Numerical simulation of 2D sloshing waves due to horizontal and vertical random excitation

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Applied ocean

Research

E L S E V I E R Applied Ocean Reseaich 28 (2006) 19-32

www.elsevier.com/locate/apor

Numerical simulation of 2D sloshing waves due to horizontal and vertical'

random excitation

V. Sriram, S.A. Sannasiraj*, V. Sundar

Department of Ocean Engineering, Indian Institute of Technology Madras, Cliennai 600 036, India Received 17 October 2005; received in revised forai 13 January 2006; accepted 14 January 2006

Available online 5 June 2006

Abstract

The motion of sloshing waves under random excitation in the sway and heave modes have been simulated in a numerical wave tank. The fully nonlinear wave is numerically simulated using the finite element method with the cubic spline and finite difference approximations, in which the need for smoothing and regridding is minimal. The present model predictions are compared with that of Frandsen [Frandsen JB. Sloshing motions in the excited tanks. J Comput Phys 2004;196:53-87] for the regular wave excitation in the vertical and horizontal modes. The sloshing due to the simulated random excitation with different peak frequencies relative to the natural sloshing frequency has been subjected to frequency domain analysis. The results showed that iiTespective of peak excitation frequency, the peaks appear at the natural frequencies of the system and the peak magnitude appears close to the natural frequency for the sway excitation. The higher magnitude is seen when the excitation frequency is equal to the first mode of natural frequency, due to the resonance condition. I n the case of heave excitation, even though the peaks appear at the natural frequencies, the magnitude of the spectral peak remains the same for different excitation frequencies.

Keywords: Sway; Heave; Fully nonlinear wave; Finite element method; Cubic spline approximation

1. Introduction

Sloshing is a phenomenon w h i c h still needs considerable understanding i n its behavior related to various engineering problems and is a composition of highly nonlinear waves that may lead to stmctural damage o f the side walls o f the container and/or may destabilize the ship. Numerous studies have been canied out i n understanding sloshing waves using analytical, experimental and numerical approaches. The numerical approach has been canied out using t w o -dimensional and thuree- -dimensional tanks. I n the case o f two-dimensional tanks, Faltinsen [5,6] derived the analytical solution f o r sway and roll/pitch motions using the perturbation approach; Okamoto and Kawahara [19] and A r m e n i o and L a Rocca [1] compared the results o f numerical and experimental data; Nakayama and Washizu [18] presented a numeiical approach f o r forced pitching oscillation o f the l i q u i d tank;

* Conesponding author. Tel.: +91 44 2257 4817; fax: +91 44 2257 4802/4801.

E-mail address: sasraj@iitm.ac.in (S.A. Sannasiraj).

and Frandsen [10] analysed the sloshing motions i n the vertical, horizontal and combined motions o f the tank using analytical and numerical approaches. For the case o f the three-dimensional tank, W u et al. [24] numerically simulated the sloshing waves and Huang and Hsiung [12] used the shallow water equations f o r the flow on the ship deck. Recently, Faltinsen et al. [8] performed the experimental investigation i n a three-dimensional tank and compared the experimental results w i t h the asymptotic modal system f o r longitudinal and diagonal excitations. Steady state regime f o r diagonal excitations versus water depth, excitation frequency and amphtude are classified.

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20 V. Sriram et al./Applied Ocean Research 28 (2006) 19-32

the vertical excitation. To have an i n i t i a l perturbation i n the free surface inside the container, horizontal motions need to be excited before the vertical excitation. A detailed review on sloshing o f Faraday waves has been reported by M i l e s and Henderson [ 1 7 ] . Jiang et al. [15] and Bredmose et al. [3] have carried out experiments f o r the Faraday waves. A considerable amount o f w o r k has been carried out based on sloshing due to the horizontal excitation. The focus o f many studies has been o n the earthquake induced sloshing. I b r a h i m et al. [13] gives a comprehensive review on sloshing motion predictions w i t h more than 1000 references.

