Authors

John Z. Shi

Abstract

Qian (Tsien) Jian (1939–2018), a Chinese theoretical physicist and fluid dynamicist, devoted the second part of his scientific life to the physical understanding of small-scale turbulence to the exclusion of all else. To place Qian’s contribution in an appropriate position in the field of small-scale turbulence, a historical overview and a state-of-the art review are attempted. Qian developed his own statistical theory of small-scale turbulence based on the Liouville [“Sur l’équation aux différences partielles,” J. Math. Pures Appl. 18, 71–72 (1853)] equation and a perturbation variational approach to non-equilibrium statistical mechanics, which is compatible with the Kolmogorov–Oboukhov energy spectrum. Qian’s statistical theory of small-scale turbulence, which appears mathematically and physically valid, successfully led to his contributions to (i) the closure problem of turbulence; (ii) one-dimensional turbulence; (iii) two-dimensional turbulence; (iv) the turbulent passive scalar field; (v) the cascade model of turbulence; (vi) the universal equilibrium range of turbulence; (vii) a simple model of the bump phenomenon; (viii) universal constants of turbulence; (ix) the intermittency of turbulence; and perhaps most importantly, and (x) the effect of the Taylor microscale Reynolds number (Rλ) on both the width of the inertial range of finite Rλ turbulence and the scaling exponents of velocity structure functions. In particular, Qian found that the inertial range cannot exist when Rλ≪2000. In contrast to the prevailing intermittency models, he discovered that normal scaling is valid in the real Kolmogorov inertial range when Rλ approaches infinity while the anomalous scaling observed in experiments reflects the finite Rλ effect (Qe). He then made a correction to the famous Kolmogorov [“Dissipation of energy in the locally isotropic turbulence,” Dokl. Akad. Nauk SSSR 32(1), 19–21 (1941c) (in Russian); reprinted in Proc. R. Soc. London A 434, 15–17 (1991)] equation and obtained the finite Rλ effect equation or the Kolmogorov–Novikov–Qian equation. He also independently derived the decay law of the finite Rλ effect. Qian steered all of us along the right path to an improved understanding of small-scale turbulence and solutions to its problems. Qian is credited with his contribution to enhanced knowledge about the finite Rλ effect of turbulence, which has profoundly shaped and stimulated thinking about the K41 turbulence, the K62 turbulence, and the finite Rλ turbulence.

Citation

  • Journal: Physics of Fluids
  • Year: 2021
  • Volume: 33
  • Issue: 4
  • Pages:
  • Publisher: AIP Publishing
  • DOI: 10.1063/5.0043566

BibTeX

@article{Shi_2021,
  title={{Qian Jian (1939–2018) and his contribution to small-scale turbulence studies}},
  volume={33},
  ISSN={1089-7666},
  DOI={10.1063/5.0043566},
  number={4},
  journal={Physics of Fluids},
  publisher={AIP Publishing},
  author={Shi, John Z.},
  year={2021}
}

Download the bib file

References

  • Essai d’une Nouvelle Théorie de la Résistance des Fluides (1752)
  • Anselmet, F., Gagne, Y., Hopfinger, E. J. & Antonia, R. A. High-order velocity structure functions in turbulent shear flows. Journal of Fluid Mechanics vol. 140 63–89 (1984) – 10.1017/s0022112084000513
  • Antonia, R. A. Some small scale properties of boundary layer turbulence. The Physics of Fluids vol. 16 1198–1206 (1973) – 10.1063/1.1694498
  • Antonia, R. A., Anselmet, F. & Chambers, A. J. Assessment of local isotropy using measurements in a turbulent plane jet. Journal of Fluid Mechanics vol. 163 365–391 (1986) – 10.1017/s0022112086002331
  • Small-scale turbulence: How universal is it?. Proceedings of the 15th Australasian Fluid Mechanics Conference (2004)
  • ANTONIA, R. A. & BURATTINI, P. Approach to the 4/5 law in homogeneous isotropic turbulence. Journal of Fluid Mechanics vol. 550 175 (2006) – 10.1017/s0022112005008438
  • Antonia, R. A., Chambers, A. J. & Satyaprakash, B. R. Reynolds number dependence of high-order moments of the streamwise turbulent velocity derivative. Boundary-Layer Meteorology vol. 21 159–171 (1981) – 10.1007/bf02033934
  • Antonia, R. A., Djenidi, L., Danaila, L. & Tang, S. L. Small scale turbulence and the finite Reynolds number effect. Physics of Fluids vol. 29 (2017) – 10.1063/1.4974323
  • Antonia, R. A., Satyaprakash, B. R. & Chambers, A. J. Reynolds number dependence of velocity structure functions in turbulent shear flows. The Physics of Fluids vol. 25 29–37 (1982) – 10.1063/1.863624
  • Antonia, R. A., Tang, S. L., Djenidi, L. & Danaila, L. Boundedness of the velocity derivative skewness in various turbulent flows. Journal of Fluid Mechanics vol. 781 727–744 (2015) – 10.1017/jfm.2015.539
  • K41 versus K62: Recent developments. (2019)
  • Antonia, R. A., Tang, S. L., Djenidi, L. & Zhou, Y. Finite Reynolds number effect and the 4/5 law. Physical Review Fluids vol. 4 (2019) – 10.1103/physrevfluids.4.084602
  • Arenas, A. & Chorin, A. J. On the existence and scaling of structure functions in turbulence according to the data. Proceedings of the National Academy of Sciences vol. 103 4352–4355 (2006) – 10.1073/pnas.0600482103
  • Barenblatt, G. I. & Chorin, A. J. Meccanica vol. 33 445–468 (1998) – 10.1023/a:1004312409376
  • BATCHELOR, G. K. Double Velocity Correlation Function in Turbulent Motion. Nature vol. 158 883–884 (1946) – 10.1038/158883a0
  • Batchelor, G. K. Kolmogoroff’s theory of locally isotropic turbulence. Mathematical Proceedings of the Cambridge Philosophical Society vol. 43 533–559 (1947) – 10.1017/s0305004100023793
  • Batchelor, G. K. Energy decay and self-preserving correlation functions in isotropic turbulence. Quarterly of Applied Mathematics vol. 6 97–116 (1948) – 10.1090/qam/28162
  • Batchelor, G. K. Pressure fluctuations in isotropic turbulence. Mathematical Proceedings of the Cambridge Philosophical Society vol. 47 359–374 (1951) – 10.1017/s0305004100026712
  • The Theory of Homogeneous Turbulence (1953)
  • Batchelor, G. K. Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity. Journal of Fluid Mechanics vol. 5 113 (1959) – 10.1017/s002211205900009x
  • Favre, The dynamics of homogeneous turbulence: Introductory remarks. (1962)
  • Kolmogorov’s work on turbulence. Bull. London Math. Soc. (1990)
  • Decay of vorticity in isotropic turbulence. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences vol. 190 534–550 (1947) – 10.1098/rspa.1947.0095
  • Decay of isotropic turbulence in the initial period. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences vol. 193 539–558 (1948) – 10.1098/rspa.1948.0061
  • Decay of turbulence in the final period. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences vol. 194 527–543 (1948) – 10.1098/rspa.1948.0095
  • The nature of turbulent motion at large wave-numbers. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences vol. 199 238–255 (1949) – 10.1098/rspa.1949.0136
  • Barenblatt, G. I. & Chorin, A. J. New Perspectives in Turbulence: Scaling Laws, Asymptotics, and Intermittency. SIAM Review vol. 40 265–291 (1998) – 10.1137/s0036144597320047
  • Bazdenkov, S. V. & Kukharkin, N. N. On the variational method of closure in the theory of turbulence. Physics of Fluids A: Fluid Dynamics vol. 5 2248–2254 (1993) – 10.1063/1.858563
  • Benzi, R. & Biferale, L. Homogeneous and Isotropic Turbulence: A Short Survey on Recent Developments. Journal of Statistical Physics vol. 161 1351–1365 (2015) – 10.1007/s10955-015-1323-9
  • Benzi, R., Biferale, L., Ciliberto, S., Struglia, M. V. & Tripiccione, R. Generalized scaling in fully developed turbulence. Physica D: Nonlinear Phenomena vol. 96 162–181 (1996) – 10.1016/0167-2789(96)00018-8
