CFD aided modelling of anti-rolling tanks towards more accurate ship dynamics

Ocean Engineering 92 (2014) 296-303

ELSEVIER

Contents lists available at ScienceDirect

Ocean Engineering

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / o c e a n e n g

CFD aided modelling of anti-rolling tanks towards more accurate

ship dynamics

Bhushan Uday Taskar'', Debabrata DasGupta*", Vishwanath Nagarajan

Suman Chakraborty ^, Anindya Chatterjee ^ Om Prakash Sha ^

^ Ocean Engineering and Naval ArchiteclMre, Indian Institute of Teclinology, Kharagpur 721302, India ''Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India

'^Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India

CrossMairk

A R T I C L E I N F O

Article history: Received 30 April 2013 Accepted 27 September 2014 Available online 13 November 2014 Keywords:

Passive anti-roll tank Roll damping Anti-roll tank equation ART damping coefficient CFD

Oscillating flow inside U-tube

A B S T R A C T

Passive anti-roll tanks (ART) have been studied extensively to optimize their various design parameters. However, not much attention has been given to the accurate estimation o f t h e i r damping. Here, we have developed the motion equation of water inside ART and proposed a CFD-based methodology to find the ART damping coefficient. Since our aim is to improve the modelling of flow inside ART, our analysis focuses on ART ( w i t h o u t ship) to remove ship related complexities. Key findings regarding the damping are: (a) It is quadratic i n the velocity and the damping coefficient can be expressed using a non-dimensional number, (b) one main part of the dissipation is bend losses, computed here using steady-state turbulent CFD, and (c) a second part of the dissipation is due to turbulence near the free surface, estimated here using a simple formula. We have calculated damping coefficients for five different designs of ARTs, and validated our results through both transient CFD simulations and experiments. The effect of these designs on ship m o t i o n has been discussed. The proposed model w i l l help in better prediction of ship dynamics as far as inaccuracies due to ART modelling are concerned.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Anti-rolling tanks (ARTs) are one of the oldest roll stabilizing mechanisms used on board ships. Their advantages over other roll stabilizers are low maintenance, ease of construction, and low cost. ARTs can be active or passive, depending on whether or not the water movement is actively controlled through pumps and valves. Different configurations of ARTs have been proposed, chiefly free surface tanks and U-type tanks: of these, we have worked w i t h passive U-type ARTs i n this paper.

ARTs are expected to function in a variety of sea conditions. Additionally, there are typically size, weight, and location con-straints based on considerations of safety and economics (Goodrich, 1969). These issues make its design both challenging and important.

An ART essentially works as a tuned vibration damper for roll oscillations. Accordingly, the natural frequency of the ART must be close to that of the ship's natural roll oscillation frequency.

•Corresponding author. Tel.: +91 3222 283782, Mob.: +91 8016727912; fax: +91 3222 282284/-i-91 3222 283782.

E-mail address: vishwanath_n®naval.iitkgp.ernet.in (V. Nagarajan).

Additionally, the damper must have its damping coefficient w i t h i n a useful range. Tuned vibration absorbers that are too lightly damped merely shift the large resonant response to two different nearby frequencies (Field and Martin, 1976), while those that are too heavily damped cannot respond strongly enough to be effective. An additional complication for ARTs is that their o w n inherent damping is not well known f r o m empirical formulas, especially because the fluid flow i n them tends to be turbulent. Admittedly, for designing ARTs for ships, thumbrule type f o r m u -lae are available in the literature (Lewandowski, 2004). However, most of these formulae have somewhat ad hoc predictions of the ART'S inherent damping coefficient. Damping can be evaluated experimentally to avoid the errors. Such experiments were carried out by Field and Martin (1976) on a series of ARTs. However, only a limited number of ARTs can be tested.

ART response can also be derived entirely using unsteady CFD simulation (Van Daalen et al., 2001; Thanyamanta and Molyneux, 2012). CFD can accurately predict the behaviour of the ART but simulating different models using unsteady CFD simulations can be very time consuming. Here, we propose an improved numerical model by taking limited inputs f r o m steady CFD simulation, which is less time consuming as compared to the f u l l unsteady simulation.

http;//dx.doi.org/10.1016/j.oceaneng.2014.09.035 0029-8018/© 2014 Elsevier Ltd. All rights reserved.

