Research ArticleNumerical Simulation of Sloshing in 2D Rectangular TanksBased on the Prediction of Free Surface

Haitao Zhang and Beibei Sun

School of Mechanical Engineering Southeast University Nanjing 211189 China

Correspondence should be addressed to Beibei Sun bbsunseueducn

Received 31 March 2014 Revised 19 July 2014 Accepted 22 July 2014 Published 12 August 2014

Academic Editor Haranath Kar

Copyright copy 2014 H Zhang and B SunThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A finite difference method for analyzing 2D nonlinear sloshing waves in a tank has been developed based on the potential flowtheory After 120590-transformation the free surface is predicted by the kinematic condition and nonlinear terms are approximatedthe governing equation and boundary conditions are discretized to linear equations in the iterative process of time Simulations ofstanding waves and sloshing in horizontally excited tanks are presented The results are compared with analytical and numericalsolutions in other literatures which demonstrate the effectiveness and accuracy of this numericalmethodThebeating phenomenonof sloshing in the tank with different aspect ratios is studied The relationship between sloshing force and aspect ratio under thesame external excitation is also discussed

1 Introduction

Sloshing is a liquid vibration phenomenon caused by themovement of the tankWhen the liquid cargo is in transit thesloshing would affect stability of the system severely leadingto damage or fatigue of the structure So it is necessary todiminish the impact of sloshing and avoid large amplituderesonance Sloshing has been studied for many years byanalytical numerical and experimental methods In manyearly studies the analyticalmethodwas dominant Abramson[1] used potential theory to study the linear sloshing ofsmall amplitude His work was systematic and influentialConsidering the importance of nonlinear effects in thesloshing response Faltinsen [2] analyzed nonlinear sloshingby perturbation theory Later numerical simulation of largeamplitude sloshing has been a hot topic in research Compli-cated 2D or 3D models have been calculated and fully non-linear theory has been developed to get the accurate solutionsof sloshing Faltinsen [3] presented a nonlinear numericalmethod of 2D sloshing in tanks Nakayama and Washizu[4] adopted the boundary element method to analyze thesloshing in a tank which is subjected to pitching oscillationWu et al [5] calculated sloshing waves in a 3D tank by usinga finite element method based on fully nonlinear potentialtheory Chen and Nokes [6] simulated complete 2D sloshing

motion by a finite difference method The primitive 2DNavier-Stokes equations are solved and both the nonlinearfree surface condition and fluid viscosity are consideredAfterwards Wu and Chen [7] simulated fluid sloshing ina 3D tank with the six degrees of freedom by this time-independent finite difference method Kim [8] employedSOLA scheme to solve Navier-Stokes equations and assumedthe free surface profile to be a single-valued function Heused this method to simulate sloshing flows in 2D and3D containers Celebi and Akyildiz [9] calculated nonlinearviscous liquid sloshing in a partially filled rectangular tankThey used finite difference method to solve equations andthe free surface was captured by VOF technique Wang andKhoo [10] considered 2D nonlinear sloshing under randomexcitation by the finite element method Sriram et al [11]studied random sloshing under both horizontal and verticalexcitation In the aspect of theory Faltinsen et al [12 13]developed the infinite dimensional modal analysis techniqueto describe nonlinear sloshing of incompressible fluid wherethe free surface motion was expanded in generalized Fourierseries Wu [14] analyzed second-order resonance of sloshingTheir conclusions were discussed in many other researcharticles (Hill [15] Frandsen [16] Firouz-Abadi et al [17] etc)

It is difficult to solve the sloshing equations with nonlin-ear free surface boundary conditions because free surface is

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 395107 12 pageshttpdxdoiorg1011552014395107

2 Mathematical Problems in Engineering

alwaysmoving in the sloshing process Some special methodsare used to treat the moving boundary of free surfacesuch as VOF and ALE Recently 120590-transformation has beenemployed in sloshing problems It is an efficient spacetransformationmethod especially suitable for equationswithcurved boundary The basic idea of 120590-transformation is tointroduce a new stretching variable in the vertical directionwhich transforms the computational domain into a regularshape Chern et al [18] and Frandsen et al [16 19] used 120590-transformation to map the liquid domain onto rectangularregion and then calculated horizontally and vertically forcedsloshing problems in tanks Turnbull et al [20] simulatedseveral 2D free surface wave motions by 120590-transformedfinite element model According to Frandsenrsquos analysis 120590-transformation has two major advantages in simulating freesurface flows remeshing due to the moving free surface isavoided and free surface smoothing is often not requiredThe disadvantage of it is that it is a unique transformationand restricts the wave shape to be nonoverturning In thispaper a new numerical method is proposed based on 120590-transformation In the iterative process of time a forecastscheme is used to estimate the free surface boundary andsome nonlinear terms are approximated Then the governingequation and boundary conditions are linearized and solvedby finite differencemethod Inmany numericalmethods freesurface is often updated directly by explicit schemes suchas Adams-Bashforth or Runge Kutta methods while in thismethod a predictor-corrector scheme of free surface is usedThe free surface is first predicted before the iteration so thefluid boundary is more precise for the numerical simulationafter the governing equation and boundary conditions aresolved the free surface is updated by the solutions Because ofthe predictive step of free surface it is reasonable to considerhigh accuracy is one of the advantages of this method Fur-thermore semi-implicit Crank-Nicolsen scheme is employedto discretize linear terms in free surface boundary conditionsso this method has better numerical stability than explicitschemes Nonlinear sloshing in fixed and horizontally excitedrectangular tanks has been simulated by this numericalmethod The results are compared with analytical solutionsor other numerical results which prove the efficiency of thismethod The limitation of this numerical method is alsodescribed

Many researches have focused on sloshing phenomenonin a fixed aspect ratio by different excitations In fact theaspect ratio is always changed in the process of storageor transportation of liquid cargo So it is necessary toinvestigate the influence of aspect ratio on liquid sloshingChen and Chiang [21] analyzed the effect of fluid depth onnonlinear characteristics of sloshing In this paper beatingphenomenon of sloshing in different aspect ratios is consid-ered The shape of the tank is unchangeable but the depthsof liquid are set to be many different values Under the sameexternal excitation the vibrations of free surface and sloshingforces acting on the walls are calculated and analyzedThe natural frequency of sloshing is affected by differentaspect ratios whichmay change the whole sloshing situationThe effect of nonlinear factor on sloshing force is alsodiscussed

120585

b

h

X(t)X

Z

O

z

xo

Figure 1 2D sloshing model

2 Mathematical Model

The fluid is assumed to be incompressible irrotational andinviscid water so the potential flow theory is employed todescribe the sloshing in tanks Surface tension is neglectedThe free surface is assumed to never become overturnedor broken during the sloshing process 119887 and 119889 are thelength and the width of the rectangular tank respectivelyℎ is the still water depth Two Cartesian coordinate systemsare introduced in Figure 1 One is the inertial Cartesiancoordinate system (119883 119885) fixed in space 119909-axis is horizontaland 119911-axis is vertical The other is the moving coordinatesystem (119909 119911) connected to the tank with the origin at theintersection of undisturbed free surface and the left wallof the tank The tank is expected to have a translationaloscillation along the 119909-axis so there is no flow along thewidth direction of the tank which means the sloshing ofliquid is only 2D nonlinear motion in coordinate systemsThe displacement of the tank is expressed as 119883(119905) Beingconsidered in moving coordinate system (119909 119911) fixed to thetank the sloshing equations can be gotten as

1205972120593

1205971199092+

1205972120593

1205971199112= 0 (1)

120597120593

120597119909

10038161003816100381610038161003816100381610038161003816119909=0119887

= 0 (2)

120597120593

120597119911

10038161003816100381610038161003816100381610038161003816119911=minusℎ

= 0 (3)

120597120585

120597119905

=

120597120593

120597119911

10038161003816100381610038161003816100381610038161003816119911=120585

minus

120597120585

120597119909

120597120593

120597119909

10038161003816100381610038161003816100381610038161003816119911=120585

(4)

120597120593

120597119905

10038161003816100381610038161003816100381610038161003816119911=120585

= minus119892120585 minus 11990911988310158401015840(119905) minus

1

2

[(

120597120593

120597119909

)

2

+ (

120597120593

120597119911

)

2

]

119911=120585

(5)

Equation (1) is the continuity equation of ideal fluid 120593denotes velocity potential Equations (2) and (3) are rigidwall boundary conditionsThey indicate that the componentsof the fluid relative velocity normal to the walls are equalto zero Equations (4) and (5) are boundary conditions

Mathematical Problems in Engineering 3

of the free surface One is the kinematic condition theother is the dynamic condition 120585(119909 119905) is the free surfaceelevation deviating the undisturbed water level in themovingcoordinate system (119909 119911) 119892 denotes acceleration of gravity If(1)ndash(5) are solved the pressure in the fluid can be calculatedby the incompressible Bernoulli equation

minus

119901

120588

=

120597120593

120597119905

+ 119892119911 + 11990911988310158401015840(119905) +

1

2

[(

120597120593

120597119909

)

2

+ (

120597120593

120597119911

)

2

] (6)

where 119901 is the pressure and 120588 is the density of fluid Thesloshing force is determined by integrating the liquid pressureover the tank wall area Because the tank is moving alongthe 119909-axis and the sloshing models are 2D cases only 119909components of sloshing force would have great influence onthe dynamics of the tank In this paper the sloshing forceacting on the left and right walls is considered

3 Numerical Process

According to Frandsen [16] 120590-transformation method isused

120590 =

119911 + ℎ

ℎ + 120585

Φ (119909 120590 119905) = 120593 (119909 119911 119905) (7)

A new variable 120590 is introduced The derivatives of thepotential function 120593(119909 119911 119905) with respect to 119909 119911 and 119905 aretransformed into derivatives ofΦ(119909 120590 119905)The first derivativesof 120593 can be expressed as

120597120593

120597119909

=

120597Φ

120597119909

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

120597120593

120597119911

=

1

ℎ + 120585

120597Φ

120597120590

120597120593

120597119905

=

120597Φ

120597119905

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119905

(8)

Similarly it is not difficult to obtain the second derivatives of120593(119909 119911 119905) Then (1)ndash(5) can be transformed by using variablesubstitution

1205972Φ

1205971199092minus

2120590

ℎ + 120585

120597120585

120597119909

1205972Φ

120597119909120597120590

+ [

2120590

(ℎ + 120585)2(

120597120585

120597119909

)

2

minus

120590

ℎ + 120585

1205972120585

1205971199092]

120597Φ

120597120590

+ [(

120590

ℎ + 120585

120597120585

120597119909

)

2

+

1

(ℎ + 120585)2]

1205972Φ

1205971205902= 0

(9)

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119909=0119887

minus (

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

119909=0119887

= 0 (10)

120597Φ

120597120590

10038161003816100381610038161003816100381610038161003816120590=0

= 0 (11)

120597120585

120597119905

=

1

ℎ + 120585

120597Φ

120597120590

10038161003816100381610038161003816100381610038161003816120590=1

minus

120597120585

120597119909

(

120597Φ

120597119909

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

120590=1

(12)

(

120597Φ

120597119905

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119905

)

120590=1

= minus119892120585 minus 11990911988310158401015840(119905)

minus

1

2

[(

120597Φ

120597119909

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

2

+ (

1

ℎ + 120585

120597Φ

120597120590

)

2

]

120590=1

(13)

The fluid domain of (1) is 0 lt 119909 lt 119887minusℎ lt 119911 lt 120585(119909 119905) After 120590-transformation it has changed into a fixed rectangular area0 lt 119909 lt 119887 0 lt 120590 lt 1

A finite difference method has been developed to solve(9)ndash(13) Δ119905 is set to be the step size of time In the iterativeprocess of time assuming that the values of velocity potentialfunctionΦ and the free surface function 120585 are known at time119896Δ119905 that is on 119896 time step the aim is to obtain the valuesof the two functions on 119896 + 1 time step Some difficulties areencountered in the numerical algorithm First the governingequation cannot be solved directly on 119896+ 1 time step becausefree surface 120585 which is unknown at that time is a boundarycurve of the governing equation Second there are nonlinearterms existing in free surface boundary conditions

For the first problem a prediction method has beenintroduced Considering (9)ndash(11) on 119896 + 1 time step 120585 shouldhave been assigned to the value on 119896 + 1 time step but itis unknown On the other hand (9) on 119896 + 1 time step wastransformed from

1205972120593

1205971199092+

1205972120593

1205971199112

119896+1

= 0 0 lt 119909 lt 119887 minusℎ lt 119911 lt 120585119896+1 (14)

120585 does not appear in the governing equation it is only aboundary curve An explicit scheme of (4) can be adopted tomake prediction of 120585119896+1 and then substitute it into (14) So thefour boundary curves of (14) are determined which meansthat there is only one unknown quantity 120593119896+1 in (14)

This idea should be implemented in (9)ndash(11) Because thecomputational domain is transformed an explicit scheme of(12) is used to make prediction of 120585119896+1

120585

119896+1

= 120585119896+ Δ119905 [

1

ℎ + 120585119896

120597Φ119896

120597120590

minus

120597120585119896

120597119909

(

120597Φ119896

120597119909

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120597120585119896

120597119909

)]

120590=1

(15)

120585

119896+1

denotes the prediction value Substitute it into (9) and(10) on 119896+1 time step then onlyΦ119896+1 is the unknownquantity

4 Mathematical Problems in Engineering

in the two equations Equation (11) does not have 120585 so thereis no need to do so These three equations are discretized as

1205972Φ

1205971199092minus

2120590

ℎ + 120585

120597120585

120597119909

1205972Φ

120597119909120597120590

+[

[

2120590

(ℎ + 120585)

2(

120597120585

120597119909

)

2

minus

120590

ℎ + 120585

1205972120585

1205971199092

]

]

120597Φ

120597120590

119896+1

+

[

[

(

120590

ℎ + 120585

120597120585

120597119909

)

2

+

1

(ℎ + 120585)

2

]

]

1205972Φ

1205971205902

119896+1

= 0

(16)

120597Φ119896+1

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0119887

minus (

120590

ℎ + 120585

120597Φ119896+1

120597120590

120597120585

120597119909

)

119909=0119887

= 0 (17)

120597Φ119896+1

120597120590

100381610038161003816100381610038161003816100381610038161003816120590=0

= 0 (18)

For the second problem an approximate method hasbeen introduced Consider two boundary conditions (12) and(13) on 119896 + 12 time step they are complicated nonlinearequations For example

120597120585

120597119909

(

120597Φ

120597119909

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

120590=1

(19)

is a nonlinear term it should have been assigned to the valueon 119896 + 12 time step in the calculation process but here itis assigned to the value on 119896 time step as an approximationOther nonlinear terms are dealt with by the same methodLinear terms in equations are discretized by Crank-Nicolsonscheme where 120585119896+1 and Φ119896+1 (120590 = 1) are set to be unknownquantities It is noteworthy that 120585119896+1 is set to be the unknownquantity in (12) and (13) so as to update the free surfacewhen the equations are solved Overall these two boundaryconditions are discretized as

120585119896+1

minus 120585119896

Δ119905

= [

1

ℎ + 120585119896(

1

2

120597Φ119896+1

120597120590

+

1

2

120597Φ119896

120597120590

)

minus

120597120585119896

120597119909

(

120597Φ119896

120597119909

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120597120585119896

120597119909

)]

120590=1

(20)

(

Φ119896+1

minus Φ119896

Δ119905

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120585119896+1

minus 120585119896

Δ119905

)

120590=1

= minus119892(

120585119896+ 120585119896+1

2

) minus 11990911988310158401015840(119905119896+12

)

minus

1

2

[(

120597Φ119896

120597119909

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120597120585119896

120597119909

)

2

+(

1

ℎ + 120585119896

120597Φ119896

120597120590

)

2

]

120590=1

(21)

Semi-implicit scheme equations of 120585119896+1 andΦ119896+1 (120590 = 1)have been derived Moreover (16)ndash(21) can be discretized inspace and solved by finite difference method The solution120585119896+1 will be taken as the corrected value of 120585 on 119896 + 1 timestep

After 120590-transformation (6) has been transformed as

minus

119901

120588

= (

120597Φ

120597119905

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119905

) + 119892 [(ℎ + 120585) 120590 minus ℎ] + 11990911988310158401015840(119905)

+

1

2

[(

120597Φ

120597119909

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

2

+ (

1

ℎ + 120585

120597Φ

120597120590

)

2

]

(22)

When the values of 120585 and Φ are solved it is not difficult todiscretize (22) In the numerical simulation sloshing force onthe left and right walls can be expressed as

119865119871= int

119878119871

119901119889119904 = 119889int

120585

minusℎ

1199011003816100381610038161003816119909=0

119889119911

119865119877= int

119878119877

119901119889119904 = 119889int

120585

minusℎ

1199011003816100381610038161003816119909=119887

119889119911

(23)

The resultant sloshing force acting on the tank is

119865 = 119865119877minus 119865119871 (24)

4 Numerical Results

Some examples of nonlinear sloshing are simulated by thisnumerical method They have been investigated by otherresearchers using various numerical and analytical methodsand many discussions and conclusions have been derived

A rectangular tank model is established The naturalfrequencies of linear transverse sloshing along 119909-axis are(Faltinsen [2])

120596119899= radic119892

119899120587

119887

tanh 119899120587ℎ119887

119899 = 1 2 3 (25)

41 Standing Waves The tank is fixed so the moving coor-dinate system (119909 119911) coincides with the inertial coordinatesystem (119883 119885) The initial conditions are described as

120585 (119909 0) = 119886 cos(119899120587119887

) 120593 (119909 119911 0) = 0 (26)

119886 is the amplitude of the initial wave profile which influencesthe nonlinearity of the sloshing 119899 indicates themodenumberThe convergence of this numerical simulation is investigatedIn the tank 119887 119889 and ℎ are set to be 2m 1m and 1mrespectively 119886 is set to be 02m 119899 takes value of 1 Thefirst natural frequency of the sloshing is 120596

1= 376 rads

Figure 2(a) shows the time history of the free surface at theleft wall with different meshes but the time step sizes arethe same (0005 s) Figure 2(b) shows the time history wherethe meshes are the same (40 times 40) but the time step sizes

Mathematical Problems in Engineering 5

20 40 60 80 100

00

05

10

15

minus05

minus10

Mesh 40 times 40Mesh 60 times 60

1205961t

120585a

(a) Free surface elevation at the left wall using different meshes

Step 001Step 0005

20 40 60 80 1001205961t

00

05

10

15

minus05

minus10

120585a

(b) Free surface elevation at the left wall using different time step sizes

Figure 2 Time history of free surface elevation for different meshes and time step sizes

20 40 60 80 100

00

05

10

15

minus05

minus10

1205961t

120585a

AnalyticalNumerical

(a) Comparison of free surface elevation at the left wall

05 10 15 20x

00

05

10

15

minus05

minus10

120585a

(b) Wave profiles for a half-period

20 40 60 80 100

020

022

024

026

028

030

1205961t

FL

120588gbd

h

(c) Sloshing force on the left wall

20 40 60 80 100

020

022

024

026

028

030

1205961t

FR

120588gbd

h

(d) Sloshing force on the right wall

Figure 3 Standing wave (119899 = 1)

6 Mathematical Problems in Engineering

20 40 60 80 100

00

05

10

AnalyticalNumerical

1205961t

minus05

minus10

120585a

(a) Comparison of free surface elevation at the left wall

05 10 15 20x

00

05

10

minus05

minus10

120585a

(b) Wave profiles for a half-period

Figure 4 Standing wave (119899 = 3)

are different The time and the vibration displacement in thefigures are dealt with by dimensionless method

1205851015840=

120585

119886

1199051015840= 1205961119905 (27)

It is found that the results from these meshes and timestep sizes are in good agreement so in this example a meshdistribution of 40 times 40 and time step size of 0005 s ensure areasonable converged solution

Nonlinear sloshing equations of standing waves can besolved by perturbation techniques First nondimensionalvariables are introduced to the equations Then 120598 = 119886120596

2

119899119892

is chosen as the characteristic small parameter Approximatelinear equations are derived by small parameter expansionand approximate analytical solutions can be obtained Thesecond-order approximate analytical free surface elevationfor the 119899th sloshing mode is prescribed as (Frandsen [16])

120585 (119909 119905) = 119886 ( cos (120596119899119905) cos (119896

119899119909)

+ 120598 (119860119899+ 119861119899cos (2120596

119899119905)

+119862119899cos (120596

2119899119905)) cos (2119896

119899119909))

(28)

where

119896119899=

119899120587

119887

1205962119899= radic1198922119896

119899tanh (2119896

119899ℎ)

119860119899=

1205964

119899+ 11989221198962

119899

81205964

119899

119861119899=

31205964

119899minus 11989221198962

119899

81205964

119899

minus

3

2

1205964

119899minus 11989221198962

119899

1205962

119899(41205962

119899minus 1205962

2119899)

119862119899=

1205962

1198991205962

2119899minus 1205964

119899minus 311989221198962

119899

21205962

119899(41205962

119899minus 1205962

2119899)

(29)

Figure 3(a) shows the time history of the free surfaceelevation for 119899 = 1 at the leftwall by the twomethods numer-ical solution by the finite difference method proposed in thispaper and second-order approximate analytical solution byperturbation method Initially good agreement between thetwo results is achieved but the phase difference between themincreases with time This phenomenon has been noticed anddiscussed by many researchers (Chern et al [18] Frandsen[16]) The conclusion is that the second-order analyticalsolution is not very accurate for large amplitude sloshingFigure 3(b) displays numerical wave profiles for nearly halfa period from dimensionless time 120596

1119905 = 318 to 120596

1119905 =

346 Stationary point does not exist and nonlinearity isobvious Figures 3(c) and 3(d) provide the sloshing forceson the left wall and right wall respectively (the forces arealso nondimensionalized) It shows that slight double peaksappear in the time historyThis has also been observed byWuet al [5]

The standing wave of 119899 = 3 is also calculated The initialconditions are set to be

120585 (119909 0) = 006 cos(3120587119887

) 120593 (119909 119911 0) = 0 (30)

Figure 4(a) shows the comparison of numerical solutionsand approximate analytical solutions Figure 4(b) showsnumerical wave profiles for nearly half a period Phasedifference and nonlinearity are also observed

42 Sloshing Caused by Horizontal Excitations The tankis assumed to take the horizontal oscillation the externalexcitation acting on the tank is defined as harmonic motion

119883 (119905) = 119886ℎcos (120596

ℎ119905) (31)

119886ℎand 120596

ℎare the amplitude and angular frequency of the

excited motionThe fluid is assumed to be stationary at 119905 = 0Then in the moving coordinate system the initial conditionsare described as

120585 (119909 0) = 0 120593 (119909 119911 0) = minus1199091198831015840(119905) = 0 (32)

Mathematical Problems in Engineering 7

2

1

0

minus2

minus1

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(a) Example 1 120596ℎ = 071205961 120598ℎ = 0036

2

0

minus2

4

minus4

6

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(b) Example 2 120596ℎ = 131205961 120598ℎ = 0072

Figure 5 Comparison of free surface elevation at the left wall

20 40 60 80 100

0

20

40

60

minus20

120585a

1205961t

(a) The first resonance mode

20 40 60 80 100

0

10

20

minus10

1205963t

120585a h

(b) The third resonance mode

Figure 6 Free surface elevation at the left wall (resonance phenomenon)

20 40 60 80 100

000

005

010

015

minus015

minus010

minus005F120588gbd

h

1205961t

(a) The first resonance mode

20 40 60 80 100

000

001

002

minus002

minus001

1205963t

F120588gbd

h

(b) The third resonance mode

Figure 7 Resultant sloshing force (resonance phenomenon)

In order to verify the precision of this numerical methodfor the sloshing models in horizontally moving tanks twoexamples are calculated and compared with the solutions ofFrandsen [16] (1) 120596

ℎ= 07120596

1 120598ℎ= 0036 (2) 120596

ℎ= 13120596

1

120598ℎ= 0072 The grid size of the numerical method is 40 times 40

and the time step size is 0006 s Figure 5(a) shows that the twosimulations are in close agreement In the second examplethe nonlinear effect is strong so there is a little differencebetween the two solutions in Figure 5(b) Neverthelessnonlinear characteristics of sloshing are captured by the two

8 Mathematical Problems in Engineering

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 005

20 40 60 80 100 120

0

20

40

60

minus20

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 8 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 12)

20 40 60 80 100

0

2

4

6

minus2

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 003

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 9 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 18)

02 04 06 08 10 12 14

02

04

06

08

10

1205961120596

lowast 1

hb

Figure 10 The function graph of the first natural frequency againstaspect ratio

numerical methods such as the higher peaks and the lessdeep troughs

421 Resonance Phenomenon Two situations of resonanceare considered the first resonance mode and the thirdresonance mode The external excitations of the tank aredescribed as 119883

1(119905) = 0007 cos(120596

1119905) 1198832(119905) = 0007 cos(120596

3119905)

The amplitudes are the same but the excited frequencies

are different The time histories of free surface elevation atthe left wall and the resultant sloshing forces are calculatedFigure 6(a) shows that the sloshing of the first resonancemode is violate the amplitude of the vibration is increasingFigure 6(b) shows that the amplitude of the third resonancemode is much smaller than that of the first resonance modeIn Figure 7 it can be concluded that the resultant sloshingforce of the third resonance mode is trivial compared withthe force of the first resonance mode In fact the first naturalfrequency of sloshing is the most influential of all the naturalfrequencies

422 The Limitation of the Numerical Method The presentnumerical method is employed to simulate nonlinear slosh-ing in moving tanks If the nonlinearity is quite strong thismethodmay lose its effectiveness Two excitation frequenciesare considered in the simulations One is 120596

ℎ= 085120596

1 the

other is 120596ℎ= 095120596

1 which indicates near resonant sloshing

situation Different excitation amplitudes 119886ℎand aspect ratios

ℎ119887 are tested and the limitation of the convergence of thisnumerical method is found out

First the aspect ratio is set to be ℎ119887 = 12 just asbefore Excitation amplitude is increased in the numericalsimulation and the sloshing becomesmore andmore violentFor 120596

ℎ= 085120596

1 if the excitation amplitude is 119886

ℎ= 006m

Mathematical Problems in Engineering 9

20 40 60 80 100

0

5

10

15

20

minus5

minus10

120585a h

120596ht

(a) Aspect ratio ℎ119887 = 05

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) Aspect ratio ℎ119887 = 075

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(c) Aspect ratio ℎ119887 = 1

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 11 Free surface elevation at the left wall (low-frequency excitation)

the simulation is divergent after several times of iteration if119886ℎ= 005m the simulation lasts for a long time So it is

considered that the limitation of 119886ℎfor nonresonant sloshing

is 005m For 120596ℎ= 095120596

1 the excitation amplitude is much

smaller The maximum limit of 119886ℎis estimated to be 0012m

The time histories of free surface elevation at the left wall inthe limit cases are shown in Figure 8

Then the aspect ratio is changed by decreasing the stillwater depth ℎ For the aspect ratio ℎ119887 = 18 the numericalmethod still works with a mesh size of 80 times 20 If the aspectratio is less than 18 the simulation is divergent As a resultthe minimum limit of aspect ratio ℎ119887 is considered to be18 In this critical aspect ratio situation for 120596

ℎ= 095120596

1

the maximum limit of excitation amplitude 119886ℎis still 0012m

while for 120596ℎ= 085120596

1 the maximum limit of 119886

ℎis 003m

Figure 9 shows the time histories of free surface elevation inthe limit cases in this critical aspect ratio The phenomena ofbeating and resonance are very obscure it can be concludedthat the sloshing in low aspect ratio is quite different from theone in normal aspect ratio

423 Beating Phenomenon in Different Aspect Ratios Slosh-ing in a fixed aspect ratio has been studied in many articlesHowever little work has been done to analyze the response ofliquid sloshing in different aspect ratios by the same external

excitation When the external excitation is close to the firstnatural frequency of sloshing the beating phenomenon canbe observed The study here is intended to investigate theeffect of different aspect ratios on beating phenomenon andsloshing force

In the numerical examples calculated before the lengthand width of the tank are set to be 2m and 1m the still waterdepth is 1m The first natural frequency of sloshing is 120596

1=

376 rads The aspect ratio is ℎ119887 = 12 Now the length 119887and the width 119889 of the tank are fixed and the aspect ratio ischanged by changing the still water depth ℎThe former waterdepth and natural frequency are set to be the characteristicquantity ℎlowast = 1m 120596lowast

1= 376 rads

The relationship between aspect ratio and the first naturalfrequency must be considered From (25) it is easy to get thegraph of the natural frequency as a function of aspect ratioFigure 10 indicates that the slope of the function curve is steepwhen the aspect ratio is small if the aspect ratio is greater than06 the function curve is nearly horizontal

Under the same external excitation the sloshing forcesof beating are affected by three factors proximity of exci-tation frequency to the natural frequency liquid mass andnonlinearity of sloshing When the aspect ratio of the liquidchanges these three factors would change along with it Inorder to discuss the effect of aspect ratio two horizontaloscillations of the tank are studied First the tank is subjected

10 Mathematical Problems in Engineering

20 40 60 80 100

00

01

02

minus01

F120588gbd

hlowast

120596ht

(a) Aspect ratio ℎ119887 = 05

005

010

015

minus005

minus010

minus015

F120588gbd

hlowast

20 40 60 80 100120596ht

000

(b) Aspect ratio ℎ119887 = 075

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(c) Aspect ratio ℎ119887 = 1

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 12 Resultant sloshing forces (low-frequency excitation)

06 07 08 09 10 11 12

019

020

021

022

LinearNonlinear

F120588gbd

hlowast

hb

Figure 13 Tendency of amplitude of sloshing force (low-frequencyexcitation)

to a low-frequency excitation The movement of the tank isdescribed as 119886

ℎ= 003m 120596

ℎ= 34 rads Figure 10 suggests

that if the aspect ratio is 03 the corresponding naturalfrequency is 34 rads Our aim is to analyze the influence ofthis low-frequency excitation on the high aspect ratio slosh-ing system Figure 11 shows the time histories of free surface

elevation at the left wall in four different aspect ratios wherethe parameters are nondimensionalized

1199051015840= 120596ℎ119905 120585

1015840=

120585

119886ℎ

(33)

