We apply a hydrodynamic approach to analyze ejecta emanating from doubly shocked liquid metals. In particular, we are interested in characterizing ejecta velocities in such situations by treating the problem as a limiting case of the Richtmyer–Meshkov instability. We find existing models for ejecta velocities do not adequately capture all the relevant physics, including compressibility, nonlinearities, and nonstandard shapes. We propose an empirical model that is capable of describing ejecta behavior across the entire parameter range of interest. We then suggest a protocol to apply this model when the donor material is shocked twice in rapid succession. Finally, the model and the suggested approach are validated using detailed continuum hydrodynamic simulations. The results provide a baseline understanding of the hydrodynamic aspects of ejecta, which can then be used to interpret experimental data from target experiments.
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Figure 1. (a) Geometric parameters associated with a flycut profile and (b) flycut and sine waveforms fitted to bubble surface from numerical simulations. The solid vertical line indicates the free surface.
Figure 2. Schematic showing the problem setup for FLASH simulations, with terminology from Ref. 41.
Figure 3. Typical x–t diagram for double-shock simulations. IS, incident shock; TS, transmitted shock; RR, reflected rarefaction; SI, shock–interface interaction. Subscripts indicate first or second shock.
Figure 4. Density contours from numerical simulations of case 1 with sinusoidal initial perturbation (khbu−−∼0.45 at the second shock).
Figure 5. Density contours from numerical simulations of case 5 with sinusoidal initial perturbation (khbu−−∼1.32 at the second shock).
Figure 6. Density contours from numerical simulations of case 6 with chevron initial perturbation (khbu−−∼1.38 at the second shock).
Figure 7. Scaled plots of (a) bubble amplitude, (b) spike amplitude, (c) bubble velocity, and (d) spike velocity vs nondimensional time for case 1, and comparison with models.
Figure 8. Scaled plots of (a) bubble amplitude, (b) spike amplitude, (c) bubble velocity, and (d) spike velocity vs nondimensional time for case 5, and comparison with models.
Figure 9. Scaled plots of (a) bubble amplitude, (b) spike amplitude, (c) bubble velocity, and (d) spike velocity vs nondimensional time for case 6, and comparison with models.
Figure 10. (a) Bubble velocities from all simulations scaled with the model of Ref. 10. (b) Spike velocities from all simulations scaled with Eq. (8).
Figure 11. Summary plot: comparison of spike velocities from all models with FLASH data for single-shock and double-shock cases, as well as reported growth rates from experiments.24,25
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