  • Benzi, R. et al. Extended self-similarity in turbulent flows. Physical Review E vol. 48 R29–R32 (1993) – 10.1103/physreve.48.r29
  • Benzi, R. & Vulpiani, A. Small-scale intermittency of turbulent flows. Journal of Physics A: Mathematical and General vol. 13 3319–3324 (1980) – 10.1088/0305-4470/13/10/026
  • The Analyst (1734)
  • Biferale, L. & Procaccia, I. Anisotropy in turbulent flows and in turbulent transport. Physics Reports vol. 414 43–164 (2005) – 10.1016/j.physrep.2005.04.001
  • Birnir, B. The Kolmogorov–Obukhov Statistical Theory of Turbulence. Journal of Nonlinear Science vol. 23 657–688 (2013) – 10.1007/s00332-012-9164-z
  • Weitere studien über das wärmegleichgewicht unter gasmolekülen. Sitzungsber. Akad. Wiss. (1872)
  • Borgas, M. S. A comparison of intermittency models in turbulence. Physics of Fluids A: Fluid Dynamics vol. 4 2055–2061 (1992) – 10.1063/1.858375
  • Borgas, M. S., Sawford, B. L., Xu, S., Donzis, D. A. & Yeung, P. K. High Schmidt number scalars in turbulence: Structure functions and Lagrangian theory. Physics of Fluids vol. 16 3888–3899 (2004) – 10.1063/1.1780550
  • Bos, W. J. T., Chevillard, L., Scott, J. F. & Rubinstein, R. Reynolds number effect on the velocity increment skewness in isotropic turbulence. Physics of Fluids vol. 24 (2012) – 10.1063/1.3678338
  • Bos, W. J. T. & Rubinstein, R. On the strength of the nonlinearity in isotropic turbulence. Journal of Fluid Mechanics vol. 733 158–170 (2013) – 10.1017/jfm.2013.405
  • Bos, W. J. T., Rubinstein, R. & Fang, L. Reduction of mean-square advection in turbulent passive scalar mixing. Physics of Fluids vol. 24 (2012) – 10.1063/1.4731302
  • Boschung, J., Gauding, M., Hennig, F., Denker, D. & Pitsch, H. Finite Reynolds number corrections of the 4/5 law for decaying turbulence. Physical Review Fluids vol. 1 (2016) – 10.1103/physrevfluids.1.064403
  • Essai Sur la Theorie Des Eaux Courantes (1877)
  • Bowman, J. C. On inertial-range scaling laws. Journal of Fluid Mechanics vol. 306 167–181 (1996) – 10.1017/s0022112096001279
  • Application of a model system to illustrate some points of the statistical theory of free turbulence. Proc. Acad. Sci. Amsterdam (1940)
  • Burgers, J. M. A Mathematical Model Illustrating the Theory of Turbulence. Advances in Applied Mechanics 171–199 (1948) doi:10.1016/s0065-2156(08)70100-5 – 10.1016/s0065-2156(08)70100-5
  • Essays: Collected and Republished (1872)
  • Cerbus, R. T., Liu, C., Gioia, G. & Chakraborty, P. Small-scale universality in the spectral structure of transitional pipe flows. Science Advances vol. 6 (2020) – 10.1126/sciadv.aaw6256
  • Champagne, F. H. The fine-scale structure of the turbulent velocity field. Journal of Fluid Mechanics vol. 86 67–108 (1978) – 10.1017/s0022112078001019
  • Chandrasekhar, S. Turbulence - a Physical Theory of Astrophysical Interest. The Astrophysical Journal vol. 110 329 (1949) – 10.1086/145210
  • Chapman, D. R. Computational Aerodynamics Development and Outlook. AIAA Journal vol. 17 1293–1313 (1979) – 10.2514/3.61311
  • The Mathematical Theory of Non-Uniform Gases (1952)
  • Chen, S. & Doolen, G. D. LATTICE BOLTZMANN METHOD FOR FLUID FLOWS. Annual Review of Fluid Mechanics vol. 30 329–364 (1998) – 10.1146/annurev.fluid.30.1.329
  • Chen, S., Doolen, G. D., Kraichnan, R. H. & Wang, L.-P. Is the Kolmogorov Refined Similarity Relation Dynamic or Kinematic? Physical Review Letters vol. 74 1755–1758 (1995) – 10.1103/physrevlett.74.1755
  • Chen, S. et al. Robert Harry Kraichnan. Physics Today vol. 61 70–71 (2008) – 10.1063/1.2930746
  • Chen, S., Eyink, G. L., Wan, M. & Xiao, Z. Is the Kelvin Theorem Valid for High Reynolds Number Turbulence? Physical Review Letters vol. 97 (2006) – 10.1103/physrevlett.97.144505
  • Chen, S., Sreenivasan, K. R. & Nelkin, M. Inertial Range Scalings of Dissipation and Enstrophy in Isotropic Turbulence. Physical Review Letters vol. 79 1253–1256 (1997) – 10.1103/physrevlett.79.1253
  • Chen, S., Sreenivasan, K. R., Nelkin, M. & Cao, N. Refined Similarity Hypothesis for Transverse Structure Functions in Fluid Turbulence. Physical Review Letters vol. 79 2253–2256 (1997) – 10.1103/physrevlett.79.2253
  • Chou, P.-Y. & Chou, R.-L. 50 Years of Turbulence Research in China. Annual Review of Fluid Mechanics vol. 27 1–16 (1995) – 10.1146/annurev.fl.27.010195.000245
  • The vorticity structure of homogeneous isotropic turbulence in its final period of decay. Chin. J. Theor. Appl. Mech. (1957)
  • Clausius, R. X. On the mean length of the paths described by the separate molecules of gaseous bodies on the occurrence of molecular motion: together with some other remarks upon the mechanical theory of heat. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science vol. 17 81–91 (1859) – 10.1080/14786445908642626
  • Collar, A. R. Arthur Fage, 1890-1977. Biographical Memoirs of Fellows of the Royal Society vol. 24 32–53 (1978) – 10.1098/rsbm.1978.0003
  • An experimental verification of local isotropy. J. Aeronaut. Sci. (1949)
  • Corrsin, S. On the Spectrum of Isotropic Temperature Fluctuations in an Isotropic Turbulence. Journal of Applied Physics vol. 22 469–473 (1951) – 10.1063/1.1699986
  • Crowdy, D. & Tanveer, S. Philip Geoffrey Saffman. 19 March 1931 — 17 August 2008. Biographical Memoirs of Fellows of the Royal Society vol. 60 375–395 (2014) – 10.1098/rsbm.2014.0021
  • DANAILA, L., ANSELMET, F., ZHOU, T. & ANTONIA, R. A. A generalization of Yaglom’s equation whichaccounts for the large-scale forcing inheated decaying turbulence. Journal of Fluid Mechanics vol. 391 359–372 (1999) – 10.1017/s0022112099005418
  • Dannevik, W. P., Yakhot, V. & Orszag, S. A. Analytical theories of turbulence and the ε expansion. The Physics of Fluids vol. 30 2021–2029 (1987) – 10.1063/1.866216
  • Day, C. Physics in China. Physics Today vol. 63 33–38 (2010) – 10.1063/1.3366238
  • Djenidi, L. & Antonia, R. A. Assessment of large-scale forcing in isotropic turbulence using a closed Kármán–Howarth equation. Physics of Fluids vol. 32 055104 (2020) – 10.1063/5.0006466
  • Djenidi, L., Antonia, R. A. & Tang, S. L. Scale invariance in finite Reynolds number homogeneous isotropic turbulence. Journal of Fluid Mechanics vol. 864 244–272 (2019) – 10.1017/jfm.2019.28
  • Djenidi, L., Antonia, R. A. & Tang, S. L. Mathematical constraints on the scaling exponents in the inertial range of fluid turbulence. Physics of Fluids vol. 33 (2021) – 10.1063/5.0039643
  • Djenidi, L., Antonia, R. A., Talluru, M. K. & Abe, H. Skewness and flatness factors of the longitudinal velocity derivative in wall-bounded flows. Physical Review Fluids vol. 2 (2017) – 10.1103/physrevfluids.2.064608
  • Dorfman, J. R. & Faller, A. J. Jan Burgers. Physics Today vol. 35 84–85 (1982) – 10.1063/1.2890021
  • Dubrulle, B. Beyond Kolmogorov cascades. Journal of Fluid Mechanics vol. 867 (2019) – 10.1017/jfm.2019.98
  • Scale invariance and scaling exponents in fully developed turbulence. J. Phys. II (1996)
  • Birds and frogs. Not. Am. Math. Soc. (2009)
  • Fourier’s transformational thinking. Nature vol. 555 413–413 (2018) – 10.1038/d41586-018-03389-w
  • Edwards, S. F. The statistical dynamics of homogeneous turbulence. Journal of Fluid Mechanics vol. 18 239 (1964) – 10.1017/s0022112064000180
  • Edwards, S. F. & McComb, W. D. Statistical mechanics far from equilibrium. Journal of Physics A: General Physics vol. 2 157–171 (1969) – 10.1088/0305-4470/2/2/003
  • A local energy transport equation for isotropic turbulence. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences vol. 325 313–321 (1971) – 10.1098/rspa.1971.0171
