ARTICLE IN PRESS

0029-8018/$ - se

doi:10.1016/j.oc

�CorrespondiE-mail addr

(M.H. Kim).

Ocean Engineering 34 (2007) 3–9

www.elsevier.com/locate/oceaneng

A parametric sensitivity study on LNG tank sloshing loadsby numerical simulations

D.H. Leea,�, M.H. Kima,�, S.H. Kwona, J.W. Kimb, Y.B. Leec

aDepartment of Civil Engineering, Texas A&M University, College Station, TX 77843-3136, USAbAmerican Bureau of Shipping, Houston, USA

cDaewoo Shipbuilding and Marine Engineering Co. Ltd., Geoje, Korea

Received 1 November 2005; accepted 15 March 2006

Available online 7 July 2006

Abstract

A series of parametric sensitivity studies on unmatched dimensionless scale parameters is carried out on the liquified natural gas

(LNG) tank sloshing loads by using a computational fluid dynamics (CFD) program. First, a brief dimensional analysis is conducted to

identify the governing and non-matched non-dimensional parameters, assuming that Froude scaling law is adopted. Then the sensitivity

of impact pressure is checked through numerical simulations against non-matched parameters, such as fluid viscosity, liquid–gas density

ratio, and ullage pressure and compressibility. The CFD simulations are also verified against experimental results. It is concluded that the

effects of viscosity and density ratio are insignificant, while the compressibility of ullage space plays an appreciable role, as was pointed

out by Bass et al. [Bass, R.L., Bowles, E.B., Trudell, R.W., Navickas, J., Peck, J.C., Yoshimura, N., Endo, S., Pots, B.F.M., 1985.

Modeling criteria for scaled LNG sloshing experiments. Transactions of the ASME 107, 272–280].

r 2006 Elsevier Ltd. All rights reserved.

Keywords: LNG tank sloshing; Impact pressure; Sensitivity study; Non-matching dimensionless parameters; Ullage pressure and compressibility;

Viscosity; Density ratio

1. Introduction

As the demand for natural gas increases, supersizeliquified natural gas (LNG) carriers are emerging as themost efficient commercial alternative for long-distancetransportation. This trend requires that those LNG carriersoperate with partially filled conditions, for which the riskof sloshing damage inside cargo tanks should carefully bechecked. Thus, the possible sloshing damage by arbitrarilyfilled LNG cargo becomes a new challenging technologicalproblem and attracts more attention in LNG industrytoday. For the analysis of the sloshing, experimentalstudies are inevitable due to the complex nature of thephenomenon. Nevertheless, analysis and numerical toolsmay also play a key role in understanding the relevantphysics.

e front matter r 2006 Elsevier Ltd. All rights reserved.

eaneng.2006.03.014

ng authors. Fax: +1979 862 8162.

esses: dlee@tamu.edu (D.H. Lee), m-kim3@tamu.edu

Until recently, there have been a number of studies forsloshing impact loads. To name a few, Bass et al. (1985)overviewed modeling criteria for scaled LNG sloshingexperiments. Lee (1997) analyzed the hydroelastic effect onthe sloshing loads. Landrini et al. (2003) and Colagrossiet al. (2004) studied sloshing in two-dimensional flowsbased on the smoothed particle hydrodynamics technique.Kim et al. (2003) developed a three-dimensional finiteelement code for the analysis of a partially filled LNG tank.Loots et al. (2004) proposed an improved volume of fluid(VOF) method for the numerical simulations of LNGsloshing. Wemmenhove et al. (2005) investigated therobustness of two-phase modeling for LNG tank sloshing.In the sloshing experiment, the Froude scaling law is

generally adopted. It is impossible to match all the relevantnon-dimensional terms, and, therefore, a distorted modelhas to be used. The unmatched parameters are fluidviscosity, density ratio between liquid and ullage gas, ullagepressure and compressibility, and the wall elasticity. Forthe unmatched parameters, scale effect or sensitivity has to

ARTICLE IN PRESSD.H. Lee et al. / Ocean Engineering 34 (2007) 3–94

be checked to interpret the model-scale experimentalresults in a better way.

