Linear stability of high-speed mixing layers

Abstract

A newly developed spectral compressible linear stability code (SPECLS) is presented for analysis of shear flow stability and applied to high-speed boundary layers and free shear flows. The formulation utilizes the first application of a staggered mesh for a compressible flow analysis by a spectral technique and a multi-domain spectral discretization (MDSPD) option to resolve highly irregular structures. An order of magnitude less number of points is needed for equivalent accuracy in disturbance growth rates compared to calculations by a finite difference formulation, and a factor of three fewer points is required by the MDSPD relative to a single-domain spectral discretization (SPSPD). The mean flow in this study consists of two parallel gases, one of which is quiescent. A sudden appearance of multiple supersonic modes is found to occur near Mach 3. Their existence has not been observed in past studies. This occurence is attributed to the boundary conditions which impose zero perturbations (reflecting boundary conditions) of all disturbances in the far field. The condition is imposed sufficiently far that subsonic disturbances are unaffected (i.e., results match the case of an unbounded free shear layer). However, at approximately Mach 2 there are radiating solutions (supersonic disturbances) both above and below the shear region. The imposed “wall” far-field condition causes a multiplicity of higher supersonic modes at Mach numbers exceeding two. The issue of relevance in studying the stability of a free shear flow is the impact of transition on fuel-air mixing efficiency in scramjet combustors. The bounded free shear layer in this study is the proper analogue of this situation. In particular the observation of a pronounced drop in mixing efficiency as Mach number is increased may be related to these multiple modes which have very low growth rates relative to the subsonic disturbances which predominate at low Mach numbers. At M ∞ = 3 it is found that increasing the temperature of the quiescent gas relative to the injected gas in the mixing layer inhibits the development of these supersonic modes for low streamwise wave numbers. Verification of the method will be given for the case of boundary layers using an existing finite difference compressible stability code, and by comparison with analytical results obtained for free shear flows.