Resonant sloshing in a square-base tank due to an angular-and-horizontal periodic forcing

Розробляється модальний метод для задачi про коливання рiдини у резервуарi квадратного перерiзу, який виконує перiодичнi горизон- тальнi та кутовi рухи малої амплiтуди. Аналiз показує, що домiнан- тна хвильова компонента виключно визначається першою гармонiкою перiодичного збурення. Еквiвалентнi гармонiчнi рухи є зворотно- поступального чи елiптичного типу. Вивчено усталенi резонанснi хви- льовi режими для таких типiв збурення. Разрабатывается модальный метод для задачи про колебания жид- кости в резервуаре квадратного сечения, который совершает пери- одические горизонтальные и угловые движения малой амплитуды. Анализ показывает, что доминантная волновая компонента исклю- чительно определяется первой гармоникой периодического возбужде- ния. Эквивалентные гармонические движения являются возвратно- поступательными или эллиптического типов. Изучено установившиеся резонансные волновые режимы для такого типа возмущений.publishedVersionCopyright Timokha A. N., 2016. Published by Institute of Mathematics, Ukrain

The Narimanov-Moiseev multimodal analysis of nonlinear sloshing in circular conical tanks

The chapter reports mathematical aspects of the Narimanov–Moiseev multimodal modelling for the liquid sloshing in rigid circular conical tanks, which perform small-magnitude oscillatory motions with the forcing frequency close to the lowest natural sloshing frequency. To derive the corresponding nonlinear modal system (of ordinary differential equations), we introduce an infinite set of the sloshing-related generalised coordinates governing the free-surface elevation but the velocity potential is posed as a Fourier series by the natural sloshing modes where the time-depending coefficients are treated as the generalised velocities. The employed approximate natural sloshing modes exactly satisfy both the Laplace equation and the zero-Neumann boundary condition on the wetted tank walls. The Lukovsky non-conformal mapping technique transforms the inner (conical) tank (physical) domain to an artificial upright circular cylinder, for which the single-valued representation of the free surface is possible. Occurrence of secondary resonances for the V-shaped truncated conical tanks is evaluated. The Narimanov–Moiseev modal equations allow for deriving an analytical steady-state (periodic) solution, whose stability is studied. The latter procedure is illustrated for the case of longitudinal harmonic excitations. Standing (planar) waves and swirling as well as irregular sloshing (chaos) are established in certain frequency ranges. The corresponding amplitude response curves are drawn and extensively discussed.acceptedVersionThis is a post-peer-review, pre-copyedit version of an article. Locked until 22.2.2020 due to copyright restrictions. The final authenticated version is available online at: https://doi.org/10.1007/978-3-319-99918-0_

An inviscid analysis of the Prandtl azimuthal mass transport during swirl-type sloshing

An inviscid analytical theory of a slow steady liquid mass rotation during the swirl-type sloshing in a vertical circular cylindrical tank with a fairly deep depth is proposed by utilising the asymptotic steady-state wave solution by Faltinsen et al. (J. Fluid Mech., vol. 804, 2016, pp. 608–645). The tank performs a periodic horizontal motion with the forcing frequency close to the lowest natural sloshing frequency. The azimuthal mass transport (first observed in experiments by Prandtl (Z. Angew. Math. Mech., vol. 29(1/2), 1949, pp. 8–9)) is associated with the summarised effect of a vortical Eulerian-mean flow, which, as we show, is governed by the inviscid Craik–Leibovich equation, and an azimuthal non-Eulerian mean. Suggesting the mass-transport velocity tends to zero when approaching the vertical wall (supported by existing experiments) leads to a unique non-trivial solution of the Craik–Leibovich boundary problem and, thereby, gives an analytical expression for the summarised mass-transport velocity within the framework of the inviscid hydrodynamic model. The analytical solution is validated by comparing it with suitable experimental data.acceptedVersion© 2019. This is the authors' accepted and refereed manuscript to the article. Locked until 27 August 2019 due to copyright restrictions. The final authenticated version is available online at: https://doi.org/10.1017/jfm.2019.9

Damped steady-state resonant sloshing in a container of circular cross-section for arbitrary periodic nonparametric forcing

Nonlinear modal Narimanov-Moiseev—type equations are investigated to study resonant sloshing in a vertical cylindrical tank. The tank moves periodically in the space with the forcing frequency close to the lowest natural sloshing frequency. We show that the considered sloshing problem can to within the higher-order asymptotic terms be reduced to the case of orbital tank motions in the horizontal plane. Analytical solutions of the secular system which couples the dominant amplitudes of the steady-state sloshing are analytically solved. Effect of viscous damping is accounted. The results are compared with experimental measurements conducted by diverse authors for longitudinal and circular orbital tank excitations. A parametric analysis of the amplitude curves is done to clarify how the steady-state wave regimes and their stability change versus the forcing frequency and the semi-axes ratio of the elliptic orbit. The main result consists of confirming the experimental disappearance of the counter-directed swirling wave mode (relative to the elliptic orbit direction) when passaging to the circular orbit. Pages of the article in the issue: 97 - 100 Language of the article: Ukrainia

Differential and variational formalism for acoustically-levitating drops

We consider the most general problem of waves on the interface of two ideal fluids regarded as an ullage gas and a liquid, respectively. Separating the fast and slow time scales, we develop the differential and variational formalism for an acoustically levitating drop and determine its time-averaged shape (vibroequilibrium state of the drop). The vibroequilibrium states of the drop may differ from the spherical shape. Stable vibroequilibria are associated with the local minima of the quasipotential energy whose analytic form is also established.submittedVersionThis is the authors' manuscript to the article (preprint)

The Bateman-type variational formalism for an acoustically-driven drop

By employing the Clebsch potentials, the Bateman-type variational formulation for a drop levitating in an acoustic field is proposed when both fluids, liquid drop and external ullage gas, are barotropic, inviscid, compressible and admit rotational flows.Використовуючи потенціали Клебша, пропонується варіаційне формулювання типу Бейтмена для краплі, що левітує в акустичному полі, коли обидві рідини, крапля рідини та зовнішній газ є баротропними, нев’язкими, стисливими та допускають вихорові рухи