Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



New perspectives in turbulence

Author: Alexandre J. Chorin
Journal: Quart. Appl. Math. 56 (1998), 767-785
MSC: Primary 76F02; Secondary 76M55
DOI: https://doi.org/10.1090/qam/1668737
MathSciNet review: MR1668737
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An analysis of the mean velocity profile in the intermediate region of wall-bounded turbulence shows that the well-known von Kármán-Prandtl logarithmic law of the wall must be jettisoned in favor of a power law. An analogous analysis of the local structure of turbulence shows that the Kolmogorov-Obukhov scaling of the second and third structure functions is exact in the limit of vanishing viscosity while, in the same limit, higher-order moments fail to exist. These results rely on advanced similarity methods and on vanishing-viscosity asymptotics, and are consistent with a near-equilibrium theory of turbulence of which a new version is presented.

References [Enhancements On Off] (What's this?)

  • [1] Grigory Isaakovich Barenblatt, Scaling, self-similarity, and intermediate asymptotics, Cambridge Texts in Applied Mathematics, vol. 14, Cambridge University Press, Cambridge, 1996. With a foreword by Ya. B. Zeldovich. MR 1426127
  • [2] G. I. Barenblatt, On the scaling laws (incomplete self-similarity with respect to Reynolds number) in the developed turbulent flow in pipes, C. R. Acad. Sci. Paris, series II, 313, 3.9-3.12 (1991)
  • [3] G. I. Barenblatt, Scaling laws for fully developed turbulent shear flows. Part 1: Basic hypotheses and analysis, J. Fluid Mech. 248, 513-520 (1993)
  • [4] G. I. Barenblatt and A. J. Chorin, Small viscosity asymptotics for the inertial range of local structure and for the wall region of wall-bounded turbulence, Proc. Nat. Acad. Sciences USA 93, 6749-6752 (1996)
  • [5] G. I. Barenblatt and Alexandre J. Chorin, Scaling laws and vanishing-viscosity limits for wall-bounded shear flows and for local structure in developed turbulence, Comm. Pure Appl. Math. 50 (1997), no. 4, 381–398. MR 1438152, https://doi.org/10.1002/(SICI)1097-0312(199704)50:4<381::AID-CPA5>3.0.CO;2-6
  • [6] G. I. Barenblatt and Alexandre J. Chorin, Scaling laws and vanishing viscosity limits in turbulence theory, Recent advances in partial differential equations, Venice 1996, Proc. Sympos. Appl. Math., vol. 54, Amer. Math. Soc., Providence, RI, 1998, pp. 1–25. MR 1492690, https://doi.org/10.1090/psapm/054/1492690
  • [7] G. I. Barenblatt and A. J. Chorin, New perspectives in turbulence: scaling laws, asymptotics, and intermittency, SIAM Rev. 40 (1998), no. 2, 265–291. MR 1624102, https://doi.org/10.1137/S0036144597320047
  • [8] G. I. Barenblatt and A. J. Chorin, A near-equilibrium statistical theory of turbulence with applications, in preparation, 1998
  • [9] G. I. Barenblatt, A. J. Chorin, and V. M. Prostokishin, Scaling laws in fully developed turbulent pipe flow: Discussion of experimental data, Proc. Nat. Acad. Sciences USA 94a, 773-776 (1997)
  • [10] G. I. Barenblatt, A. J. Chorin, and V. M. Prostokishin, Scaling laws in fully developed turbulent pipe flow, Appl. Mech. Rev. 50, 413-429 (1997)
  • [11] G. I. Barenblatt, A. J. Chorin, O. H. Hald, and V. M. Prostokishin, Structure of the zero-pressure-gradient turbulent boundary layer, Proc. Nat. Acad. Sciences USA 94, 7817-7819 (1997)
  • [12] G. I. Barenblatt and V. M. Prostokishin, Scaling laws for fully developed turbulent shear flows. II. Processing of experimental data, J. Fluid Mech. 248 (1993), 521–529. MR 1209047, https://doi.org/10.1017/S0022112093000886
  • [13] Roberto Benzi, Sergio Ciliberto, Cristophe Baudet, and Gerardo Ruiz-Chavarría, On the scaling of three-dimensional homogeneous and isotropic turbulence, Phys. D 80 (1995), no. 4, 385–398. MR 1312600, https://doi.org/10.1016/0167-2789(94)00190-2