The amplitude of free surface elevation is quite large inFigure 11(a) which shows strong nonlinearity of sloshingwhile in Figures 11(b) 11(c) and 11(d) the amplitudes arenearly the same smaller than the amplitude in Figure 11(a)so the nonlinear effects are weaker in these three sloshingsituations

Figure 12 shows the time histories of resultant sloshingforce acting on the tank (the forces are also nondimensional-ized 1198651015840 = 119865120588119892119887119889ℎlowast) The amplitudes of the sloshing forceare extracted in Figure 12 In order to study the effect ofnonlinear factor linear sloshing models by Abramson [1] arecalculated under the same condition to make comparisonsThe relationship between the amplitude of sloshing force andaspect ratio is shown in Figure 13 It is obvious that bothlinear and nonlinear results have the same trend decreasingfirst and then increasing On the one hand the rise of theaspect ratio increases the natural frequency which makesthe sloshing far from resonance and reduces sloshing forcesOn the other hand the rise of the aspect ratio increases themass of the liquid thereby increasing the sloshing forces

Mathematical Problems in Engineering 11

0

10

20120585a h

20 40 60 80 100120596ht

minus10

(a) Aspect ratio ℎ119887 = 05

0

5

10

15

minus5

120585a h

20 40 60 80 100120596ht

minus10

(b) Aspect ratio ℎ119887 = 045

0

5

10

120585a h

20 40 60 80 100120596ht

minus5

minus10

(c) Aspect ratio ℎ119887 = 04

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 035

Figure 14 Free surface elevation at the left wall (high-frequency excitation)

For the sloshing in aspect ratio ℎ119887 = 05 the excitationfrequency is close to the natural frequency If aspect ratiois increased the factor of reducing sloshing force is moredominant However for the sloshing in high aspect ratiosℎ119887 gt 075 the rise of aspect ratio brings about slight changesin the natural frequency so the factor of increasing sloshingforce is more dominant Comparing nonlinear results withlinear ones in Figure 13 some conclusions are obtained Firstthe nonlinearity of sloshing always decreases sloshing forceSecond to a certain extent the difference between linear andnonlinear amplitude reflects the effect of nonlinear factoron sloshing force For the sloshing in aspect ratio ℎ119887 =

05 the difference is the largest while for the other sloshingsituations the differences are nearly the same This resultaccordswith the conclusion in Figure 11Third nonlinearity isan important factor in large amplitude sloshing which shouldnot be neglected

Then the tank is subjected to a high-frequency excitationThe movement of the tank is described as 119886

ℎ= 0015m

120596ℎ= 4 rads Our aim is to analyze the influence of this

high-frequency excitation on the low aspect ratio sloshingsystem Figure 14 shows the time histories of free surfaceelevation at the left wall in four different aspect ratios Asthe aspect ratio decreases the amplitude and period of freesurface elevation are diminishing For the sloshing in lowaspect ratios ℎ119887 lt 05 the reduction of aspect ratio brings

about significant changes in the natural frequency so thebeating phenomenon has changed greatly even if the aspectratio is slightly decreasedThe resultant sloshing forces actingon the tank are also calculated The relationship between theamplitude of sloshing force and aspect ratio is showed inFigure 15The reduction of the aspect ratio makes the naturalfrequency far from the excitation frequency meanwhile itdecreases the mass of the liquid in the tank So both of thefactors reduce sloshing force As the aspect ratio decreasesthe difference between linear and nonlinear amplitude ofsloshing force also decreases The reason is that nonlineareffects are becoming weaker and the sloshing force itself isalso decreasing

5 Conclusions

Anewfinite differencemethod based on the prediction of freesurface has been developed and used to simulate nonlinearsloshing phenomenon in tanks by the potential flow theoryAfter 120590-transformation the fluid domain is transformed to arectangle and a predictor-corrector scheme for free surfaceis used in the iteration of time then the governing equationand boundary conditions are approximately linearized andsolved in every time step The convergence of this methodis surveyed The numerical examples of standing waves and

12 Mathematical Problems in Engineering

035 040 045 050

006

008

010

012

014

016

LinearNonlinear

F120588gbd

hlowast

hb

Figure 15 Tendency of amplitude of sloshing force (high-frequencyexcitation)

horizontally forced sloshing models have been calculatedand good agreement between the numerical simulation andresults in other literatures has been obtained This numericalmethod is simple and easy in programming with excellentaccuracy and stability and can be extended to other sloshingproblems The time histories of free surface elevation andsloshing forces in the first and third resonance situation arealso presented and analyzed

The beating phenomenon of sloshing in different aspectratios under the same excitation has been numerically simu-lated by this method The results show that the effect of theaspect ratio on the sloshing force is complicated For the low-frequency excitation acting on the high aspect ratio modelsthe rise of aspect ratio makes the sloshing far from resonanceand increases the liquid mass so the sloshing force decreasesfirst and then increases For the high-frequency excitationacting on the low aspect ratio models the reduction of aspectratiomakes the sloshing far from resonance and decreases theliquid mass so the sloshing force decreases all the way Sincethe first natural frequency of liquid sloshing in not a linearfunction of aspect ratio the change of aspect ratio producesmuch stronger effect in low aspect ratio sloshing systemsMoreover nonlinearity of sloshing always decreases sloshingforce The effect of nonlinear factor depends on the violenceof sloshing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H N Abramson ldquoThe dynamic behavior of liquids in movingcontainers nasa sp-106rdquo NASA Special Publication vol 1061966

[2] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 no 4 pp 114ndash241 1974

[3] O M Faltinsen ldquoA numerical nonlinear method of sloshing intanks with two-dimensional flowrdquo Journal of Ship Research vol22 no 3 pp 193ndash202 1978

[4] T Nakayama and K Washizu ldquoNonlinear analysis of liquidmotion in a container subjected to forced pitching oscillationrdquoInternational Journal for Numerical Methods in Engineering vol15 no 8 pp 1207ndash1220 1980

[5] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[6] B-F Chen and R Nokes ldquoTime-independent finite differenceanalysis of fully non-linear and viscous fluid sloshing in arectangular tankrdquo Journal of Computational Physics vol 209 no1 pp 47ndash81 2005

[7] C Wu and B Chen ldquoSloshing waves and resonance modesof fluid in a 3D tank by a time-independent finite differencemethodrdquoOcean Engineering vol 36 no 6-7 pp 500ndash510 2009

[8] Y Kim ldquoNumerical simulation of sloshing flows with impactloadrdquo Applied Ocean Research vol 23 no 1 pp 53ndash62 2001

[9] M S Celebi and H Akyildiz ldquoNonlinear modeling of liquidsloshing in a moving rectangular tankrdquo Ocean Engineering vol29 no 12 pp 1527ndash1553 2002

[10] C Z Wang and B C Khoo ldquoFinite element analysis of two-dimensional nonlinear sloshing problems in random excita-tionsrdquo Ocean Engineering vol 32 no 2 pp 107ndash133 2005

[11] V Sriram S A Sannasiraj and V Sundar ldquoNumerical simula-tion of 2D sloshingwaves due to horizontal and vertical randomexcitationrdquo Applied Ocean Research vol 28 no 1 pp 19ndash322006

[12] O M Faltinsen O F Rognebakke I A Lukovsky and AN Timokha ldquoMultidimensional modal analysis of nonlinearsloshing in a rectangular tank with finite water depthrdquo Journalof Fluid Mechanics vol 407 pp 201ndash234 2000

[13] O M Faltinsen and A N Timokha ldquoAn adaptive multimodalapproach to nonlinear sloshing in a rectangular tankrdquo Journalof Fluid Mechanics vol 432 pp 167ndash200 2001

[14] G XWu ldquoSecond-order resonance of sloshing in a tankrdquoOceanEngineering vol 34 no 17-18 pp 2345ndash2349 2007

[15] D F Hill ldquoTransient and steady-state amplitudes of forcedwaves in rectangular basinsrdquo Physics of Fluids vol 15 no 6 pp1576ndash1587 2003

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004

[17] RD Firouz-AbadiMGhasemi andHHaddadpour ldquoAmodalapproach to second-order analysis of sloshing using boundaryelement methodrdquo Ocean Engineering vol 38 no 1 pp 11ndash212011

[18] M J Chern A G L Borthwick and R E Taylor ldquoA pseu-dospectral 120590-transformation model of 2-D nonlinear wavesrdquoJournal of Fluids and Structures vol 13 no 5 pp 607ndash630 1999

[19] J B Frandsen and A G L Borthwick ldquoSimulation of sloshingmotions in fixed and vertically excited containers using a 2-Dinviscid120590-transformed finite difference solverrdquo Journal of Fluidsand Structures vol 18 no 2 pp 197ndash214 2003

[20] M S Turnbull A G L Borthwick and R Eatock TaylorldquoNumerical wave tank based on a 120590-transformed finite elementinviscid flow solverrdquo International Journal for Numerical Meth-ods in Fluids vol 42 no 6 pp 641ndash663 2003

[21] B Chen andH Chiang ldquoComplete 2D and fully nonlinear anal-ysis of ideal fluid in tanksrdquo Journal of Engineering Mechanicsvol 125 no 1 pp 70ndash78 1999

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

alwaysmoving in the sloshing process Some special methodsare used to treat the moving boundary of free surfacesuch as VOF and ALE Recently 120590-transformation has beenemployed in sloshing problems It is an efficient spacetransformationmethod especially suitable for equationswithcurved boundary The basic idea of 120590-transformation is tointroduce a new stretching variable in the vertical directionwhich transforms the computational domain into a regularshape Chern et al [18] and Frandsen et al [16 19] used 120590-transformation to map the liquid domain onto rectangularregion and then calculated horizontally and vertically forcedsloshing problems in tanks Turnbull et al [20] simulatedseveral 2D free surface wave motions by 120590-transformedfinite element model According to Frandsenrsquos analysis 120590-transformation has two major advantages in simulating freesurface flows remeshing due to the moving free surface isavoided and free surface smoothing is often not requiredThe disadvantage of it is that it is a unique transformationand restricts the wave shape to be nonoverturning In thispaper a new numerical method is proposed based on 120590-transformation In the iterative process of time a forecastscheme is used to estimate the free surface boundary andsome nonlinear terms are approximated Then the governingequation and boundary conditions are linearized and solvedby finite differencemethod Inmany numericalmethods freesurface is often updated directly by explicit schemes suchas Adams-Bashforth or Runge Kutta methods while in thismethod a predictor-corrector scheme of free surface is usedThe free surface is first predicted before the iteration so thefluid boundary is more precise for the numerical simulationafter the governing equation and boundary conditions aresolved the free surface is updated by the solutions Because ofthe predictive step of free surface it is reasonable to considerhigh accuracy is one of the advantages of this method Fur-thermore semi-implicit Crank-Nicolsen scheme is employedto discretize linear terms in free surface boundary conditionsso this method has better numerical stability than explicitschemes Nonlinear sloshing in fixed and horizontally excitedrectangular tanks has been simulated by this numericalmethod The results are compared with analytical solutionsor other numerical results which prove the efficiency of thismethod The limitation of this numerical method is alsodescribed

Many researches have focused on sloshing phenomenonin a fixed aspect ratio by different excitations In fact theaspect ratio is always changed in the process of storageor transportation of liquid cargo So it is necessary toinvestigate the influence of aspect ratio on liquid sloshingChen and Chiang [21] analyzed the effect of fluid depth onnonlinear characteristics of sloshing In this paper beatingphenomenon of sloshing in different aspect ratios is consid-ered The shape of the tank is unchangeable but the depthsof liquid are set to be many different values Under the sameexternal excitation the vibrations of free surface and sloshingforces acting on the walls are calculated and analyzedThe natural frequency of sloshing is affected by differentaspect ratios whichmay change the whole sloshing situationThe effect of nonlinear factor on sloshing force is alsodiscussed

120585

b

h

X(t)X

Z

O

z

xo

Figure 1 2D sloshing model

2 Mathematical Model

The fluid is assumed to be incompressible irrotational andinviscid water so the potential flow theory is employed todescribe the sloshing in tanks Surface tension is neglectedThe free surface is assumed to never become overturnedor broken during the sloshing process 119887 and 119889 are thelength and the width of the rectangular tank respectivelyℎ is the still water depth Two Cartesian coordinate systemsare introduced in Figure 1 One is the inertial Cartesiancoordinate system (119883 119885) fixed in space 119909-axis is horizontaland 119911-axis is vertical The other is the moving coordinatesystem (119909 119911) connected to the tank with the origin at theintersection of undisturbed free surface and the left wallof the tank The tank is expected to have a translationaloscillation along the 119909-axis so there is no flow along thewidth direction of the tank which means the sloshing ofliquid is only 2D nonlinear motion in coordinate systemsThe displacement of the tank is expressed as 119883(119905) Beingconsidered in moving coordinate system (119909 119911) fixed to thetank the sloshing equations can be gotten as

1205972120593

1205971199092+

1205972120593

1205971199112= 0 (1)

120597120593

120597119909

10038161003816100381610038161003816100381610038161003816119909=0119887

= 0 (2)

120597120593

120597119911

10038161003816100381610038161003816100381610038161003816119911=minusℎ

= 0 (3)

120597120585

120597119905

=

120597120593

120597119911

10038161003816100381610038161003816100381610038161003816119911=120585

minus

120597120585

120597119909

120597120593

120597119909

10038161003816100381610038161003816100381610038161003816119911=120585

(4)

120597120593

120597119905

10038161003816100381610038161003816100381610038161003816119911=120585

= minus119892120585 minus 11990911988310158401015840(119905) minus

1

2

[(

120597120593

120597119909

)

2

+ (

120597120593

120597119911

)

2

]

119911=120585

(5)

Equation (1) is the continuity equation of ideal fluid 120593denotes velocity potential Equations (2) and (3) are rigidwall boundary conditionsThey indicate that the componentsof the fluid relative velocity normal to the walls are equalto zero Equations (4) and (5) are boundary conditions

Mathematical Problems in Engineering 3

of the free surface One is the kinematic condition theother is the dynamic condition 120585(119909 119905) is the free surfaceelevation deviating the undisturbed water level in themovingcoordinate system (119909 119911) 119892 denotes acceleration of gravity If(1)ndash(5) are solved the pressure in the fluid can be calculatedby the incompressible Bernoulli equation

minus

119901

120588

=

120597120593

120597119905

+ 119892119911 + 11990911988310158401015840(119905) +

1

2

[(

120597120593

120597119909

)

2

+ (

120597120593

120597119911

)

2

] (6)

where 119901 is the pressure and 120588 is the density of fluid Thesloshing force is determined by integrating the liquid pressureover the tank wall area Because the tank is moving alongthe 119909-axis and the sloshing models are 2D cases only 119909components of sloshing force would have great influence onthe dynamics of the tank In this paper the sloshing forceacting on the left and right walls is considered

3 Numerical Process

According to Frandsen [16] 120590-transformation method isused

120590 =

119911 + ℎ

ℎ + 120585

Φ (119909 120590 119905) = 120593 (119909 119911 119905) (7)

A new variable 120590 is introduced The derivatives of thepotential function 120593(119909 119911 119905) with respect to 119909 119911 and 119905 aretransformed into derivatives ofΦ(119909 120590 119905)The first derivativesof 120593 can be expressed as

120597120593

120597119909

=

120597Φ

120597119909

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

120597120593

120597119911

=

1

ℎ + 120585

120597Φ

120597120590

120597120593

120597119905

=

120597Φ

120597119905

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119905

(8)

Similarly it is not difficult to obtain the second derivatives of120593(119909 119911 119905) Then (1)ndash(5) can be transformed by using variablesubstitution

1205972Φ

1205971199092minus

2120590

ℎ + 120585

120597120585

120597119909

1205972Φ

120597119909120597120590

+ [

2120590

(ℎ + 120585)2(

120597120585

120597119909

)

2

minus

120590

ℎ + 120585

1205972120585

1205971199092]

120597Φ

120597120590

+ [(

120590

ℎ + 120585

120597120585

120597119909

)

2

+

1

(ℎ + 120585)2]

1205972Φ

1205971205902= 0

(9)

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119909=0119887

minus (

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

119909=0119887

= 0 (10)

120597Φ

120597120590

10038161003816100381610038161003816100381610038161003816120590=0

= 0 (11)

120597120585

120597119905

=

1

ℎ + 120585

120597Φ

120597120590

10038161003816100381610038161003816100381610038161003816120590=1

minus

120597120585

120597119909

(

120597Φ

120597119909

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

120590=1

(12)

(

120597Φ

120597119905

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119905

)

120590=1

= minus119892120585 minus 11990911988310158401015840(119905)

minus

1

2

[(

120597Φ

120597119909

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

2

+ (

1

ℎ + 120585

120597Φ

120597120590

)

2

]

120590=1

(13)

The fluid domain of (1) is 0 lt 119909 lt 119887minusℎ lt 119911 lt 120585(119909 119905) After 120590-transformation it has changed into a fixed rectangular area0 lt 119909 lt 119887 0 lt 120590 lt 1

A finite difference method has been developed to solve(9)ndash(13) Δ119905 is set to be the step size of time In the iterativeprocess of time assuming that the values of velocity potentialfunctionΦ and the free surface function 120585 are known at time119896Δ119905 that is on 119896 time step the aim is to obtain the valuesof the two functions on 119896 + 1 time step Some difficulties areencountered in the numerical algorithm First the governingequation cannot be solved directly on 119896+ 1 time step becausefree surface 120585 which is unknown at that time is a boundarycurve of the governing equation Second there are nonlinearterms existing in free surface boundary conditions

For the first problem a prediction method has beenintroduced Considering (9)ndash(11) on 119896 + 1 time step 120585 shouldhave been assigned to the value on 119896 + 1 time step but itis unknown On the other hand (9) on 119896 + 1 time step wastransformed from

1205972120593

1205971199092+

1205972120593

1205971199112

119896+1

= 0 0 lt 119909 lt 119887 minusℎ lt 119911 lt 120585119896+1 (14)

120585 does not appear in the governing equation it is only aboundary curve An explicit scheme of (4) can be adopted tomake prediction of 120585119896+1 and then substitute it into (14) So thefour boundary curves of (14) are determined which meansthat there is only one unknown quantity 120593119896+1 in (14)

This idea should be implemented in (9)ndash(11) Because thecomputational domain is transformed an explicit scheme of(12) is used to make prediction of 120585119896+1

120585

119896+1

= 120585119896+ Δ119905 [

1

ℎ + 120585119896

120597Φ119896

120597120590

minus

120597120585119896

120597119909

(

120597Φ119896

120597119909

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120597120585119896

120597119909

)]

120590=1

(15)

120585

119896+1

denotes the prediction value Substitute it into (9) and(10) on 119896+1 time step then onlyΦ119896+1 is the unknownquantity

4 Mathematical Problems in Engineering

in the two equations Equation (11) does not have 120585 so thereis no need to do so These three equations are discretized as

1205972Φ

1205971199092minus

2120590

ℎ + 120585

120597120585

120597119909

1205972Φ

120597119909120597120590

+[

[

2120590

(ℎ + 120585)

2(

120597120585

120597119909

)

2

minus

120590

ℎ + 120585

1205972120585

1205971199092

]

]

120597Φ

120597120590

119896+1

+

[

[

(

120590

ℎ + 120585

120597120585

120597119909

)

2

+

1

(ℎ + 120585)

2

]

]

1205972Φ

1205971205902

119896+1

= 0

(16)

120597Φ119896+1

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0119887

minus (

120590

ℎ + 120585

120597Φ119896+1

120597120590

120597120585

120597119909

)

119909=0119887

= 0 (17)

120597Φ119896+1

120597120590

100381610038161003816100381610038161003816100381610038161003816120590=0

= 0 (18)

For the second problem an approximate method hasbeen introduced Consider two boundary conditions (12) and(13) on 119896 + 12 time step they are complicated nonlinearequations For example

120597120585

120597119909

(

120597Φ

120597119909

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

120590=1

(19)

is a nonlinear term it should have been assigned to the valueon 119896 + 12 time step in the calculation process but here itis assigned to the value on 119896 time step as an approximationOther nonlinear terms are dealt with by the same methodLinear terms in equations are discretized by Crank-Nicolsonscheme where 120585119896+1 and Φ119896+1 (120590 = 1) are set to be unknownquantities It is noteworthy that 120585119896+1 is set to be the unknownquantity in (12) and (13) so as to update the free surfacewhen the equations are solved Overall these two boundaryconditions are discretized as

120585119896+1

minus 120585119896

Δ119905

= [

1

ℎ + 120585119896(

1

2

120597Φ119896+1

120597120590

+

1

2

120597Φ119896

120597120590

)

minus

120597120585119896

120597119909

(

120597Φ119896

120597119909

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120597120585119896

120597119909

)]

120590=1

(20)

(

Φ119896+1

minus Φ119896

Δ119905

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120585119896+1

minus 120585119896

Δ119905

)

120590=1

= minus119892(

120585119896+ 120585119896+1

2

) minus 11990911988310158401015840(119905119896+12

)

minus

1

2

[(

120597Φ119896

120597119909

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120597120585119896

120597119909

)

2

+(

1

ℎ + 120585119896

120597Φ119896

120597120590

)

2

]

120590=1

(21)

Semi-implicit scheme equations of 120585119896+1 andΦ119896+1 (120590 = 1)have been derived Moreover (16)ndash(21) can be discretized inspace and solved by finite difference method The solution120585119896+1 will be taken as the corrected value of 120585 on 119896 + 1 timestep

After 120590-transformation (6) has been transformed as

minus

119901

120588

= (

120597Φ

120597119905

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119905

) + 119892 [(ℎ + 120585) 120590 minus ℎ] + 11990911988310158401015840(119905)

+

1

2

[(

120597Φ

120597119909

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

2

+ (

1

ℎ + 120585

120597Φ

120597120590

)

2

]

(22)

When the values of 120585 and Φ are solved it is not difficult todiscretize (22) In the numerical simulation sloshing force onthe left and right walls can be expressed as

119865119871= int

119878119871

119901119889119904 = 119889int

120585

minusℎ

1199011003816100381610038161003816119909=0

119889119911

119865119877= int

119878119877

119901119889119904 = 119889int

120585

minusℎ

1199011003816100381610038161003816119909=119887

119889119911

(23)

The resultant sloshing force acting on the tank is

119865 = 119865119877minus 119865119871 (24)

4 Numerical Results

Some examples of nonlinear sloshing are simulated by thisnumerical method They have been investigated by otherresearchers using various numerical and analytical methodsand many discussions and conclusions have been derived

A rectangular tank model is established The naturalfrequencies of linear transverse sloshing along 119909-axis are(Faltinsen [2])

120596119899= radic119892

119899120587

119887

tanh 119899120587ℎ119887

119899 = 1 2 3 (25)

41 Standing Waves The tank is fixed so the moving coor-dinate system (119909 119911) coincides with the inertial coordinatesystem (119883 119885) The initial conditions are described as

120585 (119909 0) = 119886 cos(119899120587119887

) 120593 (119909 119911 0) = 0 (26)

119886 is the amplitude of the initial wave profile which influencesthe nonlinearity of the sloshing 119899 indicates themodenumberThe convergence of this numerical simulation is investigatedIn the tank 119887 119889 and ℎ are set to be 2m 1m and 1mrespectively 119886 is set to be 02m 119899 takes value of 1 Thefirst natural frequency of the sloshing is 120596

1= 376 rads

Figure 2(a) shows the time history of the free surface at theleft wall with different meshes but the time step sizes arethe same (0005 s) Figure 2(b) shows the time history wherethe meshes are the same (40 times 40) but the time step sizes

Mathematical Problems in Engineering 5

20 40 60 80 100

00

05

10

15

minus05

minus10

Mesh 40 times 40Mesh 60 times 60

1205961t

120585a

(a) Free surface elevation at the left wall using different meshes

Step 001Step 0005

20 40 60 80 1001205961t

00

05

10

15

minus05

minus10

120585a

(b) Free surface elevation at the left wall using different time step sizes

Figure 2 Time history of free surface elevation for different meshes and time step sizes

20 40 60 80 100

00

05

10

15

minus05

minus10

1205961t

120585a

AnalyticalNumerical

(a) Comparison of free surface elevation at the left wall

05 10 15 20x

00

05

10

15

minus05

minus10

120585a

(b) Wave profiles for a half-period

20 40 60 80 100

020

022

024

026

028

030

1205961t

FL

120588gbd

h

(c) Sloshing force on the left wall

20 40 60 80 100

020

022

024

026

028

030

1205961t

FR

120588gbd

h

(d) Sloshing force on the right wall

Figure 3 Standing wave (119899 = 1)

6 Mathematical Problems in Engineering

20 40 60 80 100

00

05

10

AnalyticalNumerical

1205961t

minus05

minus10

120585a

(a) Comparison of free surface elevation at the left wall

05 10 15 20x

00

05

10

minus05

minus10

120585a

(b) Wave profiles for a half-period

Figure 4 Standing wave (119899 = 3)

are different The time and the vibration displacement in thefigures are dealt with by dimensionless method

1205851015840=

120585

119886

1199051015840= 1205961119905 (27)

It is found that the results from these meshes and timestep sizes are in good agreement so in this example a meshdistribution of 40 times 40 and time step size of 0005 s ensure areasonable converged solution

Nonlinear sloshing equations of standing waves can besolved by perturbation techniques First nondimensionalvariables are introduced to the equations Then 120598 = 119886120596

2

119899119892

is chosen as the characteristic small parameter Approximatelinear equations are derived by small parameter expansionand approximate analytical solutions can be obtained Thesecond-order approximate analytical free surface elevationfor the 119899th sloshing mode is prescribed as (Frandsen [16])

120585 (119909 119905) = 119886 ( cos (120596119899119905) cos (119896

119899119909)

+ 120598 (119860119899+ 119861119899cos (2120596

119899119905)

+119862119899cos (120596

2119899119905)) cos (2119896

119899119909))

(28)

where

119896119899=

119899120587

119887

1205962119899= radic1198922119896

119899tanh (2119896

119899ℎ)

119860119899=

1205964

119899+ 11989221198962

119899

81205964

119899

119861119899=

31205964

119899minus 11989221198962

119899

81205964

119899

minus

3

2

1205964

119899minus 11989221198962

119899

1205962

119899(41205962

119899minus 1205962

2119899)

119862119899=

1205962

1198991205962

2119899minus 1205964

119899minus 311989221198962

119899

21205962

119899(41205962

119899minus 1205962

2119899)

(29)

Figure 3(a) shows the time history of the free surfaceelevation for 119899 = 1 at the leftwall by the twomethods numer-ical solution by the finite difference method proposed in thispaper and second-order approximate analytical solution byperturbation method Initially good agreement between thetwo results is achieved but the phase difference between themincreases with time This phenomenon has been noticed anddiscussed by many researchers (Chern et al [18] Frandsen[16]) The conclusion is that the second-order analyticalsolution is not very accurate for large amplitude sloshingFigure 3(b) displays numerical wave profiles for nearly halfa period from dimensionless time 120596

1119905 = 318 to 120596

1119905 =

346 Stationary point does not exist and nonlinearity isobvious Figures 3(c) and 3(d) provide the sloshing forceson the left wall and right wall respectively (the forces arealso nondimensionalized) It shows that slight double peaksappear in the time historyThis has also been observed byWuet al [5]

The standing wave of 119899 = 3 is also calculated The initialconditions are set to be

120585 (119909 0) = 006 cos(3120587119887

) 120593 (119909 119911 0) = 0 (30)

Figure 4(a) shows the comparison of numerical solutionsand approximate analytical solutions Figure 4(b) showsnumerical wave profiles for nearly half a period Phasedifference and nonlinearity are also observed

42 Sloshing Caused by Horizontal Excitations The tankis assumed to take the horizontal oscillation the externalexcitation acting on the tank is defined as harmonic motion

119883 (119905) = 119886ℎcos (120596

ℎ119905) (31)

119886ℎand 120596

ℎare the amplitude and angular frequency of the

excited motionThe fluid is assumed to be stationary at 119905 = 0Then in the moving coordinate system the initial conditionsare described as

120585 (119909 0) = 0 120593 (119909 119911 0) = minus1199091198831015840(119905) = 0 (32)

Mathematical Problems in Engineering 7

2

1

0

minus2

minus1

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(a) Example 1 120596ℎ = 071205961 120598ℎ = 0036

2

0

minus2

4

minus4

6

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(b) Example 2 120596ℎ = 131205961 120598ℎ = 0072

Figure 5 Comparison of free surface elevation at the left wall

20 40 60 80 100

0

20

40

60

minus20

120585a

1205961t

(a) The first resonance mode

20 40 60 80 100

0

10

20

minus10

1205963t

120585a h

(b) The third resonance mode

Figure 6 Free surface elevation at the left wall (resonance phenomenon)

20 40 60 80 100

000

005

010

015

minus015

minus010

minus005F120588gbd

h

1205961t

(a) The first resonance mode

20 40 60 80 100

000

001

002

minus002

minus001

1205963t

F120588gbd

h

(b) The third resonance mode

Figure 7 Resultant sloshing force (resonance phenomenon)

In order to verify the precision of this numerical methodfor the sloshing models in horizontally moving tanks twoexamples are calculated and compared with the solutions ofFrandsen [16] (1) 120596

ℎ= 07120596

1 120598ℎ= 0036 (2) 120596

ℎ= 13120596

1

120598ℎ= 0072 The grid size of the numerical method is 40 times 40

and the time step size is 0006 s Figure 5(a) shows that the twosimulations are in close agreement In the second examplethe nonlinear effect is strong so there is a little differencebetween the two solutions in Figure 5(b) Neverthelessnonlinear characteristics of sloshing are captured by the two

8 Mathematical Problems in Engineering

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 005

20 40 60 80 100 120

0

20

40

60

minus20

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 8 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 12)

20 40 60 80 100

0

2

4

6

minus2

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 003

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 9 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 18)

02 04 06 08 10 12 14

02

04

06

08

10

1205961120596

lowast 1

hb

Figure 10 The function graph of the first natural frequency againstaspect ratio

numerical methods such as the higher peaks and the lessdeep troughs

421 Resonance Phenomenon Two situations of resonanceare considered the first resonance mode and the thirdresonance mode The external excitations of the tank aredescribed as 119883

