Convective instability of air flows in the exhaust shaft above a four-row finned beam

Herein, multidirectional quasiperiodic air flows in an exhaust shaft above a four-order horizontal bundle consisting of bimetallic finned tubes used to remove heat in heat exchangers are considered. Modeling of the air movement is carried out on the basis of equations for thermogravitational convection in the Boussinesq approximation. It takes into account the viscosity of the air and the dependence of the air density on the temperature. An interpretation of quasiperiodic airstreams is proposed on the basis of Rayleigh – Bénard convection, as a result of which regular structures, called Rayleigh – Bénard cells, are formed in a liquid or gas. Rayleigh – Bénard cells are an analytical solution to the problem of the stability of hydrodynamics flows in the linear approximation. The appearance of two-dimensional (convective rolls) and threedimensional (rectangular cells) is possible. To estimate the number of emerging structures, the critical Rayleigh numbers were calculated, which characterizes the transition from an unstable mode of the convective fluid flow to a stable mode. For two experiments, the experimental Rayleigh numbers are compared with their critical values. The differences between the experimental conditions and the ideal boundary conditions used in the calculations and the partial destruction of quasiperiodic structures as a result of this are also discussed.

1. Bénard H. Les tourbillons cellulaires dans une nappe liquide. Revue Générale des Sciences Pures et Appliquées, 1900, vol. 11, pp. 1261–1271, 1309–1328.

2. Rayleigh L. On convective currents in a horizontal layer of fluid when the higher temperature is on the under side. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1916, vol. 32, no. 192, pp. 529–546.

3. Gershuni G. Z., Zhuhovickij E. M. Convective Stability of an Incompressible Fluid. Moscow, Nauka Publ., 1972. 392 p. (in Russian).

4. Graham A. Shear patterns in an unstable layer of air. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 1933, vol. 232, no. 707–720, pp. 285-296. https://doi.org/10.1098/rsta.1934.0008

5. Tippelskirch H. V. Über Konvectionzellen, insbesondere im flüssigen Schwefel. Beiträge zur Physik der Atmosphäre, 1956, Bd. 29, pp. 37–54 (in German).

6. Block M. J. Surface tension as the cause of Bénard cells and surface deformation in a liquid film. Nature, 1956, vol. 178, no. 4534, pp. 650–651. https://doi.org/10.1038/178650a0

7. Landau L. D., Livshic E. M. Hydrodynamics, 3d ed. Moscow, Nauka Publ., 1986. 736 p. (in Russian).

8. Getling A. V. Formation of spatial structures in Rayleigh–Bénard convection. Soviet Physics Uspekhi , 1991, vol. 34, no. 9, pp. 737–776. https://doi.org/10.1070/pu1991v034n09abeh002470

9. Manneville P. A two-dimensional model for three-dimensional convective patterns in wide containers. Journal de Physique France, 1983, vol. 44, no. 7, pp. 759–765. https://doi.org/10.1051/jphys:01983004407075900

10. Finlayson B. A. The Galerkin method applied to convective instability problems. Journal of Fluid Mechanics, 1968, vol. 33, no. 1, pp. 201–208. https://doi.org/10.1017/s0022112068002454

11. Palymskiy I. B. Numerical modeling of complex Releigh-Bénard convections regimes. Permian, 2012. 206 p. (in Russian).

12. Pandey A., Scheel J. D., Schumacher J. Turbulent superstructures in Rayleigh-Bénard convection. Nature Communications, 2018, vol. 9, no. 1. https://doi.org/10.1038/s41467-018-04478-0

13. Nosov V. V., Grigor’ev V. M., Kovadlo P. G., Lukin V. P., Nosov E. V., Torgaev E. V. Coherent structures in a turbulent atmosphere. Experiment and theory. Solnechno-zemnaya fizika = Solar-Terrestrial Physics, 2009, iss. 14, pp. 97–113.

14. Shmerlin B. Ja., Kalashnik M. V. Rayleigh convective instability in the presence of phase transitions of water vapor. The formation of large-scale eddies and cloud structures. Physics-Uspekhi, 2013, vol. 56, no. 5, pp. 473–485. https://doi.org/10.3367/ufne.0183.201305d.0497

15. Arzhanik A. R., Mihajlichenko Ju. P., Sotoriadi R. N. Staging demonstrations of Benard cells and Taylor vortices. Fizicheskoe obrazovanie v vuzah = Physics in Higher Education, 2000, vol. 6, no. 4, pp. 60–67 (in Russian).

16. Trapeznikov D. E., Suncov A. S., Rybal’chenko T. M. On the origin of columnar separation in basalts and its analogues. Vestnik Permskogo universiteta. Geologiya = Bulletin of Perm University. Geology, 2012, iss. 2 (15), pp. 8–15 (in Russian).

17. Berdnikov V. S., Vinokurov V. V., Panchenko V. I., Solov’ev S. V. Heat transfer in the classical Czochralski method. Journal of Engineering Physics and Thermophysics, 2001, vol. 74, no. 4, pp. 1007–1014. https://doi.org/10.1023/a:1012375827219

18. Travis B., Olson P., Schubert G. The transition from two-dimensional to threedimensional planforms in infinite-Prandtlnumber thermal convection. Journal of Fluid Mechanics, 1990, vol. 216, pp. 71–91. https://doi.org/10.1017/s0022112090000349

19. Bogorodskii V. V., Gusev A. V., Doronin Yu. P. [et al.] O cean P hysics. Leningrad, Gidrometeoizdat Publ., 1978. 294 p. (in Russian).

20. Nicolis G., Prigozhin I. Self-organization in Nonequilibrium Systems. Moscow, Mir Publ., 1979. 512 p. (in Russian).

21. Mil’man O. O. An experimental study of heat transfer during natural air circulation in an air condenser model with an exhaust shaft. Teploenergetika = Thermal Engineering, 2005, no. 5, pp. 16–19 (in Russian).

22. Suhockii A. B., Marshalova G. S. Features of the gravitational flow of heated air in an exhaust shaft above a multi-row finned beam. Journal of Engineering Physics and Thermophysics, 2019, vol. 92, no. 3, pp. 596–602. https://doi.org/10.1007/s10891-019-01967-x

23. Davis S. H. Convection in a box: linear theory. Journal of Fluid Mechanics, 1967, vol. 30, no. 3, pp. 465–478. https://doi.org/10.1017/s0022112067001545

24. Stork K., Müller U. Convection in boxes: experiments. Journal of Fluid Mechanics, vol. 54, no. 4, pp. 599–611. https://doi.org/10.1017/s0022112072000898

25. Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability. Oxford at the Clarendon Press, 1961. 654 p.

26. Palymskii I. B. Modeling complex Rayleigh-Benard convection regimes. Sibirskii zhurnal vychislitelnoi matematiki = Siberian Journal of Numerical Mathematics, 2011, vol. 14, no. 2, pp. 179–204 (in Russian).

27. Shvartsblat D. L. On the spectrum of perturbations and convective instability of a plane horizontal layer of a liquid with permeable boundaries. Journal of Applied Mathematics and Mechanics, 1968, vol. 32, no. 2, pp. 266–271. https://doi.org/10.1016/0021-8928(68)90127-5