B.U Taskar et al. / Ocean Engineering 92 (2014) 296-303 297 The primary contribution of this paper is to present a

CFD-based approach to obtain the damping coefficient of a given ART; discuss a correction i n the estimated effective mass of the water in the ART; and outiine a derivation of the resulting equations of motion. The CFD analysis in this study has been done using the commercial software package (ANSYS® Fluent, Release 13.0).

We note that the ART has been mathematically modelled by several authors over almost 50 years (e.g., Stigter, 1966; Cawad et al., 2001; Holden et al., 2011; Holden and Fossen, 2012; Tuan et al., 2008). But the damping in the ART has always been taken as linear; specific verification of the dynamics of the water i n the ART alone to compare w i t h its numerical model has not been attempted. Note, also, that the ship+ART dynamics includes many ship-specific approximations and complications. Therefore the modelling of ART would benefit f r o m more ART-specific studies. W i t h these motivations, we have made the following contributions.

In this paper we have demonstrated, through theoretical arguments and CFD (steady state and transient, 2D and 3D, and for several ART geometries) that the ART damping should be quadratic. We have studied the dynamics of the ART water through transient CFD simulations. The transient CFD simulations were validated w i t h model experiments. They were also compared very favourably w i t h reduced single-degree-of-freedom model. We have offered a simplifying insight into how the dissipation i n the ART can be decomposed into a turbulence-dominated quad-ratic drag plus added dissipation near the free surface. We have discussed how the effective mass of the single-degree-of-freedom ART model is slightly smaller than that estimated i n the usual approach. And finally, for completeness, we have outiined a derivation of the coupled ship+ART equations using a Lagrangian approach.

2. Equation of motion of fluid inside anti-roll tank

There are several shapes of ART, but the two main categories are U-tube type and free surface type. In this paper we have studied the damping for a U-tube type ART. The layout of the ART has been presented i n Fig. 1. Heeling moments due to shifting of water w i t h i n individual reservoirs are much smaller than that due to the shifting of water f r o m one reservoir to another. For simplicity, therefore, we assume the water surface in the reservoir is always parallel to the base of the ART, and seek an equivalent one-dimensional description i n terms of averaged quantities at each cross section. The one-dimensional description is in the form of an unsteady Bernoulli's equation (Som et al., 2011)

P2 , V i

gjt dV

ds 0 )

In the above equation Pi, Viand fii correspond to the averaged pressure at the free surface, the velocity of the free surface, and the height of the water i n the left reservoir respectively. P2, V2 and /12 are the corresponding terms for the right reservoir, hf is the yet-unknown head loss due to the water's motion; and ds is a differential length along a streamline, which we take as the midline along the U-shaped ART, shown using a dot-dashed line in Fig. l.The velocity V w i t h i n the integral is understood to be always directed along the dot-dashed line.

The unsteady Bernoulli equation above is used for unsteady flows in pipes w i t h bends, wherein the head loss hf consists of "major" and "minor" losses. Major losses arise mostly f r o m the length of the pipe (distributed losses), while minor losses are due to localized losses near bends and changes in cross section. For the ART, however, the f l o w tends to become turbulent due to complex

h ! S Vz 2.75 m 16 m ] 1.5 m 2.76 m 2.75 m '' ' 1 ' ' 1 ' ' ' 2.76 m

1

Front view of anli roll tank

Plan view of anti roll tanf^

Fig. 1. Main dimensions of the ART. Inner geometrical details are not shown in this schemadc. The nominal water height h is a design parameter.

geometric features and large size, and i t is the so-called minor losses that dominate. The minor losses or the bend losses i n Eq. (1) can be written as

ViWtl

2g '• (2)

where for a given ART geometry the constant K is to be found using CFD as described below, and here the f o r m adopted allows Vl to be both positive and negative.

