Nonlinear transient wave motions in base-excited rectangular tanks

Journal of Fluids and Strucfures (1992) 6,471-491

NONLINEAR TRANSIENT WAVE MOTIONS IN BASE-EXCITED RECTANGULAR TANKS

A. N. WILLIAMS AND X. WANG

Department of Civil & Environmental Engineering, University of Houston Houston, TX 77204-4791, U.S.A.

(Received 26 July 1990 and in revised form 15 October 1991)

A nonlinear, dispersive, dissipative shallow water theory in two horizontal dimensions is developed to study the waves produced by the bidirectional, translational, oscillatory motion of a rectangular tank partially filled with liquid. An efficient finite difference technique is used to solve the resulting system of equations. Numerical results are presented which illustrate the form of the generated wave profile and its development in time for a range of geometric and excitation parameters. The computed results show the influence of the direction, duration and frequency of excitation, liquid depth and dispersive effects on the wave motions.

1. INTRODUCTION

THE STUDY OF THE LIQUID MOTIONS in an oscillating rigid container has been the subject of many studies in recent years, due to its frequent application in several engineering disciplines. Interest in the present case is focused on the finite-amplitude transient waves generated in an oscillating rectangular container partially filled with a liquid whose mean depth is small compared with the horizontal dimensions of the tank. The motivation for this study is the use of such liquid-containing tanks as tuned, sloshing dampers to suppress the wind-induced response of tall buildings (Kareem 1989). Clearly the success of this approach to structural motion mitigation depends crucially on the availability of a computational model which accurately and efficiently describes the nonlinear transient behavior of the oscillating liquid in the tank. It is this aspect of the problem that the current work seeks to address. The integration of the hydrodynamic model into a dynamic analysis package is currently underway and will be reported in the near future.

The problem of shallow water wave motions in oscillating rectangular tanks has been the subject of several previous investigations. Verhagen & Wijngaarden (1965) conducted both a theoretical and experimental study of the finite-amplitude, steady- state fluid motions in an oscillating shallow rectangular container. Their analytical model, which neglected the effects of frequency dispersion and dissipation, achieved only limited agreement with the experimental results. Frequency dispersion and dissipation effects were later included in the steady-state model of Chester (1968) for the shallow water waves generated by the horizontal oscillations of a closed basin. He found that although nonlinear effects were important near resonance, dispersion introduced higher harmonics into the solution spectrum. Chester & Bones (1968) carried out with a series of experiments on the waves produced in a tank oscillating horizontally with a sinusoidal motion; the results of these experiments were found to be in reasonably good agreement with the theoretical predictions of Chester (1968). The nonlinear shallow water waves produced in a rectangular tank excited at resonance

0889-9746/92/040471 + 21 so3.w 0 1992 Academic Press Limited

472 A. N. WILLIAMS AND X. WANG

have been studied theoretically by Miles (1985). The characteristics of the forced waves in this case have been found to be in qualitative agreement with those predicted by the analysis of Chester (1968).

Recently, Lepelletier & Raichlen (1988), reported a one-dimensional, nonlinear, dispersive, dissipative shallow water model to describe the liquid motions in a rectangular tank excited by a unidirectional, oscillatory, translational motion. The results from this model, which was solved by a finite element approach, were compared with laboratory measurements of free-surface elevations in an oscillating tank. Good agreement between theory and experiment was obtained for all cases investigated. Lepelletier & Raichlen also noted that, for continuous excitation at or near a resonant mode of oscillation, the linear theory becomes inadequate and the nonlinear, dispersive, dissipative theory should be used to obtain good correlation with the experimental data. Lepelletier & Raichlen (1987) have also formulated a nonlinear, dispersive, dissipative shallow water theory in two horizontal dimensions and utilized this theory to investigate the problem of the resonant excitation of harbors by tsunamis, again through a finite element solution of the governing equations.

