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An investigation of transition to turbulence in bounded oscillatory Stokes flows Part 2. Numerical simulations

Published online by Cambridge University Press:  26 April 2006

R. Akhavan ,
R. D. Kamm  and
A. H. Shapiro
R. Akhavan
Affiliation:
Department of Mechanical Engineering, The University of Michigan, Ann Arbor, MI 48109–2125 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
R. D. Kamm
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
A. H. Shapiro
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
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Abstract

The stability of oscillatory channel flow to different classes of infinitesimal and finite-amplitude two- and three-dimensional disturbances has been investigated by direct numerical simulations of the Navier–Stokes equations using spectral techniques. All infinitesimal disturbances were found to decay monotonically to a periodic steady state, in agreement with earlier Floquet theory calculations. However, before reaching this periodic steady state an infinitesimal disturbance introduced in the boundary layer was seen to experience transient growth in accordance with the predictions of quasi-steady theories for the least stable eigenmodes of the Orr–Sommerfeld equation for instantaneous ‘frozen’ profiles. The reason why this growth is not sustained in the periodic steady state is explained. Two-dimensional infinitesimal disturbances reaching finite amplitudes were found to saturate in an ordered state of two-dimensional quasi-equilibrium waves that decayed on viscous timescales. No finite-amplitude equilibrium waves were found in our cursory study. The secondary instability of these two-dimensional finite-amplitude quasi-equilibrium states to infinitesimal three-dimensional perturbations predicts transitional Reynolds numbers and turbulent flow structures in agreement with experiments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 225 , April 1991 , pp. 423 - 444
Copyright
© 1991 Cambridge University Press

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References

Akhavan, R.: 1987 An investigation of transition and turbulence in oscillatory Stokes layers. Ph.D. thesis, MIT, Cambridge, MA.
Akhavan, R., Kamm, R. D. & Shapiro, A. H., 1991 An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 1. Experiments. J. Fluid Mech. 225, 395–422.Google Scholar
Bayly, B. J.: 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163.Google Scholar
Collins, J. I.: 1963 Inception of turbulence at the bed under periodic gravity waves. J. Geophys. Res. 18, 6007–6014.Google Scholar
Cowley, S. J.: 1987 High frequency Rayleigh instability of Stokes layers. In Stability of Time Dependent and Spatially Varying Flows (ed. D. L. Dwoyer & M. Y. Hussaini), pp. 261–275. Springer.
Hall, P.: 1978 The linear stability of flat Stokes layers. Proc. R. Soc. Lond. A 359, 151–166.Google Scholar
Herbert, T.: 1976 Periodic secondary motions in a plane channel. In Proc. 5th Intl Conf. on Numerical Methods in Fluid Dynamics (ed. A. I. Van de Vooren & P. J. Zandbergen). Lecture Notes in Physics, vol. 59, p. 235. Springer.
Herbert, T.: 1983 Secondary instability of plane channel flow to subharmonic three-dimensional disturbances. Phys. Fluids 26, 871–874.Google Scholar
Herbert, T.: 1984 Analysis of the subharmonic route to transition in boundary layers. AIAA Paper 84–0009.Google Scholar
Hino, M., Kashiwayanagi, M., Nakayama, A. & Hara, T., 1983 Experiments on the turbulence statistics and the structure of a reciprocating oscillatory flow. J. Fluid Mech. 131, 363–100.Google Scholar
Hino, M., Sawamoto, M. & Takasu, S., 1976 Experiments on transition to turbulence in an oscillating pipe flow. J. Fluid Mech. 75, 193–207.Google Scholar
Von Kerczek, C. & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753–773.Google Scholar
Landman, M. J. & Saffman, P. G., 1987 The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids 30, 2339–2342.Google Scholar
Merkli, P. & Thomann, H., 1975 Transition to turbulence in oscillating pipe flow. J. Fluid Mech. 68, 567–575.Google Scholar
Monkewitz, P. A. & Bunster, A., 1987 The stability of the Stokes layer: visual observations and some theoretical considerations. In Stability of Time Dependent and Spatially Varying Flows (ed. D. L. Dwoyer & M. Y. Hussaini), pp. 244–260. Springer.
Obremski, H. J. & Morkovin, M. V., 1969 Applications of a quasi-steady stability model to periodic boundary layer flow. AIAA J. 7, 1298–1301.Google Scholar
Ohmi, M., Iguchi, M., Kakehachi, K. & Masuda, T., 1982 Transition to turbulence and velocity distribution in an oscillating pipe flow. Bull. JSME 25, 365–371.Google Scholar
Orszag, S. A.: 1971 Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Orszag, S. A. & Kells, L. C., 1980 Transition to turbulence in plane Poiseulle and plane Couette flow. J. Fluid Mech. 96, 159–205.Google Scholar
Orszag, S. A. & Patera, A. T., 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Pierrehumbert, R. T.: 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57, 2157–2159.Google Scholar
Pierrehumbert, R. T. & Widnall, S. E., 1982 The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82.Google Scholar
Sergeev, S. I.: 1966 Fluid oscillations in pipes at moderate Reynolds number. Fluid Dyn. 1, 121–122.Google Scholar