LIU Li-qin,GUO Ying,LIU Chun-yuan,TANG You-gang

(State Key Laboratory of Hydraulic Engineering Simulation and Safety,Tianjin University,Tianjin 300072,China)

Effects of Water Inside Moon Pool on the Heaving Motions of a Spar Platform by CFD Method

LIU Li-qin,GUO Ying,LIU Chun-yuan,TANG You-gang

(State Key Laboratory of Hydraulic Engineering Simulation and Safety,Tianjin University,Tianjin 300072,China)

Abstract:Based on the CFD method,this paper studied the effects of water inside the moon pool on the heaving motions of a Spar platform.The two-dimension and three-dimension numerical wave tanks were established and the stable regular waves were obtained by the numerical wave making.The damping coefficients of main hull of the platform for different opening ratios of the moon pool were calculated.The motion characteristics of the water inside the moon pool were analyzed,and the effects of opening ratios of the moon pool on the platform heaving were investigated.The results show that the damping of platform heaving increases for semi-closed moon pool of the Spar platform.The opening ratios of moon pool will affect the piston motion amplitude of water inside the moon pool and RAO of the platform heaving motions.In practice,an optimal opening ratio of the moon pool can be designed to reduce the heaving motions of the Spar platform.

Key words:Spar platform;heaving;semi-closed moon pool;CFD

0 Introduction

Moon pool is an important structures used in the Spar platform,it goes vertically through the main hull.The risers and the other important drilling equipment are located in the moon pool to prevent excessive wave and current excitations[1].According to the actual requirements,the moon pool is sometimes designed to be semi-opening,which allows water to flow freely in and out of the moon pool.If the top tension risers are used,mass of the water inside moon pool is comparable to the mass of the platform and the influence of the water inside the moon pool to the motion of the platform cannot be ignored[2].

There are two kinds of natural vibration modes of water in the moon pool,that are pistonlike motion along the depth direction of the moon pool and sloshing motion caused by the free liquid surface.Faltinsen et al[3]studied piston-like motion inside a two dimensional moon pool by the domain-decomposition scheme,experimental studies were carried out as well to compare with the analytical solution.Kristiansen and Faltinsen[4-5]investigated the influence of damping on the piston motion considering the nonlinear free surface,flow of the boundary lay-ers and flow separation.They found that the flow separation is the main reason for the discrepancy between the measured results and that estimated by linear theory.Molin[6-7]proposed a fiction boundary condition for the open bottom of the moon pool,and deduced the natural frequencies of water in the moon pool by analytical method.Sun et al[8]established a 3D(threedimension)numerical wave tank based on the Navier-Stokes equations and the VOF method,the coupling motions between the hull and the water motions inside the moon pool were analyzed.Liu et al[9]established the coupled motion equations between heaving of the Spar platform and the water motions in the moon pool,the influences of water motions on the heave motions of the Spar platform were analyzed,the model experiment was carried out to verify the analytical results.

The motions of water inside the moon pool of a Spar platform were studied by the CFD method.The damping coefficients of main hull were calculated.The motion characteristics of the water in the moon pool were analyzed,and the effects of opening ratios of the moon pool on the platform heaving were investigated.

1 The mathematic equations

1.1 Wave making

The two dimensions linear wave is generated first by the flat plate type wave generator,as demonstrated in Fig.1.

Fig.1 The flat plate type wave generator

where h is the depth of the numerical water tank;z is the vertical direction of the water tank;x is the longitudinal direction of the water tank;o is the original point of the tank.For the incompressible fluids,the velocity potential of the whole flow field can be written as:

where φ is the velocity potential function.The boundary conditions are as follows:

where η is the wave surface elevation;g is the acceleration of the gravity.Assuming the flap revolves around the z axis in sinusoidal form,the horizontal displacement of the flap plate at different water depth can be written as:

Suppose the generated waves spread on the x axis direction to infinity,the velocity potential function can be written as:

where Apand C are constants need to be determined,kpis wave number.Substituting Eq.(7)into the free surface boundary conditions Eq.(2)and Eq.(3),the following results can be obtained:

The velocity potential function can be further written as:

According to the boundary condition Eq.(6),Apand Cncan be written as:

where H is wave’s height.Substituting Eq.(11)to Eq.(13)into Eq.(14),we can get the propor-tion of target height H andas follows(Robert G and Robert A,1984):

Eq.(15)can be used to set the boundary condition of the flat plate to obtain the objective wave height.

