Radiative shock solutions with grey nonequilibrium diffusion

This study describes a semi-analytic solution of planar radiative shock waves with a grey nonequilibrium diffusion radiation model. The solution may be used to verify radiation-hydrodynamics codes. Comparisons are made with the equilibrium diffusion solutions of Lowrie and Rauenzahn (Shock Waves 16(6):445–453, 2007). The solution also gives additional insight into the structure of radiative shocks. Previous work has assumed that the material temperature reaches its maximum at the post-shock state of the embedded hydrodynamic shock (Zel’dovich spike). We show that in many cases, the temperature may continue to increase after the hydrodynamic shock and reaches its maximum at the isothermal sonic point. Also, a temperature spike may exist even in the absence of an embedded hydrodynamic shock. We also derive an improved estimate for the maximum temperature.      

Shock wave structure in a diatomic gas based on a kinetic model

The structure of a normal (direct) shock in a gas for the parameters corresponding to nitrogen is investigated with allowance for the rotational degrees of freedom on the basis of a model kinetic equation. For various Mach numbers the structure is compared with both the known experimental results and the solutions of the Navier-Stokes approximation within the framework of two-temperature hydrodynamics. The possibility of assuming the constancy of the fraction of excited rotational degrees of freedom is studied.     

The quasi-stationary structure of radiating shock waves. II. The two-temperature fluid

We solve the equations of radiation hydrodynamics in the two-temperature fluid approximation on an adaptive grid. The temperature structure depends upon the electron-ion energy exchange length, , and the electron conduction length, . Three types of radiating shock structure are observed: subcritical, where preheating of the unshocked gas is negligible; electron supercritical, where radiation preheating raises the temperature of the unshocked electron fluid to be equal to the final electron temperature; supercritical, where preheating and electron-ion energy exchange raise the preshock to their final post shock values. No supercritical shock develops when is larger than the photospheric depth of the shocked gas because a negligible amount of the ion energy is transferred to the electrons and the shock is weakly radiating. Electron conduction smooths the profile on a length scale , reducing the radiation flux. Received 21 September 1998 / Accepted 15 January 1999     

Mach number effects on shock-bubble interaction

An investigation of Mach number effects on the interaction of a shock wave with a cylindrical bubble, is presented. We have conducted simulations with the Euler equations for various incident shock Mach numbers () in the range of , using high-resolution Godunov-type methods and an implicit solver. Our results are found in a very good agreement with previous investigations and further reveal additional gasdynamic features with increasing the Mach number. At higher Mach numbers larger deformations of the bubble occur and a secondary-reflected shock wave arises upstream of the bubble. Negative vorticity forms at all Mach numbers, but the “c-shaped” vortical structure appeared at gives its place to a circular-shaped structure at higher Mach numbers. The computations reveal that the (instantaneous) displacements of the upstream, downstream and jet interfaces are not significantly affected by the incident Mach number for values (approximately) greater than . With increasing the incident Mach number, the speed of the jet (arising from the centre of the bubble during the interaction) also increases. Received 21 December 2000 / Accepted 23 April 2001     

Certain three-dimensional flows with Mach interaction of shock waves

In [1, 2] it is shown that there is a range of values of the Mach number and geometric parameters for which flow past conical bodies is realized with plane attached shocks and regular intersection of the shocks in space. In the present article we determine the class of solutions corresponding to flow past conical star-shaped bodies with a configuration of the shocks in space of the Mach type.     

Submerged shocks in conical gas flows in the presence of a mach shock-wave configuration

The results of a numerical study of a new type of singularities in the Mach shock-wave structure realized in supersonic nonsymmetric conical flows over V-wings with a bow shock attached to the leading edges are presented. Within the framework of the ideal gas model we study the changes in the shock system on transition, with increase in the sweep angle, from the region of nonsymmetric Mach interaction of the shocks attached to the leading edges of the wing to the region of special flow patterns, where on the windward cantilever surface a rarefaction flow is realized rather than a flow with an internal shock. It is shown, in particular, that in the region with special wing flow patterns a Mach system of shocks with a submerged shock proceeding from the branch point above the windward cantilever may exist.     

