Numerical shock propagation using geometrical shock dynamics

A numerical method of solution is developed for steady–state hyperbolic equations in conservation form. An upwind direction is identified and a smooth mapping is introduced to align the equations with that direction. A flux F(u) in the upwind direction is regarded as the ‘state’ variable and advanced numerically on a grid according to a second–order extension of Godunov' method. The solution u is determined from the computed flux, and a constraint on the mapping is that the Jacobian, det(∂F∂u), is of one sign. The method is applied to the equations modelling shock–wave propagation according to Whitham' theory of geometrical shock dynamics. In this context the solution vector u gives the components of the gradient of a function whose level curves determine the shock positions at successive times. Shock propagation in both uniform and non–uniform gases is considered and several example calculations are presented. A further application of the numerical method is inviscid supersonic flow, as governed by the steady Euler equations. The steady supersonic flow in a converging–diverging duct is computed to illustrate the numerical method for this latter application.

In the present paper, the exact solution of quasilinear hyperbolic system of equations governing the propagation of weak shock waves in a one-dimensional dusty adiabatic gas flow with generalized geometries is derived. Here, the density behind the shock front is assumed to vary according to a power law of the distance. An analytical expression for the total energy carried by weak shock wave in dusty gas is also derived.

A dynamic method for creation and measurement of high pressures that has been developed in Russia and in the U.S. during the past 50 years allows study of condensed material properties over a wide range of densities and temperatures that cannot be reached by any other method [1–3]. A strong shock wave, having a velocity exceeding 10 km/s, heats the compressed substance up to several tens of thousands of degrees. If the substance is transparent in its initial state, e.g., an ionic crystal or noble gas, then measurement of the radiation passing from the shock-heated material through its uncompressed layer makes it possible to obtain unique information about its properties at the high pressure. Fundamental studies of the role of radiation in strong shock waves in gases were made by Zel’dovich and Raizer [1]. Model’ made the first measurements of the brightness temperature of a shock front in a gas [4]. Those pioneering studies showed the phenomenon of a sharp decrease of the brightness temperature from the shock front to the material behind the front. A large body of experimental data on the radiating properties of shock waves in dense gases is presented in [5]. At VNIIEF, Kormer was the absolute leader in experimental study of optical properties of shock compressed, transparent, condensed substances. He and Zel’dovich gathered around themselves a team of young enthusiasts who started investigations in the 1960s and 1970s [6]. This work is still unique.

In the present work, a numerical scheme for the unsteady motion of shock waves in gases is developed based on Whitham's rule of geometrical shock dynamics (GSD). To investigate the effect of the initial shock strength on the shock reflection and diffraction on a wall, the two-dimensional focusing of cylindrical blast waves reflected from a concave elliptic wall is analyzed under various initial conditions. The results are compared with the finite difference solutions based on the piecewise linear method (PLM) as well as the experimental results. It is found that the geometrical shock dynamics is a practical and economical method to analyze the shock focusing process of the blast waves produced by exploding wires. Pressure change along the major axis of the ellipse shows good agreement when the present computations are compared with the experimental results as well as with PLM solutions.

The behavior of sinusoidally-rippled shock fronts is a fundamental research topic in the dynamics of shock waves [1]. The Whitham’s ray-shock theory, which is sometimes called the geometrical-shock-dynamics (GSD) theory, is known as a simple method for analyzing the behavior of a non-planar shock front. In this theory, narrow ray tubes corresponding to the light rays in the geometrical optics are placed perpendicularly to the every portions of a non-planar shock front, and the evolution of the shock front is calculated by tracking the shock front in each ray tube sequentially. When the behavior of an inert sinusoidally-rippled shock front is analyzed by the GSD theory and the Chester-Chisnell-Whitham’s (CCW’s) A-M relationship [2], where A is the cross-sectional area of a ray tube and M is the propagation Mach number of the shock wave in the ray tube, the amplitude of the shock-front ripple oscillates as the shock wave propagates [3]. Actually, the behavior of an inert sinusoidally-rippled shock front is influenced by the fluid motion in the shock-compressed region, and the amplitude of the shock-front ripple shows damped oscillation as the shock wave propagates [1,3,4,5].

Although the perfect analogy between hydraulic jumps in liquids and shock waves in gases does not exist, both of analyses and experiments reveal that the approximate analogy between hydraulic jumps in liquids and shock waves in gases with specific heat ratio γ=1.4 exists over a range of free-stream Mach number from 2 to 6 roughly. Furthermore, the hydraulic Jumps in liquids are roughly analysed. Lastly, the method to vary the ratio of the specific heat of gas by changing form of water tank section is generally formulated.