1(119905) = 0007 cos(120596

1119905) 1198832(119905) = 0007 cos(120596

3119905)

The amplitudes are the same but the excited frequencies

are different The time histories of free surface elevation atthe left wall and the resultant sloshing forces are calculatedFigure 6(a) shows that the sloshing of the first resonancemode is violate the amplitude of the vibration is increasingFigure 6(b) shows that the amplitude of the third resonancemode is much smaller than that of the first resonance modeIn Figure 7 it can be concluded that the resultant sloshingforce of the third resonance mode is trivial compared withthe force of the first resonance mode In fact the first naturalfrequency of sloshing is the most influential of all the naturalfrequencies

422 The Limitation of the Numerical Method The presentnumerical method is employed to simulate nonlinear slosh-ing in moving tanks If the nonlinearity is quite strong thismethodmay lose its effectiveness Two excitation frequenciesare considered in the simulations One is 120596

ℎ= 085120596

1 the

other is 120596ℎ= 095120596

1 which indicates near resonant sloshing

situation Different excitation amplitudes 119886ℎand aspect ratios

ℎ119887 are tested and the limitation of the convergence of thisnumerical method is found out

First the aspect ratio is set to be ℎ119887 = 12 just asbefore Excitation amplitude is increased in the numericalsimulation and the sloshing becomesmore andmore violentFor 120596

ℎ= 085120596

1 if the excitation amplitude is 119886

ℎ= 006m

Mathematical Problems in Engineering 9

20 40 60 80 100

0

5

10

15

20

minus5

minus10

120585a h

120596ht

(a) Aspect ratio ℎ119887 = 05

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) Aspect ratio ℎ119887 = 075

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(c) Aspect ratio ℎ119887 = 1

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 11 Free surface elevation at the left wall (low-frequency excitation)

the simulation is divergent after several times of iteration if119886ℎ= 005m the simulation lasts for a long time So it is

considered that the limitation of 119886ℎfor nonresonant sloshing

is 005m For 120596ℎ= 095120596

1 the excitation amplitude is much

smaller The maximum limit of 119886ℎis estimated to be 0012m

The time histories of free surface elevation at the left wall inthe limit cases are shown in Figure 8

Then the aspect ratio is changed by decreasing the stillwater depth ℎ For the aspect ratio ℎ119887 = 18 the numericalmethod still works with a mesh size of 80 times 20 If the aspectratio is less than 18 the simulation is divergent As a resultthe minimum limit of aspect ratio ℎ119887 is considered to be18 In this critical aspect ratio situation for 120596

ℎ= 095120596

1

the maximum limit of excitation amplitude 119886ℎis still 0012m

while for 120596ℎ= 085120596

1 the maximum limit of 119886

ℎis 003m

Figure 9 shows the time histories of free surface elevation inthe limit cases in this critical aspect ratio The phenomena ofbeating and resonance are very obscure it can be concludedthat the sloshing in low aspect ratio is quite different from theone in normal aspect ratio

423 Beating Phenomenon in Different Aspect Ratios Slosh-ing in a fixed aspect ratio has been studied in many articlesHowever little work has been done to analyze the response ofliquid sloshing in different aspect ratios by the same external

excitation When the external excitation is close to the firstnatural frequency of sloshing the beating phenomenon canbe observed The study here is intended to investigate theeffect of different aspect ratios on beating phenomenon andsloshing force

In the numerical examples calculated before the lengthand width of the tank are set to be 2m and 1m the still waterdepth is 1m The first natural frequency of sloshing is 120596

1=

376 rads The aspect ratio is ℎ119887 = 12 Now the length 119887and the width 119889 of the tank are fixed and the aspect ratio ischanged by changing the still water depth ℎThe former waterdepth and natural frequency are set to be the characteristicquantity ℎlowast = 1m 120596lowast

1= 376 rads

The relationship between aspect ratio and the first naturalfrequency must be considered From (25) it is easy to get thegraph of the natural frequency as a function of aspect ratioFigure 10 indicates that the slope of the function curve is steepwhen the aspect ratio is small if the aspect ratio is greater than06 the function curve is nearly horizontal

Under the same external excitation the sloshing forcesof beating are affected by three factors proximity of exci-tation frequency to the natural frequency liquid mass andnonlinearity of sloshing When the aspect ratio of the liquidchanges these three factors would change along with it Inorder to discuss the effect of aspect ratio two horizontaloscillations of the tank are studied First the tank is subjected

10 Mathematical Problems in Engineering

20 40 60 80 100

00

01

02

minus01

F120588gbd

hlowast

120596ht

(a) Aspect ratio ℎ119887 = 05

005

010

015

minus005

minus010

minus015

F120588gbd

hlowast

20 40 60 80 100120596ht

000

(b) Aspect ratio ℎ119887 = 075

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(c) Aspect ratio ℎ119887 = 1

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 12 Resultant sloshing forces (low-frequency excitation)

06 07 08 09 10 11 12

019

020

021

022

LinearNonlinear

F120588gbd

hlowast

hb

Figure 13 Tendency of amplitude of sloshing force (low-frequencyexcitation)

to a low-frequency excitation The movement of the tank isdescribed as 119886

ℎ= 003m 120596

ℎ= 34 rads Figure 10 suggests

that if the aspect ratio is 03 the corresponding naturalfrequency is 34 rads Our aim is to analyze the influence ofthis low-frequency excitation on the high aspect ratio slosh-ing system Figure 11 shows the time histories of free surface

elevation at the left wall in four different aspect ratios wherethe parameters are nondimensionalized

1199051015840= 120596ℎ119905 120585

1015840=

120585

119886ℎ

(33)

The amplitude of free surface elevation is quite large inFigure 11(a) which shows strong nonlinearity of sloshingwhile in Figures 11(b) 11(c) and 11(d) the amplitudes arenearly the same smaller than the amplitude in Figure 11(a)so the nonlinear effects are weaker in these three sloshingsituations

Figure 12 shows the time histories of resultant sloshingforce acting on the tank (the forces are also nondimensional-ized 1198651015840 = 119865120588119892119887119889ℎlowast) The amplitudes of the sloshing forceare extracted in Figure 12 In order to study the effect ofnonlinear factor linear sloshing models by Abramson [1] arecalculated under the same condition to make comparisonsThe relationship between the amplitude of sloshing force andaspect ratio is shown in Figure 13 It is obvious that bothlinear and nonlinear results have the same trend decreasingfirst and then increasing On the one hand the rise of theaspect ratio increases the natural frequency which makesthe sloshing far from resonance and reduces sloshing forcesOn the other hand the rise of the aspect ratio increases themass of the liquid thereby increasing the sloshing forces

Mathematical Problems in Engineering 11

0

10

20120585a h

20 40 60 80 100120596ht

minus10

(a) Aspect ratio ℎ119887 = 05

0

5

10

15

minus5

120585a h

20 40 60 80 100120596ht

minus10

(b) Aspect ratio ℎ119887 = 045

0

5

10

120585a h

20 40 60 80 100120596ht

minus5

minus10

(c) Aspect ratio ℎ119887 = 04

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 035

Figure 14 Free surface elevation at the left wall (high-frequency excitation)

For the sloshing in aspect ratio ℎ119887 = 05 the excitationfrequency is close to the natural frequency If aspect ratiois increased the factor of reducing sloshing force is moredominant However for the sloshing in high aspect ratiosℎ119887 gt 075 the rise of aspect ratio brings about slight changesin the natural frequency so the factor of increasing sloshingforce is more dominant Comparing nonlinear results withlinear ones in Figure 13 some conclusions are obtained Firstthe nonlinearity of sloshing always decreases sloshing forceSecond to a certain extent the difference between linear andnonlinear amplitude reflects the effect of nonlinear factoron sloshing force For the sloshing in aspect ratio ℎ119887 =

05 the difference is the largest while for the other sloshingsituations the differences are nearly the same This resultaccordswith the conclusion in Figure 11Third nonlinearity isan important factor in large amplitude sloshing which shouldnot be neglected

Then the tank is subjected to a high-frequency excitationThe movement of the tank is described as 119886

ℎ= 0015m

120596ℎ= 4 rads Our aim is to analyze the influence of this

high-frequency excitation on the low aspect ratio sloshingsystem Figure 14 shows the time histories of free surfaceelevation at the left wall in four different aspect ratios Asthe aspect ratio decreases the amplitude and period of freesurface elevation are diminishing For the sloshing in lowaspect ratios ℎ119887 lt 05 the reduction of aspect ratio brings

about significant changes in the natural frequency so thebeating phenomenon has changed greatly even if the aspectratio is slightly decreasedThe resultant sloshing forces actingon the tank are also calculated The relationship between theamplitude of sloshing force and aspect ratio is showed inFigure 15The reduction of the aspect ratio makes the naturalfrequency far from the excitation frequency meanwhile itdecreases the mass of the liquid in the tank So both of thefactors reduce sloshing force As the aspect ratio decreasesthe difference between linear and nonlinear amplitude ofsloshing force also decreases The reason is that nonlineareffects are becoming weaker and the sloshing force itself isalso decreasing

5 Conclusions

Anewfinite differencemethod based on the prediction of freesurface has been developed and used to simulate nonlinearsloshing phenomenon in tanks by the potential flow theoryAfter 120590-transformation the fluid domain is transformed to arectangle and a predictor-corrector scheme for free surfaceis used in the iteration of time then the governing equationand boundary conditions are approximately linearized andsolved in every time step The convergence of this methodis surveyed The numerical examples of standing waves and

12 Mathematical Problems in Engineering

035 040 045 050

006

008

010

012

014

016

LinearNonlinear

F120588gbd

hlowast

hb

Figure 15 Tendency of amplitude of sloshing force (high-frequencyexcitation)

horizontally forced sloshing models have been calculatedand good agreement between the numerical simulation andresults in other literatures has been obtained This numericalmethod is simple and easy in programming with excellentaccuracy and stability and can be extended to other sloshingproblems The time histories of free surface elevation andsloshing forces in the first and third resonance situation arealso presented and analyzed

The beating phenomenon of sloshing in different aspectratios under the same excitation has been numerically simu-lated by this method The results show that the effect of theaspect ratio on the sloshing force is complicated For the low-frequency excitation acting on the high aspect ratio modelsthe rise of aspect ratio makes the sloshing far from resonanceand increases the liquid mass so the sloshing force decreasesfirst and then increases For the high-frequency excitationacting on the low aspect ratio models the reduction of aspectratiomakes the sloshing far from resonance and decreases theliquid mass so the sloshing force decreases all the way Sincethe first natural frequency of liquid sloshing in not a linearfunction of aspect ratio the change of aspect ratio producesmuch stronger effect in low aspect ratio sloshing systemsMoreover nonlinearity of sloshing always decreases sloshingforce The effect of nonlinear factor depends on the violenceof sloshing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H N Abramson ldquoThe dynamic behavior of liquids in movingcontainers nasa sp-106rdquo NASA Special Publication vol 1061966

[2] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 no 4 pp 114ndash241 1974

[3] O M Faltinsen ldquoA numerical nonlinear method of sloshing intanks with two-dimensional flowrdquo Journal of Ship Research vol22 no 3 pp 193ndash202 1978

[4] T Nakayama and K Washizu ldquoNonlinear analysis of liquidmotion in a container subjected to forced pitching oscillationrdquoInternational Journal for Numerical Methods in Engineering vol15 no 8 pp 1207ndash1220 1980

[5] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[6] B-F Chen and R Nokes ldquoTime-independent finite differenceanalysis of fully non-linear and viscous fluid sloshing in arectangular tankrdquo Journal of Computational Physics vol 209 no1 pp 47ndash81 2005

[7] C Wu and B Chen ldquoSloshing waves and resonance modesof fluid in a 3D tank by a time-independent finite differencemethodrdquoOcean Engineering vol 36 no 6-7 pp 500ndash510 2009

[8] Y Kim ldquoNumerical simulation of sloshing flows with impactloadrdquo Applied Ocean Research vol 23 no 1 pp 53ndash62 2001

[9] M S Celebi and H Akyildiz ldquoNonlinear modeling of liquidsloshing in a moving rectangular tankrdquo Ocean Engineering vol29 no 12 pp 1527ndash1553 2002

[10] C Z Wang and B C Khoo ldquoFinite element analysis of two-dimensional nonlinear sloshing problems in random excita-tionsrdquo Ocean Engineering vol 32 no 2 pp 107ndash133 2005

[11] V Sriram S A Sannasiraj and V Sundar ldquoNumerical simula-tion of 2D sloshingwaves due to horizontal and vertical randomexcitationrdquo Applied Ocean Research vol 28 no 1 pp 19ndash322006

[12] O M Faltinsen O F Rognebakke I A Lukovsky and AN Timokha ldquoMultidimensional modal analysis of nonlinearsloshing in a rectangular tank with finite water depthrdquo Journalof Fluid Mechanics vol 407 pp 201ndash234 2000

[13] O M Faltinsen and A N Timokha ldquoAn adaptive multimodalapproach to nonlinear sloshing in a rectangular tankrdquo Journalof Fluid Mechanics vol 432 pp 167ndash200 2001

[14] G XWu ldquoSecond-order resonance of sloshing in a tankrdquoOceanEngineering vol 34 no 17-18 pp 2345ndash2349 2007

[15] D F Hill ldquoTransient and steady-state amplitudes of forcedwaves in rectangular basinsrdquo Physics of Fluids vol 15 no 6 pp1576ndash1587 2003

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004

[17] RD Firouz-AbadiMGhasemi andHHaddadpour ldquoAmodalapproach to second-order analysis of sloshing using boundaryelement methodrdquo Ocean Engineering vol 38 no 1 pp 11ndash212011

[18] M J Chern A G L Borthwick and R E Taylor ldquoA pseu-dospectral 120590-transformation model of 2-D nonlinear wavesrdquoJournal of Fluids and Structures vol 13 no 5 pp 607ndash630 1999

[19] J B Frandsen and A G L Borthwick ldquoSimulation of sloshingmotions in fixed and vertically excited containers using a 2-Dinviscid120590-transformed finite difference solverrdquo Journal of Fluidsand Structures vol 18 no 2 pp 197ndash214 2003

[20] M S Turnbull A G L Borthwick and R Eatock TaylorldquoNumerical wave tank based on a 120590-transformed finite elementinviscid flow solverrdquo International Journal for Numerical Meth-ods in Fluids vol 42 no 6 pp 641ndash663 2003

[21] B Chen andH Chiang ldquoComplete 2D and fully nonlinear anal-ysis of ideal fluid in tanksrdquo Journal of Engineering Mechanicsvol 125 no 1 pp 70ndash78 1999

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

of the free surface One is the kinematic condition theother is the dynamic condition 120585(119909 119905) is the free surfaceelevation deviating the undisturbed water level in themovingcoordinate system (119909 119911) 119892 denotes acceleration of gravity If(1)ndash(5) are solved the pressure in the fluid can be calculatedby the incompressible Bernoulli equation

minus

119901

120588

=

120597120593

120597119905

+ 119892119911 + 11990911988310158401015840(119905) +

1

2

[(

120597120593

120597119909

)

2

+ (

120597120593

120597119911

)

2

] (6)

where 119901 is the pressure and 120588 is the density of fluid Thesloshing force is determined by integrating the liquid pressureover the tank wall area Because the tank is moving alongthe 119909-axis and the sloshing models are 2D cases only 119909components of sloshing force would have great influence onthe dynamics of the tank In this paper the sloshing forceacting on the left and right walls is considered

3 Numerical Process

According to Frandsen [16] 120590-transformation method isused

120590 =

119911 + ℎ

ℎ + 120585

Φ (119909 120590 119905) = 120593 (119909 119911 119905) (7)

A new variable 120590 is introduced The derivatives of thepotential function 120593(119909 119911 119905) with respect to 119909 119911 and 119905 aretransformed into derivatives ofΦ(119909 120590 119905)The first derivativesof 120593 can be expressed as

120597120593

120597119909

=

120597Φ

120597119909

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

120597120593

120597119911

=

1

ℎ + 120585

120597Φ

120597120590

120597120593

120597119905

=

120597Φ

120597119905

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119905

(8)

Similarly it is not difficult to obtain the second derivatives of120593(119909 119911 119905) Then (1)ndash(5) can be transformed by using variablesubstitution

1205972Φ

1205971199092minus

2120590

ℎ + 120585

120597120585

120597119909

1205972Φ

120597119909120597120590

+ [

2120590

(ℎ + 120585)2(

120597120585

120597119909

)

2

minus

120590

ℎ + 120585

1205972120585

1205971199092]

120597Φ

120597120590

+ [(

120590

ℎ + 120585

120597120585

120597119909

)

2

+

1

(ℎ + 120585)2]

1205972Φ

1205971205902= 0

(9)

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119909=0119887

minus (

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

119909=0119887

= 0 (10)

120597Φ

120597120590

10038161003816100381610038161003816100381610038161003816120590=0

= 0 (11)

120597120585

120597119905

=

1

ℎ + 120585

120597Φ

120597120590

10038161003816100381610038161003816100381610038161003816120590=1

minus

120597120585

120597119909

(

120597Φ

120597119909

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

120590=1

(12)

(

120597Φ

120597119905

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119905

)

120590=1

= minus119892120585 minus 11990911988310158401015840(119905)

minus

1

2

[(

120597Φ

120597119909

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

2

+ (

1

ℎ + 120585

120597Φ

120597120590

)

2

]

120590=1

(13)

The fluid domain of (1) is 0 lt 119909 lt 119887minusℎ lt 119911 lt 120585(119909 119905) After 120590-transformation it has changed into a fixed rectangular area0 lt 119909 lt 119887 0 lt 120590 lt 1

A finite difference method has been developed to solve(9)ndash(13) Δ119905 is set to be the step size of time In the iterativeprocess of time assuming that the values of velocity potentialfunctionΦ and the free surface function 120585 are known at time119896Δ119905 that is on 119896 time step the aim is to obtain the valuesof the two functions on 119896 + 1 time step Some difficulties areencountered in the numerical algorithm First the governingequation cannot be solved directly on 119896+ 1 time step becausefree surface 120585 which is unknown at that time is a boundarycurve of the governing equation Second there are nonlinearterms existing in free surface boundary conditions

For the first problem a prediction method has beenintroduced Considering (9)ndash(11) on 119896 + 1 time step 120585 shouldhave been assigned to the value on 119896 + 1 time step but itis unknown On the other hand (9) on 119896 + 1 time step wastransformed from

1205972120593

1205971199092+

1205972120593

1205971199112

119896+1

= 0 0 lt 119909 lt 119887 minusℎ lt 119911 lt 120585119896+1 (14)

120585 does not appear in the governing equation it is only aboundary curve An explicit scheme of (4) can be adopted tomake prediction of 120585119896+1 and then substitute it into (14) So thefour boundary curves of (14) are determined which meansthat there is only one unknown quantity 120593119896+1 in (14)

This idea should be implemented in (9)ndash(11) Because thecomputational domain is transformed an explicit scheme of(12) is used to make prediction of 120585119896+1

120585

119896+1

= 120585119896+ Δ119905 [

1

ℎ + 120585119896

120597Φ119896

120597120590

minus

120597120585119896

120597119909

(

120597Φ119896

120597119909

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120597120585119896

120597119909

)]

120590=1

(15)

120585

119896+1

denotes the prediction value Substitute it into (9) and(10) on 119896+1 time step then onlyΦ119896+1 is the unknownquantity

4 Mathematical Problems in Engineering

in the two equations Equation (11) does not have 120585 so thereis no need to do so These three equations are discretized as

1205972Φ

1205971199092minus

2120590

ℎ + 120585

120597120585

120597119909

1205972Φ

120597119909120597120590

+[

[

2120590

(ℎ + 120585)

2(

120597120585

120597119909

)

2

minus

120590

ℎ + 120585

1205972120585

1205971199092

]

]

120597Φ

120597120590

119896+1

+

[

[

(

120590

ℎ + 120585

120597120585

120597119909

)

2

+

1

(ℎ + 120585)

2

]

]

1205972Φ

1205971205902

119896+1

= 0

(16)

120597Φ119896+1

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0119887

minus (

120590

ℎ + 120585

120597Φ119896+1

120597120590

120597120585

120597119909

)

119909=0119887

= 0 (17)

120597Φ119896+1

120597120590

100381610038161003816100381610038161003816100381610038161003816120590=0

= 0 (18)

For the second problem an approximate method hasbeen introduced Consider two boundary conditions (12) and(13) on 119896 + 12 time step they are complicated nonlinearequations For example

120597120585

120597119909

(

120597Φ

120597119909

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

120590=1

(19)

is a nonlinear term it should have been assigned to the valueon 119896 + 12 time step in the calculation process but here itis assigned to the value on 119896 time step as an approximationOther nonlinear terms are dealt with by the same methodLinear terms in equations are discretized by Crank-Nicolsonscheme where 120585119896+1 and Φ119896+1 (120590 = 1) are set to be unknownquantities It is noteworthy that 120585119896+1 is set to be the unknownquantity in (12) and (13) so as to update the free surfacewhen the equations are solved Overall these two boundaryconditions are discretized as

120585119896+1

minus 120585119896

Δ119905

= [

1

ℎ + 120585119896(

1

2

120597Φ119896+1

120597120590

+

1

2

120597Φ119896

120597120590

)

minus

120597120585119896

120597119909

(

120597Φ119896

120597119909

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120597120585119896

120597119909

)]

120590=1

(20)

(

Φ119896+1

minus Φ119896

Δ119905

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120585119896+1

minus 120585119896

Δ119905

)

120590=1

= minus119892(

120585119896+ 120585119896+1

2

) minus 11990911988310158401015840(119905119896+12

)

minus

1

2

[(

120597Φ119896

120597119909

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120597120585119896

120597119909

)

2

+(

1

ℎ + 120585119896

120597Φ119896

120597120590

)

2

]

120590=1

(21)

Semi-implicit scheme equations of 120585119896+1 andΦ119896+1 (120590 = 1)have been derived Moreover (16)ndash(21) can be discretized inspace and solved by finite difference method The solution120585119896+1 will be taken as the corrected value of 120585 on 119896 + 1 timestep

After 120590-transformation (6) has been transformed as

minus

119901

120588

= (

120597Φ

120597119905

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119905

) + 119892 [(ℎ + 120585) 120590 minus ℎ] + 11990911988310158401015840(119905)

+

1

2

[(

120597Φ

120597119909

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

2

+ (

1

ℎ + 120585

120597Φ

120597120590

)

2

]

(22)

When the values of 120585 and Φ are solved it is not difficult todiscretize (22) In the numerical simulation sloshing force onthe left and right walls can be expressed as

119865119871= int

119878119871

119901119889119904 = 119889int

120585

minusℎ

1199011003816100381610038161003816119909=0

119889119911

119865119877= int

119878119877

119901119889119904 = 119889int

120585

minusℎ

1199011003816100381610038161003816119909=119887

119889119911

(23)

The resultant sloshing force acting on the tank is

119865 = 119865119877minus 119865119871 (24)

4 Numerical Results

Some examples of nonlinear sloshing are simulated by thisnumerical method They have been investigated by otherresearchers using various numerical and analytical methodsand many discussions and conclusions have been derived

A rectangular tank model is established The naturalfrequencies of linear transverse sloshing along 119909-axis are(Faltinsen [2])

120596119899= radic119892

119899120587

119887

tanh 119899120587ℎ119887

119899 = 1 2 3 (25)

41 Standing Waves The tank is fixed so the moving coor-dinate system (119909 119911) coincides with the inertial coordinatesystem (119883 119885) The initial conditions are described as

120585 (119909 0) = 119886 cos(119899120587119887

) 120593 (119909 119911 0) = 0 (26)

119886 is the amplitude of the initial wave profile which influencesthe nonlinearity of the sloshing 119899 indicates themodenumberThe convergence of this numerical simulation is investigatedIn the tank 119887 119889 and ℎ are set to be 2m 1m and 1mrespectively 119886 is set to be 02m 119899 takes value of 1 Thefirst natural frequency of the sloshing is 120596

1= 376 rads

Figure 2(a) shows the time history of the free surface at theleft wall with different meshes but the time step sizes arethe same (0005 s) Figure 2(b) shows the time history wherethe meshes are the same (40 times 40) but the time step sizes

Mathematical Problems in Engineering 5

20 40 60 80 100

00

05

10

15

minus05

minus10

Mesh 40 times 40Mesh 60 times 60

1205961t

120585a

(a) Free surface elevation at the left wall using different meshes

Step 001Step 0005

20 40 60 80 1001205961t

00

05

10

15

minus05

minus10

120585a

(b) Free surface elevation at the left wall using different time step sizes

Figure 2 Time history of free surface elevation for different meshes and time step sizes

20 40 60 80 100

00

05

10

15

minus05

minus10

1205961t

120585a

AnalyticalNumerical

(a) Comparison of free surface elevation at the left wall

05 10 15 20x

00

05

10

15

minus05

minus10

120585a

(b) Wave profiles for a half-period

20 40 60 80 100

020

022

024

026

028

030

1205961t

FL

120588gbd

h

(c) Sloshing force on the left wall

20 40 60 80 100

020

022

024

026

028

030

1205961t

FR

120588gbd

h

(d) Sloshing force on the right wall

Figure 3 Standing wave (119899 = 1)

6 Mathematical Problems in Engineering

20 40 60 80 100

00

05

10

AnalyticalNumerical

1205961t

minus05

minus10

120585a

(a) Comparison of free surface elevation at the left wall

05 10 15 20x

00

05

10

minus05

minus10

120585a

(b) Wave profiles for a half-period

Figure 4 Standing wave (119899 = 3)

are different The time and the vibration displacement in thefigures are dealt with by dimensionless method

1205851015840=

120585

119886

1199051015840= 1205961119905 (27)

It is found that the results from these meshes and timestep sizes are in good agreement so in this example a meshdistribution of 40 times 40 and time step size of 0005 s ensure areasonable converged solution

Nonlinear sloshing equations of standing waves can besolved by perturbation techniques First nondimensionalvariables are introduced to the equations Then 120598 = 119886120596

2

119899119892

is chosen as the characteristic small parameter Approximatelinear equations are derived by small parameter expansionand approximate analytical solutions can be obtained Thesecond-order approximate analytical free surface elevationfor the 119899th sloshing mode is prescribed as (Frandsen [16])

120585 (119909 119905) = 119886 ( cos (120596119899119905) cos (119896

119899119909)

+ 120598 (119860119899+ 119861119899cos (2120596

119899119905)

+119862119899cos (120596

2119899119905)) cos (2119896

119899119909))

(28)

where

119896119899=

119899120587

119887

1205962119899= radic1198922119896

119899tanh (2119896

119899ℎ)

119860119899=

1205964

119899+ 11989221198962

119899

81205964

119899

119861119899=

31205964

119899minus 11989221198962

119899

81205964

119899

minus

3

2

1205964

119899minus 11989221198962

119899

1205962

119899(41205962

119899minus 1205962

2119899)

119862119899=

1205962

1198991205962

2119899minus 1205964

119899minus 311989221198962

119899

21205962

119899(41205962

119899minus 1205962

2119899)

(29)

Figure 3(a) shows the time history of the free surfaceelevation for 119899 = 1 at the leftwall by the twomethods numer-ical solution by the finite difference method proposed in thispaper and second-order approximate analytical solution byperturbation method Initially good agreement between thetwo results is achieved but the phase difference between themincreases with time This phenomenon has been noticed anddiscussed by many researchers (Chern et al [18] Frandsen[16]) The conclusion is that the second-order analyticalsolution is not very accurate for large amplitude sloshingFigure 3(b) displays numerical wave profiles for nearly halfa period from dimensionless time 120596

1119905 = 318 to 120596

1119905 =

346 Stationary point does not exist and nonlinearity isobvious Figures 3(c) and 3(d) provide the sloshing forceson the left wall and right wall respectively (the forces arealso nondimensionalized) It shows that slight double peaksappear in the time historyThis has also been observed byWuet al [5]

The standing wave of 119899 = 3 is also calculated The initialconditions are set to be

120585 (119909 0) = 006 cos(3120587119887

) 120593 (119909 119911 0) = 0 (30)

Figure 4(a) shows the comparison of numerical solutionsand approximate analytical solutions Figure 4(b) showsnumerical wave profiles for nearly half a period Phasedifference and nonlinearity are also observed

42 Sloshing Caused by Horizontal Excitations The tankis assumed to take the horizontal oscillation the externalexcitation acting on the tank is defined as harmonic motion

119883 (119905) = 119886ℎcos (120596

ℎ119905) (31)

119886ℎand 120596

ℎare the amplitude and angular frequency of the

excited motionThe fluid is assumed to be stationary at 119905 = 0Then in the moving coordinate system the initial conditionsare described as

120585 (119909 0) = 0 120593 (119909 119911 0) = minus1199091198831015840(119905) = 0 (32)

Mathematical Problems in Engineering 7

2

1

0

minus2

minus1

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(a) Example 1 120596ℎ = 071205961 120598ℎ = 0036

2

0

minus2

4

minus4

6

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(b) Example 2 120596ℎ = 131205961 120598ℎ = 0072

Figure 5 Comparison of free surface elevation at the left wall

20 40 60 80 100

0

20

40

60

minus20

120585a

1205961t

(a) The first resonance mode

20 40 60 80 100

0

10

20

minus10

1205963t

120585a h

(b) The third resonance mode

Figure 6 Free surface elevation at the left wall (resonance phenomenon)

20 40 60 80 100

000

005

010

015

minus015

minus010

minus005F120588gbd

h

1205961t

(a) The first resonance mode

20 40 60 80 100

000

001

002

minus002

minus001

1205963t

F120588gbd

h

(b) The third resonance mode

Figure 7 Resultant sloshing force (resonance phenomenon)

In order to verify the precision of this numerical methodfor the sloshing models in horizontally moving tanks twoexamples are calculated and compared with the solutions ofFrandsen [16] (1) 120596

ℎ= 07120596

1 120598ℎ= 0036 (2) 120596

ℎ= 13120596

1

120598ℎ= 0072 The grid size of the numerical method is 40 times 40

and the time step size is 0006 s Figure 5(a) shows that the twosimulations are in close agreement In the second examplethe nonlinear effect is strong so there is a little differencebetween the two solutions in Figure 5(b) Neverthelessnonlinear characteristics of sloshing are captured by the two

