Some developments in the theory of turbulence
10.1017/s002211208100150x
Crossref journal-article
Cambridge University Press (CUP)
Journal of Fluid Mechanics (56)
Abstract

This is in no way intended as a review of turbulence-the subject is far too big for adequate treatment within a reasonably finite number of pages; the monumental treatise of Monin & Yaglom (1971, 1975) bears witness to this statement. It is rather a discourse on those aspects of the problem of turbulence with which I have myself had contact over the last twenty years or so. My choice of topics therefore has a very personal bias - but that is perhaps consistent with the style and objectives of this rather unusual issue of JFM. I have approached the dynamical problem of turbulence via two simpler (but nevertheless far from trivial) problems – viz the convection and diffusion of a passive scalar field and of a passive vector field by turbulence of known statistical properties. Particular emphasis is given to the method of successive averaging (a simplified version of the renormalization-group technique) which seems to me to have considerable potential. The difficulty of extending this method to the dynamical problem is discussed. In a final section (§ 6) I have allowed myself the luxury of discussing a somewhat esoteric topic - the problem of inviscid invariants and their relationship with the topological structure of a complex vorticity field. The helicity invariant is the prototype; it is identifiable with the Hopf invariant (1931) and it may be obtained from appropriate manipulation of Noether's theorem (Moreau 1977). A suggestion is made concerning possible measurement of this fundamental measure of ‘lack of reflexional symmetry’ in a turbulent flow.

Bibliography

Moffatt, H. K. (1981). Some developments in the theory of turbulence. Journal of Fluid Mechanics, 106, 27–47.

Authors 1
  1. H. K. Moffatt (first)
References 88 Referenced 148
  1. Howells, I. D. 1960 An approximate equation for the spectrum of a conserved scalar quantity in a turbulent fluid.J. Fluid Mech. 9,104. (10.1017/S0022112060000943)
  2. Deardorff, J. W. 1970 A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers.J. Fluid Mech. 41,453. (10.1017/S0022112070000691)
  3. Batchelor, G. K. 1950 On the spontaneous magnetic field in a conducting liquid in turbulent motion.Proc. Roy. Soc. A 201,405. (10.1098/rspa.1950.0069)
  4. Saffman, P. G. 1967 The large-scale structure of homogeneous turbulence.J. Fluid Mech. 27,581. (10.1017/S0022112067000552)
  5. Moreau, J. J. 1977 Sur les intégrales premières de la dynamique d'un fluid parfait barotrope et le théorème de Helmholtz-Kelvin. Séminaire d'Analyse Convexe, Montpellier, Exposé no. 7.
  6. Edwards, S. F. 1967 Statistical mechanics with topological constraints: I.Proc. Phys. Soc. 91,513. (10.1088/0370-1328/91/3/301)
  7. Gurvich, A. S. 1960 Experimental investigation of frequency spectra of the vertical wind velocity in the atmospheric surface layer.Dokl. Akad. Nauk S.S.S.R. 132,806.
  8. Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers.Dokl. Akad. Nauk S.S.S.R. 30,299.
  9. Frisch, U. , Sulem, P.-L. & Nelkin, M. 1978 A simple dynamical model of intermittent fully developed turbulence.J. Fluid Mech. 87,719. (10.1017/S0022112078001846)
  10. Kraichnan, R. H. 1976a Diffusion of weak magnetic fields by isotropic turbulence.J. Fluid Mech. 75,657. (10.1017/S002211207600044X)
  11. Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics , vol. II.Massachusetts Institute of Technology Press.
  12. Fermigier, M. 1980 Mésures sur les écoulements laminaires et turbulents par marquage optique. Thèse du 3me cycle, Université Paris VI.