  • Local transport equations for turbulent shear flow. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences vol. 330 495–516 (1972) – 10.1098/rspa.1972.0156
  • Effinger, H. & Grossmann, S. Static structure function of turbulent flow from the Navier-Stokes equations. Zeitschrift f�r Physik B Condensed Matter vol. 66 289–304 (1987) – 10.1007/bf01305419
  • Elsinga, G. E., Ishihara, T. & Hunt, J. C. R. Extreme dissipation and intermittency in turbulence at very high Reynolds numbers. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences vol. 476 (2020) – 10.1098/rspa.2020.0591
  • Eyink, G. L. & Sreenivasan, K. R. Onsager and the theory of hydrodynamic turbulence. Reviews of Modern Physics vol. 78 87–135 (2006) – 10.1103/revmodphys.78.87
  • On the flow of air behind an inclined flat plate of infinite span. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character vol. 116 170–197 (1927) – 10.1098/rspa.1927.0130
  • An examination of turbulent flow with an ultramicroscope. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character vol. 135 656–677 (1932) – 10.1098/rspa.1932.0059
  • The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom (2009)
  • Fokker, A. D. Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Annalen der Physik vol. 348 810–820 (1914) – 10.1002/andp.19143480507
  • Zhen, F. The statistical theory of homogenous turbulence. Mechanics Research Communications vol. 9 273–282 (1982) – 10.1016/0093-6413(82)90078-7
  • Forster, D., Nelson, D. R. & Stephen, M. J. Long-Time Tails and the Large-Eddy Behavior of a Randomly Stirred Fluid. Physical Review Letters vol. 36 867–870 (1976) – 10.1103/physrevlett.36.867
  • Forster, D., Nelson, D. R. & Stephen, M. J. Large-distance and long-time properties of a randomly stirred fluid. Physical Review A vol. 16 732–749 (1977) – 10.1103/physreva.16.732
  • Théorie Analytique De La Chaleur (1822)
  • FOWLER, R. H. The Mathematical Theory of Non-Uniform Gases. Nature vol. 144 993–995 (1939) – 10.1038/144993a0
  • Frenkiel, F. N. & Klebanoff, P. S. On the lognormality of the small-scale structure of turbulence. Boundary-Layer Meteorology vol. 8 173–200 (1975) – 10.1007/bf00241336
  • Turbulence: The Legacy of A. N. Kolmogorov (1995)
  • Frisch, U., Sulem, P.-L. & Nelkin, M. A simple dynamical model of intermittent fully developed turbulence. Journal of Fluid Mechanics vol. 87 719 (1978) – 10.1017/s0022112078001846
  • Gad-el-Hak, M. & Bandyopadhyay, P. R. Reynolds Number Effects in Wall-Bounded Turbulent Flows. Applied Mechanics Reviews vol. 47 307–365 (1994) – 10.1115/1.3111083
  • Gamard, S. & George, W. K. Flow, Turbulence and Combustion vol. 63 443–477 (2000) – 10.1023/a:1009988321057
  • Zhi, G. & Qian, J. Spectral line profile of turbulent gas. Applied Optics vol. 26 1579 (1987) – 10.1364/ao.26.001579
  • George, The self-preservation of turbulent flows and its relation to initial conditions and coherent structures. Advances in Turbulence (1989)
  • George, W. K. The decay of homogeneous isotropic turbulence. Physics of Fluids A: Fluid Dynamics vol. 4 1492–1509 (1992) – 10.1063/1.858423
  • Hanjalic, Some new ideas for similarity of turbulent shear flows. (1995)
  • Farge, Reconsidering the ‘local equilibrium’ hypothesis for small scale turbulence. (2013)
  • Zero-pressure-gradient turbulent boundary layer. Appl. Rev. (1997)
  • Gibson, C. H. & Schwarz, W. H. The universal equilibrium spectra of turbulent velocity and scalar fields. Journal of Fluid Mechanics vol. 16 365 (1963) – 10.1017/s0022112063000835
  • GIBSON, M. M. Spectra of Turbulence at High Reynolds Number. Nature vol. 195 1281–1283 (1962) – 10.1038/1951281a0
  • Gotoh, T., Fukayama, D. & Nakano, T. Velocity field statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulation. Physics of Fluids vol. 14 1065–1081 (2002) – 10.1063/1.1448296
  • GRANT, H. L., MOILLIET, A. & STEWART, R. W. A Spectrum of Turbulence at Very High Reynolds Number. Nature vol. 184 808–810 (1959) – 10.1038/184808b0
  • Grant, H. L. & Nisbet, I. C. T. The inhomogeneity of grid turbulence. Journal of Fluid Mechanics vol. 2 263–272 (1957) – 10.1017/s0022112057000117
  • Grossmann, S. & Lohse, D. Intermittency Exponents. Europhysics Letters (EPL) vol. 21 201–206 (1993) – 10.1209/0295-5075/21/2/014
  • Grossmann, S. & Lohse, D. Scale resolved intermittency in turbulence. Physics of Fluids vol. 6 611–617 (1994) – 10.1063/1.868357
  • Grossmann, S. & Lohse, D. Universality in fully developed turbulence. Physical Review E vol. 50 2784–2789 (1994) – 10.1103/physreve.50.2784
  • Grossmann, S., Lohse, D., L’vov, V. & Procaccia, I. Finite size corrections to scaling in high Reynolds number turbulence. Physical Review Letters vol. 73 432–435 (1994) – 10.1103/physrevlett.73.432
  • Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I. & Shraiman, B. I. Fractal measures and their singularities: The characterization of strange sets. Physical Review A vol. 33 1141–1151 (1986) – 10.1103/physreva.33.1141
  • Hanjalić, K. & Launder, B. E. Contribution towards a Reynolds-stress closure for low-Reynolds-number turbulence. Journal of Fluid Mechanics vol. 74 593–610 (1976) – 10.1017/s0022112076001961
  • He, G.-W. Anomalous scaling for Lagrangian velocity structure functions in fully developed turbulence. Physical Review E vol. 83 (2011) – 10.1103/physreve.83.025301
  • He, G., Chen, S., Kraichnan, R. H., Zhang, R. & Zhou, Y. Statistics of Dissipation and Enstrophy Induced by Localized Vortices. Physical Review Letters vol. 81 4636–4639 (1998) – 10.1103/physrevlett.81.4636
  • He, G., Jin, G. & Yang, Y. Space-Time Correlations and Dynamic Coupling in Turbulent Flows. Annual Review of Fluid Mechanics vol. 49 51–70 (2017) – 10.1146/annurev-fluid-010816-060309
  • He, G.-W., Jin, G. & Zhao, X. Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence. Physical Review E vol. 80 (2009) – 10.1103/physreve.80.066313
  • He, G.-W. & Zhang, J.-B. Elliptic model for space-time correlations in turbulent shear flows. Physical Review E vol. 73 (2006) – 10.1103/physreve.73.055303
  • Heisenberg, W. Zur statistischen Theorie der Turbulenz. Zeitschrift für Physik vol. 124 628–657 (1948) – 10.1007/bf01668899
  • On the theory of statistical and isotropic turbulence. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences vol. 195 402–406 (1948) – 10.1098/rspa.1948.0127
  • Herring, J. R. Self-Consistent-Field Approach to Nonstationary Turbulence. The Physics of Fluids vol. 9 2106–2110 (1966) – 10.1063/1.1761579
  • Hill, R. J. Models of the scalar spectrum for turbulent advection. Journal of Fluid Mechanics vol. 88 541–562 (1978) – 10.1017/s002211207800227x
  • Turbulence (1975)
  • Molecular Theory of Gases and Liquids (1954)
  • Huang, Y. A note on Kolmogorov’s −5/3 scaling law in a non-inertial frame of reference. Journal of Turbulence vol. 6 N27 (2005) – 10.1080/14685240500331926
  • Huang, Y.-N. & Chou, P.-Y. On the solutions of Navier-Stokes equations and the theory of homogeneous isotropic turbulence. Proceedings of the Indian Academy of Sciences Section C: Engineering Sciences vol. 4 177–197 (1981) – 10.1007/bf02896740
  • Huang, Y.-N. & Durst, F. Reynolds stress under a change of frame of reference. Physical Review E vol. 63 (2001) – 10.1103/physreve.63.056305
  • The natural viscosity of turbulence. J. Turbul. (2003)
  • Huang, Y. X. et al. Second-order structure function in fully developed turbulence. Physical Review E vol. 82 (2010) – 10.1103/physreve.82.026319
  • Kolmogorov’s contributions to the physical and geometrical understanding of small-scale turbulence and recent developments. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences vol. 434 183–210 (1991) – 10.1098/rspa.1991.0088