8 Mathematical Problems in Engineering

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 005

20 40 60 80 100 120

0

20

40

60

minus20

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 8 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 12)

20 40 60 80 100

0

2

4

6

minus2

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 003

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 9 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 18)

02 04 06 08 10 12 14

02

04

06

08

10

1205961120596

lowast 1

hb

Figure 10 The function graph of the first natural frequency againstaspect ratio

numerical methods such as the higher peaks and the lessdeep troughs

421 Resonance Phenomenon Two situations of resonanceare considered the first resonance mode and the thirdresonance mode The external excitations of the tank aredescribed as 119883

1(119905) = 0007 cos(120596

1119905) 1198832(119905) = 0007 cos(120596

3119905)

The amplitudes are the same but the excited frequencies

are different The time histories of free surface elevation atthe left wall and the resultant sloshing forces are calculatedFigure 6(a) shows that the sloshing of the first resonancemode is violate the amplitude of the vibration is increasingFigure 6(b) shows that the amplitude of the third resonancemode is much smaller than that of the first resonance modeIn Figure 7 it can be concluded that the resultant sloshingforce of the third resonance mode is trivial compared withthe force of the first resonance mode In fact the first naturalfrequency of sloshing is the most influential of all the naturalfrequencies

422 The Limitation of the Numerical Method The presentnumerical method is employed to simulate nonlinear slosh-ing in moving tanks If the nonlinearity is quite strong thismethodmay lose its effectiveness Two excitation frequenciesare considered in the simulations One is 120596

ℎ= 085120596

1 the

other is 120596ℎ= 095120596

1 which indicates near resonant sloshing

situation Different excitation amplitudes 119886ℎand aspect ratios

ℎ119887 are tested and the limitation of the convergence of thisnumerical method is found out

First the aspect ratio is set to be ℎ119887 = 12 just asbefore Excitation amplitude is increased in the numericalsimulation and the sloshing becomesmore andmore violentFor 120596

ℎ= 085120596

1 if the excitation amplitude is 119886

ℎ= 006m

Mathematical Problems in Engineering 9

20 40 60 80 100

0

5

10

15

20

minus5

minus10

120585a h

120596ht

(a) Aspect ratio ℎ119887 = 05

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) Aspect ratio ℎ119887 = 075

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(c) Aspect ratio ℎ119887 = 1

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 11 Free surface elevation at the left wall (low-frequency excitation)

the simulation is divergent after several times of iteration if119886ℎ= 005m the simulation lasts for a long time So it is

considered that the limitation of 119886ℎfor nonresonant sloshing

is 005m For 120596ℎ= 095120596

1 the excitation amplitude is much

smaller The maximum limit of 119886ℎis estimated to be 0012m

The time histories of free surface elevation at the left wall inthe limit cases are shown in Figure 8

Then the aspect ratio is changed by decreasing the stillwater depth ℎ For the aspect ratio ℎ119887 = 18 the numericalmethod still works with a mesh size of 80 times 20 If the aspectratio is less than 18 the simulation is divergent As a resultthe minimum limit of aspect ratio ℎ119887 is considered to be18 In this critical aspect ratio situation for 120596

ℎ= 095120596

1

the maximum limit of excitation amplitude 119886ℎis still 0012m

while for 120596ℎ= 085120596

1 the maximum limit of 119886

ℎis 003m

Figure 9 shows the time histories of free surface elevation inthe limit cases in this critical aspect ratio The phenomena ofbeating and resonance are very obscure it can be concludedthat the sloshing in low aspect ratio is quite different from theone in normal aspect ratio

423 Beating Phenomenon in Different Aspect Ratios Slosh-ing in a fixed aspect ratio has been studied in many articlesHowever little work has been done to analyze the response ofliquid sloshing in different aspect ratios by the same external

excitation When the external excitation is close to the firstnatural frequency of sloshing the beating phenomenon canbe observed The study here is intended to investigate theeffect of different aspect ratios on beating phenomenon andsloshing force

In the numerical examples calculated before the lengthand width of the tank are set to be 2m and 1m the still waterdepth is 1m The first natural frequency of sloshing is 120596

1=

376 rads The aspect ratio is ℎ119887 = 12 Now the length 119887and the width 119889 of the tank are fixed and the aspect ratio ischanged by changing the still water depth ℎThe former waterdepth and natural frequency are set to be the characteristicquantity ℎlowast = 1m 120596lowast

1= 376 rads

The relationship between aspect ratio and the first naturalfrequency must be considered From (25) it is easy to get thegraph of the natural frequency as a function of aspect ratioFigure 10 indicates that the slope of the function curve is steepwhen the aspect ratio is small if the aspect ratio is greater than06 the function curve is nearly horizontal

Under the same external excitation the sloshing forcesof beating are affected by three factors proximity of exci-tation frequency to the natural frequency liquid mass andnonlinearity of sloshing When the aspect ratio of the liquidchanges these three factors would change along with it Inorder to discuss the effect of aspect ratio two horizontaloscillations of the tank are studied First the tank is subjected

10 Mathematical Problems in Engineering

20 40 60 80 100

00

01

02

minus01

F120588gbd

hlowast

120596ht

(a) Aspect ratio ℎ119887 = 05

005

010

015

minus005

minus010

minus015

F120588gbd

hlowast

20 40 60 80 100120596ht

000

(b) Aspect ratio ℎ119887 = 075

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(c) Aspect ratio ℎ119887 = 1

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 12 Resultant sloshing forces (low-frequency excitation)

06 07 08 09 10 11 12

019

020

021

022

LinearNonlinear

F120588gbd

hlowast

hb

Figure 13 Tendency of amplitude of sloshing force (low-frequencyexcitation)

to a low-frequency excitation The movement of the tank isdescribed as 119886

ℎ= 003m 120596

ℎ= 34 rads Figure 10 suggests

that if the aspect ratio is 03 the corresponding naturalfrequency is 34 rads Our aim is to analyze the influence ofthis low-frequency excitation on the high aspect ratio slosh-ing system Figure 11 shows the time histories of free surface

elevation at the left wall in four different aspect ratios wherethe parameters are nondimensionalized

1199051015840= 120596ℎ119905 120585

1015840=

120585

119886ℎ

(33)

The amplitude of free surface elevation is quite large inFigure 11(a) which shows strong nonlinearity of sloshingwhile in Figures 11(b) 11(c) and 11(d) the amplitudes arenearly the same smaller than the amplitude in Figure 11(a)so the nonlinear effects are weaker in these three sloshingsituations

Figure 12 shows the time histories of resultant sloshingforce acting on the tank (the forces are also nondimensional-ized 1198651015840 = 119865120588119892119887119889ℎlowast) The amplitudes of the sloshing forceare extracted in Figure 12 In order to study the effect ofnonlinear factor linear sloshing models by Abramson [1] arecalculated under the same condition to make comparisonsThe relationship between the amplitude of sloshing force andaspect ratio is shown in Figure 13 It is obvious that bothlinear and nonlinear results have the same trend decreasingfirst and then increasing On the one hand the rise of theaspect ratio increases the natural frequency which makesthe sloshing far from resonance and reduces sloshing forcesOn the other hand the rise of the aspect ratio increases themass of the liquid thereby increasing the sloshing forces

Mathematical Problems in Engineering 11

0

10

20120585a h

20 40 60 80 100120596ht

minus10

(a) Aspect ratio ℎ119887 = 05

0

5

10

15

minus5

120585a h

20 40 60 80 100120596ht

minus10

(b) Aspect ratio ℎ119887 = 045

0

5

10

120585a h

20 40 60 80 100120596ht

minus5

minus10

(c) Aspect ratio ℎ119887 = 04

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 035

Figure 14 Free surface elevation at the left wall (high-frequency excitation)

For the sloshing in aspect ratio ℎ119887 = 05 the excitationfrequency is close to the natural frequency If aspect ratiois increased the factor of reducing sloshing force is moredominant However for the sloshing in high aspect ratiosℎ119887 gt 075 the rise of aspect ratio brings about slight changesin the natural frequency so the factor of increasing sloshingforce is more dominant Comparing nonlinear results withlinear ones in Figure 13 some conclusions are obtained Firstthe nonlinearity of sloshing always decreases sloshing forceSecond to a certain extent the difference between linear andnonlinear amplitude reflects the effect of nonlinear factoron sloshing force For the sloshing in aspect ratio ℎ119887 =

05 the difference is the largest while for the other sloshingsituations the differences are nearly the same This resultaccordswith the conclusion in Figure 11Third nonlinearity isan important factor in large amplitude sloshing which shouldnot be neglected

Then the tank is subjected to a high-frequency excitationThe movement of the tank is described as 119886

ℎ= 0015m

120596ℎ= 4 rads Our aim is to analyze the influence of this

high-frequency excitation on the low aspect ratio sloshingsystem Figure 14 shows the time histories of free surfaceelevation at the left wall in four different aspect ratios Asthe aspect ratio decreases the amplitude and period of freesurface elevation are diminishing For the sloshing in lowaspect ratios ℎ119887 lt 05 the reduction of aspect ratio brings

about significant changes in the natural frequency so thebeating phenomenon has changed greatly even if the aspectratio is slightly decreasedThe resultant sloshing forces actingon the tank are also calculated The relationship between theamplitude of sloshing force and aspect ratio is showed inFigure 15The reduction of the aspect ratio makes the naturalfrequency far from the excitation frequency meanwhile itdecreases the mass of the liquid in the tank So both of thefactors reduce sloshing force As the aspect ratio decreasesthe difference between linear and nonlinear amplitude ofsloshing force also decreases The reason is that nonlineareffects are becoming weaker and the sloshing force itself isalso decreasing

5 Conclusions

Anewfinite differencemethod based on the prediction of freesurface has been developed and used to simulate nonlinearsloshing phenomenon in tanks by the potential flow theoryAfter 120590-transformation the fluid domain is transformed to arectangle and a predictor-corrector scheme for free surfaceis used in the iteration of time then the governing equationand boundary conditions are approximately linearized andsolved in every time step The convergence of this methodis surveyed The numerical examples of standing waves and

12 Mathematical Problems in Engineering

035 040 045 050

006

008

010

012

014

016

LinearNonlinear

F120588gbd

hlowast

hb

Figure 15 Tendency of amplitude of sloshing force (high-frequencyexcitation)

horizontally forced sloshing models have been calculatedand good agreement between the numerical simulation andresults in other literatures has been obtained This numericalmethod is simple and easy in programming with excellentaccuracy and stability and can be extended to other sloshingproblems The time histories of free surface elevation andsloshing forces in the first and third resonance situation arealso presented and analyzed

The beating phenomenon of sloshing in different aspectratios under the same excitation has been numerically simu-lated by this method The results show that the effect of theaspect ratio on the sloshing force is complicated For the low-frequency excitation acting on the high aspect ratio modelsthe rise of aspect ratio makes the sloshing far from resonanceand increases the liquid mass so the sloshing force decreasesfirst and then increases For the high-frequency excitationacting on the low aspect ratio models the reduction of aspectratiomakes the sloshing far from resonance and decreases theliquid mass so the sloshing force decreases all the way Sincethe first natural frequency of liquid sloshing in not a linearfunction of aspect ratio the change of aspect ratio producesmuch stronger effect in low aspect ratio sloshing systemsMoreover nonlinearity of sloshing always decreases sloshingforce The effect of nonlinear factor depends on the violenceof sloshing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H N Abramson ldquoThe dynamic behavior of liquids in movingcontainers nasa sp-106rdquo NASA Special Publication vol 1061966

[2] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 no 4 pp 114ndash241 1974

[3] O M Faltinsen ldquoA numerical nonlinear method of sloshing intanks with two-dimensional flowrdquo Journal of Ship Research vol22 no 3 pp 193ndash202 1978

[4] T Nakayama and K Washizu ldquoNonlinear analysis of liquidmotion in a container subjected to forced pitching oscillationrdquoInternational Journal for Numerical Methods in Engineering vol15 no 8 pp 1207ndash1220 1980

[5] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[6] B-F Chen and R Nokes ldquoTime-independent finite differenceanalysis of fully non-linear and viscous fluid sloshing in arectangular tankrdquo Journal of Computational Physics vol 209 no1 pp 47ndash81 2005

[7] C Wu and B Chen ldquoSloshing waves and resonance modesof fluid in a 3D tank by a time-independent finite differencemethodrdquoOcean Engineering vol 36 no 6-7 pp 500ndash510 2009

[8] Y Kim ldquoNumerical simulation of sloshing flows with impactloadrdquo Applied Ocean Research vol 23 no 1 pp 53ndash62 2001

[9] M S Celebi and H Akyildiz ldquoNonlinear modeling of liquidsloshing in a moving rectangular tankrdquo Ocean Engineering vol29 no 12 pp 1527ndash1553 2002

[10] C Z Wang and B C Khoo ldquoFinite element analysis of two-dimensional nonlinear sloshing problems in random excita-tionsrdquo Ocean Engineering vol 32 no 2 pp 107ndash133 2005

[11] V Sriram S A Sannasiraj and V Sundar ldquoNumerical simula-tion of 2D sloshingwaves due to horizontal and vertical randomexcitationrdquo Applied Ocean Research vol 28 no 1 pp 19ndash322006

[12] O M Faltinsen O F Rognebakke I A Lukovsky and AN Timokha ldquoMultidimensional modal analysis of nonlinearsloshing in a rectangular tank with finite water depthrdquo Journalof Fluid Mechanics vol 407 pp 201ndash234 2000

[13] O M Faltinsen and A N Timokha ldquoAn adaptive multimodalapproach to nonlinear sloshing in a rectangular tankrdquo Journalof Fluid Mechanics vol 432 pp 167ndash200 2001

[14] G XWu ldquoSecond-order resonance of sloshing in a tankrdquoOceanEngineering vol 34 no 17-18 pp 2345ndash2349 2007

[15] D F Hill ldquoTransient and steady-state amplitudes of forcedwaves in rectangular basinsrdquo Physics of Fluids vol 15 no 6 pp1576ndash1587 2003

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004

[17] RD Firouz-AbadiMGhasemi andHHaddadpour ldquoAmodalapproach to second-order analysis of sloshing using boundaryelement methodrdquo Ocean Engineering vol 38 no 1 pp 11ndash212011

[18] M J Chern A G L Borthwick and R E Taylor ldquoA pseu-dospectral 120590-transformation model of 2-D nonlinear wavesrdquoJournal of Fluids and Structures vol 13 no 5 pp 607ndash630 1999

[19] J B Frandsen and A G L Borthwick ldquoSimulation of sloshingmotions in fixed and vertically excited containers using a 2-Dinviscid120590-transformed finite difference solverrdquo Journal of Fluidsand Structures vol 18 no 2 pp 197ndash214 2003

[20] M S Turnbull A G L Borthwick and R Eatock TaylorldquoNumerical wave tank based on a 120590-transformed finite elementinviscid flow solverrdquo International Journal for Numerical Meth-ods in Fluids vol 42 no 6 pp 641ndash663 2003

[21] B Chen andH Chiang ldquoComplete 2D and fully nonlinear anal-ysis of ideal fluid in tanksrdquo Journal of Engineering Mechanicsvol 125 no 1 pp 70ndash78 1999

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

in the two equations Equation (11) does not have 120585 so thereis no need to do so These three equations are discretized as

1205972Φ

1205971199092minus

2120590

ℎ + 120585

120597120585

120597119909

1205972Φ

120597119909120597120590

+[

[

2120590

(ℎ + 120585)

2(

120597120585

120597119909

)

2

minus

120590

ℎ + 120585

1205972120585

1205971199092

]

]

120597Φ

120597120590

119896+1

+

[

[

(

120590

ℎ + 120585

120597120585

120597119909

)

2

+

1

(ℎ + 120585)

2

]

]

1205972Φ

1205971205902

119896+1

= 0

(16)

120597Φ119896+1

120597119909

100381610038161003816100381610038161003816100381610038161003816119909=0119887

minus (

120590

ℎ + 120585

120597Φ119896+1

120597120590

120597120585

120597119909

)

119909=0119887

= 0 (17)

120597Φ119896+1

120597120590

100381610038161003816100381610038161003816100381610038161003816120590=0

= 0 (18)

For the second problem an approximate method hasbeen introduced Consider two boundary conditions (12) and(13) on 119896 + 12 time step they are complicated nonlinearequations For example

120597120585

120597119909

(

120597Φ

120597119909

minus

1

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

120590=1

(19)

is a nonlinear term it should have been assigned to the valueon 119896 + 12 time step in the calculation process but here itis assigned to the value on 119896 time step as an approximationOther nonlinear terms are dealt with by the same methodLinear terms in equations are discretized by Crank-Nicolsonscheme where 120585119896+1 and Φ119896+1 (120590 = 1) are set to be unknownquantities It is noteworthy that 120585119896+1 is set to be the unknownquantity in (12) and (13) so as to update the free surfacewhen the equations are solved Overall these two boundaryconditions are discretized as

120585119896+1

minus 120585119896

Δ119905

= [

1

ℎ + 120585119896(

1

2

120597Φ119896+1

120597120590

+

1

2

120597Φ119896

120597120590

)

minus

120597120585119896

120597119909

(

120597Φ119896

120597119909

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120597120585119896

120597119909

)]

120590=1

(20)

(

Φ119896+1

minus Φ119896

Δ119905

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120585119896+1

minus 120585119896

Δ119905

)

120590=1

= minus119892(

120585119896+ 120585119896+1

2

) minus 11990911988310158401015840(119905119896+12

)

minus

1

2

[(

120597Φ119896

120597119909

minus

1

ℎ + 120585119896

120597Φ119896

120597120590

120597120585119896

120597119909

)

2

+(

1

ℎ + 120585119896

120597Φ119896

120597120590

)

2

]

120590=1

(21)

Semi-implicit scheme equations of 120585119896+1 andΦ119896+1 (120590 = 1)have been derived Moreover (16)ndash(21) can be discretized inspace and solved by finite difference method The solution120585119896+1 will be taken as the corrected value of 120585 on 119896 + 1 timestep

After 120590-transformation (6) has been transformed as

minus

119901

120588

= (

120597Φ

120597119905

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119905

) + 119892 [(ℎ + 120585) 120590 minus ℎ] + 11990911988310158401015840(119905)

+

1

2

[(

120597Φ

120597119909

minus

120590

ℎ + 120585

120597Φ

120597120590

120597120585

120597119909

)

2

+ (

1

ℎ + 120585

120597Φ

120597120590

)

2

]

(22)

When the values of 120585 and Φ are solved it is not difficult todiscretize (22) In the numerical simulation sloshing force onthe left and right walls can be expressed as

119865119871= int

119878119871

119901119889119904 = 119889int

120585

minusℎ

1199011003816100381610038161003816119909=0

119889119911

119865119877= int

119878119877

119901119889119904 = 119889int

120585

minusℎ

1199011003816100381610038161003816119909=119887

119889119911

(23)

The resultant sloshing force acting on the tank is

119865 = 119865119877minus 119865119871 (24)

4 Numerical Results

Some examples of nonlinear sloshing are simulated by thisnumerical method They have been investigated by otherresearchers using various numerical and analytical methodsand many discussions and conclusions have been derived

A rectangular tank model is established The naturalfrequencies of linear transverse sloshing along 119909-axis are(Faltinsen [2])

120596119899= radic119892

119899120587

119887

tanh 119899120587ℎ119887

119899 = 1 2 3 (25)

41 Standing Waves The tank is fixed so the moving coor-dinate system (119909 119911) coincides with the inertial coordinatesystem (119883 119885) The initial conditions are described as

120585 (119909 0) = 119886 cos(119899120587119887

) 120593 (119909 119911 0) = 0 (26)

119886 is the amplitude of the initial wave profile which influencesthe nonlinearity of the sloshing 119899 indicates themodenumberThe convergence of this numerical simulation is investigatedIn the tank 119887 119889 and ℎ are set to be 2m 1m and 1mrespectively 119886 is set to be 02m 119899 takes value of 1 Thefirst natural frequency of the sloshing is 120596

1= 376 rads

Figure 2(a) shows the time history of the free surface at theleft wall with different meshes but the time step sizes arethe same (0005 s) Figure 2(b) shows the time history wherethe meshes are the same (40 times 40) but the time step sizes

Mathematical Problems in Engineering 5

20 40 60 80 100

00

05

10

15

minus05

minus10

Mesh 40 times 40Mesh 60 times 60

1205961t

120585a

(a) Free surface elevation at the left wall using different meshes

Step 001Step 0005

20 40 60 80 1001205961t

00

05

10

15

minus05

minus10

120585a

(b) Free surface elevation at the left wall using different time step sizes

Figure 2 Time history of free surface elevation for different meshes and time step sizes

20 40 60 80 100

00

05

10

15

minus05

minus10

1205961t

120585a

AnalyticalNumerical

(a) Comparison of free surface elevation at the left wall

05 10 15 20x

00

05

10

15

minus05

minus10

120585a

(b) Wave profiles for a half-period

20 40 60 80 100

020

022

024

026

028

030

1205961t

FL

120588gbd

h

(c) Sloshing force on the left wall

20 40 60 80 100

020

022

024

026

028

030

1205961t

FR

120588gbd

h

(d) Sloshing force on the right wall

Figure 3 Standing wave (119899 = 1)

6 Mathematical Problems in Engineering

20 40 60 80 100

00

05

10

AnalyticalNumerical

1205961t

minus05

minus10

120585a

(a) Comparison of free surface elevation at the left wall

05 10 15 20x

00

05

10

minus05

minus10

120585a

(b) Wave profiles for a half-period

Figure 4 Standing wave (119899 = 3)

are different The time and the vibration displacement in thefigures are dealt with by dimensionless method

1205851015840=

120585

119886

1199051015840= 1205961119905 (27)

It is found that the results from these meshes and timestep sizes are in good agreement so in this example a meshdistribution of 40 times 40 and time step size of 0005 s ensure areasonable converged solution

Nonlinear sloshing equations of standing waves can besolved by perturbation techniques First nondimensionalvariables are introduced to the equations Then 120598 = 119886120596

2

119899119892

is chosen as the characteristic small parameter Approximatelinear equations are derived by small parameter expansionand approximate analytical solutions can be obtained Thesecond-order approximate analytical free surface elevationfor the 119899th sloshing mode is prescribed as (Frandsen [16])

120585 (119909 119905) = 119886 ( cos (120596119899119905) cos (119896

119899119909)

+ 120598 (119860119899+ 119861119899cos (2120596

119899119905)

+119862119899cos (120596

2119899119905)) cos (2119896

119899119909))

(28)

where

119896119899=

119899120587

119887

1205962119899= radic1198922119896

119899tanh (2119896

119899ℎ)

119860119899=

1205964

119899+ 11989221198962

119899

81205964

119899

119861119899=

31205964

119899minus 11989221198962

119899

81205964

119899

minus

3

2

1205964

119899minus 11989221198962

119899

1205962

119899(41205962

119899minus 1205962

2119899)

119862119899=

1205962

1198991205962

2119899minus 1205964

119899minus 311989221198962

119899

21205962

119899(41205962

119899minus 1205962

2119899)

(29)

Figure 3(a) shows the time history of the free surfaceelevation for 119899 = 1 at the leftwall by the twomethods numer-ical solution by the finite difference method proposed in thispaper and second-order approximate analytical solution byperturbation method Initially good agreement between thetwo results is achieved but the phase difference between themincreases with time This phenomenon has been noticed anddiscussed by many researchers (Chern et al [18] Frandsen[16]) The conclusion is that the second-order analyticalsolution is not very accurate for large amplitude sloshingFigure 3(b) displays numerical wave profiles for nearly halfa period from dimensionless time 120596

1119905 = 318 to 120596

1119905 =

346 Stationary point does not exist and nonlinearity isobvious Figures 3(c) and 3(d) provide the sloshing forceson the left wall and right wall respectively (the forces arealso nondimensionalized) It shows that slight double peaksappear in the time historyThis has also been observed byWuet al [5]

The standing wave of 119899 = 3 is also calculated The initialconditions are set to be

120585 (119909 0) = 006 cos(3120587119887

) 120593 (119909 119911 0) = 0 (30)

Figure 4(a) shows the comparison of numerical solutionsand approximate analytical solutions Figure 4(b) showsnumerical wave profiles for nearly half a period Phasedifference and nonlinearity are also observed

42 Sloshing Caused by Horizontal Excitations The tankis assumed to take the horizontal oscillation the externalexcitation acting on the tank is defined as harmonic motion

119883 (119905) = 119886ℎcos (120596

ℎ119905) (31)

119886ℎand 120596

ℎare the amplitude and angular frequency of the

excited motionThe fluid is assumed to be stationary at 119905 = 0Then in the moving coordinate system the initial conditionsare described as

120585 (119909 0) = 0 120593 (119909 119911 0) = minus1199091198831015840(119905) = 0 (32)

Mathematical Problems in Engineering 7

2

1

0

minus2

minus1

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(a) Example 1 120596ℎ = 071205961 120598ℎ = 0036

2

0

minus2

4

minus4

6

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(b) Example 2 120596ℎ = 131205961 120598ℎ = 0072

Figure 5 Comparison of free surface elevation at the left wall

20 40 60 80 100

0

20

40

60

minus20

120585a

1205961t

(a) The first resonance mode

20 40 60 80 100

0

10

20

minus10

1205963t

120585a h

(b) The third resonance mode

Figure 6 Free surface elevation at the left wall (resonance phenomenon)

20 40 60 80 100

000

005

010

015

minus015

minus010

minus005F120588gbd

h

1205961t

(a) The first resonance mode

20 40 60 80 100

000

001

002

minus002

minus001

1205963t

F120588gbd

h

(b) The third resonance mode

Figure 7 Resultant sloshing force (resonance phenomenon)

In order to verify the precision of this numerical methodfor the sloshing models in horizontally moving tanks twoexamples are calculated and compared with the solutions ofFrandsen [16] (1) 120596

ℎ= 07120596

1 120598ℎ= 0036 (2) 120596

ℎ= 13120596

1

120598ℎ= 0072 The grid size of the numerical method is 40 times 40

and the time step size is 0006 s Figure 5(a) shows that the twosimulations are in close agreement In the second examplethe nonlinear effect is strong so there is a little differencebetween the two solutions in Figure 5(b) Neverthelessnonlinear characteristics of sloshing are captured by the two

8 Mathematical Problems in Engineering

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 005

20 40 60 80 100 120

0

20

40

60

minus20

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 8 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 12)

20 40 60 80 100

0

2

4

6

minus2

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 003

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 9 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 18)

02 04 06 08 10 12 14

02

04

06

08

10

1205961120596

lowast 1

hb

Figure 10 The function graph of the first natural frequency againstaspect ratio

numerical methods such as the higher peaks and the lessdeep troughs

421 Resonance Phenomenon Two situations of resonanceare considered the first resonance mode and the thirdresonance mode The external excitations of the tank aredescribed as 119883

1(119905) = 0007 cos(120596

1119905) 1198832(119905) = 0007 cos(120596

3119905)

The amplitudes are the same but the excited frequencies

are different The time histories of free surface elevation atthe left wall and the resultant sloshing forces are calculatedFigure 6(a) shows that the sloshing of the first resonancemode is violate the amplitude of the vibration is increasingFigure 6(b) shows that the amplitude of the third resonancemode is much smaller than that of the first resonance modeIn Figure 7 it can be concluded that the resultant sloshingforce of the third resonance mode is trivial compared withthe force of the first resonance mode In fact the first naturalfrequency of sloshing is the most influential of all the naturalfrequencies

422 The Limitation of the Numerical Method The presentnumerical method is employed to simulate nonlinear slosh-ing in moving tanks If the nonlinearity is quite strong thismethodmay lose its effectiveness Two excitation frequenciesare considered in the simulations One is 120596

ℎ= 085120596

1 the

other is 120596ℎ= 095120596

1 which indicates near resonant sloshing

situation Different excitation amplitudes 119886ℎand aspect ratios

ℎ119887 are tested and the limitation of the convergence of thisnumerical method is found out

First the aspect ratio is set to be ℎ119887 = 12 just asbefore Excitation amplitude is increased in the numericalsimulation and the sloshing becomesmore andmore violentFor 120596

ℎ= 085120596

1 if the excitation amplitude is 119886

ℎ= 006m

Mathematical Problems in Engineering 9

20 40 60 80 100

0

5

10

15

20

minus5

minus10

120585a h

120596ht

(a) Aspect ratio ℎ119887 = 05

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) Aspect ratio ℎ119887 = 075

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(c) Aspect ratio ℎ119887 = 1

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 11 Free surface elevation at the left wall (low-frequency excitation)

the simulation is divergent after several times of iteration if119886ℎ= 005m the simulation lasts for a long time So it is

considered that the limitation of 119886ℎfor nonresonant sloshing

is 005m For 120596ℎ= 095120596

1 the excitation amplitude is much

smaller The maximum limit of 119886ℎis estimated to be 0012m

The time histories of free surface elevation at the left wall inthe limit cases are shown in Figure 8

Then the aspect ratio is changed by decreasing the stillwater depth ℎ For the aspect ratio ℎ119887 = 18 the numericalmethod still works with a mesh size of 80 times 20 If the aspectratio is less than 18 the simulation is divergent As a resultthe minimum limit of aspect ratio ℎ119887 is considered to be18 In this critical aspect ratio situation for 120596

ℎ= 095120596

1

the maximum limit of excitation amplitude 119886ℎis still 0012m

while for 120596ℎ= 085120596

1 the maximum limit of 119886

ℎis 003m

Figure 9 shows the time histories of free surface elevation inthe limit cases in this critical aspect ratio The phenomena ofbeating and resonance are very obscure it can be concludedthat the sloshing in low aspect ratio is quite different from theone in normal aspect ratio

423 Beating Phenomenon in Different Aspect Ratios Slosh-ing in a fixed aspect ratio has been studied in many articlesHowever little work has been done to analyze the response ofliquid sloshing in different aspect ratios by the same external

excitation When the external excitation is close to the firstnatural frequency of sloshing the beating phenomenon canbe observed The study here is intended to investigate theeffect of different aspect ratios on beating phenomenon andsloshing force

In the numerical examples calculated before the lengthand width of the tank are set to be 2m and 1m the still waterdepth is 1m The first natural frequency of sloshing is 120596

