Dispersive Shock Waves Research Articles

Shock Wave Structures in Gas-Particle Mixtures

Effects of the particle specific heat on shock wave structures in dusty gases are numerically investigated by the moment method for the Boltzmann equation. It is shown that for the small specific heat the particle temperature overshoots the Rankine-Hugoniot equilibrium value, provided a shock wave is strong, and that for the large specific heat the particle temperature increases monotonically and the gas temperature overshoots. Also it is shown that the frozen region disappears with the increasing specific heat, and the dispersed shock wave is formed.

Nonlinear coupling of polarized plasma waves

Weak nonlinear coupling between two polarized transverse waves in a weakly relativistic plasma is studied including the effect of dispersion. The frequency of the coupled waves is of the same order of magnitude as the electron plasma frequency. By using the multiple scale method it is shown that the slow modulation of the complex amplitudes is described by simultaneous nonlinear Schrödinger equations. Travelling wave solutions are then obtained for this system of equations by the analytical and the numerical methods. As the result of the wave-wave interaction, two envelope waves are, in general, composed of dispersive shock waves or non-periodic nonlinear wave-trains, while the usual solitary waves or periodic nonlinear wave-trains can exist as a special case. case. The results of analysis can readily be applied to other coupled waves with different polarizations in a nonlinear dispersive medium.

Supersonic Flow of a Dusty Gas over a Slender Wedge

Effects of solid or liquid particles contained in a gas are investigated for supersonic flow over a slender wedge by a perturbation method. The frozen shock wave, attached to the leading edge, decays with distance from the surface and bends backward. Various manners of the change in flow variables of the gas, depending on the properties of the mixture and the Mach number, in the relaxation zone behind the frozen shock wave are illustrated by examples. The development of the dispersed shock wave far from the wedge is studied too by the method of matched expansions.

The steady two-dimensional reflexion of an oblique partly dispersed shock wave from a plane wall

The steady two-dimensional problem of reflexion of an oblique partly dispersed plane shock wave from a plane wall is studied analytically. Viscosity, diffusion and heat conduction are neglected. The thermodynamic state of the gas is assumed to be determined by the instantaneous values of the specific entropy s, pressure p and a finite number of internal state variables. Results for the flow field behind the reflected shock are obtained by a perturbation method which is based on the assumption that the influence of relaxation is relatively weak.

The non-linear acoustics of a radiating gas. II. Weak shock waves

Sound pulses in which the disturbed region has small or large optical thickness are considered. An analogy is drawn between the influence on the non-stationary velocity field structure of radiant energy transport and of chemical transformations occurring in reacting mixtures. This analogy embraces both completely dispersed shock waves, and waves including discontinuities with incomplete dispersion. The present paper continues the section and figure numbering used in part I [1]. Our aim is to apply the approximate equations derived in [I] to the study of the structure of weak shock waves in a radiating gas.

Dynamic Initial Slip in a Linear Chain

A linear Frenkel-Kontorova model is used to study, within the harmonic approximation, dynamic effects on the Peierls stress that occur when a dislocation is subjected to a shock wave. At large distances into the lattice, a small reduction in the Peierls stress can occur due to dynamic amplification of the peak stress associated with a dispersive shock wave. A larger reduction in the Peierls stress is shown to occur far behind the head of the incident wave and is due to the frequency-of-dislocation motion falling within the spectrum of frequencies that contribute to the stress wave. The reduction is the same as that which would occur if the dislocation were subject to a sudden step jump in stress. Finally, the effect of wave reflections along a slip plane is considered, and the unacceptable consequence of using a one-dimensional model in this last problem is demonstrated.

Real-gas effects in very weak shock waves in the atmosphere and the structure of sonic bangs

An approximate expression is given for the thickness of weak fully dispersed shock waves. Using available data on the thermodynamic properties of air, it is shown that shocks of the strength expected in sonic bangs are fully dispersed. Estimated relaxation times for dry and humid air lead to wide variations in possible thickness, varying from millimetres to metres.

