Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-20T14:37:57.666Z Has data issue: false hasContentIssue false
  • English
  • Français

The wake behind a cylinder rolling on a wall at varying rotation rates

Published online by Cambridge University Press:  07 April 2010

B. E. STEWART ,
M. C. THOMPSON ,
T. LEWEKE  and
K. HOURIGAN
B. E. STEWART*
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia Institut de Recherche sur les Phénomènes Hors Equilibre (IRPHE), CNRS/Universités Aix-Marseille, 49 rue Frédéric Joliot-Curie, BP 146, F-13384 Marseille cedex 13, France
M. C. THOMPSON
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia
T. LEWEKE
Affiliation:
Institut de Recherche sur les Phénomènes Hors Equilibre (IRPHE), CNRS/Universités Aix-Marseille, 49 rue Frédéric Joliot-Curie, BP 146, F-13384 Marseille cedex 13, France
K. HOURIGAN
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia Division of Biological Engineering, Monash University, Melbourne, Victoria 3800, Australia
*
†Email address for correspondence: stewart.bronwyn01@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

A study investigating the flow around a cylinder rolling or sliding on a wall has been undertaken in two and three dimensions. The cylinder motion is specified from a set of five discrete rotation rates, ranging from prograde through to retrograde rolling. A Reynolds number range of 20–500 is considered. The effects of the nearby wall and the imposed body motion on the wake structure and dominant wake transitions have been determined. Prograde rolling is shown to destabilize the wake flow, while retrograde rotation delays the onset of unsteady flow to Reynolds numbers well above those observed for a cylinder in an unbounded flow.

Two-dimensional simulations show the presence of two recirculation zones in the steady wake, the lengths of which increase approximately linearly with the Reynolds number. Values of the lift and drag coefficient are also reported for the steady flow regime. Results from a linear stability analysis show that the wake initially undergoes a regular bifurcation from a steady two-dimensional flow to a steady three-dimensional wake for all rotation rates. The critical Reynolds number Rec of transition and the spanwise wavelength of the dominant mode are shown to be highly dependent on, but smoothly varying with, the rotation rate of the cylinder. Varying the rotation from prograde to retrograde rolling acts to increase the value of Rec and decrease the preferred wavelength. The structure of the fully evolved wake mode is then established through three-dimensional simulations. In fact it is found that at Reynolds numbers only marginally (~5%) above critical, the three-dimensional simulations indicate that the saturated state becomes time dependent, although at least initially, this does not result in a significant change to the mode structure. It is only at higher Reynolds numbers that the wake undergoes a transition to vortex shedding.

