Among a few kinds of recognized fluid instabilities there is the instability concerning two fluids of different densities superposed one over another. We can distinguish two cases
The Rayleigh-Taylor instability is of a fingering type on an interface between two fluids of different densities, which occurs when the light fluid is pushing the heavy fluid. Let us assume that in Fig. 1 denser fluid is colored in dark gray and lighter fluid in light gray and a small stochastic perturbation is introduced. Regardless of fluids interface preparation the layers will deviate from planarity by some small amount. Then, portions of fluid, which lay higher than initial interface level, experience higher pressure than is needed for their support. Therefore, they push aside heavier fluid and begin to rise. Similarly neighboring portions of fluid, where the surface is a little lower than the initial interface level, will require more than average pressure to support them and begin to fall. The initial irregularities increase in magnitude resulting first in wrinkles (ripples in Fig.1b) then in bubbles-and-spikes formation and finally in turbulent mixing.
The Rayleigh-Taylor instability was thoroughly investigated
analytically, experimentally and via computer simulations. Its role in
natural phenomena investigations (e.g., supernovae explosions, evolution
of galaxies, plasma physics) and technological applications (e.g., laser
and electromagnetic implosions) is unbeatable and results in numerous
papers and books already published
. The linear theory based
on both Eulerian and Lagrangian fluid dynamics models and stochastic
theory is well motivated and approved by computer and physical
experiments
. Nevertheless, the most interesting is non-linear
regime in which a mixing layer is formed. The mixing layer has three
principal regions
: the edge where bubbles of light fluid are
penetrating into the heavy fluid (bubble regime), the edge where spikes
of heavy fluid are penetrating into the light fluid (spike regime), and
the connecting region (mixing layer). The bubble regime has been most
carefully studied
and has the simplest properties.
The stages of mixing can be described in more details:
:
is the Atwood number
appears
reach the
amplitude of 0.5 its exponential growth rate slows down and
longer wave disturbances begin to grow rapidly. They merge, in turn,
yielding larger and larger structures. If initialized perturbations are
small enough it is presumed
that these large structures evolve
from the non-linear interactions between the smaller ones rather than
from an initial disturbance of the corresponding wavelength. When the
dominant wavelength exceeds about 10 the memory of the
initial conditions is lost which finishes the stage 2 of mixing.
because memory of initial
conditions has been lost.
Numerous factors, which influence the development of a Rayleigh-Taylor
instability in a simple fluid were investigated
Other factors like material properties, state equations of fluid, heat
conduction, diffusion, several mass components, phase transition remain
very difficult to investigate and still need more effort. The same
concerns the most difficult nonlinear aspects of Rayleigh-Taylor
instability: interactions of perturbations of different frequency, the
effects of statistically distributed heterogenities, break-up of spikes
and bubble amalgamation. It seems that existing analytical
approaches
to solve some of these problems are insufficient.
Moreover, widely used numerical models based on the mass, momentum and
energy continuity are too coarse to solve them via computer simulation.
An explanation of this presumption on the base of Church-Turing
hypothesis and the universal Turing Machine model
is given below.
Theoretical approaches are concerned with devising shorter calculations
to reproduce the outcome of complex systems. Let us assume that
there exists a class of systems that can be predicted using a formula
describing its future state. This means that the calculations performed
by using the formula must be more sophisticated than those performed by
the physical system itself. That means that the formal system must be
more powerful than the corresponding physical system. However, for some
physical systems which are as powerful as those performed by Universal
Turing Machine, no shortcuts are possible. Moreover, some chaotic
physical processes can not be viewed as computations. It is not
possible to predict the future accurately just because small
perturbations in the initial conditions will grow exponentially in time.
If no general predictive procedure is possible this computation is
called computationally irreducible. If the Church-Turing hypothesis is
true (universal computers are as powerful in their computational
capabilities as any physically realizable system can be, so they can
simulate any physical system) no physical system can shortcut
a computationally irreducible process. Simulation is the only way out.
Existing analytical solutions and numerical simulations of
Rayleigh-Taylor instability, which are based on the Eulerian and/or
Lagrangian models
, represent in fact the shortcuts of the
process, whose nature comes from particle paradigm, i.e., direct
interactions of particles in fluid. The continuous models represent
only the transformation module, which processes the signal (initial
perturbation) given on its input yielding its response on its output.
Linear theory of the first stages of mixing can give some answers
concerning, e.g. the growth rate of idealized (sinusoidal) perturbations of
given wavelengths but fails for statistically distributed
heterogenities. Moreover the statistics and the nature of these
heterogenities is unknown. In the fluid, assuming only thermal
fluctuations, the wavelength resulting from fluctuations is much lower
than the critical mixing wavelength. Therefore, in accordance with the
theory they have no chance to evolve and mixing will never occur.
The shape of input signals assumed is usually fully speculative and has nothing in common with physical phenomena, which take place at the beginning of mixing process. Therefore, it cannot be treated as any shortcut of real physics of the process. Simply, the existing models both analytical and numerical, do not cover the initial process of perturbation creation. Nevertheless, we cannot say that such the model does not exist. It does and this is the particle model.
At the end of the second stage of mixing the memory about initial conditions of the system is getting lost. In this region, however, the analytical model is poor and computer simulations (along with physical experiments) begin to play crucial role. When the process becomes more and more chaotic, the continuous computational model, in turn, is insufficient (bubbles agglomeration, spikes break-out) due to pre-dominant role of shorter and shorter wavelength scale phenomena in the system behavior in whole. Prediction of calculations are not good enough and the continuous model ceases to be an useful shortcut for the real atomistic model.
One of the main problems encoutered when one is using particle model is the time-space scale, limited by the power of computers. However, the situation is now much better than yet ten years ago. Powerful multiprocessor systems and parallel computing paradigms enable simulations of the evolution of millions of particles in million of time steps what in physical units reflects samples of micrometer in size (for 2-D simulations) and simulation time of a few nanoseconds. Of course it is far away from macroscopic world phenomena scales, but too interesting to be neglected. In this paper the authors try to answer the following questions: