On weak shock layer stability in the continuum limit

Abstract

While accurate description of the internal structure of strong shock layers is beyond the scope of the Navier–Stokes equations [2, 4, 18, 30], continuum equations may still be used to describe weak shock layers at low Mach numbers. The present work extends the steady laminar base flow model of [11] and the two-dimensional modal linear stability analysis of [8] to a full three-component base flow and to two- and three-dimensional perturbations, solving the resulting eigenvalue problem by two structurally independent strategies: a matrix-forming Chebyshev collocation and an orthonormalized shooting method. The main findings are: (i) the full-spectrum-structure map, $$d(\omega )=0$$ d ( ω ) = 0 , built semi-analytically from the far-field characteristic exponents [12], locating the entire continuous spectrum over the complex frequency plane and used throughout as the reference against which the two discretization strategies are compared; (ii) no discrete eigenvalues and no unstable members of the continuous spectrum found, consistent with the results of [8] and in line with the recent analytical, Evans-function-based stability results [13]; (iii) the spectrum is proved to depend on the transverse wavenumbers $$k_y,k_z$$ k y , k z only through their magnitude $$k_t=\sqrt{k_y^2+k_z^2}$$ k t = k y 2 + k z 2 , an exact SO(2) covariance of the linearized operator and an analog of Squire’s theorem, from which three-dimensional perturbations follow to be damped by exactly the same mechanism as, and always at least as strongly as, their two-dimensional counterparts; (iv) the least-damped continuous branches become less damped as the disturbance wavelength grows toward macroscopic scales, without ever crossing into instability; (v) the least-damped eigenfunctions display a damped oscillatory decay on the hot (downstream) side and, more generally, separate by branch into shock-localized acoustic disturbances and non-localized entropy/vorticity disturbances carried downstream; (vi) a general three-component steady base flow necessarily reduces, under a boundedness constraint, to the plane oblique shock, whose frequency spectrum is a rigid Doppler shift of its normal shock counterparts, damping rates remaining unchanged.