A computational study of shock speeds in high-performance shock tubes

This paper describes U2DE, a finite-volume code that numerically solves the Euler equations. The code was used to perform multi-dimensional simulations of the gradual opening of a primary diaphragm in a shock tube. From the simulations, the speed of the developing shock wave was recorded and compared with other estimates. The ability of U2DE to compute shock speed was confirmed by comparing numerical results with the analytic solution for an ideal shock tube. For high initial pressure ratios across the diaphragm, previous experiments have shown that the measured shock speed can exceed the shock speed predicted by one-dimensional models. The shock speeds computed with the present multi-dimensional simulation were higher than those estimated by previous one-dimensional models and, thus, were closer to the experimental measurements. This indicates that multi-dimensional flow effects were partly responsible for the relatively high shock speeds measured in the experiments. Received 15 November 1996 / Accepted 3 February 1997     

Transonic shock oscillations on NACA0012 aerofoil

The mechanism of the origin of shock oscillations on NACA0012 aerofoils is investigated using a moving grid thin layer Navier Stokes code. The method used to understand the mechanism is to initiate the shock oscillations on an aerofoil by moving the aerofoil from a regime of steady transonic flow into a regime of periodic flow by a change in airflow incidence. The results indicate that the shock induced bubble plays a leading role in the origin of shock oscillations and the trailing edge has an affect on its amplitude. Received 1 April 1997 / Accepted 1 December 1997     

Pressure measurements on cone surface in 3-D shock reflection processes

In a shock tube the pressure distribution was measured on a cone with an angle of attack when a shock wave passed the cone. The cone has a semi-apex angle of 35°, the angle of attack varied from 0° to 25° and the shock Mach numbers from 1.05 to 3.0. A series of pressure distributions on the cone circumference are given. Received 17 November 1997 / Accepted 5 December 1997     

Experimental and computational studies focusing processes of shock waves reflected from parabolic reflectors

This paper describes experimental and numerical studies of the focusing process of shock waves reflected from various shapes of a parabolic reflector. The effect of incident shock strength on the focusing process was also investigated. Experiments were carried out in a conventional shock tube and a test gas was air for incident shock Mach numbers ranging from 1.1 to 2.0. In the experiments, the process of shock focusing was visualized by schlieren method. Numerical simulations were conducted for incident shock Mach numbers up to 3.0 by solving the two-dimensional unsteady Euler equations. The numerical results were compared with experiment for various parabolic reflector shapes and for various incident shock Mach numbers. Based on the experimental and computational results, the pattern of shock focusing and shock focusing mechanism are discussed.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Enhancement of shock waves

The flow field developed behind a shock wave propagating inside a constant cross-section conduit is solved numerically for two different cases. First, when the density of the ambient gas into which the shock propagates has a logarithmic change with distance. In the second, and the more practical case, the ambient gas is composed of pairs of air–helium layers having a continually decreasing width. It is shown that in both cases meaningful pressure amplification can be reached behind the transmitted shock wave. It is especially so in the second case. By proper choice of the number of air–helium layers and their width reduction ratio, pressure amplification as high as 7.5 can be obtained.      

Temperature measurements of shock heated materials using multispectral pyrometry: Application to bismuth

A new device for temperature measurements of shock heated materials has been developed in order to refine their equations of state. We present temperature measurements for bismuth samples in contact with a lithium fluoride window. The shock pressure of these experiments ranges from 18 to 97 GPa. Received 2 June 1997 / Accepted 26 January 1998     

Formation of a jet and vortices behind a shock wave reflected from a concave body

A jet and vortices have been observed when a plane shock wave reflects from a concave body in a shock tube. If the cavity is deep enough then two reflected shocks appear near its edges. Air, carbon tetrafluoride (CF) and dichlorodifluoromethane (CClF) were chosen as test gases. The flow was visualized with the aid of a conventional shadow technique. Pressure measurements at the body surface were also obtained. Numerical studies have been conducted using a two-dimensional inviscid model. There is a good qualitative agreement between the experimental and numerical results. Received 8 February 1996 / Accepted 30 June 1997     

Linear relationship between Hugoniot temperature and shock Mach number

It is the purpose of this publication to discuss further the apparent validity of a linear relationship between the Hugoniot temperature and the shock Mach number, when used as an independent variable in the thermodynamics of very high pressures. Additional evidence for seventeen different materials is presented. Some of the materials discussed might present phase transitions within the ranges of pressure and temperature here studied. The case of molybdenum is discussed in particular because experimental data on phase transitions are available within the ranges of pressure and temperature considered. Equation of state results for a few materials, obtained using an exact analytical equation of state, are compared with those computed employing an approximate form of the equation, consequence of the linear relationship between the Hugoniot temperature and shock Mach number. The excellent agreement shows that this approximate and very simple equation of state can be very reliable and useful. Received 17 June 1997 / Accepted 4 November 1997     

