Abstract
The flow driven by a rotating disk of a thin fluid layer in a fixed cylindrical casing is studied by direct numerical simulation and experimental flow visualizations. The characteristics of the flow are first briefly discussed but the focus of this work is to understand the transition to the primary instability. The primary bifurcation is 3D and appears as spectacular sharp-cornered polygonal patterns located along the shroud. The stability diagram is established experimentally in a (Re, G plane, where G is the aspect ratio of the cavity and Re the rotational Reynolds number and confirmed numerically. The number of vortices scales well with the Ekman number based on the water depth, which confirms the existence of a Stewartson layer along the external cylinder. The critical mixed Reynolds number is found to be constant as in other rotating flows involving a shear-layer instability. Hysteresis cycles are observed highlighting the importance of the spin-up and spin-down processes. In some particular cases, a crossflow instability appears under the form of high azimuthal wave number spiral patterns, similar to those observed in a rotor-stator cavity with throughflow and coexists with the polygons. The DNS calculations confirm the experimental results under the flat free surface hypothesis.
Similar content being viewed by others
References
A.J. Faller, J. Fluid Mech. 230, 245 (1991).
B.E. Launder, S. Poncet, E. Serre, Annu. Rev. Fluid Mech. 42, 229 (2010).
M. Ghil, P.L. Read, S.A. Smith, Astron. Geophys. 51, 28 (2010).
B.M. Lewis, H.F. Hawkins, Bull. Am. Meteorol. Soc. 63, 1294 (1982).
J. Sommeria, Courr. CNRS 77, 97 (1991).
A.C. Barbosa Aguiar, P.L. Read, R.D. Wordsworth, T. Salter, Y.H. Yamazaki, Icarus 206, 755 (2010).
R. Hide, C.W. Titman, J. Fluid Mech. 29, 39 (1967).
M. Rabaud, Y. Couder, J. Fluid Mech. 136, 291 (1983).
J.M. Chomaz, M. Rabaud, C. Basdevant, Y. Couder, J. Fluid Mech. 187, 115 (1988).
K. Bergeron, E.A. Coutsias, J.P. Lynov, A.H. Nielsen, J. Fluid Mech. 402, 255 (2000).
F.V. Dolzhanskii, V.A. Krymov, D.Y. Manin, J. Fluid Mech. 241, 705 (1992).
S.C. Abrahamson, J.K. Eaton, D.J. Koga, Phys. Fluids A 1, 241 (1989).
J. Herrero, F. Giralt, J.A.C. Humphrey, Phys. Fluids 11, 88 (1999).
R.F. Huang, M.K. Hsieh, Exp. Fluids 51, 1529 (2011).
J.A. Van de Konijnenberg, A.H. Nielsen, J.J. Rasmussen, B. Stenum, J. Fluid Mech. 387, 177 (1999).
J.M. Lopez, J.E. Hart, F. Marques, S. Kittelman, J. Shen, J. Fluid Mech. 462, 383 (2002).
F. Moisy, O. Doaré, T. Pasutto, O. Daube, M. Rabaud, J. Fluid Mech. 507, 175 (2004).
L. Tophøj, J. Mougel, T. Bohr, D. Fabre, Phys. Rev. Lett. 110, 194502 (2013).
T.R.N. Jansson, M.P. Haspang, K.H. Jensen, P. Hersen, T. Bohr, Phys. Rev. Lett. 96, 174502 (2006).
R. Bergmann, L. Tophøj, T.A.M. Homan, P. Hersen, A. Andersen, T. Bohr, J. Fluid Mech. 679, 415 (2011).
G.H. Vatistas, H. Ait Abderrahmane, M.H.K. Siddiqui, Phys. Rev. Lett. 100, 174503 (2008).
M. Amaouche, H. Ait Abderrahmane, G.H. Vatistas, Eur. J. Mech. B/Fluids 41, 133 (2013).
S. Poncet, M.P. Chauve, J. Flow Vis. Image Process. 14, 85 (2007).
L. Kahouadji, L. Martin-Witkowski, P. Le Quéré, Méc. Ind. 11, 339 (2010).
L. Kahouadji, Analyse de stabilité linéaire d'écoulements tournants en présence de surface libre, PhD thesis, Université Pierre et Marie Curie - Paris VI (2011).
L. Kahouadji, L. Martin-Witkowski, N. Peres, S. Poncet, P. Le Quéré, E. Serre, Primary bifurcation of a thin-layer flow driven by a rotating disk in a fixed open cylindrical cavity: flat free surface, in 4th International Symposium Bifurcations and instabilities in fluid dynamics (Barcelona, 2011).
S. Poncet, Polygons on a rotating thin-water layer: a combined experimental and numerical approach, in Topical Problems of Fluid Mechanics (Prague, 2014).
H.P. Greenspan, The Theory of Rotating Fluids (Cambridge University Press, 1968).
S. Poncet, E. Serre, P. Le Gal, Phys. Fluids 21, 064106 (2009).
N. Peres, S. Poncet, E. Serre, J. Comput. Phys. 231, 6290 (2012).
S. Leibovich, K. Stewartson, J. Fluid Mech. 126, 335 (1983).
H. Niino, N. Misawa, J. Atmos. Sci. 41, 1992 (1984).
N. Schaeffer, P. Cardin, Phys. Fluids 17, 104111 (2005).
K. Stewartson, J. Fluid Mech. 3, 17 (1957).
S. Poncet, M.P. Chauve, Int. J. Fluid Mech. Res. 37, 343 (2010).
S. Poncet, M.P. Chauve, J. Fluid Mech. 545, 281 (2005).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Poncet, S. The stability of a thin water layer over a rotating disk revisited. Eur. Phys. J. Plus 129, 167 (2014). https://doi.org/10.1140/epjp/i2014-14167-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2014-14167-2