneuverability problems,such as the encounter situation,underway replenishment work and obstacle-avoidance strategy for multiple ships,have aroused great concerns from researchers.If two ships are advancing in close proximity,the suction force and drifting moment,caused by the interaction effects between ships,exert great difficulty on the ship maneuverability.In the severe cases,this kind of interaction effects would lead to the collision accidents of ships.From the consensuses above,we can see that the hydrodynamic forces and their laws are needed to be study further for the ships’advancing parallel in close proximity.

Early researches on the hydrodynamic interaction of two floating bodies are mainly dominated by the model test and potential flow theory.In the aspect of model test,the overtaking problem of two ships in deep water was explored by Newton[1]and the interaction effect of the advancing ship to a mooring ship was discussed by Remery[2].Based on the slender body theory and boundary element method,the hydrodynamics of two ships in open water and the interaction force of encounter and overtaking problems in finite depth water were probed by Tuck[3],Davis[4]and Zhang[5].

The research on hydrodynamic interaction of two ships has entered into a new stage with the improving of model test technique and the perfection of mathematical models,which is mainly based on the potential flow theory.By the captive model tests,Vantorre[6]obtained the interaction forces and moments of two ships in a variety of conditions.Through asymptotic expansion method in the slender body theory,the heave and longitudinal trim of two ships were explored in the deep and finite-depth open water by Gourlay T et al[7].Additionally,the interaction forces of ships in restricted water were numerically analyzed using the potential flow theory by Zhou et al[8].

The gradual maturing of CFD technique and enhancement of computer performance has made it possible to numerically solve the viscous flows around two ships by applying the RANS equations.By dynamic mesh and sliding interface technique,the influence rules of transversal separation,water depth,advancing velocity and ship length exerted on ship-ship interaction effects were investigated in the process of encounter by Zhang[9].A two-dimension simplified model to obtain ship’s six degrees of freedom was applied to discuss the effects of ship speed,relative distance and ship principal dimensions on the hydrodynamic forces in the overtaking problem for two ships[10].

Currently,the research of two-ship interaction is mainly focused on the cases as two ships are advancing parallel in a stand-on course and being overtaking or overtaken.However,the hydrodynamic laws of two ships with drift angle remain to be further confirmed and the influence of relative location on interaction force is needed to be verified by more effective data from model tests and mathematical models.

On the basis of effective validation of the proposed model,present paper analyzes the comparison of hydrodynamic performance between single ship and two ships.The influence of drift angle and relative position on the ship resistance,lateral force and yaw moment for two ships advancing parallel in clam water are also discussed.It is expected that present work contributes to provide some valuable insight to the multi-ship encounter and replenishment problems.

1 Mathematical model and boundary conditions

1.1 Governing equations and turbulence model

The Reynolds averaged N-S equations,namely RANS equations,are the governing equa-tions to represent the kinematics and dynamics of viscous fluid,which are the kernel to calculate the 3-D viscous flow around ships in present paper.The specific form of RANS equations are shown as follows:

where ρ,μ,p and fiare the fluid density,viscosity coefficient,hydrostatic pressure and mass force per unit mass respectively,uiand ujare the fluid velocity components.

SST k-ω model is chosen as the turbulence model in present paper,whose detailed derivation process and parameter selection can be seen in Ref.[11].The mathematical expressions of SST k-ω model are written as Eqs.(2)and(3).

where Γκand Γωrepresent the effective diffusion coefficients of κ and ω,Gκis the turbulence kinetic energy caused by the average flow velocity gradient,Gωindicates the generation of dissipation ω of special turbulence kinetic energy,Yκand Yωare the dissipative coefficients of κ and ω,Sκand Sωare the user-defined source terms.

The pressure equation is numerically discretized by the standard discrete format.And the second upwind scheme is applied to the discretization for the momentum equation,turbulent flow equation and transport equations of Renolds stressed.For the pressure-velocity coupling problem,SIMPLEC algorithm is adopted numerically.

1.2 Objects and coordinate definition

The objects to study are two displacement ships illustrated in Fig.1 and their main dimensions are listed in Tab.1.In terms of coordinate system,Fig.2 gives the definition of position,drift angle,force and moment for two ships advancing in close proximity,in which dx and dy are longitudinal and transversal separation.And these for single ship are defined identically.

Tab.1 Main dimensions of Ship-A and Ship-B

Fig.1 3-D models of Ship-A and Ship-B

1.3 Boundary conditions and mesh discretization

The computational domain of ship-ship models is presented as Fig.3 and the settings of boundary conditions are defined as follows.

