A shock dynamics theory based analytical solution of double Mach reflections

The shock-dynamics based governing equations of double Mach reflection (DMR) wave configurations were developed and solved. The analytical results were compared to experimental results from various sources. Very good to excellent agreement was evident.     

An experimental and theoretical study of converging polygonal shock waves

An experimental investigation was carried out to explore the possibility of producing converging polygonal shocks in an essentially two-dimensional cavity. Previous calculations by Apazidis and Lesser (1996) suggested that such configurations could be produced by reflecting a cylindrical outgoing shock from a smoothly altered circular boundary, the alteration having n-gonal symmetry. In the experiments the outgoing shock was produced by a spark discharge which yielded shocks in the Mach number range from 1.1 to 1.7 at a radius just prior to the reflection. Polygonal shocks were observed as predicted by using a modified form of geometrical shock dynamics, derived in the above paper. In addition, the modified theory was used to calculate the results of an experiment carried out by Sturtevant and Kulkarny (1976). The results of the numerical calculations were found to be in substantial agreement with both experiments, suggesting that the modifications in geometrical shock dynamics for non-uniform flow ahead of an advancing shock are useful in the case of shock focusing. The experiment also showed that the polygonal shapes were stable in the examined range of shock Mach numbers, a result that may be of importance for a number of practical situations in which shock focusing is present. Received 9 October 2001 / Accepted 7 January 2002 – Published online 11 June 2002     

Experimental and numerical studies of underwater shock wave attenuation

The attenuation of an underwater shock wave by a thin porous layer is studied both experimentally and numerically. The shock waves are generated by exploding 10 mg silver azide pellets and the pressures at different distances from the explosion center are measured. Measurements are also carried out with a gauze layer placed between the explosion source and the pressure gauge. The results with and without the gauze layer are compared evaluating the shock wave attenuation. Numerical simulations of the phenomenon are also carried out for a simple wave attenuation model. The results are compared with the experimental data. Despite the simple mathematical model of wave attenuation, the agreement between the experimental and numerical results is reasonable.Received: 22 October 2002, Accepted: 17 June 2003, Published online: 5 August 2003PACS: 47.11.+j, 47.40.Nm, 47.55.Mh     

Focusing of strong shocks in an elliptic cavity

Focusing of strong shock waves in a gas-filled thin chamber with an elliptic boundary is investigated theoretically and numerically. The process of reflection and convergence of cylindrical shocks generated at one of the focal points of the chamber is studied by means of Whitham's theory of geometrical shock dynamics (1957, 1959). The current calculations are based on the modified version of this theory, which takes into account the non-homogeneous flow ahead of the advancing shock, Apazidis and Lesser (1996). There are two main results of the present study indicating a qualitatively different behavior of strong shocks as compared to the linear acoustic case. (1) - the form of the reflected shock is not cylindrical and (2) - the reflected, converging shock does not focus at the geometrical second focal point of the chamber. The dependence of the form of the reflected shock and the location of its focusing point on the strength of the shock and the eccentricity of the chamber is investigated.     

The formation of a secondary shock wave behind a shock wave diffracting at a convex corner

This paper deals with the formation of a secondary shock wave behind the shock wave diffracting at a two-dimensional convex corner for incident shock Mach numbers ranging from 1.03 to 1.74 in air. Experiments were carried out using a 60 mm 150 mm shock tube equipped with holographic interferometry. The threshold incident shock wave Mach number () at which a secondary shock wave appeared was found to be = 1.32 at an 81° corner and = 1.33 at a 120° corner. These secondary shock waves are formed due to the existence of a locally supersonic flow behind the diffracting shock wave. Behind the diffracting shock wave, the subsonic flow is accelerated and eventually becomes locally supersonic. A simple unsteady flow analysis revealed that for gases with specific heats ratio the threshold shock wave Mach number was = 1.346. When the value of is less than this, the vortex is formed at the corner without any discontinuous waves accompanying above the slip line. The viscosity was found to be less effective on the threshold of the secondary shock wave, although it attenuated the pressure jump at the secondary shock wave. This is well understood by the consideration of the effect of the wall friction in one-dimensional duct flows. In order to interpret the experimental results a numerical simulation using a shock adaptive unstructured grid Eulerian solver was also carried out. Received 1 May 1996 / Accepted 12 September 1996     

