A numerical investigation of energy dissipation with a shallow depth sloshing absorber

Applied Mathematical Modelling 34 (2010) 2941–2957

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

A numerical investigation of energy dissipation with a shallow depthsloshing absorber

Adam Marsh a, Mahesh Prakash b,*, Eren Semercigil a, Özden F. Turan a

a Victoria University, School of Architectural, Civil and Mechanical Engineering, Melbourne, Victoria 3011, Australiab CSIRO Mathematical and Information Sciences, Private Bag 33, Melbourne, Victoria 3168, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 May 2009Received in revised form 22 December 2009Accepted 7 January 2010Available online 13 January 2010

Keywords:Liquid sloshingStructural controlEnergy dissipationEnergy transferSmoothed Particle Hydrodynamics

0307-904X/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.apm.2010.01.004

* Corresponding author. Tel.: +61 0395458010; faE-mail address: [email protected] (M. Pr

A liquid sloshing absorber consists of a container, partially filled with liquid. The absorberis attached to the structure to be controlled, and relies on the structure’s motion to excitethe liquid. Consequently, a sloshing wave is produced at the liquid free surface within theabsorber, possessing energy dissipative qualities. The behaviour of liquid sloshing absorb-ers has been well documented, although their use in structural control applications hasattracted considerably less attention.

Generally it is accepted that sloshing absorbers with shallow liquid levels are more effec-tive energy dissipaters than those with higher levels, although there has not yet been astudy to reveal an ‘optimum’ design mechanism. The main limitation of numerically mod-elling such circumstances is the inherent complexity in the free surface behaviour, predic-tions of which are limited when using grid-based modelling techniques. Considering suchlimitations, Smoothed Particle Hydrodynamics (SPH) is used in this study to model a 2-dimensional rectangular liquid sloshing absorber. SPH is a Lagrangian method of solvingthe equations of fluid flow that is suitable to model liquid sloshing due to its grid-free nat-ure, and inherent ability to deal with complex free surface behaviour.

The primary objective of this paper is to numerically demonstrate the effect of tuning acontainer’s width, to complement previous work on the effect of liquid depth [B. Guzel, M.Prakash, S.E. Semercigil, Ö.F. Turan, Energy dissipation with sloshing for absorber design,in: International Mechanical Engineering Congress and Exposition, 2005, IMECE2005-79838]. This study is an attempt to reveal geometry that enables both effective energytransfer to sloshing liquid and to dissipate this energy quickly.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

Sloshing is the oscillation of a liquid within a partially full container. In study of sloshing, efforts are usually made in thedirection of suppression, due to the damaging effects it can impose [2–4]. On the other hand, sloshing has an inherent abilityto dissipate large amounts of kinetic energy via shearing of the fluid. For this reason, it is possible to employ liquid sloshingas an effective energy sink in structural control applications, providing protection for structures exposed to excessive levelsof vibration. Examples of effectiveness, along with the successful application of sloshing absorbers can be found in [5–7].

Liquid sloshing in rectangular tanks have long been an area of study [2,3,8]. However, the complex free surface behaviour(such as wave breaking and collision) exhibited by such a system has limited the numerical prediction attempts. As men-

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2942 A. Marsh et al. / Applied Mathematical Modelling 34 (2010) 2941–2957

tioned earlier, such behaviour is of great interest in structural control applications due to the intrinsic ability to dissipatekinetic energy quickly.

Various studies focused on using liquid sloshing as a structural control mechanism can be found in the literature. Theeffects of implementing flexible container walls [9], a net for the fluid to pass through [10] and the employment of baffles[11] on the energy dissipative characteristics of deep liquid level absorbers have been analysed, all shown to improve energydissipation qualities. However, the principal standing waveform observed at these deep liquid levels is inefficient in naturerelative to the travelling waveform observed in shallow liquid levels [1,12].

Studies involving varying liquid depth in a rectangular container of constant width have been undertaken, focusing on theamount of energy dissipation produced [8,13]. Yet, the liquid depth at which the transition from travelling to standing wavedevelopment is made has not been investigated for a given container width.

