Two-dimensional shock wave

The basic characteristics of a one-dimensional shock wave are described in Chapter 1 of this text. However, the shock waves in supersonic flow propagate not only one-dimensionally but also two- or three-dimensionally in space. For example, the shock waves formed at the air-intake of a ducted rocket are two- or three-dimensional in shape. Expansion waves are also formed in supersonic flow. The pressure downstream of an expansion wave is reduced and the flow velocity is increased. With reference to Chapter 1, brief descriptions of the characteristics of a two-dimensional shock wave and of an expansion wave are given here.Ii-5]... 

Oran et al. [218,219] developed a global parameterized model which describes the chemical induction time as a function of temperature and pressure. Parameters of the induction time function were determined for stoichiometric hydrogen and methane in air mixtures. The parameters were fitted to numerical results obtained from the simulations based on detailed reaction mechanisms. This technique allowed a 22-times faster calculation of the induction time and reduced the simulation time in a onedimensional model by a factor of 7.5. The fitted model was used in two-dimensional shock-wave simulations. 

Xinghai, F. Measurement of Two-Dimensional Shock Wave Velocity and Composite Probe, Proc. 8th Symposium (International) on Detonation, NSWC MP 8 194, Albuquerque, NM, 1985, pp. 85 93. 

Brandani, S., Rapagna, S., Foscolo, P.U. and Gibilaro, L.G. (1996). Jump conditions for one-dimensional two-phase shock waves in fluidized beds the effect of the jump in fluid pressure. Chem. Eng. Sci., 51, 4639. 

In the case of most nonporous minerals at sufficiently low-shock stresses, two shock fronts form. The first wave is the elastic shock, a finite-amplitude essentially elastic wave as indicated in Fig. 4.11. The amplitude of this shock is often called the Hugoniot elastic limit Phel- This would correspond to state 1 of Fig. 4.10(a). The Hugoniot elastic limit is defined as the maximum stress sustainable by a solid in one-dimensional shock compression without irreversible deformation taking place at the shock front. The particle velocity associated with a Hugoniot elastic limit shock is often measured by observing the free-surface velocity profile as, for example, in Fig. 4.16. In the case of a polycrystalline and/or isotropic material at shock stresses at or below HEL> the lateral compressive stress in a plane perpendicular to the shock front... 

Computational methods have played an exceedingly important role in understanding the fundamental aspects of shock compression and in solving complex shock-wave problems. Major advances in the numerical algorithms used for solving dynamic problems, coupled with the tremendous increase in computational capabilities, have made many problems tractable that only a few years ago could not have been solved. It is now possible to perform two-dimensional molecular dynamics simulations with a high degree of accuracy, and three-dimensional problems can also be solved with moderate accuracy. 

On the other hand, the formation of the high pressure phase is preceded by the passage of the first plastic wave. Its shock front is a surface on which point, linear and two-dimensional defects, which become crystallization centers at super-critical pressures, are produced in abundance. Apparently, the phase transitions in shock waves are always similar in type to martensite transitions. The rapid transition of one type of lattice into another is facilitated by nondilTusion martensite rearrangements they are based on the cooperative motion of many atoms to small distances. ... 

Shock Wave Propagation in a Two-Dimensional Flow Field... 

When a two-dimensional wedge is placed in a supersonic flow, a shock wave that... 

Figure C-3. Oblique shock wave formed by a two-dimensional wedge.

Delayed-, After-, or Post-Reactions in Detonation. There are two general types those which occur within a confined space such as in a closed bomb, and those which involve reaction with external air and are known as "afterburning". Accdg to classical one-dimensional detonation theory, chemical equilibrium is achieved and reaction ceases at the CJ (Chapman-Jouguer) plane, which terminates the reaction zone. In some cases, however, as noted by Craig (Ref 3, p 863), the sharp shock wave and the reaction zone of falling pressure are followed by a further rapid pressure drop which is not predicted by an extrapolation of the one-dimensional theory... 

Another early attempt to incorporate chemieal reactions into molecular dynamics of shock waves was the use of the LEPS (London, Eyring, Polanyi, Sato) potential [4], originally developed in the 1930 s to model the H3 potential energy surface. This method can be applied to systems in which each atom interacts with exactly two nearest neighbors, and is therefore suitable for modeling one-dimensional reactive chains [5-6]. It provides a more realistic treatment of energy release as a function of bond formation but is not readily extended to more complex systems. 

Our discussion has been restricted to shock waves in which the flow is perpendicular to the wave that is the only kind which can occur in onedimensional flow. Such waves are called normal shock waves or normal shocks because of the perpendicular relationship. In two-dimensional flow, another kind, called an oblique shock., occurs, in which the flow is not perpendicular to the shock wave. Oblique shock waves form at the leading edge of the wings of supersonic aircraft and cause sonic booms for a discussion of them see Shapiro [5]. 

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