Damping of liquid sloshing by floating balls

Liquid sloshing is the free-surface oscillation of liquid in a partially filled container subjected to external excitation. Sloshing motion depends on the amplitude and frequency of the excitation. Liquid depth also plays a key role in producing different wave systems.^1,2 When the excitation frequency is close to the frequency of the free-surface wave, resonant occurs, and sloshing amplitude becomes large and causes breaking waves, spilling, splashing, and slamming effects.³ Large amplitude sloshing leads to significant pressure forces on the container's walls and can be a cause for damage of container.⁴ Liquid sloshing involves in a spectrum of engineering applications, not limited to cargo ships,^5–7 transportation of oil and liquefied gas in tankers,^8,9 tuned liquid dampers,^10–13 fuel tanks in space vehicles,¹⁴ nuclear liquid waste storage tank,¹⁵ oscillating water column (OWC) wave energy device,¹⁶ and piezoelectric energy harvester.¹⁷ The investigation of mechanisms that significantly mitigate sloshing motion is of practical importance as there is an increasing demand to mitigate severe liquid sloshing in large containers. Anti-sloshing techniques are also of great significance for the safety of floating liquified natural gas platforms.¹⁸ 

In order to dampen sloshing amplitude, various methods have been proposed by using elastic membranes,^19–21 baffles,^22–26 and elastic wall.²⁷ However, these methods may be difficult to implement in large containers, and they demand continuous maintenance.²⁸ Floating materials for damping of liquid sloshing in large vessels are promising solution over to the fixed baffles in terms of design and maintenance.

There were much research studies, which focused on the damping of sloshing waves. From theoretical studies^28,29 on the damping of sloshing standing waves by bubbles and foam, it was found that the viscous damping due to the tank walls was the main source of damping. These theoretical studies concluded that the damping rate depended on the depth of the layer of bubbles or foam. Further, the theoretical investigation of a single-dominant modal sloshing system,³⁰ based on machine learning of the unknown a priori viscous damping, reported that viscous damping rates of the first natural sloshing mode could depend on the steady-state wave amplitude.

Recent research^31,32 on air–oil–water systems reported that the damping rate of gravity waves depended on the thickness of the oil layer. Fish-oil film was used in suppressing wind generated waves.³³ From a very recent numerical study³⁴ on sloshing mitigation using fixed and floating baffles, it was reported that the floating baffles effectively mitigate the first sloshing mode than the fixed baffles, for both intermediate and finite water depths.

Different floating materials were used in the previous studies^18,35–37 to comprehend the damping mechanisms. In-extensible covers³⁵ were used to dampen the monochromatic surface gravity waves in a flume. It was reported that the damping was due to no-slip boundary conditions. Plastic foam spheres¹⁸ were used to dampen the sloshing waves in a tank. It was reported that as the number of layers increases, the first modal sloshing frequency decreases slightly and further, the nonlinear effects diminish. In a recent study, spherical hydrogel beads³⁶ were utilized to dampen the sloshing waves in a rectangular tank. It was observed that the decay in the amplitude of the sloshing waves showed early time exponential damping, followed by a finite-time arrest approaching zero amplitude. Floating polystyrene spheres³⁸ were used to dampen the standing gravity Faraday wave in a rectangular vessel. It was shown that the use of the highly concentrated polystyrene particle layer modified the wave dynamics and suppressed the breaking waves. Floating plastic foam spheres³⁷ were used to mitigate water sloshing in a recent numerical study using ABAQUS software. For simplicity, in this study, the foam–foam, foam–wall, and foam–water friction forces across their interface are defined via the Coulomb friction model with a coefficient of 0.5. It has been reported that interactions between floating foams play a major role in the energy dissipation rate.

The results and analysis of the damping of sloshing water waves by floating particles could shed light on the mechanism for the decay of ocean waves propagating into the marginal ice zone.^36,39 However, the physical processes pertaining to damping mechanisms are still not explored fully. Further, there are open questions concerning the damping mechanisms, which are listed as follows:

• Can floating balls be effective for damping of either shallow or intermediate or finite water-depth sloshing?

• Can floating balls suppress the nonlinear behavior of sloshing waves?

• What could be the effects of floating balls in damping of higher sloshing modes?

• What could be the main sources of damping?

The present study aims to answer these questions.

