ResearchArticle Nonlinear Finite Element Analysis of Sloshing

Hindawi Publishing CorporationAdvances in Numerical AnalysisVolume 2013 Article ID 571528 10 pageshttpdxdoiorg1011552013571528

Research ArticleNonlinear Finite Element Analysis of Sloshing

Siva Srinivas Kolukula and P Chellapandi

Structural Mechanics Laboratory Reactor Design Group Indira Gandhi Center for Atomic Research Kalpakkam 603102Tamilnadu India

Correspondence should be addressed to Siva Srinivas Kolukula allwayzitzmegmailcom

Received 28 November 2012 Accepted 10 January 2013

Academic Editor Zhangxing Chen

Copyright copy 2013 S S Kolukula and P Chellapandi This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The disturbance on the free surface of the liquid when the liquid-filled tanks are excited is called sloshing This paper examinesthe nonlinear sloshing response of the liquid free surface in partially filled two-dimensional rectangular tanks using finite elementmethod The liquid is assumed to be inviscid irrotational and incompressible fully nonlinear potential wave theory is consideredand mixed Eulerian-Lagrangian scheme is adopted The velocities are obtained from potential using least square method foraccurate evaluation The fourth-order Runge-Kutta method is employed to advance the solution in time A regridding techniquebased on cubic spline is employed to avoid numerical instabilities Regular harmonic excitations and random excitations are usedas the external disturbance to the container The results obtained are compared with published results to validate the numericalmethod developed

1 Introduction

It is common everyday knowledge to each of us that any smallcontainer filled with liquid must be moved or carried verycarefully to avoid spills For example one has to be carefulwhile carrying a cup of coffee while moving because themotion of the person makes coffee spill Such a motion onthe free surface of the liquid due to external excitation in theliquid-filled containers is called sloshing Sloshing is likelyto be seen whenever we have a liquid with a free surface inthe presence of gravity At equilibrium the free surface of theliquid is static and coincideswith a gravitational equipotentialsurface When the surface is perturbed an oscillation is setup in which the energy oscillates between kinetic energyand gravitational potential energy The phenomenon calledsloshing occurs in a variety of engineering applications suchas sloshing in liquid-propellant launch vehicles sloshingin liquids used in industries to store oil water chemicalsliquefied natural gases and so forth and sloshing in thenuclear reactors of pool type nuclear fuel storage tanksunder earthquake The liquid sloshing may cause huge lossof human economic and environmental resources owing

to unexpected failure of the container for example thespillage of toxic chemicals stored in tanks in industries cancause contamination of soil and the environment Thusunderstanding the dynamic behaviour of liquid free surfaceis essential As a result the problem of sloshing has attractedmany researchers and engineers targeting to understand thecomplex behaviour of sloshing and to design the structures towithstand its effects

Abundant research has been made on the sloshing phe-nomenon and the literature is vast with wide varieties ofnumerical methods analytical solutions and experimentsThe position of the free surface of liquid is not known apriori and obeys a dynamic boundary condition which isnonlinear andmakes the problem of sloshing that a nonlinearboundary value problem Although the sloshing problem isnonlinear by assuming the free-surface elevation to be smalland applying the linearized free-surface boundary conditiona linear theory of sloshing is developed [1 2] This lineartheory is acceptable in few cases when the amplitude ofexternal excitation is small and not in the neighborhoodof the sloshing natural frequency When the above men-tioned conditions do not hold linear theory fails to predict

2 Advances in Numerical Analysis

the sloshing response accurately Hence nonlinear analysisbecomes inevitable for accurate and reliable evaluation ofliquid sloshing

Nonlinear sloshing problem is difficult to solve analyti-cally because of its nonlinear boundary conditions implyinga numerical modeling is necessary Faltinsen [3 4] solvedthe nonlinear sloshing problem numerically and derived theanalytical solution using perturbation approach with two-dimensional flow Nakayama and Washizu [5] employed theboundary element method for the problem Wu and Taylor[6 7] applied finite element analysis for twodimensional non-linear transient waves Chen et al [8] applied finite differencemethod to simulate large-amplitude sloshing under seismicexcitations Turnbull et al [9] used sigma-transformed finiteelement inviscid flow solver for the problem Frandsen [10]analysed the nonlinear sloshing motions of liquid undervertical horizontal and combined motions of the tankusing analytical and numerical methods The author usedperturbation methods for analytical solution and modifiedsigma-transformed finite difference method for numericalsolution Cho and Lee [11] used semi-Lagrangian nonlinearfinite element approximation to analyse the large amplitudesloshing in two-dimensional tanks Wang and Khoo [12]analysed the nonlinear sloshing in two-dimensional tanksunder random excitations Sriram et al [13] analysed non-linear sloshing in two-dimensional tanks using finite elementand finite difference method Biswal et al [14] analysed thenonlinear sloshing response in tanks with baffles using finiteelement method Ibrahim et al [15] give an excellent reviewof sloshing phenomenonwith extensive number of referencesavailable in the literature