The free (damped) oscillatory motion of the fluid inside the ART is governed by (see Appendix A for details)

Axf2h w\.. K.... . ,0.

where A; and A3 are the horizontal and the vertical cross section areas of the reservoir and the duct respectively, w is w i d t h of the duct, X is the difference in the levels of water i n t w o reservoirs, h is the design water level of ART (refer Fig. 1 for the definitions). Eq. (3) w i l l i n general hold for moderately large oscillations of the water. For extremely small motions, the quad-ratic dissipation w i l l be tiny and linear viscous dissipation w i l l dominate, but we are naturally not interested in that regime o f operation because it corresponds to a ship w i t h no significant roll oscillations, where the ART is not needed. Eq. (3) can be w r i t t e n i n terms of Xi (refer Fig. 1) and multiplied throughout by 2;9Aig to yield:

m x i + c X i | x , | + 2pAigx, = 0 where m = pA](^^+^)

pAxK

and c = - (4)

The coupled equations of motion for the ship and ART, are obtained using Lagrange's equations and given i n Appendix B for completeness.

298 B.U. Taskar et al. / Ocean Engineering 92 (20:4) 296-303 The focus of this paper is on estimating the ART damping

coefficient K along w i t h the discussion of the equivalent mass of water inside the ART for accurate modelling of ART. We now turn to these taslcs. "

3. Determination of damping coefficient of ART

Assuming quadratic damping inside the ART, /C(of Eq. (3)) can be calculated using transient CFD simulation or model testing, both of which are time consuming. Here, we have estimated i t using a simple, direct and reasonably efficient approach, system-atically described below.

In our approach, though the actual flow inside the ART is oscillating, we estimate K by first simulating a sustained or steady state turbulent f l o w f r o m one reservoir to another. The reservoirs of the ART are extended in length, to avoid the effect of possible back f l o w at the outlet. Fig. 2 presents the discretized domain of the model ART. A finer mesh is used near the sharp edges, for better results.

Steady f l o w was set up using the K-e Model w i t h an enhanced wall treatment, which is a near-wall modelling method that combines a two-layer model w i t h so-called enhanced wall func-tions. This obviates the need of using a very fine mesh near the wall. Refer ANSYS® Fluent, Release 13.0 for further details. Unstructured hexahedral and tetrahedral meshes were used for 2D and 3D simulations, respectively. Pressure inlet and pressure outlet were used as boundary conditions' w i t h second order discretization scheme. A no-slip boundary condition was used at the other domain boundaries.

After running the simulation—pressure, velocity and the kinetic energy correction factor averaged over the section were found at two imaginary surfaces placed at the height of the design water level in ART. These surfaces can be seen as green planes i n Fig. 3.

Fig. 2. Surface mesh-denser mesh near the corners of the ART.

Velocily

18

12

J

Fig. 3. Velocity profile inside extended ART. Water flow is downward on the left side and upwards on the right side.

In case of 2D simulations, these quantities were averaged over a I D line segment instead of 2D surface.

The important role played by inner and outer corners i n the ART damping is evident f r o m the velocity distribution seen in Fig. 3. Keeping the corners sharp, reducing the cross section area of the duct, and introducing stiffeners inside the tank that obstruct the flow- w i l l lead to increase i n the ART damping coefficient. However, CFD simulation is necessary to quantitatively determine these effects.

The procedure for calculating K is described below.

Energy equation between two surfaces can be written as follows:

V _{t avg}

2 g

P 2 _ ^ „ Vlavg

h + hf (5)

Since both planes are at the same height (hi =/i2). Eq. (5) is simplified to V = ^ + « 2 -Pg 2 avg V 2g K-1 avg 2g (6)

Here, a is a kinetic energy correction factor, introduced to make the kinetic energy expressible i n terms of the average velocity over a section. It is defined as follows:

fV^dA ^ avg"

(7)

where A is the area of the cross section. Vis velocity at an arbitrary location w i t h i n the cross section, and Vavg is average velocity over cross section. This quantity a can be calculated using the custom field functions i n ANSYS® Fluent, Release 13.0.

