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Variations on Kolmogorov flow: turbulent energy dissipation and mean flow profiles

Published online by Cambridge University Press:  22 February 2011

B. ROLLIN ,
Y. DUBIEF  and
C. R. DOERING
B. ROLLIN*
Affiliation:
Los Alamos National Laboratory, Los Alamos, NM 87544, USA
Y. DUBIEF
Affiliation:
School of Engineering, University of Vermont, Burlington, VT 05405, USA
C. R. DOERING
Affiliation:
Department of Mathematics, Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109-1107, USA
*
†Email address for correspondence: bertrand@lanl.gov
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Abstract

The relation between the form of a body force driving a turbulent shear flow and the dissipation factor β = ϵℓ/U3 is investigated by means of rigorous upper bound analysis and direct numerical simulation. We consider unidirectional steady forcing functions in a three-dimensional periodic domain and observe that a rigorous infinite Reynolds number bound on β displays the same qualitative behaviour as the computationally measured dissipation factor at finite Reynolds number as the force profile is varied. We also compare the measured mean flow profiles with the Stokes flow profile for the same forcing. The mean and Stokes flow profiles are strikingly similar at the Reynolds numbers obtained in the numerical simulations, lending quantitative credence to the notion of a turbulent eddy viscosity.

JFM classification

Turbulent Flows: Turbulence modelling Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 670 , 10 March 2011 , pp. 204 - 213
Creative Commons
This is a work of the U.S. Government and is not subject to copyright protection in the United States.
Copyright
Copyright © Cambridge University Press 2011. This is a work of the U.S. Government and is not subject to copyright protection in the United States.

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References

REFERENCES

Borue, V. & Orszag, S. A. 1996 Numerical study of three-dimensional Kolmogorov flow at high Reynolds numbers. J. Fluid Mech. 306, 293–323.CrossRefGoogle Scholar
Busse, F. H. 1978 The optimum theory of turbulence. Adv. Appl. Mech. 18, 77–121.CrossRefGoogle Scholar
Cheskidov, A., Doering, C. R. & Petrov, N. P. 2007 Energy dissipation in fractal-forced flow. J. Math. Phys. 48, 065208.CrossRefGoogle Scholar
Childress, S., Kerswell, R. R. & Gilbert, A. D. 2001 Bounds on dissipation for Navier–Stokes flow with Kolmogorov forcing. Physica D 158 (1–4), 105–128.CrossRefGoogle Scholar
Dascaliuc, R., Foias, C. & Jolly, M. S. 2009 On the asymptotic behavior of average energy and enstrophy in 3D turbulent flows. Physica D 238, 725–736.CrossRefGoogle Scholar
Doering, C. R. & Constantin, P. 1992 Energy dissipation in shear driven turbulence. Phys. Rev. Lett. 69 (11), 1648–1651.CrossRefGoogle ScholarPubMed
Doering, C. R. & Constantin, P. 1994 Variational bounds on energy dissipation in incompressible flows: shear flow. Phys. Rev. E 49 (5), 4087–4099.CrossRefGoogle ScholarPubMed
Doering, C. R., Eckhardt, B. & Schumacher, J. 2003 Energy dissipation in body-forced plane shear flow. J. Fluid Mech. 494, 275–284.CrossRefGoogle Scholar
Doering, C. R. & Foias, C. 2002 Energy dissipation in body-forced turbulence. J. Fluid Mech. 467, 289–306.CrossRefGoogle Scholar
Foias, C. 1997 What do the Navier–Stokes equations tell us about turbulence? Contemp. Maths 208, 151–180.CrossRefGoogle Scholar
Foias, C., Manley, O. P. & Temam, R. 1993 Bounds for the mean dissipation of the 2D enstrophy and 3D energy in turbulent flows. Phys. Lett. A 174, 210–215.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Howard, L. N. 1972 Bounds on flow quantities. Annu. Rev. Fluid Mech. 4, 473–494.CrossRefGoogle Scholar
Kerswell, R. R. 1998 Unification of variational principles for turbulent shear flows: the background method of Doering–Constantin and the mean-fluctuation formulation of Howard–Busse. Physica D 121 (1–2), 175–192.CrossRefGoogle Scholar
Kolmogorov, A. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301–305.Google Scholar
Petrov, N. P., Lu, L. & Doering, C. R. 2005 Variational bounds on the energy dissipation rate in body-forced shear flow. J. Turbul. 6, Art. No. 17, doi:10.1080/14685240500228262.CrossRefGoogle Scholar
Richardson, L. F. 1922 Weather Prediction by Numerical Process. Cambridge University Press.Google Scholar
Sarris, I. E., Jeanmart, H., Carati, D. & Winckelmans, G. 2007 Box-size dependence and breaking of translational invariance in the velocity statistics computed from three-dimensional turbulent Kolmogorov flows. Phys. Fluids 19, 095101.CrossRefGoogle Scholar
Sreenivasan, K. R. 1984 On the scaling of the turbulence energy dissipation rate. Phys. Fluids 27, 1048–1051.CrossRefGoogle Scholar
Sreenivasan, K. R. 1998 An update on the energy dissipation rate in isotropic turbulence. Phys. Fluids 10, 528–529.CrossRefGoogle Scholar
Taylor, G. I. 1938 The spectrum of turbulence. Proc. R. Soc. Lond. A 164 (919), 476–490.CrossRefGoogle Scholar
Wang, X. 1997 Time-averaged energy dissipation rate for shear driven flows in Rn. Physica D 99 (4), 555–563.CrossRefGoogle Scholar