Generally, container width is tuned so that the natural sloshing frequency coincides with the natural frequency of thestructure to be controlled [14]. In liquid depths producing strong travelling wave development, this may not be as important.Instead, the focus can be on increasing energy transfer and dissipation through tuning.

Due to complex free surface behaviour, Smoothed Particle Hydrodynamics (SPH) is used as a numerical tool in this studyto model 2-dimensional rectangular liquid sloshing absorbers. SPH is a Lagrangian method of solving the equations of fluidflow that is suitable to model liquid sloshing due to its grid-free nature, and inherent ability to capture free surface behaviouraccurately [15–18]. SPH has been successfully applied to a wide range of industrial fluid flow applications [19–23].

In this paper, transfer and dissipation of kinetic energy within a rectangular tank has been investigated while varying li-quid depth, and container width. The objective is, hence, an attempt to reveal liquid depth and absorber width that optimisesenergy transfer and dissipation in a rectangular container, subject to a specific sinusoidal excitation. Physical events that en-hance energy dissipation are commented on. Results are presented in the form of numerical case studies.

2. Methodology

2.1. Smoothed Particle Hydrodynamics

Smoothed Particle Hydrodynamics (SPH) is a mesh-free particle method (MPM) of modelling fluid flows. The fluid beingmodelled is discretized into fluid elements or particles, the properties of which are attributed to their centres. SPH is aLagrangian continuum method used for solving systems of partial differential equations. The method works by tracking par-ticles and approximating them as moving interpolation points. These fluid particles (or moving interpolation points) have aspatial distance over which field variables such as density, velocity and energy are smoothed. This is achieved via an inter-polation kernel function.

The fundamental concept of the integral representation of a function used in the SPH method comes from the identityshown in Eq. (1). The identity simply implies that a function can be represented in integral form. Where f(x) is a functionof x and d(x � x0) is the dirac delta function. V is the volume of the integral that contains x.

f ðxÞ ¼Z

Vf ðx0Þdðx� x0Þdx0: ð1Þ

The integral representation in Eq. (1) is exact since the delta function is used, providing that f(x) is defined and continuous inV [24]. In SPH the dirac delta function is replaced by the smoothing function W(x � x0, h) so that the integral representation off(x) is specified as:

hf ðxÞi ¼Z

Vf ðx0ÞWðx� x0; hÞdx0: ð2Þ

W is the interpolation kernel, h is the smoothing length that defines the region in which the smoothing function operates. Acubic smoothing kernel has been used here for W(x � x0, h), approximating the shape of a Gaussian profile but having com-pact support, so that W(x � x0, h) = 0 for x � x0 > h.

The integral representation of the spatial derivative of a function in SPH is performed in the same way. f(x) is substitutedfor r � f(x) in Eq. (2) to produce:

hr � f ðxÞi ¼Z

Vr � f ðx0ÞWðx� x0;hÞdx0: ð3Þ

From the identity in Eq. (4) below, we obtain Eq. (5),

½r � f ðx0Þ�Wðx� x0;hÞ ¼ r � ½f ðx0ÞWðx� x0;hÞ� � f ðx0Þ � rWðx� x0; hÞ; ð4Þ

hr � f ðxÞi ¼Z

V½r � f ðx0ÞWðx� x0; hÞ�dx0 �

ZV

f ðx0Þr �Wðx� x0;hÞdx0: ð5Þ

The first integral on the right hand side of Eq. (5) is converted using the divergence theorem of Gauss to obtain Eq. (6).

hr � f ðxÞi ¼Z

Sf ðx0ÞWðx� x0; hÞ � n

*dS�

ZV

f ðx0Þr �Wðx� x0; hÞdx0: ð6Þ

A. Marsh et al. / Applied Mathematical Modelling 34 (2010) 2941–2957 2943

S is the surface of the integration domain V ; n*

is the unit normal to the domain surface S. When the support domain of W islocated within the problem domain the first integral on the right hand side of Eq. (6) is zero. However, when the support do-main overlaps with the problem domain W is truncated by the problem domain boundary, hence, the surface integral is nolonger zero. For all points in space whose support domain lies within the problem domain, Eq. (6) simplifies to Eq. (7) below.