1=

376 rads The aspect ratio is ℎ119887 = 12 Now the length 119887and the width 119889 of the tank are fixed and the aspect ratio ischanged by changing the still water depth ℎThe former waterdepth and natural frequency are set to be the characteristicquantity ℎlowast = 1m 120596lowast

1= 376 rads

The relationship between aspect ratio and the first naturalfrequency must be considered From (25) it is easy to get thegraph of the natural frequency as a function of aspect ratioFigure 10 indicates that the slope of the function curve is steepwhen the aspect ratio is small if the aspect ratio is greater than06 the function curve is nearly horizontal

Under the same external excitation the sloshing forcesof beating are affected by three factors proximity of exci-tation frequency to the natural frequency liquid mass andnonlinearity of sloshing When the aspect ratio of the liquidchanges these three factors would change along with it Inorder to discuss the effect of aspect ratio two horizontaloscillations of the tank are studied First the tank is subjected

10 Mathematical Problems in Engineering

20 40 60 80 100

00

01

02

minus01

F120588gbd

hlowast

120596ht

(a) Aspect ratio ℎ119887 = 05

005

010

015

minus005

minus010

minus015

F120588gbd

hlowast

20 40 60 80 100120596ht

000

(b) Aspect ratio ℎ119887 = 075

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(c) Aspect ratio ℎ119887 = 1

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 12 Resultant sloshing forces (low-frequency excitation)

06 07 08 09 10 11 12

019

020

021

022

LinearNonlinear

F120588gbd

hlowast

hb

Figure 13 Tendency of amplitude of sloshing force (low-frequencyexcitation)

to a low-frequency excitation The movement of the tank isdescribed as 119886

ℎ= 003m 120596

ℎ= 34 rads Figure 10 suggests

that if the aspect ratio is 03 the corresponding naturalfrequency is 34 rads Our aim is to analyze the influence ofthis low-frequency excitation on the high aspect ratio slosh-ing system Figure 11 shows the time histories of free surface

elevation at the left wall in four different aspect ratios wherethe parameters are nondimensionalized

1199051015840= 120596ℎ119905 120585

1015840=

120585

119886ℎ

(33)

The amplitude of free surface elevation is quite large inFigure 11(a) which shows strong nonlinearity of sloshingwhile in Figures 11(b) 11(c) and 11(d) the amplitudes arenearly the same smaller than the amplitude in Figure 11(a)so the nonlinear effects are weaker in these three sloshingsituations

Figure 12 shows the time histories of resultant sloshingforce acting on the tank (the forces are also nondimensional-ized 1198651015840 = 119865120588119892119887119889ℎlowast) The amplitudes of the sloshing forceare extracted in Figure 12 In order to study the effect ofnonlinear factor linear sloshing models by Abramson [1] arecalculated under the same condition to make comparisonsThe relationship between the amplitude of sloshing force andaspect ratio is shown in Figure 13 It is obvious that bothlinear and nonlinear results have the same trend decreasingfirst and then increasing On the one hand the rise of theaspect ratio increases the natural frequency which makesthe sloshing far from resonance and reduces sloshing forcesOn the other hand the rise of the aspect ratio increases themass of the liquid thereby increasing the sloshing forces

Mathematical Problems in Engineering 11

0

10

20120585a h

20 40 60 80 100120596ht

minus10

(a) Aspect ratio ℎ119887 = 05

0

5

10

15

minus5

120585a h

20 40 60 80 100120596ht

minus10

(b) Aspect ratio ℎ119887 = 045

0

5

10

120585a h

20 40 60 80 100120596ht

minus5

minus10

(c) Aspect ratio ℎ119887 = 04

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 035

Figure 14 Free surface elevation at the left wall (high-frequency excitation)

For the sloshing in aspect ratio ℎ119887 = 05 the excitationfrequency is close to the natural frequency If aspect ratiois increased the factor of reducing sloshing force is moredominant However for the sloshing in high aspect ratiosℎ119887 gt 075 the rise of aspect ratio brings about slight changesin the natural frequency so the factor of increasing sloshingforce is more dominant Comparing nonlinear results withlinear ones in Figure 13 some conclusions are obtained Firstthe nonlinearity of sloshing always decreases sloshing forceSecond to a certain extent the difference between linear andnonlinear amplitude reflects the effect of nonlinear factoron sloshing force For the sloshing in aspect ratio ℎ119887 =

05 the difference is the largest while for the other sloshingsituations the differences are nearly the same This resultaccordswith the conclusion in Figure 11Third nonlinearity isan important factor in large amplitude sloshing which shouldnot be neglected

Then the tank is subjected to a high-frequency excitationThe movement of the tank is described as 119886

ℎ= 0015m

120596ℎ= 4 rads Our aim is to analyze the influence of this

high-frequency excitation on the low aspect ratio sloshingsystem Figure 14 shows the time histories of free surfaceelevation at the left wall in four different aspect ratios Asthe aspect ratio decreases the amplitude and period of freesurface elevation are diminishing For the sloshing in lowaspect ratios ℎ119887 lt 05 the reduction of aspect ratio brings

about significant changes in the natural frequency so thebeating phenomenon has changed greatly even if the aspectratio is slightly decreasedThe resultant sloshing forces actingon the tank are also calculated The relationship between theamplitude of sloshing force and aspect ratio is showed inFigure 15The reduction of the aspect ratio makes the naturalfrequency far from the excitation frequency meanwhile itdecreases the mass of the liquid in the tank So both of thefactors reduce sloshing force As the aspect ratio decreasesthe difference between linear and nonlinear amplitude ofsloshing force also decreases The reason is that nonlineareffects are becoming weaker and the sloshing force itself isalso decreasing

5 Conclusions

Anewfinite differencemethod based on the prediction of freesurface has been developed and used to simulate nonlinearsloshing phenomenon in tanks by the potential flow theoryAfter 120590-transformation the fluid domain is transformed to arectangle and a predictor-corrector scheme for free surfaceis used in the iteration of time then the governing equationand boundary conditions are approximately linearized andsolved in every time step The convergence of this methodis surveyed The numerical examples of standing waves and

12 Mathematical Problems in Engineering

035 040 045 050

006

008

010

012

014

016

LinearNonlinear

F120588gbd

hlowast

hb

Figure 15 Tendency of amplitude of sloshing force (high-frequencyexcitation)

horizontally forced sloshing models have been calculatedand good agreement between the numerical simulation andresults in other literatures has been obtained This numericalmethod is simple and easy in programming with excellentaccuracy and stability and can be extended to other sloshingproblems The time histories of free surface elevation andsloshing forces in the first and third resonance situation arealso presented and analyzed

The beating phenomenon of sloshing in different aspectratios under the same excitation has been numerically simu-lated by this method The results show that the effect of theaspect ratio on the sloshing force is complicated For the low-frequency excitation acting on the high aspect ratio modelsthe rise of aspect ratio makes the sloshing far from resonanceand increases the liquid mass so the sloshing force decreasesfirst and then increases For the high-frequency excitationacting on the low aspect ratio models the reduction of aspectratiomakes the sloshing far from resonance and decreases theliquid mass so the sloshing force decreases all the way Sincethe first natural frequency of liquid sloshing in not a linearfunction of aspect ratio the change of aspect ratio producesmuch stronger effect in low aspect ratio sloshing systemsMoreover nonlinearity of sloshing always decreases sloshingforce The effect of nonlinear factor depends on the violenceof sloshing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H N Abramson ldquoThe dynamic behavior of liquids in movingcontainers nasa sp-106rdquo NASA Special Publication vol 1061966

[2] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 no 4 pp 114ndash241 1974

[3] O M Faltinsen ldquoA numerical nonlinear method of sloshing intanks with two-dimensional flowrdquo Journal of Ship Research vol22 no 3 pp 193ndash202 1978

[4] T Nakayama and K Washizu ldquoNonlinear analysis of liquidmotion in a container subjected to forced pitching oscillationrdquoInternational Journal for Numerical Methods in Engineering vol15 no 8 pp 1207ndash1220 1980

[5] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[6] B-F Chen and R Nokes ldquoTime-independent finite differenceanalysis of fully non-linear and viscous fluid sloshing in arectangular tankrdquo Journal of Computational Physics vol 209 no1 pp 47ndash81 2005

[7] C Wu and B Chen ldquoSloshing waves and resonance modesof fluid in a 3D tank by a time-independent finite differencemethodrdquoOcean Engineering vol 36 no 6-7 pp 500ndash510 2009

[8] Y Kim ldquoNumerical simulation of sloshing flows with impactloadrdquo Applied Ocean Research vol 23 no 1 pp 53ndash62 2001

[9] M S Celebi and H Akyildiz ldquoNonlinear modeling of liquidsloshing in a moving rectangular tankrdquo Ocean Engineering vol29 no 12 pp 1527ndash1553 2002

[10] C Z Wang and B C Khoo ldquoFinite element analysis of two-dimensional nonlinear sloshing problems in random excita-tionsrdquo Ocean Engineering vol 32 no 2 pp 107ndash133 2005

[11] V Sriram S A Sannasiraj and V Sundar ldquoNumerical simula-tion of 2D sloshingwaves due to horizontal and vertical randomexcitationrdquo Applied Ocean Research vol 28 no 1 pp 19ndash322006

[12] O M Faltinsen O F Rognebakke I A Lukovsky and AN Timokha ldquoMultidimensional modal analysis of nonlinearsloshing in a rectangular tank with finite water depthrdquo Journalof Fluid Mechanics vol 407 pp 201ndash234 2000

[13] O M Faltinsen and A N Timokha ldquoAn adaptive multimodalapproach to nonlinear sloshing in a rectangular tankrdquo Journalof Fluid Mechanics vol 432 pp 167ndash200 2001

[14] G XWu ldquoSecond-order resonance of sloshing in a tankrdquoOceanEngineering vol 34 no 17-18 pp 2345ndash2349 2007

[15] D F Hill ldquoTransient and steady-state amplitudes of forcedwaves in rectangular basinsrdquo Physics of Fluids vol 15 no 6 pp1576ndash1587 2003

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004

[17] RD Firouz-AbadiMGhasemi andHHaddadpour ldquoAmodalapproach to second-order analysis of sloshing using boundaryelement methodrdquo Ocean Engineering vol 38 no 1 pp 11ndash212011

[18] M J Chern A G L Borthwick and R E Taylor ldquoA pseu-dospectral 120590-transformation model of 2-D nonlinear wavesrdquoJournal of Fluids and Structures vol 13 no 5 pp 607ndash630 1999

[19] J B Frandsen and A G L Borthwick ldquoSimulation of sloshingmotions in fixed and vertically excited containers using a 2-Dinviscid120590-transformed finite difference solverrdquo Journal of Fluidsand Structures vol 18 no 2 pp 197ndash214 2003

[20] M S Turnbull A G L Borthwick and R Eatock TaylorldquoNumerical wave tank based on a 120590-transformed finite elementinviscid flow solverrdquo International Journal for Numerical Meth-ods in Fluids vol 42 no 6 pp 641ndash663 2003

[21] B Chen andH Chiang ldquoComplete 2D and fully nonlinear anal-ysis of ideal fluid in tanksrdquo Journal of Engineering Mechanicsvol 125 no 1 pp 70ndash78 1999

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

20 40 60 80 100

00

05

10

15

minus05

minus10

Mesh 40 times 40Mesh 60 times 60

1205961t

120585a

(a) Free surface elevation at the left wall using different meshes

Step 001Step 0005

20 40 60 80 1001205961t

00

05

10

15

minus05

minus10

120585a

(b) Free surface elevation at the left wall using different time step sizes

Figure 2 Time history of free surface elevation for different meshes and time step sizes

20 40 60 80 100

00

05

10

15

minus05

minus10

1205961t

120585a

AnalyticalNumerical

(a) Comparison of free surface elevation at the left wall

05 10 15 20x

00

05

10

15

minus05

minus10

120585a

(b) Wave profiles for a half-period

20 40 60 80 100

020

022

024

026

028

030

1205961t

FL

120588gbd

h

(c) Sloshing force on the left wall

20 40 60 80 100

020

022

024

026

028

030

1205961t

FR

120588gbd

h

(d) Sloshing force on the right wall

Figure 3 Standing wave (119899 = 1)

6 Mathematical Problems in Engineering

20 40 60 80 100

00

05

10

AnalyticalNumerical

1205961t

minus05

minus10

120585a

(a) Comparison of free surface elevation at the left wall

05 10 15 20x

00

05

10

minus05

minus10

120585a

(b) Wave profiles for a half-period

Figure 4 Standing wave (119899 = 3)

are different The time and the vibration displacement in thefigures are dealt with by dimensionless method

1205851015840=

120585

119886

1199051015840= 1205961119905 (27)

It is found that the results from these meshes and timestep sizes are in good agreement so in this example a meshdistribution of 40 times 40 and time step size of 0005 s ensure areasonable converged solution

Nonlinear sloshing equations of standing waves can besolved by perturbation techniques First nondimensionalvariables are introduced to the equations Then 120598 = 119886120596

2

119899119892

is chosen as the characteristic small parameter Approximatelinear equations are derived by small parameter expansionand approximate analytical solutions can be obtained Thesecond-order approximate analytical free surface elevationfor the 119899th sloshing mode is prescribed as (Frandsen [16])

120585 (119909 119905) = 119886 ( cos (120596119899119905) cos (119896

119899119909)

+ 120598 (119860119899+ 119861119899cos (2120596

119899119905)

+119862119899cos (120596

2119899119905)) cos (2119896

119899119909))

(28)

where

119896119899=

119899120587

119887

1205962119899= radic1198922119896

119899tanh (2119896

119899ℎ)

119860119899=

1205964

119899+ 11989221198962

119899

81205964

119899

119861119899=

31205964

119899minus 11989221198962

119899

81205964

119899

minus

3

2

1205964

119899minus 11989221198962

119899

1205962

119899(41205962

119899minus 1205962

2119899)

119862119899=

1205962

1198991205962

2119899minus 1205964

119899minus 311989221198962

119899

21205962

119899(41205962

119899minus 1205962

2119899)

(29)

Figure 3(a) shows the time history of the free surfaceelevation for 119899 = 1 at the leftwall by the twomethods numer-ical solution by the finite difference method proposed in thispaper and second-order approximate analytical solution byperturbation method Initially good agreement between thetwo results is achieved but the phase difference between themincreases with time This phenomenon has been noticed anddiscussed by many researchers (Chern et al [18] Frandsen[16]) The conclusion is that the second-order analyticalsolution is not very accurate for large amplitude sloshingFigure 3(b) displays numerical wave profiles for nearly halfa period from dimensionless time 120596

1119905 = 318 to 120596

1119905 =

346 Stationary point does not exist and nonlinearity isobvious Figures 3(c) and 3(d) provide the sloshing forceson the left wall and right wall respectively (the forces arealso nondimensionalized) It shows that slight double peaksappear in the time historyThis has also been observed byWuet al [5]

The standing wave of 119899 = 3 is also calculated The initialconditions are set to be

120585 (119909 0) = 006 cos(3120587119887

) 120593 (119909 119911 0) = 0 (30)

Figure 4(a) shows the comparison of numerical solutionsand approximate analytical solutions Figure 4(b) showsnumerical wave profiles for nearly half a period Phasedifference and nonlinearity are also observed

42 Sloshing Caused by Horizontal Excitations The tankis assumed to take the horizontal oscillation the externalexcitation acting on the tank is defined as harmonic motion

119883 (119905) = 119886ℎcos (120596

ℎ119905) (31)

119886ℎand 120596

ℎare the amplitude and angular frequency of the

excited motionThe fluid is assumed to be stationary at 119905 = 0Then in the moving coordinate system the initial conditionsare described as

120585 (119909 0) = 0 120593 (119909 119911 0) = minus1199091198831015840(119905) = 0 (32)

Mathematical Problems in Engineering 7

2

1

0

minus2

minus1

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(a) Example 1 120596ℎ = 071205961 120598ℎ = 0036

2

0

minus2

4

minus4

6

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(b) Example 2 120596ℎ = 131205961 120598ℎ = 0072

Figure 5 Comparison of free surface elevation at the left wall

20 40 60 80 100

0

20

40

60

minus20

120585a

1205961t

(a) The first resonance mode

20 40 60 80 100

0

10

20

minus10

1205963t

120585a h

(b) The third resonance mode

Figure 6 Free surface elevation at the left wall (resonance phenomenon)

20 40 60 80 100

000

005

010

015

minus015

minus010

minus005F120588gbd

h

1205961t

(a) The first resonance mode

20 40 60 80 100

000

001

002

minus002

minus001

1205963t

F120588gbd

h

(b) The third resonance mode

Figure 7 Resultant sloshing force (resonance phenomenon)

In order to verify the precision of this numerical methodfor the sloshing models in horizontally moving tanks twoexamples are calculated and compared with the solutions ofFrandsen [16] (1) 120596

ℎ= 07120596

1 120598ℎ= 0036 (2) 120596

ℎ= 13120596

1

120598ℎ= 0072 The grid size of the numerical method is 40 times 40

and the time step size is 0006 s Figure 5(a) shows that the twosimulations are in close agreement In the second examplethe nonlinear effect is strong so there is a little differencebetween the two solutions in Figure 5(b) Neverthelessnonlinear characteristics of sloshing are captured by the two

8 Mathematical Problems in Engineering

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 005

20 40 60 80 100 120

0

20

40

60

minus20

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 8 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 12)

20 40 60 80 100

0

2

4

6

minus2

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 003

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 9 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 18)

02 04 06 08 10 12 14

02

04

06

08

10

1205961120596

lowast 1

hb

Figure 10 The function graph of the first natural frequency againstaspect ratio

numerical methods such as the higher peaks and the lessdeep troughs

421 Resonance Phenomenon Two situations of resonanceare considered the first resonance mode and the thirdresonance mode The external excitations of the tank aredescribed as 119883

1(119905) = 0007 cos(120596

1119905) 1198832(119905) = 0007 cos(120596

3119905)

The amplitudes are the same but the excited frequencies

are different The time histories of free surface elevation atthe left wall and the resultant sloshing forces are calculatedFigure 6(a) shows that the sloshing of the first resonancemode is violate the amplitude of the vibration is increasingFigure 6(b) shows that the amplitude of the third resonancemode is much smaller than that of the first resonance modeIn Figure 7 it can be concluded that the resultant sloshingforce of the third resonance mode is trivial compared withthe force of the first resonance mode In fact the first naturalfrequency of sloshing is the most influential of all the naturalfrequencies

422 The Limitation of the Numerical Method The presentnumerical method is employed to simulate nonlinear slosh-ing in moving tanks If the nonlinearity is quite strong thismethodmay lose its effectiveness Two excitation frequenciesare considered in the simulations One is 120596

ℎ= 085120596

1 the

other is 120596ℎ= 095120596

1 which indicates near resonant sloshing

situation Different excitation amplitudes 119886ℎand aspect ratios

ℎ119887 are tested and the limitation of the convergence of thisnumerical method is found out

First the aspect ratio is set to be ℎ119887 = 12 just asbefore Excitation amplitude is increased in the numericalsimulation and the sloshing becomesmore andmore violentFor 120596

ℎ= 085120596

1 if the excitation amplitude is 119886

ℎ= 006m

Mathematical Problems in Engineering 9

20 40 60 80 100

0

5

10

15

20

minus5

minus10

120585a h

120596ht

(a) Aspect ratio ℎ119887 = 05

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) Aspect ratio ℎ119887 = 075

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(c) Aspect ratio ℎ119887 = 1

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 11 Free surface elevation at the left wall (low-frequency excitation)

the simulation is divergent after several times of iteration if119886ℎ= 005m the simulation lasts for a long time So it is

considered that the limitation of 119886ℎfor nonresonant sloshing

is 005m For 120596ℎ= 095120596

1 the excitation amplitude is much

smaller The maximum limit of 119886ℎis estimated to be 0012m

The time histories of free surface elevation at the left wall inthe limit cases are shown in Figure 8

Then the aspect ratio is changed by decreasing the stillwater depth ℎ For the aspect ratio ℎ119887 = 18 the numericalmethod still works with a mesh size of 80 times 20 If the aspectratio is less than 18 the simulation is divergent As a resultthe minimum limit of aspect ratio ℎ119887 is considered to be18 In this critical aspect ratio situation for 120596

ℎ= 095120596

1

the maximum limit of excitation amplitude 119886ℎis still 0012m

while for 120596ℎ= 085120596

1 the maximum limit of 119886

ℎis 003m

Figure 9 shows the time histories of free surface elevation inthe limit cases in this critical aspect ratio The phenomena ofbeating and resonance are very obscure it can be concludedthat the sloshing in low aspect ratio is quite different from theone in normal aspect ratio

423 Beating Phenomenon in Different Aspect Ratios Slosh-ing in a fixed aspect ratio has been studied in many articlesHowever little work has been done to analyze the response ofliquid sloshing in different aspect ratios by the same external

excitation When the external excitation is close to the firstnatural frequency of sloshing the beating phenomenon canbe observed The study here is intended to investigate theeffect of different aspect ratios on beating phenomenon andsloshing force

In the numerical examples calculated before the lengthand width of the tank are set to be 2m and 1m the still waterdepth is 1m The first natural frequency of sloshing is 120596

1=

376 rads The aspect ratio is ℎ119887 = 12 Now the length 119887and the width 119889 of the tank are fixed and the aspect ratio ischanged by changing the still water depth ℎThe former waterdepth and natural frequency are set to be the characteristicquantity ℎlowast = 1m 120596lowast

1= 376 rads

The relationship between aspect ratio and the first naturalfrequency must be considered From (25) it is easy to get thegraph of the natural frequency as a function of aspect ratioFigure 10 indicates that the slope of the function curve is steepwhen the aspect ratio is small if the aspect ratio is greater than06 the function curve is nearly horizontal

Under the same external excitation the sloshing forcesof beating are affected by three factors proximity of exci-tation frequency to the natural frequency liquid mass andnonlinearity of sloshing When the aspect ratio of the liquidchanges these three factors would change along with it Inorder to discuss the effect of aspect ratio two horizontaloscillations of the tank are studied First the tank is subjected

10 Mathematical Problems in Engineering

20 40 60 80 100

00

01

02

minus01

F120588gbd

hlowast

120596ht

(a) Aspect ratio ℎ119887 = 05

005

010

015

minus005

minus010

minus015

F120588gbd

hlowast

20 40 60 80 100120596ht

000

(b) Aspect ratio ℎ119887 = 075

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(c) Aspect ratio ℎ119887 = 1

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 12 Resultant sloshing forces (low-frequency excitation)

06 07 08 09 10 11 12

019

020

021

022

LinearNonlinear

F120588gbd

hlowast

hb

Figure 13 Tendency of amplitude of sloshing force (low-frequencyexcitation)

to a low-frequency excitation The movement of the tank isdescribed as 119886

ℎ= 003m 120596

ℎ= 34 rads Figure 10 suggests

that if the aspect ratio is 03 the corresponding naturalfrequency is 34 rads Our aim is to analyze the influence ofthis low-frequency excitation on the high aspect ratio slosh-ing system Figure 11 shows the time histories of free surface

elevation at the left wall in four different aspect ratios wherethe parameters are nondimensionalized

1199051015840= 120596ℎ119905 120585

1015840=

120585

119886ℎ

(33)

The amplitude of free surface elevation is quite large inFigure 11(a) which shows strong nonlinearity of sloshingwhile in Figures 11(b) 11(c) and 11(d) the amplitudes arenearly the same smaller than the amplitude in Figure 11(a)so the nonlinear effects are weaker in these three sloshingsituations

Figure 12 shows the time histories of resultant sloshingforce acting on the tank (the forces are also nondimensional-ized 1198651015840 = 119865120588119892119887119889ℎlowast) The amplitudes of the sloshing forceare extracted in Figure 12 In order to study the effect ofnonlinear factor linear sloshing models by Abramson [1] arecalculated under the same condition to make comparisonsThe relationship between the amplitude of sloshing force andaspect ratio is shown in Figure 13 It is obvious that bothlinear and nonlinear results have the same trend decreasingfirst and then increasing On the one hand the rise of theaspect ratio increases the natural frequency which makesthe sloshing far from resonance and reduces sloshing forcesOn the other hand the rise of the aspect ratio increases themass of the liquid thereby increasing the sloshing forces

Mathematical Problems in Engineering 11

0

10

20120585a h

20 40 60 80 100120596ht

minus10

(a) Aspect ratio ℎ119887 = 05

0

5

10

15

minus5

120585a h

20 40 60 80 100120596ht

minus10

(b) Aspect ratio ℎ119887 = 045

0

5

10

120585a h

20 40 60 80 100120596ht

minus5

minus10

(c) Aspect ratio ℎ119887 = 04

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 035

Figure 14 Free surface elevation at the left wall (high-frequency excitation)

For the sloshing in aspect ratio ℎ119887 = 05 the excitationfrequency is close to the natural frequency If aspect ratiois increased the factor of reducing sloshing force is moredominant However for the sloshing in high aspect ratiosℎ119887 gt 075 the rise of aspect ratio brings about slight changesin the natural frequency so the factor of increasing sloshingforce is more dominant Comparing nonlinear results withlinear ones in Figure 13 some conclusions are obtained Firstthe nonlinearity of sloshing always decreases sloshing forceSecond to a certain extent the difference between linear andnonlinear amplitude reflects the effect of nonlinear factoron sloshing force For the sloshing in aspect ratio ℎ119887 =

05 the difference is the largest while for the other sloshingsituations the differences are nearly the same This resultaccordswith the conclusion in Figure 11Third nonlinearity isan important factor in large amplitude sloshing which shouldnot be neglected

Then the tank is subjected to a high-frequency excitationThe movement of the tank is described as 119886

ℎ= 0015m

120596ℎ= 4 rads Our aim is to analyze the influence of this

high-frequency excitation on the low aspect ratio sloshingsystem Figure 14 shows the time histories of free surfaceelevation at the left wall in four different aspect ratios Asthe aspect ratio decreases the amplitude and period of freesurface elevation are diminishing For the sloshing in lowaspect ratios ℎ119887 lt 05 the reduction of aspect ratio brings

about significant changes in the natural frequency so thebeating phenomenon has changed greatly even if the aspectratio is slightly decreasedThe resultant sloshing forces actingon the tank are also calculated The relationship between theamplitude of sloshing force and aspect ratio is showed inFigure 15The reduction of the aspect ratio makes the naturalfrequency far from the excitation frequency meanwhile itdecreases the mass of the liquid in the tank So both of thefactors reduce sloshing force As the aspect ratio decreasesthe difference between linear and nonlinear amplitude ofsloshing force also decreases The reason is that nonlineareffects are becoming weaker and the sloshing force itself isalso decreasing

5 Conclusions

Anewfinite differencemethod based on the prediction of freesurface has been developed and used to simulate nonlinearsloshing phenomenon in tanks by the potential flow theoryAfter 120590-transformation the fluid domain is transformed to arectangle and a predictor-corrector scheme for free surfaceis used in the iteration of time then the governing equationand boundary conditions are approximately linearized andsolved in every time step The convergence of this methodis surveyed The numerical examples of standing waves and

12 Mathematical Problems in Engineering

035 040 045 050

006

008

010

012

014

016

LinearNonlinear

F120588gbd

hlowast

hb

Figure 15 Tendency of amplitude of sloshing force (high-frequencyexcitation)

horizontally forced sloshing models have been calculatedand good agreement between the numerical simulation andresults in other literatures has been obtained This numericalmethod is simple and easy in programming with excellentaccuracy and stability and can be extended to other sloshingproblems The time histories of free surface elevation andsloshing forces in the first and third resonance situation arealso presented and analyzed

The beating phenomenon of sloshing in different aspectratios under the same excitation has been numerically simu-lated by this method The results show that the effect of theaspect ratio on the sloshing force is complicated For the low-frequency excitation acting on the high aspect ratio modelsthe rise of aspect ratio makes the sloshing far from resonanceand increases the liquid mass so the sloshing force decreasesfirst and then increases For the high-frequency excitationacting on the low aspect ratio models the reduction of aspectratiomakes the sloshing far from resonance and decreases theliquid mass so the sloshing force decreases all the way Sincethe first natural frequency of liquid sloshing in not a linearfunction of aspect ratio the change of aspect ratio producesmuch stronger effect in low aspect ratio sloshing systemsMoreover nonlinearity of sloshing always decreases sloshingforce The effect of nonlinear factor depends on the violenceof sloshing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H N Abramson ldquoThe dynamic behavior of liquids in movingcontainers nasa sp-106rdquo NASA Special Publication vol 1061966

[2] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 no 4 pp 114ndash241 1974

[3] O M Faltinsen ldquoA numerical nonlinear method of sloshing intanks with two-dimensional flowrdquo Journal of Ship Research vol22 no 3 pp 193ndash202 1978

[4] T Nakayama and K Washizu ldquoNonlinear analysis of liquidmotion in a container subjected to forced pitching oscillationrdquoInternational Journal for Numerical Methods in Engineering vol15 no 8 pp 1207ndash1220 1980

[5] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[6] B-F Chen and R Nokes ldquoTime-independent finite differenceanalysis of fully non-linear and viscous fluid sloshing in arectangular tankrdquo Journal of Computational Physics vol 209 no1 pp 47ndash81 2005

[7] C Wu and B Chen ldquoSloshing waves and resonance modesof fluid in a 3D tank by a time-independent finite differencemethodrdquoOcean Engineering vol 36 no 6-7 pp 500ndash510 2009

[8] Y Kim ldquoNumerical simulation of sloshing flows with impactloadrdquo Applied Ocean Research vol 23 no 1 pp 53ndash62 2001

[9] M S Celebi and H Akyildiz ldquoNonlinear modeling of liquidsloshing in a moving rectangular tankrdquo Ocean Engineering vol29 no 12 pp 1527ndash1553 2002

[10] C Z Wang and B C Khoo ldquoFinite element analysis of two-dimensional nonlinear sloshing problems in random excita-tionsrdquo Ocean Engineering vol 32 no 2 pp 107ndash133 2005

[11] V Sriram S A Sannasiraj and V Sundar ldquoNumerical simula-tion of 2D sloshingwaves due to horizontal and vertical randomexcitationrdquo Applied Ocean Research vol 28 no 1 pp 19ndash322006