Shock reaction of hexane at 77, 193, and 273 K with special reference to shock pressure

Shock waves generated by projectile impacts were transmitted into hexane and the shocked hexane was analyzed by TCD-GC, FID-GC, GCMS, and FABMS for produced aliphatic hydrocarbons. The projectile length and its velocity were varied from 10 to 40 mm and from 220 to 1040 m/s, respectively. The initial temperature of the hexane was 77, 193 and 273 K. The major products detected throughout the reactions were hydrogen, light alkanes from C to C, and light alkenes from C to C. The minor products were heavy alkanes from C to C and soot-like materials. Experiments with varied projectile length revealed that the shock reaction occurred only while the shock wave was transmitted through hexane (about seconds). This short reaction time may be responsible for a lower yield of branched products in the shock reaction compared with yield produced by hexane pyrolysis in previous studies. In the shock reaction of hexane, the dehydrogenation was one of the important reactions and the recombination of hexyl radicals might play a role in the formation of -C. Experiments with varied initial temperature suggested that the molar yield of products depends not on the shock temperature but on the shock pressure, and that the reaction mechanisms for solid hexane and for liquid hexane are not identical. As the shock pressure increased, the relative yield of heavy products increased while that of light products decreased. This could be interpreted mainly by considering the activation volumes of the reaction involved. Received 12 September 1997 / Accepted 14 October 1997     

Modelling a shock wave source for seismic well survey

The shock wave seismic source has specific advantages for reservoir survey, particularly in the technique of vertical seismic profiling. This paper is concerned with the modelling of the characteristics of a shock wave device under conditions typical of its use as a surface seismic source. Such modelling necessitates numerical solution of the time-dependent conservation equations for axi-symmetric geometry under conditions where the fluid in the surface borehole may be air or water. A version of the piecewise parabolic method has been used to take account of the two-phase behaviour based on the formal properties of the well-known Tait equation of state for water. Results are presented which enable predictions to be made for use in the field of the pressure and velocity signatures of the shock wave source in terms of axial and radial profiles. Such information is significant in the assessment of the degrees of compressional- and shear-wave energies delivered by the source in vertical seismic profiling surveys of reservoirs. Received 6 November 1997 / Accepted 3 March 1998     

Revisiting shock waves in metals

We cast Wallace's theory of thermoplastic flow in conservative form. We point out the difference between our formulation, which accounts for contact with an external energy reservoir, and previous formulations of thermoplastic flow. The theory is exploited to show that the experiments of Johnson and Barker on 6062-T6 Al can be interpreted as a weak shock wave that splits into an infinite sequence of “infinitesimal”, shocks, caused by increasing plasticity, leading to the observed smooth temporal velocity profile (a dispersed wave). We predict that overdriven shock waves in metals will split as well. We also re-examine the need for invoking a heat dissipation mechanism for overdriven shocks. It is briefly pointed out that our approach of casting the theory of thermoplastic flow in divergence form can be generalized easily to account for heat release in energetic solids. Received 25 March 1996 / Accepted 20 August 1996     

Numerical simulation of shock wave focusing over parabolic reflectors

The problem of a plane shock wave that propagates in an air media and then is reflected from a parabolic concave reflector and focuses at some region is considered. The shock focusing can greatly magnify the pressure and the temperature. The purpose of this study is to numerically simulate the shock focusing process of the reflection of shock waves from the parabolic reflectors with different depths and to analyze their associated flow fields in detail. The present solver developed is to solve the Euler equations using an improved, implicit, upwind Total Variation Diminishing scheme in a finite-volume approach. The effects of reflectors with different depths and of the incident shock Mach numbers on shock focusing are investigated. The real-gas effect is taken into account through a proper correction of the specific heat ratio of air, when high temperature occurs due to shock focusing.     

Reflection of a ring shock wave from a rigid wall

A new type of an implosion has been observed experimentally and simulated in a numerical calculation: the formation of a quasi-spherical converging shock wave after the reflection of a ring shock wave from a solid wall. The conversion of the ring shock wave into the quasi-spherical converging shock wave intensifies the local implosion properties.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

A numerical study of shock wave reflections on low density foam

A continuum mixture theory is used to describe shock wave reflections on low density open-cell polyurethane foam. Numerical simulations are compared to the shock tube experiments of Skews (1991) and detailed wave fields are shown of a shock wave interacting with a layer of foam adjacent to a rigid wall boundary. These comparisons demonstrate that a continuum mixture theory describes well the shock interactions with low density foam.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Over forty years of continuous research at UTIAS on nonstationary flows and shock waves