When we calculate energy dissipation using steady state CFD simulation as described above, only minor losses (i.e. bend losses) can be taken into account. However, in case of transient flow, additional dissipation occurs due to turbulence near the free surface.

Accordingly, energy loss upto surface A (Fig. 4) is the one calculated using above mentioned steady CFD simulation. In this simulation, uneven velocity distribution is seen i n the right reservoir (like on surface A, Fig. 4). The influence of this uneven velocity distribution can also be observed in actual oscillating flow conditions. For example, during the first oscillation, the free surface i n the right reservoir rises w i t h an unequal velocity and wave type oscillations are observed (schematically shown as recirculation between surfaces A and B i n Fig. 4). This causes additional dissipation i n the hypothetical region between these surfaces A and B. The additional damping can be calculated as follows:

We take A and B to be two notional surfaces, where the difference i n potential energy is small compared to the difference i n the kinetic energy.

Kinetic energy at A and B can be written as:

KEA = a2- KEb V 2 avg

2g (8) I I I I I B A t t f t t I I I I I B A l l l l l Ü Ü Ü t t t

t

t

B.U. Taskar et al. / Ocean Engineering 92 (2014) 296-303 299 Therefore, energy loss between A and B is:

Head loss due to dissipation near free surface = hfs = ( « 2 - 1 ) -""^

Energy equation can be w r i t t e n by considering this additional damping as follows-(Note that K in this equation only includes bend losses)

^ + « , 4 ^ =

pg ' 2g pg 2g -2g " ^

Therefore, total K can be written as: 1 /2(Pl - P z )

2 ,,2

A

'^=•(72 1 + « 1 ^ 1 a v g - ^ 2 avg I (10)

" 1 avg ^ / This K consists of both minor losses and losses due to turbulence near free surface.

The last t w o terms contribute an added correction to the value of K that seems to be dependent mainly on the principal dimen-sions of the ART, independent of velocity. It does not vary significantly due to changes i n the minor details of the ART like rounded corners and constrictions. This correction seems minor except in case of lightly damped ARTs. We w i l l discuss this further below. Considering the structural requirements i n the ship, sig-niflcant numbers of stiffeners and girders can be present inside the reservoirs and duct of the ART especially w h e n it is an integral part of the hull. These stiffening members w i l l obstruct the flow of water and create turbulence leading to a higher damping coeffi-cient. These internal structures might be absent i n a model scale ship. Moreover, tank edges might be curved in the full scale and sharp in model scale. Hence, to study the quantitative effect of such changes in internal structures on the damping of ART and eventually on the motion response of the ship, we have analyzed four ARTs w i t h different internal structures as seen in Fig. 5. Note that all these ARTs have same principal dimensions. Inner and outer corners of ART have been given 0.5 m radius of curvature in design (a) while designs (c) and (d) are provided w i t h a constric-tion of 0.3 m opening and 0.1 m w i d t h .

We found the damping coefficients for these designs using 2D steady state CFD simulations (as described above). Their values can be seen in Table 1. Steady simulations to analyze these four ARTs were done i n 2D to save computational time. It was compared w i t h 3D simulation for design (b). The damping coefficient is found to increase w i t h the degree of obstruction to the flow. Influence of different turbulence models on value of K was also studied. The variation of K w i t h change i n the turbulence models, tabulated in Table 1, is found to be very small.

a

0

Fig. 5. Different designs of

We observed that, as the damping of ART increases, the contribution of minor losses increases while losses due to turbu-lence near the free surface play small role. As we increase the obstructions to the flow, value of the damping increases f r o m /C=9 for the ART w i t h curved edges (design (a)) to /<'=216 for the one w i t h two constrictions attached (design (d)) as seen i n Table 1. Moreover, we considered one more design of ART i n which baffle plates w i t h holes were attached to both ends of duct. In this case, damping coefficient increased to K=1670 due to increased obstruction to the flow. This design can be seen i n Fig. 6, which is an implemented design of ART i n the ship whose details are given i n Appendix C.

We emphasize that there is a large variation i n the values of K w i t h i n the range of practically implementable designs of ARTs. Later in this paper, this variation in K has been shown to cause significant changes in ship dynamics.