An analysis of the three-dimensional transition indicates that it is unlikely to be due to a centrifugal instability despite the superficial similarity to the flow over a backward-facing step, for which the transition mechanism has been speculated to be centrifugal. However, the attached elongated recirculation region and distribution of the spanwise perturbation vorticity field, and the similarity of these features with those of the flow through a partially blocked channel, suggest the possibility that the mechanism is elliptic in nature. Some analysis which supports this conjecture is undertaken.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 648 , 10 April 2010 , pp. 225 - 256
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Armaly, B. F., Durst, F., Pereira, J. C. F. & Schönung, B. 1983 Experimental and theoretical investigation of backward-facing step flow. J. Fluid Mech. 127, 473–496.CrossRefGoogle Scholar
Arnal, M. P., Goering, D. J. & Humphrey, J. A. C. 1991 Vortex shedding from a bluff body adjacent to a plane sliding wall. Trans. ASME: J. Fluids Engng 113, 384–398.Google Scholar
Barkley, D., Gomes, M. G. M. & Henderson, R. D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167–190.CrossRefGoogle Scholar
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.CrossRefGoogle Scholar
Barkley, D., Tuckerman, L. S. & Golubitsky, M. 2000 Bifurcation theory for three-dimensional flow in the wake of a circular cylinder. Phys. Rev. E 61 (5), 5247–5252.CrossRefGoogle ScholarPubMed
Barnes, F. H. 2000 Vortex shedding in the wake of a rotating circular cylinder at low Reynolds numbers. J. Phys. D 33, L141–L144.CrossRefGoogle Scholar
Bayly, B. J. 1988 Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows. Phys. Fluids 31 (1), 56–64.CrossRefGoogle Scholar
Bearman, P. W. & Zdravkovich, M. M. 1978 Flow around a circular cylinder near a plane boundary. J. Fluid Mech. 89, 33–47.CrossRefGoogle Scholar
Beaudoin, J.-F., Cadot, O., Aider, J.-L. & Wesfreid, J. E. 2004 Three-dimensional stationary flow over a backward-facing step. Eur. J. Mech B 23, 147–155.CrossRefGoogle Scholar
Chen, Y.-M., Ou, Y.-R. & Pearlstein, A. J. 1993 Development of the wake behind a circular cylinder impulsively started into a rotatory and rectilinear motion. J. Fluid Mech. 253, 449–484.CrossRefGoogle Scholar
Cheng, M. & Luo, L.-S. 2007 Characteristics of two-dimensional flow around a rotating circular cylinder near a plane wall. Phys. Fluids 19, 063601-1–063601-17.CrossRefGoogle Scholar
Chorin, A. J. 1968 Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762.CrossRefGoogle Scholar
Dennis, S. C. R. & Chang, G.-Z. 1970 Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. J. Fluid Mech. 42, 471–489.CrossRefGoogle Scholar
Díaz, F., Gavaldà, J., Kawakk, J. G., Keffer, J. F. & Giralt, F. 1983 Vortex shedding from a spinning cylinder. Phys. Fluids 26 (12), 3454–3460.CrossRefGoogle Scholar
Dipankar, A. & Sengupta, T. K. 2005 Flow past a circular cylinder in the vicinity of a plane wall. J. Fluids Struct. 20 (3), 403–423.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Fornberg, B. 1985 Steady viscous flow past a circular cylinder up to Reynolds number 600. J. Comput. Phys. 61, 297–320.CrossRefGoogle Scholar
Ghia, K. N., Osswald, G. A. & Ghia, U. 1989 Analysis of incompressible massively separated viscous flows using unsteady Navier–Stokes equations. Intl J. Numer. Methods Fluids 9, 1025–1050.CrossRefGoogle Scholar
Griffith, M. D., Thompson, M. C., Leweke, T., Hourigan, K. & Anderson, W. P. 2007 Wake behaviour and instability of flow through a partially blocked channel. J. Fluid Mech. 582, 319–340.CrossRefGoogle Scholar
Henderson, R. D. 1997 Nonlinear dynamics and pattern formation in turbulent wake transition. J. Fluid Mech. 352, 65–112.CrossRefGoogle Scholar
Huang, W.-X. & Sung, H. J. 2007 Vortex shedding from a circular cylinder near a moving wall. J. Fluids Struct. 23, 1064–1076.CrossRefGoogle Scholar
Ingham, D. B. 1983 Steady flow past a rotating cylinder. Comp. Fluids 11 (4), 351–366.CrossRefGoogle Scholar
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in the flow past variously shaped bodies. J. Fluid Mech. 182, 23–45.CrossRefGoogle Scholar
Jaminet, J. F. & Van Atta, C. W. 1969 Experiments on vortex shedding from rotating circular cylinders. AIAA J. 7, 1817–1819.CrossRefGoogle Scholar
Kang, S., Choi, H. & Lee, S. 1999 Laminar flow past a rotating circular cylinder. Phys. Fluids 11 (11), 3312–3321.CrossRefGoogle Scholar
Kano, I. & Yagita, M. 2002 Flow around a rotating circular cylinder near a moving plane wall. Japanese Soc. Mech. Eng. Intl J B 45 (2), 259–268.Google Scholar
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97, 414–443.CrossRefGoogle Scholar