  • Errata. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences vol. 435 674–674 (1991) – 10.1098/rspa.1991.0171
  • Turbulence and stochastic processes: Kolmogorov’s ideas 50 years on. Proc. R. Soc. London A (1991)
  • Ishihara, T., Gotoh, T. & Kaneda, Y. Study of High–Reynolds Number Isotropic Turbulence by Direct Numerical Simulation. Annual Review of Fluid Mechanics vol. 41 165–180 (2009) – 10.1146/annurev.fluid.010908.165203
  • Iyer, K. P., Sreenivasan, K. R. & Yeung, P. K. Scaling exponents saturate in three-dimensional isotropic turbulence. Physical Review Fluids vol. 5 (2020) – 10.1103/physrevfluids.5.054605
  • Grattan-Guinness, George Berkeley, The analyst (1734). Landmark Writings in Western Mathematics, 1640–1940 (2005)
  • JIMÉNEZ, J. Intermittency and cascades. Journal of Fluid Mechanics vol. 409 99–120 (2000) – 10.1017/s0022112099007739
  • The contributions of A. N. Kolmogorov to the theory of turbulence. Arbor (2004)
  • Kadanoff, L. P. A Model of Turbulence. Physics Today vol. 48 11–11 (1995) – 10.1063/1.2808151
  • Kaneda, Y. & Ishihara, T. High-resolution direct numerical simulation of turbulence. Journal of Turbulence vol. 7 N20 (2006) – 10.1080/14685240500256099
  • Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K. & Uno, A. Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Physics of Fluids vol. 15 L21–L24 (2003) – 10.1063/1.1539855
  • Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K. & Uno, A. Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Physics of Fluids vol. 15 L21–L24 (2003) – 10.1063/1.1539855
  • Davidson, Small-scale statistics and structure of turbulence—In the light of high resolution direct numerical simulation. Ten Chapters in Turbulence (2013)
  • Kaneda, Y., Yoshino, J. & Ishihara, T. Examination of Kolmogorov’s 4/5 Law by High-Resolution Direct Numerical Simulation Data of Turbulence. Journal of the Physical Society of Japan vol. 77 064401 (2008) – 10.1143/jpsj.77.064401
  • de Karman, T. & Howarth, L. On the Statistical Theory of Isotropic Turbulence. Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences vol. 164 192–215 (1938) – 10.1098/rspa.1938.0013
  • Kerr, R. M. Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. Journal of Fluid Mechanics vol. 153 31 (1985) – 10.1017/s0022112085001136
  • Kim, J. & Antonia, R. A. Isotropy of the small scales of turbulence at low Reynolds number. Journal of Fluid Mechanics vol. 251 219–238 (1993) – 10.1017/s0022112093003398
  • Kistler, A. L. & Vrebalovich, T. Grid turbulence at large Reynolds numbers. Journal of Fluid Mechanics vol. 26 37 (1966) – 10.1017/s0022112066001071
  • Local structure of turbulence in an incompressible viscous fluid at very large Reynolds numbers. Dokl. Akad. Nauk SSSR (1941)
  • Kolmogorov, A. N. LOCAL STRUCTURE OF TURBULENCE IN AN INCOMPRESSIBLE VISCOUS FLUID AT VERY HIGH REYNOLDS NUMBERS. Soviet Physics Uspekhi vol. 10 734–746 (1968) – 10.1070/pu1968v010n06abeh003710
  • The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR (1941)
  • The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences vol. 434 9–13 (1991) – 10.1098/rspa.1991.0075
  • On the generation of isotropic turbulence in an incompressible viscous fluid. Dokl. Acad. Nauk SSSR (1941)
  • Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR (1941)
  • Dissipation of energy in the locally isotropic turbulence. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences vol. 434 15–17 (1991) – 10.1098/rspa.1991.0076
  • Kolmogorov, A. N. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. Journal of Fluid Mechanics vol. 13 82–85 (1962) – 10.1017/s0022112062000518
  • Mathematics and Mechanics (in Russian) (1985)
  • KOVASZNAY, L. S. G. Spectrum of Locally Isotropic Turbulence. Journal of the Aeronautical Sciences vol. 15 745–753 (1948) – 10.2514/8.11707
  • Kovasznay, L. S. G., Kibens, V. & Blackwelder, R. F. Large-scale motion in the intermittent region of a turbulent boundary layer. Journal of Fluid Mechanics vol. 41 283–325 (1970) – 10.1017/s0022112070000629
  • Kraichnan, R. H. Irreversible Statistical Mechanics of Incompressible Hydromagnetic Turbulence. Physical Review vol. 109 1407–1422 (1958) – 10.1103/physrev.109.1407
  • Kraichnan, R. H. The structure of isotropic turbulence at very high Reynolds numbers. Journal of Fluid Mechanics vol. 5 497 (1959) – 10.1017/s0022112059000362
  • Kraichnan, R. H. Kolmogorov’s Hypotheses and Eulerian Turbulence Theory. The Physics of Fluids vol. 7 1723–1734 (1964) – 10.1063/1.2746572
  • Inertial-range spectrum of hydrodynamic turbulence. Phys. Fluids (1965)
  • Kraichnan, R. H. Inertial Ranges in Two-Dimensional Turbulence. The Physics of Fluids vol. 10 1417–1423 (1967) – 10.1063/1.1762301
  • Kraichnan, R. H. Small-Scale Structure of a Scalar Field Convected by Turbulence. The Physics of Fluids vol. 11 945–953 (1968) – 10.1063/1.1692063
  • Kraichnan, R. H. An almost-Markovian Galilean-invariant turbulence model. Journal of Fluid Mechanics vol. 47 513–524 (1971) – 10.1017/s0022112071001204
  • Rice, Some modern developments in the statistical theory of turbulence. (1972)
  • Kraichnan, R. H. On Kolmogorov’s inertial-range theories. Journal of Fluid Mechanics vol. 62 305–330 (1974) – 10.1017/s002211207400070x
  • Kraichnan, R. H. Lagrangian velocity covariance in helical turbulence. Journal of Fluid Mechanics vol. 81 385–398 (1977) – 10.1017/s0022112077002110
  • Kraichnan, R. H. & Herring, J. R. A strain-based Lagrangian-history turbulence theory. Journal of Fluid Mechanics vol. 88 355–367 (1978) – 10.1017/s0022112078002153
  • Kraichnan, R. H. An interpretation of the Yakhot–Orszag turbulence theory. The Physics of Fluids vol. 30 2400–2405 (1987) – 10.1063/1.866130
  • Turbulent cascade and intermittency growth. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences vol. 434 65–78 (1991) – 10.1098/rspa.1991.0080
  • Kraichnan, R. H. & Chen, S. Is there a statistical mechanics of turbulence? Physica D: Nonlinear Phenomena vol. 37 160–172 (1989) – 10.1016/0167-2789(89)90126-7
  • Kraichnan, R. H. & Herring, J. R. A strain-based Lagrangian-history turbulence theory. Journal of Fluid Mechanics vol. 88 355–367 (1978) – 10.1017/s0022112078002153
  • Kuo, A. Y.-S. & Corrsin, S. Experiments on internal intermittency and fine-structure distribution functions in fully turbulent fluid. Journal of Fluid Mechanics vol. 50 285–319 (1971) – 10.1017/s0022112071002581
  • Hydrodynamics (1895)
  • Sur la théorie de mouvement Brownien. C. R. Acad. Sci. (Paris) (1908)
  • On the problem of turbulence. C. R. Acad. Sci. URSS (1944)
  • Fluid Mechanics, Volume 6 of Course of Theoretical Physics (1959)
  • Lee, T. D. Note on the Coefficient of Eddy Viscosity in Isotropic Turbulence. Physical Review vol. 77 842–843 (1950) – 10.1103/physrev.77.842
  • Interview of Keith Moffatt: Magnetohydrodynamic attraction. Newsletter of Institute for Mathematical Sciences (2006)
  • Developments in the Theory of Turbulence (1973)
  • Liao, Z.-J. & Su, W.-D. Kolmogorov’s hypotheses and global energy spectrum of turbulence. Physics of Fluids vol. 27 (2015) – 10.1063/1.4916964
  • Goldstine, The representation of small-scale turbulence in numerical simulation experiments. (1967)
  • Lindborg, E. Correction to the four-fifths law due to variations of the dissipation. Physics of Fluids vol. 11 510–512 (1999) – 10.1063/1.869924
  • Sur l’équation aux différences partielles. J. Math. Pures Appl. (1853)
  • Lohse, D. Crossover from High to Low Reynolds Number Turbulence. Physical Review Letters vol. 73 3223–3226 (1994) – 10.1103/physrevlett.73.3223