1.2 The governing equations of the liquid motions

Ignoring the influence of the turbulent fluctuation under Cartesian coordinate system,assuming that the fluid density is constant,the continuous equations of fluid motion and Navier-Stokes equations are as follows:

where ρ is the fluid density;u is the dynamic viscosity coefficient;uiand ujare the time-average velocity component;are the pulsating quantity of the velocity components;p is the time-average stress;Siis the generalized source term of the momentum equation.

1.3 The VOF method

The VOF(Volume of Fluid)multiphase flow model is used to track the free surface.For the region contains of air and water,defining the volume fraction functionthe value ofwill be 1 if the region is full of water,while the value will be 0 if the region is occupied by air.When the value ofis between 0 and 1,the space point is considered as interface of water and air.Therefore,represents the distribution of air and water in the entire flow field.

The open source software OpenFoam is used in the following part to study the water motion in the moon pool of the Spar platform.

2 The parameters of the platform

The analysis is based on the parameters of the Horn Mountain Spar platform.In the original platform,buoyancy tank is used to support the riser and the moon pool is closed[10]in this paper,the top tensioner risers are used,and three different forms of moon pool bottom are considered,namely 0%opening ratio(closed),30%opening ratio and 70%opening ratio,as shown in Fig.2,where the diagonal filling part denotes opening and the cross section of the moon pool is square.

In order to compare with the results of model experiment,a reduced scale of 1:130 is used in the calculation.The influence of the mooring and riser are not considered here.The main structure parameters of the platform are shown in Tab.1 and the platform model is shown in Fig.3.

Fig.2 Opening ratio of the moon pool

Fig.3 Platform model

Tab.1 Platform model parameters

3 The results and analysis

3.1 The numerical tank and wave making

The control domain is a cuboid with 15 m long,2.5 m wide and 3.5 m high in which the depth of the water is 3 m,x axis is along the length direction,y axis is along the height direction and z axis is in the direction of the width.The center of the platform is 9 m from the original point and the starting point of wave dissipation is 10 m from the original point.By recompiling the interDyMFoam solver,the linear damping wave absorption is proposed.

First,the 2D(two-dimension)waves are used to test the parameters of the numerical waves,such as time step,grid sizes and the meshing.The length of the tank is divided into Nx divisions uniformly.In the height direction,the vicinity of the wave surface is divided into Ny divisions and the size of grid is gradually increased from the surface to the bottom,as Fig.4 shown.

膨润土资源开发具有原矿易开采、加工工艺多样、资源综合利用率高的显著特点，而蒙脱石的选矿也有别于传统的选矿工艺，实际选矿工艺仅用在成分简单的少量高纯产品的加工领域，而且主要是湿法的重选分离工艺，大部分膨润土资源加工是结合成分组成特点的各种矿物综合开发利用工艺，单从综合利用率指标考察，目前国内部分先进加工企业已实现100%的膨润土资源综合利用，无废矿外排。

Fig.4 Mesh

For Nx=500 and Ny=15,the wave form when t=40 s for different time step are shown in Fig.5(a).It is shown that when time step changes from 0.001 s to 0.008 s,the wave form is very stable.In the following part,the time step 0.004 s is used to making the numerical waves.For Ny=15 and time step 0.004 s,the wave form when t=40 s for different Nx are shown in Fig.5(b).It is shown that when Nx is changed from 500 to 1 500,the wave form is almost steady.The 750 divisions are divided in the length direction in the following the numerical waves making.

