Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-19T17:43:47.740Z Has data issue: false hasContentIssue false
  • English
  • Français

Cellular Stokes flow induced by rotation of a cylinder in a closed channel

Published online by Cambridge University Press:  26 April 2006

Mustapha Hellou  and
Madeleine Coutanceau
Mustapha Hellou
Affiliation:
Laboratoire de Mécanique des Fluides, Université de Poitiers, 40, Avenue du Recteur Pineau, 86022 Poitiers, France Laboratoire de Géotechnique, Thermique et Matériaux, Institut National des Sciences Appliquées. 20 Avenue des Buttes de Coësmes, 35043 Rennes, France.
Madeleine Coutanceau
Affiliation:
Laboratoire de Mécanique des Fluides, Université de Poitiers, 40, Avenue du Recteur Pineau, 86022 Poitiers, France
Get access Rights & Permissions [Opens in a new window]

Abstract

The evolution of the cellular structure of the two-dimensional creeping flow induced by a rotating circular cylinder set in the centre of a rectangular channel is studied numerically and experimentally when the aspect ratio A increases from 1 to 7. In the calculations, depending on the value of A, either only series in terms of polar coordinates, or both matched polar and Cartesian coordinates series are employed to represent the stream function and an efficient least-squares method, very easy to program, is selected to satisfy some of the boundary conditions. For the experiments, a special technique which visualizes intermittently the paths of solid tracers during long times of exposure permits us to observe the fluid motion in the whole domain, even in the regions where the velocities are very small. An excellent measure of agreement between the numerical and experimental results is found. Thus it is clearly shown how, in the region beyond the rotating flow directly driven by the cylinder, the two main corner cells visualized at A = 1, develop with increasing A and then coalesce, to finally merge and give rise to a single central cell. This central cell develops in its turn, tending finally to the unbounded channel reference cell, after passing through a maximum length however. Owing to the very high precision of the calculations, many details of the flow development have been clearly shown, in particular the periodicity, with increasing A, of all the different phases, progressively inducing a succession of cells. The prediction that the angle of separation of the fluid boundaries of the cells tends towards the theoretical limit of 58.61° when the aspect ratio becomes large is also confirmed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 236 , March 1992 , pp. 557 - 577
Copyright
© 1992 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

