The Effect of Spherical Convergence on the Dynamics of Instabilities (U)

E. J, Chapyak and R, P, Godwin Los Alamos National Laboratory

Q S T I We extend previous work on instability growth in spherically converging layered fluid systems by characterizing linearized perturbation dynamics on the collapse of a spherical gas cavity surrounded by an incompressible fluid We also investigate energy damping mechanisms related to fluid compressibility. Numerical examples are presented where water at STP initially surrounds a gas cavity filled with low pressure air. (U)

Introduction Investigations of instability growth in

otherwise spherically converging, layered fluid systems have been of interest to the weapons and fluid dynamics communities for more than fifty years. Bell (1951) derived equations governing the growth of linearized perturbations in an incompressible fluid-void system. Bell generalized his results to account for spatially uniform fluid compressibi1ity.l Plesset (1954) obtained equations describing the linearized interfacial perturbations for two incompressible fluids. Later, Fisher (1982) integrated Plesset's and Bell's work to describe linearized perturbation dynamics in a two-fluid system with spatially uniform compressibility in each fluid. More recently, Mikaelian (1990) generalized Plesset's results to a layered system of N incompressible fluids with N-1 interfaces and discussed implications for turbulent mix modeling. All these efforts describe interface position with the equation: S(r, 8, $, t ) = 0 = r - R(t)

- C a n m ( t > Y n m ( O * $ ) , where R(t) is the mean radius as a function of time and Y: (8, $) are spherical harmonics. Because the m-index turns out to be redundant, the perturbation amplitudes, a,, ( t ) can be expressed as a, ( t ) with mode number n. The perturbation equations are ordinary differential

l o n e cannot account for the removal (radiation) of energy from interface regions by including only spatially uniform compressibility effects. The propagation of acoustic or shock waves away from interface regions is a potentially important dissipation mechanism that we investigate in this paper.

equations that describe the evolution in time of a, ( t ) , provided thatR(t) is a known function of time.

Birkhoff (1956) developed a stability analysis that was directly applicable to the perturbation equations developed by Bell and Plesset. A surprising consequence of Birkhoff's analysis is that unstable amplitude growth is predicted under spherical convergence, renardless of the siF of the (radial) acceleration. This contrasts with classical (planar) Rayleigh- Taylor (R-T) theory (Taylor, 1950), which predicts unstable growth only for acceleration in the direction from the lighter fluid toward the heavier fluid. For spherical expansion, Birkhoff's analysis is qualitatively consistent with planar R-T theory.

perturbation dynamics for the Rayleigh Problem, the collapse of a spherical void surrounded by an incompressible fluid. The radial acceleration in this problem is always directed inward and therefore is classically R-T stable. Plesset and Mitchell developed an approximate solution to the perturbation equations using the WKB method (Kramers, 1926). They showed that, as the void collapses, perturbations grow Iarger in amplitude and oscillate in time with increasing frequency. This is the nature of Birkhoff instability for the Rayleigh Problem. Although unstable, perturbations do not grow exponentially in time, but rather geometrically, and they oscillate while doing so.

Here, we extend the Plesset-Mitchell approach to perturbation growth during the collapse of an incompressible fluid surrounding a nominally spherical, adiabatic gas cavity. With gas cushioning the collapse, radial acceleration changes sign near the point of maximum compression and a brief period of exponential

Plesset and Mitchell (1956) analyzed the

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DISCLAIMER

This report was prepared as a n account of work sponsored by a n agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or as~mes any legal liabili- ty or responsibility for the accuracy, completeness, or usefulness of any information, appa- ratus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessar- ily state or reflect those of the United States Government or any agency thereof.