Fig.5 Effect analyzing of parameters on the wave form

The different Ny are also considered.Through a series of calculating Ny=15 is chosen.The 3D(three-dimension)wave making is based on the parameters of the 2D waves.The width ofthe tank is divided into Nz divisions uniformly.The Nz=20 and Nz=40 are calculated,respectively,and the results for t=25 s are compared with the results of 2D waves,as Fig.6 shown.It is found that both Nz=20 and Nz=40 agree well with 2D condition.So Nz=20 is chosen for the 3D waves making.The wave elevations for the 2D waves and 3D waves are compared at the point,that is 9 m away from the original point in the longitudinal direction of the tank,the results are shown in Fig.7.The time histories of wave elevation for 3D also agree well with those of 2D.

Fig.6 Wave form for 2D and 3D with different Nz(t=25 s)

Fig.7 Wave elevation for 2D and 3D

Based on the above testing work,the parameters used to making the numerical waves are determined,that are Nx=750,Ny=15,Nz=20 and time step 0.004 s.In the following part,the control domains,grid divisions and boundary conditions are based on these parameters.

3.2 The influence of opening ratio on the viscous damping

The influences of moon pool’s opening ratio on the viscous damping of the platform are analyzed in this section.Three cases are selected,that are 0%opening ratio(totally closed),30%opening ratio and 70%opening ratio.There are many research work on the damping of heaving plates of the Spar platform[11-12].To save the calculating time,only the cylinder(the main hull of the platform)is considered here.The cylinder is moved as sinusoidal curve in the vertical direction,the damping forces are calculated in the time domain and the damping coefficient are integrated by following formula[13],

where Cdis the damping coefficient;ω is the motion frequency of the structure;ρ is the fluid density;D is the characteristic scale of moon pool of the platform;Umaxis the maximum velocity of the structure;is the damping force calculated by the OpenFoam.

The damping coefficients are calculated for different KC numbers for the three cases,respectively,the results are shown in Fig.8.It is shown that with the increase of KC number,thedamping coefficients of the platform decrease gradually for 30%opening ratio and 70%opening ratio.For 0%opening ratio,it is changed slightly when KC number is more than 0.2.For the same KC number,damping coefficients of the main hull are larger in the cases of 30%opening ratio and 70%opening ratio than those of 0%opening ratio.So,the semi-closed moon pool will afford more heave damping than the total-closed moon pool,and the damping of 30%opening ratio is larger than that of 70%opening ratio here.For KC number is 0.4,the damping coefficient for 30%opening ratio is about 1.85 times of that for 70%opening ratio and is about 4.2 times of that for 0%opening ratio.

Fig.8 Damping coefficients for different opening ratios

3.3 The motion characteristics of the water inside the moon pool

The main hull of the platform is considered here.The 30%opening ratio and 70%opening ratio are calculated to study the influence of opening ratio on the water motions in the moon pool of the platform.Six observation points are set at the water surface,as shown in Fig.9.Where,four points are set at midpoints of the four boundaries of the moon pool,that are A,B,C and D;one point E is set at center of the moon pool;one point F is set outside the platform.The C,D,E and F should be equidistant from the original point o,while the A,B and E should be equidistant from the both length boundaries.The difference of wave elevations between A and B can be considered as the sloshing motions in the x direction of water in the moon pool,the difference of wave elevations between C and D is considered as sloshing motion in the y direction of water in the moon pool,and wave elevation at E is considered as piston motion of water in the moon pool[14].

Fig.9 Observation points

Fig.10 Motions of water in the moon pool for 30%opening ratio

Fig.11 Motions of water in the moon pool for 70%opening ratio

The model moves in the heave direction and the motions in the other directions are limited.For the wave period 1.28 s,which is close to the piston natural period of the water in the moon pool,the piston motions and the sloshing motions of water in the moon pool for 30%opening ratio and 70%opening ratio are shown in Fig.10 and Fig.11.

3.4 The coupled motions between platform heave and water motions in the moon pool

The platform structure including main hull,three heaving plates and soft tank are also considered here.Three cases are selected,that are 0%opening ratio(totally closed),30%opening ratio and 70%opening ratio.The structure is excited by the wave forces made by the wave making and moves in the vertical direction.

