Abstract
An energy balance equation for plane-parallel flows of a vibrationally excited diatomic gas described by a two-temperature relaxation model is derived within the framework of the nonlinear energy theory of hydrodynamic stability. A variational problem of calculating critical Reynolds numbers Recr determining the lower boundary of the possible beginning of the laminar-turbulent transition is considered for this equation. Asymptotic estimates of Recr are obtained, which show the characteristic dependences of the critical Reynolds number on the Mach number, bulk viscosity, and relaxation time. A problem for arbitrary wave numbers is solved by the collocation method. In the realistic range of flow parameters for a diatomic gas, the minimum critical Reynolds numbers are reached on modes of streamwise disturbances and increase approximately by a factor of 2.5 as the flow parameters increase.
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References
Yu. N. Grigor’ev and I. V. Ershov, “Effect of Bulk Viscosity on Kelvin-Helmholtz Instability,” Prikl. Mekh. Tekh. Fiz. 49(3), 73–84 (2008) [Appl. Mech. Tech. Phys. 49 (3), 407–416 (2008)].
Yu. N. Grigor’ev and I. V. Ershov, “Energy Estimate of the Critical Reynolds Numbers in a Compressible Couette Flow. Effect of Bulk Viscosity,” Prikl. Mekh. Tekh. Fiz. 41(5), 59–67 (2010) [Appl. Mech. Tech. Phys. 51 (5), 669–675 (2010)].
V. M. Zhdanov and M. E. Alievskii, Transfer and Relaxation Processes in Molecular Gases (Nauka, Moscow, 1989) [in Russian].
E. A. Nagnibeda and E. V. Kustova, Kinetic Theory of Transfer and Relaxation Processes in Nonequilibrium Reactive Gas Flows (Izd. St.-Petersburg. Gos. Univ., St. Petersburg, 2003) [in Russian].
M. A. Goldshtik and V. N. Shtern, Hydrodynamic Stability and Turbulence (Nauka, Novosibirsk, 1977) [in Russian].
Ya. D. Tamarkin, Some General Problems of the Theory of Ordinary Differential Equations (Tipografiya M. P. Frolovoi, Petrograd, 1917) [in Russian].
I. N. Bronshtein and K. A. Semendyuaev, Reference Book in Mathematics for Engineers and Students of Technical Institutes (Nauka, Moscow, 1969) [in Russian].
M. N. Naimark, Linear Differential Operators (Nauka, Moscow, 1969) [in Russian].
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, Berlin, 1988).
G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York (1961).
C. B. Moler and G. W. Stewart, “An Algorithm for Generalized Matrix Eigenvalue Problems,” SIAM J. Numer. Anal. 10(2), 241–256 (1973).
Yu. N. Grigor’ev, I. V. Ershov, and E. E. Ershova, “Influence of Vibrational Relaxation on the Pulsation Activity in Flows of an Excited Diatomic Gas,” Prikl. Mekh. Tekh. Fiz. 45(3), 15–23 (2004) [Appl. Mech. Tech. Phys. 45 (3), 321–327 (2004)].
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Original Russian Text © Yu.N. Grigor’ev, I. V. Ershov.
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 53, No. 4, pp. 57–73, July–August, 2012.
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Grigor’ev, Y.N., Ershov, I.V. Critical Reynolds number of the couette flow of a vibrationally excited diatomic gas energy approach. J Appl Mech Tech Phy 53, 517–531 (2012). https://doi.org/10.1134/S0021894412040062
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DOI: https://doi.org/10.1134/S0021894412040062