In the present case, a nonlinear, dispersive, dissipative shallow water theory in two horizontal dimensions is developed to study the waves produced by the bidirectional, translational, oscillatory motion of a rectangular tank partially filled with liquid. The formulation is based on the above model of Lepelletier & Raichlen (1987); however, in the present case, an efficient finite difference technique is used to solve the resulting system of equations. The temporal discretization of the continuity equation is carried out using a Leapfrog scheme while, for the momentum equation, a Crank-Nicholson scheme is used on the linear terms and the second-order Adams-Bashforth scheme on the nonlinear terms. All spatial derivatives are evaluated by a fourth-order compact differencing method. Utilizing this approach, numerical results are presented which illustrate the form of the generated wave profile and its development in time for a range of geometric and excitation parameters. These results show the influence of the direction, duration and frequency of excitation, the liquid depth and dispersive and dissipative effects on the wave motions.

2. THEORETICAL FORMULATION

Figure 1 shows a rigid, rectangular tank of length a and width 6 partially filled with a liquid of uniform density p and kinematic viscosity Y. The still liquid depth is denoted by h. Cartesian coordinates are employed, the system OXYZ denotes a fixed Newtonian reference frame, while oxyz is a parallel system which is attached to, and moves with, the tank. The tank is subjected to a horizontal base motion whose components in the x and y directions are xb and y,, respectively.

For a liquid possessing a relatively small viscosity, the effect of viscous forces will be negligible, except near the solid boundaries. Therefore, the liquid domain may be decomposed into two regions: (i) a boundary layer region adjacent to the solid boundaries in which viscous effects are important and (ii) an exterior region consisting of the main body of the liquid in which the viscous terms may be neglected. The boundary layer region will be analysed later to provide estimates of the dissipation due to viscous effects along the solid boundaries.

Considering, for the moment, only the boundary layer at the bottom of the tank, the

TRANSIENT WAVE MOTIONS IN TANKS 473

#I

0 Gi

NM’ ,

X

a

Figure 1. Definition sketch.

equations of motion in the exterior domain are

a~ au au au 1 aP ~+u~+u-+wz= ----ib,

3Y p ax

av av av av i ap ~+u~+v-+waz= ----jj,,

aY P ay (1)

aw aw aw aw i ap -$+u-g+v-+wet= ----g.

aY P az

where u, v, and w are the liquid particle velocities in the X, y, and z directions in the uxyz system (i.e., relative to the tank), P denotes the liquid pressure, g denotes the acceleration due to gravity, and t is time. In addition, the continuity equation,

ati av aw -+-+-=o, ax aY az (2)

is valid throughout the liquid domain. Associated with the motion are boundary conditions on the tank bottom and

free-surface. The bottom boundary conditions are

(u, 21, w) = (0, 0, 0) on z = -h + 6(x, y, t), (3)

where the boundary layer thickness 6(x, y, t) is typically a few per cent of the length (or width) of the tank. The kinematic free-surface condition, which expresses the fact that the liquid velocity component normal to the surface is equal to the local velocity of the surface, is written as

ar ar ar dt+u~+v-=w

aY on z = IY(x, y, t), (4)

where z = I@, y, t) denotes the equation of the free-surface. The dynamic free-surface

474

conditions are given by

A. N. WILLIAMS AND X. WANG

P=O on 2 = T(x, y, t),

al4 -0 at- on 2 = l-(x, y, t), (5)

au -_= a2

0 on 2 = T(x, y, t),

and state, respectively, that the free surface is a surface of constant pressure and that no shear forces exist there.

The following analysis will be restricted to shallow water oscillations, such that 0(/z/a) and O((h/b) are much less than unity, and to cases where the tank’s horizontal dimensions are such that O(a/6) is unity. The characteristic length, L, of the wave motion in the x direction will be taken to be of the same order of magnitude as that in the y direction and is assumed to be large compared to the liquid depth, h. Also, the characteristic height, H, of the wave elevation is-taken to be small compared to h.