In the present study, floating rigid plastic balls are used to dampen sloshing amplitudes. The water depth is varied for achieving shallow, intermediate, and finite-water depth regimes. The study aims to comprehend the effect of floating balls on the reduction of sloshing amplitudes. The motion of the liquid has an infinite number of natural frequencies. However, the lowest few modes are most likely to be excited by the motion of a container.¹ Therefore, the present study accounts for first, third, fifth, seventh, and ninth sloshing modes. We build on recent results using a shake-table to investigate liquid sloshing motion.^39,40

The main contribution of this paper is in proposing a new experimental model to investigate damping of liquid sloshing in a large tank by floating balls. A comprehensive analysis of physical processes involved in damping mechanisms and larger experimental data for different water depths, and higher sloshing modes benefit new value to the field of liquid sloshing dynamics.

The outline of the present study is as follows: Sec. II presents the experimental setup and procedure. The experimental results are discussed in Sec. III. In Sec. IV, the physical processes associated with damping mechanisms are detailed. Finally, in Sec. V, the salient conclusions of the present study are highlighted.

A top view of the experimental sloshing tank as in Fig. 1 shows that the floating balls occupy their position on the free-surface in such a way that the space among balls is not uniform. The available spatial distance among balls causes horizontal, heave, and rotational motions. The placement of the wave probe does not significantly affect the motions of the balls lying in the vicinity of the probe, due to the free space available between the tank wall and the balls. Table I provides specific details about the balls.

A shake-table is used to conduct the forced sloshing experiments, and a schematic is shown in Fig. 2. The shake-table has a 1 × 1 m² steel platform, which moves only in a horizontal direction. As the table is designed for testing of large structures, it is not an issue to conduct sloshing experiments for different water depths. The table generates harmonic motion for different excitation frequencies and amplitudes. A linear variable differential transformer (LVDT) is fixed at an edge of the table to ensure the accuracy of the forcing frequency. A tank of length L = 0.965 m, width W = 0.439 m, and height H = 0.482 m is fabricated using an acrylic material of 0.015 m thickness. The tank is kept open on the top and filled with water to a desired depth (h). A video camera is used to record the sloshing wave patterns. The vertical displacement of the free-surface at a location near the tank wall is measured by a wave probe, and the placement of a wave probe is shown in Fig. 2. The probe consists of two 0.003 m diameter stainless steel wires spaced 0.01 m apart. It has been calibrated for each test, and it is positioned 0.005 m from the tank wall. From repeated static measurements, it has been found that the measurements of the probe have less than a 1% deviation.

The water depths of h = 0.3, 0.2, and 0.1 m are considered for the experimental investigations. The water-depth ratios h/L = 0.310 88, 0.207 25, and 0.103 62 are referred to finite, intermediate, and shallow water depths, respectively. We can provide a criterion to define the shallow, intermediate, and deep-water sloshing based on the ratio of water depth to tank length $( h / L )$⁠. The ranges $h / L ≥ 0.24 , 0.1 ≤ h / L ≤ 0.24$⁠, and $h / L ≤ 0.1$ indicate the finite water-depth sloshing, intermediate water-depth sloshing, and shallow-water sloshing regimes, respectively. More details can be found in the text books.^1,4 The excitation amplitude (A) of 0.003 m is taken for all tests. The excitation amplitude ratio is A/L = 0.0031. Table II provides the experimental test cases. The sampling frequency of 100 Hz is taken for the data acquisition to capture sharp fluctuations of the free-surface. The sloshing responses of all test cases are post-processed using MATLAB.

Forced sloshing experiments were conducted with and without balls on the free-surface. After balls were added on the free-surface, some ambient water was removed so as to maintain the constant water depth for both sloshing experiments with and without balls. This was carefully done so that the sloshing frequency was not affected by an increase in water depth due to the addition of the balls. For a given excitation amplitude and frequency, the horizontal motion of the shake-table generates sloshing waves in the partially filled water tank. The excitation frequency was varied for a wide range starting from a small value to a higher value of frequency so that it covered many sloshing modes. For each excitation frequency, the free-surface response was measured by the wave probe.

The motivation of the present study was to understand the damping of large amplitude sloshing waves by floating balls. Therefore, in the first place, the amplitudes of sloshing waves without floating balls were measured for a range of excitation frequencies, for a given excitation amplitude and water depth. For each excitation frequency, the forced sloshing tests were carried for about 500 s. Similarly, for the same range of excitation frequencies, the amplitudes of sloshing waves with floating balls were measured. The above-mentioned procedure was followed for all water depths.