  • Lohse, D. & Müller-Groeling, A. Bottleneck Effects in Turbulence: Scaling Phenomena inrversuspSpace. Physical Review Letters vol. 74 1747–1750 (1995) – 10.1103/physrevlett.74.1747
  • Long, R. R. A new theory of the energy spectrum. Boundary-Layer Meteorology vol. 24 137–160 (1982) – 10.1007/bf00121665
  • Long, R. R. Environmental Fluid Mechanics vol. 3 109–127 (2003) – 10.1023/a:1022086815714
  • Long, R. R. & Chen, T.-C. Experimental evidence for the existence of the ‘mesolayer’ in turbulent systems. Journal of Fluid Mechanics vol. 105 19 (1981) – 10.1017/s0022112081003108
  • Lumley, J. L. & Yaglom, A. M. Flow, Turbulence and Combustion vol. 66 241–286 (2001) – 10.1023/a:1012437421667
  • Lundgren, T. S. Kolmogorov two-thirds law by matched asymptotic expansion. Physics of Fluids vol. 14 638–642 (2002) – 10.1063/1.1429965
  • Lundgren, T. S. Kolmogorov turbulence by matched asymptotic expansions. Physics of Fluids vol. 15 1074–1081 (2003) – 10.1063/1.1558332
  • Lundgren, T. S. Response to “Comment on ‘Kolmogorov turbulence by matched asymptotic expansions’” [Phys. Fluids 17, 059101 (2005)]. Physics of Fluids vol. 17 (2005) – 10.1063/1.1923027
  • L’vov, V. & Procaccia, I. Cornerstones of a theory of anomalous scaling in turbulence. Physica Scripta vol. T67 131–135 (1996) – 10.1088/0031-8949/1996/t67/026
  • MACPHAIL, D. C. An Experimental Verification of the Isotropy of Turbulence Produced by a Grid. Journal of the Aeronautical Sciences vol. 8 73–75 (1940) – 10.2514/8.10481
  • The Fractal Geometry of Nature (1982)
  • MARTÍNEZ, D. O. et al. Energy spectrum in the dissipation range of fluid turbulence. Journal of Plasma Physics vol. 57 195–201 (1997) – 10.1017/s0022377896005338
  • Maxwell, J. C. V. Illustrations of the dynamical theory of gases.—Part I. On the motions and collisions of perfectly elastic spheres. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science vol. 19 19–32 (1860) – 10.1080/14786446008642818
  • XIII The Bakerian Lectur .—On the viscosity or internal friction of air and other gases. Philosophical Transactions of the Royal Society of London vol. 156 249–268 (1866) – 10.1098/rstl.1866.0013
  • McComb, D. Scale-invariance and the inertial-range spectrum in three-dimensional stationary, isotropic turbulence. Journal of Physics A: Mathematical and Theoretical vol. 42 125501 (2009) – 10.1088/1751-8113/42/12/125501
  • McComb, W. D. A local energy-transfer theory of isotropic turbulence. Journal of Physics A: Mathematical, Nuclear and General vol. 7 632–649 (1974) – 10.1088/0305-4470/7/5/013
  • McComb, W. D. The inertial-range spectrum from a local energy-transfer theory of isotropic turbulence. Journal of Physics A: Mathematical and General vol. 9 179–184 (1976) – 10.1088/0305-4470/9/1/023
  • McComb, W. D. A theory of time-dependent, isotropic turbulence. Journal of Physics A: Mathematical and General vol. 11 613–632 (1978) – 10.1088/0305-4470/11/3/023
  • McComb, W. D. Reformulation of the statistical equations for turbulent shear flow. Physical Review A vol. 26 1078–1094 (1982) – 10.1103/physreva.26.1078
  • The Physics of Fluid Turbulence (1990)
  • McComb, W. D. Theory of turbulence. Reports on Progress in Physics vol. 58 1117–1205 (1995) – 10.1088/0034-4885/58/10/001
  • Homogeneous, Isotropic Turbulence (2014)
  • McComb, W. D., Berera, A., Salewski, M. & Yoffe, S. Taylor’s (1935) dissipation surrogate reinterpreted. Physics of Fluids vol. 22 (2010) – 10.1063/1.3450299
  • Energy transfer and dissipation in forced isotropic turbulence. Phys. Rev. (2015)
  • McComb, W. D. & Fairhurst, R. B. The dimensionless dissipation rate and the Kolmogorov (1941) hypothesis of local stationarity in freely decaying isotropic turbulence. Journal of Mathematical Physics vol. 59 (2018) – 10.1063/1.5019925
  • Eulerian spectral closures for isotropic turbulence using a time-ordered fluctuation-dissipation relation. Phys. Rev. (2005)
  • McComb, W. D. & Shanmugasundaram, V. Fluid turbulence and the renormalization group: A preliminary calculation of the eddy viscosity. Physical Review A vol. 28 2588–2590 (1983) – 10.1103/physreva.28.2588
  • Mccomb, W. D. & Shanmugasundaram, V. Numerical calculation of decaying isotropic turbulence using the LET theory. Journal of Fluid Mechanics vol. 143 95–123 (1984) – 10.1017/s0022112084001270
  • Mccomb, W. D., Shanmugasundaram, V. & Hutchinson, P. Velocity-derivative skewness and two-time velocity correlations of isotropic turbulence as predicted by the LET theory. Journal of Fluid Mechanics vol. 208 91–114 (1989) – 10.1017/s0022112089002788
  • McComb, W. D. & Yoffe, S. R. A formal derivation of the local energy transfer (LET) theory of homogeneous turbulence. Journal of Physics A: Mathematical and Theoretical vol. 50 375501 (2017) – 10.1088/1751-8121/aa8379
  • Spectral analysis of structure functions and their scaling exponents in forced isotropic turbulence. Phys. Rev. (2014)
  • Meldi, M., Djenidi, L. & Antonia, R. Reynolds number effect on the velocity derivative flatness factor. Journal of Fluid Mechanics vol. 856 426–443 (2018) – 10.1017/jfm.2018.717
  • Meneveau, C. & Sreenivasan, K. R. Simple multifractal cascade model for fully developed turbulence. Physical Review Letters vol. 59 1424–1427 (1987) – 10.1103/physrevlett.59.1424
  • Meneveau, C. & Sreenivasan, K. R. The multifractal nature of turbulent energy dissipation. Journal of Fluid Mechanics vol. 224 429–484 (1991) – 10.1017/s0022112091001830
  • Mestayer, P. Local isotropy and anisotropy in a high-Reynolds-number turbulent boundary layer. Journal of Fluid Mechanics vol. 125 475 (1982) – 10.1017/s0022112082003450
  • Mi, J., Xu, M. & Zhou, T. Reynolds number influence on statistical behaviors of turbulence in a circular free jet. Physics of Fluids vol. 25 (2013) – 10.1063/1.4811403
  • Moffatt, H. K. G.K. BATCHELOR AND THEHOMOGENIZATION OFTURBULENCE. Annual Review of Fluid Mechanics vol. 34 19–35 (2002) – 10.1146/annurev.fluid.34.081701.134821
  • Moffatt, H. K. George Keith Batchelor. 8 March 1920 – 30 March 2000. Biographical Memoirs of Fellows of the Royal Society vol. 48 25–41 (2002) – 10.1098/rsbm.2002.0002
  • MOFFATT, H. K. George Batchelor: a personal tribute, ten years on. Journal of Fluid Mechanics vol. 663 2–7 (2010) – 10.1017/s0022112010004167
  • Moffatt, H. K. Homogeneous turbulence: an introductory review. Journal of Turbulence vol. 13 N39 (2012) – 10.1080/14685248.2012.721559
  • Self-Exciting Fluid Dynamos (2019)
  • Moisy, F., Tabeling, P. & Willaime, H. Kolmogorov Equation in a Fully Developed Turbulence Experiment. Physical Review Letters vol. 82 3994–3997 (1999) – 10.1103/physrevlett.82.3994
  • Lumley, Statistical Fluid Mechanics: Mechanics of Turbulence (1971)
  • Lumley, Statistical Fluid Mechanics: Mechanics of Turbulence (1975)
  • Mydlarski, L. & Warhaft, Z. On the onset of high-Reynolds-number grid-generated wind tunnel turbulence. Journal of Fluid Mechanics vol. 320 331 (1996) – 10.1017/s0022112096007562
  • Mémoire sur les lois du mouvement des fluides. Mémoires de L’Académie Royale Des Sciences (de L’Institut de France (1823)
  • Nelkin, M. Universality and scaling in fully developed turbulence. Advances in Physics vol. 43 143–181 (1994) – 10.1080/00018739400101485
  • Philosophae Naturalis Principia Mathematica (1687)
  • On the energy spectrum of the turbulent flow of an incompressible fluid. Dokl. Akad. Nauk SSSR (1961)
  • Random force method in turbulence theory. Sov. Phys. JETP (1963)
  • Functionals and the random-force method in turbulence theory. Sov. Phys. (1965)
  • Intermittency and scale similarity in the structure of a turbulent flow. Prikl. Mat. Mekh. (1971)