[12] O M Faltinsen O F Rognebakke I A Lukovsky and AN Timokha ldquoMultidimensional modal analysis of nonlinearsloshing in a rectangular tank with finite water depthrdquo Journalof Fluid Mechanics vol 407 pp 201ndash234 2000

[13] O M Faltinsen and A N Timokha ldquoAn adaptive multimodalapproach to nonlinear sloshing in a rectangular tankrdquo Journalof Fluid Mechanics vol 432 pp 167ndash200 2001

[14] G XWu ldquoSecond-order resonance of sloshing in a tankrdquoOceanEngineering vol 34 no 17-18 pp 2345ndash2349 2007

[15] D F Hill ldquoTransient and steady-state amplitudes of forcedwaves in rectangular basinsrdquo Physics of Fluids vol 15 no 6 pp1576ndash1587 2003

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004

[17] RD Firouz-AbadiMGhasemi andHHaddadpour ldquoAmodalapproach to second-order analysis of sloshing using boundaryelement methodrdquo Ocean Engineering vol 38 no 1 pp 11ndash212011

[18] M J Chern A G L Borthwick and R E Taylor ldquoA pseu-dospectral 120590-transformation model of 2-D nonlinear wavesrdquoJournal of Fluids and Structures vol 13 no 5 pp 607ndash630 1999

[19] J B Frandsen and A G L Borthwick ldquoSimulation of sloshingmotions in fixed and vertically excited containers using a 2-Dinviscid120590-transformed finite difference solverrdquo Journal of Fluidsand Structures vol 18 no 2 pp 197ndash214 2003

[20] M S Turnbull A G L Borthwick and R Eatock TaylorldquoNumerical wave tank based on a 120590-transformed finite elementinviscid flow solverrdquo International Journal for Numerical Meth-ods in Fluids vol 42 no 6 pp 641ndash663 2003

[21] B Chen andH Chiang ldquoComplete 2D and fully nonlinear anal-ysis of ideal fluid in tanksrdquo Journal of Engineering Mechanicsvol 125 no 1 pp 70ndash78 1999

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

20 40 60 80 100

00

05

10

AnalyticalNumerical

1205961t

minus05

minus10

120585a

(a) Comparison of free surface elevation at the left wall

05 10 15 20x

00

05

10

minus05

minus10

120585a

(b) Wave profiles for a half-period

Figure 4 Standing wave (119899 = 3)

are different The time and the vibration displacement in thefigures are dealt with by dimensionless method

1205851015840=

120585

119886

1199051015840= 1205961119905 (27)

It is found that the results from these meshes and timestep sizes are in good agreement so in this example a meshdistribution of 40 times 40 and time step size of 0005 s ensure areasonable converged solution

Nonlinear sloshing equations of standing waves can besolved by perturbation techniques First nondimensionalvariables are introduced to the equations Then 120598 = 119886120596

2

119899119892

is chosen as the characteristic small parameter Approximatelinear equations are derived by small parameter expansionand approximate analytical solutions can be obtained Thesecond-order approximate analytical free surface elevationfor the 119899th sloshing mode is prescribed as (Frandsen [16])

120585 (119909 119905) = 119886 ( cos (120596119899119905) cos (119896

119899119909)

+ 120598 (119860119899+ 119861119899cos (2120596

119899119905)

+119862119899cos (120596

2119899119905)) cos (2119896

119899119909))

(28)

where

119896119899=

119899120587

119887

1205962119899= radic1198922119896

119899tanh (2119896

119899ℎ)

119860119899=

1205964

119899+ 11989221198962

119899

81205964

119899

119861119899=

31205964

119899minus 11989221198962

119899

81205964

119899

minus

3

2

1205964

119899minus 11989221198962

119899

1205962

119899(41205962

119899minus 1205962

2119899)

119862119899=

1205962

1198991205962

2119899minus 1205964

119899minus 311989221198962

119899

21205962

119899(41205962

119899minus 1205962

2119899)

(29)

Figure 3(a) shows the time history of the free surfaceelevation for 119899 = 1 at the leftwall by the twomethods numer-ical solution by the finite difference method proposed in thispaper and second-order approximate analytical solution byperturbation method Initially good agreement between thetwo results is achieved but the phase difference between themincreases with time This phenomenon has been noticed anddiscussed by many researchers (Chern et al [18] Frandsen[16]) The conclusion is that the second-order analyticalsolution is not very accurate for large amplitude sloshingFigure 3(b) displays numerical wave profiles for nearly halfa period from dimensionless time 120596

1119905 = 318 to 120596

1119905 =

346 Stationary point does not exist and nonlinearity isobvious Figures 3(c) and 3(d) provide the sloshing forceson the left wall and right wall respectively (the forces arealso nondimensionalized) It shows that slight double peaksappear in the time historyThis has also been observed byWuet al [5]

The standing wave of 119899 = 3 is also calculated The initialconditions are set to be

120585 (119909 0) = 006 cos(3120587119887

) 120593 (119909 119911 0) = 0 (30)

Figure 4(a) shows the comparison of numerical solutionsand approximate analytical solutions Figure 4(b) showsnumerical wave profiles for nearly half a period Phasedifference and nonlinearity are also observed

42 Sloshing Caused by Horizontal Excitations The tankis assumed to take the horizontal oscillation the externalexcitation acting on the tank is defined as harmonic motion

119883 (119905) = 119886ℎcos (120596

ℎ119905) (31)

119886ℎand 120596

ℎare the amplitude and angular frequency of the

excited motionThe fluid is assumed to be stationary at 119905 = 0Then in the moving coordinate system the initial conditionsare described as

120585 (119909 0) = 0 120593 (119909 119911 0) = minus1199091198831015840(119905) = 0 (32)

Mathematical Problems in Engineering 7

2

1

0

minus2

minus1

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(a) Example 1 120596ℎ = 071205961 120598ℎ = 0036

2

0

minus2

4

minus4

6

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(b) Example 2 120596ℎ = 131205961 120598ℎ = 0072

Figure 5 Comparison of free surface elevation at the left wall

20 40 60 80 100

0

20

40

60

minus20

120585a

1205961t

(a) The first resonance mode

20 40 60 80 100

0

10

20

minus10

1205963t

120585a h

(b) The third resonance mode

Figure 6 Free surface elevation at the left wall (resonance phenomenon)

20 40 60 80 100

000

005

010

015

minus015

minus010

minus005F120588gbd

h

1205961t

(a) The first resonance mode

20 40 60 80 100

000

001

002

minus002

minus001

1205963t

F120588gbd

h

(b) The third resonance mode

Figure 7 Resultant sloshing force (resonance phenomenon)

In order to verify the precision of this numerical methodfor the sloshing models in horizontally moving tanks twoexamples are calculated and compared with the solutions ofFrandsen [16] (1) 120596

ℎ= 07120596

1 120598ℎ= 0036 (2) 120596

ℎ= 13120596

1

120598ℎ= 0072 The grid size of the numerical method is 40 times 40

and the time step size is 0006 s Figure 5(a) shows that the twosimulations are in close agreement In the second examplethe nonlinear effect is strong so there is a little differencebetween the two solutions in Figure 5(b) Neverthelessnonlinear characteristics of sloshing are captured by the two

8 Mathematical Problems in Engineering

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 005

20 40 60 80 100 120

0

20

40

60

minus20

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 8 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 12)

20 40 60 80 100

0

2

4

6

minus2

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 003

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 9 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 18)

02 04 06 08 10 12 14

02

04

06

08

10

1205961120596

lowast 1

hb

Figure 10 The function graph of the first natural frequency againstaspect ratio

numerical methods such as the higher peaks and the lessdeep troughs

421 Resonance Phenomenon Two situations of resonanceare considered the first resonance mode and the thirdresonance mode The external excitations of the tank aredescribed as 119883

1(119905) = 0007 cos(120596

1119905) 1198832(119905) = 0007 cos(120596

3119905)

The amplitudes are the same but the excited frequencies

are different The time histories of free surface elevation atthe left wall and the resultant sloshing forces are calculatedFigure 6(a) shows that the sloshing of the first resonancemode is violate the amplitude of the vibration is increasingFigure 6(b) shows that the amplitude of the third resonancemode is much smaller than that of the first resonance modeIn Figure 7 it can be concluded that the resultant sloshingforce of the third resonance mode is trivial compared withthe force of the first resonance mode In fact the first naturalfrequency of sloshing is the most influential of all the naturalfrequencies

422 The Limitation of the Numerical Method The presentnumerical method is employed to simulate nonlinear slosh-ing in moving tanks If the nonlinearity is quite strong thismethodmay lose its effectiveness Two excitation frequenciesare considered in the simulations One is 120596

ℎ= 085120596

1 the

other is 120596ℎ= 095120596

1 which indicates near resonant sloshing

situation Different excitation amplitudes 119886ℎand aspect ratios

ℎ119887 are tested and the limitation of the convergence of thisnumerical method is found out

First the aspect ratio is set to be ℎ119887 = 12 just asbefore Excitation amplitude is increased in the numericalsimulation and the sloshing becomesmore andmore violentFor 120596

ℎ= 085120596

1 if the excitation amplitude is 119886

ℎ= 006m

Mathematical Problems in Engineering 9

20 40 60 80 100

0

5

10

15

20

minus5

minus10

120585a h

120596ht

(a) Aspect ratio ℎ119887 = 05

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) Aspect ratio ℎ119887 = 075

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(c) Aspect ratio ℎ119887 = 1

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 11 Free surface elevation at the left wall (low-frequency excitation)

the simulation is divergent after several times of iteration if119886ℎ= 005m the simulation lasts for a long time So it is

considered that the limitation of 119886ℎfor nonresonant sloshing

is 005m For 120596ℎ= 095120596

1 the excitation amplitude is much

smaller The maximum limit of 119886ℎis estimated to be 0012m

The time histories of free surface elevation at the left wall inthe limit cases are shown in Figure 8

Then the aspect ratio is changed by decreasing the stillwater depth ℎ For the aspect ratio ℎ119887 = 18 the numericalmethod still works with a mesh size of 80 times 20 If the aspectratio is less than 18 the simulation is divergent As a resultthe minimum limit of aspect ratio ℎ119887 is considered to be18 In this critical aspect ratio situation for 120596

ℎ= 095120596

1

the maximum limit of excitation amplitude 119886ℎis still 0012m

while for 120596ℎ= 085120596

1 the maximum limit of 119886

ℎis 003m

Figure 9 shows the time histories of free surface elevation inthe limit cases in this critical aspect ratio The phenomena ofbeating and resonance are very obscure it can be concludedthat the sloshing in low aspect ratio is quite different from theone in normal aspect ratio

423 Beating Phenomenon in Different Aspect Ratios Slosh-ing in a fixed aspect ratio has been studied in many articlesHowever little work has been done to analyze the response ofliquid sloshing in different aspect ratios by the same external

excitation When the external excitation is close to the firstnatural frequency of sloshing the beating phenomenon canbe observed The study here is intended to investigate theeffect of different aspect ratios on beating phenomenon andsloshing force

In the numerical examples calculated before the lengthand width of the tank are set to be 2m and 1m the still waterdepth is 1m The first natural frequency of sloshing is 120596

1=

376 rads The aspect ratio is ℎ119887 = 12 Now the length 119887and the width 119889 of the tank are fixed and the aspect ratio ischanged by changing the still water depth ℎThe former waterdepth and natural frequency are set to be the characteristicquantity ℎlowast = 1m 120596lowast

1= 376 rads

The relationship between aspect ratio and the first naturalfrequency must be considered From (25) it is easy to get thegraph of the natural frequency as a function of aspect ratioFigure 10 indicates that the slope of the function curve is steepwhen the aspect ratio is small if the aspect ratio is greater than06 the function curve is nearly horizontal

Under the same external excitation the sloshing forcesof beating are affected by three factors proximity of exci-tation frequency to the natural frequency liquid mass andnonlinearity of sloshing When the aspect ratio of the liquidchanges these three factors would change along with it Inorder to discuss the effect of aspect ratio two horizontaloscillations of the tank are studied First the tank is subjected

10 Mathematical Problems in Engineering

20 40 60 80 100

00

01

02

minus01

F120588gbd

hlowast

120596ht

(a) Aspect ratio ℎ119887 = 05

005

010

015

minus005

minus010

minus015

F120588gbd

hlowast

20 40 60 80 100120596ht

000

(b) Aspect ratio ℎ119887 = 075

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(c) Aspect ratio ℎ119887 = 1

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 12 Resultant sloshing forces (low-frequency excitation)

06 07 08 09 10 11 12

019

020

021

022

LinearNonlinear

F120588gbd

hlowast

hb

Figure 13 Tendency of amplitude of sloshing force (low-frequencyexcitation)

to a low-frequency excitation The movement of the tank isdescribed as 119886

ℎ= 003m 120596

ℎ= 34 rads Figure 10 suggests

that if the aspect ratio is 03 the corresponding naturalfrequency is 34 rads Our aim is to analyze the influence ofthis low-frequency excitation on the high aspect ratio slosh-ing system Figure 11 shows the time histories of free surface

elevation at the left wall in four different aspect ratios wherethe parameters are nondimensionalized

1199051015840= 120596ℎ119905 120585

1015840=

120585

119886ℎ

(33)

The amplitude of free surface elevation is quite large inFigure 11(a) which shows strong nonlinearity of sloshingwhile in Figures 11(b) 11(c) and 11(d) the amplitudes arenearly the same smaller than the amplitude in Figure 11(a)so the nonlinear effects are weaker in these three sloshingsituations

Figure 12 shows the time histories of resultant sloshingforce acting on the tank (the forces are also nondimensional-ized 1198651015840 = 119865120588119892119887119889ℎlowast) The amplitudes of the sloshing forceare extracted in Figure 12 In order to study the effect ofnonlinear factor linear sloshing models by Abramson [1] arecalculated under the same condition to make comparisonsThe relationship between the amplitude of sloshing force andaspect ratio is shown in Figure 13 It is obvious that bothlinear and nonlinear results have the same trend decreasingfirst and then increasing On the one hand the rise of theaspect ratio increases the natural frequency which makesthe sloshing far from resonance and reduces sloshing forcesOn the other hand the rise of the aspect ratio increases themass of the liquid thereby increasing the sloshing forces

Mathematical Problems in Engineering 11

0

10

20120585a h

20 40 60 80 100120596ht

minus10

(a) Aspect ratio ℎ119887 = 05

0

5

10

15

minus5

120585a h

20 40 60 80 100120596ht

minus10

(b) Aspect ratio ℎ119887 = 045

0

5

10

120585a h

20 40 60 80 100120596ht

minus5

minus10

(c) Aspect ratio ℎ119887 = 04

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 035

Figure 14 Free surface elevation at the left wall (high-frequency excitation)

For the sloshing in aspect ratio ℎ119887 = 05 the excitationfrequency is close to the natural frequency If aspect ratiois increased the factor of reducing sloshing force is moredominant However for the sloshing in high aspect ratiosℎ119887 gt 075 the rise of aspect ratio brings about slight changesin the natural frequency so the factor of increasing sloshingforce is more dominant Comparing nonlinear results withlinear ones in Figure 13 some conclusions are obtained Firstthe nonlinearity of sloshing always decreases sloshing forceSecond to a certain extent the difference between linear andnonlinear amplitude reflects the effect of nonlinear factoron sloshing force For the sloshing in aspect ratio ℎ119887 =

05 the difference is the largest while for the other sloshingsituations the differences are nearly the same This resultaccordswith the conclusion in Figure 11Third nonlinearity isan important factor in large amplitude sloshing which shouldnot be neglected

Then the tank is subjected to a high-frequency excitationThe movement of the tank is described as 119886

ℎ= 0015m

120596ℎ= 4 rads Our aim is to analyze the influence of this

high-frequency excitation on the low aspect ratio sloshingsystem Figure 14 shows the time histories of free surfaceelevation at the left wall in four different aspect ratios Asthe aspect ratio decreases the amplitude and period of freesurface elevation are diminishing For the sloshing in lowaspect ratios ℎ119887 lt 05 the reduction of aspect ratio brings

about significant changes in the natural frequency so thebeating phenomenon has changed greatly even if the aspectratio is slightly decreasedThe resultant sloshing forces actingon the tank are also calculated The relationship between theamplitude of sloshing force and aspect ratio is showed inFigure 15The reduction of the aspect ratio makes the naturalfrequency far from the excitation frequency meanwhile itdecreases the mass of the liquid in the tank So both of thefactors reduce sloshing force As the aspect ratio decreasesthe difference between linear and nonlinear amplitude ofsloshing force also decreases The reason is that nonlineareffects are becoming weaker and the sloshing force itself isalso decreasing

5 Conclusions

Anewfinite differencemethod based on the prediction of freesurface has been developed and used to simulate nonlinearsloshing phenomenon in tanks by the potential flow theoryAfter 120590-transformation the fluid domain is transformed to arectangle and a predictor-corrector scheme for free surfaceis used in the iteration of time then the governing equationand boundary conditions are approximately linearized andsolved in every time step The convergence of this methodis surveyed The numerical examples of standing waves and

12 Mathematical Problems in Engineering

035 040 045 050

006

008

010

012

014

016

LinearNonlinear

F120588gbd

hlowast

hb

Figure 15 Tendency of amplitude of sloshing force (high-frequencyexcitation)

horizontally forced sloshing models have been calculatedand good agreement between the numerical simulation andresults in other literatures has been obtained This numericalmethod is simple and easy in programming with excellentaccuracy and stability and can be extended to other sloshingproblems The time histories of free surface elevation andsloshing forces in the first and third resonance situation arealso presented and analyzed

The beating phenomenon of sloshing in different aspectratios under the same excitation has been numerically simu-lated by this method The results show that the effect of theaspect ratio on the sloshing force is complicated For the low-frequency excitation acting on the high aspect ratio modelsthe rise of aspect ratio makes the sloshing far from resonanceand increases the liquid mass so the sloshing force decreasesfirst and then increases For the high-frequency excitationacting on the low aspect ratio models the reduction of aspectratiomakes the sloshing far from resonance and decreases theliquid mass so the sloshing force decreases all the way Sincethe first natural frequency of liquid sloshing in not a linearfunction of aspect ratio the change of aspect ratio producesmuch stronger effect in low aspect ratio sloshing systemsMoreover nonlinearity of sloshing always decreases sloshingforce The effect of nonlinear factor depends on the violenceof sloshing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H N Abramson ldquoThe dynamic behavior of liquids in movingcontainers nasa sp-106rdquo NASA Special Publication vol 1061966

[2] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 no 4 pp 114ndash241 1974

[3] O M Faltinsen ldquoA numerical nonlinear method of sloshing intanks with two-dimensional flowrdquo Journal of Ship Research vol22 no 3 pp 193ndash202 1978

[4] T Nakayama and K Washizu ldquoNonlinear analysis of liquidmotion in a container subjected to forced pitching oscillationrdquoInternational Journal for Numerical Methods in Engineering vol15 no 8 pp 1207ndash1220 1980

[5] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[6] B-F Chen and R Nokes ldquoTime-independent finite differenceanalysis of fully non-linear and viscous fluid sloshing in arectangular tankrdquo Journal of Computational Physics vol 209 no1 pp 47ndash81 2005

[7] C Wu and B Chen ldquoSloshing waves and resonance modesof fluid in a 3D tank by a time-independent finite differencemethodrdquoOcean Engineering vol 36 no 6-7 pp 500ndash510 2009

[8] Y Kim ldquoNumerical simulation of sloshing flows with impactloadrdquo Applied Ocean Research vol 23 no 1 pp 53ndash62 2001

[9] M S Celebi and H Akyildiz ldquoNonlinear modeling of liquidsloshing in a moving rectangular tankrdquo Ocean Engineering vol29 no 12 pp 1527ndash1553 2002

[10] C Z Wang and B C Khoo ldquoFinite element analysis of two-dimensional nonlinear sloshing problems in random excita-tionsrdquo Ocean Engineering vol 32 no 2 pp 107ndash133 2005

[11] V Sriram S A Sannasiraj and V Sundar ldquoNumerical simula-tion of 2D sloshingwaves due to horizontal and vertical randomexcitationrdquo Applied Ocean Research vol 28 no 1 pp 19ndash322006

[12] O M Faltinsen O F Rognebakke I A Lukovsky and AN Timokha ldquoMultidimensional modal analysis of nonlinearsloshing in a rectangular tank with finite water depthrdquo Journalof Fluid Mechanics vol 407 pp 201ndash234 2000

[13] O M Faltinsen and A N Timokha ldquoAn adaptive multimodalapproach to nonlinear sloshing in a rectangular tankrdquo Journalof Fluid Mechanics vol 432 pp 167ndash200 2001

[14] G XWu ldquoSecond-order resonance of sloshing in a tankrdquoOceanEngineering vol 34 no 17-18 pp 2345ndash2349 2007

[15] D F Hill ldquoTransient and steady-state amplitudes of forcedwaves in rectangular basinsrdquo Physics of Fluids vol 15 no 6 pp1576ndash1587 2003

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004

[17] RD Firouz-AbadiMGhasemi andHHaddadpour ldquoAmodalapproach to second-order analysis of sloshing using boundaryelement methodrdquo Ocean Engineering vol 38 no 1 pp 11ndash212011

[18] M J Chern A G L Borthwick and R E Taylor ldquoA pseu-dospectral 120590-transformation model of 2-D nonlinear wavesrdquoJournal of Fluids and Structures vol 13 no 5 pp 607ndash630 1999

[19] J B Frandsen and A G L Borthwick ldquoSimulation of sloshingmotions in fixed and vertically excited containers using a 2-Dinviscid120590-transformed finite difference solverrdquo Journal of Fluidsand Structures vol 18 no 2 pp 197ndash214 2003

[20] M S Turnbull A G L Borthwick and R Eatock TaylorldquoNumerical wave tank based on a 120590-transformed finite elementinviscid flow solverrdquo International Journal for Numerical Meth-ods in Fluids vol 42 no 6 pp 641ndash663 2003

[21] B Chen andH Chiang ldquoComplete 2D and fully nonlinear anal-ysis of ideal fluid in tanksrdquo Journal of Engineering Mechanicsvol 125 no 1 pp 70ndash78 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

2

1

0

minus2

minus1

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(a) Example 1 120596ℎ = 071205961 120598ℎ = 0036

2

0

minus2

4

minus4

6

20 40 60 80 1001205961t

120585a h

PresentBy Frandsen

(b) Example 2 120596ℎ = 131205961 120598ℎ = 0072

Figure 5 Comparison of free surface elevation at the left wall

20 40 60 80 100

0

20

40

60

minus20

120585a

1205961t

(a) The first resonance mode

20 40 60 80 100

0

10

20

minus10

1205963t

120585a h

(b) The third resonance mode

Figure 6 Free surface elevation at the left wall (resonance phenomenon)

20 40 60 80 100

000

005

010

015

minus015

minus010

minus005F120588gbd

h

1205961t

(a) The first resonance mode

20 40 60 80 100

000

001

002

minus002

minus001

1205963t

F120588gbd

h

(b) The third resonance mode

Figure 7 Resultant sloshing force (resonance phenomenon)

In order to verify the precision of this numerical methodfor the sloshing models in horizontally moving tanks twoexamples are calculated and compared with the solutions ofFrandsen [16] (1) 120596

ℎ= 07120596

1 120598ℎ= 0036 (2) 120596

ℎ= 13120596

1

120598ℎ= 0072 The grid size of the numerical method is 40 times 40

and the time step size is 0006 s Figure 5(a) shows that the twosimulations are in close agreement In the second examplethe nonlinear effect is strong so there is a little differencebetween the two solutions in Figure 5(b) Neverthelessnonlinear characteristics of sloshing are captured by the two

8 Mathematical Problems in Engineering

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 005

20 40 60 80 100 120

0

20

40

60

minus20

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 8 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 12)

20 40 60 80 100

0

2

4

6

minus2

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 003

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 9 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 18)

02 04 06 08 10 12 14

02

04

06

08

10

1205961120596

lowast 1

hb

Figure 10 The function graph of the first natural frequency againstaspect ratio

numerical methods such as the higher peaks and the lessdeep troughs

421 Resonance Phenomenon Two situations of resonanceare considered the first resonance mode and the thirdresonance mode The external excitations of the tank aredescribed as 119883

1(119905) = 0007 cos(120596

1119905) 1198832(119905) = 0007 cos(120596

3119905)

The amplitudes are the same but the excited frequencies

are different The time histories of free surface elevation atthe left wall and the resultant sloshing forces are calculatedFigure 6(a) shows that the sloshing of the first resonancemode is violate the amplitude of the vibration is increasingFigure 6(b) shows that the amplitude of the third resonancemode is much smaller than that of the first resonance modeIn Figure 7 it can be concluded that the resultant sloshingforce of the third resonance mode is trivial compared withthe force of the first resonance mode In fact the first naturalfrequency of sloshing is the most influential of all the naturalfrequencies

422 The Limitation of the Numerical Method The presentnumerical method is employed to simulate nonlinear slosh-ing in moving tanks If the nonlinearity is quite strong thismethodmay lose its effectiveness Two excitation frequenciesare considered in the simulations One is 120596

ℎ= 085120596

1 the

other is 120596ℎ= 095120596

1 which indicates near resonant sloshing

situation Different excitation amplitudes 119886ℎand aspect ratios

ℎ119887 are tested and the limitation of the convergence of thisnumerical method is found out

First the aspect ratio is set to be ℎ119887 = 12 just asbefore Excitation amplitude is increased in the numericalsimulation and the sloshing becomesmore andmore violentFor 120596

ℎ= 085120596

1 if the excitation amplitude is 119886

ℎ= 006m

Mathematical Problems in Engineering 9

20 40 60 80 100

0

5

10

15

20

minus5

minus10

120585a h

120596ht

(a) Aspect ratio ℎ119887 = 05

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) Aspect ratio ℎ119887 = 075

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(c) Aspect ratio ℎ119887 = 1

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 11 Free surface elevation at the left wall (low-frequency excitation)

the simulation is divergent after several times of iteration if119886ℎ= 005m the simulation lasts for a long time So it is

considered that the limitation of 119886ℎfor nonresonant sloshing

is 005m For 120596ℎ= 095120596

1 the excitation amplitude is much

smaller The maximum limit of 119886ℎis estimated to be 0012m

The time histories of free surface elevation at the left wall inthe limit cases are shown in Figure 8

Then the aspect ratio is changed by decreasing the stillwater depth ℎ For the aspect ratio ℎ119887 = 18 the numericalmethod still works with a mesh size of 80 times 20 If the aspectratio is less than 18 the simulation is divergent As a resultthe minimum limit of aspect ratio ℎ119887 is considered to be18 In this critical aspect ratio situation for 120596

ℎ= 095120596

1

the maximum limit of excitation amplitude 119886ℎis still 0012m

while for 120596ℎ= 085120596

1 the maximum limit of 119886

ℎis 003m

Figure 9 shows the time histories of free surface elevation inthe limit cases in this critical aspect ratio The phenomena ofbeating and resonance are very obscure it can be concludedthat the sloshing in low aspect ratio is quite different from theone in normal aspect ratio

423 Beating Phenomenon in Different Aspect Ratios Slosh-ing in a fixed aspect ratio has been studied in many articlesHowever little work has been done to analyze the response ofliquid sloshing in different aspect ratios by the same external

excitation When the external excitation is close to the firstnatural frequency of sloshing the beating phenomenon canbe observed The study here is intended to investigate theeffect of different aspect ratios on beating phenomenon andsloshing force

In the numerical examples calculated before the lengthand width of the tank are set to be 2m and 1m the still waterdepth is 1m The first natural frequency of sloshing is 120596

1=

376 rads The aspect ratio is ℎ119887 = 12 Now the length 119887and the width 119889 of the tank are fixed and the aspect ratio ischanged by changing the still water depth ℎThe former waterdepth and natural frequency are set to be the characteristicquantity ℎlowast = 1m 120596lowast

1= 376 rads

The relationship between aspect ratio and the first naturalfrequency must be considered From (25) it is easy to get thegraph of the natural frequency as a function of aspect ratioFigure 10 indicates that the slope of the function curve is steepwhen the aspect ratio is small if the aspect ratio is greater than06 the function curve is nearly horizontal

Under the same external excitation the sloshing forcesof beating are affected by three factors proximity of exci-tation frequency to the natural frequency liquid mass andnonlinearity of sloshing When the aspect ratio of the liquidchanges these three factors would change along with it Inorder to discuss the effect of aspect ratio two horizontaloscillations of the tank are studied First the tank is subjected

10 Mathematical Problems in Engineering

20 40 60 80 100

00

01

02

minus01

F120588gbd

hlowast

120596ht

(a) Aspect ratio ℎ119887 = 05

005

010

015

minus005

minus010

minus015

F120588gbd

hlowast

20 40 60 80 100120596ht

000

(b) Aspect ratio ℎ119887 = 075

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(c) Aspect ratio ℎ119887 = 1

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 12 Resultant sloshing forces (low-frequency excitation)

06 07 08 09 10 11 12

019

020

021

022

LinearNonlinear

F120588gbd

hlowast

hb

Figure 13 Tendency of amplitude of sloshing force (low-frequencyexcitation)

to a low-frequency excitation The movement of the tank isdescribed as 119886

ℎ= 003m 120596

ℎ= 34 rads Figure 10 suggests

that if the aspect ratio is 03 the corresponding naturalfrequency is 34 rads Our aim is to analyze the influence ofthis low-frequency excitation on the high aspect ratio slosh-ing system Figure 11 shows the time histories of free surface

elevation at the left wall in four different aspect ratios wherethe parameters are nondimensionalized

1199051015840= 120596ℎ119905 120585

1015840=

120585

119886ℎ

(33)

The amplitude of free surface elevation is quite large inFigure 11(a) which shows strong nonlinearity of sloshingwhile in Figures 11(b) 11(c) and 11(d) the amplitudes arenearly the same smaller than the amplitude in Figure 11(a)so the nonlinear effects are weaker in these three sloshingsituations