In Eq. (1), ω_n is the angular frequency of sloshing; g is the acceleration due to gravity and is taken to be 9.81 ms⁻²; and h is the water depth; and $k n = n π / L$ is the wave-number for $n = 1 , 2 , 3 …$⁠. For the experiments, the excitation frequencies are taken to be the modal frequencies obtained from Eq. (1). The shake-table is excited at one of these frequencies, and the corresponding wave amplitude is measured by the wave probe. For each measured wave, the sloshing frequency is calculated from the location of the appropriate peak of the fast Fourier transform of the corresponding time history. For tank length L = 0.965 m and water depth h = 100 mm, the measured sloshing frequencies are compared with the theoretical frequencies as given in Table III. Similarly, for the water depths of 200 and 300 mm, the comparisons are given in Tables IV and V, respectively. It is found that the measured and theoretical frequencies match closely. The reason for the small discrepancies is the limitations of the shake-table, which could only be tuned by an increment or decrement of 0.02 Hz making exact tuning impossible. Another reason for the discrepancies is the limitations of the linear theory, which discard surface tension, viscosity, and other dissipating processes on the tank walls.

Forced sloshing response time histories are presented in Figs. 3 and 4. For each excitation frequency, the response was obtained for about 500 s. Figure 3 shows the response time histories of sloshing without balls, for the water depth of 300 mm and excitation amplitude of 3 mm. The excitation frequencies are chosen in the neighborhood of the first natural frequency, with a range from 0.48 to 1.22 Hz. As the excitation frequency increases, the wave amplitude increases gradually until resonant takes place [Fig. 3(a)]. Further increase in the excitation frequency leads to resonant behavior and beating phenomenon as seen in Fig. 3(b). The resonance is observed when the excitation frequency equals 0.78 Hz. An increase in the excitation frequency away from the first natural frequency results in a drastic decrease in amplitude as seen in Fig. 3(c).

The response time histories of sloshing with balls, for the water depth of 300 mm and excitation amplitude of 3 mm, are presented in Fig. 4. Excitation frequencies are the same as considered in Fig. 3. As the excitation frequency increases, the wave amplitude increases and the resonance is observed when the excitation frequency equals 0.76 Hz. In the case of sloshing without balls [Fig. 3(b)], a strong beat phenomenon is noticed as this phenomenon is expected to occur when the excitation frequency is closer to the natural frequency.^1,4 The amplitude of transient beat response decreased as it reached the steady state. Beat phenomenon has not been observed in the case of sloshing with balls (Fig. 4), and hence, the sloshing amplitudes have not shown the decreasing trend compared with the corresponding cases without floating balls. In fact, the floating balls act like an added mass to the liquid free-surface and the free-surface becomes relatively stiffer.

On one to one comparison of Figs. 3 and 4, we identify the change in sloshing behavior and shift in the resonant frequency due to floating balls. These findings were similar to the previous results obtained by using floating foams¹⁸ and polystyrene particles.³⁸ Moreover, interestingly, the resonant beating effects were suppressed by floating balls. In Secs. IV A–IV E, we discuss the damping of shallow, intermediate, and deep-water sloshing in detail with a consideration of higher natural modes.

Shallow-water sloshing experiments are performed for a wide range of excitation frequencies, starting from relatively low to higher frequencies, which cover the first to ninth modal sloshing frequencies. For each excitation frequency, the maximum value of the sloshing amplitudes is measured and plotted as a function of excitation frequency. Figure 5 compares the frequency response curves of sloshing with and without balls for the water depth of 100 mm. In Fig. 5, the vertical axis shows the maximum amplitudes of sloshing waves, and the horizontal axis shows the excitation frequency. The dotted line represents the frequency response curve of sloshing without balls, whereas the solid line represents the frequency response curve of sloshing with balls. The peaks in the frequency response curves indicate the primary resonant frequencies. The peaks correspond to the first, third, fifth, seventh, and ninth modal frequencies. The solid line shows that the peak amplitudes decrease as the frequency increases. This decreasing trend indicates that sloshing energy reduces for higher modal frequencies. The smaller peaks at 0.7783 Hz (between first and third modes) and at 1.4623 Hz (between third and fifth modes) exhibit the secondary resonance of sloshing without balls.

The amplitudes of the peaks in the response curve of sloshing with balls are much lower than those of sloshing without balls. A greater difference between peak amplitudes clearly shows that the floating balls dampen sloshing waves effectively, for all odd modes of sloshing. The dotted curve also shows that the amplitudes of the peaks reduce as the frequency increases. This indicates that the sloshing energy reduces for higher modal frequencies. Unlike, in the response curve of sloshing without balls, there are no significant secondary peaks observed in the response curve of sloshing with balls. This implies that the floating balls suppress the nonlinear behavior of sloshing. Moreover, the peak frequencies of the response curve of sloshing with balls are shifted and slightly lower than those of sloshing without balls. This phenomenon exhibits the softening type behavior of sloshing with balls, and this behavior is clearly seen for higher modal frequencies.