In the present paper a numerical approach based onmixed Eulerian-Lagrangian scheme is adopted The freesurface nodes behave like Lagrangian particles and interiornodes behave like Eulerian particles The nonlinear sloshinganalysis is carried out using finite element method A four-noded isoparametric element is used in the analysis Thecalculation of velocities from velocity potential is an impor-tant step to study the sloshing behaviour Thus velocitiesmust be calculated accurately for accurate sloshing analysisThe velocity field is interpolated from the velocity potentialaccording to least square method [16] Fourth-order Runge-Kutta method is employed to advance the solution in timeAs the time proceeds in the simulation due to Lagrangianbehaviour of the free surface nodes these nodes move closerand develop a steep gradient leading to numerical instabilityTo get rid of this problem a cubic spline interpolation is usedfor the regridding the free surface uniformly In the presentsimulation the tank is assumed to be rigid with aspect ratio(ℎ119871) of 05 ℎ is depth of fluid119871 is length of the tank Sloshingresponse is simulated when the external excitation frequencyis in resonance and off resonance region For horizontal exci-tations the free surface undergoes resonance when excitationfrequency is equal to fundamental slosh frequency and showsbeating phenomenon when excitation frequency is close tofundamental slosh frequencyThe present numericalmodel isvalidated with Frandsen [10] numerical results for harmonicexcitations and then applied for sloshing response due torandom excitation

119911

Γ119904 120601(119909 119911 119905)

Ω

120577(119909 119905)

119899119899

119899119871

ℎ

119909nabla

nabla2120601 = 0

Γ119861 120597120601

119889119899= 0

Figure 1 Sloshing wave tank in moving coordinate system

2 Governing Equations

Consider a rectangular tank fixed in Cartesian coordinatesystem 119874119909119911 which is moving with respect to inertial Carte-sian coordinate system119874

011990901199110 The origins of this system are

at the left end of the tank wall at the free surface and pointingupwards in 119911 directionThese two Cartesian systems coincidewhen the tank is at rest Figure 1 shows the tank in themovingCartesian coordinate system 119874119909119911 along with the prescribedboundary conditions The tank is assumed to get displacedalong 119909-axes and the tank position at time is given by

119883 = 119883119905(119905) (1)

Fluid is assumed to be inviscid incompressible and irrota-tional Therefore the fluid motion is governed by Laplacersquosequation with the unknown as velocity potential 120601

nabla2120601 = 0 (2)

The fluid obeysNeumann boundary conditions at the walls ofthe container and Dirichlet boundary condition at the liquidfree surface In the moving coordinate system the velocitycomponent of the fluid normal to the walls is zero Henceon the bottom and on the walls of the tank (Γ

119861) we have

120597120601

120597119899

10038161003816100381610038161003816100381610038161003816119909=0119871= 0

120597120601

120597119899

10038161003816100381610038161003816100381610038161003816119911=minusℎ= 0 (3)

On the free surface (Γ119904) dynamic and kinematic conditions

hold they are given as

120597120601

120597119905

10038161003816100381610038161003816100381610038161003816119911=120577+

1

2nabla120601 sdot nabla120601 + 119892120577 + 119909119883

10158401015840

119905

120597120577

120597119905+

120597120601

120597119909

120597120577

120597119909minus

120597120601

120597119911= 0

(4)

Rewrite the above equations in the Lagrangian form [16]

119889120601

119889119905

10038161003816100381610038161003816100381610038161003816119911=120577=

1

2nabla120601 sdot nabla120601 minus 119892120577 minus 119909119883

10158401015840

119905 (5)

119889119909

119889119905=

120597120601

120597119909

119889119911

119889119905=

120597120601

120597119911 (6)

