Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-07T08:17:39.314Z Has data issue: false hasContentIssue false
  • English
  • Français

On rotating disk flow

Published online by Cambridge University Press:  21 April 2006

John F. Brady  and
Louis Durlofsky
John F. Brady
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Present address: Department of Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA.
Louis Durlofsky
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The relationship of the axisymmetric flow between large but finite coaxial rotating disks to the von Kármán similarity solution is studied. By means of a combined asymptotic – numerical analysis, the flow between finite disks of arbitrarily large aspect ratio, where the aspect ratio is defined as the ratio of the disk radii to the gap width separating the disks, is examined for two different end conditions: a ‘closed’ end (shrouded disks) and an ‘open’ end (unshrouded or free disks). Complete velocity and pressure fields in the flow domain between the finite rotating disks, subject to both end conditions, are determined for Reynolds number (based on gap width) up to 500 and disk rotation ratios between 0 and – 1. It is shown that the finite-disk and similarity solutions generally coincide over increasingly smaller portions of the flow domain with increasing Reynolds number for both end conditions. In some parameter ranges, the finite-disk solution may not be of similarity form even near the axis of rotation. It is also seen that the type of end condition may determine which of the multiple similarity solutions the finite-disk flow resembles, and that temporally unstable similarity solutions may qualitatively describe steady finite-disk flows over a portion of the flow domain. The asymptotic – numerical method employed has potential application to related rotating-disk problems as well as to a broad class of problems involving flow in regions of large aspect ratio.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 175 , February 1987 , pp. 363 - 394
Copyright
© 1987 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

Adams, M. L. & Szeri, A. Z. 1982 Incompressible flow between finite disks. Trans. ASME E: J. Appl. Mech. 49, 1–9Google Scholar
Batchelor, G. K. 1951 Note on a class of solutions of the Navier–Stokes equations representing steady rotationally-symmetric flow. Q. J. Mech. Appl. Maths. 4, 29–41.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bien, F. & Penner, S. S. 1970 Velocity profiles in steady and unsteady rotating flows for a finite cylindrical geometry. Phys. Fluids 13, 1665–1671.Google Scholar
Bödewadt, U. T. 1940 Die drehstromung uber festem grunde. Z. angew. Math. Mech. 20, 241–253.Google Scholar
Brady, J. F. 1984 Flow development in a porous channel and tube. Phys. Fluids 27, 1061–1067.Google Scholar
Brady, J. F. & Acrivos, A. 1982 Closed-cavity laminar flows at moderate Reynolds number. J. Fluid Mech. 115, 427–442.Google Scholar
Chen, K. K. & Libby, P. A. 1968 Boundary layers with small departures from the Falkner–Skan profile. J. Fluid Mech. 33, 273–282.Google Scholar
Cochran, W. G. 1934 The flow due to a rotating disk. Proc. Camb. Phil. Soc. 30, 365–375.Google Scholar
Dijkstra, D. & Heijst, G. J. F. van 1983 The flow between two finite rotating disks enclosed by a cylinder. J. Fluid Mech. 128, 123–154.Google Scholar
Durlofsky, L. 1986 Topics in fluid mechanics: I. Flow between finite rotating disks II. Simulation of hydrodynamically interacting particles in Stokes flow. Ph.D. thesis, Massachusetts Institute of Technology.
Durlofsky, L. & Brady, J. F. 1984 The spatial stability of a class of similarity solutions. Phys. Fluids 27, 1068–1076.Google Scholar
Greenspan, H. P. 1980 The Theory of Rotating Fluids. Cambridge University Press.
Harriott, G. M. & Brown, R. A. 1984 Flow in a differentially rotated cylindrical drop at moderate Reynolds number. J. Fluid Mech. 144, 403–418.Google Scholar
Holodniok, M., Kubicek, M. and Hlavacek, V. 1977 Computation of the flow between rotating coaxial disks. J. Fluid Mech. 81, 689–699.Google Scholar
Holodniok, M., Kubicek, M. & Hlavacek, V. 1981 Computation of the flow between rotating coaxial disks: multiplicity of steady-state solutions. J. Fluid Mech. 108, 227–240.Google Scholar
Kármán, T. von 1921 Uber laminare und turbulente reibung. Z. angew. Math. Mech. 1, 233–252.Google Scholar
Lance, G. N. & Rogers, M. H. 1962 The axially symmetric flow of viscous fluid between two infinite rotating disks. Proc. R. Soc. Lond. A266, 109–121.Google Scholar
Lugt, N. J. & Haussling, H. J. 1973 Development of flow circulation in a rotating tank. Acta Mech. 18, 255–272.Google Scholar
Mellor, G. L., Chapple, P. J. & Stokes, V. K. 1968 On the flow between a rotating and a stationary disk. J. Fluid Mech. 31, 95–112.Google Scholar
Nguyen, N. D., Ribault, J. P. & Florent, P. 1975 Multiple solutions for flow between coaxial disks. J. Fluid Mech. 68, 369–388.Google Scholar
Pao, H. P. 1970 A numerical computation of a confined rotating flow. Trans. ASME E: J. Appl. Mech. 37, 480–487Google Scholar
Pao, H. P. 1972 Numerical solution of the Navier–Stokes equations for flows in the disk–cylinder system. Phys. Fluids 15, 4–11.Google Scholar
Parter, S. V. 1982 On the swirling flow between rotating coaxial disks: a survey. MRC Technical Summary Rep. No. 2332.
Pearson, C. E. 1965 Numerical solutions for the time-dependent viscous flow between two rotating coaxial disks. J. Fluid Mech. 21, 623–633.Google Scholar
Picha, K. G. & Eckert, E. R. G. 1958 Study of the air flow between coaxial disks rotating with arbitrary velocities in an open or enclosed space. In Proc. 3rd US Natl. Cong. Appl. Mech. pp. 791–798.
Roberts, S. M. & Shipman, J. S. 1976 Computation of the flow between a rotating and a stationary disk. J. Fluid Mech. 73, 53–63.Google Scholar
Schultz-Grunow, F. 1935 Der reibungswiderstand rotierender scheiben in gehausen. Z. angew. Math. Mech. 14, 191–204.Google Scholar
Stewartson, K. 1953 On the flow between two rotating coaxial disks. Proc. Camb. Phil. Soc. 49, 333–341.Google Scholar
Szeri, A. Z., Giron, A., Schneider, S. J. & Kaufman, H. N. 1983a Flow between rotating disks. Part 2. Stability. J. Fluid Mech. 134, 133–154.Google Scholar
Szeri, A. Z., Schneider, S. J., Labbe, F. & Kaufman, H. N. 1983b Flow between rotating disks. Part 1. Basic flow. J. Fluid Mech. 134, 103–131.Google Scholar
Szeto, R. K.-H. 1978 The flow between rotating coaxial disks. Ph.D. thesis, California Institute of Technology.
Zandbergen, P. J. & Dijkstra, D. 1987 Von Kármán swirling flows. Ann. Rev. Fluid Mech. 19, (to appear).Google Scholar