  13. Kolmogorov, A. N. 1949 On the disintegration of drops in turbulent flow.Dokl. Akad. Nauk S.S.S.R. 66,825.
  14. Frisch, U. , Pouquet, A. , Léorat, J. & Mázure, A. 1975 Possibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulence.J. Fluid Mech. 68,769. (10.1017/S002211207500122X)
  15. Britter, R. E. , Hunt, J. C. R. & Mumford, J. C. 1979 The distortion of turbulence by a circular cylinder.J. Fluid Mech. 92,269. (10.1017/S0022112079000628)
  16. Elsasser, W. M. 1946 Induction effects in terrestrial magnetism, I. Theory.Phys. Rev. 69,106. (10.1103/PhysRev.69.106)
  17. Taylor, G. I. 1935 Statistical theory of turbulence. Parts 1-4.Proc. Roy. Soc. A 151,421. (10.1098/rspa.1935.0158)
  18. Saffman, P. G. 1960 On the effect of molecular diffusivity in turbulent diffusion.J. Fluid Mech. 8,273. (10.1017/S0022112060000591)
  19. Moffatt, H. K. 1967b On the suppression of turbulence by a uniform magnetic field.J. Fluid Mech. 28,571. (10.1017/S0022112067002307)
  20. Townsend, A. A. 1970 Entrainment and the structure of turbulent flow.J. Fluid Mech. 41,13. (10.1017/S0022112070000514)
  21. Ibbetson, A. & Tritton, D. J. 1975 Experiments on turbulence in a rotating fluid.J. Fluid Mech. 68,639. (10.1017/S0022112075001164)
  22. Deissler, R. G. 1961 Effects of inhomogeneity and of shear flow in weak turbulent fields.Phys. Fluids 4,1187. (10.1063/1.1706194)
  23. Kraichnan, R. H. 1973 Helical turbulence and absolute equilibrium.J. Fluid Mech. 59,745. (10.1017/S0022112073001837)
  24. Edwards, S. F. 1964 The statistical dynamics of homogeneous turbulence.J. Fluid Mech. 18,239. (10.1017/S0022112064000180)
  25. Kelvin, Lord (then W. Thomson) 1868 Trans. Roy. Soc. Edin. 25,217. (10.1017/S0080456800028179)
  26. Landahl, M. T. 1967 A wave-guide model for turbulent shear flow.J. Fluid Mech. 29,441. (10.1017/S0022112067000953)
  27. Kraichnan, R. H. 1976b Diffusion of passive-scalar and magnetic fields by helical turbulence.J. Fluid Mech. 77,753. (10.1017/S0022112076002875)
  28. Fournier, J. D. 1977 Quelques méthodes systématiques de développement en turbulence homogène. Thèse de 3ème cycle, Université de Nice.
  29. Novikov, E. A. 1961 Energy spectrum of turbulent flow of an incompressible fluid.Dokl. Akad. Nauk S.S.S.R. 139,331.
  30. Pouquet, A. , Frisch, U. & Leaorat, J. 1976 Strong MHD turbulence and the nonlinear dynamo effect.J. Fluid Mech. 77,321. (10.1017/S0022112076002140)
  31. Hunt, J. C. R. 1978 A review of the theory of rapidly distorted turbulent flows and its applications.Fluid Dynamics Trans. 9,121.
  32. Brissaud, A. , Frisch, U. , Léorat, J. , Mazure, A. , Pouquet, A. , Sadourny, R. & Sulem, P. L. 1973 Catastrophe énergétique et nature de la turbulence.Ann. Geophys. 29,539.
  33. Mandelbrot, B. B. 1974 Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier.J. Fluid Mech. 62,331. (10.1017/S0022112074000711)
  34. Edwards, S. F. 1967 Statistical mechanics with topological constraints: II.J. Phys. A (Proc. Phys. Soc.) ser. 2,1,15. (10.1088/0305-4470/1/1/303)
  35. Snyder, W. H. & Lumley, J. L. 1971 Some measurements of particle velocity autocorrelation functions in turbulent flow.J. Fluid Mech. 48,41. (10.1017/S0022112071001460)
  36. Whitehead, J. H. C. 1947 An expression of Hopf's invariant in an integral.Proc. Nat. Acacad. Sci. USA 33,117. (10.1073/pnas.33.5.117)
  37. Roberts, P. H. & Stix, M. 1971 The turbulent dynamo: a translation of a series of papers by F. Krause, K.-H. Rädler & M. Steenbeck . NCAR, Boulder, Colorado, Tech. Note 60.
  38. Moreau, J. J. 1961 Constantes d'un îlot tourbillonnaire en fluid parfait barotrope.C. R. Akad. Sci. Paris 252,2810.
  39. Steenbeck, M. & Krause, F. 1966 The generation of stellar and planetary magnetic fields by turbulent dynamo action.Z. Naturforsch. 21a,1285. (10.1515/zna-1966-0813)
  40. Hunt, J. C. R. & Graham, J. M. R. 1977 Free-stream turbulence near plane boundaries.J. Fluid Mech. 84,209. (10.1017/S0022112078000130)
  41. Pouquet, A. & Patterson, G. S. 1978 Numerical simulation of helical magnetohydrodynamic turbulence.J. Fluid Mech. 85,305. (10.1017/S0022112078000658)
  42. Townsend, A. A. 1956 The Structure of Turbulent Shear Flow .Cambridge University Press.
  43. Townsend, A. A. 1976 The Structure of Turbulent Shear Flow ,2nd edn. Cambridge University Press.
  44. Steenbeck, M. , Krause, F. & Rädler, K.-H. 1966 A calculation of mean electromotive force in an electrically conducting fluid in turbulent motion, under the influence of Coriolis forces. Z. Naturforsch . 21a,369.
