On finding a generalized analytical solution to the Hagen-Poiseuille problem for a turbulent flow mode

DOI: 10.34759/trd-2021-118-02

Аuthors

Khatuntseva O. N.

e-mail: olga.khatuntseva@rsce.ru

Abstract

The modification of the Navier-Stokes equations by accounting for the additional degrees of freedom associated with the excitation of stochastic pulsations in the fluid flow allowed distinguishing two solutions to the fluid flow problem in a circular pipe (the Hagen-Poisel problem). One of these solutions is implemented for any values of Reynolds number and corresponds to the laminar flow regime, the second one is implemented only for large enough values of the Reynolds number and corresponds to the turbulent flow regime.

However, the boundary conditions, i.e. of the liquid «sticking» on the walls of the pipe with the of a linear velocity profile formation near the walls along the length of the viscous layer are V+ ~ y+. The zero derivative of the velocity in its center herewith allows obtaining a «smooth» solution to the problem for an arbitrary Reynolds number only in the case of a laminar flow regime.

Boundary conditions that arbitrarily «fixing» a solution at two or more points generally do not guarantee existence of a smooth solution to ordinary differential equations (ODE) or partial differential equations (PDE), even if these equations obey Cauchy’s existence and uniqueness theorem.

The absence of a smooth solution of the ODE or the PDE in the entire domain under study can be considered from the point of view of the existence of two or more asymptotes of the solution, as well as areas of uncertainty between them. A function of two summand, each of which is the product of two functions can be considered as a generalized (in the sense defined in the article) an ODE or a PDE solution. One function defines one of the solution asymptotes, while the other one defines the degree of this asymptote effect of the overall solution in each point of the area under study.

From this point of view, the presented work considers a generalized solution of the Hagen-Poiseuille problem for the turbulent flow regime of a liquid. One asymptote of the solution satisfies the boundary condition of liquid «sticking» on the pipe wall, while the second asymptote of the solution is a constant setting zero velocity derivative on its axis. A comparison with experimental data for the universal velocity profile in the near-wall flow region is presented.

Keywords:

stochastic systems, probability density, turbulence, Hagen-Poiseuille problem

References

1. Khatuntseva O.N. Trudy MAI, 2018, no. 100. URL: http://trudymai.ru/eng/published.php?ID=93311

2. Khatuntseva O.N. Trudy MAI, 2019, no. 108. URL: http://trudymai.ru/eng/published.php?ID=109382. DOI: 10.34759/trd-2019-108-3

3. Khatuntseva O.N. Trudy MAI, 2019, no. 106. URL: http://trudymai.ru/eng/published.php?ID=105673

4. Khatuntseva O.N. Trudy MAI, 2019, no. 104. URL: http://trudymai.ru/eng/published.php?ID=102091

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

6. Larina E.V., Kryukov I.A., Ivanov I.E. Trudy MAI, 2016, no. 91. URL: http://trudymai.ru/eng/published.php?ID=75540

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

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

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

10. Do S.Z. Trudy MAI, 2014, no. 75. URL: http://trudymai.ru/eng/published.php?ID=49670

11. Krupenin A.M., Martirosov M.I. Trudy MAI, 2014, no. 75. URL: http://trudymai.ru/eng/published.php?ID=49676

12. Krupenin A.M., Martirosov M.I. Trudy MAI, 2013, no. 69. URL: http://trudymai.ru/eng/published.php?ID=43066

13. Makhrov V.P., Glushchenko A.A., Yur’ev A.I. Trudy MAI, 2013, no. 64. URL: http://trudymai.ru/eng/published.php?ID=36423

14. Gorbatenko S.A., Makhrov V.P., Yur’ev A.I. Trudy MAI, 2013, no. 64. URL: http://trudymai.ru/eng/published.php?ID=36459

15. Dehaeze F., Barakos G.N., Batrakov A.S., Kusyumov A.N., Mikhailov S.A. Trudy MAI, 2012, no. 59. URL: http://trudymai.ru/eng/published.php?ID=34840

16. Dauchot O., Daviaud F. Finite-amplitude perturbation and spots growth mechanism in plane Couette flow, Physics of Fluids, 1995, no. 7, pp. 335 — 343.

17. Bottin S., Daviaud F., Manneville P., Dauchot O. Discontinuous transition to spatiotemporal intermittency in plane Couette flow, Europhysics Letters, 1998, no. 43, pp. 171 — 176.

18. Tuckerman Laurette S., Kreilos T, Schrobsdorff H., Schneider Tobias M., Gibson John F. Turbulent-laminar patterns in plane Poiseuille flow, Physics of Fluids, 2014, no. 26 (11). DOI: 10.1063/1.4900874

19. 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.

20. Pavlovskii V.A., Khitrykh D.P., Malamanov S.Yu. Morskie intellektual’nye tekhnologii, 2018, vol. 1, no. 2 (40), pp. 138 — 143.

21. Landau L.D., Lifshitz E.M. Fluid Mechanics, 1987, vol. 6 (2nd ed.). Butterworth-Heinemann. ISBN 978-0-08-033933-7. — 731 p.

22. Schlichting H. Boundary layer theory, London, 1955, 535 p.

23. Drazin F. Vvedenie v teoriyu gidrodinamicheskoi ustoichivosti (Introduction to hydrodynamic stability), Moscow, Fizmalit, 2005, 288 p.

24. Monin A. S., Yaglom A.M. Statistical Fluid Mechanics: The Mechanics of Turbulence. Part I, 1971, Part II, 1976.

25. Nikitin N.V. Izvestiya RAN. Mekhanika zhidkosti i gaza, 1994, no. 6, pp. 14 — 26.

26. Nikitin N.V., Pimenov V.O. Izvestiya RAN. Mekhanika zhidkosti i gaza, 2015, no. 5, pp. 64 — 75.

    
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