Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-19T22:07:42.437Z Has data issue: false hasContentIssue false
  • English
  • Français

On the role of vorticity stretching and strain self-amplification in the turbulence energy cascade

Published online by Cambridge University Press:  02 July 2021

Perry L. Johnson Open the ORCID record for Perry L. Johnson [Opens in a new window]
Perry L. Johnson*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The tendency of turbulent flows to produce fine-scale motions from large-scale energy injection is often viewed as a scale-wise cascade of kinetic energy driven by vorticity stretching. This has been recently evaluated by an exact, spatially local relationship (Johnson, P.L. Phys. Rev. Lett., vol. 124, 2020, p. 104501), which also highlights the contribution of strain self-amplification. In this paper, the role of these two mechanisms is explored in more detail. Vorticity stretching and strain amplification interactions between velocity gradients filtered at the same scale account for approximately half of the energy cascade rate, directly connecting the restricted Euler dynamics to the energy cascade. Multiscale strain amplification and vorticity stretching are equally important, however, and more closely resemble eddy viscosity physics. Moreover, ensuing evidence of a power-law decay of energy transfer contributions from disparate scales supports the notion of an energy cascade, albeit a ‘leaky’ one. Besides vorticity stretching and strain self-amplification, a third mechanism of energy transfer is introduced and related to the vortex thinning mechanism important for the inverse cascade in two dimensions. Simulation results indicate this mechanism also provides a net source of backscatter in three-dimensional turbulence, in the range of scales associated with the bottleneck effect. Taken together, these results provide a rich set of implications for large-eddy simulation modelling.

JFM classification

Turbulent Flows: Turbulent Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 922 , 10 September 2021 , A3
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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

Alexakis, A. & Chibbaro, S. 2020 Local energy flux of turbulent flows. Phys. Rev. Fluids 5, 094604.CrossRefGoogle Scholar
Aluie, H. & Eyink, G.L. 2009 Localness of energy cascade in hydrodynamic turbulence. II. Sharp spectral filter. Phys. Fluids 21 (11), 115108.CrossRefGoogle Scholar
Aoyama, T., Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. 2005 Statistics of energy transfer in high-resolution direct numerical simulation of turbulence in a periodic box. J. Phys. Soc. Japan 74 (12), 3202–3212.CrossRefGoogle Scholar
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 (8), 2343–2353.CrossRefGoogle Scholar
Ballouz, J.G., Johnson, P.L. & Ouellette, N.T. 2020 Temporal dynamics of the alignment of the turbulent stress and strain rate. Phys. Rev. Fluids 5, 114606.CrossRefGoogle Scholar
Ballouz, J.G. & Ouellette, N.T. 2018 Tensor geometry in the turbulent cascade. J. Fluid Mech. 835, 1048–1064.CrossRefGoogle Scholar
Ballouz, J.G. & Ouellette, N.T. 2020 Geometric constraints on energy transfer in the turbulent cascade. Phys. Rev. Fluids 5, 034603.CrossRefGoogle Scholar