Table VI compares the peak amplitudes of sloshing with and without balls. The difference (%) between the peak amplitudes for each mode increases with an increasing sloshing modal frequency. For the first and ninth modes, differences of 78.269% and 91.962% are, respectively, observed.

When the floating balls present on the free-surface, the resonant sloshing frequencies are slightly shifted and are lower than the resonant frequencies of sloshing without balls. Table VII provides the difference (%) between the resonant frequencies of sloshing with and without balls, and the difference monotonically decreases with an increase in the modal frequency. It indicates that the floating balls greatly affect the lower sloshing modes than the higher modes. Further, it can also be inferred that the balls make the sloshing system to behave as a softening system, in comparison with the sloshing without balls.

Sloshing of intermediate water depth and its damping behavior are discussed here. Figure 6 compares the frequency response curves of sloshing with and without balls, for the water depth of 200 mm. The dotted line shows the response curve of sloshing without balls, and the solid line exhibits the response curve of sloshing with balls. In the case of sloshing without balls, the response curve shows a non-strict monotonic decrease in peak amplitudes as the frequency increases. The peak amplitude of the first mode of sloshing is larger than that of the other modes of sloshing. The peak amplitude of the third mode has reduced in comparison with the first mode. Further, it is seen that the peak amplitude of the fifth mode has increased sufficiently in comparison with the third mode. Subsequently, the peak amplitudes correspond to the seventh and ninth modes that have reduced successively. Peak amplitudes of fifth, seventh, and ninth modes follow a monotonic decreasing trend. A smaller secondary peak at 1.7817 Hz between the third and fifth mode is observed due to secondary resonant sloshing.

The nonlinear behavior of sloshing waves causes a non-monotone trend of the peak amplitudes in the response curve of sloshing without balls. While contacting the experiments, it had been observed that sloshing response became three-dimensional wave systems with run-up at tank corners, for the third mode resonance. In that case, the wave elevations at tank corners were of higher amplitude than the wave elevations at the wave probe. Therefore, the frequency response curve of sloshing without balls showed a drastic drop in the peak amplitude around the third mode. The reason for the non-linear three-dimensional waves could be that the excitation frequency (1.5268 Hz) of the third mode (3, 0) is relatively close to the frequency (1.3388 Hz) of the coupled mode (1, 1) and very close to the frequency (1.5185 Hz) of the coupled mode (2, 1). The excitation frequency might have excited the third mode and the coupled modes simultaneously. However, non-linear three-dimensional waves were not observed for the third mode of sloshing with balls. It is worth to mention that the frequencies of the coupled sloshing modes (1, 1) and (2, 1) are calculated using the formula given in Eq. (2).

The response curve of sloshing with balls shows drastic reductions in the peak amplitudes in comparison with the response of sloshing without balls, for each sloshing mode. Further, the response of sloshing with balls exhibits a trend such that the peak amplitudes decrease monotonically with an increase in the modal frequency. The softening behavior of sloshing with balls is clearly seen for all modes. The comparison of response curves of sloshing with and without balls indicates that the balls make the sloshing wave systems to behave almost linearly.

Table VIII compares the peak amplitudes of sloshing with and without balls. In general, the difference (%) between the peak amplitudes is large for higher modes, except for the third mode, in comparison with that of the first mode. A minimum difference is observed for the third mode, it is about 40.719% (the reason being non-linear three-dimensional effects), and a maximum difference of about 90.858% is noted for the seventh mode. Nevertheless, the differences observed for the fifth, seventh, and ninth modes do not vary significantly.

The resonant frequencies of sloshing with and without balls are compared in Table IX. The resonant frequencies of sloshing with balls are observed to be slightly lower than those of sloshing without balls. A maximum difference of about 3.45% is observed for the first mode, and a minimum difference of about 1.563% is noted for the third mode. The table also indicates that the first and ninth modal frequencies of sloshing with balls reduced largely compared to other modes. Unlike as observed for shallow water depth of 100 mm, the difference (%) between the resonant frequencies of sloshing with and without balls does not follow the decreasing trend with an increase in the modal frequency, for intermediate water depth of 200 mm. Hence, it is evident from Figs. 5 and 6 and Tables VI–IX that the sloshing behavior is mainly characterized by water depth.