Advances in Numerical Analysis 3

minus1

minus08

minus06

minus04

minus02

0

02

119911(m

)

119909 (m)0 05 1 15 2

Figure 2 A typical mesh of the liquid domain using isoparametricfour-noded elements

where 120577 is the free surface elevationmeasured vertically abovestill water level11988310158401015840

119905is the horizontal acceleration of the tank

and 119892 is the acceleration due to gravityEquations (1)ndash(6) give the complete behaviour of nonlin-

ear sloshing in fluids The position of the fluid free surface isnot known a priori to solve the problem the fluid is assumedto be at rest Thus the initial conditions for the free surface inthemovingCartesian system at 119905 = 0 and 119911 = 0 can bewrittenas

120601 (119909 0 0) = minus119909119889119883119905(119905)

119889119905 (7)

120577 (119909 0) = 0 (8)

Using these initial conditions Laplace equation (2) is solvedand the free surface elevation and potential are updated forthe subsequent time steps using (5)-(6)

3 Numerical Procedure

31 Finite Element Formulation The solution of the non-linear sloshing boundary value problem is obtained usingfinite elementmethodThe entire liquid domain is discretizedby using four-noded isoparametric quadrilateral elements Atypical mesh structure is shown in Figure 2 By introducingthe finite element shape functions the liquid velocity potentialcan be approximated as

120601 (119909 119911) =

119899

sum119895=1

119873119895(119909 119911) 120601

119895 (9)

where 119873119895is the shape function 119899 is the number of nodes in

the element and 120601119895is nodal velocity potential

On applying Galerkin residual method to the Laplaceequation we get

119870120601 = 0 (10)

with matrix 119870 defined by

119870 = intΩ

(nabla119873)119879(nabla119873) 119889Ω (11)

Yes

Free surface boundary conditions

(FEM)

Compute velocity(FEM)

Update free surface (RK4 method)

Mesh stability

Regridding

Φ(119909 0)∣119905=0 and 120577(119909 0) = 0

Evaluate Φ for interior nodes

Φ(119909 119911)∣119905+Δ119905 and 120577(119909 119911)119905 + Δ119905

No

Figure 3 Numerical procedure for nonlinear sloshing simulation

Matrix 119870 is analogous to stiffness matrix in structural prob-lems Using (10) from the known free-surface velocity poten-tial the velocity potential in the interior nodes is calculated

32 Velocity Recovery To track the free surface (4) needvelocities which are computed from the calculated potentialfield using

120584 = nabla sdot 120601119895 (12)

The velocities calculated using (12) are the velocities at theGauss integration points and they do not possess interele-ment continuity and have a low accuracy at nodes andelement boundaries Utmost care should be taken to calculatethe velocities a small error in the velocity recovery willaffect the accuracy of free surface updating and gets accu-mulated with time and leads to underestimation of sloshingresponse In order to derive a smoothed and continuousvelocity patch recovery technique [17] is applied In patchrecovery technique the continuous velocity field is obtainedby considering the linear interpolation of the velocities at theGauss integration points

120584 = 1198861+ 1198862120585 + 1198863120578 + 1198864120585120578 (13)

where 120584 is any velocity component (120584119909or 120584119910) 120585 120578 are the

Gauss locations and 1198861 1198862 1198863 1198864are unknowns which need

to be evaluated To evaluate these unknowns a least square fitis considered between 120584 and 120584

119865 (119886) =

4

sum119894=1

120584 (120585119894 120578119894) minus 120584 (120585

119894 120578119894)2 (14)

4 Advances in Numerical Analysis

minus1

minus08

minus06

minus04

minus02

0

02

0 05 1 15 2

Mode 1 1205961 = 37607 rads

(a)

0 05 1 15 2

minus1

minus08

minus06

minus04

minus02

0

02

Mode 2 1205962 = 55456 rads

(b)

0 05 1 15 2

minus1

minus08

minus06

minus04

minus02

0

02

Mode 3 1205963 = 68092 rads

(c)

0 05 1 15 2

minus1

minus08

minus06

minus04

minus02

0

02

Mode 4 1205964 = 78715 rads

(d)

Figure 4 First four mode shapes of sloshing

where 119894 is 2times2 order Gauss integration pointsThen the fourunknown coefficients are determined from four simultaneousequations obtained from

120597119865 (119886)

