Abstract
A bathtub vortex in a cylindrical tank rotating at a constant angular velocity Ω is studied by means of a laboratory experiment, a numerical experiment and a boundary layer theory. The laboratory and numerical experiments show that two regimes of vortices in the steady-state can occur depending on Ω and the volume flux Q through the drain hole: when Q is large and Ω is small, a potential vortex is formed in which angular momentum outside the vortex core is constant in the non-rotating frame. However, when Q is small or Ω is large, a vortex is generated in which the angular momentum decreases with decreasing radius. Boundary layer theory shows that the vortex regimes strongly depend on the theoretical radial volume flux through the bottom boundary layer under a potential vortex : when the ratio of Q to the theoretical boundary-layer radial volume flux Q b (scaled by \({2\pi R^2 ( \Omega \nu )^\frac{1}{2}}\)) at the outer rim of the vortex core is larger than a critical value (of order 1), the radial flow in the interior exists at all radii and Regime I is realized, where R is the inner radius of the tank and ν the kinematic viscosity. When the ratio is less than the critical value, the radial flow in the interior nearly vanishes inside a critical radius and almost all of the radial volume flux occurs only in the boundary layer, resulting in Regime II in which the angular momentum is not constant with radius. This criterion is found to explain the results of the laboratory and numerical experiments very well.
Similar content being viewed by others
References
Wurman J., Gill S.: Finescale radar observations of the Dimmitt, Texas (2 June 1995), Tornado. Mon. Wea. Rev. 128, 2135–2164 (2000)
Lewellen W.S.: A solution for three-dimensional vortex flows with strong circulation. J. Fluid Mech. 14, 420–432 (1962)
Turner J.S.: The constraints imposed on tornado-like vortices by the top and bottom boundary conditions. J. Fluid Mech. 25, 377–400 (1966)
Lundgren J.: The vortical flow above the drain-hole in a rotating vessel. J. Fluid Mech. 155, 381–412 (1985)
Mory M., Yurichenko N.: Vortex generation by suction in a rotating tank. Eur. J. Mech. B/Fluids 12, 729–747 (1993)
Echavez G., McCann E.: An experimental study on the free surface vertical vortex. Exp. Fluids 33, 414–421 (2002)
Andersen A., Bohr T., Stenum B., Rasmussen J.J., Lautrup B.: The bathtub vortex in a rotating container. J. Fluid Mech. 556, 121–146 (2006)
Burggraf O.R., Stewartson K., Belcher R.: Boundary layer induced by a potential vortex. Phys. Fluids. 14, 1821–1833 (1971)
Andersen A.T., Lautrup B., Bohr T.: An averaging method for nonlinear laminar Ekman layers. J. Fluid Mech. 487, 81–90 (2003)
Jacquin, L., Fabre, D., Geffroy, P., Coustols, E.: The properties of a transport aircraft wake in the extended near field—an experimental study. In: Proceedings of the AIAA, Aerospace Sciences Meeting and Exhibit, 39th, Reno (2001)
Jacquin, L., Fabre, D., Sipp, D., Coustols, E.: Unsteadiness, instability and turbulence in trailing vortices—comptes rendus-Physique (2005)
Noguchi, T., Yukimoto, S., Kimura, R., Niino, H.: Structure and instability of a sink vortex. In: Proceedings of PSFVIP-4 (2003)
Yukimoto, S.: Structure of a Suction Vortex: Importance of the Bottom Boundary Layer. Doctoral Dissertation, Dep. Earth Planet. Sci., The University of Tokyo, 77 p. (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Aref
Rights and permissions
About this article
Cite this article
Yukimoto, S., Niino, H., Noguchi, T. et al. Structure of a bathtub vortex: importance of the bottom boundary layer. Theor. Comput. Fluid Dyn. 24, 323–327 (2010). https://doi.org/10.1007/s00162-009-0128-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00162-009-0128-3