Achieving an optimal shock-wave mitigation inside open channels with cavities for weak shock waves: A computational study

This paper deals with a numerical study of weak shock-waves propagation and their attenuation in channel flow having different heights and exhibiting a hollow circular cavities with different depths and diffraction angles inside. The effect of initial diffraction angle and cavity depth on the shock mitigation is investigated. A better shock attenuation is achieved with diffraction angle θ_w = 90° by a factor of approximately 17% in terms of shock-Mach number and 38% in terms of total energy. The obtained results show also, in addition to the initial diffraction angle and cavity depth, the importance of reducing the channel heights as well as the position of the reduced section in achieving an optimal shock-wave attenuation. The presence of a cavity inside the channel helps to attenuate faster the shock wave. The underlying physics relies on the shock diffraction phenomenon that generates large amount of vortical structures capable of dissipating part of the shock energy by inducing a pressure loss behind it. A subtle arrangement of channel position/height and a cavity location leads to an efficient pressure attenuation by approximately a factor of 57% for M_s = 1.6 and 16% for M_s = 1.1.     

Instability theory of shock wave in a shock tube

The instability theory of shock wave in a shock tube including the effects of tube wall and contact surface is studied. The experimental data of unstable shock wave coincide with one of instability criteria derived in the present paper.     

The isotope effect of gaseous hydrogen under shock compression

The shock compression method has been used to measure the Hugoniot data and shock temperature for gaseous hydrogen samples, covering the pressure range of 55-140 MPa and the temperature range of 3400-4500 K and with the initial conditions of P ₀ = 0.6 MPa, 1.2 MPa and T ₀ at room temperature. Spectral radiance histories emitted from shocked D ₂ and H ₂ + D ₂ (equimolar mixture) are monitored by a pyrometer system with seven wavelength channels. Theoretical calculations based on the Saha model with Debye-Hückel correction for the shock compression behavior of shocked gaseous samples are in good agreement with the measured Hugoniot data, but show slightly higher values for the shock temperature when comparing with experiments. An isotope effect relevant to these shocked hydrogen species has been found in the linear shock velocity vs particle velocity relation, in which the correlation factor between these hydrogen isotopes or hydrogen mixtures is simply of initial density dependence.Received: 8 December 2002, Accepted: 8 May 2003, Published online: 2 September 2003PACS: 62.50 + p, 31.30.GS, 51.90. + r     

Application of underwater shock wave and laser-induced liquid jet to neurosurgery

Paper deals with applications of underwater shock waves to medicine. A historical development of underwater shock wave generation by using pulsed Ho:YAG laser beam irradiation in water is briefly described and an overview is given regarding potential applications of shock waves to neuro-surgery. The laser beam irradiation in a liquid-filled catheter produces water vapor bubble and shock waves intermittently produces micro-liquid jets in a controlled fashion from the exit of the catheter. Correlations between shock dynamics and bubble dynamics are emphasized. To optimize the jet motion, results of basic parametric studies are briefly presented. The liquid jet discharged from the catheter exit has an impulse high enough to clearly exhibit effectiveness for various medical purposes. In liquid jets we observed reasonably strong shock waves and hence invented a compact shock generator aiming to apply to microsurgery. We applied it to a rat's bone window and developed an effective method of brain protection against shock loading. The insertion of Gore-Tex® sheet is found to attenuate shock waves drastically even for very short stand off distance and its physical mechanism is clarified. The laser-induced liquid jet (LILJ) is successfully applied to soft tissue dissection. Animal experiments were performed and results of histological observations are presented in details. Results of animal experiments revealed that LILJ can sharply dissect soft tissue with a minimum amount of liquid consumption, while blood vessels larger than 0.2 mm in diameter are preserved. Shock waves and LILJ have a potential to be indispensable tools in neuro-surgery.This paper was based on work that was presented at the 2nd International Symposium on Interdisciplinary Shock Wave Research, Sendai, Japan, March 1–3, 2005.Communicated by K. Takayama PACS 42.62.Be, 47.40.-x, 42.62.-b     

