^{1}

### Abstract

We consider the effect of a shock passing through an arbitrarily shaped interface between two fluids. The evolution of the interface into a new shape, written formally as , is found by applying the linear, classical Richtmyer–Meshkov instability result to each mode in the Fourier expansion of the original interface. We provide several examples where the new shape can be found analytically. For any interface we define an associated dual interface and show that . Representing a shock by a new mathematical operator we find how , and transform under the effect of a shock. Kink-singularities are found in when and where has a discontinuous change in its first derivative. These are the locations where jetting occurs. We briefly discuss the effects of nonlinearity, compressibility, viscosity, etc., all of which suppress kink-singularities, and present hydrocode simulations of shock tube and high-explosive-driven experiments to highlight the influence of compressibility, nonlinearity, and material strength.

This work was performed under the auspices of the U.S. Department of Energy by the University of California Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.

I. INTRODUCTION AND GENERAL RESULTS

II. EXAMPLES

A. Example 1

B. Example 2

C. Example 3

D. Example 4

E. Example 5

III. OTHER EFFECTS

IV. NUMERICAL SIMULATIONS

V. SUMMARY AND CONCLUDING REMARKS

### Key Topics

- Richtmyer Meshkov instabilities
- 23.0
- Shock wave effects
- 16.0
- Viscosity
- 9.0
- Kelvin Helmholtz instability
- 8.0
- Shock tubes
- 7.0

## Figures

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.

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.

Same as Fig. 1 for example 2, Eqs. (17a) and (17b) , with and .

Same as Fig. 1 for example 2, Eqs. (24a) and (24b) , with and .

Same as Fig. 1 for example 3, Eqs. (26a) and (26b) , with .

Same as Fig. 1 for example 4, Eqs. (29a) and (29b) .

Same as Fig. 1 for example 5, Eqs. (30a) and (30b) .

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 ).

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.

(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.

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.

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 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.

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 .

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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