Energy estimate of critical Reynolds numbers in the supersonic Couette flow of a vibrationally excited diatomic gas

I. V. Ershov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 6 (2014), pp. 66-79

Résumé

The supersonic plane Couette flow of a vibrationally excited diatomic gas is investigated within the energy theory of hydrodynamic stability. The flow was described by a system of equations of two-temperature aerodynamics, which takes into account the dependence of the transport coefficients on the flow temperature. The corresponding spectral problem for the critical Reynolds number $\mathrm{Re_{cr}}$ determining the possible start of the laminar-turbulent transition was solved numerically using the method of collocations and $\mathrm{QZ}$-algorithm. The calculations showed that in the supersonic range, when $M > 1$, the calculated values $\mathrm{Re_{cr}}$ may exceed the corresponding values for subsonic Mach numbers $M > 1$ by about two orders of magnitude. Investigation of how $\mathrm{Re_{cr}}$ is affected by changes in the degree of vibrational energy excitation of gas molecules, vibrational relaxation time, bulk viscosity, and Mach number showed that the greatest impact on the increase in $\mathrm{Re_{cr}}$ at $M>1$ is exerted by the growth of the Mach number (compressibility). In the range of $M = 2\div5$, critical Reynolds numbers increase more than by an order of magnitude. However, the excitation of vibrational modes of gas molecules and the vibrational relaxation time which determine main effects at $M \leqslant 1$ have an effect at the same level with the transition to the supersonic regime.

Détail

Keywords: energy theory, hydrodynamic stability, equations of two-temperature aerodynamics, vibrational relaxation, critical Reynolds number.

Bibliographie

[1] Ershov I. V., “Ustoychivost' techeniya Kuetta kolebatel'no-neravnovesnogo molekulyarnogo gaza. Energeticheskiy podkhod”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2013, no. 3(25), 76–88 (in Russian)

[2] Gol'dshtik M. A., Shtern V. N., Gidrodinamicheskaya ustoychivost' i turbulentnost', Nauka Publ., Novosibirsk, 1977, 367 pp. (in Russian)

[3] Duck P. W., Erlebacher G., Hussaini M. Y., “On the linear stability of compressible plane Couette flow”, J. of Fluid Mech., 258 (1994), 131–165 | DOI | MR | Zbl

[4] Hu S., Zhong X., “Linear stability of viscous supersonic plane Couette flow”, Phys. of Fluids, 10:3 (1998), 709–729 | DOI | MR | Zbl

[5] Malik M., Dey J., Alam M., “Linear stability, transient energy growth, and the role of viscosity stratification in compressible plane Couette flow”, Physical Rev. E, 77:3 (2008), 036322(1)–036322(15) | DOI | MR

[6] Chepmen S., Kauling T., Matematicheskaya teoriya neodnorodnykh gazov, IL Publ., M., 1960, 510 pp. (in Russian) | MR

[7] Zhdanov V. M., Alievskiy M. E., Protsessy perenosa i relaksatsii v molekulyarnykh gazakh, Nauka Publ., M., 1989, 336 pp. (in Russian) | Zbl

[8] Nagnibeda E. A., Kustova E. V., Kineticheskaya teoriya protsessov perenosa i relaksatsii v potokakh neravnovesnykh reagiruyushchikh gazov, St. Petersburg Univ. Publ., St. Petersburg, 2003, 272 pp. (in Russian)

[9] Grigor'ev Yu. N., Ershov I. V., Ustoychivost' techeniy relaksiruyushchikh molekulyarnykh gazov, Izd-vo SO RAN, Novosibirsk, 2012, 230 pp. (in Russian)

[10] Ershov I. V., “Energeticheskaya otsenka kriticheskikh chisel Reynol'dsa v techenii Kuetta kolebatel'no-neravnovesnogo molekulyarnogo gaza”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2012, no. 2(20), 99–112 (in Russian)

[11] Canuto C., Hussaini M. Y., Quarteroni A., Zang T. A., Spectral methods in fluid dynamics, Springer series in Computational Physics, Springer-Verlag, Berlin, 1988, 564 pp. | MR | Zbl

[12] Trefethen L. N., Spectral methods in Matlab, Soc. for Industr. and Appl. Math., Philadelphia, 2000, 160 pp. | MR

[13] Korn G., Korn T., Spravochnik po matematike, Nauka Publ., M., 1970, 720 pp. (in Russian) | MR

[14] Moler C. B., Stewart G. W., “An algorithm for generalized matrix eigenvalue problems”, SIAM J. Numer. Anal., 10:2 (1973), 241–256 | DOI | MR | Zbl