Stability of Step Shocks

Abstract

The hydrodynamic stability of a steady, plane, step shock through a fluid medium with arbitrary equation of state is investigated through consideration of the initial-value problem for the time-dependent hydrodynamic equations, linearized in perturbations from the steady flow. If the stability function, Fs = 1 + κ - κ2(v1 - v) pS/T (with κ the Mach number, v the specific volume, T the absolute temperature, pS the entropy derivative of pressure at constant volume, and subscript 1 referring to the preshock state) is negative, disturbances grow exponentially with time and the shock is unstable. The character of the shock Hugoniot with respect to stability is discussed.

Galloping instability of viscous shock waves

“…rectangular) crosssection; "spinning" instabilities for a circular cross-section, with one or more "hot spots", or "combustion heads", moving spirally along the duct. Stability of shocks and detonations may be studied within a unified mathematical framework; see, e.g., [Er1,Er2,Ko1,Ko2,D,BT,FD,LS,T,K,M1,M2,M3,FM,Me,CJLW] in the inviscid case (1.1), (1.5), and [S,Go1,Go2,KM,KMN,MN,L1,L3,GX,SX,GZ,ZH,Br1,Br2,BrZ,BDG,KK,Z1,Z2,Z3,Z4,MaZ2,MaZ3,MaZ4,GMWZ1,GWMZ2,GMWZ3,HZ,BL,LyZ1,LyZ2,JLW,…”

Unstable blast shocks in dilute granular flows

“…This problem has received much attention, starting with the work of Taylor and the work of Sedov, who predicted, independently from each other, the temporal evolution of the blast radius versus time in the case of energy conserving blasts [10,11]. The instability of such fronts has been studied since the 1960s with ongoing work as of the present date [12][13][14]. Several issues have to be examined such as the nature of the surrounding gas, the presence or not of different dissipation mechanisms as may happen in the presence of radiative processes, for example, and the exact details of the shell as well as the geometry of the blast [15][16][17][18].…”

Multidimensional Stability of Large-Amplitude Navier–Stokes Shocks

“…The mathematical study of shock stability, connected with properties of compressibility, initially lagged behind the study of other forms of hydrodynamic stability connected with incompressible flow, partly perhaps due to complexity of the problem and partly perhaps because shocks under most circumstances are extremely stable [14]. However, since its rigorous formulation by Erpenbeck [21] and others in the 1960's, the problem of shock stability has been the object of intensive investigation by a number of different authors. We mention in particular the "inviscid" studies of Erpenbeck, Majda, and Metivier [21,52,53,54,57], largely settling the question in the case where second-order transport effects of viscosity, heat conduction, magnetic resistivity, etc.…”

“…However, since its rigorous formulation by Erpenbeck [21] and others in the 1960's, the problem of shock stability has been the object of intensive investigation by a number of different authors. We mention in particular the "inviscid" studies of Erpenbeck, Majda, and Metivier [21,52,53,54,57], largely settling the question in the case where second-order transport effects of viscosity, heat conduction, magnetic resistivity, etc. are neglected, and their striking corollary, "Majda's Theorem:" for a polytropic ideal gas equation of state, inviscid shock waves are spectrally and nonlinearly stable, independent of shock amplitude, in one and multiple space dimensions.…”

“…Before starting, we make the standard observation [21,52] that, by rotational invariance of the Navier-Stokes equations, together with the assumed rotational symmetry (since constant) of a planar shock wave in transverse directions (i.e., transverse to the direction of propagation), it is sufficient in the study of spectral stability to restrict to the case of dimension two. Specifically, taking the Fourier transform in transverse directions, and denoting the resulting frequency as ξ, one finds that the eigenvalue equations depend only on |ξ|, hence reduce to the two-dimensional case.…”