(1)Strictly speaking,the whole flow field should be an unbounded domain.For the sake of simplicity,the front surface of the domain is 1.0LAahead of the front ship model,where LAis the length of Ship-A.The length of bottom,left and right surface to the adjacent ship’s gravity center are all the value of LA.A velocity-inlet condition is imposed on all the boundary surfaces above.In particular,the oblique flow is obtained by the change of inflow direction.

Fig.2 Coordinate definition for two ships

Fig.3 Computational domain of ship-ship models

(2)The outlet surface,set as the pressure-outlet condition,is 3.0LAaway from the back ship model.The initial pressure is the undisturbed boundary pressure.

(3)The upper surface of the computational flow is no-slip and impenetrable type,which can be treated as the symmetry boundary condition.

(4)The hull condition is assumed as the no-slip wall assumption.

The flow domain and boundary conditions of single-ship model are the same as that of ship-ship models.The total number of computed cells is about 1.97 million for single-ship model and 4.07 million for ship-ship models.For advancing speed of the ships is relatively low,the free water surface condition is not considered.

1.4 Definition of notation and non-dimensional parameters

The non-dimensional coefficients,such as R′,Y′and N′,are presented below

where R,Y and N are the resistance,lateral force and yaw moment of the ship,respectively,V is the inflow velocity and L is the ship length.

The longitudinal and transversal separations between two ships are transformed to the dimensionless ones as follows:

where LA,LB,BA,BB,dx and dy are length of model A,length of model B,breadth of model A,breadth of model B,longitudinal and transversal separation between two ship models,respectively.

2 Calculation results and analysis

2.1 Validation

To verify the effectiveness of presented model,the calculated results are compared with the tested values of ship resistance and lateral force coefficients for the Ship-A,with the speed of 1.021 m/s and drift angle range of-6°～6°.The case of single ship is illustrated in Fig.4 and Fig.5,and two ships with εdx=0 and εdy=2.784 in Fig.6 and Fig.7.Comparative analysis shows that the calculated resistance coefficient is a little less than the experimental data with a margin of error of 6 percent.However,the calculation and test results of lateral force coefficient are in good agreement.Therefore,the proposed model is feasible to quantitatively predict the mechanic regularity for single ship and two ships advancing in close proximity.

Fig.4 Comparison of resistance coefficient for Ship-A advancing individually

Fig.5 Comparison of lateral force coefficient for Ship-A advancing individually

Fig.6 Comparison of resistance coefficient for Ship-A advancing with Ship-B in close proximity

Fig.7 Comparison of lateral force coefficient for Ship-A advancing with Ship-B in close proximity

2.2 Comparison between advancing single ship and two ships

In order to acquire the hydrodynamic difference between advancing single ship and two ships,the resistance,lateral force and yaw moment of Ship-A are compared between the single-ship and two-ship cases,whose speed is 1.021 m/s.If Ship-A and Ship-B are advancing parallelly,εdx is set as 0 and εdy as 2.784.In addition,the flow field around two ships is analyzed especially.

2.2.1 Ship resistance

As seen in Fig.4 and Fig.6,there is some difference in quantity for the ship resistance values in individual and parallel advancing cases.To analyze the resistance components intuitively and deeply,Fig.8 illustrates the frictional and pressure resistances for single ship and two ships with drift angle.If two ships are advancing parallelly,the frictional resistance coefficient Cfof Ship-A is lager by about 4 percent than that with no Ship-B around.On the other hand,the pressure resistance coefficient Cpvaries relatively large over the drift angle.For the Ship-A advancing with Ship-B,the pressure resistance of Ship-A,located in the back-flow side,is smaller than the individual advancing case but it is opposite when the Ship-A is located in the up-flow side.At different drift angles,the direction of force caused by the low pressure zone between two-ship sides is not identical.The relative position between the low pressure zone and ship model leads to the load difference when Ship-A located in the up-flow and back-flow sides.

Fig.8 Comparison of frictional and pressure resistance coefficient for Ship-A at different drift angles

2.2.2 Lateral force

By the comparison of lateral force coefficient at the drift angle range of-6°～6°in Fig.9,it is very significant that the Ship-A is acted on an additional lateral force when advancing with Ship-B,whose direction is pointing to Ship-B.This additional lateral force,mainly caused by the low pressure zone between two-ship sides,is a kind of suction and its magnitude for Ship-A is about 14 percent of its resistance with β=0°.In the stand-on course,the pressure values of 15 points in Fig.10 are presented as Fig.11 for the individual ship and two ships cases.As seen in Fig.11,the pressure between two ships’inner sides is characterized by a significant low-pressure zone.