M o s t o f the work related to sloshing waves is based o n the regular- excitation o f the container This is quite useful f o r understanding the physical phenomenon o f sloshing waves. However, these waves are not regular- but random i n nature. Recently, Wang and K l i o o [22] have investigated the sloshing i n a container due to random excitation i n the horizontal directions. This w o r k explored the knowledge towards the analysis f o r random excitation.

The need f o r numerical modeling arises due to the significant importance o f the higher order effects f o r the sloshing waves w h i c h are nonhnear, so neither the linear nor the second order potential considerations are sufficient enough to describe steep waves [10]. The advantage o f numerical m o d e l l i n g is the fiexibility i n the simulation o f the real sea state situation that has been carried out herein. The numerical approach o f the present paper is similar to that o f the f o r m u l a t i o n adopted b y W u and Eatock Taylor [23] i n two dimensional numeiical wave tank simulations. Herein, the finite element method ( F E M ) is used to find out the velocity potential, whereas, f o r the recoveiy o f the velocity, a new f o r m u l a t i o n based on a cubic spline and finite difference approach has been adopted. The m a i n advantage o f this modification is that i t reduces the need f o r smoothing or regridding f o r small amphmde waves. Sriram et al. [20] gives the details o f its validity w i t h the error estimation f o r the present scheme.

The viscous effects play an important role after a critical depth [ 7 ] . Based on the dimensions o f the taiilc, i n the case o f horizontal excitation, W u et al. [24] reported the transition f r o m standing wave f o r m to progressive waves i n the f o r m o f a bore. Hence, the dimensions o f the tank are the influential parameters i n the study o f sloshing waves. A s this paper mainly deals w i t h the effect of random horizontal and vertical excitations, the dimension o f the tank is assumed to be of constant length ( L ) o f 2 m and depth (/:) o f 1 m , such that the depth aspect ratio (h/L) is 0.5. The present numeiical model is i n i t i a l l y vahdated w i t h the numeiical w o r k of Frandsen [10] f o r regular motions and then applied f o r the computation o f the sloshing due to random excitation, w h i c h is elaborated i n detail. I t is w o r t h w h i l e to note that the results presented herein give rise to more questions, w h i c h motivated the authors f o r further investigations.

2. Mathematical formulation

Let Oo'^-'O^O be the fixed coordinate system and Oxz be the m o v i n g coordinate system fixed w i t h the tank. These t w o coordinate systems coincide w i t h each other when the tank is at

rest. The origins of this system are at the l e f t end o f the tank w a l l at the free surface and pointing upwards i n z direction. These t w o coordinates along w i t h the prescribed boundary conditions i n each coordinate are represented i n F i g . 1(a) and (b).

The displacements o f the tank are governed by the directions o f axes as,

Xt =[xt{t),zt{t)l (1) O n the assumption that the fluid is governed by potential flow theory, the velocity potential rj) satisfies the Laplace equation, V^(* = 0 i n the fluid domain, Ü. (2) The component o f the water particle velocity normal to the walls o f the tank is equal to the tanlc velocity

dn

= U.n i n the side walls, FB

₍₃₎

where U = ^ is the velocity o f the tank and n is outward normal to the tank walls.

The dynamic and kinematic free surface boundary condition i n the fixed coordinate system can be written as,

- V ( ^ . V 0 + g)?o = 0 dt 2 9'?o , 9'?o _0. (4) (5) dt dxQ 9x0 9^0

The free surface m o t i o n can be described i n the m o v i n g coordinate system as,

M _ V < / . . ^ + + gin + z,) = Q

dt At 2 di) (dip Ax,\ di] Jt ~^ \ dx dt

)

' dx

dip dzt

97 " d7 = 0

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(7) w h i c h is obtained by substituting Eqs. (8) and (9) i n Eqs. (4) and (5), d t ) , _ dt dXt " d T V (8) (9)

on z = n, where i] = i]^ — zt is the free surface elevation i n the m o v i n g system

Oxz-Now, let the velocity potential be decomposed into.