Figure 12 shows the time histories of resultant sloshingforce acting on the tank (the forces are also nondimensional-ized 1198651015840 = 119865120588119892119887119889ℎlowast) The amplitudes of the sloshing forceare extracted in Figure 12 In order to study the effect ofnonlinear factor linear sloshing models by Abramson [1] arecalculated under the same condition to make comparisonsThe relationship between the amplitude of sloshing force andaspect ratio is shown in Figure 13 It is obvious that bothlinear and nonlinear results have the same trend decreasingfirst and then increasing On the one hand the rise of theaspect ratio increases the natural frequency which makesthe sloshing far from resonance and reduces sloshing forcesOn the other hand the rise of the aspect ratio increases themass of the liquid thereby increasing the sloshing forces

Mathematical Problems in Engineering 11

0

10

20120585a h

20 40 60 80 100120596ht

minus10

(a) Aspect ratio ℎ119887 = 05

0

5

10

15

minus5

120585a h

20 40 60 80 100120596ht

minus10

(b) Aspect ratio ℎ119887 = 045

0

5

10

120585a h

20 40 60 80 100120596ht

minus5

minus10

(c) Aspect ratio ℎ119887 = 04

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 035

Figure 14 Free surface elevation at the left wall (high-frequency excitation)

For the sloshing in aspect ratio ℎ119887 = 05 the excitationfrequency is close to the natural frequency If aspect ratiois increased the factor of reducing sloshing force is moredominant However for the sloshing in high aspect ratiosℎ119887 gt 075 the rise of aspect ratio brings about slight changesin the natural frequency so the factor of increasing sloshingforce is more dominant Comparing nonlinear results withlinear ones in Figure 13 some conclusions are obtained Firstthe nonlinearity of sloshing always decreases sloshing forceSecond to a certain extent the difference between linear andnonlinear amplitude reflects the effect of nonlinear factoron sloshing force For the sloshing in aspect ratio ℎ119887 =

05 the difference is the largest while for the other sloshingsituations the differences are nearly the same This resultaccordswith the conclusion in Figure 11Third nonlinearity isan important factor in large amplitude sloshing which shouldnot be neglected

Then the tank is subjected to a high-frequency excitationThe movement of the tank is described as 119886

ℎ= 0015m

120596ℎ= 4 rads Our aim is to analyze the influence of this

high-frequency excitation on the low aspect ratio sloshingsystem Figure 14 shows the time histories of free surfaceelevation at the left wall in four different aspect ratios Asthe aspect ratio decreases the amplitude and period of freesurface elevation are diminishing For the sloshing in lowaspect ratios ℎ119887 lt 05 the reduction of aspect ratio brings

about significant changes in the natural frequency so thebeating phenomenon has changed greatly even if the aspectratio is slightly decreasedThe resultant sloshing forces actingon the tank are also calculated The relationship between theamplitude of sloshing force and aspect ratio is showed inFigure 15The reduction of the aspect ratio makes the naturalfrequency far from the excitation frequency meanwhile itdecreases the mass of the liquid in the tank So both of thefactors reduce sloshing force As the aspect ratio decreasesthe difference between linear and nonlinear amplitude ofsloshing force also decreases The reason is that nonlineareffects are becoming weaker and the sloshing force itself isalso decreasing

5 Conclusions

Anewfinite differencemethod based on the prediction of freesurface has been developed and used to simulate nonlinearsloshing phenomenon in tanks by the potential flow theoryAfter 120590-transformation the fluid domain is transformed to arectangle and a predictor-corrector scheme for free surfaceis used in the iteration of time then the governing equationand boundary conditions are approximately linearized andsolved in every time step The convergence of this methodis surveyed The numerical examples of standing waves and

12 Mathematical Problems in Engineering

035 040 045 050

006

008

010

012

014

016

LinearNonlinear

F120588gbd

hlowast

hb

Figure 15 Tendency of amplitude of sloshing force (high-frequencyexcitation)

horizontally forced sloshing models have been calculatedand good agreement between the numerical simulation andresults in other literatures has been obtained This numericalmethod is simple and easy in programming with excellentaccuracy and stability and can be extended to other sloshingproblems The time histories of free surface elevation andsloshing forces in the first and third resonance situation arealso presented and analyzed

The beating phenomenon of sloshing in different aspectratios under the same excitation has been numerically simu-lated by this method The results show that the effect of theaspect ratio on the sloshing force is complicated For the low-frequency excitation acting on the high aspect ratio modelsthe rise of aspect ratio makes the sloshing far from resonanceand increases the liquid mass so the sloshing force decreasesfirst and then increases For the high-frequency excitationacting on the low aspect ratio models the reduction of aspectratiomakes the sloshing far from resonance and decreases theliquid mass so the sloshing force decreases all the way Sincethe first natural frequency of liquid sloshing in not a linearfunction of aspect ratio the change of aspect ratio producesmuch stronger effect in low aspect ratio sloshing systemsMoreover nonlinearity of sloshing always decreases sloshingforce The effect of nonlinear factor depends on the violenceof sloshing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H N Abramson ldquoThe dynamic behavior of liquids in movingcontainers nasa sp-106rdquo NASA Special Publication vol 1061966

[2] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 no 4 pp 114ndash241 1974

[3] O M Faltinsen ldquoA numerical nonlinear method of sloshing intanks with two-dimensional flowrdquo Journal of Ship Research vol22 no 3 pp 193ndash202 1978

[4] T Nakayama and K Washizu ldquoNonlinear analysis of liquidmotion in a container subjected to forced pitching oscillationrdquoInternational Journal for Numerical Methods in Engineering vol15 no 8 pp 1207ndash1220 1980

[5] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[6] B-F Chen and R Nokes ldquoTime-independent finite differenceanalysis of fully non-linear and viscous fluid sloshing in arectangular tankrdquo Journal of Computational Physics vol 209 no1 pp 47ndash81 2005

[7] C Wu and B Chen ldquoSloshing waves and resonance modesof fluid in a 3D tank by a time-independent finite differencemethodrdquoOcean Engineering vol 36 no 6-7 pp 500ndash510 2009

[8] Y Kim ldquoNumerical simulation of sloshing flows with impactloadrdquo Applied Ocean Research vol 23 no 1 pp 53ndash62 2001

[9] M S Celebi and H Akyildiz ldquoNonlinear modeling of liquidsloshing in a moving rectangular tankrdquo Ocean Engineering vol29 no 12 pp 1527ndash1553 2002

[10] C Z Wang and B C Khoo ldquoFinite element analysis of two-dimensional nonlinear sloshing problems in random excita-tionsrdquo Ocean Engineering vol 32 no 2 pp 107ndash133 2005

[11] V Sriram S A Sannasiraj and V Sundar ldquoNumerical simula-tion of 2D sloshingwaves due to horizontal and vertical randomexcitationrdquo Applied Ocean Research vol 28 no 1 pp 19ndash322006

[12] O M Faltinsen O F Rognebakke I A Lukovsky and AN Timokha ldquoMultidimensional modal analysis of nonlinearsloshing in a rectangular tank with finite water depthrdquo Journalof Fluid Mechanics vol 407 pp 201ndash234 2000

[13] O M Faltinsen and A N Timokha ldquoAn adaptive multimodalapproach to nonlinear sloshing in a rectangular tankrdquo Journalof Fluid Mechanics vol 432 pp 167ndash200 2001

[14] G XWu ldquoSecond-order resonance of sloshing in a tankrdquoOceanEngineering vol 34 no 17-18 pp 2345ndash2349 2007

[15] D F Hill ldquoTransient and steady-state amplitudes of forcedwaves in rectangular basinsrdquo Physics of Fluids vol 15 no 6 pp1576ndash1587 2003

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004

[17] RD Firouz-AbadiMGhasemi andHHaddadpour ldquoAmodalapproach to second-order analysis of sloshing using boundaryelement methodrdquo Ocean Engineering vol 38 no 1 pp 11ndash212011

[18] M J Chern A G L Borthwick and R E Taylor ldquoA pseu-dospectral 120590-transformation model of 2-D nonlinear wavesrdquoJournal of Fluids and Structures vol 13 no 5 pp 607ndash630 1999

[19] J B Frandsen and A G L Borthwick ldquoSimulation of sloshingmotions in fixed and vertically excited containers using a 2-Dinviscid120590-transformed finite difference solverrdquo Journal of Fluidsand Structures vol 18 no 2 pp 197ndash214 2003

[20] M S Turnbull A G L Borthwick and R Eatock TaylorldquoNumerical wave tank based on a 120590-transformed finite elementinviscid flow solverrdquo International Journal for Numerical Meth-ods in Fluids vol 42 no 6 pp 641ndash663 2003

[21] B Chen andH Chiang ldquoComplete 2D and fully nonlinear anal-ysis of ideal fluid in tanksrdquo Journal of Engineering Mechanicsvol 125 no 1 pp 70ndash78 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 005

20 40 60 80 100 120

0

20

40

60

minus20

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 8 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 12)

20 40 60 80 100

0

2

4

6

minus2

120585a h

120596ht

(a) 120596ℎ = 0851205961 119886ℎ = 003

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) 120596ℎ = 0951205961 119886ℎ = 0012

Figure 9 Free surface elevation at the left wall of the maximum 119886ℎ(ℎ119887 = 18)

02 04 06 08 10 12 14

02

04

06

08

10

1205961120596

lowast 1

hb

Figure 10 The function graph of the first natural frequency againstaspect ratio

numerical methods such as the higher peaks and the lessdeep troughs

421 Resonance Phenomenon Two situations of resonanceare considered the first resonance mode and the thirdresonance mode The external excitations of the tank aredescribed as 119883

1(119905) = 0007 cos(120596

1119905) 1198832(119905) = 0007 cos(120596

3119905)

The amplitudes are the same but the excited frequencies

are different The time histories of free surface elevation atthe left wall and the resultant sloshing forces are calculatedFigure 6(a) shows that the sloshing of the first resonancemode is violate the amplitude of the vibration is increasingFigure 6(b) shows that the amplitude of the third resonancemode is much smaller than that of the first resonance modeIn Figure 7 it can be concluded that the resultant sloshingforce of the third resonance mode is trivial compared withthe force of the first resonance mode In fact the first naturalfrequency of sloshing is the most influential of all the naturalfrequencies

422 The Limitation of the Numerical Method The presentnumerical method is employed to simulate nonlinear slosh-ing in moving tanks If the nonlinearity is quite strong thismethodmay lose its effectiveness Two excitation frequenciesare considered in the simulations One is 120596

ℎ= 085120596

1 the

other is 120596ℎ= 095120596

1 which indicates near resonant sloshing

situation Different excitation amplitudes 119886ℎand aspect ratios

ℎ119887 are tested and the limitation of the convergence of thisnumerical method is found out

First the aspect ratio is set to be ℎ119887 = 12 just asbefore Excitation amplitude is increased in the numericalsimulation and the sloshing becomesmore andmore violentFor 120596

ℎ= 085120596

1 if the excitation amplitude is 119886

ℎ= 006m

Mathematical Problems in Engineering 9

20 40 60 80 100

0

5

10

15

20

minus5

minus10

120585a h

120596ht

(a) Aspect ratio ℎ119887 = 05

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) Aspect ratio ℎ119887 = 075

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(c) Aspect ratio ℎ119887 = 1

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 11 Free surface elevation at the left wall (low-frequency excitation)

the simulation is divergent after several times of iteration if119886ℎ= 005m the simulation lasts for a long time So it is

considered that the limitation of 119886ℎfor nonresonant sloshing

is 005m For 120596ℎ= 095120596

1 the excitation amplitude is much

smaller The maximum limit of 119886ℎis estimated to be 0012m

The time histories of free surface elevation at the left wall inthe limit cases are shown in Figure 8

Then the aspect ratio is changed by decreasing the stillwater depth ℎ For the aspect ratio ℎ119887 = 18 the numericalmethod still works with a mesh size of 80 times 20 If the aspectratio is less than 18 the simulation is divergent As a resultthe minimum limit of aspect ratio ℎ119887 is considered to be18 In this critical aspect ratio situation for 120596

ℎ= 095120596

1

the maximum limit of excitation amplitude 119886ℎis still 0012m

while for 120596ℎ= 085120596

1 the maximum limit of 119886

ℎis 003m

Figure 9 shows the time histories of free surface elevation inthe limit cases in this critical aspect ratio The phenomena ofbeating and resonance are very obscure it can be concludedthat the sloshing in low aspect ratio is quite different from theone in normal aspect ratio

423 Beating Phenomenon in Different Aspect Ratios Slosh-ing in a fixed aspect ratio has been studied in many articlesHowever little work has been done to analyze the response ofliquid sloshing in different aspect ratios by the same external

excitation When the external excitation is close to the firstnatural frequency of sloshing the beating phenomenon canbe observed The study here is intended to investigate theeffect of different aspect ratios on beating phenomenon andsloshing force

In the numerical examples calculated before the lengthand width of the tank are set to be 2m and 1m the still waterdepth is 1m The first natural frequency of sloshing is 120596

1=

376 rads The aspect ratio is ℎ119887 = 12 Now the length 119887and the width 119889 of the tank are fixed and the aspect ratio ischanged by changing the still water depth ℎThe former waterdepth and natural frequency are set to be the characteristicquantity ℎlowast = 1m 120596lowast

1= 376 rads

The relationship between aspect ratio and the first naturalfrequency must be considered From (25) it is easy to get thegraph of the natural frequency as a function of aspect ratioFigure 10 indicates that the slope of the function curve is steepwhen the aspect ratio is small if the aspect ratio is greater than06 the function curve is nearly horizontal

Under the same external excitation the sloshing forcesof beating are affected by three factors proximity of exci-tation frequency to the natural frequency liquid mass andnonlinearity of sloshing When the aspect ratio of the liquidchanges these three factors would change along with it Inorder to discuss the effect of aspect ratio two horizontaloscillations of the tank are studied First the tank is subjected

10 Mathematical Problems in Engineering

20 40 60 80 100

00

01

02

minus01

F120588gbd

hlowast

120596ht

(a) Aspect ratio ℎ119887 = 05

005

010

015

minus005

minus010

minus015

F120588gbd

hlowast

20 40 60 80 100120596ht

000

(b) Aspect ratio ℎ119887 = 075

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(c) Aspect ratio ℎ119887 = 1

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 12 Resultant sloshing forces (low-frequency excitation)

06 07 08 09 10 11 12

019

020

021

022

LinearNonlinear

F120588gbd

hlowast

hb

Figure 13 Tendency of amplitude of sloshing force (low-frequencyexcitation)

to a low-frequency excitation The movement of the tank isdescribed as 119886

ℎ= 003m 120596

ℎ= 34 rads Figure 10 suggests

that if the aspect ratio is 03 the corresponding naturalfrequency is 34 rads Our aim is to analyze the influence ofthis low-frequency excitation on the high aspect ratio slosh-ing system Figure 11 shows the time histories of free surface

elevation at the left wall in four different aspect ratios wherethe parameters are nondimensionalized

1199051015840= 120596ℎ119905 120585

1015840=

120585

119886ℎ

(33)

The amplitude of free surface elevation is quite large inFigure 11(a) which shows strong nonlinearity of sloshingwhile in Figures 11(b) 11(c) and 11(d) the amplitudes arenearly the same smaller than the amplitude in Figure 11(a)so the nonlinear effects are weaker in these three sloshingsituations

Figure 12 shows the time histories of resultant sloshingforce acting on the tank (the forces are also nondimensional-ized 1198651015840 = 119865120588119892119887119889ℎlowast) The amplitudes of the sloshing forceare extracted in Figure 12 In order to study the effect ofnonlinear factor linear sloshing models by Abramson [1] arecalculated under the same condition to make comparisonsThe relationship between the amplitude of sloshing force andaspect ratio is shown in Figure 13 It is obvious that bothlinear and nonlinear results have the same trend decreasingfirst and then increasing On the one hand the rise of theaspect ratio increases the natural frequency which makesthe sloshing far from resonance and reduces sloshing forcesOn the other hand the rise of the aspect ratio increases themass of the liquid thereby increasing the sloshing forces

Mathematical Problems in Engineering 11

0

10

20120585a h

20 40 60 80 100120596ht

minus10

(a) Aspect ratio ℎ119887 = 05

0

5

10

15

minus5

120585a h

20 40 60 80 100120596ht

minus10

(b) Aspect ratio ℎ119887 = 045

0

5

10

120585a h

20 40 60 80 100120596ht

minus5

minus10

(c) Aspect ratio ℎ119887 = 04

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 035

Figure 14 Free surface elevation at the left wall (high-frequency excitation)

For the sloshing in aspect ratio ℎ119887 = 05 the excitationfrequency is close to the natural frequency If aspect ratiois increased the factor of reducing sloshing force is moredominant However for the sloshing in high aspect ratiosℎ119887 gt 075 the rise of aspect ratio brings about slight changesin the natural frequency so the factor of increasing sloshingforce is more dominant Comparing nonlinear results withlinear ones in Figure 13 some conclusions are obtained Firstthe nonlinearity of sloshing always decreases sloshing forceSecond to a certain extent the difference between linear andnonlinear amplitude reflects the effect of nonlinear factoron sloshing force For the sloshing in aspect ratio ℎ119887 =

05 the difference is the largest while for the other sloshingsituations the differences are nearly the same This resultaccordswith the conclusion in Figure 11Third nonlinearity isan important factor in large amplitude sloshing which shouldnot be neglected

Then the tank is subjected to a high-frequency excitationThe movement of the tank is described as 119886

ℎ= 0015m

120596ℎ= 4 rads Our aim is to analyze the influence of this

high-frequency excitation on the low aspect ratio sloshingsystem Figure 14 shows the time histories of free surfaceelevation at the left wall in four different aspect ratios Asthe aspect ratio decreases the amplitude and period of freesurface elevation are diminishing For the sloshing in lowaspect ratios ℎ119887 lt 05 the reduction of aspect ratio brings

about significant changes in the natural frequency so thebeating phenomenon has changed greatly even if the aspectratio is slightly decreasedThe resultant sloshing forces actingon the tank are also calculated The relationship between theamplitude of sloshing force and aspect ratio is showed inFigure 15The reduction of the aspect ratio makes the naturalfrequency far from the excitation frequency meanwhile itdecreases the mass of the liquid in the tank So both of thefactors reduce sloshing force As the aspect ratio decreasesthe difference between linear and nonlinear amplitude ofsloshing force also decreases The reason is that nonlineareffects are becoming weaker and the sloshing force itself isalso decreasing

5 Conclusions

Anewfinite differencemethod based on the prediction of freesurface has been developed and used to simulate nonlinearsloshing phenomenon in tanks by the potential flow theoryAfter 120590-transformation the fluid domain is transformed to arectangle and a predictor-corrector scheme for free surfaceis used in the iteration of time then the governing equationand boundary conditions are approximately linearized andsolved in every time step The convergence of this methodis surveyed The numerical examples of standing waves and

12 Mathematical Problems in Engineering

035 040 045 050

006

008

010

012

014

016

LinearNonlinear

F120588gbd

hlowast

hb

Figure 15 Tendency of amplitude of sloshing force (high-frequencyexcitation)

horizontally forced sloshing models have been calculatedand good agreement between the numerical simulation andresults in other literatures has been obtained This numericalmethod is simple and easy in programming with excellentaccuracy and stability and can be extended to other sloshingproblems The time histories of free surface elevation andsloshing forces in the first and third resonance situation arealso presented and analyzed

The beating phenomenon of sloshing in different aspectratios under the same excitation has been numerically simu-lated by this method The results show that the effect of theaspect ratio on the sloshing force is complicated For the low-frequency excitation acting on the high aspect ratio modelsthe rise of aspect ratio makes the sloshing far from resonanceand increases the liquid mass so the sloshing force decreasesfirst and then increases For the high-frequency excitationacting on the low aspect ratio models the reduction of aspectratiomakes the sloshing far from resonance and decreases theliquid mass so the sloshing force decreases all the way Sincethe first natural frequency of liquid sloshing in not a linearfunction of aspect ratio the change of aspect ratio producesmuch stronger effect in low aspect ratio sloshing systemsMoreover nonlinearity of sloshing always decreases sloshingforce The effect of nonlinear factor depends on the violenceof sloshing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H N Abramson ldquoThe dynamic behavior of liquids in movingcontainers nasa sp-106rdquo NASA Special Publication vol 1061966

[2] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 no 4 pp 114ndash241 1974

[3] O M Faltinsen ldquoA numerical nonlinear method of sloshing intanks with two-dimensional flowrdquo Journal of Ship Research vol22 no 3 pp 193ndash202 1978

[4] T Nakayama and K Washizu ldquoNonlinear analysis of liquidmotion in a container subjected to forced pitching oscillationrdquoInternational Journal for Numerical Methods in Engineering vol15 no 8 pp 1207ndash1220 1980

[5] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[6] B-F Chen and R Nokes ldquoTime-independent finite differenceanalysis of fully non-linear and viscous fluid sloshing in arectangular tankrdquo Journal of Computational Physics vol 209 no1 pp 47ndash81 2005

[7] C Wu and B Chen ldquoSloshing waves and resonance modesof fluid in a 3D tank by a time-independent finite differencemethodrdquoOcean Engineering vol 36 no 6-7 pp 500ndash510 2009

[8] Y Kim ldquoNumerical simulation of sloshing flows with impactloadrdquo Applied Ocean Research vol 23 no 1 pp 53ndash62 2001

[9] M S Celebi and H Akyildiz ldquoNonlinear modeling of liquidsloshing in a moving rectangular tankrdquo Ocean Engineering vol29 no 12 pp 1527ndash1553 2002

[10] C Z Wang and B C Khoo ldquoFinite element analysis of two-dimensional nonlinear sloshing problems in random excita-tionsrdquo Ocean Engineering vol 32 no 2 pp 107ndash133 2005

[11] V Sriram S A Sannasiraj and V Sundar ldquoNumerical simula-tion of 2D sloshingwaves due to horizontal and vertical randomexcitationrdquo Applied Ocean Research vol 28 no 1 pp 19ndash322006

[12] O M Faltinsen O F Rognebakke I A Lukovsky and AN Timokha ldquoMultidimensional modal analysis of nonlinearsloshing in a rectangular tank with finite water depthrdquo Journalof Fluid Mechanics vol 407 pp 201ndash234 2000

[13] O M Faltinsen and A N Timokha ldquoAn adaptive multimodalapproach to nonlinear sloshing in a rectangular tankrdquo Journalof Fluid Mechanics vol 432 pp 167ndash200 2001

[14] G XWu ldquoSecond-order resonance of sloshing in a tankrdquoOceanEngineering vol 34 no 17-18 pp 2345ndash2349 2007

[15] D F Hill ldquoTransient and steady-state amplitudes of forcedwaves in rectangular basinsrdquo Physics of Fluids vol 15 no 6 pp1576ndash1587 2003

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004

[17] RD Firouz-AbadiMGhasemi andHHaddadpour ldquoAmodalapproach to second-order analysis of sloshing using boundaryelement methodrdquo Ocean Engineering vol 38 no 1 pp 11ndash212011

[18] M J Chern A G L Borthwick and R E Taylor ldquoA pseu-dospectral 120590-transformation model of 2-D nonlinear wavesrdquoJournal of Fluids and Structures vol 13 no 5 pp 607ndash630 1999

[19] J B Frandsen and A G L Borthwick ldquoSimulation of sloshingmotions in fixed and vertically excited containers using a 2-Dinviscid120590-transformed finite difference solverrdquo Journal of Fluidsand Structures vol 18 no 2 pp 197ndash214 2003

[20] M S Turnbull A G L Borthwick and R Eatock TaylorldquoNumerical wave tank based on a 120590-transformed finite elementinviscid flow solverrdquo International Journal for Numerical Meth-ods in Fluids vol 42 no 6 pp 641ndash663 2003

[21] B Chen andH Chiang ldquoComplete 2D and fully nonlinear anal-ysis of ideal fluid in tanksrdquo Journal of Engineering Mechanicsvol 125 no 1 pp 70ndash78 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

20 40 60 80 100

0

5

10

15

20

minus5

minus10

120585a h

120596ht

(a) Aspect ratio ℎ119887 = 05

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(b) Aspect ratio ℎ119887 = 075

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(c) Aspect ratio ℎ119887 = 1

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 11 Free surface elevation at the left wall (low-frequency excitation)

the simulation is divergent after several times of iteration if119886ℎ= 005m the simulation lasts for a long time So it is

considered that the limitation of 119886ℎfor nonresonant sloshing

is 005m For 120596ℎ= 095120596

1 the excitation amplitude is much

smaller The maximum limit of 119886ℎis estimated to be 0012m

The time histories of free surface elevation at the left wall inthe limit cases are shown in Figure 8

Then the aspect ratio is changed by decreasing the stillwater depth ℎ For the aspect ratio ℎ119887 = 18 the numericalmethod still works with a mesh size of 80 times 20 If the aspectratio is less than 18 the simulation is divergent As a resultthe minimum limit of aspect ratio ℎ119887 is considered to be18 In this critical aspect ratio situation for 120596

ℎ= 095120596

1

the maximum limit of excitation amplitude 119886ℎis still 0012m

while for 120596ℎ= 085120596

1 the maximum limit of 119886

ℎis 003m

Figure 9 shows the time histories of free surface elevation inthe limit cases in this critical aspect ratio The phenomena ofbeating and resonance are very obscure it can be concludedthat the sloshing in low aspect ratio is quite different from theone in normal aspect ratio

423 Beating Phenomenon in Different Aspect Ratios Slosh-ing in a fixed aspect ratio has been studied in many articlesHowever little work has been done to analyze the response ofliquid sloshing in different aspect ratios by the same external

excitation When the external excitation is close to the firstnatural frequency of sloshing the beating phenomenon canbe observed The study here is intended to investigate theeffect of different aspect ratios on beating phenomenon andsloshing force

In the numerical examples calculated before the lengthand width of the tank are set to be 2m and 1m the still waterdepth is 1m The first natural frequency of sloshing is 120596

1=

376 rads The aspect ratio is ℎ119887 = 12 Now the length 119887and the width 119889 of the tank are fixed and the aspect ratio ischanged by changing the still water depth ℎThe former waterdepth and natural frequency are set to be the characteristicquantity ℎlowast = 1m 120596lowast

1= 376 rads

The relationship between aspect ratio and the first naturalfrequency must be considered From (25) it is easy to get thegraph of the natural frequency as a function of aspect ratioFigure 10 indicates that the slope of the function curve is steepwhen the aspect ratio is small if the aspect ratio is greater than06 the function curve is nearly horizontal

Under the same external excitation the sloshing forcesof beating are affected by three factors proximity of exci-tation frequency to the natural frequency liquid mass andnonlinearity of sloshing When the aspect ratio of the liquidchanges these three factors would change along with it Inorder to discuss the effect of aspect ratio two horizontaloscillations of the tank are studied First the tank is subjected

10 Mathematical Problems in Engineering

20 40 60 80 100

00

01

02

minus01

F120588gbd

hlowast

120596ht

(a) Aspect ratio ℎ119887 = 05

005

010

015

minus005

minus010

minus015

F120588gbd

hlowast

20 40 60 80 100120596ht

000

(b) Aspect ratio ℎ119887 = 075

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(c) Aspect ratio ℎ119887 = 1

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 12 Resultant sloshing forces (low-frequency excitation)

06 07 08 09 10 11 12

019

020

021

022

LinearNonlinear

F120588gbd

hlowast

hb

Figure 13 Tendency of amplitude of sloshing force (low-frequencyexcitation)

to a low-frequency excitation The movement of the tank isdescribed as 119886

ℎ= 003m 120596

ℎ= 34 rads Figure 10 suggests

that if the aspect ratio is 03 the corresponding naturalfrequency is 34 rads Our aim is to analyze the influence ofthis low-frequency excitation on the high aspect ratio slosh-ing system Figure 11 shows the time histories of free surface

elevation at the left wall in four different aspect ratios wherethe parameters are nondimensionalized

1199051015840= 120596ℎ119905 120585

1015840=

120585

119886ℎ

(33)

The amplitude of free surface elevation is quite large inFigure 11(a) which shows strong nonlinearity of sloshingwhile in Figures 11(b) 11(c) and 11(d) the amplitudes arenearly the same smaller than the amplitude in Figure 11(a)so the nonlinear effects are weaker in these three sloshingsituations

Figure 12 shows the time histories of resultant sloshingforce acting on the tank (the forces are also nondimensional-ized 1198651015840 = 119865120588119892119887119889ℎlowast) The amplitudes of the sloshing forceare extracted in Figure 12 In order to study the effect ofnonlinear factor linear sloshing models by Abramson [1] arecalculated under the same condition to make comparisonsThe relationship between the amplitude of sloshing force andaspect ratio is shown in Figure 13 It is obvious that bothlinear and nonlinear results have the same trend decreasingfirst and then increasing On the one hand the rise of theaspect ratio increases the natural frequency which makesthe sloshing far from resonance and reduces sloshing forcesOn the other hand the rise of the aspect ratio increases themass of the liquid thereby increasing the sloshing forces

Mathematical Problems in Engineering 11

0

10

20120585a h

20 40 60 80 100120596ht

minus10

(a) Aspect ratio ℎ119887 = 05

0

5

10

15

minus5

120585a h

20 40 60 80 100120596ht

minus10

(b) Aspect ratio ℎ119887 = 045

0

5

10

120585a h

20 40 60 80 100120596ht

minus5

minus10

(c) Aspect ratio ℎ119887 = 04

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 035

Figure 14 Free surface elevation at the left wall (high-frequency excitation)

For the sloshing in aspect ratio ℎ119887 = 05 the excitationfrequency is close to the natural frequency If aspect ratiois increased the factor of reducing sloshing force is moredominant However for the sloshing in high aspect ratiosℎ119887 gt 075 the rise of aspect ratio brings about slight changesin the natural frequency so the factor of increasing sloshingforce is more dominant Comparing nonlinear results withlinear ones in Figure 13 some conclusions are obtained Firstthe nonlinearity of sloshing always decreases sloshing forceSecond to a certain extent the difference between linear andnonlinear amplitude reflects the effect of nonlinear factoron sloshing force For the sloshing in aspect ratio ℎ119887 =

05 the difference is the largest while for the other sloshingsituations the differences are nearly the same This resultaccordswith the conclusion in Figure 11Third nonlinearity isan important factor in large amplitude sloshing which shouldnot be neglected

Then the tank is subjected to a high-frequency excitationThe movement of the tank is described as 119886