  • [14] Alexandre Joel Chorin, Lattice vortex models and turbulence theory, Wave motion: theory, modelling, and computation (Berkeley, Calif., 1986) Math. Sci. Res. Inst. Publ., vol. 7, Springer, New York, 1987, pp. 1–14. MR 920830, https://doi.org/10.1007/978-1-4613-9583-6_1
  • [15] Alexandre J. Chorin, Vorticity and turbulence, Applied Mathematical Sciences, vol. 103, Springer-Verlag, New York, 1994. MR 1281384
  • [16] Alexandre J. Chorin, Turbulence as a near-equilibrium process, Dynamical systems and probabilistic methods in partial differential equations (Berkeley, CA, 1994) Lectures in Appl. Math., vol. 31, Amer. Math. Soc., Providence, RI, 1996, pp. 235–249. MR 1363031
  • [17] A. J. Chorin, Turbulence cascades across equilibrium spectra, Phys. Rev. E 54, 2615-2619 (1996)
  • [18] H. Gould and J. Tobochnik, An Introduction to Computer Simulation Methods, Applications to Physical Systems, part 2, Addison-Wesley, Reading, MA, 1988
  • [19] P. Kailasnath, A. Migdal, K. Sreenivasan, V. Yakhot, and L. Zubair, The 5/4 Kolmogorov law and odd moments of the velocity difference in turbulence, unpublished, 1992
  • [20] Th. von Kármán, Mechanische Aehnlichkeit und Turbulenz, Nach. Ges. Wiss. Goettingen Math-Phys. Klasse, 1932, pp. 58-76
  • [21] A. N. Kolmogorov, Local structure of turbulence in incompressible fluid at a very high Reynolds number, Dokl. Acad. Sci. USSR 30, 299-302 (1941)
  • [22] R. Kupferman and A. J. Chorin, Numerical study of the Kosterlitz-Thouless phase transition, submitted for publication, 1997
  • [23] L. D. Landau and E. M. Lifshitz, Fluid mechanics, Translated from the Russian by J. B. Sykes and W. H. Reid. Course of Theoretical Physics, Vol. 6, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1959. MR 0108121
  • [24] Jonathan Miller, Statistical mechanics of Euler equations in two dimensions, Phys. Rev. Lett. 65 (1990), no. 17, 2137–2140. MR 1074119, https://doi.org/10.1103/PhysRevLett.65.2137
  • [25] A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics, Vol. 1, MIT Press, Boston, 1971
  • [26] J. Nikuradze, Gesetzmaessigkeiten der turbulenten Stroemung in glatten Rohren, VDI Forschungheft, No. 3.8, 1932
  • [27] A. M. Obukhov, Spectral energy distribution in turbulent flow, Dokl. Akad. Nauk USSR 1, 22-24 (1941)
  • [28] L. Prandtl, Zur turbulenten Stroemung in Rohren und laengs Platten, Ergeb. Aerodyn. Versuch., Series 4, Goettingen, 1932
  • [29] A. Praskovsky and S. Oncley, Measurements of the Kolmogorov constant and intermittency exponents at very high Reynolds numbers, Phys. Fluids A 7, 2778-2784 (1994)
  • [30] Hermann Schlichting, Boundary layer theory, McGraw-Hill, New York; Pergamon Press, London; Verlag G. Braun, Karlsruhe, 1955. Translated by J. Kestin. MR 0076530
  • [31] Katepalli R. Sreenivasan, On the universality of the Kolmogorov constant, Phys. Fluids 7 (1995), no. 11, 2778–2784. MR 1361356, https://doi.org/10.1063/1.868656
  • [32] H. Tennekes and J. Lumley, A First Course in Turbulence, MIT Press, Cambridge, Mass., 1990
  • [33] H. Weber and P. Minnhagen, Monte-Carlo determination of the critical temperature for the two-dimensional XY model, Phys. Rev. B 37, 5986-5989 (1988)
  • [34] M. V. Zagarola, A. J. Smits, S. A. Orszag, and V. Yakhot, Experiments in high Reynolds number turbulent pipe flow, AIAA paper 96-0654, Reno, Nevada, 1996

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 76F02, 76M55

Retrieve articles in all journals with MSC: 76F02, 76M55

Additional Information

DOI: https://doi.org/10.1090/qam/1668737
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society