Research on the Mechanism and Characteristic of Added Resistance of Moonpool with Recess

LÜ Hong-gun,GU Ji-yng,,LIU Wei-min,TAO Yn-wu,LI Wen-jun

(a.Marine Equipment and Technology Institute;b.School of Naval Architecture and Marine Engineering,Jiangsu University of Science and Technology,Zhenjiang 212003,China)

Abstract:Moonpool,the common structure is employed widely on drilling ships and drilling platforms,but it could seriously handicap the powering performance of these structures.The present research payed attention to,mainly based on related theoretical analyses and numerical simulations,the mechanisms as well as characteristics of added resistance of moonpool for three variables including Froude number(Fn),dimensionless draft(DD)and ratio of length to width of moonpool(RoLW).The mechanism of three stages so-called generation,transportation and action stage of vortex was discussed in detail.The results show that the high Fn can magnify the added resistance because of more serious situation of vortex movement,and that the DD of moonpool plays an important role in the increasing of added resistance with its value being the range of 0.75 to 1.15,and that the added resistance can be obviously enlarged when the RoLW is in a range of less than 3.5.

Key words:drillship;moonpool;added resistance;boundary layer theory;vortex shedding

0 Introduction

Moonpool,a structure installed on the hull of a drilling ship from keel to deck,has great negative effect to the hydrodynamic performance and powering performance of a drilling ship.Approaches to research a moonpool can be divided into two types(Fig.1).One way is in incident wave with motionless rigid body,which attracts the attention of researcher on the behaviors of water column by,mainly,using potential flow theory.Another way is in still water with forward speed,which yields the phenomenon of so-called vortex shedding.As is confirmed by the first way,there are two dominant types of water motions,namely,piston motion and sloshing motion,occurring in a moonpool all the time[1-8].

As is shown in Fig.2,piston motion appears along the vertical direction of a moonpool,while sloshing motion,which is totally different from piston motion,appears along horizontal direction of a moonpool from one side to another side.The most significant achievement of the first way is the natural mode of water motions which were obtained with the linearized potential flow theory and,in some particular situations,the approximate formulas can be deduced in simple form[1-3,6].For sloshing motion or piston motion,they mainly affect the hydrodynamic performance of a drilling ship.There is another phenomenon,however,so-called vortex shedding in a moonpool,which only can be observed by numerical simulations and flow visualization technique.This phenomenon,compared with the sloshing motion and piston motion,mainly affects the powering performance of a drilling ship.The shedding vortex will change the pressure field in a moonpool,thus changing the longitudinal pressure gradient of the hull and,from another viewpoint,resulting the increase of energy consumption(added resistance).

Fig.1 Two types of excitation in a moonpool

Fig.2 Piston motion and sloshing motion in a moonpool

It is hardly to study the added resistance by pure theoretical methods since added resistance is a complex multi-factor coupling phenomenon,especially in the case of strong nonlinear viscous effect.So the physical experiment and numerical simulation can be more reasonable approaches to this problem.Gaillarde and Cotteleer[9]attempted to reveal the mechanisms of water motion in a moonpool with combination of empirical and theoretical approach.They analyzed the reason of flow separation at leading edge and discussed the phase-locking phenomenon in a moonpool.Furthermore,they analyzed and discussed some devices which can be arranged on a moonpool to reduce the water motion.Veer and Tholen[10]conducted a series of model tests of the added resistance of some piston-oscillation-type moonpools in calm water based on some previous experimental data of varying moonpool shapes at Delft University of Technology.The results of their experiments show that the added resistance from resonant moving water is found to be very significant.Resistance increases in the order of 30%at low or moderate speed to as large as 100%at high forward speed.But one of obvious limitations of their work is that the experiment results are referable only within the scope of piston-oscillation-type moonpool.

Numerical simulations also can be carried out widely to simulate the fluid behavior in a moonpool[11-15].It can be obviously observed that strong vortex shedding exists in a moonpool and thus changes the pressure gradient of the drilling ship.It must be realized,however,that vortex shedding in a moonpool is a violent non-linear phenomenon in time as well as in space,so unsteady-and three-dimensional-based codes are needed to fulfill numerical simulation.Hammargren[16]and Heo[17]et al,based on two-dimensional numerical model,respectively studied the effect of a moonpool on the total resistance of a drilling ship and the influences of vortex to vertical motion of a moonpool under forced heave motion.Dimensional nonlinear effect is neglected in their work,which may lead to some subtle different results such as the structure of vortex,the resistance of drilling ship and the behaviors of streamline in the flow field of a moonpool.

