Microgravity Science and Technology

, Volume 30, Issue 4, pp 383–392 | Cite as

Inertial Waves and Steady Flows in a Liquid Filled Librating Cylinder

  • Stanislav SubbotinEmail author
  • Veronika Dyakova
Original Article
Part of the following topical collections:
  1. Topical Collection on Non-Equilibrium Processes in Continuous Media under Microgravity


The fluid flow in a non-uniformly rotating (librating) cylinder about a horizontal axis is experimentally studied. In the absence of librations the fluid performs a solid-body rotation together with the cavity. Librations lead to the appearance of steady zonal flow in the whole cylinder and the intensive steady toroidal flows near the cavity corners. If the frequency of librations is twice lower than the mean rotation rate the inertial waves are excited. The oscillating motion associated with the propagation of inertial wave in the fluid bulk leads to the appearance of an additional steady flow in the Stokes boundary layers on the cavity side wall. In this case the heavy particles of the visualizer are assembled on the side wall into ring structures. The patterns are determined by the structure of steady flow, which in turn depends on the number of reflections of inertial wave beams from the cavity side wall. For some frequencies, inertial waves experience spatial resonance, resulting in inertial modes, which are eigenmodes of the cavity geometry. The resonance of the inertial modes modifies the steady flow structure close to the boundary layer that is manifested in the direct rebuilding of patterns. It is shown that the intensity of zonal flow, as well as the intensity of steady flows excited by inertial waves, is proportional to the square of the amplitude of librations.


Librations Inertial waves Steady flows Pattern formation 



The research was supported by the Russian Foundation for Basic Research (project Nos. 16-31-60099 mol_a_dk and16-31-00169 mol_a) and grant of the President of the Russian Federation for the support of Leading Scientific Schools of the Russian Federation (grant NSh-9176.2016.1). We are grateful to Professor V.G. Kozlov for a fruitful discussion of the experimental results.


