Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-06-05T17:48:07.424Z Has data issue: false hasContentIssue false
  • English
  • Français

Splitting, merging and wavelength selection of vortices in curved and/or rotating channel flow due to Eckhaus instability

Published online by Cambridge University Press:  26 April 2006

Y. Guo  and
W. H. Finlay
Y. Guo
Affiliation:
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8
W. H. Finlay
Affiliation:
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8
Get access Rights & Permissions [Opens in a new window]

Abstract

In channels with rotation (about their spanwise axis) or curvature or both, steady two-dimensional vortices develop above a critical Reynolds number Rec, owing to centrifugal or Coriolis effects. The stability of these streamwise oriented roll cells to two-dimensional, spanwise-periodic perturbations (i.e. Eckhaus stability) is examined numerically using linear stability theory and spectral methods. The results are then confirmed by nonlinear flow simulations. In channels with curvature or rotation or both, the Eckhaus stability boundary is found to be a small closed loop. Within the boundary, two-dimensional vortices are stable to spanwise perturbations. Outside the boundary, Eckhaus instability is found to cause the vortex pairs to split apart or merge together. For all channels examined, two-dimensional vortices are always unstable when Re > 1.7 Rec. Usually, the most unstable spanwise perturbations are subharmonic disturbances, which cause two pairs of vortices with small wave-numbers to be split apart by the formation of a new vortex pair, but cause two pairs of vortices with large wavenumber to merge into a single pair. Recent experimental observations of splitting and merging of vortex pairs are discussed. When Re is not too high (Re < 4.0 Rec), the wavenumbers of vortices are selected by Eckhaus instability and most experimentally observed wavenumbers are close to the ones that are least unstable to spanwise perturbations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 228 , July 1991 , pp. 661 - 691
Copyright
© 1991 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

Alfredsson, P. A. & Persson, H. 1989 Instabilities in channel flow with system rotation. J. Fluid Mech. 202, 543–557.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. 27, 417–430.
Bippes, H. 1978 Experimental study of the laminar—turbulent transition of a concave wall in a parallel flow. NASA TM. 75243.Google Scholar
Bland, S. B. & Finlay, W. H. 1991 Transitions towards turbulence in a curved channel.. Phys. Fluids A 3, 106–114.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection. J. Fluid Mech. 65, 625–645.Google Scholar
Diprima, R. C. & Swinney, H. L. 1985 Instabilities and transition in flow between concentric rotating cylinders. In Hydrodynamic Instabilities and the Transition to Turbulence, 2nd edn (ed. H. L. Swinney & J. P. Gollub), Topics in Applied Physics, vol. 45, pp. 139–180. Springer.
Dominguez-Lerma, M. A., Cannell, D. S. & Ahler, G. 1986 Eckhaus boundary and wave-number selection in Rotating Couette–Taylor flow.. Phys. Rev. A 34, 4956–4970.Google Scholar
Eckhaus, W. 1965 Studies in Nonlinear Stability Theory. Springer.
Finlay, W. H. 1990 Transition to oscillatory motion in rotating channel flow. J. Fluid Mech. 215, 209–227.Google Scholar
Finlay, W. H., Keller, J. B. & Ferziger, J. H. 1988 Instability and transition in curved channel flow. J. Fluid Mech. 194, 417–456.Google Scholar
Fox, L. & Parker, I. B. 1968 Chebyshev Polynomials in Numerical Analysis. Oxford University Press.
Gardner, D. R., Trogdon, S. A. & Douglass, R. W. 1989 A modified Tau spectral method that eliminates spurious eigenvalues. J. Comp. Phys. 80, 133–167.Google Scholar
Jones, C. A. 1981 Nonlinear Taylor vortices and their stability. J. Fluid Mech. 157, 135–162.Google Scholar
Jones, C. A. 1985 Numerical methods for the transition to wavy Taylor vortices. J. Comp. Phys. 61, 321–344.Google Scholar
Kelleher, M. D., Flentie, D. L. & McKee, R. J. 1980 An experimental study of the secondary flow in a curved rectangular channel. Trans. ASME I: J. Fluids Engng. 102, 92–96.Google Scholar
Ligrani, P., Kendall, M. R. & Longest, J. E. 1990 Appearance, disappearance and spanwise wavenumber selection of Dean vortex pairs in a curved rectangular channel. Phys. Fluids A (Submitted).Google Scholar
Ligrani, P. & Niver, R. D. 1988 Flow visualization of Dean vortices in a curved channel with 40 to 1 aspect ratio. Phys. Fluids 31, 3605–3618.Google Scholar
Masuda, S. & Matsubara, M. 1989 Visual study of boundary layer transition on rotating flat plate IUTAM Symp. on Laminar–Turbulent Transition. Springer.
Matsson, J. O. E. & Alfredsson, P. H. 1990 Curvature- and rotation-induced instabilities in channel flow. J. Fluid Mech. 202, 543–557.Google Scholar
Meyer-Spasche, R. & Keller, H. B. 1985 Some bifurcation diagrams for Taylor vortex flows. Phys. Fluids 28, 1248–1252.Google Scholar
Moser, R. D. & Moin, P. 1984 Direct numerical simulation of curved turbulent channel flow. NASA TM 85974.Google Scholar
Moser, R. D. & Moin, P. 1987 The effects of curvature in wall-bounded turbulent flow. J. Fluid Mech. 175, 497–510.Google Scholar
Moser, R. D., Moin, P. & Leonard, A. 1983 A spectral numerical method for the Navier–Stokes equations with applications to Taylor–Couette flow. J. Comp. Phys. 52, 524–544.Google Scholar
Nagata, M. 1986 Bifurcations in Couette flow between almost corotating cylinders. J. Fluid Mech. 169, 229–250.Google Scholar
Nagata, M. 1988 On wavy instabilities of the Taylor-vortex flow between corotating cylinders. J. Fluid Mech. 188, 585–598.Google Scholar
Nagata, M. & Busse, F. H. 1983 Three-dimensional tertiary motion in a plane shear layer. J. Fluid Mech. 135, 1–26.Google Scholar
Paap, H. & Riecke, H. 1990 Wave-number restriction and mode interaction in Taylor vortex flow: Appearance of a short-wavelength instability. Phys. Rev. A 41, 1943–1951.Google Scholar
Riecke, H. & Paap, H. 1986 Stability and wave-vector restriction of axisymmetric Taylor vortex flow.. Phys. Rev. A 33, 547–553.Google Scholar
Stuart, J. T. & DiPrima, R. C. 1978 The Eckhaus and Benjamin–Feir resonance mechanism.. Proc. R. Soc. Lond. A 362, 27–41.Google Scholar
Tritton, D. J. & Davies, P. A. 1985 Instabilities in geophysical fluid dynamics. In Hydrodynamic Instabilities and the Transition to Turbulence, 2nd edn (ed. H. L. Swinney & J. P. Gollub), Topics in Applied Physics, vol. 45, pp. 229–270. Springer.
Zebib, A. 1984 A Chebyshev method for the solution of boundary value problems. J. Comp. Phys. 53, 443–455.Google Scholar