Skip to main content

Optimal fluxes and Reynolds stresses

  • Javier Jiménez

It is remarked that fluxes in conservation laws, such as the Reynolds stresses in the momentum equation of turbulent shear flows, or the spectral energy flux in anisotropic turbulence, are only defined up to an arbitrary solenoidal field. While this is not usually significant for long-time averages, it becomes important when fluxes are modelled locally in large-eddy simulations, or in the analysis of intermittency and cascades. As an example, a numerical procedure is introduced to compute fluxes in scalar conservation equations in such a way that their total integrated magnitude is minimised. The result is an irrotational vector field that derives from a potential, thus minimising sterile flux ‘circuits’. The algorithm is generalised to tensor fluxes and applied to the transfer of momentum in a turbulent channel. The resulting instantaneous Reynolds stresses are compared with their traditional expressions, and found to be substantially different. This suggests that some of the alleged shortcomings of simple subgrid models may be representational artefacts, and that the same may be true of the intermittency properties of the turbulent stresses.

Corresponding author
Email address for correspondence:
Hide All
del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135144.
Antonia, R. A. & Atkinson, J. D. 1973 High-order moments of Reynolds shear stress fluctuations in a turbulent boundary layer. J. Fluid Mech. 58, 581593.
Bardina, J.1983 Improved turbulence models based on large eddy simulation of homogeneous, incompressible, turbulence flows. PhD thesis, Thermosciences Division, Department of Mechanical Engineering, Stanford University.
Barut, A. O. 1980 Electrodynamics and Classical Theory of Fields and Particles. Dover.
Cimarelli, A. & De Angelis, E. 2011 Analysis of the Kolmogorov equation for filtered wall-turbulent flows. J. Fluid Mech. 676, 376395.
Cimarelli, A. & De Angelis, E. 2012 Anisotropic dynamics and sub-grid energy transfer in wall-turbulence. Phys. Fluids 24, 015102.
Dar, G., Verma, M. K. & Eswaran, V. 2001 Energy transfer in two-dimensional magnetohydrodynamic turbulence: formalism and numerical results. Physica D 157, 207225.
Domaradzki, J. A. & Rogallo, R. S. 1990 Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence. Phys. Fluids A 2, 413426.
Gelfand, I. M. & Fomin, S. V. 1963 Calculus of Variations. Prentice-Hall.
Germano, M., Piomelli, U., Moin, P. & Cabot, W. 1991 A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A 3, 17601765.
Hill, R. J. 2002 Exact second-order structure–function relationships. J. Fluid Mech. 468, 317326.
Jackson, J. D. 2002 From Lorenz to Coulomb and other explicit gauge transformations. Am. J. Phys. 70, 917928.
Jiménez, J. 2012 Cascades in wall-bounded turbulence. Annu. Rev. Fluid Mech. 44, 2745.
Jiménez, J. 2013a How linear is wall-bounded turbulence? Phys. Fluids 25, 110814.
Jiménez, J. 2013b Near-wall turbulence. Phys. Fluids 25, 101302.
Kim, J., Moin, P. & Moser, R. D. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers. Dokl. Akad. Nauk. SSSR 30, 301305; reprinted in Proc. R. Soc. Lond. A 434, 9–13 (1991).
Kraichnan, R. H. 1971 Inertial range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47, 525535.
Landau, L. D. & Lifshitz, E. M. 1958 Fluid Mechanics, 2nd edn, chap. 10, Addison-Wesley.
Lozano-Durán, A., Flores, O. & Jiménez, J. 2012 The three-dimensional structure of momentum transfer in turbulent channels. J. Fluid Mech. 694, 100130.
Lozano-Durán, A. & Jiménez, J. 2014 Time-resolved evolution of coherent structures in turbulent channels: characterization of eddies and cascades. J. Fluid Mech. 759, 432471.
Lu, S. S. & Willmarth, W. W. 1973 Measurements of the structure of the Reynolds stress in a turbulent boundary layer. J. Fluid Mech. 60, 481511.
Lund, T. S. & Novikov, E. A. 1993 Parameterization of subgrid-scale stress by the velocity gradient tensor. In CTR Annual Research Briefs, pp. 2743. Stanford University.
Perot, B. & Moin, P. 1996 A new approach to turbulence modelling. In Proceeding of the CTR Summer Program, pp. 3546. Stanford University.
Townsend, A. A. 1961 Equilibrium layers and wall turbulence. J. Fluid Mech. 11, 97120.
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge Uiversity Press.
Van Atta, C. W. & Wyngaard, J. C. 1975 On higher-order spectra of turbulence. J. Fluid Mech. 72, 673694.
Verma, M. K. 2004 Statistical theory of magnetohydrodynamic turbulence: recent results. Phys. Rep. 401, 229380.
Wallace, J. M., Eckelmann, H. & Brodkey, R. S. 1972 The wall region in turbulent shear flow. J. Fluid Mech. 64, 3948.
Wu, J., Zhou, Y., Lu, X. & Fan, M. 1999 Turbulent force as a diffusive field with vortical forces. Phys. Fluid 11, 627635.
Wu, J., Zhou, Y. & Wu, J.1996 Reduced stress tensor and dissipation and the transport of Lamb vector. Rep. 96-21. ICASE.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 5
Total number of PDF views: 302 *
Loading metrics...

Abstract views

Total abstract views: 379 *
Loading metrics...

* Views captured on Cambridge Core between 15th November 2016 - 25th May 2018. This data will be updated every 24 hours.