Fig.9 Comparison of lateral force coefficient for Ship-A at different drift angles

Fig.10 Uniform distribution of extracted points in the flow field

Fig.11 Comparison of the pressure values at zero drift angle for individual ship and two ships cases

Fig.12 Comparison of the yaw moment coefficient on Ship-A at different drift angle

2.3.3 Yaw moment

If two ships are advancing parallelly in close proximity,the yaw moment acted on the ship is different from that of individual ship case.In Fig.12,the yaw moment of Ship-A is presented with the drift angle.By comparison,the absolute value of yaw moment acted on the upflow side of Ship-A with Ship-B nearby is less than that of individual ship case.However,the moment on the back-flow side has no great difference at the two cases.From the pressure contour in Fig.13,there is a great difference if Ship-A is located at up-flow side,namely with a drift angle of 6°.If the drift angle is down to-6°,the pressure distribution of Ship-A is similar at the two cases.

Fig.13 Comparison of pressure contour on Ship-A at different drift angle

2.3 Effects of drift angle on forces

If two ships are advancing obliquely,great differences of ship resistance,lateral force and yaw moment are existing at the up-flow and back-flow sides.For the convenience of analysis,the force characteristics at two flow sides are illustrated with example in the case of ship speed 1.021 m/s,longitudinal separation εdx=0 and transversal separation εdy=2.784.

2.3.1 Ship resistance

From the resistance on Ship-A and Ship-B at drift angle of ±3°and ±6°shown in Fig.14,the force value at the back-flow side is 5%～10%less than that at up-flow side,which is due to the different force direction of the low pressure zone acted on the sides of two ships.From the Figs.15-17,the distribution form of low pressure zone is related to the drift angle of navigation.If the drift angle is negative,the component of the additional force in the XEdirection,caused by the low pressure zone,is positive for Ship-A,but is negative for Ship-B.However,the regularity above is exactly opposite if the drift angle is positive.

2.3.2 Lateral force

For two ships moving with a drift angle,the force direction will not change although there is a low pressure zone.In order to compare the force difference on the up-flow and back-flow sides intuitively,Fig.18 illustrates the absolute value of lateral force coefficients.As seen in Fig.18,the lateral force on up-flow side is larger than that on back-flow side.Within the range of small drift angle,the absolute value of lateral force difference between the two-flow sides,which is almost the same for Ship-A and Ship-B,varies little with the drift angle.

Fig.14 Comparison of resistance coefficient at different drift angles

Fig.15 Pressure distribution at the inner sides of two ships with-6°drift angle

Fig.16 Pressure distribution at the inner sides of two ships with 0°drift angle

Fig.17 Pressure distribution at the inner sides of two ships with 6°drift angle

Fig.18 Comparison of lateral force coefficien t at different drift angles

Fig.19 Comparison of yaw moment coefficient at different drift angles

2.3.3 Yaw moment

Similarly to the lateral force,the direction of yaw moment at two ships advancing case is the same as that at individual ship sailing case.Because of the low pressure zone at two-ship sides,the yaw moment on each ship performs differently in magnitude.The yaw moment coefficient is presented in Fig.19 at different drift angle.In the small range of drift angle,the absolute value of yaw moment coefficient difference changes little with drift angle,that is to say a relatively fixed difference between the up-flow and back-flow sides is in present for each ship at different drift angle.Due to the different pressure distribution,which is illustrated in Fig.20 and Fig.21,the difference of Ship-B is larger than that of Ship-A.

Fig.20 Pressure contour of two ships at 6°drift angle

Fig.21 Pressure contour of two ships at-6°drift angle

2.4 Effects of relative location on forces

To probe into the force rule by the relative location of two ships,the resistance,lateral force and yaw moment coefficients of Ship-A are achieved at four transversal separations and nine longitudinal separations,which are εdy=2.189,2.486,2.784,3.079 and εdx=-1.05,-0.79,-0.52,-0.26,0,0.26,0.52,0.79,1.05.From the computed force results at different shipship locations in Figs.22-24,the transversal and longitudinal separations pose a great impact to the ship loads,which indicates that reasonable location between two ships is a good way to avoid the risk of collision accident.