For all other points in space, modifications need to be made to treat the boundary effects if the surface integral is to beequated to zero [24].

hr � f ðxÞi ¼ �Z

Vf ðx0Þr �Wðx� x0;hÞdx0: ð7Þ

Eq. (7) states that the spatial gradient of a function is determined from the values of the function and the derivative of thesmoothing function, rather than the derivative of the function itself.

Discretization is performed by converting the integral representations in Eqs. (2) and (7) into summations over all theparticles that lie within the support domain of W. This is achieved by replacing the infinitesimal volume dx0 by the finite vol-ume of particle j, DVj. The mass of particle j (mj) is then related to this volume by:

mj ¼ qjDVj; ð8Þ

where qj is the density of particle j. The discretized particle approximation can then be written as:

hf ðxiÞi ¼XN

j¼1

mj

qjf ðxjÞWðxi � xj;hÞ: ð9Þ

Eq. (9) states that the value of a function at particle i is approximated using the average of the same function at all j particleswithin the support domain of particle i, weighted according to the smoothing function.

The same approach is used to produce the particle approximation of the spatial derivative of a function:

hr � f ðxiÞi ¼ �XN

j¼1

mj

qjf ðxjÞ � rWðxi � xj;hÞ: ð10Þ

rW is taken with respect to particle j in Eq. (10), when taken with respect to particle i the negative sign is removed,producing:

hr � f ðxiÞi ¼XN

j¼1

mj

qjf ðxjÞ � riWðxi � xj;hÞ: ð11Þ

The SPH approximations in Eqs. (9) and (11) are applied to the field variables and their derivatives within the Lagrangianequations of fluid flow. This yields the continuity (12) and momentum (13) equations.

dqi

dt¼XN

j¼1

mj

qjv ij � riWij; ð12Þ

where Wij = W(rij, h) and is evaluated for the distance jrijj. rij is the position vector from particle ‘j’ to particle ‘i’ and is equal to ri � rj.

dv i

dt¼ �

XN

j¼1

mjPj

q2j

þ Pi

q2i

!� f

qiqj

4lilj

ðli þ ljÞv ijrij

r2ij þ g2

" #riWij þ g; ð13Þ

where Pi and li are the pressure and viscosity of particle ‘i’, the same applies for particle ‘j’. vij = vi � vj. g is a parameter usedto smooth out the singularity at rij = 0, and g is the body force acceleration due to gravity. f is a factor which has a theoreticalvalue of 4, but has been modified empirically [25].

The code uses a compressible method for determining the fluid pressure. It is operated near the incompressible limit byselecting a speed of sound that is much larger than the velocity scale expected in the fluid flow. Eq. (14) shows the equationof state used to govern the relationship between particle density and fluid pressure.

P ¼ P0qq0

� �c

� 1� �

; ð14Þ

where P is the magnitude of the pressure and q0 is the reference density. For water, c = 7 is generally used [26]. P0 is thereference pressure. The pressure the equation of state solves for P, is then used in the SPH momentum equation governingthe particle motion.

The time stepping in this code is explicit and is limited by the Courant condition modified for the presence of viscosity,

Dt ¼mina

0:5h= cs þ2fla

hqa

� �� ; ð15Þ

where cs is the local speed of sound.


2.2. Validation of numerical model

In order to validate the SPH code used here, predicted free surface shapes are compared to experimental observations in[27] SPH is found to provide an accurate prediction of fluid behaviour for depths as shallow as 2.75 mm. To give the reader anunderstanding of the capabilities of SPH to predict the physics of such flows, representative snapshots have been taken fromthis work and are presented in Fig. 1.

The left hand column of Fig. 1 shows experimental observations of a sloshing absorber controlling a structure. The righthand column shows the SPH predictions of the observations, shown to the left. These instances show the evolution of anenergetic hydraulic jump. Such behaviour is particularly difficult to capture numerically, due to large free surface deforma-tion and the discontinuous nature of the flow field. SPH is seen to produce exceptional results.

Along with this, the code has been subject to the benchmark test of a free surface dam break flow impacting on a rigidcolumn. The predicted net force exerted on the column is compared to experimental findings. Results in excellent agreementare obtained [28].