(but not infinitesimal)

Kelvin’s theorem the therefore, a potential

Since viscous forces are neglected in the exterior region, by flow in this region will remain irrotational if it was originally so; function @(x, y, z, t) exists such that

(6)

2.1. DIMENSIONLESS EQUATIONS

The governing equations and boundary conditions in the exterior region may be written in dimensionless form by introducing the following normalizations:

(x y)Jx*jY*) * , L ’

,A h'

t=r*l@ L ’

@* h a=------

Ll@H’ p=E_

mh ’ (7)

tu*, v*) (4 v)= f& *

w* L

W=TFH’ where the starred quantities are the original dimensional variables. The above scaling ensures that all dimensionless variables are of order unity and is based on the classical shallow-water wave theory. The following dimensionless parameters appear in the dimensionless equations:

These quantities will be referred to as the nonlinearity and dispersion parameters,


respectively. In dimensionless form, the governing equations become

t a~ I ap i

,pwa+(Y3+-=o. z z LY

The dimensionless boundary conditions are

w=o

ar ar ar -g+(uu-$+MI-‘W

ay

P=O

au -0

z-

av -0

57

on z = -1 + 6/h,

on 2 = CUT,

on 2 = d-.

on 2 = a,

on z = a.

(10)

Finally, the relationship between the velocity components and the velocity potential is given in dimensionless form as

(u,v, W)’ c aa aa i aa -,- -- ax ay ’ p aZ 1

. (11)

Turning now to the boundary-layer region, for the moment only the boundary layer at the bottom of the tank will be considered. The effects of the side walls on the fluid damping will be addressed later. The primary objective in solving the boundary-layer equations is to obtain expressions for the shear stress components, tX and t,,, in the x and y directions at the boundary. Expressions for tX and tY at z = -h may be obtained analytically from the linearized form of the laminar boundary-layer equations by a Laplace transform technique, assuming that the fluid starts from rest at t = 0. The shear stress components at the bottom of the tank are found to be [see, e.g., Lamb (19431

(12)

2.2. DEYIX-AVERAGED MOMENTUM EQUATIONS

Recognizing the shallow-water nature of the flow, i.e., /3 << 1, the momentum equations in the horizontal directions will be averaged over the liquid depth. Utilizing


equations (12) and including the viscous dissipation term from the bottom boundary layer, these equations have the following dimensionless form:

au au au 1 ap -++cnc++-++(YW++(Y+fjih

ay

au au au I ap -++~+cm-+a~-+--+j&

9 az say

where E is a dissipation parameter defined by

1 YL ( 1

l/2

&=i Jr* .

(13)

(14)

The relative importance of nonlinearity, as measured by the parameter (Y, and dispersion, as measured by 6, is best examined through use of the Ursell number ZJ, = a//3. For U, < 0(l), nonlinear effects are unimportant and the linear dispersive, dissipative shallow-water theory (a = 0) may be used. When cl, > Q(l), dispersion has relatively little influence and the nonlinear, nondispersive dissipative theory (j3 = 0) should be applied. If U, = Q(l), both nonlinear and dispersive effects are important and the weakly nonlinear dispersive dissipative theory must be used. Thus, in the present case, it will be assumed that U, = O(l), i.e., I = S(j3) and, moreover, the dissipative terms are also of the same small order, so that 0(‘(a) = 19(/l) = 0’(c) < 0(l).

Retaining terms to the first order in (Y, fi, and E, the continuity and depth-averaged horizontal momentum equations may be expressed in terms of depth-averaged velocity components defined by

u dz, v dz.

The depth-averaged continuity equation may be written as

(15)

Then, writing the vertical momentum equation in terms of fi and V leads to the following form for the liquid pressure,

(17)

where the first term on the right-hand side corresponds to the hydrostatic pressure. The


horizontal depth-averaged momentum equations are then obtained as

s!+rm;+as*+ar+xb+P ax

ay

1 ---_-~ 3 1 at a”Li ax2 1 a% + 1 _ a-b

2ataxay 6atay2 I

+& I ,rg (x, y, t - p) 3 = O(c$, p2, EB, E2),

1 i a% 1 a-k g+arig+av2!+?K+L,+p ---_-~ i a3v (18)

_ a~ ay 3 at ay2 2ataxay+6ataxz I

The computation of the viscous dissipation terms as they appear in equations (18) is a time-consuming and awkward task. These expressions may be simplified by choosing U and ii to be harmonic in time with angular frequency o; that is U = %{Uge-iw’} and 6 = %{ U,e-‘“‘}, where %{ } denotes the real part of a complex expression. It is now proposed to seek expressions for the bottom shear stresses in the following forms:

