Turbulence and Self-Organization pp 295-371 | Cite as

# Stochastic-Thermodynamic Modeling of Developed Structured Turbulence

## Abstract

The context is devoted to the development of a phenomenological model for the developed turbulence in a compressible homogeneous medium by taking into account nonlinear cooperative processes. The representation of a turbulized fluid motion as a thermodynamic system consisting of two continua, the subsystem of averaged motion and the subsystem of turbulent chaos, which, in turn, is considered as a conglomerate of vortex structures of various spatiotemporal scales, serves as the primary concept. We develop the ideas of a stationary-nonequilibrium state of the dissipatively active subsystem of turbulent chaos emerging due to the influx of negentropy from the external medium (the subsystem of averaged motion) and the appearance of relatively stable coherent vortex structures in the system when varying the flow control parameters. This allows some of the turbulent field rearrangement processes to be considered as self-organization processes in an open system. The methods of the stochastic theory of irreversible processes and extended irreversible thermodynamics are used to derive the defining relations for the turbulent fluxes and forces that close the system of averaged hydrodynamic equations and describe the transport and self-organization processes in the stationary-nonequilibrium case with completeness sufficient for practice.

The original approach to the stochastic-thermohydrodynamic modeling of the subsystem of turbulent chaos we developed is based on the introduction of a set of random variables into the model—fluctuating internal coordinates (like the turbulent energy dissipation rates, the intrinsic vorticities of the field of velocity fluctuations referring to mesoscale vortex structures, etc.) characterizing the structure and temporal evolution of the fluctuating field of hydrodynamic flow parameters. This makes it possible to model the Richardson–Kolmogorov cascade process and to derive the kinetic Fokker–Planck–Kolmogorov (FPK) equations designed to describe the evolution of the probability density function for small-scale turbulence characteristics by thermodynamic methods. In particular, these equations serve as a basis in analyzing the Markovian diffusion processes of the transition from one stationary-nonequilibrium state to another in the space of internal coordinates through a successive loss of stability (and increase in supercriticality) by the subsystem of turbulent chaos far from complete chaos of thermodynamic equilibrium. Such transitions can be described as nonequilibrium “second-order phase transitions” in a vortex continuum, causing the internal coordinates at bifurcation points to change abruptly.

An alternative approach to investigating the mechanisms of such a transition that is based on stochastic Langevin equations closely related to the derived kinetic FPK equations is also considered here. We analyze a cardinal problem of the approach being developed—the possibility of the existence of asymptotically stable stationary-nonequilibrium states in the subsystem of turbulent chaos. We propose a nonequilibrium thermodynamic potential for the stochastic internal coordinates of turbulent chaos that generalizes the well-known Boltzmann–Planck relation for equilibrium states to stationary-nonequilibrium states of the ensemble representing chaos and show that this potential is the Lyapunov function for stationary-nonequilibrium states of the ensemble corresponding to the subsystem of turbulent chaos.

The last section of this chapter is devoted to the thermodynamic derivation of generalized fractional FPK equations describing the evolution of the internal coordinates of the subsystem of turbulent chaos based on fractional dynamics. The introduction of fractional time derivatives into the kinetic FPK equation allows the effects of intermittency in time with which the presence of turbulent bursts against the background of less intense low-frequency background turbulence oscillations are associated to be taken into account in the context of unified mathematical formalism.

## Keywords

Lyapunov Function Vortex Structure Structure Turbulence Inertial Range Conditional Probability Density## References

