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On the elimination of the sweeping interactions from theories of hydrodynamic turbulence. (English) Zbl 1115.76034
Summary: We revisit the claim that the Eulerian and quasi-Lagrangian time correlation tensors are equal. This statement allows us to transform the results of an MSR [P. Martin, E. Siggia and H.Rose, Phys. Rev. A 8, 423–437 (1973)] quasi-Lagrangian statistical theory of hydrodynamic turbulence back to the Eulerian representation. We define a hierarchy of homogeneity symmetries between incremental homogeneity and global homogeneity. It is shown that both the elimination of the sweeping interactions and the derivation of the 4/5-law require a homogeneity assumption stronger than incremental homogeneity but weaker than global homogeneity. The quasi-Lagrangian transformation, on the other hand, requires an even stronger homogeneity assumption which is many-time rather than one-time but still weaker than many-time global homogeneity. We argue that it is possible to relax this stronger assumption and still preserve the conclusions derived from theoretical work based on the quasi-Lagrangian transformation.

MSC:
76F05 Isotropic turbulence; homogeneous turbulence
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[1] Richardson, L., Weather prediction by numerical process, (1922), Cambridge University Press Cambridge · JFM 48.0629.07
[2] Kolmogorov, A., The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. akad. nauk. SSSR, 30, 301-305, (1941), English translation published in volume 434 of Proc. R. Soc. Lond. Ser. A · JFM 67.0850.06
[3] Kolmogorov, A., Dissipation of energy in the locally isotropic turbulence, Dokl. akad. nauk. SSSR, 32, 16-18, (1941), English translation published in volume 434 of Proc. R. Soc. Lond. Ser. A · Zbl 0063.03292
[4] Batchelor, G., Kolmogorov’s theory of locally isotropic turbulence, Proc. Cambridge philos. soc., 43, 533-559, (1947) · Zbl 0029.28405
[5] Grant, H.; Stewart, R.; Moilliet, A., Turbulence spectra from a tidal channel, J. fluid mech., 12, 241-263, (1962) · Zbl 0101.43101
[6] Gibson, M., Spectra of turbulence in a round jet, J. fluid mech., 15, 161-173, (1962) · Zbl 0109.43902
[7] Frisch, U., Turbulence: the legacy of A.N. Kolmogorov, (1995), Cambridge University Press Cambridge
[8] Sreenivasan, K.; Antonia, R., The phenomenology of small-scale turbulence, Annu. rev. fluid mech., 29, 435-472, (1997)
[9] Kolmogorov, A., A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. fluid mech., 13, 237, (1962) · Zbl 0112.42003
[10] Oboukhov, A., Some specific features of the atmospheric turbulence, J. fluid mech., 13, 77-81, (1962) · Zbl 0112.42002
[11] L’vov, V.; Procaccia, I., Hydrodynamic turbulence: A 19th century problem with a challenge for the 21st century, () · Zbl 0886.76039
[12] Frisch, U.; Bec, J.; Aurell, E., Locally homogeneous turbulence: Is it a consistent framework?, Phys. fluids, 17, 081706, (2005) · Zbl 1187.76166
[13] L’vov, V.; Procaccia, I., Exact resummations in the theory of hydrodynamic turbulence. I. the ball of locality and normal scaling, Phys. rev. E, 52, 3840-3857, (1995)
[14] Nelkin, M., Universality and scaling in fully developed turbulence, Adv. phys., 43, 143-181, (1994)
[15] Sreenivasan, K., Fluid turbulence, Rev. modern phys., 71, S383-S395, (1999)
[16] Taylor, G., Statistical theory of turbulence. parts 1-4, Proc. roy. soc. A, 151, 421-478, (1935) · JFM 61.0926.02
[17] Taylor, G., Statistical theory of turbulence. part 5, Proc. roy. soc. A, 156, 307-317, (1936) · JFM 62.1587.02
[18] Karman, T.; Howarth, L., On the statistical theory of isotropic turbulence, Proc. roy. soc. A, 164, 192-215, (1938) · JFM 64.1453.03
[19] Robertson, H., The invariant theory of isotropic turbulence, Proc. Cambridge philos. soc., 36, 209-223, (1940) · Zbl 0023.42604
[20] Batchelor, G., The theory of homogeneous turbulence, (1953), Cambridge University Press Cambridge · Zbl 0053.14404
[21] Leslie, D., Developments in the theory of turbulence, (1972), Clarendon Press Oxford