Bouard, R. 1983 Etude de l’écoulement autour d' cylindre soumis à une translation uniforme après un départ impulsif, pour des nombres de Reynolds allant de o à 104. Thèse de doctorat ès Sciences Physiques, Poitiers.
Bouard, R. & Coutanceau, M. 1986 Etude théorique et expérimentale de l’écoulement engendré par un cylindre en translation uniforme dans un fluide visqueux en régime de Stokes. Z. Angew. Math. Phys. 37, 673–684.Google Scholar
Bourot, J. M. 1969 Sur l' d' méthode de moindres carrés à la résolution approchée du problème aux limites, pour certaines catégories d’écoulements. J. Méc. 8, 301–322.Google Scholar
Bourot, J. M. 1984 Sur la structure cellulaire des écoulements plans de Stokes, à débit moyen nul, en canal indéfini à parois parallèles. C. R. Acad. Sci. Paris II 298, 161–164.Google Scholar
Bourot, J. M. & Moreau, F. 1987 Sur l' de la série cellulaire pour le calcul d’écoulements plans de Stokes en canal indéfini: application au cas d' cylindre circulaire en translation. Mech. Res. Commun. 14, 187–197.Google Scholar
Buchwald, V. T. 1964 Eigenfunctions of plane elastostatics I. The strip, Proc. R. Soc. Lond. A 277, 385–400.Google Scholar
Burgraff, O. R. 1966 Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech. 24, 113–151.Google Scholar
Collins, W. M. & Dennis, S. C. R. 1976 Viscous eddies near a 90° and a 45° corner in flow through a curved tube of triangular cross-section. J. Fluid Mech. 76, 417–432.Google Scholar
Coutanceau, M., Bouard, R., Bourot, J. M. & Hellou, M. 1984 Sur la visualisation de la structure cellulaire de l’écoulement plan de Stokes engendré par la rotation d' cylindre dans un canal. C. R. Acad. Sci. Paris II 298, 767–770.Google Scholar
Coutanceau, M. & Thizon, P. 1981 Wall effect on the bubble behaviour in highly viscous liquids. J. Fluid Mech. 107, 339–373.Google Scholar
Davis, A. M. J. & O', M. E. 1977 Separation in a slow linear shear flow past a cylinder and a plane. J. Fluid Mech. 81, 551–564.Google Scholar
Dean, W. R. & Montagnon, P. E. 1949 On the study motion of viscous liquid in a corner. Proc. Camb. Phil. Soc. 45, 389–394.Google Scholar
Hasimoto, H. & Sano, O. 1980 Stokeslests and eddies in creeping flow. Ann. Rev. Fluid Mech. 12, 335–363.Google Scholar
Hellou, M. 1983 Etude theórique et expérimentale de l’écoulement plan de Stokes autour d' cylindre en rotation dans un canal. Mise en évidence des courants secondaires et de leur structure cellulaire. Diplome d’études approfondies de l'é de Poitiers.
Hellou, M. 1988 Etude numérique et expérimentale de l’écoulement à structure cellulaire. engendré par la rotation d' cylindre dans un canal. Thèse de l'é de Poitiers.
Higdon, J. J. L. 1985 Stokes flow in arbitrary two-dimensional domains: shear flow over ridges and cavities. J. Fluid Mech. 159, 195–226.Google Scholar
Joseph, D. D. & Sturges, L. 1975 The free surface on a liquid filling a trench heated from its side. J. Fluid Mech. 69, 565–589.Google Scholar
Lewis, E. 1979 Steady flow between a rotating circular cylinder and fixed square cylinder. J. Fluid Mech. 95, 497–513.Google Scholar
Maalouf, A. & Bouard, R. 1987 Etude de l’écoulement plan d' fluide visqueux et incompressible autour et au travers de coques cylindriques poreuses, à faibles nombres de Reynolds. Z. Angew. Math. Phys. 38, 522–541.Google Scholar
Mehta, U. B. & Lavan, Z. 1969 Flow in a two-dimensional channel with a rectangular cavity. Trans. ASME E: J. Appl. Mech. 36, 897–901.Google Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.Google Scholar
Moreau, F. 1985 Sur la structure cellulaire des écoulements plans de Stokes, à débit moyen nul, en canal à parois parallèles. Diplome d’études approfondies de l'é de Poitiers.
O', V. 1972 Closed streamlines associated with channel flow over a cavity. Phys. Fluids 15, 2089–2097.Google Scholar
O', M. E. 1983 On angles of separation in Stokes flows. J. Fluid Mech. 133, 427–442.Google Scholar
Pan, F. & Acrivos, A. 1967 Steady flows in rectangular cavities. J. Fluid Mech. 28, 643–660.Google Scholar
Rybicki, A. & Floryan, J. M. 1987 Thermocapillary effects in liquid bridges. I. Thermocapillary convection. Phys. Fluids 30, 1956–1972.Google Scholar
Sanders, J., O', V. & Joseph, D. D. 1980 Stokes flow in a driven sector by two different methods. Trans. ASME E: J. Appl. Mech. 47, 482–484.Google Scholar
Shen, C. & Floryan, J. M. 1985 Low Reynolds number flow over cavities. Phys. Fluids 28, 3191–3202.Google Scholar
Sigli, D. 1970 Contribution à la mise au point d' technique de résolution approchée du problème aux limites, pour un écoulement de révolution en régime de Stokes. Thèse de Doctorat de Troisième Cycle, Poitiers, France.
Taneda, S. 1979 Visualization of separating Stokes flow. J. Phys. Soc. Japan 49, 1935–1942.Google Scholar
Weiss, R. F. & Florsheim, B. H. 1965 Flow in a cavity at low Reynolds number. Phys. Fluids 8, 1631–1635.Google Scholar