2.3. Numerical model and procedure

A rigid container 400 mm high, of varying width (W), and varying liquid depth (L) is subjected to a horizontal sinusoidaldisplacement function of pure translation with amplitude of 50 mm and a frequency of 1.4 Hz for one complete cycle. Thenumerical model is shown in Fig. 2(a) along with the excitation in Fig. 2(b). The fluid being modelled is water with a density

Fig. 1. Free surface comparisons of water within a sloshing absorber controlling a structure. Left column shows experimental observations. Right columnshows numerical predictions obtained with SPH. Fixed velocity scale shows fluid particle velocities in metres per second.

Fig. 2. Showing (a) numerical model and (b) history of base excitation.


of 1000 kg m�3 and dynamic viscosity of 0.001 Pa s. A particle represents a 0.8 mm � 0.8 mm area of water within thecontainer.

The resolution of 0.8 mm was chosen for the fluid and boundary particles following a detailed analysis ascertaining res-olution independence carried out for inter-particle spacing between 0.25 and 1 mm2 [27]. For the 5 mm liquid depth approx-imately 2500 fluid particles were needed at this resolution, whereas for the 100 mm liquid depth approximately 50,000 fluidparticles were used. Around 750 particles were used to represent the boundary. A typical time step was of the order of1 � 10�6 s. The CPU time for the simulation varied between approximately 1 day for the lowest fluid height and aroundtwo weeks for the deepest liquid level used. All simulations were performed on a single node of a 3.2 GHz Intel Xeonprocessor.

The fluid within the container is left to settle under gravity for a period of one second so that it reaches a zero energyinitial state. After this it is subjected to the sinusoidal excitation. The fluid is then allowed to respond naturally to the exci-tation within the container. The total simulation time is 20 s. Due to the base excitation imposing increasing amounts of ki-netic energy to the fluid as its mass increases, all analysis has been undertaken on a per unit mass basis.

3. Observations

An efficient absorber transfers a substantial amount of energy from the structure in which it is employed, to the workingfluid. This energy is to be dissipated as quickly as possible. The primary means of energy dissipation seems to be shearing ofthe fluid. Hence, energy dissipation is enhanced as the velocity gradients in the flow become steeper. Two main events pro-duce steep velocity gradients in a sloshing liquid, namely wave-to-wave and wave-to-wall interactions. In general, wave-to-wave interactions seem to be more effective. Travelling and/or standing waveforms can exist in a sloshing liquid. A travellingwave is capable of producing both type of interactions mentioned, whereas standing waves primarily produce wave-to-wallinteractions.

Histories of the sum of kinetic energy per unit mass for cases simulated in a container of 400 mm width are shown inFig. 3. Fig. 3(a)–(h) correspond to the liquid depths of 5, 10, 15, 20, 40, 60, 80 and 100 mm, in descending order. Positionsindicated with a red mark specify the point of maximum kinetic energy, the magnitude of which signifies the energy transferfrom the base excitation to the sloshing liquid on a per unit mass basis. All liquid depths were also simulated in containerwidths of 200 mm and 600 mm, histories of which are not shown here for brevity. Similar behaviour was observed for allcontainer widths.

Peaks in the kinetic energy histories correspond to high particle velocities, usually when the free surface shape is rela-tively flat. Troughs in the kinetic energy histories are observed when the fluid is close to largest free surface deformation,corresponding to high potential energy. In cases where periodic behaviour is observed in the history of kinetic energy,two wave-to-wall interactions take place per sloshing cycle, producing two peaks and two troughs. In these cases, the dom-inant liquid sloshing frequency predicted compares well with the exact solution defined in [29].

Depths of 20 mm and lower produce desirable energy dissipative characteristics. Liquid levels in Fig. 3(a)–(d) produceirregular histories, along with an inherent steeper rate of dissipation relative to deeper levels. At these shallow depths pre-dominant travelling wave formation is observed.

Fig. 3. Kinetic energy dissipation histories for different water depths of (a) 5 mm, (b) 10 mm, (c) 15 mm, (d) 20 mm, (e) 40 mm, (f) 60 mm, (g) 80 mm and(h) 100 mm. Constant container width of 400 mm. Note y-axes values vary significantly.