Bershadskii, A. 2008 Near-dissipation range in nonlocal turbulence. Phys. Fluids 20 (8), 085103.CrossRefGoogle Scholar
Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1 (05), 497–504.CrossRefGoogle Scholar
Borue, V. & Orszag, S.A. 1998 Local energy flux and subgrid-scale statistics in three-dimensional turbulence. J. Fluid Mech. 366, 1–31.CrossRefGoogle Scholar
Buaria, D., Pumir, A., Bodenschatz, E. & Yeung, P.-K. 2019 Extreme velocity gradients in turbulent flows. New J. Phys. 21 (4), 043004.CrossRefGoogle Scholar
Buaria, D. & Sreenivasan, K.R. 2020 Dissipation range of the energy spectrum in high Reynolds number turbulence. Phys. Rev. Fluids 5 (9), 092601(R).CrossRefGoogle Scholar
Buzzicotti, M., Linkmann, M., Aluie, H., Biferale, L., Brasseur, J. & Meneveau, C. 2018 Effect of filter type on the statistics of energy transfer between resolved and subfilter scales from a-priori analysis of direct numerical simulations of isotropic turbulence. J. Turbul. 19 (2), 167–197.CrossRefGoogle Scholar
Cantwell, B.J. 1992 Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids 4 (4), 782–793.CrossRefGoogle Scholar
Cantwell, B.J. 1993 On the behavior of velocity gradient tensor invariants in direct numerical simulations of turbulence. Phys. Fluids 5 (8), 2008–2013.CrossRefGoogle Scholar
Carbone, M. & Bragg, A.D. 2020 Is vortex stretching the main cause of the turbulent energy cascade? J. Fluid Mech. 883, R2.CrossRefGoogle Scholar
Cardesa, J.I., Vela-Martín, A., Dong, S. & Jiménez, J. 2015 The temporal evolution of the energy flux across scales in homogeneous turbulence. Phys. Fluids 27 (11), 111702.CrossRefGoogle Scholar
Chan, W.H.R, Johnson, P.L. & Moin, P. 2020 a The turbulent bubble break-up cascade. Part 1. Theoretical developments. J. Fluid Mech. 912, A42.CrossRefGoogle Scholar
Chan, W.H.R., Johnson, P.L., Moin, P. & Urzay, J. 2020 b The turbulent bubble break-up cascade. Part 2. Numerical simulations of breaking waves. J. Fluid Mech. 912, A43.CrossRefGoogle Scholar
Chen, S., Doolen, G., Herring, J.R., Kraichnan, R.H., Orszag, S.A. & She, Z.S. 1993 Far-dissipation range of turbulence. Phys. Rev. Lett. 70, 3051–3054.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle ScholarPubMed
Chong, M.S., Soria, J., Perry, A.E., Chacin, J., Cantwell, B.J. & Na, Y. 1998 Turbulence structures of wall-bounded shear flows found using DNS data. J. Fluid Mech. 357, 225–247.CrossRefGoogle Scholar
Chorin, A.J. 1988 Scaling laws in the vortex lattice model of turbulence. Commun. Math. Phys. 114 (1), 167–176.CrossRefGoogle Scholar
Clark, R.A., Ferziger, J.H. & Reynolds, W.C. 1979 Evaluation of subgrid-scale models using an accurately simulated turbulent flow. J. Fluid Mech. 91 (part 1), 1–16.CrossRefGoogle Scholar
Constantin, P., E, W. & Titi, E.S. 1994 Onsager's conjecture on the energy conservation for solutions of Euler's equation. Commun. Math. Phys. 165 (1), 207–209.CrossRefGoogle Scholar
Danish, M. & Meneveau, C. 2018 Multiscale analysis of the invariants of the velocity gradient tensor in isotropic turbulence. Phys. Rev. Fluids 3, 044604.CrossRefGoogle Scholar
Doan, N.A.K., Swaminathan, N., Davidson, P.A. & Tanahashi, M. 2018 Scale locality of the energy cascade using real space quantities. Phys. Rev. Fluids 3, 084601.CrossRefGoogle Scholar