Thermal Decomposition of N2O4 and NO2 by Shock Waves

A study of the reactions N2O4 + M⇄2NO2 + M and 2NO2 + M⇄2NO + O2 + M was carried out using the shock-wave method. For the first of these two reactions the rate equation was taken to be d[N2O4] / dt = −kD1[N2O4][M] + kR1[NO2]2[M] based on previous results. For argon as the inert gas M, with a reactant mole fraction less than 0.1, the rate constant was found to be kD1 = 2.2 × 1014exp(− 11 000 / RT) liter/mole·sec. Incorporated in these results was the new application of fully dispersed shock waves and these weak waves were found to produce more accurate results than the conventional use of partly dispersed shock waves. For the second reaction, the rate equation was taken to be d[NO2] / dt = −kD2[NO2]2 − kD3[NO2][M], and with argon as the inert gas the dissociation rate constants were found to be kD2 = 3.0 × 109exp(− 26 900 / RT) liter/mole·sec and kD3 = 3.6 × 1013exp(− 65 400 / RT) liter/mole·sec. An analytical description of the flow fields agreed well with the experimental values which were determined by a light absorption technique.

Bildung schwacher stationärer Stosswellen in relaxierenden Gasgemischen

The formation of a fully dispersed shock wave in a binary mixture of relaxing gases is investigated. It is assumed that the gas is bounded by a piston at the left, and at timet=0 is in static thermodynamic equilibrium. The piston velocity changes from zero att=0 to a constant non-zero value att>0. A uniformly valid approximation for the resulting wave motion is found by the method of matched asymptotic expansions. It turns out that the formation of a steady shock wave depends on the temperature dependence of the vibrational specific heats. If the coefficient Ω of the non-linear term in Burgers equation is greater than zero, a steady shock wave is formed. However, for Ω≤0 this is not the case. One finds that the cases Ω≤0 are not realized in nature. Numerical results for the shock formation in relaxing air indicate, how the accuracy of acoustic theory decreases due to non-linear effects as time proceeds.

Unsteady shock propagation in a relaxing gas

A study is made of the motion of the shock wave produced when a piston is impulsively set in motion with constant velocity into a diatomic gas at rest, conditions being such that behind the shock there is a zone of vibrational or dissociational relaxation. The dissociation case is treated by the method of linearized characteristics, using a form for the rate equation due to Freeman (1958), while the vibrational case is included in a discussion in which the third-order equation for acoustic disturbances derived by Chu (1958) is applied to the flow behind the shock. It is shown that in either case the initial speed of the shock is that corresponding to frozen flow between it and the piston, but that the lower speed calculated from equilibrium thermodynamics is approached at large times. Distributions of pressure and velocity between the piston and the shock are found: at long times after the start of the motion these are precisely those given by the Bethe-Teller (1941) theory for partly dispersed shock waves. Some applications to other shock motions are discussed.

On fully-dispersed shock waves in carbon dioxide

A dispersed shock wave may be defined as one in which finite changes occur over distances large compared to the mean free path in the gas. In contrast a shock wave in air extends over only a few mean free paths. When internal motions in a molecule are excited rather slowly by collisions, as is the case for molecular vibration, the shock wave may be partly dispersed; then, the sharp shock front is followed by a diffuse tail leading to complete thermal equilibrium. Alternatively, it may be fully dispersed, so that adjustments in the energy in all the degrees of freedom proceed slowly and in parallel. The purpose of this note is to point out that within a narrow speed range, from a shock Mach number of 1 to 1.042, shocks in carbon dioxide are fully-dispersed in the above sense. Such waves have been observed experimentally using a shock tube and interferometer. The possible existence of such waves was first pointed out by Bethe & Teller (1941) purely as a matter of academic interest. This note treats the problem in the same spirit.