Periodic behaviour is observed as liquid depth increases with the energy dissipation becoming more gradual. The distincttransition from seemingly stochastic behaviour to periodicity is noticeable from Fig. 3(d)–(f). A transitionary phase betweenthe two behaviours is seen in Fig. 3(e). The physical significance of this transition is the change in principal waveform fromtravelling to standing.

Overall time taken to dissipate the total energy increases with liquid depth. The shallowest case of 5 mm dissipates itsenergy the quickest. The deepest case of 100 mm has the slowest rate of energy dissipation. The rate of dissipation decreasesslightly between the depths of 5 mm and 20 mm, yet there is an approximate 5-fold increase in the amount of energy trans-ferred from the excitation to the working fluid. Energy transfer increases with liquid depth, however, cases deeper than20 mm generate standing waves, sacrificing energy dissipation performance.

Probability densities of the kinetic energy histories in Fig. 3 are shown in Fig. 4. Such densities show the amount of timespent within a given kinetic energy range (each range is represented by a vertical bar) as a fraction of the total. An ideal en-ergy dissipater would spend all its time in the left most bar, corresponding to the lowest possible kinetic energy.

Fig. 4(a) through to (h) correspond to the liquid depths of 5, 10, 15, 20, 40, 60, 80 and 100, in descending order. Liquiddepths from 5 mm up to 20 mm possess good energy dissipative qualities relative to deeper levels. These simulations spendwell over half their time within the left most kinetic energy bar. The 20 mm liquid depth remains at the lowest possible ki-netic energy for almost 70% of the total time.

Energy dissipation performance decreases as liquid depth is increased. Probability densities spread to the right as thewater level gets deeper, illustrating the introduction of mid-range kinetic energy. Standing wave development is the causeof this. The transition from travelling to standing wave formation is seen by comparing Fig. 4(d) and (f). In contrast to 20 mmdepth, 100 mm depth spends only 18% of the total time at the lowest possible energy.

A summary of the settling times for varying liquid depths in the 200 mm wide container is shown in Fig. 5. In the figure,the horizontal axis indicates liquid depth whereas the vertical axis represents settling time in s. Settling time is defined asthe time taken for the kinetic energy within the fluid to reach a certain percentage of the maximum amount introduced intothe system. In the figure this percentage is set to 1%, 2%, 5%, 10% and 25%.

There is no significant difference in settling time for depths up to and including 20 mm. Settling time is low at thesedepths, due to the dominant travelling wave development observed. A step change in the settling time is seen betweenthe depths of 20 mm and 60 mm resulting in a steep increase of around 3-fold. Settling times do not change significantlyafter a liquid depth of 60 mm. Principal standing wave development is observed at these deep liquid levels. The transitionfrom travelling to standing wave formation is responsible for the step change in the settling time.

The same analysis in Fig. 5 is shown in Fig. 6 for the 400 mm wide container. As in a 200 mm wide container, settling timeremains relatively low up until and including the 20 mm depth. At 40 mm depth the evolution in waveform from travellingto standing causes a step change in settling time. At the deepest liquid level of 100 mm, 2% residual energy is not reachedwithin the total observation time of 20 s.

Summary of the settling times for the 600 mm wide container is shown in Fig. 7. As in narrower containers, the settlingtime remains relatively low up to and including 20 mm liquid depth. From here, the magnitude increases sharply withliquid depth. At the liquid levels of 80 mm and 100 mm, 2% residual energy is not reached within the total simulation timeof 20 s.

The maximum kinetic energy transferred from the moving base to the fluid for varying liquid depths and for containerwidths of 200 mm, 400 mm and 600 mm, is shown in Fig. 8. In a container of 200 mm width, an optimal energy transfer con-dition is predicted at a depth of around 60 mm. For container widths of 400 mm and 600 mm, more energy is transferred intothe fluid on a per unit mass basis as liquid level increases. A similar optimal energy transfer condition to that seen in 200 mmcontainer width is expected in these wider containers, occurring at liquid depths higher than those simulated.

