On One Possibility of Closuring the Chain of Equations for Statistical Moments in Turbulence Theory

Affiliation(s)

The Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia.

The Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia.

ABSTRACT

The paper concerns the problem on statistical description of the turbulent velocity pulsations by using the method of characteristic functional. The equations for velocity covariance and Green’s function, which describes an average velocity response to external force action, have been obtained. For the nonlinear term in the equation for velocity covariance, it has been obtained an exact representation in the form of two terms, which can be treated as describing a momentum transport due to turbulent viscosity and action of effective random forces (within the framework of traditional phenomenological description, the turbulent viscosity is only accounted for). Using a low perturbation theory approximation for high statistical moments, a scheme of closuring the chain of equations for statistical moments is proposed. As the result, we come to a closed set of equations for velocity covariance and Green’s function, the solution to which corresponds to summing up a certain infinite subsequence of total perturbation series.

Cite this paper

E. Teodorovich, "On One Possibility of Closuring the Chain of Equations for Statistical Moments in Turbulence Theory,"*Journal of Modern Physics*, Vol. 4 No. 1, 2013, pp. 56-63. doi: 10.4236/jmp.2013.41010.

E. Teodorovich, "On One Possibility of Closuring the Chain of Equations for Statistical Moments in Turbulence Theory,"

References

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[2] M. I. Vishik and A. V. Fursikov, “Mathematical Problems in Statistical Hydrodynamics (in Russian),” Nauka, Moscow, 1980.

[3] L. V. Keller und A. A. Friedman, “Differentialgleichungen fur die Turbulente Bewegung einer kompressiblen Flussigkeit,” Proceedings of the 1st International Congress for Applied Mechanics, Delft, 1924, pp. 395-405.

[4] N. N. Bogolyubov, “Problems of Dynamic Theory in Statistical Physics (in Russian),” GITTL, Kiev, 1946.

[5] Y. Ogura, “Energy Transport in a Normally Distributed and Isotropic Turbulent Velocity Field in Two Dimensions,” Physics of Fluids, Vol. 5, No. 4, 1962, pp. 395- 401. doi:10.1063/1.1706631

[6] R. H. Kraichnan, “The Structure of Isotropic Turbulence at Very High Reynolds Numbers,” Journal of Fluid Mechanics, Vol. 5, No. 4, 1959, pp. 497-543. doi:10.1017/S0022112059000362

[7] S. F. Edwards, “The Statistical Dynamics of Homogeneous Turbulence,” Journal of Fluid Mechanics, Vol. 18, No. 2, 1964, pp. 239-293. doi:10.1017/S0022112064000180

[8] H. W. Wyld, “Formulation of the Theory of Turbulence in an Incompressible Fluid,” Annals of Physics, Vol. 14, No. 2, 1961, pp. 143-165. doi:10.1016/0003-4916(61)90056-2

[9] E. V. Teodorovich, “The Use of the Renormalization Group Method to Describe Turbulence (a Review),” Izvestia, Amospheric and Oceanic Physics, Vol. 29, No. 2, 1993, pp. 149-163.

[10] L. T. Adzhemyan, N. V. Antonov and A. N. Vasil’ev, “Quantum-Field Renormalization Group in the Theory of Hydrodynamic Turbulence,” Physics, Uspekhi, Vol. 39, No. 12, 1996, pp. 1193-1219. doi:10.1070/PU1996v039n12ABEH000183

[11] A. N. Kolmogorov, “Local Structure of Turbulence in Incompressible Fluid at Very Large Reynolds Numbers (in Russian)”, Soviet Physics, Doklady, Vol. 39, No. 4, 1941, pp. 299-303.

[12] B. B. Kadomtsev, “The Plasma Turbulence,” Academic Press, London, New York, 1965.

[13] V. I. Belinicher and V. S. L’vov, “Scale-Invariant Theory of Developed Hydrodynamic Turbulence,” Journal of Experimental and Theoretical Physics (JETP), Vol. 66, No. 2, 1987, pp. 303-313.

[14] E. V. Teodorovich, “Role of Local and Nonlocal Interactions in the Formation of the Developed Turbulence Regime,” Fluid Dynamics, Vol. 25, No. 4, 1990, pp. 522- 528. doi:10.1007/BF01049856

[15] R. M. Lewis and R. H. Kraichnan, “A Space-Time Functional Formalism for Turbulence,” Communication on Pure and Applied Mathematics, Vol. 15, No. 2, 1962, pp. 397-411. doi:10.1002/cpa.3160150403

[16] E. V. Teodorovich, “Application of the Methods from Quantum-Field Theory,” In the Monograph: A. S. Monin and A. M. Yaglom, Statistical Hydrodynamics. Theory of Turbulence. V.2, Sec. 29.4 (in Russian), St-Petersburg, Hydrometeoizdat, 1996.

[17] E. V. Teodorovoch, “Diagram Equations of the Theory of Fully Developed Turbulence,” Theoretical and Mathematical Physics, Vol. 101, No. 1, 1994, pp. 1177-1183.

