Example 1, Eqs. (7a) and (7b) : (a) the original interface ; (b) the associated dual interface ; (c) the asymptotic shape ; and (d) the negative derivative of the original interface. Shocking any of the shapes evolves it into the shape below it or, equivalently, into the derivative of the shape to its right. The arrows indicate kink-singularities. See text for explanation.
Snapshots from a CALE simulation of a 26 cm wide shock tube: A Mach 1.2 downward-moving shock in helium strikes the He/ air interface shaped like and located 122 cm away from the endwall . The amplitude is 0.7 cm and the wavelength is 13 cm. The interface moves at and therefore is at by 4.2 ms. Snapshots are shown at 0, 0.2, 0.4, 0.6, 0.8, and 1.0 ms in (a), and at 1.5, 3.0, and 4.2 ms in (b). Note that the vertical scales are different in the two frames. Reshock occurs shortly after 4.2 ms (see Fig. 8 ).
(Color). The density field of the two gases, air and helium, inside the shock tube described in Fig. 7 . The time is 4.6 ms, shortly after reshock. In frame (a) the initial shape was (its early evolution was discussed in Ref. 22 ); upon reshock the bubbles at and 19.5 cm turn into spikes. In frame (b) the initial shape was (see Fig. 7 for its early evolution); upon reshock the bubbles bifurcate. The shading indicates density from (light) to (dark). The patterns seen in the lower gas (air) are a result of the rarefaction generated in it when the last interface in Fig. 7 (4.2 ms) was hit by the upward-moving reshock.
An explosive-driven system to study the evolution of perturbations in a shocked aluminum plate. Assuming that the high explosive is LX-14, the plate moves at constant velocity a distance of 3.5 cm in 15 μs. Figures 10–13 show the evolution of various perturbations machined onto the free surface of the Al plate.
Initial perturbations, all having and , imposed at the lower free surface of the Al plate in Fig. 9 . Four different shapes ( , quadratic, and ) are machined for two periods each. Note that for clarity we have magnified the vertical scale 0.13 cm compared with the horizontal scale 8 cm.
The evolution of perturbations, shown to scale, starting with the initial shapes displayed in Fig. 10 . The snapshots are taken at 12, 15, 18, and 22 μs. In this simulation the Al plate, which starts to move at , is assumed to have no strength.
The Al free surface (dark continuous line) at 23 μs compared with its initial shape (light dashed line), assuming that Al has a material strength and shear modulus given by the Steinberg–Guinan model (Ref. 28 ). Phase reversal has occurred, but no amplification. Note that the vertical scale has been greatly magnified for clarity.
Time evolution of , the root-mean-square deviation of the Al free surface, for different assumptions about its strength: Curve assumes the SG model (Ref. 28 ); curve assumes constant (i.e., pressure and strain independent) and ; curve assumes and ; curve assumes and ; and for curve we assumed . Curve shows phase reversal but no amplification—see Fig. 12 . Curve shows healthy growth—see Fig. 11 .
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