Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-19T15:43:37.163Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of stationary endwall boundary layers during spin-down

Published online by Cambridge University Press:  26 April 2006

J. M. Lopez  and
P. D. Weidman
J. M. Lopez
Affiliation:
Department of Mathematics and Earth System Science Center, The Pennsylvania State University, University Park, PA 16802, USA
P. D. Weidman
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80302, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Since Bödewadt's (1940) seminal work on the boundary layer flow produced by a fluid in solid-body rotation over a stationary disk of infinite radius there has been much interest in determining the stability of such flows. To date, it appears that there is no theoretical study of the stability of Bödewadt's self-similar solution to perturbations that are not self-similar. Experimental studies have been compromised due to the difficulty in establishing these steady flows in the laboratory. Savaç (1983, 1987) has studied the endwall boundary layers of flow in a circular cylinder following impulsive spin-down. During the first few radians of rotation, the endwall boundary layers have a structure very similar to Bödewadt layers. For certain conditions, SavaÇ has observed a series of axisymmetric waves travelling radially inwards in the endwall boundary layers. The conjecture is that these waves represent a mode of instability of the Bödewadt layer. Within a few radians of rotation however, the centrifugal instability of the sidewall layer dominates the spin-down process and the endwall waves are difficult to examine further.

Here, the impulsive spin-down problem is examined numerically for Savaç’ (1983, 1987) conditions and good agreement with his experiments is achieved. New experimental results are also presented, which include quantitative space-time information regarding the axisymmetric waves. These agree well with both the numerics and the earlier experimental work. Further, a related problem is considered numerically. This flow is also initially in solid-body rotation, but only the endwalls are impulsively stopped, keeping the sidewall rotating. This results in a flow virtually identical to the usual spin-down flow for the first few radians of rotation, except in the immediate vicinity of the sidewall. The sidewall layer is no longer centrifugally unstable and the circular waves on the endwalls are observed without the influence of the sidewall instability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 326 , 10 November 1996 , pp. 373 - 398
Copyright
© 1996 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

Benton, E. R. 1966 On the flow due to a rotating disk. J. Fluid Mech. 4, 781–800.Google Scholar
Bödewadt, U. T. 1940 Die Drehströmung über festem Grunde. Z. Angew. Math. Mech. 20, 241–245.Google Scholar
Bragg, S. L. & Hawthorne, W. R. 1950 Some exact solutions of the flow through annular cascade actuator discs. J. Aeronaut. Sci. 17, 243–249.Google Scholar
Brown, G. L. & Lopez, J. M. 1990 Axisymmetric vortex breakdown. Part 2. Physical mechanisms. J. Fluid Mech. 221, 553–576.Google Scholar
Davidson, P. A. 1989 The interaction between swirling and recirculating velocity components in unsteady, inviscid flow. J. Fluid Mech. 209, 35–55.Google Scholar
Davis, S. H. 1976 The stability of time-periodic flows. Ann. Rev. Fluid Mech. 8, 57–74.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Kármán, Th. von 1921 über laminare und turbulente Reibung. Z. Angew. Math. Mech. 1, 233–252.Google Scholar
Khalili, A. & Rath, H. J. 1994 Analytical solution for a steady flow of enclosed rotating disks. Z. Angew Math. Phys. 45, 670–680.Google Scholar
Lopez, J. M. 1990 Axisymmetric vortex breakdown. Part 1. Confined swirling flow. J. Fluid Mech. 221, 533–552.Google Scholar
Lopez, J. M. & Perry, A. D. 1992 Axisymmetric vortex breakdown. Part 3. Onset of periodic flow and chaotic advection. J. Fluid Mech. 234, 449–471.Google Scholar
Mathis, D. M. & Neitzel, G. P. 1985 Experiments on impulsive spin-down to rest. Phys. Fluids 28, 449–454.Google Scholar
Maxworthy, T. 1971 A simple observational technique for the investigation of boundary-layer stability and turbulence. In Turbulence Measurements in Liquids (ed. G. K. Paterson & J. L. Zakin), pp. 32–37. Dept. Chemical Engineering, University of Missouri—Rolla.
Moore, F. K. 1956 Three-dimensional boundary layer theory. Adv. Appl. Mech. 4, 159–228.Google Scholar
Neitzel, G. P. & Davis, S. H. 1980 Energy stability theory of decelerating swirl flows. Phys. Fluids 23, 432–437.Google Scholar
Neitzel, G. P. & Davis, S. H. 1981 Centrifugal instabilities during spin-down to rest in finite cylinders. Numerical experiments. J. Fluid Mech. 102, 329–352.Google Scholar
Rogers, M. H. & Lance, G. N. 1964 The boundary layer on a disc of finite radius in a rotating fluid. Q. J. Mech. Appl. Maths 17, 319–330.Google Scholar
Rott, N. & Lewellen, W. S. 1967 Boundary layers and their interactions in rotating flows. Prog. Aeronaut. Sci. 7, 111–144.Google Scholar
Savaç, Ö. 1983 Circular waves on a stationary disk in rotating flow. Phys. Fluids 26, 3445–3448.Google Scholar
SavaÇ, Ö. 1985 On flow visualization using reflective flakes. J. Fluid Mech. 152, 235–248.Google Scholar
Savaç, Ö. 1987 Stability of Bödewadt flow. J. Fluid Mech. 183, 77–94.Google Scholar
Schwiderski, E. W. & Lugt, H. J. 1964 Rotating flows of von Kármán and Bödewadt. Phys. Fluids 7, 867–875.Google Scholar
Sweet, R. A. 1974 A generalized cyclic reduction algorithm. SIAM J. Num. Anal. 10, 506–592.Google Scholar
Valentine, D. T. & Miller, K. D. 1994 Generation of ring vortices in axisymmetric spin-down: A numerical investigation. Phys. Fluids 6, 1535–1547.Google Scholar
Weidman, P. D. 1976a On the spin-up and spin-down of a rotating fluid. Part 1. Extending the Wedemeyer model. J. Fluid Mech. 77, 685–708.Google Scholar
Weidman, P. D. 1976b On the spin-up and spin-down of a rotating fluid. Part 2. Measurements and stability. J. Fluid Mech. 77, 709–735.Google Scholar
Weidman, P. D. 1989 Measurement techniques in laboratory rotating flows. In Advances in Fluid Mechanics Measurements (ed. M. Gad-el-Hak). Lecture Notes in Engineering, vol. 45, pp. 401–534. Springer.