[22] Kraichnan, R., Relation of fourth order to second order moments in stationary isotropic turbulence, Phys. rev., 107, 1485-1490, (1957) · Zbl 0078.17801
[23] Kraichnan, R., Irreversible statistical mechanics of incompressible hydromagnetic turbulence, Phys. rev., 109, 1407-1422, (1958)
[24] Kraichnan, R., The structure of isotropic turbulence at the very high Reynolds numbers, J. fluid mech., 5, 497-543, (1959) · Zbl 0093.41202
[25] Kraichnan, R., Kolmogorov’s hypothesis and Eulerian turbulence theory, Phys. fluids, 7, 1723-1734, (1964) · Zbl 0151.41701
[26] Kraichnan, R., Lagrangian history closure approximation for turbulence, Phys. fluids, 8, 575-598, (1965)
[27] Kraichnan, R., Isotropic turbulence and inertial range structure, Phys. fluids, 9, 1728-1752, (1966) · Zbl 0147.46004
[28] Hopf, E., Statistical hydromechanics and functionals calculus, J. ratl. mech. anal., 1, 87-123, (1952) · Zbl 0049.41704
[29] Rosen, G., Turbulence theory and functional integration. I, Phys. fluids, 3, 519-524, (1960) · Zbl 0097.41401
[30] Rosen, G., Turbulence theory and functional integration. II, Phys. fluids, 3, 525-528, (1960) · Zbl 0097.41401
[31] Novikov, E., Functionals and the random force method in turbulence, Sov. phys. JETP, 20, 1290, (1965)
[32] Moiseev, S.; Tur, A.; Yanovskii, V., Spectra and expectation methods of turbulence in a compressible fluid, Sov. phys. JETP, 44, 556-561, (1976)
[33] Sazontov, A., The similarity relation and turbulence spectra in a stratified medium, Izv. atmos. Ocean. phys., 15, 566-570, (1979)
[34] Lewis, R.; Kraichnan, R., A space – time functional formalism for turbulence, Comm. pure appl. math., 15, 397-411, (1962) · Zbl 0112.19305
[35] Wyld, H., Formulation of the theory of turbulence in an incompressible fluid, Ann. phys., 14, 143-165, (1961) · Zbl 0099.42003
[36] Martin, P.; Siggia, E.; Rose, H., Statistical dynamics of classical systems, Phys. rev. A, 8, 423-437, (1973)
[37] Phythian, R., The functional formalism of classical statistical dynamics, J. phys. A, 10, 777-789, (1977)
[38] Katz, N.; Pavlovic, N., A cheap caffarelli – kohn – nirenberg inequality for the navier – stokes equation with hyper-dissipation, Geom. funct. anal., 12, 355-379, (2002) · Zbl 0999.35069
[39] Li, Y., On the true nature of turbulence, (2005)
[40] L’vov, V.; Procaccia, I., Exact resummations in the theory of hydrodynamic turbulence: part 0. line-resummed diagrammatic perturbation approach, ()
[41] Eyink, G., Turbulence noise, J. statist. phys., 83, 955-1019, (1996) · Zbl 1081.76556
[42] Andersen, H., Functional and graphical methods for classical statistical dynamics. I. A formulation of the martin – siggia – rose method, J. math. phys., 41, 1979-2020, (2000) · Zbl 0977.82027
[43] Yakhot, V., Ultraviolet dynamic renormalization group: small-scale properties of a randomly-stirred fluid, Phys. rev. A, 23, 1486-1497, (1981)
[44] Belinicher, V.; L’vov, V., A scale invariant theory of fully developed hydrodynamic turbulence, Sov. phys. JETP, 66, 303-313, (1987)
[45] L’vov, V., Scale invariant theory of fully developed hydrodynamic turbulence—hamiltonian approach, Phys. rep., 207, 2-47, (1991)
[46] L’vov, V.; Procaccia, I., Exact resummations in the theory of hydrodynamic turbulence. II. A ladder to anomalous scaling, Phys. rev. E, 52, 3858-3875, (1995)
[47] L’vov, V.; Procaccia, I., Exact resummations in the theory of hydrodynamic turbulence. III. scenarios for anomalous scaling and intermittency, Phys. rev. E, 53, 3468-3490, (1996)
[48] L’vov, V.; Procaccia, I., Fusion rules in turbulent systems with flux equilibrium, Phys. rev. lett., 76, 2898-2901, (1996)
[49] L’vov, V.; Procaccia, I., Viscous lengths in hydrodynamic turbulence are anomalous scaling functions, Phys. rev. lett., 77, 3541-3544, (1996)
[50] L’vov, V.; Procaccia, I., Towards a nonperturbative theory of hydrodynamic turbulence: fusion rules, exact bridge relations, and anomalous viscous scaling functions, Phys. rev. E, 54, 6268-6284, (1996)