K is a non-dimensional number, i t depends almost entirely on

the shape of the ART. As described i n Section 2, the damping of liquid motion is mainly due to bends, changes i n cross section and internal structural arrangements. The contribution of skin friction on damping is low. Therefore, scale effects on damping coefficient are expected to be marginal. We have verified this and have found no noticeable scale effects, i n that the value of K computed for a full-scale design of ART matches w i t h the one computed for model scale.

4. Validation of fluid motion in ART

The estimated value of K can be verified experimentally. However, in the experiments, only macroscopic details can be seen whereas details like inner flow structure cannot be studied. In addition, taking measurements is difficult and the extent to which various parameters contribute to the result is not easy to quantify. Therefore, we have done investigation of our estimation of K by additional calculations using transient simulations. We have used the volume of fluid (VOF) method w i t h K-e turbulence model for

Table 1

Components of ART damping. ART design K with only minor losses Total K (K-e) K for transient simulation TotalK (K-co) Total K RNG (K-e) Total K RSIVI (a) 2.6 9.19 . 9 10.67 9.50 9.04 (b) 12 26.21 31 26.02 26.01 25.75 (c) 92 120.17 120 101.21 119.68 114.26 (d) 187 216.15 240 207.80 214.64 224.89

b

300 B.U. Taskar et al. / Ocean Engineenng 92 (2014) 296-303

Fig. 6. ART with baffle plates attached at both the ends of duct.

3D - - » - 2D

Fig. 8. Initial and boundary conditions for VOF simulation (at time t=0).

different designs of ART. Principal dimensions of the ARTs con-sidered for the analysis have been given in Appendix C.

3D transient simulation was performed for design (b) and compared w i t h 2D simulation of the same design. The 2D tions were performed on well-resolved grids but for 3D simula-tions, mesh dependence studies are essential to determine the optimum resolution for accurate simulation results while at the same time keeping computations tractable. In Appendix D, we present the results of the mesh independence study for ART design (b) where very small variation in the value of K is observed. Very good match between the 2D and 3D simulations can be seen in Fig. 7. Hence, other cases were also simulated using 2D.

In the transient simulations, both the reservoirs of ART were assumed to be open to the atmosphere; thus, the boundary condition of atmospheric pressure was applied to inlet and outiet. No slip boundary condition was applied to the walls of the ART. Initial height difference between water levels i n first and second reservoir was 1.7 m. Initial and boundary conditions are shown in Fig. 8. Snapshots of simulations can be seen i n Fig. 9. The transient simulations were benchmarked using experimental studies for the ART tank seen i n design (a) (VishwanathNagarajan, 2014). The benchmarking results i n Fig. 10 show excellent match. Moreover, transient CFD simulations performed for design (a) w i t h different turbulence models produced almost identical results.

From these simulations, average height difference between free surfaces (x) was found at different times using picture frames obtained from ANSYS CFD-Post. These time series were compared with the numerical solution of Eq. (4) where, ART damping was estimated using the procedure mentioned in Section 3. Accuracy of estimated damping coefficients can be seen from the comparisons in Fig. 11.

Furthermore, we also calculated the values of K and m from transient simulations by curve fitting. These values were compared w i t h the ones used in derived equation,

We observed that by adjusting only the mass term we could match the derived equation w i t h transient simulation. Thus, the estimation of damping coefficient is accurate (within 10% error) but the mass term needs some correction. The estimated error in the equivalent mass term in each case can be seen in Table 2. These comparison plots can be seen in Fig. 11 for various designs of ARTs.

Inaccuracy in the mass term can be due to difference between the assumed flow structure and the one seen in 2D transient simulations, (e.g., not all water in the reservoir is moving in veitical direction and some amount of water in right reseivoir is static).

Viilocily ltns*-1|

B.U. Taskar et al. / Ocean Engineering 92 (2014) 296-303 301 • Experiment • CFD k epsilon -0.04 0.00 0.50 1.00 1.50 Time (sec) 2.00 2.50 3.00

Fig. 10. Benchmarking the transient simulation with experiment for model scale ART of design (a).