The effectiveness of a sloshing absorber increases as free surface motion becomes more violent. For a low excitation level,the water in a sloshing absorber of high depth does not slosh as much as that in an absorber of low depth. For a high exci-tation level, the energy transmitted is large and even absorbers with relatively deep liquid levels produce wave breaking[14]. For this reason, the liquid depth at which the transition from travelling wave-to standing wave development takesplace would vary relative to the amount of energy imparted on the absorber from the excitation.

Visualisations of 20 mm liquid depth in a 400 mm wide container are shown in Fig. 9. The scale shown below each frameindicates fluid velocity in metres per second. As in all other cases the fluid is allowed to settle under gravity for a period ofone second, after which the sinusoidal base excitation is applied.

Container motion commences from right to left, producing a travelling wave (travelling wave 1) in the same direction,shown in Fig. 9(a) at t = 1.27 s. The fluid velocity is high at the wavefront, creating steep velocity gradients. As the prescribedmotion changes direction a second travelling wave develops (travelling wave 2) at the left hand container wall. At t = 1.42 s,travelling wave 2 is seen in Fig. 9(b), moving from left to right opposing the direction of travelling wave 1. The total kineticenergy increases steadily from when the first travelling wave is generated, up until maximum is observed at t = 1.57 s. Trav-elling waves 1 and 2 are close to collision at this point, shown in Fig. 9(c).

Immediately after peak kinetic energy is observed, the two opposing travelling waves collide producing a wave-to-waveinteraction near the left side container wall. This interaction is responsible for the first sharp decline in kinetic energy afterthe maximum is observed. A break in the free surface is seen just after as a result of this at t = 1.61 s, shown in Fig. 9(d).

The observed kinetic energy continues to decrease until it reaches the first trough after maximum kinetic energy isreached, seen in Fig. 3(d) at t = 1.71 s. The predicted free surface is shown in Fig. 9(e). Complex behaviour is observed includ-

Fig. 4. Probability densities of kinetic energy shown in Fig. 3.


ing fluid fragmenting and leaving the free surface. Note that the low kinetic energy corresponds to large free surface defor-mation (high potential energy). The fluid velocity range in Fig. 9(e) is almost one half of that in Fig. 9(d).

14

12

10

8

6

4

2

00 20

25%10%5%2%1%

40 60 10080Liqud depth (mm)

Fig. 5. A summary of settling times for different water depths with a constant container width of 200 mm. Legend indicates the level of residual energy as apercentage of the maximum.

Fig. 6. A summary of settling times for different water depths. Constant container width of 400 mm. Legend indicates the level of residual energy as apercentage of the maximum.


Kinetic energy increases slightly until the second peak is observed in Fig. 3(d), at t = 1.80 s. Fluid behaviour at this instantis shown in Fig. 9(f). A travelling wave (travelling wave 3) has been generated at the right hand container wall as a result ofthe excitation motion changing direction, moving from right to left.

Travelling wave 3 collides with the product of travelling waves 1 and 2, shown in Fig. 9(g). This interaction results in arapid decline in kinetic energy at t = 2.10 s. Following this, a wave-to-wall interaction occurs at the right side of the con-tainer, transferring energy from kinetic to potential forms. As a result, the kinetic energy of the fluid is minimal at t = 2.59 s.

The wave-to-wall interaction generates a travelling wave (travelling wave 4) moving from right to left, kinetic energy in-creases steadily. Travelling wave 4 is shown just prior to colliding with the left hand container wall at t = 3.22 s in Fig. 9(h).This point in time corresponds to the fourth peak in kinetic energy in Fig. 3(d). Shortly after a wave-to-wall interaction oc-curs, rapid decrease in kinetic energy ensues.

Formation of travelling waves causes severe deflections and breaking of the free surface with complex velocity patterns. Atravelling wavefront has a discontinuous geometry, which is the product of steep velocity gradients. The wavefront is there-fore the location of high shear stress, responsible for the viscous dissipation of kinetic energy.