  • Novikov, E. A. Intermittency and scale similarity in the structure of a turbulent plow. Journal of Applied Mathematics and Mechanics vol. 35 231–241 (1971) – 10.1016/0021-8928(71)90029-3
  • Novikov, E. A. The effects of intermittency on statistical characteristics of turbulence and scale similarity of breakdown coefficients. Physics of Fluids A: Fluid Dynamics vol. 2 814–820 (1990) – 10.1063/1.857629
  • Novikov, E. A new approach to the problem of turbulence, based on the conditionally averaged Navier-Stokes equations. Fluid Dynamics Research vol. 12 107–126 (1993) – 10.1016/0169-5983(93)90108-m
  • Intermittency of turbulence and spectrum of fluctuations in energy-dissipation. Izv. Akad. Nauk, Ser. Geophys. (1964)
  • Obligado, M. & Vassilicos, J. C. The non-equilibrium part of the inertial range in decaying homogeneous turbulence. EPL (Europhysics Letters) vol. 127 64004 (2019) – 10.1209/0295-5075/127/64004
  • On the energy distribution in the spectrum of a turbulent flow. Dokl. Akad. Nauk SSSR (1941)
  • On the theory of atmospheric turbulence. Izv. Akad. Nauk. Ser. Fiz. (1942)
  • Turbulentnost’ v temperaturnoj–neodnorodnoj atmosfere (Turbulence in an atmosphere with a non-uniform temperature). Trudy Inst. Theor. Geofiz. SSSR (1946)
  • Structure of the temperature field in a turbulent flow. Izv. Akad. Nauk, SSSR Ser. Geogr. I Geofiz (1949)
  • Local structure of atmospheric turbulence. Dokl. Akad. Nauk, SSSR (1949)
  • Oboukhov, A. M. Some specific features of atmospheric tubulence. Journal of Fluid Mechanics vol. 13 77–81 (1962) – 10.1017/s0022112062000506
  • Okamura, M. Closure model for homogeneous isotropic turbulence in the Lagrangian specification of the flow field. Journal of Fluid Mechanics vol. 841 521–551 (2018) – 10.1017/jfm.2018.98
  • The distribution of energy in turbulence (Abstract). Phys. Rev. (1945)
  • Orszag, S. A. Analytical theories of turbulence. Journal of Fluid Mechanics vol. 41 363–386 (1970) – 10.1017/s0022112070000642
  • Ortiz-Suslow, D. G., Wang, Q., Kalogiros, J. & Yamaguchi, R. A Method for Identifying Kolmogorov’s Inertial Subrange in the Velocity Variance Spectrum. Journal of Atmospheric and Oceanic Technology vol. 37 85–102 (2020) – 10.1175/jtech-d-19-0028.1
  • Pao, Y.-H. Structure of Turbulent Velocity and Scalar Fields at Large Wavenumbers. The Physics of Fluids vol. 8 1063–1075 (1965) – 10.1063/1.1761356
  • PEARSON, B. R. & ANTONIA, R. A. Reynolds-number dependence of turbulent velocity and pressure increments. Journal of Fluid Mechanics vol. 444 343–382 (2001) – 10.1017/s0022112001005511
  • Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (1917)
  • Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandlungen des III. Internationalen Mathematiker Kongresses, Heidelberg, 1904 (1905)
  • Prandtl, L. 7. Bericht über Untersuchungen zur ausgebildeten Turbulenz. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik vol. 5 136–139 (1925) – 10.1002/zamm.19250050212
  • Praskovsky, A. A. Experimental verification of the Kolmogorov refined similarity hypothesis. Physics of Fluids A: Fluid Dynamics vol. 4 2589–2591 (1992) – 10.1063/1.858446
  • Praskovsky, A. & Oncley, S. Measurements of the Kolmogorov constant and intermittency exponent at very high Reynolds numbers. Physics of Fluids vol. 6 2886–2888 (1994) – 10.1063/1.868435
  • On the integral equations in the theory of mixed gas transport. Acta Phys. Sin. (1964)
  • Qian, J. Variational approach to the closure problem of turbulence theory. The Physics of Fluids vol. 26 2098–2104 (1983) – 10.1063/1.864414
  • Qian, J. Numerical experiments on one-dimensional model of turbulence. The Physics of Fluids vol. 27 1957–1965 (1984) – 10.1063/1.864850
  • Qian, J. Universal equilibrium range of turbulence. The Physics of Fluids vol. 27 2229–2233 (1984) – 10.1063/1.864902
  • Qian, J. Nonequilibrium statistical mechanics of two-dimensional turbulence. The Physics of Fluids vol. 27 2412–2417 (1984) – 10.1063/1.864545
  • Qian, J. Nonequilibrium statistical mechanics of one-dimensional turbulence. The Physics of Fluids vol. 27 2967–2969 (1984) – 10.1063/1.864583
  • Qian, J. A passive scalar field convected by turbulence. The Physics of Fluids vol. 28 1299–1304 (1985) – 10.1063/1.865014
  • Qian, J. A closure theory of intermittency of turbulence. The Physics of Fluids vol. 29 2165–2171 (1986) – 10.1063/1.865553
  • Qian, J. Turbulent passive scalar field of a small Prandtl number. The Physics of Fluids vol. 29 3586–3589 (1986) – 10.1063/1.865785
  • Qian, J. Inverse energy cascade in two-dimensional turbulence. The Physics of Fluids vol. 29 3608–3611 (1986) – 10.1063/1.865788
  • Qian, J. Cascade model of turbulence. The Physics of Fluids vol. 31 2865–2874 (1988) – 10.1063/1.866995
  • Spectral dynamics of passive scalar convected by homogeneous two-dimensional turbulence. Sci. China (1989)
  • Qian, J. The spectrum of a turbulent passive scalar in the viscous–convective range. Journal of Fluid Mechanics vol. 217 203–212 (1990) – 10.1017/s0022112090000684
  • Qian, J. Relation between universal constants of turbulence. Physics of Fluids A: Fluid Dynamics vol. 2 634–635 (1990) – 10.1063/1.857713
  • Qian, J. Real and Pseudo Kolmogorov Constant. Journal of the Physical Society of Japan vol. 62 926–932 (1993) – 10.1143/jpsj.62.926
  • A new method of determining eddy transport coefficients of turbulence. Sci. China, Ser. A (1993)
  • Jian, Q. Skewness factor of turbulent velocity derivative. Acta Mechanica Sinica vol. 10 12–15 (1994) – 10.1007/bf02487653
  • Qian, J. Generalization of the Kolmogorov -5/3 law of turbulence. Physical Review E vol. 50 611–613 (1994) – 10.1103/physreve.50.611
  • Turbulent Prandtl number. Sci. China (1994)
  • Jian, Q. Some issues of turbulence statistics. Acta Mechanica Sinica vol. 11 122–128 (1995) – 10.1007/bf02487619
  • Qian, J. Viscous range of turbulent scalar of large Prandtl number. Fluid Dynamics Research vol. 15 103–112 (1995) – 10.1016/0169-5983(95)91431-6
  • Qian, J. Correlation coefficients between the velocity difference and local average dissipation of turbulence. Physical Review E vol. 54 981–984 (1996) – 10.1103/physreve.54.981
  • Qian, J. Experimental Values of Kolmogorov Constant of Turbulence. Journal of the Physical Society of Japan vol. 65 2502–2505 (1996) – 10.1143/jpsj.65.2502
  • On entropy method of turbulence closure problem. J. Phys. (1996)
  • Qian, J. Inertial range and the finite Reynolds number effect of turbulence. Physical Review E vol. 55 337–342 (1997) – 10.1103/physreve.55.337
  • Qian, J. Scaling exponents of the second-order structure function of turbulence. Journal of Physics A: Mathematical and General vol. 31 3193–3204 (1998) – 10.1088/0305-4470/31/14/008
  • Qian, J. Normal and anomalous scaling of turbulence. Physical Review E vol. 58 7325–7329 (1998) – 10.1103/physreve.58.7325
  • Qian, J. Slow decay of the finite Reynolds number effect of turbulence. Physical Review E vol. 60 3409–3412 (1999) – 10.1103/physreve.60.3409
  • Qian, J. Closure Approach to High-Order Structure Functions of Turbulence. Physical Review Letters vol. 84 646–649 (2000) – 10.1103/physrevlett.84.646
  • QIAN, J. QUASI-CLOSURE AND SCALING OF TURBULENCE. International Journal of Modern Physics B vol. 15 1085–1116 (2001) – 10.1142/s0217979201004514
  • Qian, J. Scaling of structure functions in homogeneous shear-flow turbulence. Physical Review E vol. 65 (2002) – 10.1103/physreve.65.036301
  • Qian, J. An equality about the velocity derivative skewness in turbulence. Physics of Fluids vol. 15 1005–1011 (2003) – 10.1063/1.1556675
  • Non-Gaussian self-similarity in the inertial range of turbulence. J. Hydrodyn. (2006)
  • Qian, J. Non-Gaussian statistical model of turbulence. Journal of Turbulence vol. 7 N24 (2006) – 10.1080/14685240600604412
  • Advances in Fluid Mechanics VI. WIT Transactions on Engineering Sciences, Vol 52 (2006) doi:10.2495/afm06 – 10.2495/afm06
  • New Sedov-type solution of isotropic turbulence. Chin. Phys. Lett. (2009)
  • One exactly soluble model in isotropic turbulence. Adv. Appl. Fluid Mech. (2009)
  • Ran, Z. Multiscales and cascading in isotropic turbulence. Chinese Science Bulletin vol. 56 (2011) – 10.1007/s11434-011-4675-9
  • Ran, Z. & Yuan, X.-J. Nonlinear dynamical systems and bistability in linearly forced isotropic turbulence. Acta Mechanica Sinica vol. 29 823–826 (2013) – 10.1007/s10409-013-0085-3
  • Comte-Bellot, Reynolds number and Prandtl number influence on the determination of isotropic velocity and temperature turbulent length scales. (1987)
  • XXIX. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Philosophical Transactions of the Royal Society of London vol. 174 935–982 (1883) – 10.1098/rstl.1883.0029
  • Weather Prediction by Numerical Process (1922)
  • Ruetsch, G. R. & Maxey, M. R. The evolution of small-scale structures in homogeneous isotropic turbulence. Physics of Fluids A: Fluid Dynamics vol. 4 2747–2760 (1992) – 10.1063/1.858333
  • Ryndina, E. Family Lines Sketched in the Portrait of Lev Landau. Physics Today vol. 57 53–59 (2004) – 10.1063/1.1688070
  • Saddoughi, S. G. & Veeravalli, S. V. Local isotropy in turbulent boundary layers at high Reynolds number. Journal of Fluid Mechanics vol. 268 333–372 (1994) – 10.1017/s0022112094001370
  • Saffman, P. G. On the fine-scale structure of vector fields convected by a turbulent fluid. Journal of Fluid Mechanics vol. 16 545–572 (1963) – 10.1017/s0022112063000987
  • Saffman, P. G. The large-scale structure of homogeneous turbulence. Journal of Fluid Mechanics vol. 27 581–593 (1967) – 10.1017/s0022112067000552
  • Homogeneous Turbulence Dynamics. (2018)
  • Formules et tables nouvelles pour la solution des problèmes relatifs aux eaux courantes. (1851)
  • SATO, Y., YAMAMOTO, K. & MIZUSHINA, T. Empirical equations for the structure of isotropic turbulence. JOURNAL OF CHEMICAL ENGINEERING OF JAPAN vol. 16 273–280 (1983) – 10.1252/jcej.16.273
  • SATO, Y., YAMAMOTO, K. & MIZUSHINA, T. Comparison of empirical equation and previous hypotheses for the energy transfer of isotropic turbulence. JOURNAL OF CHEMICAL ENGINEERING OF JAPAN vol. 17 209–211 (1984) – 10.1252/jcej.17.209
  • Schmitt, F. G. Turbulence from 1870 to 1920: The birth of a noun and of a concept. Comptes Rendus. Mécanique vol. 345 620–626 (2017) – 10.1016/j.crme.2017.06.003
  • Schumacher, J. et al. Small-scale universality in fluid turbulence. Proceedings of the National Academy of Sciences vol. 111 10961–10965 (2014) – 10.1073/pnas.1410791111
  • She, Z.-S. Intermittency and non-gaussian statistics in turbulence. Fluid Dynamics Research vol. 8 143–158 (1991) – 10.1016/0169-5983(91)90039-l
  • She, Z.-S. Physical model of intermittency in turbulence: Near-dissipation-range non-Gaussian statistics. Physical Review Letters vol. 66 600–603 (1991) – 10.1103/physrevlett.66.600
  • She, Z.-S., Chen, S., Doolen, G., Kraichnan, R. H. & Orszag, S. A. Reynolds number dependence of isotropic Navier-Stokes turbulence. Physical Review Letters vol. 70 3251–3254 (1993) – 10.1103/physrevlett.70.3251
  • She, Z.-S. & Jackson, E. On the universal form of energy spectra in fully developed turbulence. Physics of Fluids A: Fluid Dynamics vol. 5 1526–1528 (1993) – 10.1063/1.858591
  • She, Z.-S., Jackson, E. & Orszag, S. A. Scale-dependent intermittency and coherence in turbulence. Journal of Scientific Computing vol. 3 407–434 (1988) – 10.1007/bf01065179
  • She, Z.-S., Jackson, E. & Orszag, S. A. Intermittent vortex structures in homogeneous isotropic turbulence. Nature vol. 344 226–228 (1990) – 10.1038/344226a0
  • Structure and dynamics of homogeneous turbulence: Models and simulations. Proc. R. Soc. London (1991)
  • She, Z.-S. & Leveque, E. Universal scaling laws in fully developed turbulence. Physical Review Letters vol. 72 336–339 (1994) – 10.1103/physrevlett.72.336
  • She, Z.-S. & Orszag, S. A. Physical model of intermittency in turbulence: Inertial-range non-Gaussian statistics. Physical Review Letters vol. 66 1701–1704 (1991) – 10.1103/physrevlett.66.1701
  • Dr. Qian Jian: In memoriam (in Chinese). Chin. J. Theor. Appl. Mech. (2019)
  • Shi, J. Z. George Keith Batchelor’s Interaction with Chinese Fluid Dynamicists and Inspirational Influence: a historical perspective. Notes and Records: the Royal Society Journal of the History of Science vol. 75 461–502 (2020) – 10.1098/rsnr.2019.0034
  • Kolmogorov and the turbulence. Miscellanea (1999)
  • Siggia, E. D. Origin of intermittency in fully developed turbulence. Physical Review A vol. 15 1730–1750 (1977) – 10.1103/physreva.15.1730
  • Siggia, E. D. Numerical study of small-scale intermittency in three-dimensional turbulence. Journal of Fluid Mechanics vol. 107 375 (1981) – 10.1017/s002211208100181x
  • Spalding, D. B. The Molecular Theory of Gases and Liquids.J. O. Hirschfelder, C. F. Curtiss and R. B. Bird. John Wiley,New York. Chapman & Hall, London, 1954. 1,219 pp.Diagrams. 160s. The Journal of the Royal Aeronautical Society vol. 59 228–228 (1955) – 10.1017/s0368393100117833
  • Small-scale intermittency in turbulence. Proceedings of the Twelfth Australian Fluid Mechanics Conference (1995)
  • Sreenivasan, K. R. On the universality of the Kolmogorov constant. Physics of Fluids vol. 7 2778–2784 (1995) – 10.1063/1.868656
  • Sreenivasan, K. R. An update on the energy dissipation rate in isotropic turbulence. Physics of Fluids vol. 10 528–529 (1998) – 10.1063/1.869575
  • Sreenivasan, K. R. Fluid turbulence. Reviews of Modern Physics vol. 71 S383–S395 (1999) – 10.1103/revmodphys.71.s383
  • Sreenivasan, K. R. & Antonia, R. A. THE PHENOMENOLOGY OF SMALL-SCALE TURBULENCE. Annual Review of Fluid Mechanics vol. 29 435–472 (1997) – 10.1146/annurev.fluid.29.1.435
  • SREENIVASAN, K. R. & BERSHADSKII, A. Finite-Reynolds-number effects in turbulence using logarithmic expansions. Journal of Fluid Mechanics vol. 554 477 (2006) – 10.1017/s002211200600913x
  • Sreenivasan, K. R. & Dhruva, B. Is There Scaling in High-Reynolds-Number Turbulence? Progress of Theoretical Physics Supplement vol. 130 103–120 (1998) – 10.1143/ptps.130.103
  • Sreenivasan, K. R. & Meneveau, C. The fractal facets of turbulence. Journal of Fluid Mechanics vol. 173 357–386 (1986) – 10.1017/s0022112086001209
  • Stewart, R. W. Triple velocity correlations in isotropic turbulence. Mathematical Proceedings of the Cambridge Philosophical Society vol. 47 146–157 (1951) – 10.1017/s0305004100026451
  • Similarity and self-preservation in isotropic turbulence. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences vol. 243 359–386 (1951) – 10.1098/rsta.1951.0007
  • Stewart, R. W., Wilson, J. R. & Burling, R. W. Some statistical properties of small scale turbulence in an atmospheric boundary layer. Journal of Fluid Mechanics vol. 41 141–152 (1970) – 10.1017/s002211207000054x
  • On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Cambridge Philos. Soc. (1845)
  • On the effect of the internal friction of fluids on the motion of Pendulums. Trans. Cambridge Philos. Soc. (1851)
  • Stolovitzky, G., Kailasnath, P. & Sreenivasan, K. R. Kolmogorov’s refined similarity hypotheses. Physical Review Letters vol. 69 1178–1181 (1992) – 10.1103/physrevlett.69.1178
  • Tang, S. L., Antonia, R. A., Djenidi, L., Danaila, L. & Zhou, Y. Finite Reynolds number effect on the scaling range behaviour of turbulent longitudinal velocity structure functions. Journal of Fluid Mechanics vol. 820 341–369 (2017) – 10.1017/jfm.2017.218
  • Tang, S. L., Antonia, R. A., Djenidi, L., Danaila, L. & Zhou, Y. Reappraisal of the velocity derivative flatness factor in various turbulent flows. Journal of Fluid Mechanics vol. 847 244–265 (2018) – 10.1017/jfm.2018.307
  • Tang, S. L., Antonia, R. A., Djenidi, L. & Zhou, Y. Scaling of the turbulent energy dissipation correlation function. Journal of Fluid Mechanics vol. 891 (2020) – 10.1017/jfm.2020.171
  • I. Eddy motion in the atmosphere. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character vol. 215 1–26 (1915) – 10.1098/rsta.1915.0001
  • Diffusion by continuous movements. Proc. London Math. Soc. (1921)
  • Taylor, G. I. Statistical theory of turbulenc. Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences vol. 151 421–444 (1935) – 10.1098/rspa.1935.0158
  • Taylor, G. I. Statistical theory of turbulence-II. Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences vol. 151 444–454 (1935) – 10.1098/rspa.1935.0159
  • Statistical theory of turbulence III. Proc. R. Soc. London A (1935)
  • Taylor, G. I. Statistical theory of turbulence IV-Diffusion in a turbulent air stream. Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences vol. 151 465–478 (1935) – 10.1098/rspa.1935.0161
  • Statistical theory of turbulence V. Proc. R. Soc. London A (1936)
  • Taylor, G. I. The Spectrum of Turbulence. Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences vol. 164 476–490 (1938) – 10.1098/rspa.1938.0032
  • Tchen, C. M. Random Flight with Multiple Partial Correlations. The Journal of Chemical Physics vol. 20 214–217 (1952) – 10.1063/1.1700381
  • Tchen, C. M. On the spectrum of energy in turbulent shear flow. Journal of Research of the National Bureau of Standards vol. 50 51 (1953) – 10.6028/jres.050.009
  • Tchen, C.-M. Transport Processes as Foundations of the Heisenberg and Obukhoff Theories of Turbulence. Physical Review vol. 93 4–14 (1954) – 10.1103/physrev.93.4
  • Tchen, C. M. Repeated cascade theory of homogeneous turbulence. The Physics of Fluids vol. 16 13–30 (1973) – 10.1063/1.1694158
  • Tchen, C. M. Cascade theory of turbulence in a stratified medium. Tellus A: Dynamic Meteorology and Oceanography vol. 27 1 (1975) – 10.3402/tellusa.v27i1.9878
  • Tchoufag, J., Sagaut, P. & Cambon, C. Spectral approach to finite Reynolds number effects on Kolmogorov’s 4/5 law in isotropic turbulence. Physics of Fluids vol. 24 (2012) – 10.1063/1.3678334
  • Thomson, W. XXXIV. Stability of motion (continued from the May, June, and August Numbers).—Broad river flowing down an inclined plane bed. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science vol. 24 272–278 (1887) – 10.1080/14786448708628094
  • Thomson, W. XLV. On the propagation of laminar motion through a turbulently moving inviscid liquid. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science vol. 24 342–353 (1887) – 10.1080/14786448708628110
  • Thoroddsen, S. T. & Van Atta, C. W. Experimental evidence supporting Kolmogorov’s refined similarity hypothesis. Physics of Fluids A: Fluid Dynamics vol. 4 2592–2594 (1992) – 10.1063/1.858447
  • Selected Works of A. N. Kolmogorov, Volume 1 Mathematics and Mechanics (1991)
  • Townsend, A. A. The measurement of double and triple correlation derivatives in isotropic turbulence. Mathematical Proceedings of the Cambridge Philosophical Society vol. 43 560–570 (1947) – 10.1017/s030500410002380x
  • Townsend, A. A. & Taylor, G. Experimental evidence for the theory of local isotropy. Mathematical Proceedings of the Cambridge Philosophical Society vol. 44 560–565 (1948) – 10.1017/s0305004100024567
  • Local isotropy in the turbulent wake of a cylinder. Aust. J. Sci. Res. (1948)
  • Vainshtein, S. I. & Sreenivasan, K. R. Kolmogorov’s ⅘th Law and Intermittency in Turbulence. Physical Review Letters vol. 73 3085–3088 (1994) – 10.1103/physrevlett.73.3085
  • Van Atta, C. W. & Antonia, R. A. Reynolds number dependence of skewness and flatness factors of turbulent velocity derivatives. The Physics of Fluids vol. 23 252–257 (1980) – 10.1063/1.862965
  • Atta, C. W. V. & Chen, W. Y. Structure functions of turbulence in the atmospheric boundary layer over the ocean. Journal of Fluid Mechanics vol. 44 145–159 (1970) – 10.1017/s002211207000174x
  • Weizsäcker, C. F. v. Das Spektrum der Turbulenz bei großen Reynoldsschen Zahlen. Zeitschrift für Physik vol. 124 614–627 (1948) – 10.1007/bf01668898
  • The Analects (2000)
  • Wang, J., Wan, M., Chen, S. & Chen, S. Kinetic energy transfer in compressible isotropic turbulence. Journal of Fluid Mechanics vol. 841 581–613 (2018) – 10.1017/jfm.2018.23
  • Wang, L.-P., Chen, S., Brasseur, J. G. & Wyngaard, J. C. Examination of hypotheses in the Kolmogorov refined turbulence theory through high-resolution simulations. Part 1. Velocity field. Journal of Fluid Mechanics vol. 309 113–156 (1996) – 10.1017/s0022112096001589
  • Warner, M. Sir Sam Edwards. 1 February 1928 — 7 July 2015. Biographical Memoirs of Fellows of the Royal Society vol. 63 243–271 (2017) – 10.1098/rsbm.2016.0028
  • Williams, R. M. & Paulson, C. A. Microscale temperature and velocity spectra in the atmospheric boundary layer. Journal of Fluid Mechanics vol. 83 547–567 (1977) – 10.1017/s0022112077001335
  • Wilson, K. G. The renormalization group and critical phenomena. Reviews of Modern Physics vol. 55 583–600 (1983) – 10.1103/revmodphys.55.583
  • The Poetical Works of William Wordsworth (1894)
  • Wu, J. Z., Fang, L., Shao, L. & Lu, L. P. Theories and applications of second-order correlation of longitudinal velocity increments at three points in isotropic turbulence. Physics Letters A vol. 382 1665–1671 (2018) – 10.1016/j.physleta.2018.04.021
  • Yaglom, A. M. Alexander Mikhailovich Obukhov, 1918–1989. Boundary-Layer Meteorology vol. 53 v–xi (1990) – 10.1007/bf00122458
  • Yaglom, A. M. A. N. Kolmogorov as a Fluid Mechanician and Founder of a School in Turbulence Research. Annual Review of Fluid Mechanics vol. 26 1–23 (1994) – 10.1146/annurev.fl.26.010194.000245
  • Yakhot, V. & Orszag, S. A. Renormalization group analysis of turbulence. I. Basic theory. Journal of Scientific Computing vol. 1 3–51 (1986) – 10.1007/bf01061452
  • Renormalization-group analysis of turbulence. Phys. Rev. Lett. (1987)
  • Yakhot, V., Orszag, S. A. & She, Z.-S. Space-time correlations in turbulence: Kinematical versus dynamical effects. Physics of Fluids A: Fluid Dynamics vol. 1 184–186 (1989) – 10.1063/1.857486
  • Yakhot, V., She, Z.-S. & Orszag, S. A. Deviations from the classical Kolmogorov theory of the inertial range of homogeneous turbulence. Physics of Fluids A: Fluid Dynamics vol. 1 289–293 (1989) – 10.1063/1.857445
  • Yang, P.-F., Pumir, A. & Xu, H. Generalized self-similar spectrum and the effect of large-scale in decaying homogeneous isotropic turbulence. New Journal of Physics vol. 20 103035 (2018) – 10.1088/1367-2630/aae72d
  • Yoffe, S. R. & McComb, W. D. Onset criteria for freely decaying isotropic turbulence. Physical Review Fluids vol. 3 (2018) – 10.1103/physrevfluids.3.104605
  • Zhu, Y., Antonia, R. A. & Hosokawa, I. Refined similarity hypotheses for turbulent velocity and temperature fields. Physics of Fluids vol. 7 1637–1648 (1995) – 10.1063/1.868482
  • Zhou, Y. Renormalization group theory for fluid and plasma turbulence. Physics Reports vol. 488 1–49 (2010) – 10.1016/j.physrep.2009.04.004
  • ZHOU, T. & ANTONIA, R. A. Reynolds number dependence of the small-scale structure of grid turbulence. Journal of Fluid Mechanics vol. 406 81–107 (2000) – 10.1017/s0022112099007296