  45. Mandelbrot, B. B. 1975 On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars.J. Fluid Mech. 72,401. (10.1017/S0022112075003047)
  46. Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence .Cambridge University Press.
  47. Taylor, G. I. 1921 Diffusion by continuous movements.Proc. Lond. Math. Soc. A 20,196. (10.1112/plms/s2-20.1.196)
  48. Woltjer, L. 1956 Astrophysical evidence for strong magnetic fields.Ann. New York Acad. Sci. 257,76. (10.1111/j.1749-6632.1975.tb36078.x)
  49. Alemany, A. , Moreau, R. , Sulem, P.-L. & Frisch, U. 1979 Influence of an external magnetic field on homogeneous MHD turbulence.J. de Mécanique 18,277.
  50. Kraichnan, R. H. 1965 Lagrangian-history closure approximation for turbulence.Phys. Fluids 8,575. (10.1063/1.1761271)
  51. Pearson, J. R. A. 1959 The effect of uniform distortion on weak homogeneous turbulence.J. Fluid Mech. 5,274. (10.1017/S0022112059000192)
  52. Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids .Cambridge University Press.
  53. André, J. D. & Lesieur, M. 1977 Evolution of high Reynolds number isotropic three-dimensional turbulence; influence of helicity.J. Fluid Mech. 81,187. (10.1017/S0022112077001979)
  54. Krause, F. & Raudler, K.-H. 1981 Mean Field Magnetohydrodynamics and Dynamo Theory .Pergamon. (10.1515/9783112729694)
  55. Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number.J. Fluid Mech. 13,82. (10.1017/S0022112062000518)
  56. Scheffer, V. 1976 Turbulence and Hausdorff dimension. In Turbulence and Navier-Stokes Equation . Lecture Notes in Mathematics, no. 565 (ed. A. Dold & B. Eckmann ).Springer. (10.1007/BFb0091455)
  57. Pouquet, A. , Fournier, J. D. & Sulem, P.-L. 1978 Is helicity relevant for large scale steady state three-dimensional turbulence?J. Physique Lettres 39,199. (10.1051/jphyslet:019780039013019900)
  58. Oboukhov, A. M. 1962 Some specific features of atmospheric turbulence.J. Fluid Mech. 13,77. (10.1017/S0022112062000506)
  59. Patterson, G. S. & Corrsin, S. 1966 Computer experiments on a random walk with both Eulerian and Lagrangian statistics. In Dynamics of Fluids and Plasmas (ed. S. I. Pai et al.),p.275.Academic.
  60. Moffatt, H. K. 1969 The degree of knottedness of tangled vortex lines.J. Fluid Mech. 35,117. (10.1017/S0022112069000991)
  61. Vologodskii, A. V. , Lukashin, A. V. , Frank-Kamentskii, M. D. & Anshelevich, V. V. 1974 The knot problem in statistical mechanics of polymer chains.Sov. Phys. J. Exp. Theor. Phys. 39,1059.
  62. Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics .Interscience.
  63. Kraichnan, R. H. 1977 Eulerian and Lagrangian renormalization in turbulence theory.J. Fluid Mech. 83,349. (10.1017/S0022112077001232)
  64. Grant, H. L. , Stewart, R. W. & Moilliet, A. 1962 Turbulence spectra from a tidal channel.J. Fluid Mech. 12,241. (10.1017/S002211206200018X)
  65. Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics , vol. I.Massachusetts Institute of Technology Press.
