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Transition to turbulence in a rotating channel

Published online by Cambridge University Press:  26 April 2006

W. H. Finlay
Affiliation:
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8

Abstract

Direct numerical simulation is used to determine the flows that occur as the Reynolds number, Re, is increased in a plane channel undergoing system rotation about a spanwise axis. (Plane Poiseuille flow occurs for zero rotation rate and low Re.) A constant system rotation speed of 0.5, non-dimensionalized with respect to the bulk streamwise velocity and channel full width, is used throughout. The spectral numerical method solves the three-dimensional, time-dependent, incompressible Navier-Stokes equations using periodic boundary conditions in the streamwise and spanwise directions. On increasing the Reynolds number above the temporally periodic wavy vortex regime, near Re = 4Rec (Rec = 88.6 is the critical Re for development of vortices), a second temporal frequency, ω2, occurs in the flow that corresponds to slow, constant, spanwise motion of the vortices, superposed on the much faster, constant, streamwise motion of the wavy vortex waves. Curiously, ω2 is always frequency locked with the wavy vortex frequency ω1 for the parameter range explored, although the locking ratio varies. At the slightly higher Re of 4.1 Rec, ω2 is replaced by a new frequency ω′2 that corresponds to a modulation of the wavy vortices like that seen in modulated wavy Taylor vortex flow. However, unlike the Taylor-Couette geometry, the modulation frequency here can become frequency locked with the wavy vortex frequency. Increasing Re further to Re = 4.2 Rec results in the appearance of a second incommensurate modulation frequency ω3, yielding a quasi-periodic three-frequency flow, although there are only two frequencies (ωω′2 and ω3) present in the reference frame moving with the travelling wave associated with ω1. At still higher Re (Re = 4.5 Rec), weak temporal chaos occurs. This flow is not turbulent however. Calculations of the instantaneous largest Lyapunov exponent, λ(t), and the spatial structure of small perturbations to the flow show that the chaos is driven by spanwise shear instability of the streamwise velocity component. At the highest Re of 6.7 Rec considered, quasi-coherent turbulent boundary layer structures occur as transient, secondary streamwise-oriented vortices in the viscous sublayer near the inviscidly unstable (high-pressure) wall. Calculations of λ(t) and the spatial structure of small perturbations to the flow show that the coherent structures are not caused by the local growth of small disturbances to the flow.