(p + XII + ZU) (10) i.e., the velocity potential i n the fixed coordinates system contains the velocity potential i n the m o v i n g coordinates and the direction o f the excitation w i t h the corresponding velocity i n that direction, where u and w are the velocity components i n the X and z directions. Substituting Eq. (10) i n Eqs. (2), (3), (6) and (7) leads to.

V ^ ^ = 0 i n the fluid domain, Q

da>

— = 0 i n the side walls, FB •

dn

(3)

(a) Fixed coordinate system.

The dynamic and Idnematic free surface condition becomes,

^ 0 (13)

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d(p 1

— + - . V ^ . V ^ + gn + x 4 + I)ZT d,)

(b) Moving coordinate system. Fig. 1. Sloshing wave tank modal domain.

1.4 1.2

d(p dl) dtp

R e w r i t i n g the above equations i n Lagrangian f o r m o f m o t i o n f o l l o w i n g [ 1 6 ] , dtp 1 - = -Vtp.Vtp •z'^it))t)-xx'^(t) dx dr dcp a 7 dz dr and — = — dtp a7 w h e r e 4 ( r ) = ^ , 4 ( r ) = | c .

The i n i t i a l condition o f the system can be assumed as,

(p{x^,z„t)^0 'Ï0 ('^'o. 0 = ^0 where r = ^ = 0; (15) (16) (17) (18) In m o v i n g coordinates, at / = 0 and z = 0, the above equations become.

<p{x,0,fy) = -XU

i)(x, 0) = 0 f o r horizontal excitation i){x, 0) = f o r vertical excitation

(19) (20a) (20b) where is the non-physical condition, specifying the i n i t i a l elevation.

Based on the i n i t i a l condition, the boundary value problem is solved and the free surface elevation and potential values are updated at the subsequent time steps. The E q . (12), (15) and (16), f o r m the boundary conditions. Thus, this f o r m of boundary conditions reduces the computational burden o f creating a finer mesh stmcture along the input boundary, since the tank excitation is incorporated i n the kinematic and dynamic free surface boundary condition.

3. Numerical procedure

3.1. FEM formulation

The solution f o r the above boundary value problem is sought using a finite element scheme i n this paper Formulating the

z(m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x(m)

Fig. 2. A typical mesh structure of the computational domain using three noded triangular element.

governing Laplace's equation constrained w i t h the associated boundary conditions w o u l d lead to the f o l l o w i n g finite element systems o f the equation [ 2 0 ] ,

^NiJ2'Pj^Njdn\jj^r^

r J"

= - lyNiJ2<Pj^Njd%er.j^r. (21)

where is the total number o f nodes i n the domain and the potential [(p(x, z)] inside an element can be expressed i n terms of its nodal potentials as tpj

n

(p(x,z) = ^ ( p j N j ( x , z ) . (22) J = l

Herein, Nj is the shape f u n c t i o n and n is the number o f nodes i n an element. The above formulation does not have any singularity effect at the intersection point between the free surface and the boundaries, compared to other methods tike B E M or B I E M [ 2 3 ] . A linear triangular element is used. The mesh structure is shown i n F i g . 2.

3.2. Cubic spline and finite dijference approximations

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22 V. Si irain et al. /Applied Ocean Research 28 (2006) 19-32

several investigators [11,24,22]. However, frequent smoothing or regridding w o u l d lead to non-conservation o f energy and there is a need to l i m i t the approximation procedures. I n the present study, the horizontal velocity is calculated b y f i t t i n g a cubic spline f o r 'x'-coordinates and (p(x,z) values. The end conditions are considered as the natural spline condition. To evaluate the derivative at the (th node, five nodes are considered (two nodes on either side o f the / t h node), i n order to m i n i m i z e the effect o f boundary constraints (natural spline condition). I t ensures the smooth first derivatives. The computational time is also f o u n d to be efficient.