The model experiments were used to study the effect of the moon pool water on the heave motions of the truss Spar platform,the testing layout is shown in Fig.12.The experiments were carried out in Tianjin University’s wave tank with size of 137 m long,7 m wide,and 3.5 m deep and is equipped with a flap type wave maker.The wave lengths made by the wave maker are from 2 m to 12 m.The model scale 1:130 was selected taking into consideration the size of the tank and the wave making capacity of the wave maker.The model experiments were explained detailed in the reference[15],the experiment results are used here to compare with the CFD calculation results.

To compare the numerical results with the results of model experiments,different wave periods are selected for the three cases.That are 2.0 s,2.1 s,2.2 s,2.3 s and 2.4 s for 0%opening ratio;2.4 s,2.5 s,2.6 s and 2.7 s for 30%opening ratio and 70%opening ratio.The responses amplitudes calculated by the numerical methods presented in this paper are shown in Fig.12(a),and those tested from the model experiments are shown in Fig.12(b).

Fig.12 Heaving amplitude for different opening ratios

Fig.12 shows that the peak values of heave RAO of the platform reduce for the semiclosed cases,the RAO of 30%opening ratio is smaller than that of 70%opening ratio.The changing trends of RAO agree well between the CFD results and the experimental results.An optimal opening ratio of the moon pool can be designed to reduce the heaving motions of the platform in practice.

The maximal response amplitudes calculated by the CFD are 2.42,2.27 and 2.37 for 0%opening ratio,30%opening ratio and 70%opening ratio,respectively.Those obtained by the experiment are 2.34,1.87 and 2.04,respectively.Comparing with the experiment results,the CFD results are about 3.4%,17.6%and 13.9%more than the experiment results for the three cases,respectively.In the experimental results,the wave periods corresponding to the maximal response amplitudes are 2.3 s,2.5 s and 2.5 s for 0%opening ratio,30%opening ratio and 70%opening ratio,respectively.While in the CFD results,they are 2.1 s,2.6 s and 2.6 s,respectively.

The reasons for the differences of peak values of heave RAO between the CFD results and the experimental results are mainly from three aspects.Firstly,in the model test,the mooring lines and truss structures are included in the model,and the model moves in six degree of freedoms;while in the CFD calculation,the mooring lines and truss structures are not included,and the model moves only in the heave degree of freedom.Secondly,the waves between the two methods are different.For the CFD method,the waves are regular with high precision;while in the model experiment the precision of wave is limited by the wave making for the longer wavelength.Thirdly,the calculation cases are not enough.If more wave periods near the peak value of RAO curves are calculated,the difference may decrease.

Fig.13 Heaving motions of the platform

The time histories of the heaving motions of the platform and corresponding wave elevations for the three cases are shown in Fig.13.It is shown that the heaving responses of the platform is stable.

4 Conclusions

The motions of water inside the moon pool of a Spar platform were studied by the CFD method.The 2-D and 3-D numerical wave tanks were established and the stable regular waves were obtained by the numerical wave making.The damping coefficients of main hull for different opening ratios of the moon pool were calculated.The motion characteristics of water in the moon pool were analyzed,the heaving motion of the platform with different opening ratios of the moon pool were investigated.The results are as follows:

(1)The damping of platform heaving increases for semi-closed moon pool of the Spar platform.Different opening ratio corresponds to different damping force,it is the largest for 30%opening ratio in the three cases of this paper.

(2)The opening ratio of moon pool will affect the piston motion amplitude of water inside the moon pool.Comparing with 30%opening ratio,it increases for 70%opening ratio.

(3)Different opening ratio leads to different RAO of the platform heaving,and it is the smallest at 30%opening in the three cases of this paper.In practice,an optimal opening ratio of the moon pool can be designed to reduce the heaving motions of the platform.

Only the heave motion is considered here.Comparing with model experiment,the RAOs obtained by CFD method are overestimated and the reason for it is explained in section 3.4.

[1]Tang Y G.Ocean engineering structural dynamics[M].Tianjin:Tianjin University Press,2008.

[2]Gupta H,Blevins R,Banon H.Effect of moonpool hydrodynamics on spar heave[C]//Proceeding of 27th International Offshore Mechanical and Arctic Engineeringrg.New York,USA,2008:275-282.