(19)

where the expressions expressions

constants C, and C, are to be obtained by ensuring that the above dissipate the same energy per unit area over a wave period as the exact given in equation (18). This condition yields C, = C, = V%&%!, and so

- -

are taken to represent the bottom shear stresses due to the presence of a laminar boundary layer at the bottom of a rectangular tank undergoing periodic translational motion. The transition to turbulence in an oscillatory flow has unanimously been observed to occur at a Reynolds number based on Stokes boundary-layer thickness (a6) of about 500 to 550, independent of the particular flow geometry (Akhavan et al.

1991a,b). In the present case $8~~ = U, S/Y, where U,, is the amplitude of the base velocity of the tank, and 6 = (2v/o)l’*. It can be shown that, in the present context, the laminar boundary layer assumption holds over a wide range of interest. Equations (20), which describe the bottom shear stresses (and hence the fluid damping due to the bottom boundary layer) as linear functions of the depth-averaged velocity components, simplify considerably the numerical treatment of the viscous long-wave equations.

The above discussion deals exclusively with the damping effect associated with the bottom boundary layer. However, there are two additional sources of energy dissipation present in the oscillating tank, due to sidewall friction and free-surface contamination. It has been verified experimentally that the observed damping of the fluid motion in a basin can be significantly larger than that computed by considering the bottom boundary layer alone (Vandorn 1966). It has been suggested (Miles 1967) that these effects can be accounted for empirically by multiplying each of the dissipation terms by a factor of the form (1 + )3w + S). The quantity A, accounts for the additional fluid damping due to the sidewall boundary layer, the friction effect induced by this layer is regarded as being the same as that associated with the bottom boundary layer. We now define the quantities A, and A,, the additional fluid damping due to the sidewall boundary layers. By analogy with the one-dimensional sloshing problem, these


quantities are chosen to be equal, i.e.,

A,=A,=A,= 2h

a sin 8 + h cos 8’ (21)

where 8 is the direction of oscillation of the tank, measured counterclockwise from the positive x-axis. The so-called surface-contamination factor, S, may vary between 0 and 2 (Miles 1967). The value S = 1, which corresponds to a fully contaminated surface, has been used by previous investigators [for example, Lepelletier & Raichlen (1988)]; this value will also be used in the present work. The preceding formulation and discussion is restricted to the case of a continuous free surface, where wave breaking is absent. However, it should be noted that, should extreme tank motions occur and wave breaking take place, this will introduce an additional source of energy dissipation.

Thus, the horizontal depth-averaged momentum equations may be written in the following simplified dimensionless forms:

in which

VW L & ox = $ ~j-J=gIl+Lx+% G,y= $$=${l+E.,,+s}. d- (23)

The following initial/boundary conditions are also prescribed, corresponding to the fluid being at rest at time t = 0 and the walls of the tank being perfectly reflective:

In order function

w, y, 0) = 0 Osxsa, Osysb,

qx, y, 0) = 0 O%xsa, Osyrb,

0(x, y, 0) = 0 Osxsa, Osysb,

qo, y, t) = 0 Osysb, t>O, (24)

ii(a, y, t) = 0 Osyrb, t>O,

iqx, 0, t) = 0 OSxSa, t>O,

fi(x, b, t) = 0 OSx%a, t>O.

to facilitate the numerical computations, the depth-averaged velocity-potential is defined as

with the corresponding velocity components

Q dz, (25)

It can be shown that ti - U = O(@, p*), and 6 - 9 = D(u$, /3*), and so the momentum equations, equations (22), can be replaced by a single equation written in terms of the


depth-averaged velocity potential, namely

where Ab is the bottom acceleration potential

and

6) = d VW L yj-J=g (1+ L + 0 (29)

The continuity equation is also transformed to a form more amenable to numerical treatment. Defining

IY = i (exp(G) - l), (30)

equation (16) becomes

;!+55fi+acij+*+?!=o. ay ax ay (31)

Equation (31) is numerically more stable than equation (16). The boundary and initial conditions given by equations (24) can now be written as

g (0, y, I) = 0

g (a, y, t) = 0

g(x, 0, t)=O

z (x, b, t) = 0

6(x, y, 0) = 0

G(x,y,O)=O

t >o, Osysb,

t >o, Osysb,

t > 0, OIxsa,

t >o, OSxla,

Osxsa, Osysb,

OSxSa, Osysb.