Consider ƒ , , ƒƒ and ƒ . " are continuous over the given interval. Based on the continuity condition,

- ^ f i - i + J 1 Sxi+i = 2, 3 (fi+i - f i ) . . . k - i . - f , + 1 Sxi i+l i f - f i - l ) (23) The above equation leads to a set o f (k - 2) linear equations f o r the 7c' unknowns and Sx is the horizontal spacing between the t w o nodes. I t is solved by using the tridiagonal system o f matrix assuming the second derivatives at the ends are zero i.e., the natural spline condition. I n the present simulation, assuming

f i = (pi, the derivatives at a particular node ( ^ , ) is f o u n d out

b y taking t w o nodes on either side {k = 5) w i t h the second derivatives {<p"_2, V^-i) at the end nodes being zero. A f t e r the evaluation o f the second derivatives, the first derivatives can be estimated using the f o l l o w i n g equation, w h i c h are derived i n the intermittent steps o f the cubic spline interpolation [14]

2 / ; " ^-!U^ = _S.Xi

fi+l - fi

Sxi (24)

O n the other hand, the vertical velocity can be estimated based on the backward finite difference ( F D ) scheme taldng the advantages o f distributing the nodes i n the vertical line during mesh generation [ 2 1 ] . Consider (pi's are the velocity potentials at the nodes coiTesponding to Zi, where / = 1, 2, 3. Then, the vertical velocity at the free surface node can be obtained as.

chp _ ( « ^ - l)(pi - a'^(p2 + (P3 dz a{a - l ) ( z i - Zl)

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where a = .

The algorithm f o r the n u m e i i c a l procedure is as follows, (1) Assume the i n i t i a l velocity potential and surface elevation

using Eqs. (19) and (20).

(2) The entire domain is subdivided into finite elements and the nodes are numbered w i t h proper element connectivity. (3) U s i n g Eq. (21) solve f o r the velocity potential inside the

domain.

(4) Recover the velocity using the cubic spline and F D approximations i n the free surface and check f o r the requirement o f regridding.

(5) Update the free sutface nodes using Eqs. (15) and (16), the integration is carried out using the f o u r t h order

Runge-Kutta method w h i c h require the evaluation o f steps 3 and 4 thi'ee more times to obtain the new position. (6) A f t e r finding out the new free surface position and velocity

potential p e r f o r m step 2 t i l l the termination time is reached.

4. I n p u t generation

4.1. Regular wave excitation

The container is assumed to take the f o l l o w i n g horizontal and vertical oscillations,

xrit) = ai,H(coi,t) zrit) =ayV{coyt)

(26a) (26b) where fl/i,u and are the characteristic amplitude and frequency o f the wave. The m o t i o n o f the container is assumed as V(x) = Hix) = c o s ( x ) .

For the rectangular container, the order o f the natural frequency is [5]

CO,, y/gk,, taiih(/c„/)) ;?. = 1 , 2 , 3 . . . (27)

where the wave number is given by k,, = nn/L, L is the length o f the container and /; is the water depth.

4.2. Random wave excitation

I n this paper, the Bretschneider spectrum is used as the input wave spectrum w h i c h is given by,

S,M) = Tzr^ ( ^ ) exp - - ( ^ ) 16aj„ V ft) / 4 V ftj /

5 "4

The free surface elevation is described by.

'1 = ^ A,' cos(ftj'V + f ' ) ( = 1

(28)

(29)

where A,- is the amplitude that is defined by A ; = y2S',,(ft)) Aft).