In this regard, the effects of fluid viscosity, liquid–gasdensity ratio, and the ullage pressure and compressibilityhave been investigated through numerical simulations.Modeling the ullage space as compressible gas newlyintroduces additional chemical and heat parameters, andmakes the numerical modeling more complicated. There-fore, in the present study, the effect of gas compressibilityhas been examined indirectly by considering two extremeconditions. When bubbles are generated, the liquidcompressibility and gas–liquid volume fraction may alsoplay an important role, which is not taken into considera-tion in this paper. The wall vibration due to its elasticityincreases or decreases the impact load by, respectively,increasing or decreasing the relative velocity betweenthe liquid and the wall, which is also not considered inthis paper.

The numerical simulations of sloshing were carried outby using a commercial viscous flow analysis program,FLOW3D, which uses the VOF method for free surfacemodeling. The numerical modeling was verified bycomparison with available experimental results. Thepresent sensitivity study by a series of numerical simula-tions shows that the viscosity and density ratio areinsignificant, and gas compressibility is important in peakpressures and the rising time of impact load.

2. Dimensional analysis

The relevant physical variables for the LNG sloshingloads can be categorized as shown in Table 1.

Table 1

Physical variables for sloshing loads

Variables Symbols Description

Geometric variables l Tanker length

T Tanker height

h Liquid depth

a Wall angle (at MWL)

Liquid variables rL Density

m Viscosity

Ullage variables rU Ullage density

EV Compressibility (bulk

modulus)

PU Ullage pressure

Kinematic variables g Gravity

o Oscillation frequency

A Oscillation amplitude

Wall E Wall elasticity

Pressure Pmax Sloshing impact

pressure

Surface tension effect is not important, so is not included.

Liquid compressibility, bubble generation, and chemical and thermo-

dynamic properties are not considered here.

According to dimensional analysis, the sloshing impactpressure is governed by the following non-dimensionalparameters:

Pmax

rLo2A2¼ f

a;h

L;A

L;T

L

zfflfflfflfflfflffl}|fflfflfflfflfflffl{geometric

;

oAffiffiffiffigLp ; rLoAL

m ;

rUrL; PU

rLo2A2 ;

oAffiffiffiffiEVrL

q ; oAffiffiffiffiErL

p

26666666666664

37777777777775, (1)

where first four parameters are related to geometricsimilarity and the fifth parameter denotes Froude number.When Froude scaling is applied to model tests, the dynamicsimilitude hinges on matching all the remaining non-dimensional parameters:

(a)

rLoALm : Reynolds number,

(b)

rUrL: density ratio,

(c)

PU

rLo2A2: ullage pressure,

(d)

oAffiffiffiffiEVrL

q : Gas (and/or liquid) compressibility, and

(e)

oAffiffiffiffiErL

p : wall elasticity.

This paper focuses on the effects of fluid viscosity,density ratio, and ullage pressure and compressibility onimpact pressure by numerical simulations of LNG sloshingby using a viscous flow solver. It should be noted that thecompressibility of ullage gas has not been fully modeled inthe present calculations, although it is a significantparameter on the impact pressure. In this regard, the roleof the gas compressibility will be discussed rather indirectlyby considering two extreme (incompressible and fullycompressible) cases. The hydroelastic effect of a flexible wallcausing greater or smaller relative velocity has not beenconsidered in the present simulations by assuming that thewall is very rigid. The liquid compressibility is not consideredeither by assuming that bubble effects are minimal.

3. Numerical model

Numerical simulations of sloshing have been carried outby using a viscous flow analysis program, FLOW3D, whichis based on the finite difference method (FDM) and thefinite volume method (FVM), and uses the VOF methodfor free surface problems (FLOW-3D, 1999). A non-inertial reference frame has been introduced to describe theprescribed roll excitations.

ARTICLE IN PRESSD.H. Lee et al. / Ocean Engineering 34 (2007) 3–9 5

The continuity equation and the momentum equationsare as follows:

qrqtþ r � ðrUÞ ¼ 0, (2)

qUqtþU � rU ¼ �

1

rrpþG�

1

rDs� KU� F, (3)

where G is the gravity and non-inertial body acceleration, s isthe viscous stress tensor, KU is the drag from porous baffles,obstacles, etc., and F denotes other forces such as surfacetension, electric forces, and externally imposed forces.

