Mixing and Chaotic Microstructure

Yuefan Deng, James Glimm, Mixing and and David H. Sharp Chaotic Microstructure ixing is a process in which performance of pellets in laser-fu- nonlinear, and they typically do not distinct fluids intermingle sion experiments. admit a small parameter for useful Min a complex way. Chaotic Without doubt the problem of expansions. Together these features microstructure refers, more broadly, chaotic microstructure, in its many lead to an extremely complex phe- to all small-scale stochastic or ramifications, is the most fundamen- nomenology. chaotic behavior that affects dynam- tal in classical continuum physics. We propose here a systematic ics at large length scales. Problems Why is this? First, because the program for addressing this class of that involve mixing and chaotic mi- problem is pervasive, and second problems. This program follows a crostructure are of fundamental im- because the only systematic solution line of development that will be fa- portance in both basic science and occurs at the engineering level miliar to many-body theorists. The engineering: They are the central through the method of conservative first step is the identification and issue in turbulence, pipeline flow, overdesign, or experimental trial and analysis of elementary modes, or co- and the dynamics of supernovae. error. The classical methods of sci- herent structures. For example, in The microstructure of porous rocks, ence—theory, computation, and ex- turbulence the elementary modes are which is stochastic on all length periment—have so far delivered vortices; in phase transitions, den- scales, is a dominant aspect of the much less than is needed either for drites; in Hele-Shaw cells, viscous geology of aquifers and oil reser- engineering purposes or for scientif- fingers; in solid materials, lattice voirs. Chaotic microstructure also ic understanding. dislocations. The second step is to plays a key role in determining the These problems are difficult for understand the pairwise interaction properties of common materials such three reasons. They possess an ex- of elementary modes. This step is as metals and plastics. Mixing is a ceedingly large number of active followed, not by an analysis of dominant phenomenon limiting the degrees of freedom, they are usually multi-mode interactions, but by the 124 Los Alamos Science Number 21 1993 Mixing and Chaotic Microstructure Mixing fig1 3/27/93 (a) statistical analysis of an ensemble of regularities are inevitably present at Ceiling modes governed by pairwise interac- the interface. Those portions of the tions. From this analysis we want to fluid at the interface that lie higher Water derive continuum-level constitutive than the average experience more laws (such as equations of state), pressure than is necessary for their which are then used (fifth step) in support. They begin to rise, pushing Air macroscopically averaged flow aside water as they do so. Neigh- equations such as the Navier-Stokes boring portions of the fluid, where equation. Thus we have a five-step the surface hangs a little lower than Floor procedure relating microscopic dy- average, require more than average namics to macroscopic observables. pressure for their support. They Although each of the steps may begin to fall. The air cannot supply (b) seem rather obvious, their integra- the specific variations in pressure tion is less commonly discussed. A from place to place necessary to pre- large fraction of work in classical vent the interface irregularities from physics is related to this program in growing. The initial irregularities Ripples that it addresses one or another of therefore increase in magnitude, ex- its steps. ponentially in time at the beginning. In this paper we will illustrate this The water falls to the floor. program by discussing its implemen- The same layer of water lying on tation in examples drawn from our the floor would have been perfectly work and that of our collaborators. stable. Irregularities die out. Thus This work as well as collateral work we can infer a simple criterion for (c) Bubbles of others can be traced from the fur- the onset of Rayleigh-Taylor insta- ther reading given at the end of the bility at the interface between two article. fluids of different densities: If the heavy fluid pushes the light fluid, the interface is stable. If the light Rayleigh-Taylor Instability fluid pushes the heavy fluid, the in- Spikes terface is unstable. Our first example concerns Rayleigh-Taylor instability occurs Rayleigh-Taylor instability. The oc- in diverse situations. Let us take a currence of this instability can be quick look at one example from re- Figure 1. An Example of Rayleigh- understood in the following simple search into inertial-confinement fu- Taylor Instability way. Imagine the ceiling of a room sion. Figure 2 shows