120597119886119896

= 0 119896 = 1 2 3 4 (15)

Substituting the obtained 119886119896rsquos in (13) gives the velocity

values for individual elements and these are averaged for thecommon nodes Finally a smoothed velocity field which isinterelement continuous is constructed by interpolating thefinite element shape functions used in (9) and nodal averagedvelocities The global continuous velocity field is given as

120584 = 119873 sdot 120584 (16)

where 120584 is velocity component (120584119909or 120584119910) Using the patch

recovery technique velocity components 120584119909and 120584119910are calcu-

lated

33 Numerical Time Integration and Free Surface TrackingAfter calculating the velocity at a time step 119905 we need tocalculate the position of free surface from (6) and determinethe potential on the free surface using (5) for the next timestep 119905 + nabla119905 As a result the liquid mesh and the boundarycondition required for the next-time step are establishedThis is done using fourth-order Runge-Kutta explicit timeintegrationmethodThe nodal coordinates of the free surfaceand the associated velocity potential at a current time step 119894

are known and can be represented in a single variable as

119904119894= (119909119894 119911119894 120601119894) (17)

where

119909119894= (1199091 1199092 119909

119873119883+1)119894

119911119894= (1199111 1199112 119911

119873119883+1)119894

119909119894= (1206011 1206012 120601

119873119883+1)119894

(18)

Advances in Numerical Analysis 5

0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

2

3

4

PresentFrandsen

1199051205961

120577(0ℎ)119886 ℎ

(a)

0 10 20 30 40 50 60 70 80 90 100minus4

minus3

minus2

minus1

0

1

2

3

4

120577(0ℎ)119886 ℎ

PresentFrandsen

1199051205961

(b)

minus3 minus2 minus1 0 1 2 3minus3

minus2

minus1

0

1

2

3

120597120577120597119905119886 ℎ1205961

120577119886ℎ

(c)

minus3 minus2 minus1 0 1 2 3minus3

minus2

minus1

0

1

2

3

120597120577120597119905119886 ℎ1205961

120577119886ℎ

(d)

Figure 5 Time history response and associated phase plane plot for case 1

where 119873119883 is number of segments along the free surfaceSimilarly the time derivative can be written as

119863119904119894

119863119905= 119865 (119905

119894 119904119894) = 119865119894 (19)

The free surface position and associated velocity potential atthe next time step 119894 + 1 can be expresses as

119904119894+1

= 119904119894+

1199041

6+

1199042

3+

1199043

3+

1199044

6 (20)

where

1199041= nabla119905119865 (119905

119894 119904119894)

1199042= nabla119905119865 (119905

119894+

nabla119905

2 119904119894+

1199041

2)

1199043= nabla119905119865 (119905

119894+

nabla119905

2 119904119894+

1199042

2)

1199044= nabla119905119865 (119905

119894+ nabla119905 119904

119894+ 1199043)

(21)

After obtaining the new positions and potential of the freesurface the liquid domain is remeshed based on theseobtained new coordinate positions

6 Advances in Numerical Analysis

0 20 40 60 80 100 120 140 160 180minus15

minus10

minus5

0

5

10

15

PresentLinear solution

1199051205961

120577(0ℎ)119886 ℎ

(a)

0 20 40 60 80 100 120 140 160 180minus10

minus5

0

5

10

15

1199051205961

120577(0ℎ)119886 ℎ

(b)

Figure 6 Time history response for case 2

34 Regridding Algorithm At the beginning of the numericalsimulation the free surface nodes are uniformly distributedalong the 119909-direction with zero surface elevation As the timeproceeds the free surface nodes are spaced unequally andcluster into a steep gradient leading to numerical instabilityThis problem occurs for a long time simulation to avoidthis instability an automatic regridding condition using cubicspline is employed when the movement of the nodes is 75more or less then the initial grid spacing For the regriddingfirst the free surface length 119871

119891is obtained Then the free

surface is divided into 119873119883 segments with the identical arclength The coordinates of node is denoted as (119909

119897 119910119897) (119897 =

1 2 119873119883+1) and let the arc length between two successivepoints 119897 and 119897 + 1 be 119878

119897 Being a uniform regridding 119878

119897can be

expressed as

119878119897=

119897119871119891

119873119883 (22)

