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

  • Odd M. Faltinsen (a1) and Alexander N. Timokha (a1) (a2)


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.


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Bouvard, J., Herreman, W. & Moisy, F. 2017 Mean mass transport in an orbitally shaken cylindrical container. Phys. Rev. Fluids 2, 084801.10.1103/PhysRevFluids.2.084801
van den Bremer, T. S. & Breivik, Ø. 2017 Stokes drift. Phil. Trans. R. Soc. Lond. A 376 (2111), A 2018 376 20170104.
Bühler, O. 2009 Waves and Mean Flows. Cambridge University Press.10.1017/CBO9780511605499
Craik, A. D. 1986 Wave Interaction and Fluid Flows. Cambridge University Press.10.1017/CBO9780511569548
Craik, A. D. & Leibovich, S. 1976 A rational model for Langmuir circulations. J. Fluid Mech. 73 (3), 401426.
Ducci, A. & Weheliye, W. H. 2014 Orbitally shaken bioreactors-viscosity effects on flow characteristics. AIChE J. 60 (11), 39513968.10.1002/aic.14608
Faltinsen, O. M., Lukovsky, I. A. & Timokha, A. N. 2016 Resonant sloshing in an upright annular tank. J. Fluid Mech. 804, 608645.10.1017/jfm.2016.539
Faltinsen, O. M. & Timokha, A. N. 2009 Sloshing. Cambridge University Press.
Hutton, R. E. 1964 Fluid-particle motion during rotary sloshing. Trans. ASME J. Appl. Mech. 31 (1), 145153.10.1115/1.3629532
Kochin, N. E., Kibel, I. A. & Roze, N. V. 1965 Theoretical Hydromechanics. Wiley.
Leibovich, S. 1980 On wave-current interaction theories of Langmuir circulations. J. Fluid Mech. 99 (4), 715724.10.1017/S0022112080000857
Moiseev, N. N. 1958 On the theory of nonlinear vibrations of a liquid of finite volume. J. Appl. Math. Mech. 22 (5), 860872.
Prandtl, L. 1949 Erzeugung von Zirkulation beim Schütteln von Gefässen. Z. Angew. Math. Mech. 29 (1/2), 89.10.1002/zamm.19490290106
Reclari, M.2013 Hydrodynamics of orbital shaken bioreactors. Thesis no 5759 (2013), Ècole Polytechnique Federale de Lausanne.
Reclari, M., Dreyer, M., Tissot, S., Obreschkow, D., Wurm, F. M. & Farhat, M. 2014 Surface wave dynamics in orbital shaken cylindrical containers. Phys. Fluids 26, 052104.
Riley, N. 2001 Steady streaming. Annu. Rev. Fluid Mech. 33, 4365.
Royon-Lebeaud, A., Hopfinger, E. J. & Cartellier, A. 2007 Liquid sloshing and wave breaking in circular and square-base cylindrical containers. J. Fluid Mech. 577, 467494.
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Faltinsen and Timokha supplementary material
Faltinsen and Timokha supplementary material 1

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