The frequency response curves of sloshing with and without balls, for the water depth of 300 mm, are presented in Fig. 7. The response curve of sloshing without balls shows that the peak amplitude of the first mode is larger than the other modes. The peak amplitude of the third mode is relatively smaller than the first and fifth modes. It had been experimentally observed that sloshing response became non-linear three-dimensional wave systems for the third mode resonance. In that case, the wave elevations at tank corners were of higher amplitude than the wave elevations at the wave probe. The reason for the three-dimensional waves could be that the excitation frequency (1.5534 Hz) of the third mode (3, 0) is relatively close to the frequency (1.3853 Hz) of the coupled mode (1, 1) and very close to the frequency (1.5459 Hz) of the coupled mode (2, 1). From the fifth mode, the peak amplitudes have reduced monotonically till the ninth mode. Between third and fifth modes, there is also a small peak at 1.8053 Hz due to secondary resonance.

The response curve of sloshing with balls has smaller peak amplitudes than that of sloshing without balls. Due to balls, a substantial reduction in peak amplitudes has been achieved for third, fifth, seventh, and ninth modes. Unlike in the case of sloshing without balls, the non-linear three-dimensional sloshing wave systems were not observed for the third mode of sloshing with balls. Between first and third modal peaks, a secondary resonant peak at 1.2736 Hz is observed. Further, the softening behavior of sloshing system with balls is clearly observed for higher modes.

Table X compares the peak amplitudes of sloshing modes with and without balls. The difference between the peak amplitudes increases monotonically with an increase in the modal frequency. A minimum difference of about 27.835% and a maximum difference of about 92.031% are observed for the first and the ninth modes, respectively. Nevertheless, the differences observed for seventh and ninth modes do not vary significantly.

The difference between the resonant frequencies of sloshing with and without balls is given in Table XI. A maximum difference of about 3.032% is observed for the first mode of sloshing, and a minimum difference of about 1.539% is noticed for the third mode. The differences observed for the fifth and seventh modes are close.

From Tables VI–XI, we emphasize the following:

• Water depth plays a key role in sloshing dynamics. Floating balls greatly suppress the peak amplitudes of the sloshing modes, for shallow, intermediate, and finite water depths.

• Floating balls significantly suppress the peak amplitude of the first mode of sloshing for the water depth of 100 mm (h/L = 0.1036), in comparison with the other water depths of 200 mm (h/L = 0.2072) and 300 mm (h/L = 0.3108). It indicates that the floating balls act as an effective dampening device for shallow water sloshing than intermediate and finite water-depth sloshing waves.

• For shallow water sloshing, the amplitude reduction was observed to be 78.269%, 78.572%, 83.504%, 86.206%, and 91.962% for the first, third, fifth, seventh, and ninth modes, respectively. For the intermediate water depth, the amplitude reduction was observed to be 65.272%, 40.719%, 88.518%, 90. 858%, and 90.508% for the first, third, fifth, seventh, and ninth sloshing modes, respectively. Further, for the finite water depth, the amplitude reduction was observed to be 27.835%, 66.937%, 83.268%, 91.261%, and 92.031% for the first, third, fifth, seventh, and ninth modes, respectively.

• The floating balls drastically lower the first modal sloshing frequency with a reduction of about 13.05% for the water depth of 100 mm (h/L = 0.1036).

• For all water depths considered, it is observed that the floating balls largely reduce the frequency of the first mode than that of the higher modes.

• For shallow water sloshing, the frequency shift was observed to be 13.050%, 6.893%, 2.343%, 2.185%, and 1.814% for first, third, fifth, seventh, and ninth modes, respectively. In the case of intermediate water depth, the frequency shift was observed to be 3.450%, 1.563%, 2.270%, 2.285%, and 2.721% for first, third, fifth, seventh, and ninth modes, respectively. Further, for finite water depth, the frequency shift was observed to be 3.032%, 1.539%, 2.270%, 2.285%, and 1.830 for first, third, fifth, seventh, and ninth modes, respectively.

The previous studies on sloshing damping by foams²⁸ and polystyrene particle³⁸ shown that the foams and polystyrene particles had not significantly affected the resonant frequency of the finite water-depth sloshing. However, in the present study, it is observed that the resonant frequency is reduced about 13.05% for shallow water sloshing. This reduction is quite significant. The reason for the frequency shift is the following. As the floating balls produce added mass to the sloshing system and the floating balls do not affect the restoring force of the sloshing response, the added mass reduces the resonant sloshing frequencies. It is worth to mention that the past studies also reported that the use of floating³⁴ and fixed baffles⁴² inside the sloshing tank results in a lower first mode natural frequency than that of the tank with no internal structures.