Ö 2D transient simulation

Derived equation with damping obteined using CFD Derived equation with adjusted mass

2 : 1.5 1 • K=<6 2 : 1.5 1 0.5 • 0 • -0.5 • 0.5 • 0 • -0.5 • 0.5 • 0 • -0.5 • -1 • -1.5 • -2 -1 -1 • -1.5 • -2 -1 ^,^•^1 ^

_{_}

Fig. 11. Comparison of derived ART equation with transient simulations. The plots w i t h adjusted mass match very well w i t h the transient simulation. Thus, derived 1D model correctly captures the physical phenomena of motion of water inside ART; assumed quadratic f o r m of damping is correct and its prediction is accurate. The difference i n the ratios of successive peaks of the plots also confirm the nonlinear nature of ART damping.

In all these cases, corrected mass was observed to be w i t h i n 70% to 80% of the mass obtained from the derived equation. This error has to be accepted since it is difficult to capture all the effects in our I D lumped parameter model.

As we increase the damping coefficient of the ART, our predic-tions match better w i t h transient simulation results. Damping coefficients of ARTs considered here range f r o m 9 to 216. The

Table 2

Comparison of equivalent mass obtained from derived equation and transient simulation.

ART design m obtained from derived equation

Value of m needed to match transient simulation (a) 34.8 X 10" 24.3 X 10" (b) 34.8 X 10" 26.0 X 10" (c) 34.8 X 10" 27.4 X 10" (d) 34.8 X 10" 28.3 X 10" K = l K = 60 K = 400 K = 1500 K=6000 withoiLt ART

A A

\ ^

1 \

Fig. 12. Roll response of ship at different damping coefficients of ART in Beam Sea with 2.5 m wave height.

derived equation solved w i t h /<'=216 demonstrates a good match w i t h the transient simulation even without any mass correction (as seen in Fig. 11). In addition, later we have shown that the optimum value of K for which a ship of interest has m i n i m u m roll response is around 400. Hence, in the range of optimum designs of ART, our I D model w i t h estimated damping provides reasonably accurate results.

5. Motion of ship with ART

We have obtained the motion response of ship for different ART damping coefficients i n order to see its effect on the ship dynamics. Derivation of coupled equations of ship-i-ART using Lagrangian approach along w i t h ship particulars have been pre-sented in Appendices B and C.

5.J. Effect of damping coefficient of ART on beam sea rolling Different designs of ART and its effect on the damping coeffi-cient of ART were described i n Section 3. Fig. 12, obtained by numerically solving coupled equations of Ship-i-ART, describes the effect of variation in the ART damping on the roll motion of the ship.

• There is a significant difference between the motion response of a ship w i t h ART i n presence (/C=1670) and absence (/<'=31) of the internal structure. Hence, i f model scale ship lacks the internal structures present i n the full-scale ship then model experiments may wrongly predict the motion of ship.

• Optimum motion response is obtained when two peaks i n the motion spectrum start disappearing while gradually increasing the damping ( f f = 4 0 0 in this case). Hence, an ART design w i t h right amount of damping can drastically reduce the roll motions (almost 75% reduction i n maximum roll angle).

302 B.U. Taskar et al. / Ocean Engineering 92 (20J4) 296-303

• A n ART may increase tire motions of sliip i f not designed properly w i t h o u t estimating damping as seen i n the plot for /C=l.

6. Conclusion

This study has demonstrated the importance of accurately finding the damping coefficient of ARTs. A ship's motion response can differ considerably for practically possible variations in the internal structure of ART ( w i t h same principal dimensions). The methodology developed here to find the damping of ART has the potential to deliver the results w i t h i n acceptable range of accuracy. Different configurations of ARTs can be analyzed in this way to obtain a design having optimum damping. Then confirma-tory model tests can be performed w i t h the final design to save time, money and efforts. Further w o r k in this direction might seek to combine optimal damping w i t h minimal overall weight.