Nw is the number o f sinusoidal wave components, OJ' and i/f'

are the frequency and the phase angle, wherein the frequency ranges f r o m 0 to n/dt and phase angle (a random variable based on the fixed seed number is used i n this paper) ranges f r o m 0 to 2jt. Herein, ft)/ and Hs are the peak frequency and significant wave height. I n this paper, the range o f frequency has been taken up to the c u t o f f frequency that is assumed to be five times the natural frequency o f the container; since the higher frequency w i l l not have much influence o n the generated waves. Based on the spectrum, the random waves are generated w h i c h are given as the input to the oscillation o f the container; thus, E q . (29) is assumed to be

Nw

XT it) = ^ Ai cos{üj't + f ) r=\

f o r horizontal random oscillation

ZT(t)^^AiCOS{oJt + i f ' ) (•=1

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(a) 3 r — — — — — I (b) 3

Fig. 3. Free surface elevation at the left wall due to regulai'liorizontal excitation of frequencies, ft)/, = 0.7ftJi. (a) fl/,ftj,^ = 0.0036g; (b) fl/,ft)^ = 0.036g. (—present numerical, • • • Frandsen [10] numeiical).

0 1 2 3 0 1 2 3

C D / C O l C O / C O i

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/

/ /

_/

/

1

/

/ /

/

/ 1

/

/ 0 0.5 1 1.5 2 X{m)

Fig. 5. Generated mesh due to horizontal excitation after every 5 s interval for a;, = 0.005/i and a)/, is 13.

number of nodes in .y direction is 31 and in z direction

5. Results and discussion

5.1. Regular wave excitation

5.1.1. Horizontal excitation

Tiie n u m e i i c a l simulation o f regular waves using the present methodology is compared w i t h the numeiical results o f Frand-sen [10]. The displacement o f the container is given by the E q . (26a) resuhs i n the excitation velocity as - f l / , ( U / , sin(ft)/,r). This leads the i n i t i a l condition f o r velocity potential (Eq. (19)) to zero and the suiface elevation is considered as i n Eq. (20a) f o r this mode o f excitation. For only horizontal excitation to exist, the vertical acceleration Zj (t) is assumed to be zero. Compar-isons have been made f o r a smaller and a steeper wave w i t h the n u m e i i c a l simulation o f Frandsen [10] (Fig. 3). I t can be observed that both simulations are i n close agreement f o r the f o r c i n g frequency ( « / , ) equal to seventy percent o f the first mode o f natural frequency ( « i ) . There are 31 nodes i n the x direction (free surface) and 13 nodes i n the z dhection and the time step adopted is 0.01 s f o r both simulations. A n automatic regridding condition as proposed by D o n n n e i m u t h and Yue [4] is adopted i n the present model when the movement o f the nodes is 7 5 % more or less than the i n i t i a l g r i d spacing. Smooth-ing is not adopted i n the present simulation. I t is f o u n d that no regridding is required f o r the simulation o f small amplitude waves. I n the case o f steep waves, the need f o r regridding arises about 20 times f o r a simulated duration o f 61.44 s.

The induced sloshing time series is subjected to frequency domain analysis f o r a range o f excitation frequencies w i t h excitation amplitude o f aj, = 0.005/) f r o m w h i c h the occurrence o f spectral peaks are identified. The spectral

density o f free surface elevation i n the tank f o r the excitation frequencies, &>/, = 0 . 3 5 o j i , coj, — O.lScoi, co;, = « i , coi, =

1.56J1, « / , = l.Ooji and ojj, — co^, are depicted i n F i g . 4. The m a x i m u m spectral peak occurs at the excitation frequency when the excitation frequency is less than the natural frequency and a secondary peak occurs at the container natural frequency as can be seen i n F i g . 4(a) and (b). W h i l e the excitation frequency is more than the container natural frequency and up to the second modal frequency [coi < coi, < 0J2 {—l.Scoi)], the primary peak occurs at the first mode f o l l o w e d by a secondary peak at the third natural frequency ( F i g . 4(d)). For excitation frequencies equal to or greater than 1.8a)i ( = « 3 ) , the third modal frequency dominates the sloshing motion. F r o m F i g . 4(c) and ( f ) , the sloshing m o t i o n is more violent at the natural frequency o f the container when the excitation frequency is equal to the first mode rather than at the third mode, w h i c h is a w e l l - k n o w n resonance phenomenon. The typical m o v i n g mesh generated after every 5 s interval is shown i n F i g . 5 f o r the above said resonance condition.