- Kolmogorov, A.N.: Refining the views about the local structure of turbulence in an incompressible viscous fluid at large reynolds numbers, Mecanique de la turbulence: Colloq. Intern. CNRS, Marseille, aout, Sept. 1961, 447 (in Russian) (1962)Google Scholar
- Landau, L.D., Lifshitz, V.M.: Hydrodynamics. Nauka, Moscow (1988a) (in Russian)Google Scholar
- Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Applications. Springer, New York (1976)MATHCrossRefGoogle Scholar
- Kolesnichenko, A.V.: Stefan-Maxwell Relations and Heat Flow for Turbulent Multicomponent Continuous Media in Problems of Modern Mechanics, p. 52. Moscow State University, Moscow (in Russian) (1998a)Google Scholar
- Batchelor, G.K., Townsend, A.A.: The nature of turbulent motion at high wave number. Proc. Roy. Soc. London
**A199**, 238 (1949)ADSGoogle Scholar - Blackadar, A.K.: Extension of the laws of thermodynamics to turbulent system. J. Meteorol.
**12**, 165 (1955b)CrossRefGoogle Scholar - Boiko, A.V., Grek, G.R., Dovgal, A.V., Kozlov, V.V.: Physical Mechanisms of the Transition to Turbulence in Open Flows. “Regular and Chaotic Dynamics” Research Center, Institute of Computer Research, Moscow-Izhevsk (in Russian) (2006)Google Scholar
- Brown, G.L., Roshko, A.: On density effects and large structures in turbulent mixing layers. J. Fluid Mech.
**64**, 775 (1974b)ADSCrossRefGoogle Scholar - Crow, S.C., Champagne, F.H.: Orderly structures in jet turbulence. J. Fluid Mech.
**48**, 547 (1971b)ADSCrossRefGoogle Scholar - Demidovich, B.P.: Lectures on the Mathematical Theory of Stability. Moscow State University, Moscow (in Russian) (1998)Google Scholar
- Druden, H.I.: Recent advances in the mechanics of boundary layer flow. Adv. Appl. Mech.
**1**, 1 (1948)CrossRefGoogle Scholar - Ebeling, W.: Formation of Structures in Irreversible Processes: An Introduction to the Theory of Dissipative Structures. Institute of Computer Research, “Regular and Chaotic Dynamics” Research Center, Moscow-Izhevsk (in Russian) (2004)Google Scholar
- Ebeling, W., Engel, A., Feistel, R.: Physics of Evolution Processes. A Synergetic Approach, Editorial USSR, Moscow (in Russian) (2001)Google Scholar
- Frisch, U., Sulem, P.L., Nelkin, M.: A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech.
**87**, 719 (1978)ADSMATHCrossRefGoogle Scholar - Graham, R.: Stochastic Methods in Nonequilibrium Thermodynamics in Stochastic Nonlinear Systems in Physics, Chemistry, and Biology. In: Arnold, L., Lefever, R. (eds.) Proceedings of the Workshop, Bielefeld, Fed. Rep. of Germany, Springer, Heidelberg, 202, 5–11 October 1980 (1981)Google Scholar
- Khlopkov, Y.I., Zharov, V.A., Gorelov, S.L.: Coherent Structures in a Turbulent Boundary Layer. Moscow Physicotechnical Institute, Moscow (in Russian) (2002b)Google Scholar
- Klimontovich, Y.L.: Statistical Theory of Open Systems. Janus, Moscow (in Russian) (1995a)MATHCrossRefGoogle Scholar
- Klimontovich, Y.L.: Introduction to the Physics of Open Systems. Janus-K, Moscow (in Russian) (2002b)Google Scholar
- Klyatskin, V.I., Tatarskii, V.I.: The approximation of a diffusion random process in some nonstationary stochastic problems of physics. Usp. Fiz. Nauk