In contrast to the 20 mm liquid depth case, 100 mm depth possesses a dominant standing wave form. Energy dissipationis poor as a result of this. Visualisations of 100 mm liquid depth in a 400 mm wide container are shown in Fig. 10. As exci-tation commences from right to left, fluid rises at the right hand container wall, generating a travelling wave in the samedirection, out of phase from the prescribed motion.

Fig. 7. A summary of settling times for different water depths. Constant container width of 600 mm. Legend indicates the level of residual energy as apercentage of the maximum.

Fig. 8. Maximum energy transferred from base excitation to the working fluid per unit mass of fluid for container widths of 200 mm, 400 mm and 600 mm,implementing liquid depths of 5 mm, 10 mm, 15 mm, 20 mm, 40 mm, 60 mm, 80 mm and100 mm.


As the excitation slows and changes direction, fluid rises at the left hand wall. The travelling wave merges with the raisedsection of fluid, generating a breaking travelling wave of high elevation moving from left to right. Fig. 10(a) shows the for-mation of this travelling wave at the left hand container wall at t = 1.58 s. This large free surface deformation corresponds toa low kinetic energy level, approaching the first trough seen in Fig. 3(h).

The curling of the wave produces a section of fluid that separates from the main body, then free falls, making contactagain in the middle of the container. The free surface settles to an almost flat level as shown in Fig. 10(b), correspondingto maximum kinetic energy, observed at t = 1.82 s. The free surface motion develops into periodic behaviour, exhibitingsmaller free surface velocities than the 20 mm depth case. A standing wave is the mechanism in which energy is being trans-ferred between kinetic and potential forms.

Kinetic energy decreases rapidly from this point. Fig. 10(c) shows the free surface shape whilst the system is experiencinga steep rate of change, at t = 1.95 s. A standing wave is rising at the right hand container wall at this point in time. At t = 2.07 speak free surface displacement is seen in Fig. 10(d). The first trough in kinetic energy after the maximum is observed at thisinstant.

Fig. 10(e) and (h) correspond to the 2nd (t = 2.86 s) and 25th (t = 14.55 s) peaks in kinetic energy respectively, after themaximum is observed. These instants in time correspond to close to flat free surfaces. Fig. 10(f) and (g) show two of thewave-to-wall interactions occurring at t = 3.10 s and t = 5.68 s. Large free surface deflections are seen at these instantsand kinetic energy is at a minimum.

Fig. 9. Still frames at instants of interest for 20 mm liquid depth and 400 mm wide container. Fixed velocity scale is from 0 to 1.1 metres per second.


As opposed to the discontinuous nature of the surface flow in shallow cases, motion observed in 100 mm depth assumes aperiodic behaviour. Oscillation occurs between large free surface deformation (high potential energy), and relatively flat free

Fig. 10. Still frames at instants of interest for 100 mm liquid depth and 400 mm wide container. Fixed velocity scale is from 0 to 2.1 metres per second.


Fig. 11. Kinetic energy dissipation histories for different container widths of (a) 5 mm, (b) 50 mm, (c) 75 mm, (d) 100 mm, (e) 125 mm, (f) 150 mm, (g)200 mm and (h) 400 mm. Constant water depth of 20 mm. Note y-axes values vary significantly.


Fig. 12. Probability densities of kinetic energy shown in Fig. 11.


Fig. 13. A summary of settling times for different container widths with a constant liquid depth of 20 mm. Legend indicates the level of residual energy as apercentage of the maximum.


surface shapes (high kinetic energy). This produces less complex surface velocity patterns of smaller magnitude. The inher-ent periodic motion of the standing wave inhibits the formation of travelling waves and wave-to-wave interactions, conse-quently restricting dissipation capabilities.

The effects of tuning have been analysed at constant liquid depth of 20 mm. This depth has been investigated due to beingthe deepest level possessing a predominant travelling waveform, and therefore, its desirable energy dissipative qualities.Histories of kinetic energy for all cases simulated with a constant liquid depth of 20 mm are shown in Fig. 11. Fig. 11(a)–(h) correspond to container widths of 25, 50, 75, 100, 125, 150, 200 and 400 mm, in descending order. Positions indicatedwith a red arrow mark the point of maximum kinetic energy; text indicates magnitude.