In the pure sloshing states, viscous dissipation induced by the motion of liquid free-surface against the walls is the main source of damping.^28,30,43 Nevertheless, many physical mechanisms were observed while conducting the sloshing experiments with balls. The following mechanisms play a key role in the damping processes of waves in the tank:

• Decomposition of liquid free-surface into many small free-surfaces, as gaps between any two balls.

• Coupled heave, rotational, and horizontal motions of the balls on the liquid free-surface.

• Motions of liquid shear flow, between the tank walls and balls, due to sliding motions of the balls on the tank walls.

• Collisions among the balls due to horizontal motions of them.

• Oscillations of liquid–liquid interface that exists in the gap between any two balls.

• Oscillations of the liquid boundary layer present around the perimeter of each ball.

• Acoustic noise due to ball–ball collisions and sliding motions of the balls.

The contact mechanism between balls and tank wall is presented in Fig. 8. There is a small gap, on a more microscopic level, between a ball and the tank wall, by which water passes upward and causes capillary rise. The contact angle between liquid free-surface and tank walls is not orthogonal due to the presence of balls. This shows the ability of water to flow up on the tank walls.

Figure 9 exhibits the schematic diagram of the motions of the liquid shear layers on the walls. Figure 9(a) presents the vertical motion of the dome of the shear layer, which undergoes up and down motions. Figure 9(b) presents the horizontal motion of the neck of the shear layer, which undergoes inward shrinking and outward expansion motions. When the dome of the shear layer undergoes upward motion, the neck of the shear layer undergoes inward shrinking motion; when the dome of the shear layer undergoes downward motion, the neck of the shear layer undergoes outward expansion motion. Figure 9(c) represents the rotational motions of the shear layer; the motion would be either in the clockwise or counterclockwise directions depending upon the ball motions. However, the reason for change in the direction of the rotational motion is not investigated here.

The schematic diagram of the top view of the motions of the balls on the liquid free-surface is presented in Fig. 10. At the left end, there is a small rectangular strip representing a wave probe. Figure 10(a) presents the horizontal motions of a ball in the longitudinal and lateral directions. Figure 10(b) presents the circular liquid boundary layers around the perimeter of the two balls and the gap between them. In the experiments, very small oscillations of liquid are observed in the gap between any two balls. Figure 10(c) presents an almost idle ball, which undergoes insignificant motions, and the idle balls are observed to lie in the nodal points of the sloshing modes. Figure 10(d) shows the scenario where two balls collide each other and so the liquid boundary layer around the perimeter of one ball strongly couples with the liquid boundary layer around the perimeter of another ball. Small oscillations of liquid boundary layers are also observed. Both the phenomena shown in Figs. 10(b) and 10(d) contribute significant dissipative effects around the perimeter of each ball. In fact, it has been observed that the liquid boundary layer around the perimeter of a ball strongly interacts with the liquid boundary layers of the neighboring balls. Finally, Fig. 10(e) presents the rotational motion of a ball, and it would be of either clockwise or counterclockwise directions. However, the reason for change in the direction of the rotational motion of balls has not been found from the present study.

Similar to the previous study,²⁸ where bubbles close to the walls had a significant impact on the dissipation of energy, the floating balls close to the tank walls were active in motion, and hence, they significantly dissipated the wave energy. Also, it is worth to mention that the balls on the anti-nodal points of the sloshing modes [refer Figs. 11(d) and 11(e)] were observed to undergo predominant heave motion.

Experimental snapshots of the anti-symmetric sloshing modes are presented in Fig. 11 for the water depth of 300 mm with the amplitude of excitation of 3 mm. The free-surface wave elevation measured at a fixed location does not provide a complete information on the sloshing state. Hence, these snapshots will be useful in getting global information. For example, from Fig. 11, one can visually see the liquid mass participation in the sloshing modes. For the first mode, the amount of liquid mass participation in the sloshing motion is higher than the third mode. The layer of balls on the free-surface follows the wave pattern of sloshing modes as predicted by the potential flow based theoretical models.^1,4,44 The anti-symmetric sloshing mode shapes are unaffected by the floating balls.