The marker VOF function moves according to thevelocity field in the fluid. The VOF advection equation is

qF

qtþ

qqxðFuÞ þ

qqyðFvÞ þ

qqzðFwÞ

� �¼ FDIF þ FSOR;

(4)

where FDIF and FSOR are diffusion of fluid fraction andfluid source/sink, respectively. The variables in each meshcell and their locations are shown in Fig. 1.

4. Numerical results

Fig. 2 shows the computational model, which is the sameas the experimental model given by Lee (1997). The lengthof the box is 1.0m and its height is 0.6m.

0.6 m

1.0 m

0.3 m

R.C

P5

P4

P3

P2

P1

Fig. 2. Computational model.

F, VF, p

x

y

z

u, Ax

v, Ay

w, Az

Fig. 1. Location of variables in a mesh cell.

Solid circles denote the measurement points, which arelocated at a height of 15, 30, 45, and 52.5 cm from thebottom of the model. P5 is located at the top, 7.5 cm fromthe left corner.The roll motions of the rectangular tank were excited

with the following conditions. The center of rotation waslocated at the center of the box.

�

Ele

vatio

n / 0

.5Lθ

Fig

fill

Filling ratio: 25% and 50%,

� Exciting amplitude: 61, � Exciting period: 0.85Tn,

where Tn denotes the resonance period of the lowestsloshing motion.From this point on, the numerical results will be based

on a two-phase model, both as incompressible fluids, unlessmentioned otherwise. Wemmenhove et al. (2005) recentlyshowed that the two-phase model of incompressible fluidsis robust and compares reasonably with sloshing experi-ments if there is no air trapping or bubble collapsing.For the verification of the present numerical scheme,

surface elevations near the left wall of the tank arecomputed and compared with the experiments by Lee(1997). Fig. 3 shows the maximum surface elevation nearthe wall as a function of oscillation frequency, where thefilling ratio is 50%, and f and fn denote excitation andnatural frequencies, respectively. They show good agree-ment.In addition to the free-surface elevation, the peak

pressures at the measurement points P1 and P2 are alsocalculated and compared with the experimental data inFig. 4. Considering the variability and complexity of thesloshing phenomenon, the agreement between numericaland experimental results is pretty good. It is observed thatthe peak pressures at P1 and P2 show a trend similar tothat as the surface elevation.Fig. 5 shows the time histories of the pressure acting on

the wall near the mean waterline for different fill ratios.The pressure signal is random and non-repeatable. It is

0

1

2

3

4

5

6

7Exp

Cal

0.5 0.7

f / fn

0.9 1.1 1.3 1.5

. 3. Maximum surface elevation near the left wall (L, tank width; 50%

ratio).

ARTICLE IN PRESS

0 1 2 3 4 5

0

t/T

Pre

ssur

e (b

ar)

25%50%

-0.2

0.2

0.4

0.6

0.8

Fig. 5. Pressure time histories from according to fill ratios (T=Tn ¼ 0:85).

99969 100252 100534 100817 101099 101382 101664

contourspressure

contourspressure

contourspressure

0.4

0.0N

-0.4

0.4

0.0N

-0.4

0.4

0.0N

-0.4

-0.60 -0.36 -0.12 0.12

X

0.36 0.60

-0.60 -0.36 -0.12 0.12

X

0.36 0.60

-0.60 -0.36 -0.12 0.12

X

0.36 0.60

99953 101586 103220 104853 106487 108121 109754

99967 100322 100676 101030 101385 101739 102094

(a)

(b)

(c)

Fig. 6. Snapshots of the second impact. (a) t ¼ 4.08 s; (b) t ¼ 4.21 s;

(c) t ¼ 4.29 s.

0.8 10

1

2

3

4

5

6

pres

sure

/ 0.

5ρg

Lθ

Cal (P1) Exp (P1) Cal (P2) Exp (P2)

0.5 0.6 0.7

f / fn

0.9 1.1 1.2

Fig. 4. Peak pressure at P1 and P2 (50 % fill ratio).

