Controlling the dual cascade of two-dimensional turbulence

• Published in: Journal of fluid mechanics Vol. 668; pp. 202 - 222
• Main Authors: FARAZMAND, M. M., KEVLAHAN, N. K.-R., PROTAS, B.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 10.02.2011
• Subjects: Computational fluid dynamics; Energy; Exact sciences and technology; Fluid dynamics; Fluid flow; Fluid mechanics; Fundamental areas of phenomenology (including applications); Inertial; Mathematical models; Numerical analysis; Physics; Reynolds number; Turbulence; Turbulence control; Turbulence simulation and modeling; Turbulent flow; Turbulent flows, convection, and heat transfer; Two dimensional; turbulence simulation; turbulence control; turbulence theory; Turbulent flow; Energy spectra; Optimal control; Enstrophy; Digital simulation; Scaling laws; Boundary conditions; Modelling; Incompressible fluid; Turbulence structure
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/S0022112010004635

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.