  66. Batchelor, G. K. 1959 Small scale variation of convected quantities like temperature in turbulent fluid, I. General discussion and the case of small conductivity.J. Fluid Mech. 5,113. (10.1017/S002211205900009X)
  67. Nelkin, M. 1975 Scaling theory of hydrodynamic turbulence.Phys. Rev. A 11,1737. (10.1103/PhysRevA.11.1737)
  68. Parker, E. N. 1971 The generation of magnetic fields in astrophysical bodies. III. Turbulent diffusion of fields and efficient dynamos.Astrophys. J. 163,279. (10.1086/150766)
  69. Kuznetsov, E. A. & Mikhailov, A. V. 1980 On topological meaning of canonical Clebsch variables.Phys. Lett. 77A,37. (10.1016/0375-9601(80)90627-1)
  70. Kraichnan, R. H. 1959 The structure of isotropic turbulence at very high Reynolds numbers.J. Fluid Mech. 5,497. (10.1017/S0022112059000362)
  71. Rose, H. A. & Sulem, P. L. 1978 Fully developed turbulence and statistical mechanics.J. de Phys. 39,441. (10.1051/jphys:01978003905044100)
  72. Saffman, P. G. 1962 Some aspects of the effects of the molecular diffusivity in turbulent diffusion. In Mécanique de la Turbulence .Ed. du CNRS, no. 108. Paris
  73. Grant, H. L. 1958 The large eddies of turbulent motion.J. Fluid Mech. 4,149. (10.1017/S0022112058000379)
  74. Parker, E. N. 1979 Cosmical Magnetic Fields .Clarendon.
  75. Hopf, H. 1931 Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche.Math. Annalen 104,637. (10.1007/BF01457962)
  76. Siggia, E. D. & Patterson, G. S. 1978 Intermittency effects in a numerical simulation of stationary three-dimensional turbulence.J. Fluid Mech. 86,567. (10.1017/S0022112078001287)
  77. Rose, H. A. 1977 Eddy diffusivity, eddy noise and subgridscale modelling.J. Fluid Mech. 81,719. (10.1017/S0022112077002316)
  78. Townsend, A. A. 1951 On the fine-scale structure of turbulence.Proc. Roy. Soc. A 208,534. (10.1098/rspa.1951.0179)
  79. Forster, D. , Nelson, D. R. & Stephen, M. J. 1977 Large distance and long-time properties of a randomly stirred fluid.Phys. Rev. A 16,732. (10.1103/PhysRevA.16.732)
  80. Moffatt, H. K. 1967a The interaction of turbulence with strong wind shear. In Atmospheric Turbulence and Radio Wave Propagation (ed. A. M. Yaglom & V. I. Tatarski ),p.139.Moscow:Nauka.
  81. Heisenberg, W. 1948 On the theory of statistical and isotropic turbulence.Proc. Roy. Soc. A 195,402. (10.1098/rspa.1948.0127)
  82. Batchelor, G. K. & Proudman, I. 1956 The large-scale structure of homogeneous turbulence.Phil. Trans. Roy. Soc. A 248,369. (10.1098/rsta.1956.0002)
  83. Batchelor, G. K. & Townsend, A. A. 1956 Turbulent diffusion. In Surveys in Mechanics (ed. G. K. Batchelor & R. M. Davies ),p.352.Cambridge University Press.
  84. Kraichnan, R. H. 1978 Negative diffusivity. Woods Hole Oceanographic Inst. Tech. Rep. WHO I-78-67,136.
  85. Phythian, R. & Curtis, W. D. 1978 The effective long-time diffusivity for a passive scalar in a Gaussian model fluid flow.J. Fluid Mech. 89,241. (10.1017/S0022112078002591)
  86. Clugreanu, G. 1959 L'Integrale de Gauss et l'Analyse des nœuds tridimensionelles.Rev. Math. Pures Appl. 4,5.
  87. Hunt, J. R. C. 1973 A theory of turbulent flow round two-dimensional bluff bodies.J. Fluid Mech. 61,625. (10.1017/S0022112073000893)
  88. Nelkin, M. 1974 Turbulence, critical phenomena and intermittency.Phys. Rev. A 9,388. (10.1103/PhysRevA.9.388)
Dates
Type When
Created 19 years, 4 months ago (April 20, 2006, 4:49 a.m.)
Deposited 2 months, 1 week ago (June 21, 2025, 1:43 a.m.)
Indexed 2 months ago (June 22, 2025, 12:02 a.m.)
Issued 44 years, 3 months ago (May 1, 1981)
Published 44 years, 3 months ago (May 1, 1981)
Published Online 19 years, 4 months ago (April 20, 2006)
Published Print 44 years, 3 months ago (May 1, 1981)
Funders 0

None

@article{Moffatt_1981, title={Some developments in the theory of turbulence}, volume={106}, ISSN={1469-7645}, url={http://dx.doi.org/10.1017/s002211208100150x}, DOI={10.1017/s002211208100150x}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Moffatt, H. K.}, year={1981}, month=may, pages={27–47} }