(32)

3. FINITE DIFFERENCE DISCRETIZATION

Equations (27) and (31) subject to the initial/boundary conditions given by equations (32) are now solved by a finite difference technique. The continuity equation is discretized temporally utilizing the Leapfrog scheme,

G n+L _ G”-’

2a At + G;rY’ + G;V” + ii; + V; = 0, (33a)

or, on rearranging, by

G n+l = G”-’ -2a At(G;ii” + G;V + U: + V;), (33b)


As for the temporal discretization of the momentum equation, the Crank-Nicholson scheme is used on all the linear terms, while for the nonlinear terms, the second-order Adams-Bashforth scheme is utilized, namely

@&gnn_“@ 3 3 YY

@ -n+1+*!n

At + co 2

+~3(~~)2+3(6;)2-(~-1)2-(~;-1)2+rv2+’+,~+A~+1+A;:

2 2 2 2 = 0, (34a)

or, equivalently,

_ At~3(&;)2 + 3(&;)2 - (+‘=-‘)’ - (&;-‘)’ At

2 2 - y (m+’ + r” + A;+’ + A;), (34b)

where the free-surface elevation, r, is given by equation (30). Since the above model requires information at more than one previous timestep, to

start the calculations the first time step is divided into ten intervals with a sub-timestep At, = At/lo. In the first sub-timestep, the Euler-forward scheme is used and then for the next nine sub-timesteps, the Leapfrog scheme is applied. Once the information at t = At is obtained, the calculation reverts to the normal timestep, At, and proceeds.

All the spatial derivatives are evaluated by the fourth-order compact differencing method. By utilizing this approach, the phase error can be reduced considerably compared to classical finite differencing schemes. In this method, the first and second derivatives of a function, f(g), are given in terms of the function values by

(35)

where A5 is the step length. Equations (35) are each solved using a tridiagonal system solver.

The time-marching scheme used herein results in a Poisson equation from the momentum equation at each new time step, which can be written as

3(@92+3(W2- (W’)2-(q-1)2+,.+1 +r” +A;+1 +A;: . 2 1 (36)

This equation is solved by a fourth-order IMSL fast Poisson solver, which is also based on the compact differencing method.

In summary, the numerical schemes used herein to solve equations (27) and (31) subject to the initial/boundary conditions given by equations (32) are second-order accurate in time and fourth-order accurate in space.


4. NUMERICAL RESULTS AND DISCUSSION

First, the influence of mesh size on the numerical results was investigated. An example tank of dimensions, a = 2-O m, 6 = 1-O m, h = O-3 m, was considered. The amplitude of the base oscillation was taken to be O-01 m occurring at an angle 8 = 30” to the positive x-axis. The lowest natural frequency of the waves in a one-dimensional tank of length L as predicted by linearized theory is given by Lepelletier & Raichlen (1988) as

‘%= L(l - 7c2p/6) ’ (37)

where L is a characteristic length associated with the wave motion. Four separate studies were carried out for mesh sizes 32 x 64, 64 x 128 and 128 x 256 with a constant dimensionless timestep At = 0.005. The calculations were carried out up to t = 25, i.e., 5,000 time-steps. The frequency of excitation w = 0”. The numerical results showed a difference of 3.5% in the computed maximum free-surface elevations at t = 25 between the 32 x 64 and the 64 x 128 meshes, whereas the differences in the maximum free-surface elevations at this time calculated utilizing the 64 x 128 and 128 x 256 meshes was 1.2%. There was a 100% increase in computational effort at each time-step between the 32 x 64 and the 64 x 128 meshes, and a 280% increase in computational effort at each time-step between the 64 x 128 and 128 x 256 meshes. Therefore, the 64 x 128 mesh size was chosen for the parametric study on this tank geometry, with the typical loss of about 1% accuracy after 5,000 time-steps.

