Linear stability in the internal structure of finite-thickness planar weak shock layers of ideal gases at low Knudsen numbers*

The internal structure of a steady (laminar) fully-resolved viscous shock layer of ideal gases in the continuum regime has been described by Gilbarg and Paolucci (1952). Somewhat surprisingly, the only work known to-date to have addressed linear stability of this flow has been that of Duck and Balakumar (1992). These authors, henceforth referred to as DB92, derived two-dimensional temporal linear stability equations (describing shock corrugation on the shock plane) and reported that no discrete exponentially-growing two-dimensonal eigenmodes could be identified. DB92 also commenced analysis of the corresponding initial-value problem and provided an analytic demonstration of the long-time structure of perturbations, as products of arbitrary functions of time and the exponentially decaying (in time) factor of modal analysis. However, complete solutions of the initial-value problem could not be found analytically, due to contamination of the spectrum by spurious numerical modes. The present work has revisited the DB92 analysis, also in a temporal context, extended the linearized disturbance equations in the third spatial dimension (into the plane along which the base flow is homogeneous) and re-introduced the Reynolds number (which was absorbed into the governing equations by DB92) explicitly into the governing equations. The three-dimensional eigenvalue problem has been solved for a sufficiently large number of discrete values of the pertinent parameters (combinations of transverse and lateral wavenumbers at given values of the Mach and Reynolds numbers) for the structure of the spectrum to be established and the influence of the wavenumber parameters to be understood, followed by a systematic scan of Mach and Reynolds numbers which addressed the effect of these parameters on weak shock layer stability; the corresponding initial-value problems were also solved numerically to reveal the early-time response of the shock layer to small-amplitude perturbations. Results obtained will be discussed at the time of the conference.