The coordinates of the nodes (119909119897 119910119897) is a function of the arc

length 119878119897

(119909119897 119910119897) = 119891 (119878

119897) (23)

The cubic spline interpolation is used to calculate the coor-dinates (119909

119897 119910119897) and the velocity potential on the new uniform

free surface is also obtained in a similar fashion

35 Complete Algorithm for Nonlinear Sloshing Including allthe steps above the algorithm for numerical simulation ofnonlinear sloshing is as shown in Figure 3

4 Numerical Results and Discussion

A code is developed following the above numerical formu-lation for computing sloshing response The liquid sloshing

response inside a 20m wide rectangular rigid container withliquid filled to a depth of 1m is simulated Aspect ratio of 05is maintained In the numerical simulations 40 nodes alongthe 119909-direction and 20 nodes along the 119911-direction are taken

41 Free Vibration Analysis To validate the code for stiffnessmatrix formulation a free vibration problem is solved firstA mass matrix 119872 (shown in (24)) for the free surface of theliquid is formed

119872 =1

119892intΓ119904

119873119879119873119889Γ119904 (24)

If 120596119899denotes the 119899th natural slosh frequency of the coupled

system and Ψ119899 the corresponding mode shape the free

vibration problem to be solved is

(119870 minus 1205962

119899119872) Ψ

119899 = 0 (25)

The natural slosh frequencies obtained from (25) arecompared with Faltinsenrsquos analytical solution [4] For arectangular tank the order of natural sloshing frequency is[4]

120596119899= radic119892119896

119899tanh (119896

119899ℎ) (119899 = 1 2 3 ) (26)

where 119896119899is the wave number given by 119896

119899= 119899120587119871 119871 is the

length of the container and ℎ is thewater depth Table 1 showsthe slosh frequencies in rads obtained for the present tankusing finite elementmethod and the above analytical formulaBoth the results are in good match Figure 4 shows the firstfour mode shapes obtained

42 Sloshing Response under Harmonic Excitation In thissection sloshing response is simulated when the tank is

Advances in Numerical Analysis 7

0 10 20 30 40 50 60 70 80 90 100minus60

minus40

minus20

0

20

40

60

80

PresentFrandsen

1199051205961

120577(0ℎ)119886 ℎ

(a)

0 10 20 30 40 50 60 70 80 90 100minus40

minus20

0

20

40

60

80

PresentFrandsen

1199051205961

120577(0ℎ)119886 ℎ

(b)

minus60 minus40 minus20 0 20 40 60 80minus80

minus60

minus40

minus20

0

20

40

60

80

120597120577120597119905119886 ℎ1205961

120577119886ℎ

(c)

minus30 minus20 minus10 0 10 20 30 40 50 60minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50120597120577120597119905119886 ℎ1205961

120577119886ℎ

(d)

Figure 7 Time history response and associated phase plane plots for case 3

Table 1 Slosh frequencies compared with analytical solution

Mode no Present (rads) Equation (26) (rads) Error 1 37607 37594 003532 55456 55411 008013 68092 67986 015534 78715 78510 026205 88126 87777 03983

excited with harmonic motion The tank is assumed tobe subjected to the following forced harmonic horizontalmotion

119883119905(119905) = 119886

ℎcos (120596

ℎ119905) (27)

where 119886ℎis horizontal forcing amplitude 119905 is time and 120596

ℎ

is the angular frequency of the forced horizontal motionEquation (27) gives excitation velocity as minus119886

ℎ120596ℎsin(120596ℎ119905)

which leads to a zero-free surface velocity potential initialcondition from (7) In the present numerical simulation40 nodes along the 119909-direction and 20 nodes along the 119911-direction are taken and a time step of 0003 s is adoptedThe sloshing response is evaluated for various excitation fre-quencies which are closer away and equal to the fundamentalsloshing frequency The free surface behaviour is examinedfor smaller and steeper wave according to Frandsen [10] Thesimulation cases considered are shown in Table 2The resultsobtained are compared with numerical results of Frandsen[10]

8 Advances in Numerical Analysis

119905 = 3 s

(a)

119905 = 36 s

(b)

119905 = 6 s

(c)

119905 = 9 s

(d)

119905 = 12 s

(e)

119905 = 15 s

(f)

Figure 8 Generated mesh in case 3 for large forcing amplitude at various time steps