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3000 1 2000 H M 1000 H 15 on 0.5

I Spectral peak at 1st mode I Spectral peak at excitation mode } Spectral peak at 3id mode

-region I

1

region I I

1

region I I I Fig (a)

-1 1 -

1 1, ,

1.5 Q)|/COi 0.2 0.3 0.4 0.5 0.6 0.7 (Oh/CO, 2.5 0.9 2740 1 _{1 1 1 1} Zoomed view of region I 1 Fig (b) 1 _ 1 1 in 1 I 1 1 -1.1 t=; c^l

?

1.1 1.15 1.2 Zoomed view of region I I 1.25 1.3 1.35 (B,/CO| Fig (c) 11.1 liJI 1.4 1.45 1.5 1.55 H c-l

?

OT

60 40 20 0 450 485 156 162 102 > k J V

hllll ,1

II

1

1 1 1 1 Zoomed view of region I I I

i l ,

1

1 1 Fig (d) _ , 1 ^ 1.6 1.7 1.; 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6

Fig. 6. Bar chart showing the magnitude of different spectral peaks of sloshing waves due to horizontal excitation with frequencies (0)2 = 1.5a)i, 013 = I.8&J1, « 5 = 2.334(Di) (For interpretation of color legends in the figure, reader can refer the online version of this paper).

Fig. 6(b). The response component at the first modal frequency is observed to increase as the frequency increases. W h e n the excitation frequency is greater than the first mode, the sloshing dominates at the first modal frequency up to second mode ( « 2 = l . S w i ) and till this frequency, the secondaiy peak is observed at the excitation frequency as can be seen i n Fig. 6(c). W h e n the excitation frequency is greater than CDI, the domination o f the first mode reduces w i t h an increase i n the frequency ratio. W i t h a further increase i n the excitation frequency ratio (co/, > = l.5coi), the spectral peak at third mode increases rapidly to 450 and 485 at a>i, — l.llcoi and

u>k = 1.8cui respectively. This is clearly seen f r o m F i g . 6(d). Furthermore, f o r the range o f excitation frequency ratio 1.5 to

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26 V. Sriram et al/Applied Ocean Research 28 (2006) 19-32

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 SO

tCD, tCO.

Fig. 7. Free surface elevation at the left wall due to regular vertical excitation of frequency, = 0.789coi, OUOJI = Q.Sg, (a) E = 0.0014 (b) s = 0.288, (— present numerical, Frandsen [10] numerical).

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1 0.8 0.6 • 0.4 • 0.2 • 0 -(a) co^,=0.75co, - — / \ . 1 1 , 1 I (d) to„=2co, C O / C O i C O / C O l

Fig. 9. Power spectta of fiee surface sloshing waves at the left corner of the wall due to regular vertical excitation for an initial steepness of 0.288, o„ = 0.005/; ( r i = 2 7 r / c « i ) .

excitation is equal to the first mode and the next largest being 5.65 times lesser occuriing at CÜT,.

5.1.2. Vertical excitation

To simulate the condition of vertical excitation f o r the container, x'^ is set to zero i n Eqs. (13) and (15). The tank is assumed to be periodically excited w i t h the displacement given by Eq. (26b), w h i c h leads to a velocity o f the f o r m - f l u W u sin(ft)uf). Thus, the i n i t i a l velocity potential becomes zero according to E q . (19). The i n i t i a l condition f o r the surface elevation is an important parameter as there should be some i n i t i a l perturbation i n the system f o r the generation o f waves due to vertical excitation. The linear solution and the stability criteria o f Faraday waves are given by B e n j a m i n and U r s e l l [2]. I n the numerical simulation, i t is quite usual to adopt an i n i t i a l free surface perturbation o f ?; = fo = acosiknx). I t should be noted that this condition does not necessarily arise i n

