Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-12-01T02:30:39.808Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Basdevant, C., Legras, B. & Sadourny, R. 1981 A study of barotropic model flows: intermittency, waves and predictability. J. Atmos. Sci. 38, 2305–2326.2.0.CO;2>CrossRefGoogle Scholar
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids Suppl. II, 233–239.CrossRefGoogle Scholar
Bewley, T. R. 2001 Flow control: new challenges for a renaissance. Prog. Aerosp. Sci. 37, 21–58.CrossRefGoogle Scholar
Boffetta, G. 2007 Energy and enstrophy fluxes in the double cascade of two-dimensional turbulence. J. Fluid Mech. 589, 253–260.CrossRefGoogle Scholar
Boffetta, G., Celani, A., Musacchio, S. & Vergassola, M. 2002 Intermittency in two-dimensional Ekman–Navier–Stokes turbulence. Phys. Rev. E 66, 026304.CrossRefGoogle ScholarPubMed
Boffetta, G. & Musacchio, S. 2010 Evidence for the double cascade scenario in two-dimensional turbulence. Phys. Rev. E 82, 016307.CrossRefGoogle ScholarPubMed
Borue, V. 1993 Spectral exponents of enstrophy cascade in stationary two-dimensional homogeneous turbulence. Phys. Rev. Lett. 71, 3967.CrossRefGoogle ScholarPubMed
Bracco, A. & McWilliams, J. C. 2010 Reynolds-number dependency in homogeneous, stationary two-dimensional turbulence. J. Fluid Mech. 646, 517–526.CrossRefGoogle Scholar
Bruneau, C. H. & Kellay, H. 2005 Experiments and direct numerical simulations of two-dimensional turbulence. Phys. Rev. E 71, 046305.CrossRefGoogle ScholarPubMed
Chen, S., Ecke, R. E., Eyink, G. L., Wang, X. & Xiao, Z. 2003 Physical mechanism of the two-dimensional enstrophy cascade. Phys. Rev. Lett. 91, 214501.CrossRefGoogle ScholarPubMed
Clercx, H. J. H. & van Heijst, G. J. F. 2009 Two-dimensional Navier–Stokes turbulence in bounded domains. Appl. Mech. Rev. 62, 020802.CrossRefGoogle Scholar
Constantin, P., Foias, C. & Manley, O. 1994 Effects of the forcing function spectrum on energy spectrum in 2-D turbulence. Phys. Fluids 6, 427–429.CrossRefGoogle Scholar
Edwards, W. S., Tuckerman, L. S., Friesner, R. A. & Sorensen, D. C. 1994 Krylov methods for the incompressible Navier–Stokes equations. J. Comp. Phys. 110, 82–102.CrossRefGoogle Scholar
Fjørtoft, R. 1953 On the changes in the spectral distribution of kinetic energy for two dimensional, nondivergent flow. Tellus 5, 225–230.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Frisch, U. & Sulem, P. L. 1984 Numerical simulation of inverse cascade in two-dimensional turbulence. Phys. Fluids 27, 1921–1923.CrossRefGoogle Scholar
Gage, K. S. & Nastrom, G. D. 1985 On the spectrum of the atmospheric velocity fluctuations seen by MST/ST radar and their interpretation. Radio Sci. 20, 1339–1374.CrossRefGoogle Scholar
Gioia, G. & Chakraborty, P. 2006 Turbulent friction in rough pipes and the energy spectrum of the phenomenological theory. Phys. Rev. Lett. 96, 044502.CrossRefGoogle ScholarPubMed
Gkioulekas, E. & Tung, K. K. 2007 A new proof on net upscale energy cascade in two-dimensional and quasi-geostrophic turbulence. J. Fluid Mech. 576, 173–189.CrossRefGoogle Scholar
Gunzburger, M. D. 2003 Perspectives in Flow Control and Optimization. SIAM.Google Scholar
Kolmogorov, A. K. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk. 30, 301–305.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.CrossRefGoogle Scholar
Kraichnan, R. H. 1971 Inertial-range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47, 525–535.CrossRefGoogle Scholar
Lebedev, L. P. & Vorovich, I. I. 2002 Functional Analysis in Mechanics. Springer.Google Scholar