D.H. Lee et al. / Ocean Engineering 34 (2007) 3–96

found that the case of 25% fill ratio has a larger andsharper peak pressure than that of 50% fill ratio. Thistrend coincides with that of experimental results (Lee et al.,2004).

To have an insight into the trend of the pressure signal,the snapshots of the liquid sloshing motion including theimpact moment for 25% fill ratio are given in Fig. 6. Thesharper and higher peaks in the pressure signal for 25% fillratio are associated with the impact by the bore-type free-surface motions. The corresponding snapshots of pressuredistribution along the vertical wall are shown in Fig. 7.A high pressure can be observed near the impact area. It isalso found that the highest impact pressure occurred at thethird stage near the mean waterline.

As the numerical simulation has been reasonably verifiedagainst sloshing experimental results, we use the numericaltools to investigate the sensitivity of impact pressureagainst various non-matched scale parameters.

4.1. Sensitivity against turbulence

Fig. 8 shows pressure time histories from two different(laminar and turbulent) viscous models. In this calculation,

a two-fluid model is used and the fluids are assumed to beincompressible (no air trap). It is found that the impactcharacteristic is insensitive to viscous models (laminar vs.turbulent models).

4.2. Sensitivity against viscosity

Fig. 9 shows the effect of changing kinematic viscosityon impact pressures. A two-fluid turbulent model is used inthe simulation. The pressure of the ullage space is set to beatmospheric pressure. The original kinematic viscosity in

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-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.25

-0.2

-0.15

-0.1

-0.05

0

Pressure Variation (bar)

Wal

l coo

rdin

ate

(cm

)

t=4.08 s

t=4.21 s

t=4.29 s

Water Line

Fig. 7. Pressure distributions along the vertical wall (25% fill ratio).

0 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

time (s)

P1

(bar

)

laminar model

turbulence model

Fig. 8. Time histories of sloshing pressure for laminar and turbulent

models (25% fill ratio).

0 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

time (s)

P1

(bar

)

ν0*0.1

ν0*10

Fig. 9. Time histories of pressure for different kinematic viscosity values

(25% fill ratio).

4 4.2 4.4 4.6 4.8 5 5.2 5.4

0

2

4

6

8x10-4

time (s)

P1

(bar

/ ρ)

water (ρ~1000 kg/m3)

LNG (ρ~400 kg/m3)

Fig. 10. Time histories of pressure for two different density ratios (25%

fill ratio).

D.H. Lee et al. / Ocean Engineering 34 (2007) 3–9 7

Fig. 8 is increased or decreased 10-fold. As shown in thefigure, the numerically simulated pressure time histories arealmost the same. This conclusion (the insensitivity ofimpact pressure to liquid viscosity) is also supported byBass et al. (1985).

4.3. Sensitivity against density ratio

In order to check the sensitivity against the liquid–gasdensity ratio on the impact pressure, two liquid-gas modelsare used, i.e., water–air model and LNG–ullage gas model(LNG density ¼ 400 kg/m3). The calculated results areplotted in Fig. 10, where pressure on the y-axis is dividedby liquid density. According to this numerical example, thedensity ratio is not a significant parameter on the impactpressure.

4.4. Sensitivity against ullage pressure and compressibility

It is reported by Bass et al. (1985) that reducing ullagepressure has a small effect on impact pressure until ratherlow pressures close to the vapor pressure are reached. Inthis region (PUo7 kPa absolute pressure), a significantincrease in impact pressure was observed. However, it isnot yet clearly understood whether the ullage compres-sibility or ullage gas condensation (thermodynamic effect)is the main cause of the drastic rise in pressure. Thegeneration and collapsing of bubbles may also be related tothe increase.In the present case, the ullage pressure is varied

(decreased five-fold) while keeping the assumption of gasincompressibility. (This cannot be assumed at much lowerullage pressure, as mentioned above.) Fig. 11 shows that

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4 5 6 7 8

0

0.2

0.4

0.6

0.8

time (s)

P1

(bar

)

Pu =1e5 Pa

Pu = 2e4 Pa

Fig. 11. Time histories of sloshing pressure for two different ullage

pressures (25% fill ratio).

0 2 4 6 8 10-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (s)

P1

(bar

)

Incompressible air

vacuum

Fig. 12. Time histories of sloshing pressure for two different ullage space

models (25% fill ratio).