Type
Research Article
Copyright
© 1992 Cambridge University Press

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References

Alfredsson, P. A. & Persson, H. 1989 Instabilities in channel flow with system rotation. J. Fluid Mech. 202, 543.Google Scholar
Arnold, V. I. 1965 Small denominators I. Mappings of the circumference onto itself. Am. Math. Soc. Transl. 46, 213.Google Scholar
Aubry, N., Holmes, P. Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structure in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115.Google Scholar
Benettin, G., Galgani, L. & Strelcyn, J. 1976 Kolmogorov entropy and numerical experiments. Phys. Rev. A 14, 2338.Google Scholar
BergeU, P., Pomeau, Y. & Vidal, C. Order Within Chaos. John Wiley & Sons.
Biringen, S. & Peltier, L. J. 1990 Numerical simulation of 3-D BeAnard convection with gravitational modulation. Phys. Fluids A2, 754.Google Scholar
Bland, S. B. & Finlay, W. H. 1991 Transitions toward turbulence in a curved channel. Phys. Fluids A3, 106.Google Scholar
Cantwell, B. J. 1981 Organized motion in turbulent flow. Ann. Rev. Fluid Mech. 13, 457.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.
Coughlin, K. T. 1990 Quasiperiodic Taylor-Couette flow. Ph.D. thesis. Harvard University.
Coughlin, K. T. & Marcus, P. S. 1992a Modulated waves in Taylor-Couette flow. Part 1. Analysis. J. Fluid Mech. 234, 234.Google Scholar
Coughlin, K. T. & Marcus, P. S. 1992b Modulated waves in Taylor-Couette flow. Part 2. Numerical simulation. J. Fluid Mech. 234, 234.Google Scholar
Curry, J. H., Herring, J. R., Loncaric, J. & Orszag, S. A. 1984 Order and disorder in two- and three-dimensional BeAnard convection. J. Fluid Mech. 147, 1.Google Scholar
Fenstermacher, P. R., Swinney, H. L. & Gollub, J. P. 1979 Dynamical instabilities and the transition to chaotic Taylor vortex flow. J. Fluid Mech. 94, 103.Google Scholar
Finlay, W. H. 1989 Perturbation expansion and weakly nonlinear analysis for two-dimensional vortices in curved or rotating channels. Phys. Fluids A 1, 854.Google Scholar
Finlay, W. H. 1990 Transition to oscillatory motion in rotating channel flow. J. Fluid Mech. 215, 209227.Google Scholar
Finlay, W. H., Keller, J. B. & Ferziger, J. H. 1988 Instability and transition in curved channel flow. J. Fluid Mech. 194, 417.Google Scholar
Finlay, W. H. & Kandakumar, K. 1990 Onset of two-dimensional cellular flow in finite curved channels of large aspect ratio. Phys. Fluids A 2, 209227.Google Scholar
Gollub, J. P. & Benson, S. V. 1980 Many routes to turbulent convection. J. Fluid Mech. 100, 449.Google Scholar
Gorman, M. & Swinney, H. L. 1982 Spatial and temporal characteristics of modulated waves in the circular Couette system. J. Fluid Mech. 117, 123.Google Scholar
Guezennec, Y. G., Piomelli, U. & Kim, J. 1989 On the shape and dynamics of wall structures in turbulent channel flow. Phys. Fluids A1, 764.Google Scholar
Guo, Y. & Finlay, W. H. 1991 Splitting, merging and wavelength selection of vortices in curved and/or rotating channels due to Eckhaus instability. J. Fluid Mech. 228, 661.Google Scholar
Hall, P. 1988 The nonlinear development of GoUrtler vortices in growing boundary layers. J. Fluid Mech. 193, 243.Google Scholar
Herbert, Th. 1988 Secondary stability of boundary layers. Ann. Rev. Fluid Mech. 20, 487.Google Scholar
JimeAnez, J. 1990 Transition to turbulence in two-dimensional Poiseuille flow. J. Fluid Mech. 218, 265.Google Scholar
JimeAnez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213.Google Scholar
Johnston, J. P., Halleen, R. M. & Lezius, D. K. 1972 Effects of spanwise rotation on the structure of two-dimensional fully developed turbulent channel flow. J. Fluid Mech. 56, 533.Google Scholar
Kleiser, L. & Zang, T. 1991 Numerical simulations of transition in wall-bounded flows. Ann. Rev. Fluid Mech. 23, 495.Google Scholar
Kuz'minskii, L. V., Smirnov, E. M. & Yurkin, S. V.1983 Longitudinal cellular structures of Taylor-GoUrtler type vortices on the high-pressure side of rotating channels. J. Appl. Mech. Tech. Phys. 24, 882.Google Scholar
Malik, M. R. & Hussaini, M. Y. 1990 Numerical simulation of interactions between GoUrtler vortices and Tollmien-Schlichting waves. J. Fluid Mech. 210, 183.Google Scholar
Matsson, O. J. & Alfredsson, P. H. 1990 Curvature- and rotation-induced instabilities in channel flow. J. Fluid Mech. 210, 537.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. Comput. Phys. 52, 524.Google Scholar
Olinger, D.J. & Sreenivasan 1988 Konlinear dynamics of the wake of an oscillating cylinder. Phys. Rev. Lett. 60, 797.Google Scholar
Rand, D. 1982 Dynamics and symmetry. Predictions for modulated waves in rotating fluids. Arch. Rat. Mech. Anal. 79, 1.Google Scholar
Reed, H. L. 1988 Wave interactions in swept-wing flows. Phys. Fluids 30, 3419.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Ann. Rev. Fluid Mech. 23, 601.Google Scholar
Robinson, S. K., Kline, S. J. & Spalart, P. R. 1988 Quasi-coherent structures in the turbulent boundary-layer: Part II. Verification and new information from a numerically simulated flat-plate layer. In Near-Wall Turbulence (ed. S. J. Kline & N. H. Afgan), p. 218. Hemisphere.
Smirnov, E. M. & Yurkin, S. V. 1983 Fluid flow in a rotating channel of square section. Fluid Dyn. 18, 850.Google Scholar
Tafti, D. K. & Vanka, S. P. 1991 A numerical study of the effects of spanwise rotation on turbulent channel flow. Phys. Fluids A3, 642.Google Scholar
Tritton, D. J. & Davies, P. A. 1985 Instabilities in geophysical fluid dynamics. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub), p. 229. Springer.
Tryggvason, G. & Unverdi, S. O. 1990 Computations of three-dimensional Rayleigh-Taylor instability. Phys. Fluids A 2, 656.Google Scholar
Vastano, J. A., Moser, R. D. & Keefe, L. 1989 A proposed mechanism for the transition to chaos in Taylor-Couette flow. Bull. Amer. Phys. Soc. 34, 2264.Google Scholar
Yang, K.-S. & Kim, J. 1991 Numerical investigation of instability and transition in rotating plane poiseuille flow. Phys. Fluids A 3, 633.Google Scholar