the real situation because both horizontal and vertical motions co-exist. For experimental purposes, the i n i t i a l perturbation is introduced i n the system, by exciting the tank horizontally for the prescribed time. Two typical sloshing simulations are canied out, f o l l o w i n g small amphtude m o t i o n and steep wave excitation as shown i n Fig. 7. The steepness parameter depends on the adopted i n i t i a l condition, e = aa>}jg. The present simulation shows a close agreement w i t h the numerical simulation o f Frandsen [ 1 0 ] . The grid size and the time step adopted are the same as that f o r horizontal m o t i o n .

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28 V. Sriram et al./Applied Ocean Research 28 (2006) 19-32

100 110

10 20 30 40 50 60 70 80 90 100 110

tO), 0 10 20 30 40 50 60 _tco, 70 80 90 100 110

Fig, 10. (a) Typical excitation spectrum with Hs = O.Ol/i and &)p = . (b) Displacement generated from the spectrum, (c) Free surface elevation at tlie left corner of the wall due to horizontal motions prescribed by (b). (d) Free surface elevation at the left corner of the wall due to vertical motions prescribed by (b).

and 0.288 w i t h constant excitation amplitude are presented i n Figs. 8 and 9 respectively. Such plots have been reported f o r = 0.75a)i,a>i, \.5a>\,2oj\ and 2.5oJi. A comparison o f the results presented i n the above t w o figures reveals the f o l l o w i n g . I n the case o f lesser i n i t i a l steepness, a single peak o f the dimensionless sloshing energy is observed at frequency ratio o f 1, iirespective o f the magnitude o f the excitation frequency. I n the case o f the higher i n i t i a l steepness adopted i n the study, the dimensionless sloshing frequency spectra exhibits secondary peaks at ID/coi = 1.5 and 2 f o r all the ojy tested. The magnitude o f the secondary peak is insignificant compared to the p r i m a r y peak at the resonance frequency. Frandsen [10] noticed a similar existence o f peaks i n the wave spectra. 5.2. Random wave excitation

5.2.1. General

I n order to understand the sloshing phenomenon, the container is subjected to random excitation under the real random sea state, w i t h different peak frequency by keeping the total supplied energy to the system as constant. A typical input excitation spectrum and the coiresponding displacement time history are given i n F i g . 10(a) and (b) respectively. The horizontal and vertical container displacements are obtained f r o m this spectrum (Eqs. (30) or (31)). The smface elevations at the l e f t w a l l o f the container due to horizontal (ri/li)H and vertical ()]//?)y excitation are shown i n F i g . 10(c) and (d) respectively. The time step adopted is 0.06 s and the duration o f the simulation is 61.44 s corresponding to 1024 data points, w h i c h is used to analyse the spectrum herein.

5.2.2. Horizontal excitation

For a significant wave height o f O.OO6/7, the tank is excited w i t h random waves o f different peak frequency (cop = 0.35a)i, ojp — 0.75(01, COp = coi,(Op = 1.5 0Ji,cOp = oj^). The power spectra o f free surface sloshing elevation at the l e f t corner o f the tank w a l l f o r various excitation peak frequencies are shown i n F i g . 11. I t can be seen that w h i l e the excitation peak frequency is less than the natural frequency, the sloshing spectral peaks appear only at natural frequencies (first, third and fifth mode) o f the tank and no peak is visible at the excitation peak frequency as noticed i n the case o f regular excitation. The primary spectral peak lies at the first mode. W h e n the excitation peak frequency is greater than the first mode o f the tank system, there were some h i g h frequency waves i n the container (Fig. 11(d), (e) and ( f ) ) . Thus, i n horizontal excitation, the spectral peaks appear only at the first or higher mode natural frequency, irrespective o f the excitation peak frequency. A sinnlar trend i n the occurrence o f spectral peaks is also noticed by Wang and K h o o [ 2 2 ] . The spectrum f o r forces and surface elevation o f the tank w a l l was examined using F E M adopting isoparametric elements. It has been shown that the energy m a i n l y concentrates at the natural frequency o f the container and is f o u n d to domhiate at the iih mode o f the container when the peak frequency is close to the (th mode.

