Accounting for entropy production in the liouville equation and the derivation of a "MODIFIED" system of navier-stokes equations from it


Аuthors

Khatuntseva O. N.

e-mail: olga.khatuntseva@rsce.ru

Abstract

Turbulent and laminar flow regimes of a liquid or gas are indistinguishable on the scale of thermal motion of molecules. However, there are significant differences between them on the meso- and macro-scales. The turbulent regime has the features of a stochastic time–irreversible process at all scales of consideration, moreover, stochastic pulsations in the turbulent regime at different scales are correlated – they have a collective character. In contrast, the laminar regime is deterministic and time-reversible at all scales significantly exceeding the scale of thermal motion of molecules. There are ranges of parameters above some critical values at which both laminar and turbulent modes can be realized and exist with different probabilities. Transitions between them occur abruptly, irreversibly, that is, the reverse transition when changing parameters in the opposite direction can occur (and usually does) at other parameter values. Thus, an equation describing both of these modes should allow for a non-unique solution, with an ill-smooth and ambiguously defined transition between them.

Earlier, studies were conducted on the possibility of describing both laminar and turbulent fluid flow based on the same “modified” Navier-Stokes equations, which take into account entropy production in the turbulent regime due to the excitation of stochastic disturbances at different flow scales [1-4].

Solutions corresponding to laminar and turbulent flow regimes of an incompressible non-thermally conductive liquid were analytically obtained for the Hagen-Poiseuille problems, the Poiseuille plane flow and the Couette plane flow. Experimental and analytical solutions for different values of the Reynolds number are compared.

This paper shows the possibility of moving from the Liouville equation, which takes into account the production of entropy at different scales (the “modified” Liouville equation) to the “modified” Boltzmann equation through a chain of “modified” Bogolyubov equations. Based on these equations, a “modified” system of Navier-Stokes equations is derived.

Keywords:

Liouville, Boltzmann, Navier-Stokes equations, turbulent flow, laminar-turbulent transition

References

  1. Khatuntseva O.N. Trudy MAI, 2021, no. 118. URL: http://trudymai.ru/eng/published.php?ID=158211. DOI: 10.34759/trd-2021-118-02

  2. Khatuntseva O.N. Trudy MAI, 2022, no. 122. URL: http://trudymai.ru/eng/published.php?ID=164194. DOI: 10.34759/trd-2022-122-07

  3. Khatuntseva O.N. Trudy MAI, 2022, no. 123. URL: https://trudymai.ru/eng/published.php?ID=165492. DOI: 10.34759/trd-2022-123-08

  4. Khatuntseva O.N. Trudy MAI, 2023, no. 131. URL: https://trudymai.ru/eng/published.php?ID=175916. DOI: 10.34759/trd-2023-131-10

  5. Larina E.V., Kryukov I.A., Ivanov I.E. Trudy MAI, 2016, no. 91. URL: http://trudymai.ru/eng/published.php?ID=75565

  6. Kudimov N.F., Safronov A.V., Tret'yakova O.N. Trudy MAI, 2013, no. 70. URL: http://trudymai.ru/eng/published.php?ID=44440

  7. Kravchuk M.O., Kudimov N.F., Safronov A.V. Trudy MAI, 2015, no. 82. URL: http://trudymai.ru/eng/published.php?ID=58536

  8. Vu M.Kh., Popov S.A., Ryzhov Yu.A. Trudy MAI, 2012, no. 53. URL: http://trudymai.ru/eng/published.php?ID=29361

  9. Dehaeze F., Barakos G.N., Batrakov A.S., Kusyumov A.N., Mikhailov S.A. Simulation of flow around aerofoil with DES model of turbulence, Trudy MAI, 2012, no. 59. URL: http://trudymai.ru/eng/published.php?ID=34840

  10. Dauchot O., Daviaud F. Finite-amplitude perturbation and spots growth mechanism in plane Couette flow, Physics of Fluids, 1995, no. 7, pp. 335-343. DOI: 10.1209/0295-5075/28/4/002

  11. Orszag Steven A., Kells Lawrence C. Transition to turbulence in plane Poiseuille and plane Couette flow, Journal of Fluid Mechanics, 1980, no. 96, pp. 59-205. DOI: 10.1017/S0022112080002066

  12. Menter F.R. Zonal two equation k-w turbulence models for aerodynamic flows, AIAA Paper, 1993, N93-2906, pp. 21. DOI: 10.2514/6.1993-2906

  13. Shih T.-H., Liou W.W., Shabbir A., Yang Z., and Zhu J. A New k-e Eddy-Viscosity Model for High Reynolds Number Turbulent Flows – Model Developmentand Validation, Computers Fluids, 1995, vol. 24, no. 3, pp. 227-238.

  14. Launder B.E., Reece G.J., Rodi W. Progress in the Development of a Reynolds-Stress Turbulence Closure, Journal of Fluid Mechanics, April 1975, vol. 68, no. 3, pp. 537-566. DOI: 10.1017/S0022112075001814

  15. Spalart P.R. Strategies for turbulence modeling and simulation, International Journal of Heat and Fluid Flow, 2000, vol. 21, no. 3, pp. 252-263. DOI: 10.1016/S0142-727X(00)00007-2

  16. Berezko M.E., Nikitchenko Yu.A. Trudy MAI, 2020, no. 110. URL: https://trudymai.ru/eng/published.php?ID=112842. DOI: 10.34759/trd-2020-110-8

  17. Nikitchenko Yu.A. Aerospace MAI Journal, 2014, vol. 21, no. 4, pp. 39-48.

  18. Yakhot V., Orszag S.A., Thangam S., Gatski T.B., Speziale C.G. Development of turbulence models for shear flows by a double expansion technique, Physics of Fluids, 1992, vol. 4, no. 7, pp. 510–520. DOI: 10.1063/1.858424

  19. Shelest A.V. Metod Bogolyubova v dinamicheskoi teorii kineticheskikh uravnenii (Bogolyubov's method in the dynamic theory of kinetic equations), Moscow, Nauka, 1990, 159 p.

  20. Landau L.D., Lifshits E.M. Fluid Mechanics, Butterworth-Heinemann, 1987, vol. 6, 558 p

  21. Lifshits E.M., Pitaevsky L.P. Physical kinetics, Butterworth-Heinemann, 2002, vol. 10, 536 p.

  22. Khatuntseva O.N. Trudy MAI, 2018, no. 102. URL: http://trudymai.ru/eng/published.php?ID=98854


Download

mai.ru — informational site MAI

Copyright © 2000-2024 by MAI

 Вход