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A simple dynamical system that mimics open-flow turbulence

Bhat, G. S. and Narashima, R. and Wiggins, S. (1990) A simple dynamical system that mimics open-flow turbulence. Physics of Fluids A, 2 (11). pp. 1983-2001. ISSN 0899-8213.

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The possible relevance of recent theories concerning the chaotic behavior of nonlinear dynamical systems to turbulence, especially in open flows, has frequently been questioned. Here, the issues that have led to this skepticism are investigated by studying a simple system that has been devised to include, albeit in an impressionistic way, the major mechanisms that are widely considered to operate in a broad class of turbulent flows. The variables in the system seek to represent the amplitudes of large- and small-eddy motion, respectively, and are governed by equations that allow for a Landau–Stuart nonlinear growth, a one-step Richardson cascade, and a specified time-dependent driving force. It is found that the critical value (at the onset of chaos) of the Reynolds-number-like control parameter (nu−1) in the system depends on the character and magnitude of the driving force; and it is analytically demonstrated using the Melnikov technique that, with an appropriate choice of model parameters, chaos can persist at all sufficiently high values of the model Reynolds number (unlike as in most other low-dimensional models). The routes to chaos in the system when the forcing is increased at fixed nu are different from those when nu is decreased at fixed forcing, the latter being found to be more relevant to the case of streamwise-developing flows like a boundary layer. The observed routes are sensitive to the presence of even small stochastic components in the forcing. Computed spectral evolutions in the model show qualitative similarities with observations in boundary layer flow under different disturbance environments. It is concluded that many of the gross features of open-flow turbulence can be understood as dynamical chaos.

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Additional Information:Copyright © 1990 American Institute of Physics. (Received 7 June 1989; accepted 18 June 1990) We are grateful to Professor A. Roshko for his constant encouragement and support, and have benefited from many discussions with him, Professor H.W. Liepmann, and Professor Phillip Holmes. The present version of this paper was prepared during a visit by RN to Cambridge, where he thanks Professor M. Gaster for interesting discussions and hospitality. The continued support of Professor A. Prabhu and Professor V.H. Arakeri at Bangalore is gratefully acknowledged. This work has been supported in India by a grant from the Department of Science and Technology and in the U.S. by Contract No. N00014-85-K-0205 from the Office of Naval Research. The support received from the BOYS-CAST scheme of DST, and the Millikan Fund from Caltech, enabling visits to the United States by GSB and RN, respectively, were crucial for carrying out the project.
Issue or Number:11
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Deposited On:29 Oct 2007
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