ℎ= 0015m

120596ℎ= 4 rads Our aim is to analyze the influence of this

high-frequency excitation on the low aspect ratio sloshingsystem Figure 14 shows the time histories of free surfaceelevation at the left wall in four different aspect ratios Asthe aspect ratio decreases the amplitude and period of freesurface elevation are diminishing For the sloshing in lowaspect ratios ℎ119887 lt 05 the reduction of aspect ratio brings

about significant changes in the natural frequency so thebeating phenomenon has changed greatly even if the aspectratio is slightly decreasedThe resultant sloshing forces actingon the tank are also calculated The relationship between theamplitude of sloshing force and aspect ratio is showed inFigure 15The reduction of the aspect ratio makes the naturalfrequency far from the excitation frequency meanwhile itdecreases the mass of the liquid in the tank So both of thefactors reduce sloshing force As the aspect ratio decreasesthe difference between linear and nonlinear amplitude ofsloshing force also decreases The reason is that nonlineareffects are becoming weaker and the sloshing force itself isalso decreasing

5 Conclusions

Anewfinite differencemethod based on the prediction of freesurface has been developed and used to simulate nonlinearsloshing phenomenon in tanks by the potential flow theoryAfter 120590-transformation the fluid domain is transformed to arectangle and a predictor-corrector scheme for free surfaceis used in the iteration of time then the governing equationand boundary conditions are approximately linearized andsolved in every time step The convergence of this methodis surveyed The numerical examples of standing waves and

12 Mathematical Problems in Engineering

035 040 045 050

006

008

010

012

014

016

LinearNonlinear

F120588gbd

hlowast

hb

Figure 15 Tendency of amplitude of sloshing force (high-frequencyexcitation)

horizontally forced sloshing models have been calculatedand good agreement between the numerical simulation andresults in other literatures has been obtained This numericalmethod is simple and easy in programming with excellentaccuracy and stability and can be extended to other sloshingproblems The time histories of free surface elevation andsloshing forces in the first and third resonance situation arealso presented and analyzed

The beating phenomenon of sloshing in different aspectratios under the same excitation has been numerically simu-lated by this method The results show that the effect of theaspect ratio on the sloshing force is complicated For the low-frequency excitation acting on the high aspect ratio modelsthe rise of aspect ratio makes the sloshing far from resonanceand increases the liquid mass so the sloshing force decreasesfirst and then increases For the high-frequency excitationacting on the low aspect ratio models the reduction of aspectratiomakes the sloshing far from resonance and decreases theliquid mass so the sloshing force decreases all the way Sincethe first natural frequency of liquid sloshing in not a linearfunction of aspect ratio the change of aspect ratio producesmuch stronger effect in low aspect ratio sloshing systemsMoreover nonlinearity of sloshing always decreases sloshingforce The effect of nonlinear factor depends on the violenceof sloshing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H N Abramson ldquoThe dynamic behavior of liquids in movingcontainers nasa sp-106rdquo NASA Special Publication vol 1061966

[2] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 no 4 pp 114ndash241 1974

[3] O M Faltinsen ldquoA numerical nonlinear method of sloshing intanks with two-dimensional flowrdquo Journal of Ship Research vol22 no 3 pp 193ndash202 1978

[4] T Nakayama and K Washizu ldquoNonlinear analysis of liquidmotion in a container subjected to forced pitching oscillationrdquoInternational Journal for Numerical Methods in Engineering vol15 no 8 pp 1207ndash1220 1980

[5] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[6] B-F Chen and R Nokes ldquoTime-independent finite differenceanalysis of fully non-linear and viscous fluid sloshing in arectangular tankrdquo Journal of Computational Physics vol 209 no1 pp 47ndash81 2005

[7] C Wu and B Chen ldquoSloshing waves and resonance modesof fluid in a 3D tank by a time-independent finite differencemethodrdquoOcean Engineering vol 36 no 6-7 pp 500ndash510 2009

[8] Y Kim ldquoNumerical simulation of sloshing flows with impactloadrdquo Applied Ocean Research vol 23 no 1 pp 53ndash62 2001

[9] M S Celebi and H Akyildiz ldquoNonlinear modeling of liquidsloshing in a moving rectangular tankrdquo Ocean Engineering vol29 no 12 pp 1527ndash1553 2002

[10] C Z Wang and B C Khoo ldquoFinite element analysis of two-dimensional nonlinear sloshing problems in random excita-tionsrdquo Ocean Engineering vol 32 no 2 pp 107ndash133 2005

[11] V Sriram S A Sannasiraj and V Sundar ldquoNumerical simula-tion of 2D sloshingwaves due to horizontal and vertical randomexcitationrdquo Applied Ocean Research vol 28 no 1 pp 19ndash322006

[12] O M Faltinsen O F Rognebakke I A Lukovsky and AN Timokha ldquoMultidimensional modal analysis of nonlinearsloshing in a rectangular tank with finite water depthrdquo Journalof Fluid Mechanics vol 407 pp 201ndash234 2000

[13] O M Faltinsen and A N Timokha ldquoAn adaptive multimodalapproach to nonlinear sloshing in a rectangular tankrdquo Journalof Fluid Mechanics vol 432 pp 167ndash200 2001

[14] G XWu ldquoSecond-order resonance of sloshing in a tankrdquoOceanEngineering vol 34 no 17-18 pp 2345ndash2349 2007

[15] D F Hill ldquoTransient and steady-state amplitudes of forcedwaves in rectangular basinsrdquo Physics of Fluids vol 15 no 6 pp1576ndash1587 2003

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004

[17] RD Firouz-AbadiMGhasemi andHHaddadpour ldquoAmodalapproach to second-order analysis of sloshing using boundaryelement methodrdquo Ocean Engineering vol 38 no 1 pp 11ndash212011

[18] M J Chern A G L Borthwick and R E Taylor ldquoA pseu-dospectral 120590-transformation model of 2-D nonlinear wavesrdquoJournal of Fluids and Structures vol 13 no 5 pp 607ndash630 1999

[19] J B Frandsen and A G L Borthwick ldquoSimulation of sloshingmotions in fixed and vertically excited containers using a 2-Dinviscid120590-transformed finite difference solverrdquo Journal of Fluidsand Structures vol 18 no 2 pp 197ndash214 2003

[20] M S Turnbull A G L Borthwick and R Eatock TaylorldquoNumerical wave tank based on a 120590-transformed finite elementinviscid flow solverrdquo International Journal for Numerical Meth-ods in Fluids vol 42 no 6 pp 641ndash663 2003

[21] B Chen andH Chiang ldquoComplete 2D and fully nonlinear anal-ysis of ideal fluid in tanksrdquo Journal of Engineering Mechanicsvol 125 no 1 pp 70ndash78 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

20 40 60 80 100

00

01

02

minus01

F120588gbd

hlowast

120596ht

(a) Aspect ratio ℎ119887 = 05

005

010

015

minus005

minus010

minus015

F120588gbd

hlowast

20 40 60 80 100120596ht

000

(b) Aspect ratio ℎ119887 = 075

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(c) Aspect ratio ℎ119887 = 1

00

01

02

minus01

minus02

F120588gbd

hlowast

20 40 60 80 100120596ht

(d) Aspect ratio ℎ119887 = 125

Figure 12 Resultant sloshing forces (low-frequency excitation)

06 07 08 09 10 11 12

019

020

021

022

LinearNonlinear

F120588gbd

hlowast

hb

Figure 13 Tendency of amplitude of sloshing force (low-frequencyexcitation)

to a low-frequency excitation The movement of the tank isdescribed as 119886

ℎ= 003m 120596

ℎ= 34 rads Figure 10 suggests

that if the aspect ratio is 03 the corresponding naturalfrequency is 34 rads Our aim is to analyze the influence ofthis low-frequency excitation on the high aspect ratio slosh-ing system Figure 11 shows the time histories of free surface

elevation at the left wall in four different aspect ratios wherethe parameters are nondimensionalized

1199051015840= 120596ℎ119905 120585

1015840=

120585

119886ℎ

(33)

The amplitude of free surface elevation is quite large inFigure 11(a) which shows strong nonlinearity of sloshingwhile in Figures 11(b) 11(c) and 11(d) the amplitudes arenearly the same smaller than the amplitude in Figure 11(a)so the nonlinear effects are weaker in these three sloshingsituations

Figure 12 shows the time histories of resultant sloshingforce acting on the tank (the forces are also nondimensional-ized 1198651015840 = 119865120588119892119887119889ℎlowast) The amplitudes of the sloshing forceare extracted in Figure 12 In order to study the effect ofnonlinear factor linear sloshing models by Abramson [1] arecalculated under the same condition to make comparisonsThe relationship between the amplitude of sloshing force andaspect ratio is shown in Figure 13 It is obvious that bothlinear and nonlinear results have the same trend decreasingfirst and then increasing On the one hand the rise of theaspect ratio increases the natural frequency which makesthe sloshing far from resonance and reduces sloshing forcesOn the other hand the rise of the aspect ratio increases themass of the liquid thereby increasing the sloshing forces

Mathematical Problems in Engineering 11

0

10

20120585a h

20 40 60 80 100120596ht

minus10

(a) Aspect ratio ℎ119887 = 05

0

5

10

15

minus5

120585a h

20 40 60 80 100120596ht

minus10

(b) Aspect ratio ℎ119887 = 045

0

5

10

120585a h

20 40 60 80 100120596ht

minus5

minus10

(c) Aspect ratio ℎ119887 = 04

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 035

Figure 14 Free surface elevation at the left wall (high-frequency excitation)

For the sloshing in aspect ratio ℎ119887 = 05 the excitationfrequency is close to the natural frequency If aspect ratiois increased the factor of reducing sloshing force is moredominant However for the sloshing in high aspect ratiosℎ119887 gt 075 the rise of aspect ratio brings about slight changesin the natural frequency so the factor of increasing sloshingforce is more dominant Comparing nonlinear results withlinear ones in Figure 13 some conclusions are obtained Firstthe nonlinearity of sloshing always decreases sloshing forceSecond to a certain extent the difference between linear andnonlinear amplitude reflects the effect of nonlinear factoron sloshing force For the sloshing in aspect ratio ℎ119887 =

05 the difference is the largest while for the other sloshingsituations the differences are nearly the same This resultaccordswith the conclusion in Figure 11Third nonlinearity isan important factor in large amplitude sloshing which shouldnot be neglected

Then the tank is subjected to a high-frequency excitationThe movement of the tank is described as 119886

ℎ= 0015m

120596ℎ= 4 rads Our aim is to analyze the influence of this

high-frequency excitation on the low aspect ratio sloshingsystem Figure 14 shows the time histories of free surfaceelevation at the left wall in four different aspect ratios Asthe aspect ratio decreases the amplitude and period of freesurface elevation are diminishing For the sloshing in lowaspect ratios ℎ119887 lt 05 the reduction of aspect ratio brings

about significant changes in the natural frequency so thebeating phenomenon has changed greatly even if the aspectratio is slightly decreasedThe resultant sloshing forces actingon the tank are also calculated The relationship between theamplitude of sloshing force and aspect ratio is showed inFigure 15The reduction of the aspect ratio makes the naturalfrequency far from the excitation frequency meanwhile itdecreases the mass of the liquid in the tank So both of thefactors reduce sloshing force As the aspect ratio decreasesthe difference between linear and nonlinear amplitude ofsloshing force also decreases The reason is that nonlineareffects are becoming weaker and the sloshing force itself isalso decreasing

5 Conclusions

Anewfinite differencemethod based on the prediction of freesurface has been developed and used to simulate nonlinearsloshing phenomenon in tanks by the potential flow theoryAfter 120590-transformation the fluid domain is transformed to arectangle and a predictor-corrector scheme for free surfaceis used in the iteration of time then the governing equationand boundary conditions are approximately linearized andsolved in every time step The convergence of this methodis surveyed The numerical examples of standing waves and

12 Mathematical Problems in Engineering

035 040 045 050

006

008

010

012

014

016

LinearNonlinear

F120588gbd

hlowast

hb

Figure 15 Tendency of amplitude of sloshing force (high-frequencyexcitation)

horizontally forced sloshing models have been calculatedand good agreement between the numerical simulation andresults in other literatures has been obtained This numericalmethod is simple and easy in programming with excellentaccuracy and stability and can be extended to other sloshingproblems The time histories of free surface elevation andsloshing forces in the first and third resonance situation arealso presented and analyzed

The beating phenomenon of sloshing in different aspectratios under the same excitation has been numerically simu-lated by this method The results show that the effect of theaspect ratio on the sloshing force is complicated For the low-frequency excitation acting on the high aspect ratio modelsthe rise of aspect ratio makes the sloshing far from resonanceand increases the liquid mass so the sloshing force decreasesfirst and then increases For the high-frequency excitationacting on the low aspect ratio models the reduction of aspectratiomakes the sloshing far from resonance and decreases theliquid mass so the sloshing force decreases all the way Sincethe first natural frequency of liquid sloshing in not a linearfunction of aspect ratio the change of aspect ratio producesmuch stronger effect in low aspect ratio sloshing systemsMoreover nonlinearity of sloshing always decreases sloshingforce The effect of nonlinear factor depends on the violenceof sloshing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H N Abramson ldquoThe dynamic behavior of liquids in movingcontainers nasa sp-106rdquo NASA Special Publication vol 1061966

[2] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 no 4 pp 114ndash241 1974

[3] O M Faltinsen ldquoA numerical nonlinear method of sloshing intanks with two-dimensional flowrdquo Journal of Ship Research vol22 no 3 pp 193ndash202 1978

[4] T Nakayama and K Washizu ldquoNonlinear analysis of liquidmotion in a container subjected to forced pitching oscillationrdquoInternational Journal for Numerical Methods in Engineering vol15 no 8 pp 1207ndash1220 1980

[5] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[6] B-F Chen and R Nokes ldquoTime-independent finite differenceanalysis of fully non-linear and viscous fluid sloshing in arectangular tankrdquo Journal of Computational Physics vol 209 no1 pp 47ndash81 2005

[7] C Wu and B Chen ldquoSloshing waves and resonance modesof fluid in a 3D tank by a time-independent finite differencemethodrdquoOcean Engineering vol 36 no 6-7 pp 500ndash510 2009

[8] Y Kim ldquoNumerical simulation of sloshing flows with impactloadrdquo Applied Ocean Research vol 23 no 1 pp 53ndash62 2001

[9] M S Celebi and H Akyildiz ldquoNonlinear modeling of liquidsloshing in a moving rectangular tankrdquo Ocean Engineering vol29 no 12 pp 1527ndash1553 2002

[10] C Z Wang and B C Khoo ldquoFinite element analysis of two-dimensional nonlinear sloshing problems in random excita-tionsrdquo Ocean Engineering vol 32 no 2 pp 107ndash133 2005

[11] V Sriram S A Sannasiraj and V Sundar ldquoNumerical simula-tion of 2D sloshingwaves due to horizontal and vertical randomexcitationrdquo Applied Ocean Research vol 28 no 1 pp 19ndash322006

[12] O M Faltinsen O F Rognebakke I A Lukovsky and AN Timokha ldquoMultidimensional modal analysis of nonlinearsloshing in a rectangular tank with finite water depthrdquo Journalof Fluid Mechanics vol 407 pp 201ndash234 2000

[13] O M Faltinsen and A N Timokha ldquoAn adaptive multimodalapproach to nonlinear sloshing in a rectangular tankrdquo Journalof Fluid Mechanics vol 432 pp 167ndash200 2001

[14] G XWu ldquoSecond-order resonance of sloshing in a tankrdquoOceanEngineering vol 34 no 17-18 pp 2345ndash2349 2007

[15] D F Hill ldquoTransient and steady-state amplitudes of forcedwaves in rectangular basinsrdquo Physics of Fluids vol 15 no 6 pp1576ndash1587 2003

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004

[17] RD Firouz-AbadiMGhasemi andHHaddadpour ldquoAmodalapproach to second-order analysis of sloshing using boundaryelement methodrdquo Ocean Engineering vol 38 no 1 pp 11ndash212011

[18] M J Chern A G L Borthwick and R E Taylor ldquoA pseu-dospectral 120590-transformation model of 2-D nonlinear wavesrdquoJournal of Fluids and Structures vol 13 no 5 pp 607ndash630 1999

[19] J B Frandsen and A G L Borthwick ldquoSimulation of sloshingmotions in fixed and vertically excited containers using a 2-Dinviscid120590-transformed finite difference solverrdquo Journal of Fluidsand Structures vol 18 no 2 pp 197ndash214 2003

[20] M S Turnbull A G L Borthwick and R Eatock TaylorldquoNumerical wave tank based on a 120590-transformed finite elementinviscid flow solverrdquo International Journal for Numerical Meth-ods in Fluids vol 42 no 6 pp 641ndash663 2003

[21] B Chen andH Chiang ldquoComplete 2D and fully nonlinear anal-ysis of ideal fluid in tanksrdquo Journal of Engineering Mechanicsvol 125 no 1 pp 70ndash78 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0

10

20120585a h

20 40 60 80 100120596ht

minus10

(a) Aspect ratio ℎ119887 = 05

0

5

10

15

minus5

120585a h

20 40 60 80 100120596ht

minus10

(b) Aspect ratio ℎ119887 = 045

0

5

10

120585a h

20 40 60 80 100120596ht

minus5

minus10

(c) Aspect ratio ℎ119887 = 04

20 40 60 80 100

0

5

10

minus5

120585a h

120596ht

(d) Aspect ratio ℎ119887 = 035

Figure 14 Free surface elevation at the left wall (high-frequency excitation)

For the sloshing in aspect ratio ℎ119887 = 05 the excitationfrequency is close to the natural frequency If aspect ratiois increased the factor of reducing sloshing force is moredominant However for the sloshing in high aspect ratiosℎ119887 gt 075 the rise of aspect ratio brings about slight changesin the natural frequency so the factor of increasing sloshingforce is more dominant Comparing nonlinear results withlinear ones in Figure 13 some conclusions are obtained Firstthe nonlinearity of sloshing always decreases sloshing forceSecond to a certain extent the difference between linear andnonlinear amplitude reflects the effect of nonlinear factoron sloshing force For the sloshing in aspect ratio ℎ119887 =

05 the difference is the largest while for the other sloshingsituations the differences are nearly the same This resultaccordswith the conclusion in Figure 11Third nonlinearity isan important factor in large amplitude sloshing which shouldnot be neglected

Then the tank is subjected to a high-frequency excitationThe movement of the tank is described as 119886

ℎ= 0015m

120596ℎ= 4 rads Our aim is to analyze the influence of this

high-frequency excitation on the low aspect ratio sloshingsystem Figure 14 shows the time histories of free surfaceelevation at the left wall in four different aspect ratios Asthe aspect ratio decreases the amplitude and period of freesurface elevation are diminishing For the sloshing in lowaspect ratios ℎ119887 lt 05 the reduction of aspect ratio brings

about significant changes in the natural frequency so thebeating phenomenon has changed greatly even if the aspectratio is slightly decreasedThe resultant sloshing forces actingon the tank are also calculated The relationship between theamplitude of sloshing force and aspect ratio is showed inFigure 15The reduction of the aspect ratio makes the naturalfrequency far from the excitation frequency meanwhile itdecreases the mass of the liquid in the tank So both of thefactors reduce sloshing force As the aspect ratio decreasesthe difference between linear and nonlinear amplitude ofsloshing force also decreases The reason is that nonlineareffects are becoming weaker and the sloshing force itself isalso decreasing

5 Conclusions

Anewfinite differencemethod based on the prediction of freesurface has been developed and used to simulate nonlinearsloshing phenomenon in tanks by the potential flow theoryAfter 120590-transformation the fluid domain is transformed to arectangle and a predictor-corrector scheme for free surfaceis used in the iteration of time then the governing equationand boundary conditions are approximately linearized andsolved in every time step The convergence of this methodis surveyed The numerical examples of standing waves and

12 Mathematical Problems in Engineering

035 040 045 050

006

008

010

012

014

016

LinearNonlinear

F120588gbd

hlowast

hb

Figure 15 Tendency of amplitude of sloshing force (high-frequencyexcitation)

horizontally forced sloshing models have been calculatedand good agreement between the numerical simulation andresults in other literatures has been obtained This numericalmethod is simple and easy in programming with excellentaccuracy and stability and can be extended to other sloshingproblems The time histories of free surface elevation andsloshing forces in the first and third resonance situation arealso presented and analyzed

The beating phenomenon of sloshing in different aspectratios under the same excitation has been numerically simu-lated by this method The results show that the effect of theaspect ratio on the sloshing force is complicated For the low-frequency excitation acting on the high aspect ratio modelsthe rise of aspect ratio makes the sloshing far from resonanceand increases the liquid mass so the sloshing force decreasesfirst and then increases For the high-frequency excitationacting on the low aspect ratio models the reduction of aspectratiomakes the sloshing far from resonance and decreases theliquid mass so the sloshing force decreases all the way Sincethe first natural frequency of liquid sloshing in not a linearfunction of aspect ratio the change of aspect ratio producesmuch stronger effect in low aspect ratio sloshing systemsMoreover nonlinearity of sloshing always decreases sloshingforce The effect of nonlinear factor depends on the violenceof sloshing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H N Abramson ldquoThe dynamic behavior of liquids in movingcontainers nasa sp-106rdquo NASA Special Publication vol 1061966

[2] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 no 4 pp 114ndash241 1974

[3] O M Faltinsen ldquoA numerical nonlinear method of sloshing intanks with two-dimensional flowrdquo Journal of Ship Research vol22 no 3 pp 193ndash202 1978

[4] T Nakayama and K Washizu ldquoNonlinear analysis of liquidmotion in a container subjected to forced pitching oscillationrdquoInternational Journal for Numerical Methods in Engineering vol15 no 8 pp 1207ndash1220 1980

[5] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[6] B-F Chen and R Nokes ldquoTime-independent finite differenceanalysis of fully non-linear and viscous fluid sloshing in arectangular tankrdquo Journal of Computational Physics vol 209 no1 pp 47ndash81 2005

[7] C Wu and B Chen ldquoSloshing waves and resonance modesof fluid in a 3D tank by a time-independent finite differencemethodrdquoOcean Engineering vol 36 no 6-7 pp 500ndash510 2009

[8] Y Kim ldquoNumerical simulation of sloshing flows with impactloadrdquo Applied Ocean Research vol 23 no 1 pp 53ndash62 2001

[9] M S Celebi and H Akyildiz ldquoNonlinear modeling of liquidsloshing in a moving rectangular tankrdquo Ocean Engineering vol29 no 12 pp 1527ndash1553 2002

[10] C Z Wang and B C Khoo ldquoFinite element analysis of two-dimensional nonlinear sloshing problems in random excita-tionsrdquo Ocean Engineering vol 32 no 2 pp 107ndash133 2005

[11] V Sriram S A Sannasiraj and V Sundar ldquoNumerical simula-tion of 2D sloshingwaves due to horizontal and vertical randomexcitationrdquo Applied Ocean Research vol 28 no 1 pp 19ndash322006

[12] O M Faltinsen O F Rognebakke I A Lukovsky and AN Timokha ldquoMultidimensional modal analysis of nonlinearsloshing in a rectangular tank with finite water depthrdquo Journalof Fluid Mechanics vol 407 pp 201ndash234 2000

[13] O M Faltinsen and A N Timokha ldquoAn adaptive multimodalapproach to nonlinear sloshing in a rectangular tankrdquo Journalof Fluid Mechanics vol 432 pp 167ndash200 2001

[14] G XWu ldquoSecond-order resonance of sloshing in a tankrdquoOceanEngineering vol 34 no 17-18 pp 2345ndash2349 2007

[15] D F Hill ldquoTransient and steady-state amplitudes of forcedwaves in rectangular basinsrdquo Physics of Fluids vol 15 no 6 pp1576ndash1587 2003

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004

[17] RD Firouz-AbadiMGhasemi andHHaddadpour ldquoAmodalapproach to second-order analysis of sloshing using boundaryelement methodrdquo Ocean Engineering vol 38 no 1 pp 11ndash212011

[18] M J Chern A G L Borthwick and R E Taylor ldquoA pseu-dospectral 120590-transformation model of 2-D nonlinear wavesrdquoJournal of Fluids and Structures vol 13 no 5 pp 607ndash630 1999

[19] J B Frandsen and A G L Borthwick ldquoSimulation of sloshingmotions in fixed and vertically excited containers using a 2-Dinviscid120590-transformed finite difference solverrdquo Journal of Fluidsand Structures vol 18 no 2 pp 197ndash214 2003

[20] M S Turnbull A G L Borthwick and R Eatock TaylorldquoNumerical wave tank based on a 120590-transformed finite elementinviscid flow solverrdquo International Journal for Numerical Meth-ods in Fluids vol 42 no 6 pp 641ndash663 2003

[21] B Chen andH Chiang ldquoComplete 2D and fully nonlinear anal-ysis of ideal fluid in tanksrdquo Journal of Engineering Mechanicsvol 125 no 1 pp 70ndash78 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

035 040 045 050

006

008

010

012

014

016

LinearNonlinear

F120588gbd

hlowast

hb

Figure 15 Tendency of amplitude of sloshing force (high-frequencyexcitation)

horizontally forced sloshing models have been calculatedand good agreement between the numerical simulation andresults in other literatures has been obtained This numericalmethod is simple and easy in programming with excellentaccuracy and stability and can be extended to other sloshingproblems The time histories of free surface elevation andsloshing forces in the first and third resonance situation arealso presented and analyzed

The beating phenomenon of sloshing in different aspectratios under the same excitation has been numerically simu-lated by this method The results show that the effect of theaspect ratio on the sloshing force is complicated For the low-frequency excitation acting on the high aspect ratio modelsthe rise of aspect ratio makes the sloshing far from resonanceand increases the liquid mass so the sloshing force decreasesfirst and then increases For the high-frequency excitationacting on the low aspect ratio models the reduction of aspectratiomakes the sloshing far from resonance and decreases theliquid mass so the sloshing force decreases all the way Sincethe first natural frequency of liquid sloshing in not a linearfunction of aspect ratio the change of aspect ratio producesmuch stronger effect in low aspect ratio sloshing systemsMoreover nonlinearity of sloshing always decreases sloshingforce The effect of nonlinear factor depends on the violenceof sloshing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H N Abramson ldquoThe dynamic behavior of liquids in movingcontainers nasa sp-106rdquo NASA Special Publication vol 1061966

[2] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 no 4 pp 114ndash241 1974

[3] O M Faltinsen ldquoA numerical nonlinear method of sloshing intanks with two-dimensional flowrdquo Journal of Ship Research vol22 no 3 pp 193ndash202 1978

[4] T Nakayama and K Washizu ldquoNonlinear analysis of liquidmotion in a container subjected to forced pitching oscillationrdquoInternational Journal for Numerical Methods in Engineering vol15 no 8 pp 1207ndash1220 1980

[5] G X Wu Q W Ma and R Eatock Taylor ldquoNumericalsimulation of sloshing waves in a 3D tank based on a finiteelement methodrdquo Applied Ocean Research vol 20 no 6 pp337ndash355 1998

[6] B-F Chen and R Nokes ldquoTime-independent finite differenceanalysis of fully non-linear and viscous fluid sloshing in arectangular tankrdquo Journal of Computational Physics vol 209 no1 pp 47ndash81 2005

[7] C Wu and B Chen ldquoSloshing waves and resonance modesof fluid in a 3D tank by a time-independent finite differencemethodrdquoOcean Engineering vol 36 no 6-7 pp 500ndash510 2009

[8] Y Kim ldquoNumerical simulation of sloshing flows with impactloadrdquo Applied Ocean Research vol 23 no 1 pp 53ndash62 2001

[9] M S Celebi and H Akyildiz ldquoNonlinear modeling of liquidsloshing in a moving rectangular tankrdquo Ocean Engineering vol29 no 12 pp 1527ndash1553 2002

[10] C Z Wang and B C Khoo ldquoFinite element analysis of two-dimensional nonlinear sloshing problems in random excita-tionsrdquo Ocean Engineering vol 32 no 2 pp 107ndash133 2005

[11] V Sriram S A Sannasiraj and V Sundar ldquoNumerical simula-tion of 2D sloshingwaves due to horizontal and vertical randomexcitationrdquo Applied Ocean Research vol 28 no 1 pp 19ndash322006

[12] O M Faltinsen O F Rognebakke I A Lukovsky and AN Timokha ldquoMultidimensional modal analysis of nonlinearsloshing in a rectangular tank with finite water depthrdquo Journalof Fluid Mechanics vol 407 pp 201ndash234 2000

[13] O M Faltinsen and A N Timokha ldquoAn adaptive multimodalapproach to nonlinear sloshing in a rectangular tankrdquo Journalof Fluid Mechanics vol 432 pp 167ndash200 2001

[14] G XWu ldquoSecond-order resonance of sloshing in a tankrdquoOceanEngineering vol 34 no 17-18 pp 2345ndash2349 2007

[15] D F Hill ldquoTransient and steady-state amplitudes of forcedwaves in rectangular basinsrdquo Physics of Fluids vol 15 no 6 pp1576ndash1587 2003

[16] J B Frandsen ldquoSloshing motions in excited tanksrdquo Journal ofComputational Physics vol 196 no 1 pp 53ndash87 2004

[17] RD Firouz-AbadiMGhasemi andHHaddadpour ldquoAmodalapproach to second-order analysis of sloshing using boundaryelement methodrdquo Ocean Engineering vol 38 no 1 pp 11ndash212011

[18] M J Chern A G L Borthwick and R E Taylor ldquoA pseu-dospectral 120590-transformation model of 2-D nonlinear wavesrdquoJournal of Fluids and Structures vol 13 no 5 pp 607ndash630 1999

[19] J B Frandsen and A G L Borthwick ldquoSimulation of sloshingmotions in fixed and vertically excited containers using a 2-Dinviscid120590-transformed finite difference solverrdquo Journal of Fluidsand Structures vol 18 no 2 pp 197ndash214 2003

[20] M S Turnbull A G L Borthwick and R Eatock TaylorldquoNumerical wave tank based on a 120590-transformed finite elementinviscid flow solverrdquo International Journal for Numerical Meth-ods in Fluids vol 42 no 6 pp 641ndash663 2003

[21] B Chen andH Chiang ldquoComplete 2D and fully nonlinear anal-ysis of ideal fluid in tanksrdquo Journal of Engineering Mechanicsvol 125 no 1 pp 70ndash78 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of