The present research,based on related theoretical analyses and numerical simulations,aims to illustrate the mechanism of the added resistance of a drilling ship induced by the moonpool structure.In Chap.1,the natural frequency of a moonpool was estimated by using the Newman formulas and was compared with the Molin formulas.In Chap.2,the evolutionary process of vortex was divided into three stages so-called generation stage,transportation stage and action stage.Moreover,the boundary layer control equations of the bottom with leading edge of a moonpol were deduced and the mechanism of the yielding of added resistance was demonstrated in detail with boundary layer theory and vortex shedding theory.Furthermore,in Chap.3,the main parameters which affect the added resistance of a drillship including Fn,DD of moonpool and RoLW of moonpool were discussed.Finally,some recommend parameters were given to design a moonpool with recess in practical engineering.

1 Natural mode of moonpool

As shown in Fig.3,the main parameters of the drillship and its moonpool structure with recess were illustrated.The Ltis the overall length of the moonpool,with being divided into two part named Lland Ls,which means respectively the length of deep part of the moonpool and the length of shallow part of the moonpool.Htis the total height of the moonpool and Hris the height of the recess.The final parameter B means the total width of the moonpool.Tab.1 shows the basic geometry information for the drillship.

Molin(2001)developed two expressions for rectangle-shape moonpool with assuming the beam and length of a barge to be infinite and the barge to be motionless.

The piston mode can be written as follows：

Fig.3 The geometry parameters of the drillship and its moonpool with recess

Tab.1 The geometry information of the drillship and its moonpool

where

The sloshing mode can be written as follows：

where

Jn0can be expressed as

where r=b/l and tanθ0=1/r.

Newman(2018)also developed an approximate equation based on the experimental re-sults carried out by Guo et al(2017).Standing-wave approximations was used and the sloshing mode equation was written as follows：

where

Piston mode equation,however,according to the conclusions from Newman(2018),was not recommended since it is better to use radiation-diffraction theory to analysis.

Fig.4 Sloshing natural frequency for varying RoLW and recess depthsFig.4(a)shows the sloshing natural frequency for varying RoLW and depths of the recess.

It can be clearly seen from Fig.4(a)that the sloshing natural frequency calculated by the Molin formula is significantly lower than,especially at n=2,by the Newman formula because the Molin formula does not consider the shallow water effect induced by the existence of recess.Molin(2017)pointed out that Guo et al(2017)mistakenly applied his formulas to calculate the natural frequency of the moonpool they studied,because the results of the Molin formula are only suitable for the moonpool without recess.Whether the moonpool has the greatest impact of recess is the so-called shallow water effect.Newman pointed out that the non-linear shallow water effect in the upper region of recess is particularly obvious,and it is critical to consider when study such problems.It also can be seen from Fig.4(a)that the first-order sloshing natural frequency calculated by Molin formula and Newman formula is relatively close and the reason of this phenomenon may be that the non-linear effect has little effect on the firstorder frequency.There are significant differences,however,in the second-order sloshing frequency,which means that the phenomenon of shallow water effect is active in this order.The curve in Fig.4(b)was extracted by the Newman formula,and it can be clearly seen from Fig.4(b)that the sloshing natural frequency increases following the recess depth.When the recess depth is large enough to a range,the sloshing natural frequency tends to be a certain value,which is due to the reduction of the shallow water effect caused by the increase of recess depth.The common recess depth range in practical engineering is about 3～6 m,and the cor-responding first to third order natural frequencies are about 0.734 6-0.769 2,1.090 5-1.144 8,1.377 0-1.434 4 with respect to the studied moonpool.

2 CFD calculation model and mesh verification

Fig.5 shows the mesh of the numerical model,the red mesh is the transverss slice mesh,and the blue mesh is the horizontal slice mesh,and the green mesh is the longitudinal slice mesh.Prism mesh is carried out to recover all of wall boundary,with the y+being 30 to 150.As is shown in Fig.6,in order to monitor the wave head in the moonpool,six virtual wave height gauges were installed in six different typical locations,which can obtain the time history curve of the wave elevations.The coefficients of piston,sloshing,and added resistance at this work were defined,respectively,as follows：

Fig.5 Grid construction of the numerical model

where Wi（i=A,B,C,D,E ）are the wave gauges shown in Fig.6 and d is the draft.

