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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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