Thermodynamic description of a developed turbulence at the registration of coherent vortex structures

A. V. Kolesnichenko

Matematičeskoe modelirovanie, Tome 16 (2004) no. 9, pp. 92-126.

Résumé

The thermodynamic approach to build-up of phenomenological macroscopic model of a developed turbulence in compressible fluid is considered at the registration of nonlinear cooperative processes, happening in her. The insert in model of a gang of variates as interior parameters of a subsystem of turbulent chaos, has enabled to receive by methods of statistical nonequilibrium thermodynamics the Fokker–Planck equation in configurational space. This equation is intended for definition of time changes of a density function of conditional probability of structural parameters concerning to cascade process of bucking of widescale vortexes and temperature inhomogeneities, and also for the analysis of Markovian stochastic processes of transition from one stationary disequilibriums in other as a result of series loss of stability at change of driving parameters. The alternate method to examination of mechanisms of similar transition, founded on the stochastic differential equation simultaneously surveyed, intimately bound with the output kinetic equation (and the given connection is traced in paper). The cardinal problem of the developed approach is parsed – an opportunity of existence asymptotically of inconvertible stationary – disequilibrium of a subsystem of turbulent chaos. The nonequilibrium thermodynamic potential for interior coordinates extending a known relation Boltzmann–Planck for equilibrium states on stationary-disequilibrium of representing band is offered and is rotined, that he is Lyapunov function for such states. The connection between an interior alternation in an inertial interval of gauges and fluctuations of energy of a dissipation was surveyed.

Bibliographie

[1] Kolesnichenko A. V., Marov M. Ya., “Methods of nonequilibrium thermodynamics for description of multicomponent turbulens gas mixture”, Arch. Mech., 37:1–2 (1985), 3–19 | MR | Zbl

[2] Kolesnichenko A. V., Mapoe M. Ya., Turbulentnost mnogokomponentnykh sred, Nauka, M., 1999, 336 pp. | MR

[3] Kolesnichenko A. V., “Sinergeticheskii podkhod k opisaniyu statsionarno-neravnovesnoi turbulentnosti”, Matematicheskoe modelirovanie, 16:1 (2004), 37–66 | MR | Zbl

[4] Kolesnichenko A. V., “Sinergeticheskii podkhod k opisaniyu statsionarno-neravnovesnoi turbulentnosti astrogeofizicheskikh sistem”, Sovremennye problemy mekhaniki i fiziki kosmosa, Fizmatlit, M., 2003, 123–162

[5] Prigozhii I., Stengers I., Poryadok iz khaosa. Novyi dialog cheloveka s prirodoi, Progress, M., 1986, 310 pp.

[6] Marov M. Ya., Kolesnichenko A. V., Mechanics of turbulence of multicomponent gases, Kluwer Academic Publishers, Dordrecht, Boston, London, 2001, 375 pp. | MR | Zbl

[7] Belotserkovskii O. M., Oparin A. M., Chislennyi eksperiment v turbulentnosti: Ot poryadka k khaosu, Nauka, M., 2000, 223 pp. | MR

[8] Blackadar A. K., “Extension of the laws of thermodynamics to turbulent system”, J. Meteorology, 12 (1955), 165–175

[9] Prigozhii I., Vvedenie v termodinamiku neobratimykh protsessov, Inostr. Lit., M., 1960, 160 pp. | MR

[10] Onsager L., “Statistical hydrodynamics”, Nuovo Cimento (Supplement), 6 (1949), 279–287 | DOI | MR

[11] Druden H. I., “Recent Advances in the Mechanic of Boundary Layes Flow”, Adv. Appl. Mech., 1 (1948), 1–40 | DOI | MR

[12] Crow S. C., Champagne F. H., “Orderly structures in jet turbulence”, J. Fluid Mech., 48 (1971), 547–591 | DOI

[13] Brown G. L., Roshko A., “On density effects and large structures in turbulent mixing layers”, J. Fluid Mech., 64 (1974), 775–816 | DOI

[14] Landau L. D., Lifshits V. M., Gidrodinamika, Nauka, M., 1988, 733 pp.