An experimental investigation of the sonic criterion for transition from regular to Mach reflection of weak shock waves

The signal speed, namely the local sound speed plus the flow velocity, behind the reflected shocks produced by the interaction of weak shock waves (M _i < 1.4) with rigid inclined surfaces has been measured for several shock strengths close to the point of transition from regular to Mach reflection. The signal speed was measured using piezo-electric transducers, and with a multiple schlieren system to photograph acoustic signals created by a spark discharge behind a small aperture in the reflecting surfaces. Both methods yielded results with equal values within experimental error. The theoretical signal speeds behind regularly reflected shocks were calculated using a non-stationary model, and these agreed with the measured results at large angles of incidence. As the angle of incidence was reduced, for the same incident shock Mach number, so as to approach the point of transition from regular to Mach reflection, the measured values of the signal speed deviated significantly from the theoretical predictions. It was found, within experimental uncertainty, that transition from regular to Mach reflection occurred at the experimentally observed sonic point, namely, when the signal speed was equal to the speed of the reflection point along the reflecting surface. This sonic condition did not coincide with the theoretical value.     

Effects of three-phase-lag on two-temperature generalized thermoelasticity for infinite medium with spherical cavity

The thermoelastic interaction for the three-phase-lag (TPL) heat equation in an isotropic infinite elastic body with a spherical cavity is studied by two-temperature generalized thermoelasticity theory (2TT). The heat conduction equation in the theory of TPL is a hyperbolic partial differential equation with a fourth-order derivative with respect to time. The medium is assumed to be initially quiescent. By the Laplace transformation, the fundamental equations are expressed in the form of a vector-matrix differential equation, which is solved by a state-space approach. The general solution obtained is applied to a specific problem, when the boundary of the cavity is subjected to the thermal loading (the thermal shock and the ramp-type heating) and the mechanical loading. The inversion of the Laplace transform is carried out by the Fourier series expansion techniques. The numerical values of the physical quantity are computed for the copper like material. Significant dissimilarities between two models (the two-temperature Green-Naghdi theory with energy dissipation (2TGN-III) and two-temperature TPL model (2T3phase)) are shown graphically. The effects of two-temperature and ramping parameters are also studied.     

On flows behind shock waves reflected from a solid wall

The flow fields associated with the interaction of a normal shock wave with a plane wall kept at a constant temperature were studied based on kinetic theory which can describe appropriately the shock structure and its reflection process. With the use of a difference scheme, the time developments of the distributions of the fluid dynamic quantities (velocity, temperature, pressure and number density of the gas) were obtained numerically from the BGK model of the Boltzmann equation subject to the condition of diffusive-reflection at the wall for several cases of incident Mach number:M ₁=1.2, 1.5, 2.0, 3.0, 4.0, 5.0 and 6.0. The reflection process of the shocks is shown explicitly together with the resulting formation of the flow fields as time goes on. The nonzero uniform velocity toward the wall occurring between the viscous boundary layer and the reflected shock wave is found to be fairly large, the magnitude of which is of the order of several percent of the velocity induced behind the incident shock, decreasing as the incident Mach number increases. It is also seen that a region of positive velocity (away from the wall) within the viscous boundary layer manifests itself in the immediate vicinity of the wall, which is distinct for larger incident Mach numbers. Some of the calculated density profiles are compared with available experimental data and also with numerical results based on the Navier-Stokes equations. The agreement between the three results is fairly good except in the region close to the wall, where the difference in the conditions of these studies and the inappropriateness of the Navier-Stokes equations manifest themselves greatly in the gas behavior.This article was processed using Springer-Verlag TEX Shock Waves macro package 1990.     