Aclaiowledgments

—* Velociiy of tank with respect to ground

1^ Velocity of A R T water with inspect io tank

Fig. B l . Motion of water in the ART. The ART is located directly above the center of gravity of the ship. 'O' is a point, about which ship rolls, 'P' is a center of mass of water in the ART

Indian Maritime University, Visakhapatnam Campus (IMU (V) POn/IITKGP/2010/001) initiated this research. Full ship data were provided by National Institute of Ocean Technology, Chennai, India. We thank these organizations.

Appendix A. Derivation of motion of water inside ART

Unsteady Bernoulli's equation can be written as:

pg 2g ' gJi dt

Considering atmospheric pressure at both free surfaces and w r i t i n g h i and fi2 in terms of x we get,

(A.1)

2g

Head loss can be written as: V i l V i l

/!ƒ = ƒ<:• 2g

(A.2)

(A.3)

Now, Vl = V2 using continuity equation. Substituting Eq. (A.3) in Eq. (A.I) and writing V in terms of x, as:

I f = 0

Also note that

, ^ f d s \ _ h + x ,

1 U .

w h - X 2 A 3 +

A2

2h w

Thus, final equation can be written as:

A1/2/1 w \ . . /C.|.

2^U+A;J^WI^I+^=°

(A.4)

(A.5)

(A.6)

Appendix B. Derivation of coupled equations of motion of Ship+ART using Lagrangian formulation

The coupled equations of the ship and ART can now be obtained using a Lagrangian approach.

The ship is assumed to perform rolling about the longitudinal centreline of the water plane area, namely point O i n Fig. B l . The nominal centre of gravity of the water in the ART is at point P, located at a distance R f r o m point O. Roll angle is denoted by 4?.

The velocity of water inside the ART consists of two compo-nents: one due to the motion of the tank w i t h respect to inertial frame, and the other due to the motion of the water relative to the ART. Using these two components, the total Kinetic Energy [KE] of the ART water can be found.

/ C £ A R T = K £ u r+/C£Du

1 /A-\ ^

= {pA^h+^pA3wi^^j ]xi + (-pAiwh+pA-^wli,)k-i^

+ (Q.i+Q.2)'P (B.I)

Where, (2i and Q2 are the moments of inertia of the water inside the duct and the reservoirs respectively w i t h respect to the roll axis of the ship. We have introduced them for brevity, anticipating that they w i l l play a small role i n what follows. We rewrite Eq. (B.I) as

1 ' "2

(B.2) The Potential Energy (PE) of the ART water is:

P E A R T = - R M w a t e r g ( l - COS (p)+pA-igx] COS ^ - p A i g w x i sin (p

(B.3) where Mwater is the mass of water inside the ART. On assuming motions to be small, Eq. (B.3) can be rewritten i n the f o r m

PE ART = ^ Q i 1 V ^a22X? + 012X1 ^6

The KE and PE expressions for the ship are as

foUows-/C£ship=J/»<i^'> PEs m'gGMcp^

(B.4)

(B.5)

where m' is the mass of ship, Ixx the moment of inertia of ship w i t h respect to the roll axis, CM is the metacentric height. Hence, the total kinetic energy of the system takes the form

/<^£Totai = ^ r n i X ? + m 2 X i < j ! . + ( Q i +(i2)4>^+±i^^ (B.6)

where we note that, for a typical ship, ((3.1 + Q 2 ) « (1/2)Ixx and so we simply absorb Q.-[+Q.2 into (\/2)Ixx. The total kinetic energy

JCfiotai

B.U. Taskar et al. / Ocean Engineering 92 (2014) 296-303 3 0 3

The total potential energy of the system is, similarly.