Domaradzki, J.A. & Carati, D. 2007 a An analysis of the energy transfer and the locality of nonlinear interactions in turbulence. Phys. Fluids 19 (8), 085112.CrossRefGoogle Scholar
Domaradzki, J.A. & Carati, D. 2007 b A comparison of spectral sharp and smooth filters in the analysis of nonlinear interactions and energy transfer in turbulence. Phys. Fluids 19 (8), 085111.CrossRefGoogle Scholar
Domaradzki, J.A., Teaca, B. & Carati, D. 2009 Locality properties of the energy flux in turbulence. Phys. Fluids 21 (2), 025106.CrossRefGoogle Scholar
Dong, S., Huang, Y., Yuan, X. & Lozano-Durán, A. 2020 The coherent structure of the kinetic energy transfer in shear turbulence. J. Fluid Mech. 892, A22.CrossRefGoogle Scholar
Donzis, D.A. & Sreenivasan, K.R. 2010 The bottleneck effect and the Kolmogorov constant in isotropic turbulence. J. Fluid Mech. 657, 171–188.CrossRefGoogle Scholar
Elsinga, G.E. & Marusic, I. 2010 Evolution and lifetimes of flow topology in a turbulent boundary layer. Phys. Fluids 22 (1), 015102.CrossRefGoogle Scholar
Eyink, G.L. 1995 Local energy flux and the refined similarity hypothesis. J. Stat. Phys. 78 (1–2), 335–351.CrossRefGoogle Scholar
Eyink, G.L. 2005 Locality of turbulent cascades. Physica D 207 (1), 91–116.CrossRefGoogle Scholar
Eyink, G.L. 2006 Multi-scale gradient expansion of the turbulent stress tensor. J. Fluid Mech. 549, 159–190.CrossRefGoogle Scholar
Eyink, G.L. 2018 Review of the Onsager ‘Ideal Turbulence’ theory. arXiv:1803.02223.Google Scholar
Eyink, G.L. & Aluie, H. 2009 Localness of energy cascade in hydrodynamic turbulence. I. Smooth coarse graining. Phys. Fluids 21 (11), 115107.CrossRefGoogle Scholar
Falkovich, G. 1994 Bottleneck phenomenon in developed turbulence. Phys. Fluids 6 (4), 1411–1414.CrossRefGoogle Scholar
Fiscaletti, D., Elsinga, G.E., Attili, A., Bisetti, F. & Buxton, O.R.H. 2016 Scale dependence of the alignment between strain rate and rotation in turbulent shear flow. Phys. Rev. Fluids 1, 064405.CrossRefGoogle Scholar
Foias, C., Manley, O. & Sirovich, L. 1990 Empirical and stokes eigenfunctions and the far-dissipative turbulent spectrum. Phys. Fluids A: Fluid Dyn. 2 (3), 464–467.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Frisch, U., Kurien, S., Pandit, R., Pauls, W., Ray, S.S., Wirth, A. & Zhu, J.-Z. 2008 Hyperviscosity, Galerkin truncation, and bottlenecks in turbulence. Phys. Rev. Lett. 101, 144501.CrossRefGoogle ScholarPubMed
Germano, M. 1992 Turbulence: the filtering approach. J. Fluid Mech. 238, 325–336.CrossRefGoogle Scholar
Gulitski, G., Kholmyansky, M., Kinzelbach, W., Luthi, B., Tsinober, A. & Yorish, S. 2007 Velocity and temperature derivatives in high-Reynolds-number turbulent flows in the atmospheric surface layer. Part 1. Facilities, methods and some general results. J. Fluid Mech. 589, 57–81.CrossRefGoogle Scholar
Hill, R.J. 2001 Equations relating structure functions of all orders. J. Fluid Mech. 434, 379–388.CrossRefGoogle Scholar
Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. 2005 Energy spectrum in the near dissipation range of high resolution direct numerical simulation of turbulence. J. Phys. Soc. Japan 74 (5), 1464–1471.CrossRefGoogle Scholar
Jiménez, J. & Wray, A.A. 1998 On the characteristics of vortex filaments in isotropic turbulence. J. Fluid Mech. 373, 255–285.CrossRefGoogle Scholar
