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Geometric constraints on energy transfer in the turbulent cascade

Joseph G. Ballouz and Nicholas T. Ouellette*

  • Department of Civil and Environmental Engineering, Stanford University, Stanford, California 94305, USA

  • *nto@stanford.edu
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Phys. Rev. Fluids 5, 034603 – Published 11 March, 2020

DOI: https://doi.org/10.1103/PhysRevFluids.5.034603

Abstract

The energy cascade is the most significant feature that separates turbulence from other unsteady flows, and results from the behavior of the nonlinear term in the Navier-Stokes equations. The mathematical form of this term, however, places constraints on exactly how it can act. Here we consider the action of the nonlinear term in physical space rather than in Fourier space, where the energy transfer between scales can be interpreted as a mechanical process where some scales do work on others. This formulation reveals the fundamental role played by geometry, as work can only be done when the eigenframes of the turbulent stress and strain rate are appropriately aligned. By comparing a direct numerical simulation of the Navier-Stokes equations, an ensemble of random solenoidal vector fields, and a random sampling of uniform eigenframe alignments, we show that this geometric alignment plays a much stronger role in determining the flux between scales than do the magnitudes of the stress and strain rate. We also show that when the alignment is effectively two dimensional, even when embedded in a three-dimensional flow, the energy flux is typically inverse, suggesting that the inverse cascade in two-dimensional turbulence may have a kinematic origin. Our results point to some potentially fruitful directions for turbulence modeling.

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References (48)