a highly (a) The pressure of the air is quite suffi- plastered uniformly with water to a schematic picture of the implosion cient to support a perfectly uniform layer depth of 1 meter (Figure 1). The of a deuterium-tritium (DT) pellet. of water 1 meter thick against the ceiling. layer of water will fall. However, it A spherical glass or metal tamper is (b) But the air pressure cannot constrain is not through lack of support from filled with DT gas. The tamper is the air-water interface to flatness. Rip- the air that the water falls. The irradiated with intense laser light, ples or irregularites will inevitably be pre- pressure of the atmosphere is equiv- which causes it to accelerate inward, sent at the interface. (c) The irregulari- alent to that of a layer of water 10 compressing the DT gas inside the ties grow, forming “bubbles” and “spikes.” meters thick, quite sufficient to hold cavity in order to bring about nu- The water falls to the floor. the water against the ceiling. But in clear fusion. During irradiation the one respect the atmosphere fails as a outer surface of the tamper is the in- sion, when the light fluid pushes the supporting medium. It fails to con- terface between a heavy fluid (glass heavy fluid. As the pellet is com- strain the air-water interface to flat- or metal) and a light fluid (vapor- pressed to perhaps 1000 times its ness. No matter how carefully the ized glass or metal) and is unstable normal density, the pressure in the water layer was prepared, small ir- during the initial phase of the implo- cavity increases until it is sufficient 1993 Number 21 Los Alamos Science 125 Mixing and Chaotic Microstructure to slow the inward motion of the interactions among bubbles and connecting interior region (mixing tamper. This phase of the implosion spikes lead to the formation of a layer). The bubble regime has been is also Rayleigh-Taylor unstable; chaotic mixing layer at the interface. the most carefully studied and has here the high-pressure light gas in Understanding the growth rate and the simplest properties. It is found the cavity is pushing the tamper. structure of this mixing layer is the in the Read and Youngs experi- Although this picture of the implo- central problem in the study of ments, as well as in numerical simu- sion of a DT pellet is oversimplified Rayleigh-Taylor instability. lations (Figure 3), that the edge of in a number of ways, it nevertheless This question has been investigat- the mixing layer in the bubble suggests quite clearly that Rayleigh- ed experimentally by Read and regime is dominated by a collection Taylor instability can have an im- Youngs. Their important findings of bubbles. portant effect on the compression of bring the questions to be understood The average height of the bubbles the pellet. into sharp focus and can be summa- relative to the mean position of the A closer look at the time evolu- rized as follows. The mixing layer interface, h(t), grows with time ac- tion of a Rayleigh-Taylor unstable has three principal regions: the edge cording to the scaling law interface reveals a complex phenom- where bubbles of light fluid are pen- enology. As the instabilities devel- etrating into the heavy fluid (bubble h(t) = Æ Agt2, op, bubbles of light fluid and spikes regime),Sharp fig2 the edge where spikes of 4/9/93 of heavy fluid form, each penetrat- heavy fluid are penetrating into the where A, the Atwood number, is a ing into the other phase. Complex light fluid (spike regime), and the modification due to buoyancy of the acceleration force (gravity) g and is Laser light defined as A = ( Ω 2 ° Ω 1)/( Ω 2 + Ω 1), in which Ω i represents the density of fluid i. It is a remarkable experimental finding of Read and Tamper Youngs that for incompressible flu- Vapor (vapor) ids the acceleration constant Æ in Tamper this equation is an approximately Solid universal constant, in that it is near- ly independent of initial conditions DT gas DT gas and of fluid properties such as density, viscosity, and surface tension. P1 P1 Finally, it is observed that the average number of bubbles at the inter- P2 P2 face decreases with time, and that P1 > P2 their average radius increases. P 3 The main objective of our work P3 > P2 has been to understand the physical properties of the Rayleigh-Taylor mixing layer. The fluid in the mixing layer is in a chaotic state, and it is necessary to have a strategy to guide Figure 2. Rayleigh-Taylor Instabilities in Laser Fusion one through the complexities of this On the left appears an extremely simplified view of a laser-fusion pellet at an early problem. The fundamental challenge stage of a fusion experiment. Laser irradiation of the solid tamper vaporizes the outer in modeling fluid chaos is to provide layer of the tamper and pushes the tamper inward.

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