Skip to main content
SpringerLink
Log in

Turbulence: An Old Challenge and New Perspectives

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

Turbulence at very large Reynolds numbers (often called developed turbulence) is widely considered to be one of the happier provinces of the turbulence realm, as it is widely thought that two of its basic results are well-established, and have a chance to enter, basically untouched, into a future complete theory of turbulence. These results are the von Kármán-Prandtl universal logarithmic law in the wall-region of wall-bounded turbulent shear flow, and the Kolmogorov-Obukhov scaling laws for the local structure of developed turbulent flow.However, doubts have been expressed over the years about the fluid mechanical assumptions that underlie these laws.

After a concise review of the problem of turbulence as a whole we will show in the present paper that the von Kármán–Prandtl universal logarithmic law is based on an assumption which,though plausible, in fact is not quite correct. We will come to the conclusion, based on theoretical considerations and on processing of experimental data, that the universal logarithmic law does not describe the real features of developed turbulent wall-bounded flow of viscous fluid; it should be jettisoned and replaced by a different law, a scaling law.

Experimental evidence for the local structure of turbulent flows is now not sufficiently well-established to allow a similarly definite conclusion. However, the application of the new approach presented here makes it very plausible that the classical, non-modified version of Kolmogorov–Obukhov ‘K-41’ laws gives an adequate description of the local features of developedturbulent flows.

Sommario.La turbolenza agli altissimi numeri di Reynolds (spesso chiamata turbolenza sviluppata) è largamente ritenuta una delle regioni felici del regno della turbolenza: si pensa infatti che due suoi risultati fondamentali siano ben assodati e che abbiano speranza di entrare senza rilevanti modifiche in una futura teoria completa della turbolenza.Questi risultati sono la legge logaritmica universale di Kármán–Prandtl per la regione di parete dei flussi turbolenti confinati e la legge di simulitudine di Kolmogorov–Obukhov per la struttura locale del flusso turbolento sviluppato. Nel corso degli anni sono stati tuttavia espressi dubbi sulle ipotesi fluidodinamiche che sottendono queste leggi. Nel presente lavoro, dopo un breve esame del problema della turbolenza nel suo insieme,dimostreremo che la legge logaritmica universale di Kármán–Prandtl è basata su un'assunzione che, per quanto plausibile, non è del tutto corretta. Giungeremo alla conclusione, basata su considerazioni teoriche esull'elaborazione di dati sperimentali, che la legge logaritmica universale non descrive le caratteristiche reali del flusso turbolento di un fluido viscoso sviluppato e confinato da una parete; essa dovrebbe essere sostituita de una legge differente, una legge di similitude. L'evidenza sperimentale per la stzuttura locale di flusso turbolento non è al momento sufficientemente assodata perpermettere una conclusione altrettanto definita. L'impiegodel nuovo approccio qui presentato, tuttavia, rende assai plausible che la classica versione non modificata della legge ‘K-41’ di Kolmogorov-Obukhov fornisca una adeguata descrizione delle caratteristiche locali del flussoturbolento sviluppato.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barenblatt, G.I., Chorin, A.J. and Prostokishin, V.M., ‘Scaling laws in turbulent pipe flow: discussion of experimental data’, Proc. Nat. Acad. Sci. USA 94 (1997) 773–776.

    Google Scholar 

  2. Barenblatt, G.I., Chorin, A.J. and Prostokishin, V.M., ‘Scaling laws in turbulent pipe flow’, Appl. Mech. Rev. 50(7) (1997) 413–429.

    Google Scholar 

  3. Barenblatt, G.I. and Chorin, A.J., ‘A near-equilibrium statistical theory of turbulence with energy and smoothness constraints’, 1998, (in preparation).

  4. Barenblatt, G.I., ‘Intermediate asymptotics, scaling laws and renormalization group in continuum mechanics’, Meccanica 28 (1993) 177–183.

    Google Scholar 

  5. Barenblatt, G.I., Scaling, Self-Similarity and Intermediate Asymptotics, Cambridge University Press, 1996.

  6. Barenblatt, G.I. and Goldenfeld, N., ‘Does fully developed turbulence exist? Reynolds number independence vs. asymptotic covariance’, Phys. Fluids A (1995) 3078–3082.

