Particle Velocity Histories

Test data are available for two experiments at different impact velocities in this configuration. In one of the tests the projectile impact velocity was 1.54 km/s, while in the second the impact velocity was 2.10 km/s. This test was simulated with the WONDY [60] one-dimensional Lagrangian wave code, and Fig. 9.21 compares calculated and measured particle velocity histories at the sample/window interface for the two tests [61]. Other test parameters are listed at the top of each plot in the figure. 

Particle Velocity Histories, Shot 3301-2-5 (Cast TNT, Gage Grooves) Distances in mm from driver-attenuator/target interface to gage centers are 0, 10.6,21.2, 32.1,42.5, and 44.1. Note that the last two records are PMMA the records from the four primary gages embedded in HE are numbered... 

Fig 3 shows the type of records obtained in Ref 14. Some discussion of the significance of particle velocity histories (such as Fig 3) will be presented in the following section Flash X-Ray Method... 

This section covers two main subjects, namely theoretical estimates of uCJ, and application of measured particle velocity histories to elucidation of initiation phenomena in detonations and to flow characteristics behind the detonation front... 

ForIsopmpylnitrate ua = 1.49km/sec (Ref 9) Kennedy et al (Ref 11) and Nunziato et al (Ref 12) used laser.interferometry to obtain the particle velocity histories in PBX 9404 shown in Fig 11 for long-duration input pulses and in Fig 12 for short-duration pulses. The PBX 9404 targets were too thin (8 is target thickness) to detonate,-but particle velocity increase denoting reaction in the targets is clearly indicated. These particle velocity histories are qualitatively similar to those obtained in Ref 16, shown in Fig 4... 

Fig 11 Particle velocity histories observed in fused-silica window in long-duration pulse (1.1 /xsec experiments. Corrections have been made for waveform distortion and transit time through the 1.5mm thick buffer of fused silica... 

Fig 12 Particle velocity histories observed in fused-silica window in short-duration pulse (0.28 /xsec) experiments, with corrections made for buffer transit time and waveform distortion... 

Initiation of detonation will always occur if the rear-boundary condition is maintained long enough to allow the peak in the Lagrange particle velocity histories to overtake the wavefront... 

Figure 6. Particle velocity histories for LX-17 impacted by a Kel-F flyer at 2.951 km/s.

Figure 9. Interface particle velocity histories for detonating LX-17 and various salt crystals.

Figure 2.4. Velocity history of the first bead on a wire. Average velocity v is identified with drift or with the particle velocity of the bead.

The P-t histories illustrated by Fig. 2.9 are not histories of a particle of material moving with the flow, because the coordinate that is fixed is x, and material is flowing past it. A more useful P-t history would use a coordinate system which is attached to the material itself, as a stress or particle velocity gauge would be. Such a coordinate system is defined in the next section. 

It is important to note that the state determined by this analysis refers only to the pressure (or normal stress) and particle velocity. The material on either side of the point at which the shock waves collide reach the same pressure-particle velocity state, but other variables may be different from one side to the other. The material on the left-hand side experienced a different loading history than that on the right-hand side. In this example the material on the left-hand side would have a lower final temperature, because the first shock wave was smaller. Such a discontinuity of a variable, other than P or u that arises from a shock wave interaction within a material, is called a contact discontinuity. Contact discontinuities are frequently encountered in the context of inelastic behavior, which will be discussed in Chapter 5. 

Contact discontinuity A spatial discontinuity in one of the dependent variables other than normal stress (or pressure) and particle velocity. Examples such as density, specific internal energy, or temperature are possible. The contact discontinuity may arise because material on either side of it has experienced a different loading history. It does not give rise to further wave motion. 

The diagnostics applied to shock experiments can be characterized as either prompt or delayed. Prompt instrumentation measures shock velocity, particle velocity, stress history, or temperature during the initial few shock transits of the specimen, and leads to the basic equation of state information on the specimen material. Delayed instrumentation includes optical photography and flash X-rays of shock-compression events, as well as post-mortem examinations of shock-produced craters and soft-recovered debris material. 

Table 3.3 summarizes the history of the development of wave-profile measurement devices as they have developed since the early period. The devices are categorized in terms of the kinetic or kinematic parameter actually measured. From the table it should be noted that the earliest devices provided measurements of displacement versus time in either a discrete or continuous mode. The data from such measurements require differentiation to relate them to shock-conservation relations, and, unless constant pressures or particle velocities are involved, considerable accuracy can be lost in data processing. 

As a blast wave passes through the air or interacts with and loads a structure or target, rapid variations in pressure, density, temperature and particle velocity occur. The properties of blast waves which are usually defined are related both to the properties which can be easily measured or observed and to properties which can be correlated with blast damage patterns. It is relatively easy to measure shock front arrival times and velocities and entire time histories of overpressures. Measurement of density variations and time histories of particle velocity are more difficult, and few reliable measurements of temperature variations exist. 

As noted above, added mass and history contributions can be neglected for large 7, especially at high Re or Rqjs- The motion is then of Type 2, with the fluid responding rapidly to changes in particle velocity. If the history term is neglected and y 1, Eq. (11-33) becomes... 

The crossing trajectory effect refers to the impact of the continuous change of the fluid eddy-particle interactions as the heavy particle trajectory might go through numerous eddies reflecting different flow properties. Hence it follows that the velocity history of heavy particles may differ from that of a marked fluid particle. Similar closure models for the drift velocity and the velocity co-variances have been derived from kinetic theory by Koch and co-workers [38, 39] and Reeks [62, 63]. 

If a motion is specified with satisfies the continuity condition, the velocity, strain, and density at each material particle are determined at each time t throughout the motion. Given the constitutive functions (e, k), c(e, k), b( , k), and a s,k) with suitable initial conditions, the constitutive equations (5.1), (5.4), and (5.11) may be integrated along the strain history of each material particle to determine its stress history. If the density, velocity, and stress histories are substituted into (5.32), the history of the body force at each particle may be calculated, which is required to sustain the motion. Any such motion is termed an admissible motion, although all admissible motions may not be attainable in practice. 

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

The upper limit of gas velocity for particulate expansion is termed the minimum bubbling velocity, umb. Determining this can present difficulties as its value may depend on the nature of the distributor, on the presence of even tiny obstructions in the bed, and even on the immediate pre-history of the bed. The ratio umb/umf, which gives a measure of the degree of expansion which may be effected, usually has a high value for fine light particles and a low value for large dense particles. 

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