Figures 11(a)–11(c) (Multimedia view) present the three important stages of the first mode. Figure 11(a) presents the first mode with the maximum elevation at the left end wall. At this stage, the liquid center of gravity (CoG) shifts horizontally toward the left. The equilibrium position of the free-surface is shown in Fig. 11(b), where the CoG remains exactly at the mid-tank. Further, Fig. 11(c) exhibits the first mode with the maximum elevation at the right wall. At this stage, CoG shifts horizontally toward the right. Both the left and right end walls are the anti-nodal points for the first mode. If we closely examine the motion of the balls near the end walls, the balls move up and down depending upon the rise and fall of the free-surface. Similar descriptions hold good for the third mode as shown in Figs. 11(d)–11(f) (Multimedia view). For the third mode, the shift in CoG is not clearly visible from these figures. However, it was reported that for the first mode the shift in CoG is much larger than for other modes.^1,43,45 The above-mentioned three stages will continue sequentially for both the modes. From Figs. 11(a) and 11(c), it is clear that the balls undergo heave motions as the free-surface moves upward and downward along the end walls. Similarly, Figs. 11(d) and 11(e) also show the heave motions of the floating balls on the anti-nodal points of the sloshing mode. Also, it is clear that the node occurs at the mid-tank, where the motions of the balls were observed to be insignificant. The arrow marks in Figs. 11(a), 11(c), 11(d), and 11(f) indicate the rise and fall of the free-surface profile at the locations of the anti-nodal points where the balls undergo predominant heave motion. This observation of heave motion of the balls is in line with the heave motion of the floating foams as reported in past studies.^18,37

In the experiments of sloshing with balls, the global sloshing waves were observed to be almost linear and two-dimensional flow. After putting the balls on the free-surface, the floating balls produce an added mass to the liquid free-surface and so the liquid mass participating in the sloshing motion (for the first mode sloshing with no balls, from theoretical calculations,¹ it has found that about 62.38% of the total liquid mass is expected to participate in the sloshing motion; however, for the third mode, only 9.97% of the total liquid mass is expected to participate) reduces and the liquid mass moving in unison with the tank increases. Therefore, the global sloshing oscillations become planar in motion. The liquid mass experiences sloshing at the free-surface, and the liquid mass that moves in unison with the tank is important parameter in developing the equivalent mechanical models for sloshing.¹ Equivalent models can be developed for sloshing with balls by a multiple degrees of freedom system.

The liquid behavior in rectangular tanks may be treated as two-dimensional or three-dimensional flow depending on the tank length and liquid depth. Also, the non-linear sloshing response is expected for shallow and finite water depth cases even for small amplitudes of excitation with resonating excitation frequencies. Theoretical models are available for small amplitude, linear, and weakly non-linear sloshing systems. One such model is based on the Duffing equation. At low amplitudes, the Duffing equation governs sloshing motion and at large amplitudes, different wave systems including wave breaking, spilling, and run up at the walls emerge and so the sloshing dynamics do not obey the Duffing equation.⁴³ Sloshing without balls exhibits three-dimensional non-linear wave systems in the case of third mode of excitation for the water depth of 300 mm with the amplitude of excitation of 3 mm. The sloshing states are presented in Fig. 12. The reason for the three-dimensional waves could be that the frequency (1.5534 Hz) of the third mode (3, 0) is relatively close to the frequencies (1.3853 and 1.5459 Hz) of the coupled modes (1, 1) and (2, 1). These coupled frequencies are calculated from the relation (2). However, three-dimensional waves, breaking waves, spilling, and run up are suppressed due to balls.

Figure 12 shows a sequence (a, b,…, l) of snapshots of the global sloshing states. In Fig. 12(a) (Multimedia view), a quasi-two-dimensional profile shows the maximum and minimum wave elevations at the left and right tank walls, respectively. The tank corners are denoted by C₁, C₂, C₃, and C₄. Figure 12(b) shows a profile of a diagonal-like wave (a non-planar unstable surface profile), which is generally observed in square and nearly square-base tanks,^40,46 in which the wave elevation at the left wall corner (C₂) has the maximum elevation (as a wave run-up on the corner) and the corresponding diagonal corner (C₄) has the minimum. Also, a sharper breaking crest is seen at mid-tank. Figure 12(c) exhibits a non-smooth profile where there are maximum and minimum surface elevations at the right and left tank walls, respectively. A three-dimensional wave system (a non-planar surface wave) is visible in Fig. 12(d). At corners of the left wall, the profile shows the maximum elevation at one corner and minimum at another corner. Similarly, at the corners of right walls, the profile shows the maximum elevation at one corner and minimum at another corner. On seeing the left and right walls, the maximum and minimum surface elevations are observed at inline corners: C₁–C₄ and C₂–C₃, respectively. Further, a wave crest is seen only at the back wall. Figure 12(e) presents a wave system with different elevations at corners.