Multiscale dislocation dynamics simulations of shock compression in copper single crystal

Ultra short pulse shock wave propagation, plastic deformation and evolution of dislocations in copper single crystals with (0 0 1), (0 1 1) and (1 1 1) orientations are investigated using multiscale dislocation dynamics plasticity analyses. The effects of peak pressure, pulse duration, crystal anisotropy and the nonlinear elastic properties on the interaction between shock wave and dislocations are investigated. The results of our calculations show that the dislocation density has a power law dependence on pressure with a power of 1.70 and that the dislocation density is proportional to pulse duration and sensitive to crystal orientation. These results are in very good agreement with the analytical predications of Meyers et al. [Meyers, M.A., Gregori, F., Kad, B.K., Schneider, M.S., Kalantar, D.H., Remington, B.A., Ravichandran G., Boehly, T., Wark, J., 2003. Laser-induced shock compression of monocrystalline copper: characterization and analysis. Acta Materialia 51, 1211–1228] and the experimental results of Murr [Murr, L.E., 1981. Residual microstructure-mechanical property relationships in shock loaded metals and alloys. In: Meyers, M.A., Murr, L.E. (Eds.), Shock Waves and High Strain Rate Phenomena in Metals. Plenum, New York, pp. 607–673]. It is shown also that incorporating the effect of crystal anisotropy in the elastic properties results in orientation dependent wave speed and peak pressure. The relaxed configurations of dislocation microstructures show the formation of microbands coincident with the slip planes.     

A study of shock wave interaction with a rotating cylinder

Shock wave reflection over a rotating circular cylinder is numerically and experimentally investigated. It is shown that the transition from the regular reflection to the Mach reflection is promoted on the cylinder surface which rotates in the same direction of the incident shock motion, whereas it is retarded on the surface that rotates to the reverse direction. Numerical calculations solving the Navier-Stokes equations using extremely fine grids also reveal that the reflected shock transition from RR MR is either advanced or retarded depending on whether or not the surface motion favors the incident shock wave. The interpretation of viscous effects on the reflected shock transition is given by the dimensional analysis and from the viewpoint of signal propagation.Received: 24 April 2002, Accepted: 16 August 2002, Published online: 25 March 2003     

Near field shock structure of dual co-axial jets

Experimental results on the shock structure of dual co-axial jets are presented. The effects of the geometric parameters of the inner nozzle, jet static pressure ratio (ratio of the exit plane static pressures of the inner and outer nozzles) and the ratio of outer to inner nozzle throat area on the shock structure were studied. A superimposed outer and inner jet structure was observed in the schlieren photographs. The inner flow is compressed by the outer flow resulting in the formation of a Mach disc and an exit shock. A parameter incorporating the effect of Mach number of the inner nozzle and jet static pressure ratio was found to correlate the observations regarding the Mach disc location.     

Instability theory of shock wave in a channel

The instability theory of shock wave was extended from the case with an infinitefront to the case of a channel with a rectangular cross section.First,themathematical formulation of the problem was given which included a system ofdisturbed equations and three kinds of boundary conditions.Then,the general solutionsof the equations upstream and downstream were given and each contained fiveconstants to be determined.Thirdly,under one boundary condition and oneassumption,it was proved that all of the disturbances in front of the shock front andone of the two acoustic disturbances behind the shock front should be zero.Theboundary condition was that all of the disturbed physical quantities should approach tozero at infinity.The assumption was that only the unstable shock wave was concernedhere.So it was reasonable to assumeω=iγ,γwas the instability growth rate andwas a positive real number.Another kind of boundary conditions was that the normaldisturbed velocities should be zero at the solid wall of the cha     

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.      

