No CrossRef data available.
Article contents
The interscale behaviour of uncertainty in three-dimensional Navier–Stokes turbulence
Published online by Cambridge University Press: 20 August 2025
Abstract
We derive the scale-by-scale uncertainty energy budget equation and demonstrate theoretically and computationally the presence of a self-similar equilibrium cascade of decorrelation in an inertial range of scales during the time range of power-law growth of uncertainty in statistically stationary homogeneous turbulence. This cascade is predominantly inverse and driven by compressions of the reference field’s relative deformation tensor and their alignments with the uncertainty velocity field. Three other subdominant cascade mechanisms are also present, two of which are forward and also dominated by compressions and one of which, the weakest and the only nonlinear one of the four, is inverse. The uncertainty production and dissipation scalings which may follow from the self-similar equilibrium cascade of decorrelation lead to power-law growths of the uncertainty integral scale and the average uncertainty energy which are also investigated. Compressions are key not only to chaoticity, as previously shown, but also to stochasticity.
Information
- Type
- JFM Papers
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press
References
Alves Portela, F., Papadakis, G. & Vassilicos, J.C. 2017 The turbulence cascade in the near wake of a square prism. J. Fluid Mech. 825, 315–352.10.1017/jfm.2017.390CrossRefGoogle Scholar
Apostolidis, A., Laval, J. & Vassilicos, J.C. 2023 Turbulent cascade in fully developed turbulent channel flow. J. Fluid Mech. 967, A22.10.1017/jfm.2023.487CrossRefGoogle Scholar
Aurell, E., Boffetta, G., Crisanti, A., Paladin, G. & Vulpiani, A. 1997 Predictability in the large: an extension of the concept of Lyapunov exponent. J. Phys. A Math. Theor. 30 (1), 1.Google Scholar
Beaumard, P., Bragança, P., Cuvier, C., Steiros, K. & Vassilicos, J.C. 2024 Scale-by-scale non-equilibrium with Kolmogorov-like scalings in non-homogeneous stationary turbulence. J. Fluid Mech. 984, A35.10.1017/jfm.2024.220CrossRefGoogle Scholar
Berera, A. & Ho, R.D.J.G. 2018 Chaotic properties of a turbulent isotropic fluid. Phys. Rev. Lett. 120 (2), 024101.10.1103/PhysRevLett.120.024101CrossRefGoogle ScholarPubMed
Boffetta, G., Cencini, M., Falcioni, M. & Vulpiani, A. 2002 Predictability: a way to characterize complexity. Phys. Rep. 356 (6), 367–474.10.1016/S0370-1573(01)00025-4CrossRefGoogle Scholar
Boffetta, G. & Musacchio, S. 2017 Chaos and predictability of homogeneous-isotropic turbulence. Phys. Rev. Lett. 119 (5), 054102.10.1103/PhysRevLett.119.054102CrossRefGoogle ScholarPubMed
Chen, J.G. & Vassilicos, J.C. 2022 Scalings of scale-by-scale turbulence energy in non-homogeneous turbulence. J. Fluid Mech. 938, A7.10.1017/jfm.2022.153CrossRefGoogle Scholar
Cimarelli, A., Angelis, E., Jimenez, J. & Casciola, C.M. 2016 Cascades and wall-normal fluxes in turbulent channel flows. J. Fluid Mech. 796, 417–436.10.1017/jfm.2016.275CrossRefGoogle Scholar
Danaila, L., Krawczynski, J.F., Thiesset, F. & Renou, B. 2012 Yaglom-like equation in axisymmetric anisotropic turbulence. Phys. D: Nonlinear Phenom. 241 (3), 216–223.10.1016/j.physd.2011.08.011CrossRefGoogle Scholar
Deissler, R.G. 1986 Is Navier–Stokes turbulence chaotic? Phys. Fluids 29 (5), 1453–1457.10.1063/1.865663CrossRefGoogle Scholar