[18] E. V. Teodorovich, “The Renormalization-Group Method in Problems of Mechanics,” Journal of Applied Mathematics and Mechanics (PMM), Vol. 68, No. 2, 2004, pp. 299-326. doi:10.1016/S0021-8928(04)90029-9

[19] E. V. Teodorovich, “Renormalization-Group Approach to Solving the Equation of Nonlinear Transfer,” Journal of Physics A: Mathematical and Theoretical, Vol. 42, No. 15, 2009, Article ID: 155202.

[1] A. S. Monin and A. M. Yaglom, “Statistical Hydrodynamics. Theory of turbulence, V.1 (in Russian),” Hydrometeoizdat, St-Petersburg, 1992.

[2] M. I. Vishik and A. V. Fursikov, “Mathematical Problems in Statistical Hydrodynamics (in Russian),” Nauka, Moscow, 1980.

[3] L. V. Keller und A. A. Friedman, “Differentialgleichungen fur die Turbulente Bewegung einer kompressiblen Flussigkeit,” Proceedings of the 1st International Congress for Applied Mechanics, Delft, 1924, pp. 395-405.

[4] N. N. Bogolyubov, “Problems of Dynamic Theory in Statistical Physics (in Russian),” GITTL, Kiev, 1946.

[5] Y. Ogura, “Energy Transport in a Normally Distributed and Isotropic Turbulent Velocity Field in Two Dimensions,” Physics of Fluids, Vol. 5, No. 4, 1962, pp. 395- 401. doi:10.1063/1.1706631

[6] R. H. Kraichnan, “The Structure of Isotropic Turbulence at Very High Reynolds Numbers,” Journal of Fluid Mechanics, Vol. 5, No. 4, 1959, pp. 497-543. doi:10.1017/S0022112059000362

[7] S. F. Edwards, “The Statistical Dynamics of Homogeneous Turbulence,” Journal of Fluid Mechanics, Vol. 18, No. 2, 1964, pp. 239-293. doi:10.1017/S0022112064000180

[8] H. W. Wyld, “Formulation of the Theory of Turbulence in an Incompressible Fluid,” Annals of Physics, Vol. 14, No. 2, 1961, pp. 143-165. doi:10.1016/0003-4916(61)90056-2

[9] E. V. Teodorovich, “The Use of the Renormalization Group Method to Describe Turbulence (a Review),” Izvestia, Amospheric and Oceanic Physics, Vol. 29, No. 2, 1993, pp. 149-163.

[10] L. T. Adzhemyan, N. V. Antonov and A. N. Vasil’ev, “Quantum-Field Renormalization Group in the Theory of Hydrodynamic Turbulence,” Physics, Uspekhi, Vol. 39, No. 12, 1996, pp. 1193-1219. doi:10.1070/PU1996v039n12ABEH000183

[11] A. N. Kolmogorov, “Local Structure of Turbulence in Incompressible Fluid at Very Large Reynolds Numbers (in Russian)”, Soviet Physics, Doklady, Vol. 39, No. 4, 1941, pp. 299-303.

[12] B. B. Kadomtsev, “The Plasma Turbulence,” Academic Press, London, New York, 1965.

[13] V. I. Belinicher and V. S. L’vov, “Scale-Invariant Theory of Developed Hydrodynamic Turbulence,” Journal of Experimental and Theoretical Physics (JETP), Vol. 66, No. 2, 1987, pp. 303-313.

[14] E. V. Teodorovich, “Role of Local and Nonlocal Interactions in the Formation of the Developed Turbulence Regime,” Fluid Dynamics, Vol. 25, No. 4, 1990, pp. 522- 528. doi:10.1007/BF01049856

[15] R. M. Lewis and R. H. Kraichnan, “A Space-Time Functional Formalism for Turbulence,” Communication on Pure and Applied Mathematics, Vol. 15, No. 2, 1962, pp. 397-411. doi:10.1002/cpa.3160150403

[16] E. V. Teodorovich, “Application of the Methods from Quantum-Field Theory,” In the Monograph: A. S. Monin and A. M. Yaglom, Statistical Hydrodynamics. Theory of Turbulence. V.2, Sec. 29.4 (in Russian), St-Petersburg, Hydrometeoizdat, 1996.

[17] E. V. Teodorovoch, “Diagram Equations of the Theory of Fully Developed Turbulence,” Theoretical and Mathematical Physics, Vol. 101, No. 1, 1994, pp. 1177-1183.

[18] E. V. Teodorovich, “The Renormalization-Group Method in Problems of Mechanics,” Journal of Applied Mathematics and Mechanics (PMM), Vol. 68, No. 2, 2004, pp. 299-326. doi:10.1016/S0021-8928(04)90029-9

[19] E. V. Teodorovich, “Renormalization-Group Approach to Solving the Equation of Nonlinear Transfer,” Journal of Physics A: Mathematical and Theoretical, Vol. 42, No. 15, 2009, Article ID: 155202.