Skip to main content
×
×
Home

Tensor geometry in the turbulent cascade

  • Joseph G. Ballouz (a1) and Nicholas T. Ouellette (a1)
Abstract

The defining characteristic of highly turbulent flows is the net directed transport of energy from the injection scales to the dissipation scales. This cascade is typically described in Fourier space, obscuring its connection to the mechanics of the flow. Here, we recast the energy cascade in mechanical terms, noting that for some scales to transfer energy to others, they must do mechanical work on them. This work can be expressed as the inner product of a turbulent stress and a rate of strain. But, as with all inner products, the relative alignment of these two tensors matters, and determines how strong the energy transfer will be. We show that this tensor alignment behaves very differently in two and three dimensions; in particular, the tensor eigenvalues affect the inner product in very different ways. By comparing the observed energy flux to the maximum possible if the tensors were in perfect alignment, we define an efficiency for the energy cascade. Using data from a direct numerical simulation of isotropic turbulence, we show that this efficiency is perhaps surprisingly low, with an average value of approximately 25 % in the inertial range, although it is spatially heterogeneous. Our results have implications for how the stress and strain-rate magnitudes influence the flux of energy between scales, and may help to explain why the energy cascades in two and three dimensions are different.

Copyright
Corresponding author
Email address for correspondence: nto@stanford.edu
References
Hide All
Ashurst, W. T., Kerstein, A. R., Kerr, R. M. & Gibson, C. H. 1987 Alignment of vorticity and scalar gradient with strain rate in simulated Navier–Stokes turbulence. Phys. Fluids 30, 23432353.
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1, 497504.
Borue, V. & Orszag, S. A. 1998 Local energy flux and subgrid-scale statistics in three-dimensional turbulence. J. Fluid Mech. 366, 131.
Chen, S., Ecke, R. E., Eyink, G. L., Rivera, M., Wan, M. & Xiao, Z. 2006 Physical mechanism of the two-dimensional inverse energy cascade. Phys. Rev. Lett. 96, 084502.
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.
Eyink, G. L. 1995 Local energy flux and the refined similarity hypothesis. J. Stat. Phys. 78, 335351.
Falkovich, G. 2009 Symmetries of the turbulent state. J. Phys. A 42, 123001.
Fang, L. & Ouellette, N. T. 2016 Advection and the efficiency of spectral energy transfer in two-dimensional turbulence. Phys. Rev. Lett. 117, 104501.
Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.
Germano, M. 1992 Turbulence: the filtering approach. J. Fluid Mech. 238, 325336.
Hamlington, P. E., Schumacher, J. & Dahm, W. J. A. 2008a Direct assessment of vorticity alignment with local and nonlocal strain rates in turbulent flows. Phys. Fluids 20, 111703.
Hamlington, P. E., Schumacher, J. & Dahm, W. J. A. 2008b Local and nonlocal strain rate fields and vorticity alignment in turbulent flows. Phys. Rev. E 77, 026303.
Higgins, C. W., Meneveau, C. & Palange, M. B. 2007 The effect of filter dimension on the subgrid-scale stress, heat flux, and tensor alignments in the atmospheric surface layer. J. Atmos. Sci. 24, 360375.
Higgins, C. W., Palange, M. B. & Meneveau, C. 2003 Alignment trends of velocity gradients and subgrid-scale fluxes in the turbulent atmospheric boundary layer. Bound.-Layer Meteorol. 109, 5983.
Li, Y., Perlman, E., Wan, M., Yang, Y., Meneveau, C., Burns, R., Chen, S., Szalay, A. & Eyink, G. 2008 A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J. Turbul. 9, 31.
Liao, Y. & Ouellette, N. T. 2013 Spatial structure of spectral transport in two-dimensional flow. J. Fluid Mech. 725, 281298.
Liao, Y. & Ouellette, N. T. 2014 Geometry of scale-to-scale energy and enstrophy transport in two-dimensional flow. Phys. Fluids 26, 045103.
Liao, Y. & Ouellette, N. T. 2015 Long-range ordering of turbulent stresses in two-dimensional flow. Phys. Rev. E 91, 063004.
Liu, S., Meneveau, C. & Katz, J. 1994 On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet. J. Fluid Mech. 275, 83119.
Marion, J. B. & Thornton, S. T. 1995 Classical Dynamics of Particles and Systems. Harcourt College Publishers.
Ni, R., Kramel, S., Ouellette, N. T. & Voth, G. A. 2015 Measurements of the coupling between the tumbling of rods and the velocity gradient tensor in turbulence. J. Fluid Mech. 766, 202225.
Ni, R., Ouellette, N. T. & Voth, G. A. 2014 Alignment of vorticity and rods with Lagrangian fluid stretching in turbulence. J. Fluid Mech. 743, R3.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Rivera, M. K., Daniel, W. B., Chen, S. Y. & Ecke, R. E. 2003 Energy and enstrophy transfer in decaying two-dimensional turbulence. Phys. Rev. Lett. 90, 104502.
Saffman, P. G. & Pullin, D. I. 1994 Anistropy of the Lundgren–Townsend model of fine-scale turbulence. Phys. Fluids 6, 802807.
Tao, B., Katz, J. & Meneveau, C. 2000 Geometry and scale relationships in high Reynolds number turbulence determined from three-dimensional holographic velocimetry. Phys. Fluids 12, 941944.
Tao, B., Katz, J. & Meneveau, C. 2002 Statistical geometry of subgrid-scale stresses determined from holographic particle image velocimetry measurements. J. Fluid Mech. 457, 3578.
Taylor, G. I. 1932 The transport of vorticity and heat through fluids in turbulent motion. Proc. R. Soc. Lond. A 135, 685702.
Taylor, G. I. 1938 Production and dissipation of vorticity in a turbulent fluid. Proc. R. Soc. Lond. A 164, 1523.
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.
Tsinober, A., Kit, E. & Dracos, T. 1992 Experimental investigatin of the field of velocity gradients in turbulent flows. J. Fluid Mech. 242, 169192.
Vincent, A. & Meneguzzi, M. 1994 The dynamics of vorticity tubes in homogeneous turbulence. J. Fluid Mech. 258, 245254.
Wallace, J. M. 2009 Twenty years of experimental and direct numerical simulation access to the velocity gradient tensor: What have we learned about turbulence? Phys. Fluids 21, 021301.
Xiao, Z., Wan, M., Chen, S. & Eyink, G. L. 2009 Physical mechanism the inverse energy cascade of two-dimensional turbulence: a numerical approach. J. Fluid Mech. 619, 144.
Xu, H., Pumir, A. & Bodenschatz, E. 2011 The pirouette effect in turbulent flows. Nat. Phys. 7, 709712.
Yang, Z. & Wang, B.-C. 2016 On the topology of the eigenframe of the subgrid-scale stress tensor. J. Fluid Mech. 798, 598627.
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? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 6
Total number of PDF views: 234 *
Loading metrics...

Abstract views

Total abstract views: 549 *
Loading metrics...

* Views captured on Cambridge Core between 29th November 2017 - 19th September 2018. This data will be updated every 24 hours.