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Nonlinear dynamics near the stability margin in rotating pipe flow

Published online by Cambridge University Press:  26 April 2006

Z. Yang  and
S. Leibovich
Z. Yang
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA Present address: Center for Modeling of Turbulence and Transition, NASA Lewis Research Center/ICOMP, Cleveland, OH 44135, USA.
S. Leibovich
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
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Abstract

The nonlinear evolution of marginally unstable wave packets in rotating pipe flow is studied. These flows depend on two control parameters, which may be taken to be the axial Reynolds number R and a Rossby number, q. Marginal stability is realized on a curve in the (R, q)-plane, and we explore the entire marginal stability boundary. As the flow passes through any point on the marginal stability curve, it undergoes a supercritical Hopf bifurcation and the steady base flow is replaced by a travelling wave. The envelope of the wave system is governed by a complex Ginzburg–Landau equation. The Ginzburg–Landau equation admits Stokes waves, which correspond to standing modulations of the linear travelling wavetrain, as well as travelling wave modulations of the linear wavetrain. Bands of wavenumbers are identified in which the nonlinear modulated waves are subject to a sideband instability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 233 , December 1991 , pp. 329 - 347
Copyright
© 1991 Cambridge University Press

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References

Akylas, T. R. & Demurger, J. P. 1984 The effect of rigid rotation on the finite-amplitude stability of pipe flow at high Reynolds number. J. Fluid Mech. 148, 193–205.Google Scholar
Batchelor, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14, 529–551.Google Scholar
Bernoff, A. J. 1988 Slowly varying fully nonlinear wavetrains in the Ginzburg–Landau equation.. Physica D 30, 363–381.Google Scholar
Cotton, F. W. & Salwen, H. 1981 Linear stability of rotating Hagen–Poiseuille flow. J. Fluid Mech. 108, 101–125.Google Scholar
Davey, A. 1978 On Itoh's finite amplitude stability theory for pipe flow. J. Fluid Mech. 86, 695–703.Google Scholar
Davey, A. & Nguyen, H. P. F. 1971 Finite-amplitude stability of pipe flow. J. Fluid Mech. 45, 701–720.Google Scholar
Eckhaus, W. 1965 Studies in Non-Linear Stability Theory. Springer.
Gill, A. E. 1965 On the stability of small disturbances to Poiseuille flow in a circular pipe. J. Fluid Mech. 21, 145–172.Google Scholar
Herbert, T. 1983 On perturbation methods in nonlinear stability theory. J. Fluid Mech. 126, 167–186.Google Scholar
Itoh, N. 1977 Nonlinear stability of parallel flows with subcritical Reynolds number. Part 2. Stability of pipe Poiseuille flow to finite axisymmetric disturbances. J. Fluid Mech. 82, 469–479.Google Scholar
Joseph, D. D. & Carmi, S. 1969 Stability of Poiseuille flow in pipes, annuli, and channels. Q. Appl. Maths 26, 575–599.Google Scholar
Keefe, L. R. 1985 Dynamics of perturbed wavetrain solutions to the Ginzburg–Landau equation. Stud. Appl. Maths 73, 91–153.Google Scholar
Landman, M. J. 1990 Time-dependent helical waves in rotating pipe flow. J. Fluid Mech. 221, 289–310.Google Scholar
Leibovich, S., Brown, S. N. & Patel, Y. 1986 Bending waves on inviscid columnar vortices. J. Fluid Mech. 173, 595–624.Google Scholar
Mackrodt, P. A. 1976 Stability of Hagen–Poiseuille flow with superimposed rigid rotation. J. Fluid Mech. 73, 153–164.Google Scholar
Mahalov, A. & Leibovich, S. 1989 Weakly nonlinear expansion for viscous rotating Hagen–Poiseuille flow. Bull. Am. Phys. Soc. 34, 2318.Google Scholar
Mahalov, A., Titi, E. & Leibovich, S. 1990 Invariant helical subspaces for the Navier–Stokes equations. Arch. Rat. Mech. Anal. 112, 193–222.Google Scholar
Mahalov, A. & Leibovich, S. 1991 Multiple bifurcation of rotating pipe flow. Theor. Comput. Fluid Mech. (in press).Google Scholar
Newell, A. C. 1974 Envelope equations. Lect. Appl. Maths 15, 157–163.Google Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Patera, A. T. & Orszag, S. A. 1981 Finite amplitude stability of axisymmetric pipe flow. J. Fluid Mech. 112, 467–474.Google Scholar
Pedley, T. J. 1968 On the instability of rapidly rotating shear flows to non-axisymmetric disturbances. J. Fluid Mech. 31, 603–607.Google Scholar
Pedley, T. J. 1969 On the instability of viscous flow in a rapidly rotating pipe. J. Fluid Mech. 35, 97–115.Google Scholar
Reynolds, O. 1883 An experimental investigation of the circumstances which determines whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. 174, 935–982.Google Scholar
Salwen, H. & Grosch, C. E. 1972 The stability of Poiseuille flow in a pipe of circular cross-section. J. Fluid Mech. 54, 93–112.Google Scholar
Sirovich, L. & Newton, P. K. 1986 Periodic solutions of the Ginzburg–Landau equation. Physica D21, 115–125.Google Scholar
Stewartson, K. & Stuart, J. T. 1971 Nonlinear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529–545.Google Scholar
Stuart, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. J. Fluid Mech. 9, 353–370.Google Scholar
Stuart, J. T. & DiPrima, R. C. 1978 The Eckhaus and Benjamin–Feir resonance mechanisms.. Proc. R. Soc. Lond. A 362, 27–41.Google Scholar
Toplosky, N. & Akylas, T. R. 1988 Nonlinear spiral waves in rotating pipe flow. J. Fluid Mech. 190, 39–54.Google Scholar
Watson, J. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow. J. Fluid Mech. 9, 371–389.Google Scholar