  7. Barenblatt, G.I., ‘Scaling laws for fully developed shear flows. Part 1: Basic hypotheses and analysis’, J. Fluid Mech. 248 (1993) 513–520.

    Google Scholar 

  8. Barenblatt, G.I. and Prostokishin, V.M., ‘Scaling laws for fully developed shear flows. Part 2: Processing of experimental data’, J. Fluid Mech. 248 (1993) 521–529.

    Google Scholar 

  9. Barenblatt, G.I. and Chorin, A.J., ‘Scaling laws and vanishing viscosity limits for wall-bounded shear flows and for local structure in developed turbulence’, Comm. Pure Appl. Math. 50 (1997) 381–398.

    Google Scholar 

  10. Barenblatt, G.I. and Chorin, A.J., ‘Small viscosity asymptotics for the inertial range of local structure and for the wall region of wall-bounded turbulent shear flow’, Proc. Nat. Acad. Sci. USA 93 (1996) 6749–6752.

    Google Scholar 

  11. Batchelor, G.K., ‘Kolmogoroff's theory of locally isotropic turbulence’, Proc. Cambr. Phil. Soc. 43(4) (1947) 533–559.

    Google Scholar 

  12. Batchelor, G.K., The Theory of Homogeneous Turbulence, Cambridge University Press, 1953.

  13. Bernard, P., Thomas, J. and Handler, R., ‘Vortex dynamics and the production of Reynolds stress’, J. Fluid Mech. 253 (1993) 385–419.

    Google Scholar 

  14. Boussinesq, J., ‘Essai sur la théorie des eaux courantes’, Mém. prés. par div. savants à l'Académie Sci. Paris. 23(1) (1877) 1–680.

    Google Scholar 

  15. Burgers, J.M., ‘Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion’, Kon. Ned. Acad. Wet. Verh. 17 (1939) 1–53.

    Google Scholar 

  16. Cebeci, T. and Smith, A.M.O., Analysis of Turbulent Boundary Layers, Academic Press, 1974.

  17. Chorin, A.J., ‘Theories of turbulence’, In: P. Bernard and T. Ratiu, (eds.), Berkeley Turbulence Seminar, Springer, New York, 1977.

    Google Scholar 

  18. Chorin, A.J., ‘Equilibrium statistics of a vortex filament with applications,’ Comm. Math. Phys. 141 (1991) 619–631.

    Google Scholar 

  19. Chorin, A.J., ‘Turbulence as a near-equilibrium process’, Lectures in Appl. Math. 31 (1996) 235–248.

    Google Scholar 

  20. Chorin, A.J., ‘Turbulence cascades across equilibrium spectra’, Phys. Rev. E 54 (1996) 2616–2619.

    Google Scholar 

  21. Chorin, A.J., ‘Onsager's contribution to turbulence theory: Vortex dynamics and turbulence in ideal flow’, In: P. Hemmer and others, (eds), L. Onsager, Collected Works with Commentary, World Scientific, 1996.

  22. Chorin, A.J., Vorticity and Turbulence, Springer, New York, 1994.

    Google Scholar 

  23. Constantin, P., ‘Navier-Stokes equations and area of interfaces’, Commun. Math. Phys. 129 (1990) 241–266.

    Google Scholar 

  24. Constantin, P., ‘Geometric and analytic studies in turbulence’, In: L. Sirovich, (ed.), Trends and Perspectives in Applied Mathematics, Springer, New York, 1991, pp. 21–54.

    Google Scholar 

  25. Constantin, P., ‘Geometric statistics in turbulence’, SIAM Rev. 36 (1994) 73–98.

    Google Scholar 

  26. Coles, D., ‘The law of the wall in the turbulent boundary layer’, J. Fluid Mech. 1 (1956) 191–226.

    Google Scholar 

  27. Corrsin, S., ‘On the spectrum of isotropic temeprature fluctuations in an isotropic turbulence’, J. Appl. Phys. 22(4) (1951) 469–473.

    Google Scholar 

  28. Ferrari, C., Lo stato attuale della teoria della turbolenza. Conferenza tenuta il 26 Maggio 1936, R. Università e R. Scuola di Ingegneria di Torino.