Leith, C. E. 1968 Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11, 671–673.CrossRefGoogle Scholar
Lilly, D. D. 1969 Numerical simulation of two-dimensional turbulence. Phys. Fluids Suppl. II, 240–249.CrossRefGoogle Scholar
Lindborg, E. 1999 Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? J. Fluid Mech. 388, 259–288.CrossRefGoogle Scholar
Lions, J. L. 1969 Contrôle Optimal des Systémes Gouvernés par des Equations aux Dérivées Partielles. Dunod (English translation, Springer).Google Scholar
Lundgren, T. S. 2003 Linearly forced isotropic turbulence. Center for Turbulence Research Annual Research Briefs, 461–473.Google Scholar
Maltrud, M. E. & Vallis, G. K. 1993 Energy and enstrophy transfer in numerical simulations of two-dimensional turbulence. Phys. Fluids A 5, 1760–1775.CrossRefGoogle Scholar
Marteau, D., Cardoso, O. & Tabeling, P. 1995 Equilibrium states of two-dimensional turbulence: an experimental study. Phys. Rev. E 51, 5124–5127.CrossRefGoogle ScholarPubMed
Merilees, P. E. & Warn, H. 1975 On energy and enstrophy exchanges in two-dimensional non-divergent flow. J. Fluid Mech. 69, 625–630.CrossRefGoogle Scholar
Nocedal, J. & Wright, S. J. 2000 Numerical Optimization. Springer.Google Scholar
Pasquero, C. & Falkovich, G. 2002 Stationary spectrum of vorticity cascade in two-dimensional turbulence. Phys. Rev. E 65, 056305.CrossRefGoogle ScholarPubMed
Protas, B., Bewley, T. R. & Hagen, G. 2004 A comprehensive framework for the regularization of adjoint analysis in multiscale PDE systems. J. Comp. Phys. 195, 49–89.CrossRefGoogle Scholar
Rosales, C. & Meneveau, C. 2005 Linear forcing in numerical simulations of isotropic turbulence: physical space implementation and convergence studies. Phys. Fluids 17, 095106.CrossRefGoogle Scholar
Schorghofer, N. 2000 Energy spectra of steady two-dimensional turbulent flows. Phys. Rev. E 61, 6575–6577.CrossRefGoogle ScholarPubMed
Schulze, J. C., Schmid, P. J. & Sesterhenn, J. L. 2009 Exponential time integration using Krylov subspaces. Intl J. Numer. Meth. Fluids 60, 591–609.CrossRefGoogle Scholar
Scott, R. K. 2007 Non-robustness of the two-dimensional turbulent inverse cascade. Phys. Rev. E 75, 046301.CrossRefGoogle Scholar
Smith, L. M. & Yakhot, V. 1993 Bose condensation and small-scale structure generation in a random force driven 2D turbulence. Phys. Rev. Lett. 71, 352–355.CrossRefGoogle Scholar
Sommeria, J. 1986 Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139–168.CrossRefGoogle Scholar
Tran, C. V. & Bowman, J. C. 2003 On the dual cascade in two dimensional turbulence. Physica D 176, 242–255.CrossRefGoogle Scholar
Tran, C. V. & Dritschel, D. G. 2006 Vanishing enstrophy dissipation in two-dimensional Navier–Stokes turbulence in the inviscid limit. J. Fluid Mech. 559, 107–116.CrossRefGoogle Scholar
Tran, C. V., Dritschel, D. G. & Scott, R. K. 2007 Revisiting Batchelor's theory of two-dimensional turbulence. J. Fluid Mech. 591, 379–391.Google Scholar
Tran, C. V. & Shepherd, T. G. 2002 Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence. Physica D 165, 199–212.CrossRefGoogle Scholar
Tsang, Y.-K., Ott, E., Antonsen, T. M. & Guzdar, P. N. 2005 Intermittency in two-dimensional turbulence with drag. Phys. Rev. E 71, 066313.CrossRefGoogle ScholarPubMed
Tsang, Y.-K. & Young, W. R. 2009 Force-dissipative two-dimensional turbulence: a scaling regime controlled by drag. Phys. Rev. E 79, 045308(R).CrossRefGoogle ScholarPubMed
Welch, W. T. & Tung, K. K. 1998 Nonlinear baroclinic adjustment and wavenumber selection in a simple case. J. Atmos. Sci. 55, 1285–1302.2.0.CO;2>CrossRefGoogle Scholar