Ge, J., Rolland, J. & Vassilicos, J.C. 2023 The production of uncertainty in three-dimensional Navier–Stokes turbulence. J. Fluid Mech. 977, A17.10.1017/jfm.2023.967CrossRefGoogle Scholar
Germano, M. 2007 A direct relation between the filtered subgrid stress and the second order structure function. Phys. Fluids 19 (3), 038102.10.1063/1.2714078CrossRefGoogle Scholar
Hill, R.J. 2002 Exact second-order structure–function relationships. J. Fluid Mech. 468, 317–326.10.1017/S0022112002001696CrossRefGoogle Scholar
Ho, R.D.J.G., Armua, A. & Berera, A. 2020 Fluctuations of Lyapunov exponents in homogeneous and isotropic turbulence. Phys. Rev. Fluids 5 (2), 024602.10.1103/PhysRevFluids.5.024602CrossRefGoogle Scholar
Kármán, T. & Howarth, L. 1938 On the statistical theory of isotropic turbulence. Proc. R. Soc. A
164 (917), 192–215.Google Scholar
Kraichnan, R.H. 1970 Instability in fully developed turbulence. Phys. Fluids 13 (3), 569–575.10.1063/1.1692962CrossRefGoogle Scholar
Leith, C.E. 1971 Atmospheric predictability and two-dimensional turbulence. J. Atmos. Sci. 28 (2), 145–161.10.1175/1520-0469(1971)028<0145:APATDT>2.0.CO;22.0.CO;2>CrossRefGoogle Scholar
Leith, C.E. & Kraichnan, R.H. 1972 Predictability of turbulent flows. J. Atmos. Sci. 29 (6), 1041–1058.10.1175/1520-0469(1972)029<1041:POTF>2.0.CO;22.0.CO;2>CrossRefGoogle Scholar
Lesieur, M. 1987 Turbulence in Fluids: Stochastic and Numerical Modelling. Nijhoff.10.1007/978-94-009-3545-7CrossRefGoogle Scholar
Lorenz, E.N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20 (2), 130–141.10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;22.0.CO;2>CrossRefGoogle Scholar
Lorenz, E.N. 1969 The predictability of a flow which possesses many scales of motion. Tellus 21 (3), 289–307.10.3402/tellusa.v21i3.10086CrossRefGoogle Scholar
Marati, N., Casciola, C.M. & Piva, R. 2004 Energy cascade and spatial fluxes in wall turbulence. J. Fluid Mech. 521, 191–215.10.1017/S0022112004001818CrossRefGoogle Scholar
Métais, O. & Lesieur, M. 1986 Statistical predictability of decaying turbulence. J. Atmos. Sci. 43 (9), 857–870.10.1175/1520-0469(1986)043<0857:SPODT>2.0.CO;22.0.CO;2>CrossRefGoogle Scholar
Mohan, P., Fitzsimmons, N. & Moser, R.D. 2017 Scaling of Lyapunov exponents in homogeneous isotropic turbulence. Phys. Rev. Fluids 2 (11), 114606.10.1103/PhysRevFluids.2.114606CrossRefGoogle Scholar
Monin, A.S. & Yaglom, A.M. 2013 Statistical fluid mechanics. Volume II: Mechanics of Turbulence, vol. 2. Courier Corporation.Google Scholar
Thalabard, S., Bec, J. & Mailybaev, A.A. 2020 From the butterfly effect to spontaneous stochasticity in singular shear flows. Commun. Phys. 3 (1), 122.10.1038/s42005-020-0391-6CrossRefGoogle Scholar
Togni, R., Cimarelli, A. & De Angelis, E. 2015 Physical and scale-by-scale analysis of Rayleigh–Bénard convection. J. Fluid Mech. 782, 380–404.10.1017/jfm.2015.547CrossRefGoogle Scholar
Valente, P.C. & Vassilicos, J.C. 2015 The energy cascade in grid-generated non-equilibrium decaying turbulence. Phys. Fluids 27 (4), 045103.10.1063/1.4916628CrossRefGoogle Scholar
Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–20.10.1017/S0022112091001957CrossRefGoogle Scholar
Yasuda, T. & Vassilicos, J.C. 2018 Spatio-temporal intermittency of the turbulent energy cascade. J. Fluid Mech. 853, 235–252.10.1017/jfm.2018.584CrossRefGoogle Scholar
Yoffe, S.R. 2013 Investigation of the transfer and dissipation of energy in isotropic turbulence, arXiv: 1306.3408.Google Scholar