**110**, 499 (1973)CrossRefGoogle Scholar - Kolesnichenko, A.V.: Synergetic approach to describing developed turbulence. Astron. Vestn.
**36**, 121 (2002a)MathSciNetGoogle Scholar - Kolesnichenko, A.V.: Thermodynamic modeling of developed structured turbulence with energy dissipation fluctuations. Astron. Vestn.
**38**, 144 (2004a)Google Scholar - Kolesnichenko, A.V., Marov, M.Y.: Methods of nonequilibrium thermodynamics for description of multicomponent turbulent gas mixtures. Arch. Mech.
**37**, 3 (1985)MathSciNetMATHGoogle Scholar - Kolmogorov, A.N.: Local structure of turbulence in an incompressible fluid at very large Reynolds numbers. Dokl. Akad. Nauk USSR
**30**, 299 (1941d)ADSGoogle Scholar - Landau, L.D., Lifshitz, V.M.: Hydrodynamics. Nauka, Moscow (in Russian) (1988b)Google Scholar
- Mandelbrot, B.B.: Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. J. Fluid Mech.
**62**, 401 (1974)CrossRefGoogle Scholar - Mandelbrot, B.: The Fractal Geometry of Nature. Freeman and Company, New York (1983)Google Scholar
- Marov, M.Y., Kolesnichenko, A.V.: Mechanics of Turbulence of Multicomponent Gases. Kluwer Academic, Dordrecht–Boston–London (2001b)MATHGoogle Scholar
- Millionshchikov, M.D.: To the theory of homogeneous and isotropic turbulence. Izv. Akad. Nauk USSR, Ser. Geogr. Geoph.
**5**, 433 (1941)Google Scholar - Monin, A.S., Yaglom, A.M.: Statistical Hydrodynamics, vol. 1. Gidrometeoizdat, St. Petersburg (in Russian) (1992d)Google Scholar
- Monin, A.S., Yaglom, A.M.: Statistical Hydrodynamics, vol. 2. Gidrometeoizdat, St. Petersburg (in Russian) (1996b)Google Scholar
- Monin, A.S., Polubarinova-Kochina, P.Y., Khlebnikov, V.I.: Cosmology, Hydrodynamics, Turbulence: A.A. Fridman and the Development of his Legacy. Nauka, Moscow (in Russian) (1989a)Google Scholar
- Munster, A.: Chemical Thermodynamics. Editorial URSS, Moscow (in Russian) (2002c)Google Scholar
- Nakhushev, A.N.: Fractional Calculus and Its Applications. Fizmatlit, Moscow (in Russian) (2003)MATHGoogle Scholar
- Nigmatullin, R.R.: The realization of generalized transfer equation in a medium with fractal geometry. Phys. Status Solidi B
**133**, 425 (1986)ADSCrossRefGoogle Scholar - Nigmatullin, R.R.: Fractional integral and its physical interpretation. Teor. Mat. Fiz.
**90**, 354 (1992)MathSciNetCrossRefGoogle Scholar - Novikov, E.A., Stewart, R.U.: Intermittency of turbulence and the spectrum of energy dissipation fluctuations. Izv. Akad. Nauk USSR, Ser. Geoph.
**3**, 408 (1964)Google Scholar - Obukhov, А.М.: Some specific features of atmospheric turbulence. J. Fluid Mech.
**13**, 77 (1962b)MathSciNetADSCrossRefGoogle Scholar - Oldham, K.B., Spanier, J.: The Fractional Calculus (Theory and Applications of Differentiation and Integration to Arbitrary Order). Acadamic, New York (1974)MATHGoogle Scholar
- Olemskii, A.I.: Theory of stochastic systems with singular multiplicative noise. Usp. Fiz. Nauk
**168**, 287 (1998)CrossRefGoogle Scholar - Olemskii, A.I., Flat, A.Y.: Using the concept of a fractal in condensed matter physics. Usp. Fiz. Nauk
**163**, 1 (1993)CrossRefGoogle Scholar - Onsager, L.: Statistical hydrodynamics. Nuovo Cimento
**6**((Suppl.)), 279 (1949)MathSciNetGoogle Scholar - Prigogine, I., Stengers, I.: Time. Chaos. Quantum. Toward Solving the Paradox of Time. Progress Publishing House, Moscow (in Russian) (1994a)Google Scholar
- Rabinovich, M.I., Sushchik, M.M.: Regular and chaotic dynamics of structures in fluid flows. Usp. Fiz. Nauk