[3]Faltinsen O M,Rognebakke O F,Timokha A N.Two-dimensional resonant piston-like sloshing in a moonpool[J].Journal of Fluid Mechanics,2007,575(3):359-397.

[4]Kristiansen T,Faltinsen O M.A two-dimensional numerical and experimental study of resonant coupled ship and pistonmode motion[J].Applied Ocean Research,2010,32(2):158-176.

[5]Kristiansen T,Faltinsen O M.Gap resonance analyzed by a new domain-decomposition method combining potential and viscous flow DRAFT[J].Applied Ocean Research,2012,34(1):198-208.

[6]Molin B.On the piston mode in moon pools[C]//Proceeding of 14th International Workshop on Water Waves&Floating Bodies.Port Huron,USA,1999:103-106.

[7]Molin B.On the sloshing modes in moon pools,or the dispersion equation for progressive waves in a channel through the ice sheet[C]//Proceeding of 15th International Workshop on Water Waves&Floating Bodies.Caesarea,Israel,2000:1-4.

[8]Sun C W,Yang J M,Lv H N.Numerical investigation on motions of vessel with moonpool in wave conditions[J].Ocean Engineering,2013,31(4):25-33.

[9]Liu L Q,I Atilla,Zhang Y H.Analysis of heave motions of a truss spar platform with semi-closed moon pool[J].Ocean Engineering,2014,94:162-174.

[10]Petter A B.Dynamic response analysis of a truss Spar in waves[D].Doctor thesis.Newcastle:University of Newcastle,2000.

[11]Shen W J,Tang Y G,Liu L Q.Research on the hydrodynamic characteristics of heave plate structure with different form edges of a spar platform[J].China Ocean Engineering,2013,26(1):177-184.

[12]Yang J M,Yang X L,Tian X L.Hydrodynamic characteristics of an oscillating circular disk under steady in-plane current conditions[J].Ocean Engineering,2014,75:53-63.

[13]Robert A,Dean,Robert G,Dalrymple.Water wave mechanics for engineers and scientist[M].New Jersey,USA,1984.

[14]Zha J J.Numerical model of wave generation and absorption based on openfoam and its applications[D].Master thesis.Shanghai:Shanghai Jiaotong University,2011.

[15]Liu L Q,Zhang Y H,Man J S.Model experimental research on the coupled motion of heave and moonpool water of the Spar platform[J].Journal of Harbin Engineering University,2015,36(3):287-291.

基于CFD方法研究月池内部流体对Spar平台垂荡运动的影响

刘利琴，郭 颖，刘春媛，唐友刚
（天津大学 水利工程仿真与安全国家重点实验室，天津300072）

文章采用CFD方法研究了Spar平台月池内部流体对平台垂荡运动的影响。建立了二维和三维的数值水池，并获得了稳定的规则波。针对月池的不对开口率计算了Spar平台的垂荡阻尼系数，分析了月池内部流体的运动特性，并研究了开口率对平台垂荡运动的影响。研究表明，半封闭月池时Spar平台的垂荡阻尼有所增大，月池的开口率影响月池内部流体的运动以及平台垂荡运动的RAOs。在实际中，对月池开口率的优化设计可以适当地降低Spar平台的垂荡运动。

Spar平台；垂荡；半开口月池；CFD

P751

A

国家自然科学基金资助项目（51179125）

刘利琴（1977-），女，天津大学副教授，E-mail：liuliqin@tju.edu.cn；郭 颖（1991-），女，天津大学硕士研究生；刘春媛（1989-），女，天津大学硕士研究生；唐友刚（1952-），男，天津大学教授，博士生导师。

10.3969/j.issn.1007-7294.2017.09.004

Article ID： 1007-7294（2017）09-1086-13

Received date：2017-03-11

Foundation item:Supported by the National Natural Science Foundation of China(Grant No.51179125)

Biography：LIU Li-qin(1977-),female,associate professor,E-mail:liuliqin@tju.edu.cn;GUO Ying(1991-),female,master student.