[51] L’vov, V.; Podivilov, E.; Procaccia, I., Temporal multiscaling in hydrodynamic turbulence, Phys. rev. E, 55, 7030-7035, (1997)
[52] L’vov, V.; Procaccia, I., Computing the scaling exponents in fluid turbulence from first principles: the formal setup, Physica A, 257, 165-196, (1998)
[53] Belinicher, V.; L’vov, V.; Procaccia, I., A new approach to computing the scaling exponents in fluid turbulence from first principles, Physica A, 254, 215-230, (1998)
[54] Belinicher, V.; L’vov, V.; Pomyalov, A.; Procaccia, I., Computing the scaling exponents in fluid turbulence from first principles: demonstration of multiscaling, J. statist. phys., 93, 797-832, (1998) · Zbl 0953.76036
[55] L’vov, V.; Procaccia, I., Analytic calculation of the anomalous exponents in turbulence: using the fusion rules to flush out a small parameter, Phys. rev. E, 62, 8037-8057, (2000)
[56] Arad, I.; L’vov, V.; Procaccia, I., Correlation functions in isotropic and anisotropic turbulence: the role of the symmetry group, Phys. rev. E, 59, 6753-6765, (1999)
[57] Biferale, L.; Procaccia, I., Anisotropy in turbulent flows and in turbulent transport, Phys. rep., 414, 43-164, (2005)
[58] McComb, W., The physics of fluid turbulence, (1990), Clarendon Press Oxford · Zbl 0748.76005
[59] Smith, L.; Woodruff, S., Renormalization-group analysis of turbulence, Annu. rev. fluid mech., 30, 275-310, (1998) · Zbl 1398.76085
[60] Kraichnan, R., An interpretation of the yakhot – orzag turbulence, Phys. fluids, 30, 2400-2405, (1987) · Zbl 0642.76069
[61] Eyink, G., The renormalization group method in statistical hydrodynamics, Phys. fluids, 6, 3063-3078, (1994) · Zbl 0830.76042
[62] Eyink, G., Lagrangian field theory, multifractals, and universal scaling in turbulence, Phys. lett. A, 172, 335-360, (1993)
[63] Eyink, G., Renormalization group and operator product expansion in turbulence: shell models, Phys. rev. E, 48, 1823-1838, (1993)
[64] Giles, M., Anomalous scaling in homogeneous isotropic turbulence, J. phys. A, 34, 4389-4435, (2001) · Zbl 1015.76034
[65] Dubois, T.; Jauberteau, F.; Temam, R., Dynamic multilevel methods and the numerical simulation of turbulence, (1999), Cambridge University Press Cambridge · Zbl 0948.76070
[66] Galdi, G., An introduction to the navier – stokes initial-boundary value problem, () · Zbl 1108.35133
[67] Altaisky, M.; Moiseev, S.; Pavlik, S., Scaling and supersymmetry in spectral problems of strong turbulence, Phys. lett. A, 147, 142-146, (1990)
[68] Gozzi, E., Hidden brs invariance in classical mechanics, Phys. lett. B, 201, 525-528, (1988)
[69] Gozzi, E.; Reuter, M.; Thacker, W., Hidden brs invariance in classical mechanics. II, Phys. rev. D, 40, 3363-3377, (1989)
[70] Gozzi, E.; Reuter, M., Lyapunov exponents, path integrals and forms, Chaos solitons fractals, 4, 1117-1139, (1994) · Zbl 0809.58013
[71] Thacker, W., A path integral for turbulence in incompressible fluids, J. math. phys., 38, 300-320, (1997) · Zbl 0868.76041
[72] Frisch, U., From global scaling, a la Kolmogorov, to local multifractal scaling in fully developed turbulence, Proc. R. soc. lond. A, 434, 89-99, (1991) · Zbl 0727.76064
[73] Monin, A.; Yaglom, A., Statistical fluid mechanics, (1975), MIT Press Cambridge, MA
[74] 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, Phys. fluids, 15, L21-L24, (2003)
[75] Hill, R., The approach of turbulence to the locally homogeneous asymptote as studied using exact structure-function equations, (2002)
[76] E. Gkioulekas, A theoretical study of the cascades of 3D, 2D, and QG turbulence, Ph.D. Thesis, University of Washington, 2006. Advisor: Ka-Kit Tung
[77] Rasmussen, H., A new proof of kolmogorov’s 4/5-law, Phys. fluids, 11, 3495-3498, (1999) · Zbl 1149.76519
[78] Monin, A., The theory of locally isotropic turbulence, Dokl. akad. nauk. SSSR, 125, 515-518, (1959)
[79] Lindborg, E., A note on kolmogorov’s third order structure function law, the local isotropy hypothesis and the pressure velocity correlation, J. fluid mech., 326, 343-356, (1996) · Zbl 0884.76027