Fig.6 The setting of virtual wavehead gauges in the moonpool

Fig.7 Mesh independence validation

where Ftis the total added resistance force;Smis the area of the moonpool below free-surface;V is the speed of the drillship.

Five sets of grids,included extreme coarse,very coarse,coarse,moderate,fine and very fine,are used to verify the mesh independence.As is shown in Fig.7,the 3M was chosen because the value of total resistance coefficient of the drillship performs little changes in the range of 3M to 5M.

3 The mechanism of added resistance of a moonpool

The added resistance of a moonpool can be divided into two parts,namely,added frictional resistance and added residual resistance.The added frictional resistance can be estimate by the Newton's shear law,while the added residual resistance is more complex because of strong non-linear phenomenon.The phenomenon,so-called boundary layer separation,is the most essential cause of the added residual resistance of a moonpool.In general,the forming process of the added residual resistance can be divided into three stages,namely,the generation stage,the transportation stage and the action stage.The generation stage yields,in certain frequency relating to the geometry of moonpool,lots of vortex continuously.The vortex generated in the first stage was transported in different direction at transportation stage.The third stage,in a sense,is the direct cause to added viscous-pressure resistance of a moonpool.The specific process and the significant phenomenon of all stages are discussed particularly in sections 3.1 to 3.3.

3.1 Generation stage

In order to deduce the relation between velocity and pressure in a boundary layer,as shown in Fig.8(a),a two-dimensional slice of the moonpool was carried out.The momentum equation of the two-dimensional turbulent boundary layer at the bottom of the drillship can be expressed as follows：

In the inner of the boundary layer,the velocity and distance of the x-y coordinate can be,obviously,written as follows：

According to the Eqs.(15)and(16),the momentum equation in turbulent boundary layer can be rewritten as follows by applying the dimensional analysis in Eqs.(13)and(14)：

Fig.8 The phnomenon of boundary layer separation

Fig.9 The process of boundary layer separation at the leading edge of a moonpool

According to the analysis above,the bottom of the drillship can be treated as a flat plate so the gradient of pressure ∂p/∂x of the boundary layer at the surface is less than zero.The structure of a moonpool,however,abruptly changes at the leading edge thus the velocity becomes discontinuous and vortex shedding will persist at a certain frequency.This situation is the so-called generation stage.

3.2 Transportation stage

The vortex yielded in the generation stage is transported to the rear wall of moonpool by following the inflow direction.During the process of transportation,the vortex can be divided into two types.As shown in the Fig.10,the part marked by the solid circle is the vortex that directly enters the middle of the moonpool and accumulates gradually,which leads to the forming of a large vortex.

Fig.10 The process of the transportation stage

The second type of vortex,marked by the dash circle,continues to be transported to the rear wall of moonpool and then the phenomenon of second separation occurs(Fig.10(b)).This part of vortex also can be divided into two types after the second separation.One,as is shown in the Fig.10(c)marked by the dash circle,is blocked by the rear wall of the moonpool,entering the moonpool vertically,and then mixing with the fluid in the moonpool(Fig.10(c)).Another is transported outside directly and then enters into the stern flow field and then participates in the movement of the flow in stern region.All of these processes are called the transportation stage.

3.3 Action stage

Part of vortex that climbs along the wall of the moonpool,described in the section3.2,will arrive at the free surface and then breaks(Fig.11).The broken vortex enters the middle of the moonpool and then mixes with the fluid in the middle,as is shown in Fig.12,which provides the source of energy for the movement of the fluid in the moonpool.It can be obviously observed that there is a large vortex locating in the moonpool near the rear wall by visualizing the streamline behavior(Fig.12).This vortex will change the gradient of pressure violently thus the added energy consumption is enlarged.

Fig.11 The phnomenon of second separation

Fig.12 The Q-criterion field in the transportation stage and the behaviors of streamline

According to the analysis of sections 3.1-3.3,there are two kinds of energy sources for the central large vortex in a moonpool.One is the part of the vortex that directly enters the middle of a moonpool when the boundary layer is separated,and the other is those vortex climbing along the rear wall,which breaks at the free-surface.The first of this process is that a core of a large vortex is formed in the middle of the moonpool,which leads to the change of the gradient of pressure in the moonpool.This core replenishes its energy through two ways.One way is the subsequent boundary layer separation and another way is to absorb the vortex broken at the free-surface.Generally,three stages can be described in a simple way as the Tab.2.