Johnson, P.L. 2020 a Energy transfer from large to small scales in turbulence by multi-scale nonlinear strain and vorticity interactions. Phys. Rev. Lett. 124, 104501.CrossRefGoogle Scholar
Johnson, P.L. 2020 b Lagrangian dynamics of the tensor diffusivity model for turbulent subfilter stresses. Center for Turbulence Research Annual Research Briefs, pp. 167–173.Google Scholar
de Kármán, T. & Howarth, L. 1938 On the statistical theory of isotropic turbulence. Proc. R. Soc. Lond. A 164 (917), 192–215.CrossRefGoogle Scholar
Kerr, R.M. 1987 Histograms of helicity and strain in numerical turbulence. Phys. Rev. Lett. 59, 783–786.CrossRefGoogle ScholarPubMed
Khurshid, S., Donzis, D.A. & Sreenivasan, K.R. 2018 Energy spectrum in the dissipation range. Phys. Rev. Fluids 3, 082601.CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 a Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16–18.Google Scholar
Kolmogorov, A.N. 1941 b The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299–303.Google Scholar
Kraichnan, R.H. 1959 The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5 (4), 497–543.CrossRefGoogle Scholar
Kraichnan, R.H. 1966 Isotropic turbulence and inertial-range structure. Phys. Fluids 9 (9), 1728–1752.CrossRefGoogle Scholar
Kraichnan, R.H. 1971 Inertial-range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47 (3), 525–535.CrossRefGoogle Scholar
Kraichnan, R.H. 1976 Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33 (8), 1521–1536.2.0.CO;2>CrossRefGoogle Scholar
Kurien, S., Taylor, M.A. & Matsumoto, T. 2004 Cascade time scales for energy and helicity in homogeneous isotropic turbulence. Phys. Rev. E 69, 066313.CrossRefGoogle ScholarPubMed
Leonard, A. 1975 Energy cascade in large-eddy simulations of turbulent fluid flows. In Turbulent Diffusion in Environmental Pollution (ed. F.N. Frenkiel & R.E. Munn), Advances in Geophysics, vol. 18, pp. 237–248. Elsevier.CrossRefGoogle Scholar
Leung, T., Swaminathan, N. & Davidson, P.A. 2012 Geometry and interaction of structures in homogeneous isotropic turbulence. J. Fluid Mech. 710, 453–481.CrossRefGoogle Scholar
Lohse, D. & Müller-Groeling, A. 1995 Bottleneck effects in turbulence: scaling phenomena in $r$ versus $p$ space. Phys. Rev. Lett. 74, 1747–1750.CrossRefGoogle Scholar
Lozano-Durán, A., Holzner, M. & Jiménez, J. 2016 Multiscale analysis of the topological invariants in the logarithmic region of turbulent channels at a friction Reynolds number of 932. J. Fluid Mech. 803, 356–394.CrossRefGoogle Scholar
Lumley, J.L. 1992 Some comments on turbulence. Phys. Fluids 4 (2), 203–211.CrossRefGoogle Scholar
Lund, T.S. & Rogers, M.M. 1994 An improved measure of strain state probability in turbulent flows. Phys. Fluids 6, 1838–1847.CrossRefGoogle Scholar
Lundgren, T.S. 1982 Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25 (12), 2193–2203.CrossRefGoogle Scholar
L'vov, V. & Falkovich, G. 1992 Counterbalanced interaction locality of developed hydrodynamic turbulence. Phys. Rev. A 46, 4762–4772.CrossRefGoogle ScholarPubMed
Manley, O.P. 1992 The dissipation range spectrum. Phys. Fluids A: Fluid Dyn. 4 (6), 1320–1321.CrossRefGoogle Scholar
Meneveau, C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43, 219–245.CrossRefGoogle Scholar
Meneveau, C. & Katz, J. 2000 Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech. 32 (1), 1–32.CrossRefGoogle Scholar
