Small-scale intermittency in randomly stirred fluids

Malay K. Nandy

doi:10.1103/PhysRevE.48.1015

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Small-scale intermittency in randomly stirred fluids

Malay K. Nandy

Department of Physics, Indian Institute of Technology, Kanpur 208016, India

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Phys. Rev. E 48, 1015 – Published 1 August, 1993

DOI: https://doi.org/10.1103/PhysRevE.48.1015

Abstract

Starting with a self-consistent renormalized diagrammatic perturbation theory as obtained from the Navier-Stokes equation with a random stirring force, we find that vertex renormalization, as it becomes increasingly important at smaller scales, introduces intermittency corrections through a renormalization of the velocity field. Intermittency is found to appear through an additional (‘‘collision’’) mechanism of formation and subsequent breakdown of large scales. The result supports the refined local-scaling laws in a modified sense. The third hypothesis of log-normality is found to be untenable.

Comments & Replies

Comment on ‘‘Small-scale intermittency in randomly stirred fluids’’

E. V. Teodorovich

Phys. Rev. E 52, 2118 (1995)

Reply to ‘‘Comment on ‘Small-scale intermittency in randomly stirred fluids’ ’’

Malay K. Nandy

Phys. Rev. E 53, 1998 (1996)

References (23)

See A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT, Cambridge, MA, 1971), Vol. 2.
The first experiment that invalidated this universality was performed by Batchelor and Townsend. For an overview of this and later experiments, see A. S. Monin and A. M. Yaglom, Ref. [1], Chap. 25.
A. N. Kolmogorov, J. Fluid Mech. 13, 82 (1962); A. M. Obukhov, ibid. 13, 77 (1962); see also A. S. Monin and A. M. Yaglom, Ref. [1], Chap. 25.
R. H. Kraichnan, J. Fluid Mech. 62, 305 (1974).
G. Stolovitzky, P. Kailasnath and K. R. Sreenivasan, Phys. Rev. Lett. 69, 1178 (1992).
B. B. Mandelbrot, J. Fluid Mech. 62, 331 (1974).
B. B. Mandelbrot, Statistical Models and Turbulence, edited by M. Rosenblatt and C. Van Atta (Springer, Berlin, 1972), p. 333.
F. Anselmet, Y. Gagne, E. J. Hopfinger and R. A. Antonia, J. Fluid Mech. 140, 63 (1984).
C. Meneveau and K. R Sreenivasan, J. Fluid Mech. 224, 429 (1991).
D. Forster, D. R. Nelson and M. J. Stephen, Phys. Rev. Lett. 36, 867 (1976); Phys. Rev. A 16, 732 (1977).
C. DeDominicis and P. C. Martin, Phys. Rev. A 19, 419 (1979).
V. Yakhot and S. A. Orszag, Phys. Rev. Lett. 57, 1722 (1986); J. Sci. Comput. 1, 3 (1986).
H. W. Wyld, Ann. Phys. (NY) 14, 143 (1961).
Further, the convergence of the diagrams, in a self-consistent solution, was proved by V. I. Belinicher and V. S. L'vov, Zh. Eksp. Teor. Fiz. 93, 533 (1987) [Sov. Phys. JETP 66, 303 (1987)].
This remark was also made by W. P. Dannevik, V. Yakhot and S. A. Orszag, Phys. Fluids 30, 2021 (1987).
For a different result, see V. I. Belinicher and V. S. L'vov, Ref. [14].
E(k) is defined by ${tint}_{0}^{\inf}$ E(k)dk= 1/2 < $v^{2}$ ( x vec ) > .
S. A. Orszag, in Fluid Dynamics, Proceedings of the 1973 Les Houches Summer School, edited by R. Balian and P. L. Peube (Gordon and Breach, New York, 1977).
R. H. Kraichnan, J. Fluid Mech. 5, 497 (1959); D. C. Leslie, Developments in the Theory of Turbulence (Clarendon, Oxford, 1973).
J. K. Bhattacharjee (private communication).
For example, if we work with the unrenormalized velocity field u, $λ^{2}$ (k) is absent in Eq. (9), while T(k,t) has the dimension of $k^{3}$ $λ^{2}$ ( k) $E^{2}$ (k)/ ω (k). This leads to E(k) app $k^{1 - 2 y / 3 - 2 σ / 3}$ and Π (k) app $k^{4 - y}$ , while Eq. (7) remains unchanged. Setting y=4 leads to spatially homogeneous flux and results in the beta model with the identification 2 σ =d-D, D being the (homogeneous) fractal dimension. See U. Frisch, P. L. Sulem and M. Nelkin, J. Fluid Mech. 87, 719 (1978), for the beta model.
This localness can be illustrated by dividing the whole d-dimensional space of size L into cubes each of size r. In this case $K_{d}$ =1. Then x vec will denote the center of a cube and alpha is allowed to take different values in different cubes, consistent with the energy conservation. This picture is the same as that used in the multifractal models. See Ref. [9], and references therein.
E. Kuznetsov, A. C. Newell and V. E. Zakharov, Phys. Rev. Lett. 67, 3243 (1991).