  29. Ferrari, C., Sui moti fluidi turbolenti. Conferenza tenuta il 4 giugno 1936 al 1 Congresso di Matematica applicata a Roma.

  30. Ferrari, C., ‘Sui moti fluidi turbolenti’, In: M. Picone, G. Krall, C. Ferrari, Questioni di Matematica Applicata, Zanichelli Editore, Bologna, 1939, pp. 133–152.

    Google Scholar 

  31. Frisch, U., Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1995.

    Google Scholar 

  32. Goldenfeld, N., Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, Reading, Mass, 1992.

    Google Scholar 

  33. Heisenberg,W., ‘Über Stabilität undTurbulenz von Flussigkeitsströmen’, Ann. Phys. 74(4) (1924) 577–627.

    Google Scholar 

  34. Hites, M., Scaling of the High-Reynolds Number Turbulent Boundary Layer, PhD Thesis, Illinois Institute of Technology, May 1997.

  35. Hopf, E., ‘A mathematical example displaying the features of turbulence’, Comm. Pure Appl. Math. 1 (1948) 303–322.

    Google Scholar 

  36. Hunt, J.C.R., Phillips, O.M. and Williams, D., (eds), ‘Turbulence and stochastic processes: Kolmogorov's ideas 50 years on’, Proc. Roy. Soc. London A434(1890) (1991) 1–240.

  37. Kailasnath, P., Reynolds Number Effects and the Momentum Flux in Turbulent Boundary Layers, PhD Thesis, Yale University, 1993.

  38. Keller, L.V. and Friedman, A.A., ‘Differentialgleichungen für die turbulente Bewegung einer kompressiblen Flüssigkeit’, In: Proc. 1st International Congress Appl. Mech. Delft, 1924, pp. 395–405.

  39. Kline, S.J., Reynolds, W.C., Schraub, F.A. and Rundstadler, P.W., ‘The structure of turbulent boundary layers’, J. Fluid Mech. 30(4) (1967) 741–745.

    Google Scholar 

  40. Kolmogorov, A.N., ‘Local structure of turbulence in incompressible fluid at a very high Reynolds number’, Dokl. Acad. Sci. USSR 30 (1941) 299–302.

    Google Scholar 

  41. Kolmogorov, A.N., ‘A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at a high Reynolds number’, J. Fluid Mech. 12 (1962) 82–85.

    Google Scholar 

  42. Landau, L.D. and Lifshitz, E.M., Fluid Mechanics, 2nd ed., Pergamon Press, New York, 1959, ibid, 1987.

    Google Scholar 

  43. Landau, L.D., ‘The origin of turbulence’, Doklady AN SSSR 44(8) (1944) 339–342.

    Google Scholar 

  44. Lighthill, Sir James, ‘Typhoons, hurricanes and fluid mechanics’, J. Engng. Math., 1998 (in press).

  45. Lin, C.C., ‘On the stability of two-dimensional parallel flows. Parts I-III’, Quart. Appl. Math. 3(2) (1945) 117–142; (3) 218–234; (4) 277–301.

    Google Scholar 

  46. Lorenz, E.N., ‘Deterministic non-periodic flow’, J. Atmos. Sci. 20 (1963) 130–141.

    Google Scholar 

  47. McComb, W.D., Physical Theories of Turbulence, Oxford University Press, Oxford, 1991.

    Google Scholar 

  48. Monin, A.S. and Yaglom, A.M., Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 1, The MIT Press, Boston, 1971, vol. 2, The MIT Press, Boston, 1975.

    Google Scholar 

  49. Nagib, H. and Hites, M., High Reynolds Number Boundary Layer Measurements in the NDF, AIAA paper 95–0786, Reno, Nevada 1995.

  50. Nikuradze, J., ‘Gesetzmässigkeiten der turbulenten Strömung in glatten Rohren’, VDI Forschungheft, no. 356 (1932).

  51. Obukhov, A.M., ‘Some specific features of atmospheric turbulence’, J. Fluid Mech. 13 (1962) 77–81.

    Google Scholar 

  52. Obukhov, A.M., ‘Structure of the temperature field in a turbulent flow’, Izv. Akad. Nauk. SSSR, Ser. Geogr. i Geofiz. 13(1) (1949) 58–69.