0 10 20 30 40 50 60 70 80 90 100minus6

minus4

minus2

0

2

4

6

PresentFrandsen

1199051205961

120577(0ℎ)119886 ℎ

(a)

0 10 20 30 40 50 60 70 80 90 100minus6

minus4

minus2

0

2

4

6

PresentFrandsen

1199051205961

120577(0ℎ)119886 ℎ

(b)

Figure 9 Time history response for case 4

Figures 5(a) and 5(b) show the free-surface elevationat the left wall for case 1 for smaller and large horizontalforcing amplitude The external excitation frequency is inoff resonance region the forcing frequency is smaller thanthe first fundamental sloshing frequency The time historiesof the sloshing response are nondimensionalised with thefirst natural sloshing frequency The results obtained are

compared with Frandsen and found to be in good agreementFigures 5(c) and 5(d) show the associated phase plane plotsThese phase plane plots display linear behaviour of the freesurface

Figures 6(a) and 6(b) show the slosh response for case2 for small and large forcing amplitude respectively Inthis case the excitation frequency is closer to the first slosh

Advances in Numerical Analysis 9

0 2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

120596 (rads)

119878 119909(120596

) (m2middots)

times10minus6

Figure 10 Spectrum of oscillation of the tank with 119867119904

= 001 h120596119901= 1205961

Table 2 Simulation cases for the sloshing response

Case Frequency 120596ℎ(rads) Amplitude 119886

ℎ(m)

Smaller wave Steeper wave1 07120596

100052 00520

2 091205961

00039 003933 120596

199 lowast 10minus4 00097

4 131205961

00015 00301

natural frequency and as expected a beating phenomenonis observed For small forcing amplitude the response iscompared with linear solution and both the results are ingood match the response is symmetric in this case displayinga linear behaviour For large forcing amplitude the systemis in nonlinear region due to nonlinearities an asymmetricbeating phenomenon is observed

Figures 7(a) and 7(b) show the slosh response at theleft wall for case 3 for small and large forcing amplituderespectively and the corresponding phase plane plots Inthis case the external excitation frequency is equal tothe fundamental slosh frequency and as expected reso-nance takes place The results obtained are in excellentagreement with the Frandsen results For small ampli-tude case the response is almost symmetric but for largeamplitude case the response is not symmetric because ofnonlinear effects The phase plane plots show clearly thedifference between the small forcing amplitude and largeforcing amplitude A moving mesh generated at differenttype steps for large forcing amplitude case is shown inFigure 8 (Animation of simulation in this case can be seenin httpwwwyoutubecomwatchv=LlwUOWMmVtc) Ithelps in understanding the sloshing flow patterns

Figures 9(a) and 9(b) show the slosh response at theleft wall for case 4 This case is also an off resonance caseas case 1 but the forcing frequency is higher than the first

natural sloshing frequency The results obtained are in goodagreement with Frandsen

43 Sloshing Response under Random Excitation In this sec-tion sloshing response is simulated for random excitationsFor simulating random sloshing first a random excitationtime history is needed The required random excitation isgenerated using Bretschneider spectrum

119878120578=

51198672119904

16120596119901

(120596119901

120596)5

exp [minus5

4(120596119901

120596)4

] (28)

where 119867119904is the significant wave height and 120596

119901is the peak

frequency Since the higher frequencies have no influence onthe sloshing waves a cut of frequency for random waves isset In this simulation the cut of frequency is taken as fivetimes the natural slosh frequency Based on this spectrumthe random waves are generated which are given as the baseexcitation to the container The displacement of the randomwave can be obtained by linear superposition of a series ofharmonic waves with random phase as a time function

119909 (119905) =

119873120596

sum119894=1

119860119894sin (120596

119894119905 + 120601119894) (29)

where 119909(119905) denotes a random horizontal oscillation that thecontainer is subjected to 119905 is time 120596

119894is the frequency of 119894th

linear wave and 119873120596is the number of all the linear harmonic

waves The frequency 120596119894ranges from 0 to 120587119889119905 119860

119894and 120601

119894

are the wave amplitude and phase of each linear harmonicwave respectively The wave amplitude is determined by thefollowing equation

119860119894= radic2119878

120578(120596) Δ120596 (30)

where Δ120596 is the frequency interval The phase 120601119894is a random

variable and uniformly distributed in the interval [0 2120587]The specified spectrum of oscillation with 119867