3.5 4 4.5 5 5.5 6 6.5

0

0.5

1

1.5

2

time (s)

P1

(bar

)

Incompressible air

Vacuum

Fig. 13. Magnified time histories of Fig. 12.

3 3.5 4 4.5 5 5.5-0.2

0

0.2

0.4

0.6

0.8

time (s)

P2

(bar

)

Incompressible airVaccum

Fig. 14. Time histories of sloshing pressure for two different ullage space

models (50% fill ratio).

D.H. Lee et al. / Ocean Engineering 34 (2007) 3–98

the free-surface movement and impact pressure are notvery sensitive to the change in ullage pressure, as shown byBass et al. (1985), before reaching a critical level. However,the drastic increase in pressure at a low ullage pressurecannot properly be modeled without considering gascompressibility and thermodynamic effect.

To indirectly observe the sudden increase in impactpressure at a very low ullage pressure (below the criticallevel), the two limiting cases of ullage space are compared,i.e., the empty space in the one-fluid model and incom-pressible air in the two-fluid model. The ullage space in thefirst empty model is assumed to be totally compressible,while that in the second model to be completely incom-pressible. These models show two extreme cases.

Fig. 12 shows the time histories of impact pressure forthe two limiting cases, which are magnified in Fig. 13. Thegeneral trend of the two signals is similar (especially attime intervals 2–3 and 8–9), but the rising time in the

incompressible-air model is larger and the initial peaksmaller than that in the model with infinitely compressibleair. The air resistance may also play a role in the reductionof liquid motion and the corresponding impact pressure.This implies that the impact pressure can be significantly

increased when the ullage pressure is much lowered, so thatthe ullage gas becomes close to gas that is infinitelycompressible, which is supported by Bass’s experiment.When this situation is reached in model tests, the gascompressibility has to be included in the correspondingnumerical modeling. On the other hand, when ullagepressure is not properly reduced in the model test, themodel gas becomes too stiff and less compressible, aspointed out by Bass et al. (1985). The exaggerated spikes inpressure histories can be moderated by using artificialdamping or smoothing techniques (Loots et al., 2004).For comparison, Fig. 14 shows the pressure time

histories for 50% fill ratio. In this case, the liquid sloshingmotion is just up and down with low modes; therefore,there is no violent liquid motion and the correspondingimpact, which results in a relatively smooth variation inliquid pressure. In this case, there is no big differencebetween the models with incompressible and infinitelycompressible ullage.

ARTICLE IN PRESSD.H. Lee et al. / Ocean Engineering 34 (2007) 3–9 9

5. Conclusion

Various scenarios of LNG tank sloshing have beensimulated by using a viscous flow analysis programFLOW3D. The effects of liquid turbulence, viscosity,density ratio, and ullage pressure and compressibility onimpact pressure were assessed. The present numericalresults generally correlate well against Lee’s and Bass’sexperimental data. The numerical results can be summar-ized as follows:

�

The effects of liquid turbulence and viscosity on impactpressures appear to be negligible. � Liquid–gas density ratio is not a significant parameter

on the dimensionless impact pressure.

� Impact pressure is not sensitive to the change of ullage

pressure before reaching a critical level, as can be seen inthe present incompressible-gas models. To simulate theincrease in impact pressure below the critical pressurelevel, a more sophisticated numerical model includinggas compressibility, bubbles, and thermodynamic prop-erties should be used.

� The comparison of a one-fluid model (with totally

compressible gas) with a two-fluid model (with incom-pressible gas) shows that the peak amplitude is greaterand the rising time faster in the former, which canindirectly explain the rapid increase in impact pressureat a very low ullage pressure.

Acknowledgments

The authors would like to express their thanks toDaewoo Shipbuilding and Marine Engineering Co. Ltd.

for financial support. This work was also supported by thePost-doctoral Fellowship Program of Korea Science &Engineering Foundation (KOSEF).

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Yoshimura, N., Endo, S., Pots, B.F.M., 1985. Modeling criteria for

scaled LNG sloshing experiments. Transactions of the ASME 107,

272–280.

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and numerical investigation of 2D sloshing with slamming. Proceed-

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