  1. A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk SSSR 30, 301 (1941).
  2. R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani, On the multifractal nature of fully developed turbulence and chaotic systems, J. Phys. A: Math. Gen. 17, 3521 (1984).
  3. G. Parisi and U. Frisch, in Turbulence and Predictability in Geophysical Fluid Dynamics, Proceedings of the International School of Physics “E. Fermi,” Course LXXXVIII, Varenna, 1985, edited by M. Ghil, R. Benzi, and G. Parisi (North-Holland, Amsterdam, 1985), p. 84.
  4. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995).
  5. A. Arnèodo et al., Structure functions in turbulence, in various flow configurations, at Reynolds numbers between 30 and 5000, using extended self-similarity, Europhys. Lett. 34, 411 (1996).
  6. A. Arnèodo et al., Universal Intermittent Properties of Particle Trajectories in Highly Turbulent Flows, Phys. Rev. Lett. 100, 254504 (2008).
  7. G. I. Taylor, The transport of vorticity and heat through fluids in turbulent motion, Proc. R. Soc. London Ser. A 135, 685 (1932).
  8. G. I. Taylor, Production and dissipation of vorticity in a turbulent fluid, Proc. R. Soc. London Ser. A 164, 15 (1938).
  9. H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, 1972).
  10. H. Xu, A. Pumir, and E. Bodenschatz, The pirouette effect in turbulent flows, Nat. Phys. 7, 709 (2011).
  11. R. Ni, N. T. Ouellette, and G. A. Voth, Alignment of vorticity and rods with Lagrangian fluid stretching in turbulence, J. Fluid Mech. 743, R3 (2014).
  12. W. T. Ashurst, A. R. Kerstein, R. M. Kerr, and C. H. Gibson, Alignment of vorticity and scalar gradient with strain rate in simulated Navier-Stokes turbulence, Phys. Fluids 30, 2343 (1987).
  13. A. Tsinober, E. Kit, and T. Dracos, Experimental investigation of the field of velocity gradients in turbulent flows, J. Fluid Mech. 242, 169 (1992).
  14. A. Vincent and M. Meneguzzi, The dynamics of vorticity tubes in homogeneous turbulence, J. Fluid Mech. 258, 245 (1994).
  15. V. Borue and S. A. Orszag, Local energy flux and subgrid-scale statistics in three-dimensional turbulence, J. Fluid Mech. 366, 1 (1998).
  16. R. Ni, S. Kramel, N. T. Ouellette, and G. A. Voth, Measurements of the coupling between the tumbling of rods and the velocity gradient tensor in turbulence, J. Fluid Mech. 766, 202 (2015).
  17. A. Tsinober, Is concentrated vorticity that important?, Eur. J. Mech. B 17, 421 (1998).
  18. M. Carbone and A. D. Bragg, Is vortex stretching the main cause of the turbulent energy cascade?, J. Fluid Mech. 883, R2 (2020).
  19. S. Cerutti and C. Meneveau, Intermittency and relative scaling of subgrid-scale energy dissipation in isotropic turbulence, Phys. Fluids 10, 928 (1998).
  20. S. Chen, R. E. Ecke, G. L. Eyink, M. Rivera, M. Wan, and Z. Xiao, Physical Mechanism of the Two-Dimensional Inverse Energy Cascade, Phys. Rev. Lett. 96, 084502 (2006).
  21. J. G. Ballouz and N. T. Ouellette, Tensor geometry in the turbulent cascade, J. Fluid Mech. 835, 1048 (2018).
  22. L. Fang and N. T. Ouellette, Advection and the Efficiency of Spectral Energy Transfer in Two-Dimensional Turbulence, Phys. Rev. Lett. 117, 104501 (2016).
  23. J. C. H. Fung and J. C. Vassilicos, Two-particle dispersion in turbulentlike flows, Phys. Rev. E 57, 1677 (1998).
  24. G. Boffetta and R. E. Ecke, Two-dimensional turbulence, Annu. Rev. Fluid Mech. 44, 427 (2012).
  25. G. Falkovich, Symmetries of the turbulent state, J. Phys. A: Math. Theor. 42, 123001 (2009).
  26. S. B. Pope, Turbulent Flows (Cambridge University Press, Cambridge, 2000).
  27. G. L. Eyink, Local energy flux and the refined similarity hypothesis, J. Stat. Phys. 78, 335 (1995).
  28. M. K. Rivera, W. B. Daniel, S. Y. Chen, and R. E. Ecke, Energy and Enstrophy Transfer in Decaying Two-Dimensional Turbulence, Phys. Rev. Lett. 90, 104502 (2003).
  29. Y. Liao and N. T. Ouellette, Spatial structure of spectral transport in two-dimensional flow, J. Fluid Mech. 725, 281 (2013).
  30. Y. Liao and N. T. Ouellette, Long-range ordering of turbulent stresses in two-dimensional flow, Phys. Rev. E 91, 063004 (2015).
  31. Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eyink, A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence, J. Turbul. 9, 31 (2008).
  32. M. Germano, Turbulence: The filtering approach, J. Fluid Mech. 238, 325 (1992).
  33. S. Liu, C. Meneveau, and J. Katz, On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet, J. Fluid Mech. 275, 83 (1994).
  34. S. Chen, R. E. Ecke, G. L. Eyink, X. Wang, and Z. Xiao, Physical Mechanism of the Two-Dimensional Enstrophy Cascade, Phys. Rev. Lett. 91, 214501 (2003).
  35. Y. Liao and N. T. Ouellette, Geometry of scale-to-scale energy and enstrophy transport in two-dimensional flow, Phys. Fluids 26, 045103 (2014).
  36. D. H. Kelley and N. T. Ouellette, Spatiotemporal persistence of spectral fluxes in two-dimensional weak turbulence, Phys. Fluids 23, 115101 (2011).
  37. B. Vreman, B. Geurts, and H. Kuerten, Realizability conditions for the turbulent stress tensor in large-eddy simulation, J. Fluid Mech. 278, 351 (1994).
  38. J. A. Domaradzki and D. Carati, A comparison of spectral sharp and smooth filters in the analysis of nonlinear interactions and energy transfer in turbulence, Phys. Fluids 19, 085111 (2007).
  39. J. A. Domaradzki and D. Carati, An analysis of the energy transfer and the locality of nonlinear interactions in turbulence, Phys. Fluids 19, 085112 (2007).
  40. H. Aluie and G. L. Eyink, Localness of energy cascade in hydrodynamic turbulence. II. Sharp spectral filter, Phys. Fluids 21, 115108 (2009).
  41. G. L. Eyink, Locality of turbulent cascades, Physica D 207, 91 (2005).
  42. G. L. Eyink and H. Aluie, Localness of energy cascade in hydrodynamic turbulence. I. Smooth coarse graining, Phys. Fluids 21, 115107 (2009).
  43. Z. Xiao, M. Wan, S. Chen, and G. L. Eyink, Physical mechanism the inverse energy cascade of two-dimensional turbulence: A numerical approach, J. Fluid Mech. 619, 1 (2009).
  44. R. H. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluids 10, 1417 (1967).
  45. C. E. Leith, Diffusion approximation for two-dimensional turbulence, Phys. Fluids 11, 671 (1967).
  46. G. K. Batchelor, Computation of the energy spectrum in homogeneous two-dimensional turbulence, Phys. Fluids 12, 11233 (1969).
  47. L. Biferale, S. Musacchio, and F. Toschi, Inverse Energy Cascade in Three-Dimensional Isotropic Turbulence, Phys. Rev. Lett. 108, 164501 (2012).
  48. C. W. Higgins, M. B. Parlange, and C. Meneveau, Alignment trends of velocity gradients and subgrid-scale fluxes in the turbulence atmospheric boundary layer, Bound.-Layer Meteorol. 109, 59 (2003).

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Phys. Rev. Fluids 5, 034603– Published 11 March, 2020
  • Received 9 September 2019
  • Accepted 18 February 2020
Article part of CHORUS
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