On the role of heat conduction in the formation of a high-temperature plasma during counter collision of rarefaction waves of solid deuterium

This paper considers the problem of counter collision of rarefaction waves of solid deuterium produced by the simultaneous incidence of two identical shock waves on free surfaces located at a certain distance from each other. The motion of deuterium is described by the equations of one-velocity two-temperature hydrodynamics. The model of electron and ion heat transfer takes into account heat-flux relaxation. The parametric properties of the problem are investigated. It is shown that with decreasing distance between the free surfaces, the maximum temperature of the plasma ceases to depend on this parameter. At moderate distances between the free surfaces, the maximum plasma temperature becomes much lower than the temperature obtained earlier in the problem for the equations of nondissipative hydrodynamics. With increasing pressures in the incident shock wave, the maximum ion temperature increases linearly, reaching a value approximately equal to 160 · 10 ⁶ K at 500 Mbar. In the case of a shock wave with a pressure of 50 Mbar at a gap of 2 mm between the free surfaces of deuterium, the yield of fusion neutrons increases roughly by a factor of 10 compared to the yield of neutrons in the case of no gap.     

Self-similar solution of the Navier-Stokes equations for a completely ionized hydrogen plasma (the plane piston problem)

The self-similar motion of a completely ionized hydrogen plasma is considered in the two-temperature hydrodynamic approximation, i.e., we consider the plane piston problem and the problem on energy release at a fixed wall. Results obtained by numerical integration of the relevant system of ordinary differential equations are quoted.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 3, pp. 34–39, May–June, 1969.     

Effect of molecular relaxation on the propagation of sonic booms through a stratified atmosphere

Nonlinear acoustic wave propagation through a stratified atmosphere is considered. The initial signal is taken to be an isolated N-wave, which is the disturbance that is generated some distance away from a supersonic body in horizontal flight. The effect of cylindrical spreading and exponential density stratification on the propagation of the disturbance is considered, with the shock structure controlled by molecular relaxation mechanisms and by thermoviscous diffusion. An augmented Burgers equation is obtained and asymptotic solutions are derived based on the limit of small dissipation and dispersion. For a single relaxation mode, the solution depends on whether relaxation alone can support the shock or whether a sub-shock arises controlled by other mechanisms. The resulting shock structures are known as fully dispersed and partly dispersed shocks, respectively. In this paper, the spatial location of the transition between fully dispersed and partly dispersed shocks is identified for shocks propagating above and below the horizontal. This phenomenon is important in understanding the character of sonic booms since the transition to a partly dispersed shock structure leads to the appearance of a shorter scale in the shock rise-time, associated with the embedded sub-shock.     

A CFD Euler solver from a physical acoustics–convection flux Jacobian decomposition

This paper introduces a continuum, i.e. non‐discrete, upstream‐bias formulation that rests on the physics and mathematics of acoustics and convection. The formulation induces the upstream‐bias at the differential equation level, within a characteristics‐bias system associated with the Euler equations with general equilibrium equations of state. For low subsonic Mach numbers, this formulation returns a consistent upstream‐bias approximation for the non‐linear acoustics equations. For supersonic Mach numbers, the formulation smoothly becomes an upstream‐bias approximation of the entire Euler flux. With the objective of minimizing induced artificial diffusion, the formulation non‐linearly induces upstream‐bias, essentially locally, in regions of solution discontinuities, whereas it decreases the upstream‐bias in regions of solution smoothness. The discrete equations originate from a finite element discretization of the characteristic‐bias system and are integrated in time within a compact block tridiagonal matrix statement by way of an implicit non‐linearly stable Runge–Kutta algorithm for stiff systems. As documented by several computational results that reflect available exact solutions, the acoustics–convection solver induces low artificial diffusion and generates essentially non‐oscillatory solutions that automatically preserve a constant enthalpy, as well as smoothness of both enthalpy and mass flux across normal shocks. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical study of the oscillations induced by shock/shock interaction in hypersonic double-wedge flows