Tab.2 The cause,phenomenon and result of three stages

4 The effect of parameters of a moonpool on added resistance

As is shown in Fig.13,the added resistance coefficient of the drilling ship increases with the increase of the Fn.At the high Fn,especially,the growth rate of the total resistance coefficient is larger than the low Fn,which is the result of more violent vortex shedding.According to the boundary layer theory,velocity is the vital parameters determined the situation of the inner region.In the higher velocity condition,the non-linear effect,especially the vortex shedding caused by the separation of the boundary layer,is more severe,so that the growth rate of the added resistance coefficient rises intensely.

It can be seen from Fig.14(a)that the added resistance coefficient of the moonpool is significantly larger than other range within the range of 0.7 to 1.1,while the added resistance coefficient of the moonpool is relatively smaller in other range.The reason for this phenomenon may be that there will be a strong phenomenon of green water on the recess deck thus increasing the added resistance when the free-surface of the moonpool is close to the recess deck.In order to confirm this deduction,the time history data of the total force of the moonpool was extracted,as shown in Fig.14(b).It can be clearly seen that the total force of 8.5 m is indeed larger than other condition,which means that there must be a range,so-call‘green water range',where the phenomenon of green water on recess deck is violent.Generally,taking a design of a moonpool should try to avoid this‘green water range'and the recommended range,confirmed by the simulation results,is shown in Fig.15.

Fig.13 Added resistance coefficient for varying Fn

Fig.14 Added resistance coefficient for varying DD and the time history of moonpool total resistance coefficient

Fig.15 The recommend range of draft

Fig.16(a)shows the added resistance coefficient for varying RoLW.When the RoLW is less than 3.5,obviously,the added resistance coefficient is large than other range.Fast Fourier Transform(FFT)is performed on the sloshing time history with different case condition of RoLW,and the frequency corresponding to the maximum point of the amplitude is extracted.Fig.16(b)shows the sloshing frequency compared with the first three order sloshing natural modes.It can be obviously seen that when the RoLW is less than 3.5 approximately,the sloshing motion frequency is closed to the natural frequency,resulting the phenomenon of resonance in the moonpool.This resonance mainly is induced by the vortex shedding and it can be avoided if the geometry of the moonpool is reasonable.As is discussed above,it is recommended that the RoLW should be larger than about 3.5,which can avoid the resonance of fluid.

Fig.16 Added resistance coefficient for varying RoLW

5 Conclusions

With theoretical analyses and numerical simulations,the mechanism and characteristic of added resistance of a drillship were analyzed and discussed in the present research.In the works of numerical simulation,the mechanism of added resistance of a drillship was illustrated by dividing added resistance into added frictional resistance and added residual resistance,while in the theoretical analysis,the boundary layer theory and natural mode equation of a moonpool were carried out to explain the essence of the added resistance.The following conclusions can be drawn：

(1)The added resistance of a moonpool can be divided into two parts,namely,added frictional resistance and added residual resistance.Added frictional resistance can be estimated by the Newton's shear law.Added residual resistance,induced by vortex shedding,also can be divided into two parts.One is caused by the change of the longitudinal pressure gradient of the drillship resulting from vortex shedding,and the other is caused by the movement of water inside the moonpool.

(2)As the discussion in Chap.3,added residual resistance is induced by the vortex shedding,which can be divided into three stages,namely,the generate stage,the transportation stage as well as the action stage.The phenomenon of vortex shedding,so-called the first stage,yielded by discontinue velocity and abrupt of pressure gradient,will generate vortex in certain frequency relating to the shape and parameters of the moonpool.The vortex generated in first stage has two situations to be,discharging outside the moonpool and coming into the middle of moonpool,which is so-called the second stage.The vortex,transported forward the rear wall in the second stage,will divided into two parts at the outflow edge.One part climbs vertically along the rear wall of the moonpool and will be dissipated after arriving the free surface.This part will interfuse itself with the larger vortex in the middle of the moonpool(Fig.12)with result being the change of the pressure gradient in the moonpool.The other part is transported along the bottom of the hull and finally enters the stern flow field.

(3)The greater the service speed of the drilling ship,the greater the added resistance,which can explain the more violent phenomenon of vortex shedding resulting from the increase of the disturbance in boundary layer.Draft mainly effects the phenomenon of green water on recess deck,which can be avoided if the DD is more than 1.20 or less than 0.60(Fig.15).RoLW,as the result shown from FFT to the time history of the force of moonpool,effects the sloshing natural frequency.RoLW should be more than 3.5 as the present research recommended.