    Google Scholar 

  53. Obukhov, A.M., ‘Spectral energy distribution in turbulent flow’, Dokl. Acad. Sci. USSR 32 (1941) 22–24.

    Google Scholar 

  54. Onsager, L., ‘Statistical hydrodynamics’, Nuovo Cimento 6(9) suppl. 2: Convegno Internazionale di Meccanica Statistical, 1949, pp. 279–287.

    Google Scholar 

  55. Prandtl, L., ‘Zur turbulenten Strömung in Röhren und langs Platten’, Ergebn. Aerodyn. Versuchanstalt, Göttingen 4 (1932) 18–29.

    Google Scholar 

  56. Praskovsky, A. and Oncley, S., ‘Measurements of the Kolmogorov constant and intermittency exponents at very high Reynolds numbers’, Phys. Fluids A6 (1994) 2886–2891.

    Google Scholar 

  57. Reynolds, O., ‘On the dynamical theory of incompressible viscous fluids and the determination of the criteria’, Phil. Trans. Roy. Soc. London 186 (1894) 123–161.

    Google Scholar 

  58. Reynolds, O., ‘An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels’, Phil. Trans. Roy. Soc. London 174 (1883) 935–892.

    Google Scholar 

  59. Schlichting, H., ‘Zur Entstehung der Turbulenz bei der Plattenströrung’, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Klasse (1933) 160–198.

  60. Schubauer, G.B. and Scramstad, H.K., ‘Laminar boundary layer oscillations and stability of laminar flow’, J. Aeronaut. Sci. 14(2) (1947) 69–78.

    Google Scholar 

  61. Spurk, J.H., Fluid Mechanics, Springer-Verlag, New York, 1997.

    Google Scholar 

  62. Sreenivasan, K.R., ‘A unified view of the origin and morphology of turbulent boundary layer structure’, In: H.W. Liepmann and R. Narasimha, (eds), IUTAM Symposium, Bangalore, 1987.

  63. Sreenivasan, K.R., ‘The turbulent boundary layer’, In: M. Gad-el-Hak, (ed.), Frontiers in Experimental Fluid Mechanics, Springer, 1989, pp. 159–209.

  64. Sreenivasan, K.R., ‘On the universality of the Kolmogorov constant’, Phys. Fluids A7 (1995) 2778–2784.

    Google Scholar 

  65. Taylor, G.I., ‘Diffusion by continuous movements’, Proc. London Math. Soc. 20(2) (1921) 196–211.

    Google Scholar 

  66. Taylor, G.I., ‘Statistical theory of turbulence, I-IV’, Proc. Roy. Soc. London A151(874) (1935) 421–478.

    Google Scholar 

  67. Tennekes, H. and Lumley, J., A First Course in Turbulence, MIT Press, Cambridge, Mass, 1990.

    Google Scholar 

  68. Tollmien, W., ‘Über die Entstehung der Turbulenz’, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Klasse (1929) pp. 21–44.

  69. vonKármán, Th. and Howarth, L., ‘On the statistical theory of isotropic turbulence’, Proc. Roy. Soc. London A164(917) (1938) 192–215.

    Google Scholar 

  70. von Kármán, Th., ‘Mechanische Ähnlichkeit und Turbulenz’, Nach. Ges. Wiss. Göttingen, Math.-Phys. Klasse (1930) (58–76.

  71. Zagarola, M.V., Smits, A.J., Orszag, S.A. and Yakhot, V., Experiments in High Reynolds Number Turbulent Pipe Flow, AIAA paper 96–0654, Reno, Nevada, 1996.

  72. Zagarola, M.V., Mean Flow Scaling in Turbulent Pipe Flow, PhD Thesis, Princeton Unversity, 1996.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Lawrence Berkeley Laboratory, University of California, Berkeley, CA, 94720–3840, U.S.A.

    G.I. Barenblatt & A.J. Chorin

Authors
  1. G.I. Barenblatt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A.J. Chorin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barenblatt, G., Chorin, A. Turbulence: An Old Challenge and New Perspectives. Meccanica 33, 445–468 (1998). https://doi.org/10.1023/A:1004312409376

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1004312409376

Search

Navigation