  1. Boisson, J., Lamriben, C., Maas, L.R.M., Cortet, P., Moisy, F.: Inertial waves and modes excited by the libration of a rotating cube. Phys. Fluids 24, 076602 (2012)CrossRefGoogle Scholar
  2. Borcia, I.D., Abouzar, G.V., Harlander, U.: Inertial wave mode excitation in a rotating annulus with partially librating boundaries. Fluid Dyn. Res. 46, 041423 (2014)MathSciNetCrossRefGoogle Scholar
  3. Brouzet, C., Sibgatullin, I.N., Scolan, H., Ermanyuk, E.V., Dauxois, T.: Internal wave attractors examined using laboratory experiments and 3D numerical simulations. J. Fluid Mech. 793, 109–131 (2016)CrossRefGoogle Scholar
  4. Brungs, S., Egli, M., Wuest, S.L., Christianen, P.C.M., Van Loon, J.J.W.A., Ngo Anh, T.J., Hemmersbach, R.: Facilities for simulation of microgravity in the ESA ground-based facility programme. Microgravity Sci. Technol. 28, 191–203 (2016)CrossRefGoogle Scholar
  5. Busse, F.H.: Mean zonal flows generated by librations of a rotating spherical cavity. J. Fluid Mech. 650, 505–512 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Busse, F.H.: Zonal flow induced by longitudinal librations of a rotating cylindrical cavity. Physica D 240, 208–211 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Calkins, M.A., Noir, J., Eldredge, J.D., Aurnou, J.M.: Axisymmetric simulations of libration-driven fluid dynamics in a spherical shell geometry. Phys. Fluids 22, 086602 (2010)CrossRefGoogle Scholar
  8. Ermanyuk, E.V., Gavrilov, N.V.: Internal-wave radiation and optical measurements in stratified fluids. Microgravity Sci. Technol. 19, 144–147 (2007)CrossRefGoogle Scholar
  9. Favier, B., Barker, A., Baruteau, C., Ogilvie, G.: Nonlinear evolution of tidally forced inertial waves in rotating fluid bodies. Mon. Not. R. Astron. Soc. 439, 845–860 (2014)CrossRefGoogle Scholar
  10. Greenspan, H.P.: The Theory of Rotating Fluids. University Press, Cambridge (1968)zbMATHGoogle Scholar
  11. Hoff, M., Harlander, U., Egbers, C.: Experimental survey of linear and nonlinear inertial waves and wave instabilities in a spherical shell. J. Fluid Mech. 789, 589–616 (2016)CrossRefGoogle Scholar
  12. Kerswell, R.: On the internal shear layers spawned by the critical regions in oscillatory Ekman boundary layers. J. Fluid Mech. 298, 311–325 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Klein, M., Seelig, T., Kurgansky, M.V., Ghasemi, A.V., Borcia, I.D., Will, A., Schaller, E., Egbers, C., Harlander, U.: Inertial wave excitation and focusing in a liquid bounded by a frustum and a cylinder. J. Fluid Mech. 751, 255–297 (2014)CrossRefGoogle Scholar
  14. Koch, S., Harlander, U., Egbers, C., Hollerbach, R.: Inertial waves in a spherical shell induced by librations of the inner sphere: Expe- rimental and numerical results. Fluid Dyn. Res. 45, 035504 (2013)CrossRefzbMATHGoogle Scholar
  15. Kozlov, V.G., Kozlov, N.V.: Vibrational dynamics of a light body in a liquid-filled rotating cylinder. Fluid Dyn. 43(1), 9–19 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Kozlov, V.G., Ivanova, A.A.: Dramatic effect of vibrations on dynamics of rotating hydrodynamic systems. Microgravity Sci. Technol. 21, 339–348 (2009)CrossRefGoogle Scholar
  17. Kozlov, V., Polezhaev, D.: Flow patterns in a rotating horizontal cylinder partially filled with liquid. Phys. Rev. E 92, 013016 (2015)MathSciNetCrossRefGoogle Scholar
  18. Kozlov, V.G., Ivanova, A.A., Vjatkin, A.A., Sabirov, R.R.: Vibrational convection of heat-generating fluid in a rotating horizontal cylinder. The role of relative cavity length. Acta Astronaut. 112, 48–55 (2015)CrossRefGoogle Scholar
  19. Kozlov, V.G., Kozlov, N.V., Subbotin, S.V.: Steady flows excited by circular oscillations of free inner core in rotating spherical cavity. Eur. J. Mech. B-Fluids 58(4), 85–94 (2016)CrossRefGoogle Scholar
  20. Le Bars, M., Cébron, D., Le Gal, P.: Flows driven by libration, precession, and tides. Annu. Rev. Fluid Mech. 47, 163–193 (2015)MathSciNetCrossRefGoogle Scholar
  21. Lopez, J.M., Marques, F.: Rapidly rotating cylinder flow with an oscillating sidewall. Phys. Rev. E. 89, 013013 (2014)CrossRefGoogle Scholar
  22. Margot, J.L., Peale, S.J., Jurgens, R.F., Slade, M.A., Holin, I.V.: Large longitude libration of Mercury reveals a molten core. Science 316, 710–714 (2007)CrossRefGoogle Scholar
  23. McEwan, A.D.: Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40, 603–639 (1970)CrossRefGoogle Scholar
  24. Messio, L., Morize, C., Rabaud, M., Moisy, F.: Experimental observation using particle image velocimetry of inertial waves in a rotating fluid. Exp. Fluids 44, 519–528 (2008)CrossRefGoogle Scholar
  25. Morize, C., Le Bars, M., Le Gal, P., Tilgner, A.: Experimental determination of zonal winds driven by tides. Phys. Rev. Lett. 104, 214501 (2010)CrossRefGoogle Scholar
  26. Noir, J., Calkins, M.A., Lasbleis, M., Cantwell, J., Aurnou, J.M.: Experimental study of libration-driven zonal flows in a straight cylinder. Phys. Earth Planet Inter. 182, 98–106 (2010)CrossRefGoogle Scholar
  27. Rieutord, M., Georgeot, B., Valdettaro, L.: Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103–144 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Sauret, A., Le Dizès, S.: Steady flow induced by longitudinal libration in a spherical shell. J. Fluid Mech. 718, 181–209 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Sauret, A., Cebron, D., Morize, C., Le Bars, M.: Experimental and numerical study of mean zonal flows generated by librations of a rotating spherical cavity. J. Fluid Mech. 662, 260–268 (2010)CrossRefzbMATHGoogle Scholar
  30. Sauret, A., Cébron, D., Le Bars, M., Le Dizès, S.: Fluid flows in a librating cylinder. Phys. Fluids 24, 026603 (2012)CrossRefGoogle Scholar
  31. Sauret, A., Le Bars, M., Le Gal, P.: Tide-driven shear instability in planetary liquid cores. Geophys. Res. Lett. 41, 6078–6083 (2014)CrossRefGoogle Scholar
  32. Tilgner, A.: Zonal wind driven by inertial modes. Phys. Rev. Lett. 99, 194501 (2007)CrossRefGoogle Scholar
  33. Thielicke, W., Stamhuis, E.J.: PIVLab – Time-resolved digital particle image velocimetry tool for MATLAB (version: 1.41). J. Open Res. Software 2(1), e30 (2014)Google Scholar
  34. Wang, C.Y.: Cylindrical tank of fluid oscillating about a steady rotation. J. Fluid Mech. 41, 581–592 (1970)CrossRefzbMATHGoogle Scholar
  35. Warnke, E., Kopp, S., Wehland, M., Hemmersbach, R., Bauer, J., Pietsch, J., Infanger, M., Grimm D.: Thyroid cells exposed to simulated microgravity conditions – comparison of the fast rotating clinostat and the random positioning machine. Microgravity Sci. Technol. 28, 247–260 (2016)CrossRefGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratory of Vibrational HydromechanicsPerm State Humanitarian Pedagogical UniversityPermRussia
  2. 2.Department of Applied PhysicsPerm National Research Polytechnic UniversityPermRussia

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