Hostname: page-component-68945f75b7-qvshk Total loading time: 0 Render date: 2024-08-06T07:26:34.219Z Has data issue: false hasContentIssue false
  • English
  • Français

Pattern formation of granules on the bottom of a differentially rotating tank

Published online by Cambridge University Press:  26 April 2006

P. J. Thomas
P. J. Thomas
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The formation of patterns of granules on the bottom of a differentially rotating tank, partially filled with a fluid, is described. The pattern formation is initiated when the tank is spun up from an initial constant rotation rate by a sufficiently large increment to another constant higher rotation rate. The granules are observed to be set into motion, slide across the bottom of the tank and organize themselves into two distinct, superposed geometrical patterns. The first pattern constitutes a system of tightly wound equi-angular spirals and is identified as being simply a visualization of the well-known stationary Class B waves of the unstable boundary layer. This pattern is not considered further here. The second pattern, which is considered here, is shown not to be a visualization of some other known stationary wave mode. It consists of a number of large-scale spiral arms extending outward towards the wall of the tank and originating at a patch of salt in the tank's centre. The presence of Class B waves is apparently a necessary condition for the spiral arm formation. The experiments show that the radius r0 of the inner salt patch is inversely proportional to the increment Δω of the rotation rate for the larger values of Δω. The angle ε(r) between the direction of the spiral arms and the tangential direction is observed to decrease with the distance r from the centre as rb where b = −0.64 ± 0.1. The relationship between the number n of spiral arms and the associated Reynolds number Re and Rossby number Ro is found from the experimental observations and dimensional considerations to be \[n^2 = Re \, Ro. \] Various aspects related to the mechanism involved in the pattern formation are discussed. However, no conclusive model for the spiral arm formation can at present be suggested.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 274 , 10 September 1994 , pp. 23 - 41
Copyright
© 1994 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, R. S. 1990 Eolian ripples as examples of self-organization in geomorphological systems. Earth-Sci. Rev. 29, 77–96.Google Scholar
Caldwell, D. R. & Van Atta, C. W., 1970 Characteristics of Ekman boundary layer instabilities. J. Fluid Mech. 44, 79–95.Google Scholar
Faller, A. J. 1963 An experimental study of the instability of the laminar Ekman boundary layer. J. Fluid Mech. 15, 560–576.Google Scholar
Faller, A. J. & Kaylor, R. E. 1965 Investigations of stability and transition in rotating boundary layers. Tech. Note BN-427. Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, USA.
Faller, A. J. & Kaylor, R. E. 1966 A numerical study of the instability of the laminar Ekman boundary layer. J. Atmos. Res. 23, 481–494.Google Scholar
Faller, A. J. & Kaylor, R. 1967 Instability of the Ekman spiral with applications to the planetary boundary layers. Phys. Fluids 10, S212–S219.Google Scholar
Federov, B. I., Plavnik, G. Z., Prokhorov, I. V. & Zhukhovitskii, L. G. 1976 Transitional flow conditions on a rotating disk. J. Engng Phys. 31, 1448–1453.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk. Phil. Trans. R. Soc. Lond. A 248, 155–199.Google Scholar
Kobayashi, R., Kohama, Y. & Takamadate, Ch., 1980 Spiral vortices in boundary layer transition regime on a rotating disk. Acta Mech. 35, 71–82.Google Scholar
Kohama, Y. 1984 Study on boundary layer transition of a rotating disk. Acta Mech. 50, 193–199.Google Scholar
Kohama, Y. 1987 Crossflow instability in rotating disk boundary layer. AIAA Paper 87-1340.
Malik, M. R., Wilkinson, S. P. & Orszag, S. A. 1981 Instability and transition in rotating disk flow. AIAA J. 19, 1131–1138.Google Scholar
Owen, P. R. 1980 The physics of sand movement. Lecture Notes of Lectures held at the international Centre for Theoretical Physics, Autumn course on Physics of flow in the oceans, atmosphere and deserts, Trieste, Italy, 30 September-28 November, 1980.
Reed, H. L. & Saric, W. S. 1989 Stability of three-dimensional boundary layers. Ann. Rev. Fluid Mech. 21, 235–284.Google Scholar
Tatro, P. R. & Mollö-Christensen, E. L. 1967 Experiments on Ekman layer instability. J. Fluid Mech. 28, 531–543.Google Scholar
Weast, R. C., (ed.) 1971 CRC Handbook of Chemistry and Physics The Chemical Rubber Co.
Wilkinson, S. P. & Malik, M. R. 1985 Stability experiments in the flow over a rotating disk. AIAA J. 23, 588–595.Google Scholar