[80] Hill, R., Applicability of kolmogorov’s and monin’s equations of turbulence, J. fluid mech., 353, 67-81, (1997) · Zbl 0903.76044
[81] Gkioulekas, E.; Tung, K., On the double cascades of energy and enstrophy in two dimensional turbulence. part 1. theoretical formulation, Discrete contin. dyn. syst. ser. B, 5, 79-102, (2005) · Zbl 1115.76033
[82] Gkioulekas, E.; Tung, K., On the double cascades of energy and enstrophy in two dimensional turbulence. part 2. approach to the KLB limit and interpretation of experimental evidence, Discrete contin. dyn. syst. ser. B, 5, 103-124, (2005) · Zbl 1115.76333
[83] Hill, R., Exact second-order structure function relationships, J. fluid mech., 468, 317-326, (2002) · Zbl 1062.76028
[84] Duchon, J.; Robert, R., Inertial range dissipation for weak solitions of incompressible Euler and navier – stokes equations, Nonlinearity, 13, 249-255, (2000) · Zbl 1009.35062
[85] Eyink, G., Local 4/5-law and energy dissipation anomaly in turbulence, Nonlinearity, 16, 137-145, (2003) · Zbl 1138.76358
[86] Taylor, M.A.; Kurien, S.; Eyink, G.L., Recovering isotropic statistics in turbulence simulations: the Kolmogorov 4/5th law, Phys. rev. E, 68, 026310, (2003)
[87] Nie, Q.; Tanveer, S., A note on the third-order structure functions in turbulence, Proc. R. soc. lond. ser. A, 455, 1615-1635, (1999) · Zbl 0952.76026
[88] Lindborg, E., Correction to the four-fifths law due to variations of the dissipation, Phys. fluids, 11, 510-512, (1999) · Zbl 1147.76445
[89] Danaila, L.; Anselmet, F.; Zhou, T.; Antonia, R.A., Turbulent energy scale budget equations in a fully developed channel flow, J. fluid mech., 430, 87-109, (2001) · Zbl 0976.76030
[90] Danaila, L.; Anselmet, F.; Antonia, R.A., An overview of the effect of large-scale inhomogeneities on small-scale turbulence, Phys. fluids, 14, 2475-2484, (2002) · Zbl 1185.76103
[91] Danaila, L.; Anselmet, F.; Zhou, T., Turbulent energy scale-budget equations for nearly homogeneous sheared turbulence, Flow turbul. combust., 72, 287-310, (2004) · Zbl 1081.76547
[92] Danaila, L.; Antonia, R.A.; Burattini, P., Progress in studying small-scale turbulence using exact two-point equations, New J. phys., 6, 128, (2004)
[93] L’vov, V.; Lebedev, V., Anomalous scaling and fusion rules in hydrodynamic turbulence, (1994)
[94] Feynman, R.; Hibbs, A., Quantum mechanics and path integrals, (1965), McGraw-Hill New York · Zbl 0176.54902
[95] Stevens, C., The six core theories of modern physics, (1995), MIT Press Cambridge, MA
[96] Boffetta, G.; Celani, A.; Vergassola, M., Inverse energy cascade in two dimensional turbulence: deviations from Gaussian behavior, Phys. rev. E, 61, 29-32, (2000)
[97] Borue, V., Inverse energy cascade in stationary two dimensional homogeneous turbulence, Phys. rev. lett., 72, 1475-1478, (1994)
[98] Danilov, S.; Gurarie, D., Non-universal features of forced two dimensional turbulence in the energy range, Phys. rev. E, 63, 020203, (2001)
[99] Danilov, S.; Gurarie, D., Forced two-dimensional turbulence in spectral and physical space, Phys. rev. E, 63, 061208, (2001)
[100] Danilov, S., Non-universal features of forced 2d turbulence in the energy and enstrophy ranges, Discrete contin. dyn. syst. ser. B, 5, 67-78, (2005) · Zbl 1115.76032
[101] Falkovich, G., Bottleneck phenomenon in developed turbulence, Phys. fluids, 6, 1411-1414, (1994) · Zbl 0865.76030
[102] Fischer, P., Multiresolution analysis for 2d turbulence. part 1: wavelets vs cosine packets, a comparative study, Discrete contin. dyn. syst. B, 5, 659-686, (2005) · Zbl 1140.76350
[103] Mou, C.; Weichman, P., Multicomponent turbulence, the spherical limit, and non-Kolmogorov spectra, Phys. rev. E, 52, 3738-3796, (1995)
[104] Donsker, M., On function space integrals, (), 17-30
[105] Furutsu, K., On the statistical theory of electromagnetic waves in a fluctuating medium, J. res. nat. bur. standards D, 67, 303-323, (1963) · Zbl 0113.22303
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