Shock formation

Numerical Work. The results of experiments described above can be better understood when compared to the results of numerical and analytical studies. Numerical studies, in particular, provide real insight into the shock formation process. Chushkin and Shurshalov (1982) and Adamczyk (1976) provide comprehensive reviews of the many studies in this field. The majority of these studies were performed for military purposes and dealt with blast from nuclear explosions, high explosives, or... 

Many numerical methods have been proposed for this problem, most of them finite-difference methods. Using a finite-difference technique, Brode (1955) analyzed the expansion of hot and cold air spheres with pressures of 2000 bar and 1210 bar. The detailed results allowed Brode to describe precisely the shock formation process and to explain the occurrence of a second shock. 

Detonation from Burning, Transition of. See under Detonation (and Explosion) Development (Transition) from Burning (Combustion) or Deflagration and the following paper by A. Macek, "Transition from Slow Burning to Detonation. A Model for Shock Formation in a Deflagrating Solid , NOLNavOrd Rept 6105(1958) [See also Andreev Belyaev(1960), 141-44]... 

The region of shock formation, represented in Fig for simplicity by a single point S, can then be constructed by the "method of characteristics described in the book of Courant Friedrichs (Ref 3). The pertinent velocities are given as equations 12,... 

This simplified picture of self-phase modulation neglects dispersion, time-delayed nonlinearities and shock formation which is all known to occur in optical fibers. While no in fused silica is at least as fast as a few fs, the GVD broadens the pulses as they travel along the fiber so that the available peak power Pa is decreased. Effective self-phase modulation however takes place when the so called dispersion length is much smaller then the nonlinear length whose ratio is given by [27]... 

Still, todays computer power does not allow us to hilly resolve the shock transition region. We can, however, explain the non-linear kinetic plasma dynamics that generate the electromagnetic fields needed for transmission of momentum between the colliding plasma populations. Here, we report on 3D PIC simulations of the shock formation in the counter-streaming region of two colliding plasma shells. 

Figure 7.1 Migration and shape evolution of an injection profile as predicted by the ideal model in the case of a convex upward isotherm. Characteristics associated with concentrations and shock formation, velocity associated with a concentration, propagation of a band (i.e., of a concentration signal), and formation of a shock (see text), (a) Injection of a pulse with a Gaussian profile. Reproduced with permission from B. Lin, S. Golshan-Shirazi, Z. Ma and G. Guiochon, Anal. Chem., 60 (1988) 2647 (Fig. 1), 1988 American Chemical Society, (b) Injection of a continuous hnear ramp. Reproduced with permission from T. Ahmad, F. Gritti, B. Lin and G. Guiochon, Anal. Chem., 76 (2004) 977 (Fig. 2). 2004 American Chemical Society.

The time to,s is the relaxation time of the shock formation [23]. It is proportional to f , hence inversely proportional to the slope of the concentration gradient. It is also inversely proportional to the product bCp that characterizes the degree of nonlinear behavior of the isotherm at the end of the experiment, when the concentration plateau becomes equal to Cq. 

Theoretical treatments of shock waves in vapour-droplet flows are rare in the literature. Partly dispersed shock waves are discussed by Marble [3] (who made some incorrect deductions concerning the magnitude of the relaxation times), Konorski [4] and Bakhtar and Yousif [5]. Fully dispersed waves dominated by just one relaxation process were treated by Petr [6], but few details of his analysis appear in the paper. No experimental measurement of shock wave structure is available, although the work by Barschdorff [1], and Schnerr [7] shows interesting shock formation patterns and has stimulated the present work. 

As discussed, we can expect shockwaves and steep saturation discontinuities to form in time, depending on the exact form and values of our fractional flow functions and initial conditions. We will assume that the particular functions do lead to piston-like shock formation very close to the borehole. The shock boundary value problem just stated can be solved in closed form, and, in fact, is the petroleum engineering analogue of the classic nonlinear signaling problem (Pt -I- c(p) Px = 0, p = po for X > 0, t = 0, and p = g(t) for t > 0, x = 0) discussed in the wave mechanics book of Whitham (1974). 

Typical saturation and pressures in Figures 21-14a,b,c for early, intermediate, and late times illustrate shock formation and propagation. The parameters were selected to cover the entire range of inertial-to-capillary force ratios. 

We emphasize that we have obtained stable numerical results, without saturation overshoots and local oscillations, all using second-order accurate spatial central differencing, without having to introduce special upwind operators. The methods developed are stable and require minimal computing since they are based on tridiagonal equations. Several subtle aspects of numerical simulation as they affect miscible diffusion and immiscible saturation shock formation are discussed in Chapter 13. 

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