Meyers, J. & Meneveau, C. 2008 A functional form for the energy spectrum parametrizing bottleneck and intermittency effects. Phys. Fluids 20 (6), 065109.CrossRefGoogle Scholar
Mininni, P.D., Alexakis, A. & Pouquet, A. 2006 Large-scale flow effects, energy transfer, and self-similarity on turbulence. Phys. Rev. E 74, 016303.CrossRefGoogle ScholarPubMed
Mininni, P.D., Alexakis, A. & Pouquet, A. 2008 Nonlocal interactions in hydrodynamic turbulence at high Reynolds numbers: the slow emergence of scaling laws. Phys. Rev. E 77, 036306.CrossRefGoogle ScholarPubMed
Misra, A. & Pullin, D.I. 1997 A vortex-based subgrid stress model for large-eddy simulation. Phys. Fluids 9 (8), 2443–2454.CrossRefGoogle Scholar
Monin, A.S. & Yaglom, A.M. 1975 Statistical Fluid Mechanics: Mechanics of Turbulence. Dover.Google Scholar
Mullin, J.A. & Dahm, W.J.A. 2006 Dual-plane stereo particle image velocimetry measurements of velocity gradient tensor fields in turbulent shear flow. II. Experimental results. Phys. Fluids 18 (3), 035102.CrossRefGoogle Scholar
Nomura, K.K. & Post, G.K. 1998 The structure and dynamics of vorticity and rate of strain in incompressible homogeneous turbulence. J. Fluid Mech. 377, 65–97.CrossRefGoogle Scholar
Novikov, E.A. 1961 Energy spectrum of turbulent flow in an incompressible fluid. Dokl. Akad. Nauk. SSSR 139 (2), 331–334.Google Scholar
Ohkitani, K. & Kishiba, S. 1995 Nonlocal nature of vortex stretching in an inviscid fluid. Phys. Fluids 7 (2), 411–421.CrossRefGoogle Scholar
Onsager, L. 1949 Statistical hydrodynamics. Il Nuovo Cimento (1943–1954) 6 (2), 279–287.CrossRefGoogle Scholar
Ooi, A., Martin, J., Soria, J. & Chong, M.S. 1999 A study of the evolution and characteristics of the invariants of the velocity-gradient tensor in isotropic turbulence. J. Fluid Mech. 381, 141–174.CrossRefGoogle Scholar
Patterson, G.S. & Orszag, S.A. 1971 Spectral calculations of isotropic turbulence: efficient removal of aliasing interactions. The Phys. Fluids 14 (11), 2538–2541.CrossRefGoogle Scholar
Paul, I., Papadakis, G. & Vassilicos, J.C. 2017 Genesis and evolution of velocity gradients in near-field spatially developing turbulence. J. Fluid Mech. 815, 295–332.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Portwood, G.D., Nadiga, B.T., Saenz, J.A. & Livescu, D. 2020 Interpreting neural network models of residual scalar flux. J. Fluid Mech. 907, A23.Google Scholar
Pullin, D.I. & Saffman, P.G. 1994 Reynolds stresses and one-dimensional spectra for a vortex model of homogeneous anisotropic turbulence. Phys. Fluids 6 (5), 1787–1796.CrossRefGoogle Scholar
Pullin, D.I. & Saffman, P.G. 1998 Vortex dynamics in turbulence. Annu. Rev. Fluid Mech. 30, 31–51.CrossRefGoogle Scholar
Qian, J. 1984 Universal equilibrium range of turbulence. Phys. Fluids 27 (9), 2229–2233.CrossRefGoogle Scholar
Richardson, L.F. 1922 Weather Prediction by Numerical Process. Cambridge University Press.Google Scholar
Saddoughi, S.G. & Veeravalli, S.V. 1994 Local isotropy in turbulent boundary layers at high reynolds number. J. Fluid Mech. 268, 333–372.CrossRefGoogle Scholar
Sanada, T. 1992 Comment on the dissipation-range spectrum in turbulent flows. Phys. Fluids A: Fluid Dyn. 4 (5), 1086–1087.CrossRefGoogle Scholar