Analysis of critical dynamics for shock-induced adiabatic explosions by means of the Cauchy problem for the shock transformation

We present a theoretical and numerical study on the induction of adiabatic explosions by accelerated curved shocks in homogeneous explosives, and pay a special attention to critical conditions for initiation. We characterize the first stage of the decomposition process, or induction, as an initial-value problem. During induction, the reaction progress-variable remains small; the induction time is given by the runaway of the dependent variables and corresponds to a logarithmic singularity in theirs material distributions. We express these distributions as first-order expansions in the progress variable about the shock. Then, the framework of our procedure is the formal Cauchy problem for quasi-linear hyperbolic sets of first-order differential equations, such as the balance laws for adiabatic flows of inviscid fluids considered in this study. When a shock front is used as data surface, the solution to the Cauchy problem yields the flow derivatives at the shock, then the induction time, as functions of the shock normal velocity and acceleration, and , and the shock total curvature C. We next derive a necessary condition for explosion as a constraint among , and C that ensures bounded values of the induction time. This criterion is akin to Semenov's, in the sense that the critical condition for explosion is that the heat-production rate must just exceed the heat-loss rate, here given by the volumetric expansion rate at the shock. The violation of the criterion defines a critical shock dynamics as a relationship among , and C that generates infinite induction times. Depending on the rear-boundary conditions, which determine the shock dynamics, this event can be interpreted as either a non-initiation, or the decoupling of the shock and of the flame front induced by the shock. We illustrate our approach by a simple solution to the problem of the initiation by impact of a noncompressible piston. From the continuity constraint in the material speed and acceleration at the contact surface of the piston and the explosive, we first derive the initial shock dynamics, and then rewrite the induction time and the initiation condition in terms of the piston speed, acceleration and curvature. We compare these theoretical predictions to those of our direct numerical simulations, and to numerical results obtained by other authors, in the case of impacts on a gaseous explosive. Received 19 October 1998 / Accepted 1 June 1999     

Nonlinear dynamic responses of twin-tube hydraulic shock absorber

Basically on the multi-body system dynamics, the virtual prototype of the hydraulic shock absorber for the bench test is developed in the ADAMS environment. Dynamic behaviors of the absorber are studied by both computer simulation and real test. Numerical predictions of dynamic responses are produced by the established virtual prototype of the absorber and compared with experimental results. It has been shown from the comparison that the vibration behaviors of the prototype with hysteretic damping characteristics are considered to be more identical with the bench test results than those of the same prototype with piecewise linear damping properties are. The current virtual prototype of the shock absorber is correct and can be a developing terrace for the optimizing design of the absorber and matching capability of the whole car.     

Measurement of laser driven shock wave transit time through thin aluminium targets by optical shadowgraphy

Laser driven shock wave transit time in thin aluminium targets was experimentally estimated by determining the shock emergence time at the rear of thin aluminium foils of varying thickness from 5 to 35 μm. A 20 J, 5 ns Nd:glass laser was focused to produce laser intensity of 10¹² to 5 × 10¹³ W/cm² on the targets which were placed in vacuum. Target foil movement was measured to an accuracy of 10 μm using optical shadowgraphy technique. This technique was used to accurately measure the shock transit time by recording the optical shadowgrams at various instants of time and thus identify the instant at which the foil is just set into motion. Shock transit time measured in foils of different thickness can give the value of shock velocity at a given laser intensity. Target motion recorded by shadowgraphy can also give the target foil velocity from which shock pressure can be estimated. Experimental values of shock transit time, shock velocity and shock pressure were observed to agree well with the values using one-dimensional multi-group radiation hydrodynamic simulations. PACS 52.50Jm; 52.50Lp; 52.25 Communicated by K. Takayama     

On the relation between particle velocity and shock wave speed for thermoelastic materials

Results of shock-wave experiments in solids often suggest a nearly-linear relation between the particle velocity behind the shock and the shock wave speed. The present note reconsiders the question of whether thermoelastic material models may be consistent with such observations. Emphasis is placed on the role played by the response of the material in severe compression, as distinguished from its response for small or moderate deformations. The details are illustrated for materials of Mie-Grüneisen type. Received 21 December 2001 / Accepted 22 April 2002 Published online 8 July 2002