The influence of the time-step, At, on the numerical results was investigated by rerunning the above case for a range of excitation frequencies with the 64 x 128 mesh, but with the time-step first halved and then increased by a factor of two from that above, i.e., with At =0X)025 and At = O-01. The difference in the maximum free- surface elevations for the At = O-005 and At = O-01 cases at t = 25 was found to be O-42%, while the computational effort associated with the At = 0.01 case was approximately half of that for At = 0.005. Also, the difference in the maximum free-surface elevations between the At = O$Kl25 and At = 0.005 cases at t = 25 was found to be O-14%, while the associated computational effort increased approximately by a factor of two for the smailer time-step. Therefore, it was decided that a value of At = O-01 could be used for the parametric study of the wave motions inside the tank at all excitation frequencies except those near resonance, o = 1*05w,,, where a refined timestep of At = 0.005 should be taken.

In order to demonstrate the correctness of the present numerical approach and the suitability of the above choices for mesh size (64 x 128) and timestep (At = OeOl), a test case was run and compared with the one-dimensional numerical model for unidirectional base excitation and the associated experimental results reported by Lepelletier & Raichlen (1988). For this case, in the present notation, h/a = O-098, a/b = 1.0, 8 = 0 and o = 1*040,. This comparison is shown in Figure 2, which presents the free-surface at various times within a wave period, starting from an initial dimensionless time of t = 29.31. It can be seen that the agreement between the present work and the theoretical and measured values of Lepelletier & Raichlen is excellent. Figure 3 shows the time evolution of the generated waveform in this case from a dimensionless time t = 29.00 to t = 33.22, i.e., throughout the time period presented in Figure 2. It should be noted that the excellent agreement between the numerical and experimental results in the above test case also validates the theoretical basis for the present approach, and in particular the various assumptions inherent in the simplified treatment of the dissipation mechanisms present in the fluid-tank system.


(a) 0) Cc)

-Experiments - Nonlinear theory O_ ------- -_-_-

0.33

77 h

Figure 2. Computed free-surface profiles at various times within a wave period starting at a dimensionless time t = t,. Comparison of (a) experimental results and (b) one-dimensional nonlinear theory, both of Lepelletier & Raichlen (1988) with (c) present solution. For this case, in the present notation, h = 6 cm,

1, = 29.3, a = 60.95 cm, (I, = 144 and the amplitude of base excitations is 0.1% cm.

A study was then carried out to investigate the influence of the various problem parameters on the form and propagation characteristics of the generated waves. Specifically, the effects of liquid depth, amount of viscous damping and the direction, frequency and duration of the base excitation were examined. In the following examples, unless stated otherwise, a = 2-O m, b = 1.0 m, h = O-3 m. For t 2 0 the tank is

1

\

.r

/

Figure 3. Time evolution of free-surface profile from a dimensionless time I = 2940 to t = 32.22. Tank and excitation data is the same as that in Figure 2.


‘3 = 1.05 -_

0 :_

I, ?U 40 ho XU ,in,

Dm~ensvmlers time

c, = I ,,I

Dtmensionless time

Figure 4. Time-histories of free-surface elevations at stations 1, 2 and 3 for a=2-Om, b= l.Om, h = 0.3 m, r3 = 0“ and G, = 0.9, 1.0, l-05, 1.1 and 2.0; continuous harmonic excitation.


, i__~ I.~_

0 20 40 60 80 i(X)

Dimensmnless time Dimensionless time

0

_-- I I 5

0 20 40 a- 80 IW

Dnnenrionless time

0 20 40 80 103

Figure 5. Time-histories of free-surface elevations at stations 1, 2 and 3 for a = 2.0 m, b = l-Om h = O-3 m, 0 = 30” and 0 = 0.9, 1.0, 1.05, 1-l and 2-O; continuous harmonic excitation.

