Stability of expanding accretion shocks for an arbitrary equation of state

We present a theoretical stability analysis for an expanding accretion shock that does not involve a rarefaction wave behind it. The dispersion equation that determines the eigenvalues of the problem and the explicit formulae for the corresponding eigenfunction profiles are presented for an arbitrary equation of state and finite-strength shocks. For spherically and cylindrically expanding steady shock waves, we demonstrate the possibility of instability in a literal sense, a power-law growth of shock-front perturbations with time, in the range of $h_c< h<1+2 {\mathcal {M}}_2$ , where $h$ is the D'yakov-Kontorovich parameter, $h_c$ is its critical value corresponding to the onset of the instability and ${\mathcal {M}}_2$ is the downstream Mach number. Shock divergence is a stabilizing factor and, therefore, instability is found for high angular mode numbers. As the parameter $h$ increases from $h_c$ to $1+2 {\mathcal {M}}_2$ , the instability power index grows from zero to infinity. This result contrasts with the classic theory applicable to planar isolated shocks, which predicts spontaneous acoustic emission associated with constant-amplitude oscillations of the perturbed shock in the range $h_c< h<1+2 {\mathcal {M}}_2$ . Examples are given for three different equations of state: ideal gas, van der Waals gas and three-terms constitutive equation for simple metals.

Thermal instability revisited

ABSTRACT Field’s linear analysis of thermal instability is repeated using methods related to Whitham’s theory of wave hierarchies, which brings out the physically relevant parameters in a much clearer way than in the original analysis. It is also used for the stability of non-equilibrium states and we show that for gas cooling behind a shock, the usual analysis is only quantitatively valid for shocks that are just able to trigger a transition to the cold phase. A magnetic field can readily be included and we show that this does not change the stability criteria. By considering steady shock solutions, we show that almost all plausible initial conditions lead to a magnetically dominated state on the unstable part of the equilibrium curve. These results are used to analyse numerical calculations of perturbed steady shock solutions and of shocks interacting with a warm cloud.

Mach stem deformation in pseudo-steady shock wave reflections

The deformation of the Mach stem in pseudo-steady shock wave reflections is investigated numerically and theoretically. The numerical simulation provides the typical flow patterns of Mach stem deformation and reveals the differences caused by high-temperature gas effects. The results also show that the wall jet, which causes Mach stem deformation, can be regarded as a branch of the mainstream from the first reflected shock. A new theoretical model for predicting the Mach stem deformation is developed by considering volume conservation. The theoretical predictions agree well with the numerical results in a wide range of test conditions. With this model, the wall-jet velocity and the inflow velocity from the Mach stem are identified as the two dominating factors that convey the influence of high-temperature thermodynamics. The mechanism of high-temperature gas effects on the Mach stem deformation phenomenon are then discussed.

The Mechanism of Three-Dimension Steady Shock Wave Interaction

The problem of three-dimensional steady shock wave interaction is a key issue for supersonic and hypersonic corner flow. Due to the complexity of shock configurations, there is no analytical theory to such problem and the mechanism of three-dimensional shock waves and boundary layer interaction has not been clearly known. In this paper, an analytical approach to the problem of three-dimensional steady shock wave interaction was exhibited to analytically interpret the mechanism of three-dimensional interaction of two oblique planar shock waves. The results showed that the problem of three-dimensional steady shock wave interaction could be transformed to that of two moving shock wave interaction in two-dimensional plane, and there are various interaction configurations such as regular interaction, Mach interaction and weak interaction. The mechanism of three-dimensional shock wave interaction is helpful to understand the complex flow mechanism induced by three-dimensional shock wave and boundary layer in hypersonic flow. The interaction of three-dimensional shock waves and boundary layer plays important role in the complex flow feature in hypersonic rudder region. The contact surface induced by three-dimensional shock waves represents a local jet. When the flow jet impinges on the boundary layer of wall surface, the jet makes the boundary layer thinner and will inevitably cause local heat flux peak. The interaction configurations of three-dimensional shock wave play important role in the gasdynamic heating mechanisms of hypersonic complex flow.

Optimal regular reflection of shock and blast waves

The regular reflection of an oblique steady shock in supersonic gas flow is considered. The static pressure extremum conditions after the point of reflection of the shock with fixed strength depending on oncoming flow Mach number are determined analytically. The obtained results are applied to solution of the mechanically equivalent problem of the reflection of a propagating shock from an inclined surface. Non-monotonic variation of the mechanical loads on the obstacle with respect to its inclination angle is shown; the obstacle slope angles that correspond to pressure minima downwards of the unsteady shock reflection point are determined analytically.