Sloshing in Upright Circular Containers

Raynovskyy, Ihor; Timokha, Alexander

doi:10.1201/9780429356711

Cited by 15 publications

(7 citation statements)

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“…Despite the fact that the experiments performed by Reclari (2013) and Reclari et al. (2014) were made for non-dimensional fluid depths and , which lie slightly beyond the applicability threshold of the multimodal theory ( should be as stated by Raynovskyy & Timokha (2020)) and imposed by the occurrence of secondary resonances, the authors found a quantitative good agreement with the experimental observations associated with hard-spring-type single-crest (SC) swirling.…”

Section: Introductionmentioning

confidence: 79%

An amplitude equation modelling the double-crest swirling in orbital-shaken cylindrical containers

2022

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Container motion along a planar circular trajectory at a constant angular velocity, i.e. orbital shaking, is of interest in several industrial applications, e.g. for fermentation processes or in cultivation of stem cells, where good mixing and efficient gas exchange are the main targets. Under these external forcing conditions, the free surface typically exhibits a primary steady-state motion through a single-crest dynamics, whose wave amplitude, as a function of the external forcing parameters, shows a Duffing-like behaviour. However, previous experiments in laboratory-scale cylindrical containers have revealed that, owing to the excitation of super-harmonics, diverse dynamics are observable in certain driving-frequency ranges. Among these super-harmonics, the double-crest dynamics is particularly relevant, as it displays a notably large amplitude response, which is strongly favoured by the spatial structure of the external forcing. In the inviscid limit and with regards to circular cylindrical containers, we formalize here a weakly nonlinear analysis via a multiple-time-scale method of the full hydrodynamic sloshing system, leading to an amplitude equation suitable for describing such a double-crest swirling motion. The weakly nonlinear prediction is shown to be in fairly good agreement with previous experiments described in the literature. Lastly, we discuss how an analogous amplitude equation can be derived by solving asymptotically for the first super-harmonic of the forced Helmholtz–Duffing equation with small nonlinearities.

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Section: Introductionmentioning

confidence: 79%

An amplitude equation modelling the double-crest swirling in orbital-shaken cylindrical containers

2022

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“…Nevertheless, the reintroduction of surface tension in the multimodal theory can be challenging. In particular, it is not clear yet how to account for static meniscus and contact line dynamics (Raynovskyy & Timokha 2020). Moreover, in these cases, the element of the Fourier basis (Bessel) functions no more coincide with the actual natural sloshing modes.…”

Section: Discussion: Present Analysis Vs the Narimanov-moiseev Multim...mentioning

confidence: 99%

“…Below a critical liquid depth, typically H cr ≈ 1.05 for circular cylinders, such amplification (secondary or internal resonances) can happen in the vicinity of the primary resonance. These cases, which require a reordering of the asymptotic scaling, can still be tackled in the framework of the multimodal theory by employing the so-called adaptive modal approach (see chapters 7-9 of Faltinsen & Timokha 2009;Raynovskyy & Timokha 2020). However, secondary resonances may also occur more broadly even far from the primary resonance zone and for H > H cr , as in the case of the DC super-harmonic resonance.…”

Section: Discussion: Present Analysis Vs the Narimanov-moiseev Multim...mentioning

confidence: 99%

“…The occurrence of the jump-down transition was to be expected as it is produced by dissipative mechanisms (see also § 6.5), which are overlooked by the present inviscid analysis. The associated damping, which is a function of the wave amplitude and of the forcing acceleration amplitude (see Raynovskyy & Timokha (2018b), Raynovskyy & Timokha (2020) and the discussion in Appendix A of Bongarzone et al 2022a), is responsible for the modulation in the phase lag between the external driving and the wave response, which was shown by Bäuerlein & Avila (2021) (for unidirectional sloshing waves in a rectangular container) to be of crucial importance for a correct prediction of the jump-down frequency.…”

Section: Wave Amplitude Saturation: Theoretical Predictions Vs Experi...mentioning

confidence: 99%

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Super-harmonically resonant swirling waves in longitudinally forced circular cylinders

2023

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Resonant sloshing in circular cylinders was studied by Faltinsen et al. (J. Fluid Mech., vol. 804, 2016, pp. 608–645), whose theory was used to describe steady-state resonant waves due to a time-harmonic container's elliptic orbits. In the limit of longitudinal container motions, a symmetry breaking of the planar wave solution occurs, with clockwise and anti-clockwise swirling equally likely. In addition to this primary harmonic dynamics, previous experiments have unveiled that diverse super-harmonic dynamics are observable far from primary resonances. Among these, the so-called double-crest (DC) dynamics, first observed by Reclari et al. (Phys. Fluids, vol. 26, no. 5, 2014, 052104) for circular sloshing, is particularly relevant, as its manifestation is the most favoured by the spatial structure of the external driving. Following Bongarzone et al. (J. Fluid Mech., vol. 943, 2022, A28), in this work we develop a weakly nonlinear analysis to describe the system response to super-harmonic longitudinal forcing. The resulting system of amplitude equations predicts that a planar wave symmetry breaking via stable swirling may also occur under super-harmonic excitation. This finding is confirmed by our experimental observations, which identify three possible super-harmonic regimes, i.e. (i) stable planar DC waves, (ii) irregular motion and (iii) stable swirling DC waves, whose corresponding stability boundaries in the forcing frequency-amplitude plane quantitatively match the present theoretical estimates.

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“…One case is a floating cage with a vertical circular cylindrical tank. A nonlinear multimodal method has been developed for prescribed lateral motion with forcing frequency in the vicinity of the lowest pair of natural sloshing frequencies of a vertical circular cylindrical tank (Faltinsen, Lukovsky & Timokha 2016; Raynovskyy & Timokha 2020). The three-dimensional steady-state sloshing in terms of swirling and chaos can in the latter case develop because of nonlinear free-surface effects.…”

Section: Discussionmentioning

confidence: 99%

Coupling between resonant sloshing and lateral motions of a two-dimensional rectangular tank

Faltinsen

Timokha

2021

J. Fluid Mech.

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Sloshing in Upright Circular Containers

Cited by 15 publications

References 0 publications

An amplitude equation modelling the double-crest swirling in orbital-shaken cylindrical containers

An amplitude equation modelling the double-crest swirling in orbital-shaken cylindrical containers

Super-harmonically resonant swirling waves in longitudinally forced circular cylinders

Coupling between resonant sloshing and lateral motions of a two-dimensional rectangular tank

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