Periodic behaviour is observed in all dissipation histories except for the 400 mm container width. This behaviour is mostnoticeable in container widths of and between 50 mm and 150 mm. Overall time taken to dissipate the total energy increasesslightly between the container widths of 25 mm and 200 mm, a significant increase is seen between 200 mm and 400 mmcontainer widths. Reducing the container width increases the number of wave-to-wall interactions that occur within a giventime period, dissipation becomes faster as a result of this.

Probability densities of the kinetic energy histories of all cases with a liquid depth of 20 mm are shown in Fig. 12.Fig. 12(a) through to (h) correspond to container widths of 25, 50, 75, 100, 125, 150, 200 and 400 mm, in descending order.The case in Fig. 12(a) spends around 95% of the total time in the very first bar, indicating the lowest possible energy level. Asthe container width increases the amount of undesirable mid-level energy rises marginally. The 125 mm case shown inFig. 12(e) exhibits a superior energy distribution than any wider container.

A summary of the settling time analysis of varying container width, employing a liquid depth of 20 mm is presented inFig. 13. The general trend shows that the settling time increases as the container width increases, due to decreasing numberof wave-to-wall interactions. However, a trough in settling time magnitude is observed at 125 mm container width.

Maximum kinetic energy transferred from the excitation to the fluid for varying container widths, but for the constantliquid depth of 20 mm is shown in Fig. 14. Optimal energy transfer occurs at the 125 mm container width, correspondingto a dip in settling time, shown in Fig. 13.

00.020.040.060.080.1

0.120.140.160.180.2

0 100 200 300 400 500Container width (mm)

Fig. 14. Maximum kinetic energy transferred for different container widths. Constant water depth of 20 mm.


Of the cases simulated, 20 mm of water in a 125 mm wide container possesses the most desirable energy transfer anddissipation characteristics. Hence, being an optimal 2-D configuration when subjected to the above-mentioned excitation.A 3-D absorber arrangement would have a container length, and width of 125 mm in order to deal with bi-directional motioneffectively.

It has been reported that the absorber mass to structure mass ratio should be at least 1/100 for the absorber to be effec-tive, if the structure possesses more than 2% (very light) damping [14]. In order to reach this mass ratio the above-mentionedgeometry would need to be part of a much bigger, compartmentalized arrangement. For this reason, the number of theseabsorbers needed to effectively control a structure would depend entirely on its mass.

4. Conclusion

The primary form of energy dissipation in a sloshing liquid is shearing of the fluid. Shear stress is increased through pro-viding steep velocity gradients. Two types of interactions are responsible for producing steep velocity gradients, those beingwave-to-wave and wave-to-wall interactions. Travelling waves have the ability to produce both of these interactions,whereas standing waves only produce wave-to-wall interactions.

A travelling wave front has a discontinuous geometry, which is the product of steep velocity gradients. The wave front istherefore the location of high shear stress, responsible for the dissipation of kinetic energy. A standing wave exhibits periodicbehaviour, oscillating between large free surface deformation (high potential energy), and relatively flat free surface shapes(high kinetic energy). Less complex velocity patterns are produced of smaller magnitude.

Travelling waves are generated in shallow liquid levels up to and including 20 mm. As the liquid level increases, a tran-sition to a standing waveform occurs. This transition produces undesirable energy dissipative characteristics resulting in anincrease in settling times.

Therefore, employing a liquid depth higher than 20 mm in the container widths simulated is not recommended whensubjected to the given excitation. An optimum energy transfer condition exists when varying container width, for a constantliquid depth of 20 mm. Of the cases simulated, this optimum condition occurs closest to 125 mm container width.

In application, 3-D absorber geometry would be used, having a container length, and width of 125 mm enabling the effec-tive treatment of bi-directional motion. In order to reach the required absorber mass to structure mass ratio outlined in [14],the absorber geometry described above would need to be part of a much bigger, compartmentalized arrangement. The sim-plest and most effective way of doing this is to stack the necessary number of absorbers in a cube like fashion to provide therequired total absorber mass. Therefore, the number of individual absorbers needed to effectively control a structure woulddepend entirely on its mass.

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A numerical investigation of energy dissipation with a shallow depth sloshing absorber