In this paper, the shock pattern oscillations induced by shock/shock interactions over double-wedge geometries in hypersonic flows were studied numerically by solving 2D inviscid Euler equations for a multi-species system. Laminar viscous effects were considered in some cases. Temperature-dependent thermodynamic properties were employed in the state and energy equations for consideration of the distinct change of the thermodynamic state. It was shown that the oscillation results in high-frequency fluctuations of heating and pressure loads over wedge surfaces. In a case with a relatively lower free-stream Mach number, the shock/shock interaction structure maintains a seven-shock configuration during the entire oscillation process. On the other hand, the oscillation is accompanied by a transition between a six-shock configuration (regular interaction) and a seven-shock configuration (Mach interaction) in a case with a higher free-stream Mach number. Numerical results also indicate that the critical wedge angle for the transition from a steady to an oscillation solution is higher compared to the corresponding value in earlier numerical research in which the perfect diatomic gas model was used.      

Mach stem hysteresis: Experiments addressing a novel explanation of clumpy astrophysical jet emission

Recent time-series observations of shock waves in stellar jets taken with the Hubble Space Telescope reveal localized bright knots that persist over nearly 15 years. While some of these features represent shock fronts caused by variable velocities in the flow, others appear at the intersection points between distinct bow shocks. Theoretically, when the angle between two intersecting shocks exceeds a certain critical value, a third shock (Mach stem) should form. Because Mach stems form perpendicular to the direction of flow, incoming particles encounter a normal shock instead of an oblique one, which results in brighter emission at this location. To study this phenomenon in a controlled laboratory setting, we have carried out experiments on the Omega laser aimed at understanding the formation, growth, and destruction of Mach stems in the warm dense plasma regime. Our experimental results indicate how the growth rate depends upon included angle, and numerical simulations indicate that it may be possible to stabilize an already-formed Mach stem below the critical angle when certain conditions are satisfied.     

Finite element analysis of two-temperature generalized magneto-thermoelasticity

A general finite element model is proposed to analyze transient phenomena in thermoelastic solids. Youssef model of two-temperature generalized magneto-thermoelasticity is selected for an homogenous, isotropic, conducting and elastic medium, which is subjected to thermal shock, and a magnetic field with constant intensity acts tangent to the bounding plane. The numerical solution of the nondimensional governing partial differential equations of the problem has been shown graphically.     

Shock wave triple-point morphology

The existence and characteristics of shock wave triple points are examined. The analysis utilizes a single flow plane for the three shocks and is local to the triple point. It applies when the flow is unsteady, three-dimensional, and the upstream flow is nonuniform. Under more restrictive conditions, a relation is also derived for the ratio of the curvature of the Mach stem to that of the reflected shock. For given values of the ratio of specific heats, γ, and the upstream Mach number, M ₁, a solution window is established. A parametric set of solutions is generated within the window for γ = 1, 1.4, and 5/3 and for 16 values of M ₁ ranging from solution onset to M ₁ = 6.A solution can be one of three types, these stem from the velocity tangency condition along the slip stream. Topics are addressed such as solution multiplicity, shock wave and slip stream orientation, the nature of the reflected wave (weak, strong, inverted, normal), the nature of the Mach stem (weak, strong, normal), and differences due to changes in γ and M ₁.     

Asymptotic theory of neutral stability of the Couette flow of a vibrationally excited gas

An asymptotic theory of the neutral stability curve for a supersonic plane Couette flow of a vibrationally excited gas is developed. The initial mathematical model consists of equations of two-temperature viscous gas dynamics, which are used to derive a spectral problem for a linear system of eighth-order ordinary differential equations within the framework of the classical linear stability theory. Unified transformations of the system for all shear flows are performed in accordance with the classical Lin scheme. The problem is reduced to an algebraic secular equation with separation into the “inviscid” and “viscous” parts, which is solved numerically. It is shown that the thus-calculated neutral stability curves agree well with the previously obtained results of the direct numerical solution of the original spectral problem. In particular, the critical Reynolds number increases with excitation enhancement, and the neutral stability curve is shifted toward the domain of higher wave numbers. This is also confirmed by means of solving an asymptotic equation for the critical Reynolds number at the Mach number M ≤ 4.