TRANSIENT WAVE

G = 2.00 r’,,‘,‘,‘,‘,‘, ,‘I,,‘,

f 7

‘:

MOTIONS IN TANKS 485

0

Figure 6. Time-histories of free-surface elevations at stations 1, 2 and 3 for a = 2.0 rn, b = l.Om, h = O-3 m, fI = 60” and ii, = 0.9, 1.05 and 2.0; continuous harmonic excitation.

:- -- _~____ -1 0 ---iiF- 20 6n 80 loo

Dimensionless tnne

Figure 7. Time-histories of free-surface elevations at stations 1, 2 and 3 for (I = 2-O m, b = 1.0 m, /I = 0.3 m, 8 = 0” and (?, = 1.05; continuous harmonic excitation. Dissipation effects have been neglected, i.e.

EO=O.


subjected to a sinusoidal base excitation of frequency o and amplitude O-01 m occurring at an angle 8 to the positive x-direction. The dimensionless bottom excitation frequencies considered varied from 0 = w/w0 = 0.9 to 2.0. All calculations were carried out up to a dimensionless time t = 100. The time-history of the free-surface elevation was calculated and is shown at three locations, denoted by stations l-3, corresponding to the points (x, y) = (a/2, b), (a, 6/2) and (a/2, 0), respectively (see

(a) E(, = 0

Figure 8. Comparison of free-surface profiles at 1= 100, (a) neglecting, and (b) including, dissipation effects, for a = 2-O m, b = 1.0 m, h = O-3 m, 0 = 30” and 61 = 1.05; continuous harmonic excitation.

TRANSIENT WAVE MOTIONS 1N TANKS 487

Figure 1). In the figures, the top profile corresponds to station 1, the middle to station 2 and the bottom to station 3. The mesh size and time-steps used were as outlined in the previous section.

First, the influence of the direction of base excitation 8 was examined. Figure 4 shows the time-history of the free-surface elevations at stations l-3 at various excitation frequencies for 8 = 0”. In this case the motion occurs only in the x-direction and the solution is equivalent to that obtained by Lepelletier & Raichlen (1988). It can be seen that the first resonant frequency for the tank occurs very close to a dimensionless frequency ii, = 1.05; at this frequency very large amplitude waves are generated in the tank. As the difference between the excitation frequency and first resonant frequency increases, the amplitude of the oscillations decreases significantly until, at (r, = 2.0, there are essentially no wave motions in the tank. Figure 5 presents the results of the numerical calculations for the same dimensionless frequencies shown in Figure 4, but with an excitation angle 0 = 30”. The two-dimensional nature of the

z = 040

. ----ho.-’ -__. ~... ~~_.

0 20 60 80 la0 Dmxnsionless flme

Figure 9. Time-histories of free-surface elevations at stations 1, 2 and 3 for (I = 2.0 m, b = 1.0 m. h = 0.3 m, B = 30” and ~5 = 0.9, 1.05 and 2.0; harmonic excitation occurs for I I 10.70, t 5 9.17 and I c: 4.81,

respectively.


oscillations is most clearly apparent from the free-surface profile at (z, = 1.0. It can be seen that the largest-free surface elevations at the side walls and end-walls of the tank occur at the same dimensionless excitation frequency of ii, = 1.05. Again, as the excitation frequency increases, the oscillation amplitudes in both directions decrease. However, the free-surface amplitudes at the side walls increase again at CT, = 2-O and are completely different from those predicted by the one-dimensional solution at the same dimensionless frequency. Figure 6 presents the numerical results for an excitation direction of 8 = 60” for selected dimensionless frequencies CT, = 0.9, 1.05 and 2.0. It is again noted that the time-histories of the free-surface elevations are completely different from both the 0 = 0 and 30” cases. Thus, it is concluded that the direction of base excitation may have a significant influence on the characteristics of the generated waves inside the tank.

Next, the influence of the hydrodynamic damping term on the generated waveform was investigated by setting the parameter co equal to zero. The data-set chosen was that corresponding to Figure 5 where 0 = 30” and, since damping effects are likely to be

0 20 40 M 80 Dimemionlcor time

Figure 10. Time-histories of free-surface elevations at stations 1, 2 and 3 for a = 2.0m, b = 1-O m, h = O-2 m, 0 = 30” and CT, = 0.9, 1.05 and 2.0; continuous harmonic excitation.