Kawaguti, M. 1966 Numerical study of a viscous fluid flow past a circular cylinder. J. Phys. Soc. Jpn 21 (10), 2055–2062.CrossRefGoogle Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.CrossRefGoogle Scholar
Landman, M. J. & Saffman, P. G. 1987 The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids 30, 2339–2342.CrossRefGoogle Scholar
Le Dizes, S. & Laporte, F. 2002 Theoretical predictions for the elliptic instability in a two-vortex flow. J. Fluid Mech. 471, 169–201.CrossRefGoogle Scholar
Le Dizes, S. & Verga, A. 2002 Viscous interaction of two co-rotating vortices before merging. J. Fluid Mech. 467, 389–410.CrossRefGoogle Scholar
Lei, C., Cheng, L., Armfield, S. W. & Kavanagh, K. 2000 Vortex shedding suppression for flow over a circular cylinder near a plane boundary. Ocean Engng 27, 1109–1127.CrossRefGoogle Scholar
Lei, C., Cheng, L. & Kavanagh, K. 1999 Re-examination of the effect of a plane boundary on force and vortex shedding of a circular cylinder. J. Wind Engng Indus. Aerodyn. 80, 263–286.CrossRefGoogle Scholar
Leontini, J. S., Thompson, M. C. & Hourigan, K. 2007 Three-dimensional transition in the wake of a transversely oscillating cylinder. J. Fluid Mech. 577, 79–104.CrossRefGoogle Scholar
Leweke, T. & Williamson, C. H. K. 1998 Three-dimensional instabilities in wake transition. Eur. J. Mech. B 17 (4), 571–586.CrossRefGoogle Scholar
Lim, T. T., Sengupta, T. K. & Chattopadhyay, M. 2004 A visual study of vortex-induced subcritical instability on a flat plate boundary layer. Exp. Fluids 37, 47–55.CrossRefGoogle Scholar
Mittal, S. & Kumar, B. 2003 Flow past a rotating cylinder. J. Fluid Mech. 476, 303–334.CrossRefGoogle Scholar
Nishino, T., Roberts, G. T. & Zhang, X. 2007 Vortex shedding from a circular cylinder near a moving ground. Phys. Fluids 19, 025103-1–025103-12.CrossRefGoogle Scholar
Provansal, M., Mathis, C. & Boyer, L. 1987 Benard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1–22.CrossRefGoogle Scholar
Roshko, A. 1954 On the development of turbulent wakes from vortex streets. Tech Rep. 1191. National Advisory Committee for Aeronautics.Google Scholar
Ryan, K., Thompson, M. C. & Hourigan, K. 2005 Three-dimensional transition in the wake of bluff elongated cylinders. J. Fluid Mech. 538, 1–29.CrossRefGoogle Scholar
Seddon, J. R. T. & Mullin, T. 2006 Reverse rotation of a cylinder near a wall. Phys. Fluids 18, 041703-1–041703-4.CrossRefGoogle Scholar
Sengupta, T. K., De, S. & Sarkar, S. 2003 Vortex-induced instability of an incompressible wall-bounded shear layer. J. Fluid Mech. 493, 277–286.CrossRefGoogle Scholar
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2003 From spheres to circular cylinders: the stability and flow structures of bluff ring wakes. J. Fluid Mech. 492, 147–180.CrossRefGoogle Scholar
Stewart, B. E., Leweke, T., Hourigan, K. & Thompson, M. C. 2008 Wake formation behind a rolling sphere. Phys. Fluids 20, 071704-1–071704-4.CrossRefGoogle Scholar
Stewart, B. E., Thompson, M. C., Leweke, T. & Hourigan, K. 2009 Numerical and experimental studies of the rolling sphere wake. J. Fluid Mech. Forthcoming.CrossRefGoogle Scholar
Tamaki, H. & Keller, H. B. 1969 Steady two-dimensional viscous flowof an incompressible fluid past a circular cylinder. Phys. Fluids Suppl. II 12 (12), II-51–II-56.Google Scholar
Taneda, S. 1956 Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers. J. Phys. Soc. Jpn 11 (3), 302–307.CrossRefGoogle Scholar
Taneda, S. 1965 Experimental investigation of vortex streets. J. Phys. Soc. Jpn 20 (9), 1714–1721.CrossRefGoogle Scholar
Tang, T. & Ingham, D. B. 1991 On steady flow past a rotating circular cylinder at Reynolds numbers 60 and 100. Comp. Fluids 19 (2), 217–230.CrossRefGoogle Scholar
Thompson, M. C., Hourigan, K., Cheung, A. & Leweke, T. 2006 Hydrodynamics of a particle impact on a wall. Appl. Math. Model. 30, 1356–1369.CrossRefGoogle Scholar
Thompson, M. C., Leweke, T. & Provansal, M. 2001 a Kinematics and dynamics of sphere wake transition. J. Fluids and Struct. 15, 575–585.CrossRefGoogle Scholar
Thompson, M. C., Leweke, T. & Williamson, C. H. 2001 b The physical mechanism of transition in bluff body wakes. J. Fluids Struct. 15, 607–616.CrossRefGoogle Scholar
Van Atta, C. W. 1997 Comments on ‘Hopf bifurcation in wakes behind a rotating and translating circular cylinder’. Phys. Fluids 9 (10), 3105–3106.CrossRefGoogle Scholar
Williamson, C. 1996 Three-dimensional wake transition. J. Fluid Mech. 328, 345–407.CrossRefGoogle Scholar
Williamson, C. H. K. 1988 The existence of two stages in the transition to three-dimensionality of a cylinder wake. Phys. Fluids 31 (11), 3165–3168.CrossRefGoogle Scholar
Zdravkovich, M. M. 1985 Forces on a circular cylinder near a plane wall. Appl. Ocean Res. 7 (4), 197–201.CrossRefGoogle Scholar