119904= 001 h

and peak frequency 120596119901

= 1205961is shown in Figure 10 To

generate random wave from the shown spectrum 119873120596is

taken as 512 1024 data points are taken and a time step of002 s is adopted The horizontal oscillation of the containercorresponding to the spectrum is shown in Figure 11(a)Figure 11(b) shows the corresponding time history of the sloshresponse at the left wall of the container

5 Conclusion

Sloshing response of liquid in 2D fixed and forced tanks isinvestigated numerically considering fully nonlinear equa-tions A mixed Eulerian-Lagrangian nonlinear finite elementnumerical model has been developed based on potential flowtheory Free-surface sloshing response is simulated underregular harmonic base excitations for small and steep wavesand then the formulation is extended to random excitationsIn the simulations the tank is assumed to be rigid withaspect ratio of ℎ119871 = 05 For accurate velocity computationthe velocity field is interpolated from the velocity potential

10 Advances in Numerical Analysis

0 20 40 60 80 100minus5

0

5times10minus3

119883119905ℎ

1199051205961

(a)

0 20 40 60 80 100minus004minus002

0002004

1199051205961

120577ℎ

(b)

Figure 11 (a) Random horizontal excitation (b) slosh response at the left wall of the container

according to least square method The fourth-order Runge-Kutta method is employed to advance the solution in timedomain and a regridding technique based on cubic splineis employed to avoid numerical instabilities The test casesconsidered here are similar to the test cases by Frandsenfor regular harmonic excitations The results obtained arein perfect match with Frandsen results In the presentsimulation no regridding is required for the simulation ofsmall amplitude waves In the case of steep waves for longtime simulation regridding should be carried out for everyfew time steps

References

[1] H N Abramson ldquoThe dynamic behaviour of liquid in movingcontainersrdquo NASA Report SP 106 1996

[2] M A Haroun ldquoDynamic analysis of liquid storage tanksrdquo TechRep EERL 80-4 California Institute of Technology 1980

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

[4] O M Faltinsen ldquoA nonlinear theory of sloshing in rectangulartanksrdquo Journal of Ship Research vol 18 pp 224ndash241 1974

[5] T Nakayama and K Washizu ldquoThe boundary element methodapplied to the analysis of two-dimensional nonlinear sloshingproblemsrdquo International Journal for Numerical Methods inEngineering vol 17 no 11 pp 1631ndash1646 1981

[6] G X Wu and R E Taylor ldquoFinite element analysis of two-dimensional nonlinear transient water wavesrdquo Applied OceanResearch vol 16 pp 363ndash372 1994

[7] G X Wu and R E Taylor ldquoTime stepping solutions of thetwo-dimensional nonlinear wave radiation problemrdquo OceanEngineering vol 8 pp 785ndash798 1995

[8] W Chen M A Haroun and F Liu ldquoLarge amplitude liquidsloshing in seismically excited tanksrdquo Journal of EarthquakeEngineering and Structural Dynamics vol 25 pp 653ndash669 1996

[9] 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

[10] J B Frandsen ldquoSloshing motions in the excited tanksrdquo Journalof Computational Physics vol 196 no 1 pp 53ndash87 2004

[11] J R Cho and H W Lee ldquoNon-linear finite element analysisof large amplitude sloshing flow in two-dimensional tankrdquoInternational Journal for Numerical Methods in Engineering vol61 no 4 pp 514ndash531 2004

[12] 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

[13] 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

[14] K C Biswal S K Bhattacharyya and P K Sinha ldquoNon-linear sloshing in partially liquid filled containers with bafflesrdquoInternational Journal for Numerical Methods in Engineering vol68 no 3 pp 317ndash337 2006

[15] R A Ibrahim V N Pilipchuk and T Ikeda ldquoRecent advancesin liquid sloshing dynamicsrdquo ASME Applied Mechanics Reviewvol 54 pp 133ndash199 2001

[16] M S Longuet-Higgins and E D Cokelet ldquoThe deformationof steep surface waves on water I A numerical method ofcomputationrdquo Proceedings of the Royal Society A vol 350 no1660 pp 1ndash26 1976

[17] O C Zienkiewicz and J Z Zhu ldquoThe superconvergent patchrecovery and a posteriori error estimates I The recoverytechniquerdquo International Journal for Numerical Methods inEngineering vol 33 no 7 pp 1331ndash1364 1992

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