A third modal wave system is shown in Fig. 12(f), with the maximum and minimum elevations at the left and right walls, respectively. The wave form has a sharper crest and a flatter trough. Further, at the left wall, the lateral wave effect is visible. Figure 12(g) shows a wave profile as depicted in Fig. 12(d), but the corners are interchanged. The corners C₂–C₃ have the maximum elevation, whereas the corners C₁–C₄ have the minimum elevation. At the front wall, a plateau peak⁴³ is observed at mid-tank. A twisted wave profile with different elevations at the corners is presented in Fig. 12(h). A crest with a spike is seen around the mid of the front wall. Figure 12(i) shows the third modal wave system (quasi-two-dimensional flow state), with the maximum and minimum elevations at the right and left walls, respectively. The wave form has a sharper crest and a flatter trough. Also, the lateral wave effect is visible at the right wall (a non-smooth wave run-up at right wall). Figure 12(j) shows the maximum elevation at corner C₄, a spiky crest is seen at the back wall, and the free-surface has significant undulations along the lateral directions too. In comparison with Fig. 12(j), Fig. 12(k) shows a relatively smooth system; however, the minimum and maximum of wave elevations are not the same along the tank side walls. Finally, Fig. 12(l) shows a wave profile, which is almost like the one presented in Fig. 12(a). Then, the wave patterns repeat the sequence. Observation on these sloshing profiles indicate the competition and interaction between longitudinal waves and lateral waves, which make the liquid motion complex and so the Duffing equation becomes invalid to model. In this case, a three-dimensional flow model should be considered.

In the sloshing tank with no balls on the free-surface, the liquid mass participating in sloshing oscillations is higher than the liquid mass moving in unison with the tank. Therefore, we may expect the non-linear motions as demonstrated in Fig. 12. Further, in sloshing tanks with no internal obstructions, there is a critical depth,¹ above which the liquid waves possess soft nonlinear spring characteristics and below that level, the nonlinearity is of the hard type. The present study revealed that the floating balls make the sloshing system to behave as a soft spring. Also, it can be proved that for sloshing tanks with floating materials on the liquid free-surface, the critical water depth will be lower than that of the corresponding clean sloshing tanks.

The present study investigates damping of sloshing due to floating balls distributed along the liquid free-surface. A new set of experiments was conducted in the present work considering shallow, intermediate, and finite water depths in a rectangular tank, which was positioned in a shake-table. Sloshing waves were generated in the tank by exciting the shake-table due to horizontal harmonic motions. As the first step of the work, experimental tests for sloshing without balls were performed. After that, investigations were focused on the experimental tests and results for sloshing with balls. The following salient conclusions were made from the present investigations:

Floating balls were more effective as a damping device for shallow and intermediate water depths compared to finite water depth. In the case of first sloshing mode, the amplitude reduction was achieved to be 78.269%, 65.272%, and 27.835% for shallow, intermediate, and finite water depth, respectively.

For all water depths, floating balls dampen the higher sloshing modes effectively compared to lower modes. In the case of shallow water sloshing, the amplitude reduction was observed to be 78.269%, 78.572%, 83.504%, 86.206%, and 91.962% for first, third, fifth, seventh, and ninth modes, respectively.

Floating balls drastically lowered the frequency of all the modes of shallow water sloshing in comparison with intermediate and finite water depths. The decrease in the first modal frequency was observed to be 13.05%, 3.45%, and 3.032% for shallow, intermediate, and finite water depth, respectively.

Heave, rotational, and horizontal motions of floating balls, ball–ball collisions, oscillations of liquid boundary-layer around the perimeter of each ball, oscillations of liquid–liquid interface present in the gap between any two balls, the motions of liquid shear flow between tank wall and balls, and acoustic noise were found to be the main sources of damping.

Moreover, three-dimensional nonlinear sloshing states, such as breaking, spilling, run up at the walls and beating phenomenon, were suppressed by floating balls.

The author would like to thank the anonymous referees for their valuable suggestions and comments that improved the manuscript. Also, the author would like to thank the Department of Ocean Engineering, Indian Institute of Technology Madras, for their experimental facilities.

The authors have no conflicts to disclose.

Saravanan Gurusamy: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.