Fig.22 Resistance coefficient of Ship-A at different transversal and longitudinal separations

Fig.23 Lateral force coefficient of Ship-A at different transversal and longitudinal separations

2.4.1 Longitudinal separation

As seen in Figs.22-24,the changing law of Ship-A force with the longitudinal separation is basically the same at different transversal separations.If the longitudinal separation is long enough,the ship-ship interaction effect is less and the hydrodynamic force at two-ship case tends to be a stable value.If Ship-A is slowly overtaking the Ship-B,that is the process from εdx=-1.05 to εdx=1.05,the Ship-A’s resistance decreases firstly and then increases and reduces finally to a value that is a little larger than that of single sailing case.The lateral force is behaved as repulsive,attractive and repulsive force with the dimensionless longitudinal separation varying from-1.05 to 1.05 and the suction force reaches its peak at εdx=0,If εdx＜0,that is to say,Ship-A is located back of Ship-B,the absolute value of yaw moment for Ship-A increases firstly and then decreases with the value εdx varying from-1.05 to 0,which would make the Ship-A’s heading course swerve to Ship-B;If Ship-A is ahead of Ship-B,the direction of Ship-A’s yaw moment is outward to the fluids between the ship sides and its value augments with the increasing longitudinal separation and then decreases to 0 if εdx＞1.05.

Fig.24 Yaw moment coefficient of Ship-A at different transversal and longitudinal separations

2.4.2 Transversal separation

According to the comparison of calculated results at different transversal separation,the force law of Ship-A can be summarized as follows:

(1)εdx=0 case.Due to the variation of flow field between two-ship inner sides,Ship-A’s resistance increases slightly with the decreasing transversal separation,which is all larger than that of single ship advancing situation.If two ships get closer,lateral force of Ship-A shows an evident uptrend,whose value is about 25 percent of ship resistance at εdy=2.189.Similarly,the yaw moment of Ship-A increases with the decreasing transversal separation and its value at εdy=2.189 is about twice as that at εdy=3.079.

(2)εdx=-0.52 case with Ship-A backward.At this case,the resistance of Ship-A decreases as transversal separation gets smaller but the lateral force and the absolute value of yaw moment increase.The lateral force is repulsive and the direction of yaw moment is inward to the ship-ship inner sides.

(3)εdx=-0.26 case with Ship-A backward.How the Ship-A resistance and yaw moment change with the transversal separation is the same to εdx=-0.52 case.However,the lateral force is attractive and its value is getting larger as transversal separation decreases.

(4)εdx=0.26 case with Ship-A forward.The lateral force of Ship-A is attractive and the yaw moment can make Ship-A swerve outward.The values of resistance,yaw moment and absolute value of lateral force of Ship-A is increasing as transversal separation decreases.

(5)εdx=0.52 case with Ship-A forward.How the Ship-A resistance and yaw moment change with the transversal separation is the same to εdx=0.26 case.However,the lateral force is repulsive and its value is getting larger as transversal separation decreases.

3 Conclusions

On the basis of the validation of proposed computation model,hydrodynamic comparison between the single ship and two ships advancing cases is carried out in present paper.Combined with the flow field information,the influence of ship resistance,lateral force and yaw moment on drift angle and relative location of two ships is discussed.The main conclusions in this paper are as follows:

(1)At low speed Froude number in calm water,the single-ship and ship-ship models without considering the free water surface can make good prediction for the hydrodynamic force at the single ship and two ships advancing cases.

(2)If the longitudinal separation between two ships in the stand-on course is zero,the ship resistance is about 5%～6%larger than that at single ship moving case.Moreover,the lateral force is performed as suction obviously and two ships are exerted on a yaw moment which is outward to the fluids between the ship sides.

(3)For the two ship models advancing parallel with opposite drift angles,there is a certain difference of the hydrodynamic pressure distributions when the ship located in the upflow and back-flow sides.

(4)If two ships are advancing with a drift angle,the resistance and lateral force at the up-flow side of ship surface are larger than those at the back-flow side but the absolute value of yaw moment is less.At the range of low drift angle,the difference value between backflow and up-flow side has no big change with drift angle for the lateral force and the absolute value of yaw moment,and this value is almost the same for the lateral force of Ship-A and Ship-B.

(5)In the stand-on course for two ships in close proximity,the relative location poses a significant influence on the hydrodynamic force.At the identical transversal separation,the force regularity by the longitudinal separation is fundamentally the same.At the identical longitudinal separation,the hydrodynamic interaction between two ships is enhanced if the transversal separation decreases.

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