Abstract
The chapter contains the results of numerical and analytical studies of linear stability of a supersonic Couette flow of a vibrationally excited gas. Properties of even and odd inviscid modes of disturbances are analyzed as functions of the Mach number, depth of excitation of vibrational levels, and characteristic relaxation time. The general structure of the spectrum of plane perturbations is studied for finite Reynolds numbers. Two most unstable acoustic viscous modes are identified. Results calculated with using the constant viscosity model and Sutherland’s law are compared. Neutral stability curves are obtained, which show that the dissipative effect of vibrational mode excitation is inherent in both models of viscosity. The relative increase in the critical Reynolds number caused by excitation is approximately 12%.
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References
Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (2004)
Romanov, V.A.: Stability of a plane-parallel Couette flow. Dokl. Akad. Nauk SSSR 196, 1049–1051 (1971) (in Russian)
Gol’dshtik, M.A., Shtern, V.N.: Hydrodynamic Stability and Turbulence. Nauka, Novosibirsk (1977) (in Russian)
Joseph, D.D.: Stability of Fluid Motion. Springer, Berlin (1976)
Hanifi, A., Henningson, D.S.: The compressible inviscid algebraic instability for streamwise independent disturbances. Phys. Fluids 10, 1784–1786 (1998)
Duck, P.W., Erlebacher, G., Hussaini, M.Y.: On the linear stability of compressible plane Couette flow. J. Fluid Mech. 258, 131–165 (1994)
Hu, S., Zhong, X.: Linear stability of viscous supersonic plane Couette flow. Phys. Fluids 10, 709–729 (1998)
Malik, M., Dey, J., Alam, M.: Linear stability, transient energy growth, and the role of viscosity stratification in compressible plane Couette flow. Phys. Rev. E 77, 036322(15) (2008)
Lin, C.C.: The Theory of Hydrodynamic Stability. University Press, New York (1966)
Nagnibeda, E.A., Kustova, E.V.: Non-equilibrium Reacting Gas Flows. Kinetic Theory of Transport and Relaxation Processes. Springer, Berlin (2009)
Grigor’ev, Yu.N., Ershov, I.V.: Stability of Flows of Relaxing Molecular Gases. Izd. Sib. Otd. Ross. Akad. Nauk, Novosibirsk (2012) (in Russian)
Ferziger, J.H., Kaper, H.G.: Mathematical Theory of Transport Processes in Gases. North Holland Publ. Comp, Amsterdam (1972)
Grigor’ev, Yu.N., Yershov, I.V.: Linear stability of an inviscid shear flow of a vibrationally excited diatomic Gas. J. Appl. Math. Mech. 75, 410–418 (2011)
Blumen, W.: Shear layer instability of an inviscid compressibled fluid. J. Fluid Mech. 40, 769–781 (1970)
Drazin, P.G., Howard, L.N.: Hydrodynamic stability of parallel flow of inviscid fluid. In: Chernyi, G.G. et al. (Eds.) Advance in Applied Mechanics, vol. 9, pp. 1–89. Academic Press, New York (1996)
Howard, L.N.: Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512 (1961)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)
Trefethen, L.N.: Spectral Methods in Matlab. Society for Industrial and Applied Mathematics. SIAM, Philadelphia (2000)
Grigor’ev, Yu.N., Ershov, I.V.: Energy estimate of the critical Reynolds numbers in a compressible Couette flow. Effect of bulk viscosity. J. Appl. Mech. Tech. Phys. 51, 669–675 (2010)
Grigor’ev, Yu.N., Ershov, I.V.: Critical Reynolds number of the Couette flow in a vibrationally excited diatomic gas. Energy approach. J. Appl. Mech. Tech. Phys. 53, 517–531 (2012)
Korn, G.A., Korn, T.M.: Mathematical Handbook for Scientists and Engineers. McGraw-Hill, New York (1961)
Michalke, A.: On the inviscid instability of the hyperbolic tangent velocity profile. J. Fluid Mech. 19, 543–556 (1964)
Morawetz, C.S.: The eigenvalues of some stability problems involving viscosity. J. Rat. Mech. Anal. 1, 79–603 (1952)
Mack, L.M.: On the inviscid acoustic-mode instability of supersonic shear flows. Part I: two-dimensional waves. Theor. Comput. Fluid Dyn. 2, 97–123 (1990)
Grigor’ev, Yu.N., Ershov, I.V.: Linear stability of the Couette flow of a vibrationally excited gas. 1. Inviscid problem. J. Appl. Mech. Tech. Phys. 55, 258–269 (2014)
Gaponov, S.A., Maslov, A.A.: Development of Perturbations in Compressible Flows. Nauka, Novosibirsk (1980) (in Russian)
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Grigoryev, Y.N., Ershov, I.V. (2017). Linear Stability of Supersonic Plane Couette Flow of Vibrationally Excited Gas. In: Stability and Suppression of Turbulence in Relaxing Molecular Gas Flows. Fluid Mechanics and Its Applications, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-55360-3_3
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DOI: https://doi.org/10.1007/978-3-319-55360-3_3
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