The damped sloshing in an upright circular tank due to an orbital forcing

Abstract

An upright circular cylindrical rigid tank performs a small-magnitude prescribed periodic horizontal motion, which is described by the two generalized coordinates 0 1 ( ) r t η and 0 2 ( ) r t η ( 0 r is the tank radius) as shown in fig. 1. Those tank motions are relevant for bioreactors [1]. In contrast to industrial containers whose dimensions are relatively large, the bioreactors have 0 5 10 r ≈ − [cm] that requires accounting for the damping associated with a laminar boundary layer and the bulk viscosity.The problem is studied in the nondimensional statement provided by the characteristic size 0 r and time 1/ σ, where σ is the forcing frequency close to the lowest natural sloshing frequency 11 σ . The nondimensional forcing magnitude is small, i.e. ( ) Fig. 1 illustrates the adopted nomenclature. The unknowns, ς and Φ (the velocity potential), are defined in the tank-fixed coordinate system and can be found from either the corresponding free-surface problem or its equivalent variational formulation. Using the Fourier-type representation (in the cylindrical coordinates), ,

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

“…With regards to the experiments of Reclari (2013) and Reclari et al. (2014), Timokha & Raynovskyy (2017) and Raynovskyy & Timokha (2018 a , b ) have applied the Narimanov–Moiseev multimodal sloshing theory (Narimanov 1957; Moiseev 1958; Dodge, Kana & Abramson 1965; Faltinsen 1974; Narimanov, Dokuchaev & Lukovsky 1977; Lukovsky 1990). The theory is capable of accurately describing the nonlinear wave dynamics near the primary harmonic resonance, when no secondary resonances occur (Faltinsen, Rognebakke & Timokha 2005; Faltinsen, Lukovsky & Timokha 2016).…”

Linear damped interfacial wave theory for an orbitally shaken upright circular cylinder

“…There, it diverges, and waves behave strongly nonlinearly under typical shaking conditions. This gap was recently filled by Timokha & Raynovskyy (2017) and Raynovskyy & Timokha (2018b) who have applied the Narimanov-Moiseev multimodal sloshing theory describing the nonlinear wave dynamics near the first resonance. They have proven that resonant wave dynamics is of the hard-spring type and could explain its frequency-dependent hysteresis, observed before by Reclari (2013).…”

Counter-rotation in an orbitally shaken glass of beer

“…Introduction. -The mean flow induced by a rotating sloshing wave in an orbitally shaken cylinder partially filled with liquid consists in a global rotation in the direction of the applied swirl, along with toroidal recirculation vortices [1][2][3][4][5]. This mean flow, commonly observed when swirling a glass of wine, is essential for mixing processes such as in bioreactors for the cultivation of biological cells [6,7].…”