TRANSIENT WAVE MOTIONS IN TANKS

(b) h = 0.3 m

Figure 11. Comparison of free-surface profiles at t = 100, for (a) h = 0.2 m, and (b) h = 0.3 m; for a = 2.0 m, b = 1.0 m, 0 = 30” and CT, = 2.0; continuous harmonic excitation.

most pronounced near resonance, a dimensionless excitation frequency 0 = 1.05 was chosen. The results of this calculation are presented in Figure 7, where it can be seen that the calculated free-surface elevations are higher than those in the corresponding case where viscous damping was included. Also noted are the differences in the forms of the time-histories of the free-surface elevations in these cases. Figure 8 presents a comparison of the instantaneous free-surface elevations at a dimensionless time t = 100 for the two cases. It can be seen that, for the undamped case, there are more than three waves visible in the free-surface profile, while, when damping is considered, only one wave can be seen at the same dimensionless time. Thus, viscous damping influences both the form and amplitude of the generated free-surface profile.

Also investigated was the influence of the duration of the base excitation of the tank. Dimensionless excitation frequencies of 0 = 0.9, l-05 and 2.0 were considered, and the angle of excitation was again taken to be 8 = 30”. In each case the base excitation was taken to occur over five periods of motion and then stopped; this corresponds to


excitation up to dimensionless times of l= 10.70, 9.17 and 4.81, respectively. The calculation was then continued to a dimensionless time of t = 100 as before. The results of these calculations are shown in Figure 9. It can be seen from the figures that the amplitudes of the waves generated in the duration-limited cases are considerably smaller than those obtained under continuous base excitation, shown in Figure 5.

A study of the influence of the dispersive effects on the generated waves was accomplished by reducing the depth of liquid in the tank, h, from 0.3 to 0.2 m. Again the calculations were performed for 8 = 30” at dimensionless frequencies ti = 0.9, I.05 and 2.0, the results are shown in Figure 10. Very large oscillation amplitudes are observed in the tank for this particular parameter combination, especially at ti = 1.05 and 2.0. Figure 11 presents a comparison of the instantaneous free-surface elevations at a dimensionless time t = 100 at a dimensionless frequency W = 2.0. It is noted that for h = 0.3 m there are essentially no waves on the free surface at this frequency whereas, at the smaller water depth, waves are clearly visible propagating in both coordinate directions.

5. CONCLUSIONS

A nonlinear, dispersive, dissipative shallow water theory in two horizontal dimensions has been developed and utilized to study the waves produced by the bidirectional, translational, oscillatory motion of a rectangular tank partially filled with liquid. An efficient finite difference technique has been used to solve the resulting system of equations. The temporal discretization of the continuity equation is carried out using a Leapfrog scheme and for the momentum equation a Crank-Nicholson scheme is used on the linear terms and a second-order Adams-Bashforth scheme on the nonlinear terms. As far as the spatial discretization is concerned, all spatial derivatives are evaluated by a fourth-order compact differencing method.

Results obtained utilizing the present approach show excellent agreement with the one-dimensional numerical model and experimental results of Lepelletier & Raichlen (1988) for the limiting case of unidirectional base excitation. Then, utilizing this approach, numerical results have been presented which illustrate the form of the generated wave profile and its development in time for a range of geometric and excitation parameters. It has been found that the direction, duration and frequency of excitation, liquid depth and dispersive and dissipative effects may have a significant influence on both the amplitude and form of the wave motions generated in the tank.

The integration of the hydrodynamic model presented herein into a dynamic analysis package for studying the use of liquid-containing tanks as tuned, sloshing dampers to suppress the wind-induced response of tall buildings is currently underway. The results of this work will be reported in the near future.

ACKNOWLEDGEMENT

The material presented herein is based upon work funded in part by the Texas Advanced Technology Program (Grant No. 3652-200). This support is gratefully acknowledged.

REFEKENCES

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Nonlinear transient wave motions in base-excited rectangular tanks