Coupling effects of foam ceramics on the flame and shock wave of gas explosion

Propagation of a strong spherical shock wave in a gravitating or non-gravitating dusty gas with exponentially varying density

Atomic simulations of shock waves in gases have been performed using both molecular dynamics and the direct simulation Monte Carlo method devised for the Boltzmann equation. The profiles of the fluid density, velocity and temperature computed across the shock are shown to agree with the Navier-Stokes description for weak shocks (Mach numbers ranging up to 2) but then on the discrepancy increases with the Mach number. The Burnett modification to the Navier-Stokes equations is compared to the simulation results .

Dynamics of unsteady compressible cavitating flows associated with the cavity shedding

This paper reports an experimental investigation, using shadowgraphs and pressure measurements, of the detailed behaviour of converging weak shock waves near three different kinds of focus. Shocks are brought to a focus by reflecting initially plane fronts from concave end walls in a large shock tube. The reflectors are shaped to generate perfect foci, arêtes and caustics. It is found that, near the focus of a shock discontinuity, a complex wave field develops, which always has the same basic character, and which is always essentially nonlinear. A diffracted wave field forms behind the non-uniform converging shock; its compressive portions steepen to form diffraction shocks, while diffracted expansion waves overtake and weaken the diffraction shocks. The diffraction shocks participate in a Mach reflexion process near the focus, whose development is determined by competition between the convergence of the sides of the focusing front and acceleration of its central portion. In fact, depending on the aperture of the convergence and the strength of the initial wave, the three-shock intersections of the Mach reflexions either cross on a surface of symmetry or remain uncrossed. In the former case, which is observed if the shock wave is relatively weak, the wavefronts emerge from focus crossed and folded, in accordance with the predictions of geometrical acoustics theory. In the latter, the strong-shock case, the fronts beyond focus are uncrossed, as predicted by the theory of shock dynamics. It is emphasized that in both cases the behaviour at the focus is nonlinear. The overtaking of the diffraction shocks by the diffracted expansions limits the amplitude of the converging wave near focus, and is the mechanism by which the maximum amplification factor observed at focus is determined. In all cases, maximum pressures are limited to rather low values.

A new theory of shock dynamics (NTSD) has been derived in the form of a finite number of compatibility conditions along shock rays. It has been used to study the growth and decay of shock strengths for spherical and cylindrical pistons starting from a non-zero velocity. Further a weak shock theory has been derived using a simple perturbation method which admits an exact solution and also agrees with the classical decay laws for weak spherical and cylindrical shocks.

The behavior of entropy across shock waves in dusty gases is calculated by using the Navier-Stokes equations for the gas phase and the particle phase. The resulting system of six nonlinear ordinary differential equations is reduced to a system of four autonomous nonlinear differential equations which are solved exactly. This solution is obtained formally by assuming that both the velocity and temperature of particles in the gasdynamic region are constant. A careful study of the equation that governs the entropy shows that the entropy profile has a maximum value within the shock region. It is also shown that the entropy is increasing continuously across the shock wave with increasing both the Mach number and the particle concentration.

In the present work the shock wave propagation in a kinked duct was investigated to understand effects of compression and expansion waves on shock reflection. A finite volume method for two‐dimensional flow with the upwind TVD scheme was applied to solve the time‐dependent Euler equations. Two types of the flow in the duct were analyzed namely, the flow observed in a shock tube and that produced by a wire explosion. The results were compared with the available experimental data on weak shock waves. The flow field produced by wire explosion in a kinked duct was simpler than that in a shock tube, so that the shock dynamics theory may be available for engineering practice as an economical method.

Parameters of emitting shock waves in gases are investigated in the limiting case when there is no screening of emission from the shock front by the precursory layer. The one-dimensional quasi-steady-state formulation of the problem with deceleration of high-speed gas flow against a plane fixed obstacle under conditions of strong emission is given. The case of the shock waves of large optical thickness is analytically considered over a wide range of variation of the obstacle reflectivity. The parameters of emitting shock waves generated in experiments in shock tubes in the inert argon gas are estimated using the methods developed and compared with the measurement results. The shock “adiabats” of optically thick shock waves are considered with allowance for the radiation energy losses. The calculations are carried out for aluminium plasma.