Silvis, M.H. & Verstappen, R. 2019 Nonlinear subgrid-scale models for large-eddy simulation of rotating turbulent flows. In Direct and Large-Eddy Simulation XI (ed. M.V. Salvetti, V. Armenio, J. Fröhlich, B.J. Geurts & H. Kuerten), pp. 129–134. Springer.CrossRefGoogle Scholar
Sirovich, L., Smith, L. & Yakhot, V. 1994 Energy spectrum of homogeneous and isotropic turbulence in far dissipation range. Phys. Rev. Lett. 72, 344–347.CrossRefGoogle ScholarPubMed
Smith, L.M. & Reynolds, W.C. 1991 The dissipation-range spectrum and the velocity-derivative skewness in turbulent flows. Phys. Fluids A: Fluid Dyn. 3 (5), 992–994.CrossRefGoogle Scholar
Soria, J., Sondergaard, R., Cantwell, B.J., Chong, M.S. & Perry, A.E. 1994 A study of the fine-scale motions of incompressible time-developing mixing layers. Phys. Fluids 6 (2), 871–884.CrossRefGoogle Scholar
Sreenivasan, K.R. 1985 On the fine-scale intermittency of turbulence. J. Fluid Mech. 151, 81–103.CrossRefGoogle Scholar
Taylor, G.I. 1938 Production and dissipation of vorticity in a turbulent fluid. Proc. R. Soc. Lond. A 164 (916), 15–23.Google Scholar
Tennekes, H. & Lumley, J.L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Townsend, A.A. 1951 On the fine-scale structure of turbulence. Proc. R. Soc. Lond. A 208, 534–542.Google Scholar
Tsinober, A. 2009 An Informal Conceptual Introduction to Turbulence, 2nd edn. Springer.CrossRefGoogle Scholar
Tsinober, A., Kit, E. & Dracos, T. 1992 Experimental investigation of the field of velocity gradients in turbulent flows. J. Fluid Mech. 242, 169–192.CrossRefGoogle Scholar
Vela-Martín, A. & Jiménez, J. 2021 Entropy, irreversibility and cascades in the inertial range of isotropic turbulence. J. Fluid Mech. 915, A36.CrossRefGoogle Scholar
Vieillefosse, P. 1982 Perfect incompressible fluids. J. Phys. 43, 837–842.CrossRefGoogle Scholar
Vieillefosse, P. 1984 Internal motion of a small element of fluid in an inviscid flow. Physica A 125 (1), 150–162.CrossRefGoogle Scholar
Vlaykov, D.G. & Wilczek, M. 2019 On the small-scale structure of turbulence and its impact on the pressure field. J. Fluid Mech. 861, 422–446.CrossRefGoogle Scholar
Vreman, B., Geurts, B. & Kuerten, H. 1994 Realizability conditions for the turbulent stress tensor in large-eddy simulation. J. Fluid Mech. 278, 351–362.CrossRefGoogle Scholar
Vreman, B., Geurts, B. & Kuerten, H. 1996 Large-eddy simulation of the temporal mixing layer using the Clark model. Theor. Comput. Fluid Dyn. 8 (4), 309–324.CrossRefGoogle Scholar
Vreman, B., Geurts, B. & Kuerten, H. 1997 Large-eddy simulation of the turbulent mixing layer. J. Fluid Mech. 339, 357–390.CrossRefGoogle Scholar
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 (2), 021301.CrossRefGoogle Scholar
Xiao, Z., Wan, M., Chen, S. & Eyink, G.L. 2009 Physical mechanism of the inverse energy cascade of two-dimensional turbulence: a numerical investigation. J. Fluid Mech. 619, 1–44.CrossRefGoogle Scholar
Xu, H., Pumir, A. & Bodenschatz, E. 2011 The pirouette effect in turbulent flows. Nat. Phys. 7 (9), 709–712.CrossRefGoogle Scholar
Zhou, Y. 1993 a Degrees of locality of energy transfer in the inertial range. Phys. Fluids 5 (5), 1092–1094.CrossRefGoogle Scholar
Zhou, Y. 1993 b Interacting scales and energy transfer in isotropic turbulence. Phys. Fluids 5 (10), 2511–2524.CrossRefGoogle Scholar