On finding the critical Reynolds number of laminar-turbulent transition in Hagen-Poiseuille problem

Аuthors

Khatuntseva O. N.

e-mail: olga.khatuntseva@rsce.ru

Abstract

In the previous article [1], two solutions for the Hagen-Poiseuille problem were analytically obtained based on the Navier-Stokes equations with account for the entropy producing caused by the stochastic perturbations. The first solution corresponds to the laminar mode, and the second relates to the turbulent flow mode. The solution related to the turbulent flow mode at the central part of the tube was specified by the logarithmic velocity profile with the multiplier inversely proportional to the von Karman constant analytically determined in [1]. The solutions corresponding to the laminar and turbulent modes differ slightly directly at the tube wall. In the present work, this circumstance allows determine the minimal Reynolds number at which the transition fr om the laminar to the turbulent flow mode is feasible. The method of “discontinuous” functions was employed for these purposes.

The method of “discontinuous” functions description was offered in [2]. It may be used to describe the processes subjected to jump-like transitions, applied to the physical processes that can be uniquely described by   functions in all domain of α, except the small subdomains, where these functions change their values (or values of derivatives). In such subdomains the functions may demonstrate random behavior (if tracking the argument variation many times). The task of the technique consists in searching for the relations linking the functions values and their derivatives at the subdomains boundaries with dimensions and positions of these subdomains relative to the considered domains of definition of α.

The laminar-turbulent transition completely satisfies for this model of “discontinuous” functions behavior. With small Reynolds number values only the laminar mode of a liquid flow can be realized, while with Reynolds number values exceeding certain critical value both laminar and turbulent flow modes can be realized. At that, with Reynolds number increasing the turbulent flow mode stability increases, while the laminar flow mode stability decreases. The laminar to turbulent mode transition at Reynolds numbers exceeding the critical occurs jump-like and randomly.

The critical Reynolds number found in the work is approximately 1970.

Keywords:

turbulence, Hagen-Poiseulle problem, laminar-turbulent transition, critical Reynolds number, method of “discontinuous” functions

References

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

2. Khatuntseva O.N. Sibirskii zhurnal vychislitel’noi matematiki, 2009, vol. 12, no. 2, pp. 231 – 241.

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

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

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

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

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

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

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

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

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

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

13. Priymak V.G., Miyazaki T. Direct numerical simulation of equilibrium spatially localized structures in pipe flow, Physics of Fluids, 2004, vol. 16, no. 12, pp. 4221 – 4234.

14. Willis A.P., Kerswell R.R. Critical behavior in the relaminarization of localized turbulence in pipe flow, Physical Review Letters, 2007, vol. 98, pp. 014501(4).

15. Avila M., Mellibovsky F., Roland N., Hof B. Streamwise Localized Solutions at the Onset of Turbulence in Pipe Flow, Physical Review Letters, 2013, vol. 110, P. 224502.

16. Drazin P.G. Introduction to hydrodynamic stability, Cambridge University Press, 2002, 258 р.

17. Monin A.S. and Yaglom A.M. Statistical Fluid Mechanics: Mechanics of Turbulence, Dover Publications, 2007, 896 p.

18. Schlichting H. Boundary layer theory, New York, McGraw-Hill, 1955, 535 p.

19. Landau L.D., Lifshitz E.M. Fluid Mechanics, Butterworth-Heinemann. 1987, vol. 6, 539 p.

20. Pavel’ev A.A., Reshmin A.I., Teplovodskii S.Kh., Fedoseev S.G. Izvestiya RAN. Mekhanika zhidkosti i gaza, 2003, no. 4. C. 35 – 43.

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