Analytical and experimental research on non-stationary shock waves, rarefaction waves and contact surfaces has been conducted continuously at UTIAS since its inception in 1948. Some unique facilities were used to study the properties of planar, cylindrical and spherical shock waves and their interactions. Investigations were also performed on shock-wave structure and boundary layers in ionizing argon, water-vapour condensation in rarefaction waves, magnetogasdynamic flows, and the regions of regular and various types of Mach reflections of oblique shock waves. Explosively-driven implosions have been employed as drivers for projectile launchers and shock tubes, and as a means of producing industrial-type diamonds from graphite, and fusion plasmas in deuterium. The effects of sonic-boom on humans, animals and structures have also formed an important part of the investigations. More recently, interest has focussed on shock waves in dusty gases, the viscous and vibrational structure of weak spherical blast waves in air, and oblique shock-wave reflections. In all of these studies instrumentation and computational methods have played a very important role. A brief survey of this work is given herein and in more detail in the relevant references.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

An analytical model for shock wave impact on compressible open-cell foam

A simple model is proposed to describe the main features of the complex phenomena that occur when a plane shock wave strikes a open-cell foam block. The complex two-phase interactions can be simplified using the experimental evidence that the foam face may be treated as a contact surface, that the length of the collapsed foam plug grows linearly in time, and that the gas pressure profile in the region between the head of the plug and the undisturbed material may be treated as being pseudo-stationary. These simplifications enable the prominent features of the compression process to be predicted for a variety of foam types. Analytical results for the reflected shock strength, foam front face velocity, transmitted wave velocity and strength, and maximum peak back wall pressures are presented.      

Three-dimensional propagation of the transmitted shock wave in a square cross-sectional chamber

An investigation into the three-dimensional propagation of the transmitted shock wave in a square cross-section chamber was described in this paper, and the work was carried out numerically by solving the Euler equations with a dispersion-controlled scheme. Computational images were constructed from the density distribution of the transmitted shock wave discharging from the open end of the square shock tube and compared directly with holographic interferograms available for CFD validation. Two cases of the transmitted shock wave propagating at different Mach numbers in the same geometry were simulated. A special shock reflection system near the corner of the square cross-section chamber was observed, consisting of four shock waves: the transmitted shock wave, two reflection shock waves and a Mach stem. A contact surface may appear in the four-shock system when the transmitted shock wave becomes stronger. Both the secondary shock wave and the primary vortex loop are three-dimensional in the present case due to the non-uniform flow expansion behind the transmitted shock.PACS: 43.40.Nm     

Attenuation of shock waves propagating in polyurethane foams

Shock wave attenuation in polyurethane foams is investigated experimentally and numerically. This study is a part of research project regarding shock propagation in polyurethane foams with high-porosities = 0.951 ~ 0.977 and low densities of ρc = 27.6 ~55.8 kg/m³. Sixty Millimeter long cylindrical foams with various cell numbers and foam insertion condition were installed in a horizontal shock tube of 50 mm i.d. and 5.4 mm in length. Results of pressure measurements in air/foam combination are compared with CFD simulation solving the one-dimensional Euler equations. In the case of a foam B fixed on shock tube wall, pressures at the shock tube end wall increases relatively slowly comparing to non-fixed foam, free to move and a foam A fixed on shock tube wall. This implies that elastic inertia hardly contributes to pressure build up. Pressures behind a foam C fixed on shock tube wall decrease indicating that shock wave is degenerated into compression wave. Dimensionless impulse and attenuation factor decrease as the initial cell number increases. The momentum loss varies depending on cell structure and cell number.     

Gradients at a curved shock in reacting flow

The inviscid equations of motion for the flow at the downstream side of a curved shock are solved for the shock–normal derivatives. Combining them with the shock–parallel derivatives yields gradients and substantial derivatives. In general these consist of two terms, one proportional to the rate of removal of specific enthalpy by the reaction, and one proportional to the shock curvature. Results about the streamline curvature show that, for sufficiently fast exothermic reaction, no Crocco point exists. This leads to a stability argument for sinusoidally perturbed normal shocks that relates to the formation of the structure of a detonation wave. Application to the deflection–pressure map of a streamline emerging from a triple shock point leads to the conclusion that, for non–reacting flow, the curvature of the Mach stem and reflected shock must be zero at the triple point, if the incident shock is straight. The direction and magnitude of the gradient at the shock of any flow quantity may be written down using the results. The sonic line slope in reacting flow serves as an example. Extension of the results – derived in the first place for plane flow – to three dimensions is straightforward. Received 12 February 1997 / Accepted 10 June 1997