UNCLASSIFIED (R-T unstable) growth is experienced. This behavior is indicated notionally in Fig. 1, where f z nk / R , and the dot denotes differentiation with respect to time. The equation governing the time evolution of the mean gas cavity radius, is the Rayleigh-Plesset Equation (Plesset and Prosperetti, 1977):

p( Rk + [3 / 2]k2) = P, - P, where Pg is the gas pressure, P is the background pressure, and p is the fluid density. We develop an approximate solution (increasingly accurate as mode number increases) to the perturbation amplitude equations, again using the WKl3 method. The zero point of the radial acceleration divides oscillatory and exponential WKB regions. WKB connection formulae are used to link behavior in the two regions. In particular, an estimate of the maximum amplitudes reached during a given compression is obtained as a function of the initial perturbation amplitude

Effect of Compressibility We also consider fluid compressibility

effects, especially the potential to propagate disturbances and energy away from the collapsing gas cavity. We speculate that the more realistic compressible situation is as shown in Fig. 2, a genedization of Fig. 1. The possibility of a perturbation growth-limiting mechanism could help explain some of the historical weapons data base. It also has implications for ICF capsule design and for sonoluminescence as well.

Numerical Examples Finally, we present two numerical

examples. These examples were generated with MESA-2D (Benson, 1992), an Eulerian code with interface reconstruction. In the first example, the collapse of a gas cavity with y =1.4 gas at 0.025 bar pressure surrounded by water at STP with a superposed n=4 perturbation, the gas cavity survives through collapse into expansion. The perturbation develops into a bubble during the later stages of collapse and into a spike during expansion that "heals" at late time, as seen in Figs. 3 and 4. In the second example, identical to the first except that the initial gas pressure is 0.0025 bar, the gas cavity disintegrates during the later stages of collapse, as shown in Fig. 5. (A control problem also was

examined where no perturbation was imposed on the spherical cavity. Sphericity was maintained through maximum compression and into expansion. At late times, non-spherical effects were observed only within a few zones of the z- axis). For reference, the maximum radial interface speed is about 100 m/s for the frrst problem, and about 1650 d s for the second problem, as predicted by solving the Rayleigh- Plesset Equation. Actual speeds are somewhat less due to the effects of water compressibility. The watersound speed is about 1500 m/s. Comparisons between the linear theory and these numerical results are also presented.

Adiabatic Gas

Classically R-T stable Birkhoff stable

Acceleration > 0

Empty Cavity (Rayleigh Problem u Birkhoff unstable

* Time

Figure 1. Notional radius-time behavior for gas cavity collapse surrounded by an incompressible fluid.

Mach # << 1 Only largest modes damped; Most modes behave WKB-like.

VJ

4 d Remaind KB-l ie .

Mach # = 1 All modes damped, provided cavity "survives" this far.

Perturbations try to grow exponentially but are damped.

Time Figure 2. Notional radius-time behavior for gas cavity collapse surrounded by a compressible fluid.

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UNCLASSIFIED Acknowledgments

CRADA between Los Alamos National Laboratory, the Oregon Medical Laser Center, and Palomar Medical Technologies, Inc.

References Bell, G. I., "Taylor Instability on Cylinders and

Spheres in the Small Amplitude Approximation," Los Alamos Scientific Laboratory, Los Alamos, NM, Report LA-

This research was supported by a DOE

" 1321 (1951). d -

0.00 0:os . 0.10 z Im) Figure 3. Shape of gas cavity (air initially at 0.025 bar pressure) surrounded by STP water with an imposed n=4 perturbation during collapse. Times are in p. s!

c-,

z fun) Figure 4. Shape of gas cavity (air initially at 0.025 bar pressure) surrounded by STP water with an imposed n=4 perturbation during expansion. Times are in p.

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Figure 5. Shape of gas cavity (air initially at 0.0025 bar pressure) surrounded by STP water with an imposed n=4 perturbation. Times are in p.

Plesset, M. S., "On the Stability of Fluid Flows with Spherical Symmetry," J. Applied

Fisher, H. N., "Instabilities in Converging Compressible Systems," Los Alamos National Laboratory, Los Alamos, NM, Memo X-1(5/82)22 (1982).

Mikaelian, K. O., "Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified spherical shells," Phys. Rev. A, 42(6), 3400-3420 (1990).

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Plesset, M. S. and Prosperetti, A., "Bubble Dynamics and Cavitation," Ann. Rev. Fluid Mech., 9,145-185 (1977).

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Phys., 25,96-98 (1954).

SOC. A , 201, 192-196 (1950). Plesset, M. S. and Mitchell, T. P., "On the

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