[c1] See A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT, Cambridge, MA, 1971), Vol. 2.

[c2] The first experiment that invalidated this universality was performed by Batchelor and Townsend. For an overview of this and later experiments, see A. S. Monin and A. M. Yaglom, Ref. [1], Chap. 25.

[c3] A. N. Kolmogorov, J. Fluid Mech. 13, 82 (1962); A. M. Obukhov, ibid. 13, 77 (1962); see also A. S. Monin and A. M. Yaglom, Ref. [1], Chap. 25.

[c4] R. H. Kraichnan, J. Fluid Mech. 62, 305 (1974).

[c5] G. Stolovitzky, P. Kailasnath and K. R. Sreenivasan, Phys. Rev. Lett. 69, 1178 (1992).

[c6] B. B. Mandelbrot, J. Fluid Mech. 62, 331 (1974).

[c7] B. B. Mandelbrot, Statistical Models and Turbulence, edited by M. Rosenblatt and C. Van Atta (Springer, Berlin, 1972), p. 333.

[c8] F. Anselmet, Y. Gagne, E. J. Hopfinger and R. A. Antonia, J. Fluid Mech. 140, 63 (1984).

[c9] C. Meneveau and K. R Sreenivasan, J. Fluid Mech. 224, 429 (1991).

[c10] D. Forster, D. R. Nelson and M. J. Stephen, Phys. Rev. Lett. 36, 867 (1976); Phys. Rev. A 16, 732 (1977).

[c11] C. DeDominicis and P. C. Martin, Phys. Rev. A 19, 419 (1979).

[c12] V. Yakhot and S. A. Orszag, Phys. Rev. Lett. 57, 1722 (1986); J. Sci. Comput. 1, 3 (1986).

[c13] H. W. Wyld, Ann. Phys. (NY) 14, 143 (1961).

[c14] Further, the convergence of the diagrams, in a self-consistent solution, was proved by V. I. Belinicher and V. S. L'vov, Zh. Eksp. Teor. Fiz. 93, 533 (1987) [Sov. Phys. JETP 66, 303 (1987)].

[c15] This remark was also made by W. P. Dannevik, V. Yakhot and S. A. Orszag, Phys. Fluids 30, 2021 (1987).

[c16] For a different result, see V. I. Belinicher and V. S. L'vov, Ref. [14].

[c17] E(k) is defined by ${tint}_{0}^{\inf}$ E(k)dk= 1/2 < $v^{2}$ ( x vec ) > .

[c18] S. A. Orszag, in Fluid Dynamics, Proceedings of the 1973 Les Houches Summer School, edited by R. Balian and P. L. Peube (Gordon and Breach, New York, 1977).

[c19] R. H. Kraichnan, J. Fluid Mech. 5, 497 (1959); D. C. Leslie, Developments in the Theory of Turbulence (Clarendon, Oxford, 1973).

[c20] J. K. Bhattacharjee (private communication).

[c21] For example, if we work with the unrenormalized velocity field u, $λ^{2}$ (k) is absent in Eq. (9), while T(k,t) has the dimension of $k^{3}$ $λ^{2}$ ( k) $E^{2}$ (k)/ ω (k). This leads to E(k) app $k^{1 - 2 y / 3 - 2 σ / 3}$ and Π (k) app $k^{4 - y}$ , while Eq. (7) remains unchanged. Setting y=4 leads to spatially homogeneous flux and results in the beta model with the identification 2 σ =d-D, D being the (homogeneous) fractal dimension. See U. Frisch, P. L. Sulem and M. Nelkin, J. Fluid Mech. 87, 719 (1978), for the beta model.

[c22] This localness can be illustrated by dividing the whole d-dimensional space of size L into cubes each of size r. In this case $K_{d}$ =1. Then x vec will denote the center of a cube and alpha is allowed to take different values in different cubes, consistent with the energy conservation. This picture is the same as that used in the multifractal models. See Ref. [9], and references therein.

[c23] E. Kuznetsov, A. C. Newell and V. E. Zakharov, Phys. Rev. Lett. 67, 3243 (1991).

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Small-scale intermittency in randomly stirred fluids

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