[15] Monin A. S., Polubarinova-Kochina P. Ya., Khlebnikov V. I., Kosmologiya, gidrodinamika, turbulentnost: A. A. Fridman i razvitie ego nauchnogo naslediya, Nauka, Gl. red. fiz.-mat. lit., M., 1989, 326 pp. | MR | Zbl

[16] Kolesnichenko A. V., “Sootnosheniya Stefana-Maksvella i potok tepla dlya turbulentnykh mnogokomponentnykh sploshnykh sred”, Problemy sovremennoi mekhaniki, k yubileyu L. I. Sedova, ed. Grigoryan S. S., MGU, M., 1998, 52–74 | MR

[17] Obukhov A. M., “Some specific features of atmospheric turbulence”, J. Fluid Mech., 13:1 (1962), 77–81 | DOI | MR

[18] Kolmogorov A. N., “Utochnenie predstavlenii o lokalnoi strukture turbulentnosti v neszhimaemoi vyazkoi zhidkosti pri bolshikh chislakh Reinoldsa”, Mecanique de la turbulence, Colloq. Intern. CNRS (Marseille aout–sept. 1961), Paris, 1962, 447–458 | MR

[19] Frish U., Turbulentnost. Nasledie Kolmogorova, Fazis, M., 1998, 343 pp.

[20] Prigozhii I., Stengers I., Vremya. Khaos. Kvant. K resheniyu paradoksa vremeni, Izdatelskaya gruppa “Progress”, M., 1994, 240 pp.

[21] Monin A. S, Yaglom A. M., Statisticheskaya gidromekhanika, t. 2, Gidrometeoizdat, S.-Pb., 1996, 742 pp.

[22] Myunster A., Khimicheskaya termodinamika, Editorial URSS, M., 2002, 295 pp.

[23] Kaizer Dzh., Statisticheskaya termodinamika neravnovesnykh protsessov, Mir, M., 1990, 607 pp. | MR

[24] Prigozhii I. Kondepudi D., Sovremennaya termodinamika. Ot teplovykh dvigatelei do dissipativnykh struktur, Mir, M., 2002, 461 pp.

[25] de Groot S, Mazur P., Neravnovesnaya termodinamika, Mir, M., 1964, 456 pp.

[26] Khaken G., Sinergetika. Ierarkhii neustoichivosti v samoorganizuyuschikhsya sistemakh i ustroistvakh, Mir, M., 1985, 419 pp. | MR

[27] Klyatskin V. I., Tatarskii V. I., “Priblizhenie diffuzionnogo sluchainogo protsessa v nekotorykh nestatsionarnykh statisticheskikh zadachakh fiziki”, Uspekhi fizicheskikh nauk, 110:4 (1973), 499–517 | MR

[28] Demidovich B. P., Lektsii po matematicheskoi teorii ustoichivosti, Izd-vo Mosk. Un-ta, M., 1998, 480 pp. | MR

[29] Graham R., “Stochastic Methods in Nonequilibrium Thermodynamics”, Stochastic Nonlinear Systems in Physics, Chemistry, and Biology, Proc. of the Workshop (Bielefeld, Fed. Rep. of Germany, October 5–11, 1980), eds. Arnold L., Lefever R., Springer-Verlag, Berlin, Heidelberg, N.Y., 1981, 202–212 | MR | Zbl

[30] Berzhe P., Pomo I., Vidal K., Poryadok v khaose. O deterministskom podkhode k turbulentnosti, “Merkurii-PRESS”, Cherepovets, 2000, 366 pp.

[31] Marsden J. E., McCracren M., “The Hopf Bifurcation and its Applications”, Applied Math. Sci., 19 (1976), Springer, Berlin, Heidelberg, New York | MR | Zbl

[32] Mandelbrot B., Fraktalnaya geometriya prirody, Institut kompyuternykh issledovanii, M., 2002, 656 pp.

[33] Batchelor G. K., Townsend A. A., “The nature of turbulent motion at high wave number”, Pr. of the Royal society of London, A199 (1949), 238–255 | DOI

[34] Betchelor Dzh., Teoriya odnorodnoi turbulentnosti, In. lit., M., 1955, 197 pp.

[35] Mandelbrot B. B., “Intermittent turbulence in self- similar cascades: divergence of high moments and dimension of the carrie”, J. Fluid mech., 62:2 (1974), 401–416 | DOI

[36] Novikov E. A., Styuart R. U., “Peremezhaemost turbulentnosti i spektr flyuktuatsii dissipatsii energii”, Izv. AN SSSR, ser. Geofiziki, 1964, no. 3, 408–413

[37] Frisch U., Sulem P. L., Nelkin M., “A simple dynamical model of intermittent fully developed turbulence”, J. Fluid Mech., 87 (1978), 719–736 | DOI | Zbl

[38] Strannye attraktory: Sbornik statei, Per. s ang., eds. Ya. G. Sinai, L. P. Shilnikov, Mir, M., 1981, 253 pp. | MR

[39] Frenkiel F. N., Klebanojj P. S., “On the log-normality of the small-scale structure of turbulence”, Bound. Layer Meteorol., 8:2 (1975), 173–200 | DOI

[40] Monin A. C., Yaglom A. M., Statisticheskaya gidromekhanika, t. 1, Gidrometeoizdat, S.-Pb., 1992, 694 pp.