**160**, 1 (1990a)MathSciNetCrossRefGoogle Scholar - Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional-Order Integrals and Derivatives and Their Applications, Nauka I Tekhnika (in Russian) (1987)Google Scholar
- Sedov, L.I.: On Promising Directions and Problems in Continuum Mechanics in Thoughts about Science and Scientists. Nauka, Moscow (in Russian) (1980b). 173Google Scholar
- Shkhanukov, M.K.: On the convergence of difference schemes for differential equations with a fractional derivative. Dokl. Akad. Nauk USSR
**348**, 746 (1996)MathSciNetGoogle Scholar - Shkhanukov-Lafishev, M.Kh., Nakhusheva, F.M.: Boundary-Value Problems for a Fractional-Order Diffusion Equation and Grid Methods of Their Solution in Nonclassical Equations of Mathematical Physics, Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 37 (in Russian) (1998)Google Scholar
- Sinai, Ya.G., Shilnikov, L.P. (eds.). Strange Attractors, a Collection of Papers, Moscow (in Russian) (1981)Google Scholar
- Stratonovich, R.L.: Nonlinear Nonequilibrium Thermodynamics. Nauka, Moscow (in Russian) (1985b)MATHGoogle Scholar
- Tikhonov, V.I., Mironov, M.A.: Markovian Processes. Sov. Radio, Moscow (in Russian) (1977a)Google Scholar
- Zaslavsky, G.M.: Physics of Chaos in Hamiltonian Systems. Institute of Computer Research, Moscow-Izhevsk (in Russian) (2004)Google Scholar
- Prigogine, I., Stengers, I. Order out of Chaos. Heinemann. London. 1984.Google Scholar
- Monin, A.S. and Yaglom, A.M. Statistical Fluid Mechanics. Vol. 2. Cambridge, Mass., USA): MIT Press, 1975Google Scholar
- Frisch, U. The Legacy of A.N. Kolmogorov. Cambridge University Press. 1995Google Scholar
- de Groot, S.R., Mazur, P. Non –Equilibrium Thermodynamics. North-Holland Publishing Company. Amsterdam. 1962.Google Scholar
- Jou, D., Casas- Vazquez, J., and Lebon, G. Extended Irreversible Thermodynamics. Springer -Verlag Berlin Heidelberg. 2001Google Scholar
- Keizer, J. Statistical Thermodynamics of Nonequilibrium Processes. Springer Verlag New York Inc. 1987Google Scholar
- Prigogine, I. Introduction to Thermodynamics of Irreversible Processes. N.Y.: John Wiley. 1967.Google Scholar
- Kondepudi, D., Prigogine, I. Modern Thermodynamics. From Heat Engines to Dissipative Structures. John Wiley and Sons. Chichester. New York. Weinheim. Brisbane. Toronto. Singapore. 1999.Google Scholar
- Keizer, J. Statistical Thermodynamics of Nonequilibrium Processes. Springer Verlag, New York Inc. 1987.CrossRefGoogle Scholar
- Haken, H. (1981). Synergetics. Springer. Heidelberg-Berlin-New York.MATHGoogle Scholar
- Haken, H. (1988). Information and Self-Organization. Springer. Berlin- Heidelberg, New York.MATHGoogle Scholar
- de Groot, S.R., Mazur, P. Non –Equilibrium Thermodynamics. North-Holland Publishing Company. Amsterdam. 1962.Google Scholar
- Batchelor, G.K. The Theory of Homogeneous Turbulence. Cambridge University Press. Cambridge. (1953).MATHGoogle Scholar
- Ferziger, J.H., Kaper, H.G.Mathematical Theory of Transport Processes in Gases. North-Holland Publishing Company. Amsterdam- London. 1972Google Scholar
- Kolmogorov, A.N. Precisions sur la structure locale de la turbulence dans un fluide visqueux aux nombres de Reynolds eleves. In: La turbulence en mecanique des fluids. Eds. A. Favre, L.S.G.Kovasznay, R. Dumas, J. Gaviglio and M. Coantic. Gauthier-VillarsGoogle Scholar