Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-07T08:47:58.169Z Has data issue: false hasContentIssue false
  • English
  • Français

Dean vortices in finite-aspect-ratio ducts

Published online by Cambridge University Press:  31 January 2013

Philip E. Haines ,
James P. Denier  and
Andrew P. Bassom
Philip E. Haines
Affiliation:
School of Mathematical Sciences, The University of Adelaide, South Australia 5005, Australia
James P. Denier*
Affiliation:
Department of Engineering Science, The University of Auckland, Auckland 1142, New Zealand
Andrew P. Bassom
Affiliation:
School of Mathematics & Statistics, The University of Western Australia, Crawley 6009, Australia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the development of Dean vortices in a curved channel of finite aspect ratio. Solutions to the axisymmetric Navier–Stokes equations are obtained through a finite-element analysis, allowing us to explore the complex and rich bifurcation pattern of the flow as the aspect ratio and Dean number vary. We demonstrate a new class of finite-amplitude vortices and discuss their relationship to similar structures seen in finite-length Taylor–Couette flow.

JFM classification

Instability: Instability Instability: Nonlinear instability
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 716 , 10 February 2013 , R8
Copyright
©2013 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.)

Footnotes

‡

Current address: Department of Mathematics, University College London.

References

Bara, B., Nandakumar, K. & Masliyah, J. 1992 An experimental and numerical study of the Dean problem: flow development towards two-dimensional multiple solutions. J. Fluid Mech. 244, 339–376.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Benjamin, T. B. 1978 Bifurcation phenomena in steady flows of a viscous fluid. I. Theory. Proc. R. Soc. Lond. A 359, 1–26.Google Scholar
Benjamin, T. B. & Mullin, T. 1982 Notes on the multiplicity of flows in the Taylor experiment. J. Fluid Mech. 121, 219–230.CrossRefGoogle Scholar
Berger, S. A. & Talbot, L. 1983 Flow in curved pipes. Annu. Rev. Fluid Mech. 15, 461–512.CrossRefGoogle Scholar
Cliffe, K. A. 1988 Numerical calculations of the primary-flow exchange process in the Taylor problem. J. Fluid Mech. 197, 57–79.CrossRefGoogle Scholar
Dean, W. R. 1928 Fluid motion in a curved channel. Proc. R. Soc. Lond. A 121, 402–420.Google Scholar
Finlay, W. & Nandakumar, K. 1990 Onset of two-dimensional cellular flow in finite curved channels of large aspect ratio. Phys. Fluids A 2, 1163–1174.CrossRefGoogle Scholar
Gresho, P. M. & Sani, R. L. 1998 Incompressible Flow and the Finite Element Method: Volume 1, Advection–Diffusion and Isothermal Laminar Flow. John Wiley & Sons.Google Scholar
Heil, M. & Hazel, A. L. 2006 oomph-lib– an object-oriented multi-physics finite-element library. In Fluid-Structure Interaction, Lecture Notes on Computational Science and Engineering (ed. Schafer, M. & Bungartz, H.-J.), pp. 19–49. Springer.Google Scholar
Hille, P., Vehrenkamp, R. & Schulz-Dubois, E. O. 1985 The development and structure of primary and secondary flow in a curved square duct. J. Fluid Mech. 151, 219–241.CrossRefGoogle Scholar
Jospeh, B., Smith, E. P. & Adler, R. J. 1975 Numerical treatment of laminar flow in helically coiled tubes of square cross section. Part I. Stationary helically coiled tubes. AIChE J. 21, 965–974.Google Scholar
Ligrani, P., Finlay, W., Fields, W., Fuqua, S. & Subramanian, C. 1992 Features of wavy vortices in a curved channel from experimental and numerical studies. Phys. Fluids A 4, 695.CrossRefGoogle Scholar
Ligrani, P. M. & Niver, R. D. 1988 Flow visualization of Dean vortices in a curved channel with 40 to 1 aspect ratio. Phys. Fluids 31, 3605–3617.CrossRefGoogle Scholar
Mees, P., Nandakumar, K. & Masliyah, J. 1996 Steady spatial oscillations in a curved duct of square cross-section. Phys. Fluids 8 (12), 3264–3270.CrossRefGoogle Scholar
Mullin, T. 1982 Mutations of steady cellular flows in the Taylor experiment. J. Fluid Mech. 121, 207–218.CrossRefGoogle Scholar
Sankar, S. R., Nandakumar, K. & Masliyah, J. H. 1988 Oscillatory flows in coiled square ducts. Phys. Fluids 31, 1348–1359.CrossRefGoogle Scholar
Seban, R. A. & McLaughlin, E. F. 1963 Heat transfer in tube coils with laminar and turbulent flow. Intl J. Heat Mass Transfer 6, 387–395.CrossRefGoogle Scholar
Winters, K. H. 1987 Bifurcation study of laminar flow in a curved tube of rectangular cross-section. J. Fluid Mech. 180, 343–369.CrossRefGoogle Scholar
Yanase, S., Kaga, Y. & Daikai, R. 2002 Laminar flows through a curved rectangular duct over a wide range of the aspect ratio. Fluid Dyn. Res. 31, 151–183.CrossRefGoogle Scholar