Linear Stability of Inviscid Plane-Parallel Flows of Vibrationally Excited Diatomic Gases

This chapter is devoted to investigations of linear stability of plane-parallel flows of an inviscid nonheat-conducting vibrationally excited gas. Some classical results of the theory of linear stability of ideal gas flows, such as the first and second Rayleigh’s theorems and Howard’s theorem, are generalized. An equation of the energy balance of disturbances is derived, which shows that vibrational relaxation generates an additional dissipative factor, which enhances flow stability. Calculations of the most unstable inviscid modes with the maximum growth rates in a free shear layer are described. It is shown that enhancement of excitation of vibrational modes leads to reduction of the growth rates of inviscid disturbances. Linear stability of plane-parallel shear flows are traditionally studied within the framework of the hydrodynamic stability theory. Such studies for an ideal incompressible fluid were performed in the classical works of Helmholtz, Kelvin, and Rayleigh. Later on, their results were extended to more realistic problems of ideal compressible gas flows and inhomogeneous stratified and conducting fluid flows in fields of various mass forces [1]. This chapter describes the results of studies of linear stability of plane-parallel shear flows of an inviscid non-heat-conducting vibrationally excited compressible gas. Linear equations for inviscid disturbances derived by linearization of the original system (1.27) with respect to a spatially homogeneous steady flow are formulated in Sect. 2.1. In Sect. 2.2, it is proved by using energy integrals that vibrational relaxation is an additional dissipation factor, which enhances flow stability. Generalization of the Rayleigh’s classical first and second theorems is obtained as necessary conditions for instability enhancement in the flows considered. Under certain conditions, a range of eigenvalues of unstable perturbations is specified in the upper complex half-plane as a counterpart of Howard’s semicircle theorem. In the limit there is a continuous transition to well-known results for an ideal fluid as the Mach number and the vibrational relaxation time τ approach zero and for an ideal compressible gas as τ approaches zero. The results of numerical calculations of eigenvalues and eigenfunctions of the most unstable inviscid modes in a free shear layer are presented in Sect. 2.3. Their © Springer International Publishing AG 2017 Y.N. Grigoryev and I.V. Ershov, Stability and Suppression of Turbulence in Relaxing Molecular Gas Flows, Fluid Mechanics and Its Applications 117, DOI 10.1007/978-3-319-55360-3_2 35 36 2 Linear Stability of Inviscid Plane-Parallel Flows of Vibrationally ... dependencies on the Mach number of the carrier flow, τ , and the degree of thermal nonequilibrium are analyzed. The vorticity eigenfunctions of thesemodes are used as initial data for numerical calculations of nonlinear evolution of theKelvin–Helmholtz waves presented in Chap.7. 2.1 Equations of the Linear Stability Theory In the (x, y) coordinate plane we consider a shear flow in which the main (carrier) flow of uniform density, ρ0, and temperature, T0, is directed along the abscissa axis x and has a velocity profile Us = Us(y). The perturbed flow is described by the system of equations of two-temperature gas dynamics (1.27) (see also [2–4]). In dimensionless variables the system has the form