5.2.3. Vertical excitation

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Fig. 11. Spectra of free suiface sloshing waves at the left comer of the wall due to horizontal random excitation {Hs = O.OO6/1, =2TX/OJI).

vertical excitation w i t h the same input characteristics as that f o r horizontal motion f o r two different i n i t i a l conditions. The i n i t i a l conditions coiTespond to small amplitude w i t h a steepness o f 0.014 and large amplitude w i t h a steepness o f 0.288. The power spectra o f waves at the l e f t corner o f the tank w a l l w i t h the above i n i t i a l conditions having steepness o f 0.014 and 0.288 are shown i n Figs. 12 and 13 respectively. I t can be seen that inspite o f the excitation frequency, the spectral peaks appear at the first modal frequency o n l y w h i l e the i n i t i a l pertarbed waves are o f small steepness. The spectral peaks appear at first, second and t w o times the first mode (parametric) frequency, i f the i n i t i a l perturbed waves are o f large steepness, w h i c h is similar to that o f the regular m o t i o n as discussed earlier f o r the heave mode. The dominating frequency appears at the first mode. However, the spectral peak magnitudes at the first modal frequency ai-e o f the same order, iiTespective o f excitation peak frequency. This phenomenon is observed iirespective of the different i n i t i a l

perturbation i n the tank. I t should be noted that no regridding is necessary f o r the small steepness case, and hence the numerical damping does not play any role f o r the above results. However, i n the case o f regular m o t i o n , the magnitude o f peak is higher at = 2 w i f o r small and high i n i t i a l steepness.

6. S u m m a r y and conclusion

This paper presents the behavior o f the sloshing waves i n a tank subjected to excitation i n the horizontal and vertical directions using the inviscid flow solver B o t h regular and random excitations have been considered. The salient conclusions drawn f r o m the study are presented below.

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30 V. Sriram et al/Applied Ocean Research 28 (2006) 19-32

1 2 C O / C O i

Fig. 12. Spectra of free suiface sloshing waves at the left comer of the wall due to vertical random excitation for an initial steepness = O.OO6/1 (Ti = 23T/a>i).

container. Tiie critical sloshing m o t i o n occurs at the w e l l k n o w n resonance condition, w h i l e the exchation frequency is equal to the first mode.

Due to regular excitation i n the vertical direction, the spectral peak appears to be at the first mode f o r small i n i t i a l perturbation and at first, second and two times of the first mode f o r higher initial perturbation f o r constant excitation amplitude. The dominating spectral peaks i n both the cases are at the first mode. The critical sloshing occurs at the excitation frequency equal to twice the first mode, w e l l k n o w n as the parametric resonance.

For the random excitation i n the horizontal direction, the spectral peaks occur only at the natural frequencies o f the container, the dominating peak appear close to r'th mode, when the excitation peak frequency is at r'th mode. The m a x i m u m intensity appears to be at the excitation peak frequency equal to the first mode that is, resonance condition.

I n the case of random excitation i n vertical direction, the dominating peak appears only at the first mode. The magnitude o f the peak is almost the same irrespective of the excitation peak frequency and i n i t i a l perturbation, contrary to the regular excitation, wherein the magnitude is large only i f the excitation frequency is equal to twice the first mode (parametric resonance) urespective o f i n i t i a l perturbation.

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2.5-^ 2 • H i : " 1.5 •

I

1 -C O 0.5 0 2.5 -(a) C0p=0.75co, 1 2 C O / C O i 2.5

Fig. 13. Spectra of free suiface sloshing waves at the left corner o f the wall due to vertical random excitation for an initial steepness of 0.288 and Hs = O.OOó/i (7"| = 2jT/a>i).

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