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Controlling the dual cascade of two-dimensional turbulence

Published online by Cambridge University Press:  30 November 2010

M. M. FARAZMAND ,
N. K.-R. KEVLAHAN  and
B. PROTAS
M. M. FARAZMAND
Affiliation:
School of Computational Engineering & Science, McMaster University, Hamilton L8S 4K1, Canada
N. K.-R. KEVLAHAN*
Affiliation:
Department of Mathematics & Statistics, McMaster University, Hamilton L8S 4K1, Canada
B. PROTAS
Affiliation:
Department of Mathematics & Statistics, McMaster University, Hamilton L8S 4K1, Canada
*
†Email address for correspondence: kevlahan@mcmaster.ca
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Abstract

The Kraichnan–Leith–Batchelor (KLB) theory of statistically stationary forced homogeneous isotropic two-dimensional turbulence predicts the existence of two inertial ranges: an energy inertial range with an energy spectrum scaling of k−5/3, and an enstrophy inertial range with an energy spectrum scaling of k−3. However, unlike the analogous Kolmogorov theory for three-dimensional turbulence, the scaling of the enstrophy range in the two-dimensional turbulence seems to be Reynolds-number-dependent: numerical simulations have shown that as Reynolds number tends to infinity, the enstrophy range of the energy spectrum converges to the KLB prediction, i.e. E ~ k−3. The present paper uses a novel optimal control approach to find a forcing that does produce the KLB scaling of the energy spectrum in a moderate Reynolds number flow. We show that the time–space structure of the forcing can significantly alter the scaling of the energy spectrum over inertial ranges. A careful analysis of the optimal forcing suggests that it is unlikely to be realized in nature, or by a simple numerical model.

JFM classification

Turbulent Flows: Turbulence control Turbulent Flows: Turbulence simulation Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 668 , 10 February 2011 , pp. 202 - 222
Copyright
Copyright © Cambridge University Press 2010

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