(B.8) Again, since ( l / 2 ) a ^ 0 ^ « ( l / 2 ) m ' g G M l ^ ^ ^ we absorb the smal-ler term into the larger one and write

(B.9) The Lagrangian for the ship+ART system is (using Eqs. (B.7) and (B.9))

i- = / < £ T o t a | - P £ T o t a l . ( B . I O ) Using the Lagrange's equation of motion, we get two coupled

equations for ship and ART w i t h o u t any external and non-conservative forces. dL_ 'd<p = 0 dt\dxj dx" Ixx'P + m2Xi+m'gGM4)-hauX-i = 0 m i X i +m2^+G22Xi +ai2<P = 0 ( B . l l ) (B.12) (B.13) (B.14) External and non-conservative forces can be incorporated using virtual work, or more simply added on right hand side of Eqs. (B.13) and (B.14). For ships, the wave exciting moment and damping force is added. Similarly, for ART, damping is added by comparing Eq. (B.14) w i t h Eq. (A.6). The coupled ship and ART roll motion equations in beam seas i n expanded f o r m is expressed as: !xx4> + ^ship<t> + m'gGMtp = -pAi wOb - h)Xi +pAigwxi + M sin (cot) (B.I 5)

+^~-x-i\x^\ + 2pAigXi = -pA^w{li,~h)ip+pA^gwtp

(B.I 6) where M is the amplitude of the external roll moment on the ship due to waves and co is the frequency of waves.

Appendix C. Ship and ART dimensions

Ship particulars

Length OA (m) 60 KG (m) 4.650

Length BP (m) 52.8 G M ( m ) 0.99 Breadth moulded (m) 11 m' (kg) 1188 X 10^ Draught existing (m) 3 Cship(kg m^/s) 1.695 X 10^

0.6180 /xx(l<g m^) 20.41 X 10*^ ART dimensions. ART dimensions A, (m^) As (m^) h ( m ) w (m) 11.55 2.52 1.7 5.75

Appendix D. Mesh dependence

Mesh dependence analysis.

No. of elements Value of K

2.4 X 10^ 4.8 X 10^ 6.2 X 10^ 8.2 X 10= 28.3 29.2 28.9 27.3 References

ANSYS Fluent, Release 13.0, help system, User's Guide, ANSYS, Inc.

Field, S.B., Martin, J.R, 1976. Comparative effects of U tube and free surface type passive roll stabilization systems. Trans. R. Inst. Nav. Architects 118, 73-92. Gawad, A.F.A., Ragab, S.A., Nayfeh, A.H., Moolc, D.T., 2001. Roll stabilization by

anti-roll passive tanks. Ocean Eng. 28, 457-469.

Goodrich, G.J., 1969. Development and design of passive roll stabilizers. Trans. RINA II, 81-95.

Holden, C, Tristan, R, Fossen, T.I., 2011. A Lagrangian approach to nonlinear modeling of anti-roll tanks. Ocean Eng. 38, 341-359.

Holden, C, Fossen, T.I., 2012. A nonlinear 7-DOF model for U-tanks of arbitrary shape. Ocean Eng. 45, 22-37.

Lewandowski, E.M., 2004. The Dynamics of Marine Craft. World Scientific. Thanyamanta, W., Molyneux, D., 2012. Prediction of stabilizing moments and

effects of U-tube anti-roll tank geometry using CFD. In: Proceedings of the ASME 2012 31st International Conference on Ocean, Offshore and Arctic Engineering. Rio de Janeiro, Brazil.

Tuan, PhanAnh.,Kuniaki Shoji, Kiyolcazu Minami, Shigeo Mita, 2008. Responses of Roll Damping on Anti-rolling Tank Devices, OCEANS 2008-MTS/IEEE Kobe Techno-Ocean. pp. 1-6.

Som, S.K., Biswas, G., Chakraborty, S., 2011. Introduction to Fluid Mechanics and Fluid Machines. Tata McGrawHill Education Pvt. Ltd..

Stigter, C, 1966. The Performance of U-tanks as a Passive Anti-rolling Device, TNG Report no. 81 S.

Van Daalen, E.RG., Kleefsman, K.M.T., Gerrits, J., Luth, H.R., Veldman A.E.P, 2001. Anti-roll Tank Simulations with a Volume of Fluid (VOF) Based Navier-Stokes Solver. In: Twenty-Third Symposium on Naval Hydrodynamics.

VishwanathNagarajan (poster), 2014. ART Experiment Video [Video] Retrieved from <https://www.youtube.com/watch7v=RWfUiAvtvno).