The high amplitude motion of the surface of a fluid displays a number of extraordinary phenomena ranging from localized structures to turbulence. Among localized structures there can appear breather [1] and kink [2] solitons [sometimes called darkons] and aspects of turbulence can be studied via the mutual scattering of ripples running along the surface of a fluid[3].
As the height of the surface of a fluid ) ,
ζ(r,t) as a function of position and time is the key parameter measurement of the above phenomena requires a technique that
can resolve large variations in ) , ζ(r,t) . In
our experiments the surface motion is made visible by suspending into the water a .04% solution of polyballs. This concentration is
sufficiently dense that light traveling through the water is so strongly scattered that it
diffuses. In this way the technique overcomes the problems presented by the appearance of caustics
in attempts to apply shadowgraphs to high amplitude fluid motion. The key criterion
for the application of diffusing light principles to resolve surface height is that the
overall depth of the fluid must be greater than the transport mean free path which in turn must
be comparable to the maximum surface heights resolved. The concentration is
sufficiently dilute that it does not affect the viscosity. Light is then incident on the
fluid from below and a charge coupled device (CCD) records the light to exit the fluid. Typically the
image of the surface is broken up into one million pixels [1024x1024] where each pixel is
capable of recording a dynamic range of 65,000 gray scales [or 16 bits]. This image is
converted into the surface height with the help of a calibration plot. This plot shows the
amount of light to exit the surface as a function of fluid depth. The deeper the fluid the smaller
is the amount of light to make it to the surface at that location. The dependence on surface
slope is weak as demonstrated from photos of jello molds in the shape of sine waves that are
formed from these water polyball solutions. The maximum slopes of these molds corresponds to
capillary wave mach numbers of about ½, This imaging technique is useful for obtaining an
instantaneous realization of the surface height when the illumination is in the
form of a short [microsecond] flash of light. The surface height as a function of time at a
single point in the fluid can be obtained by reading out the calibrated signal from a single pixel
as a function of time.
This technique has been able to resolve, the hyperbolic secant profile of a
breather soliton, the hyperbolic tangeant profile of a dark on, the unusal shear thinned
states of non-Newtonian fluids, and the Kolmogorov spectrum of ripple turbulence. This last
case constitutes the core of a proposed experiment in microgravity and will be
described further. The law for the spectrum in the inertial region can be derived in parallel with
Kolmogorov’s law of vortex turbulence. If k E denotes the ripple energy per unit
area between k and 2k then k E must be sufficiently large that nonlinear reversible
processes dominate damping of mechanical energy due to viscosity.. Furthermore the density
of modes excited must be sufficiently large that their spacing in frequency space
is less than their broadening due to nonlinear interactions. In this limit one finds for the
power spectrum of surface motion :
where q,w, k, q are energy input [at some low frequency], frequency, wavenumber,
fluid density, and surface tension. Measurements reveal reasonable agreement
with both of theses formulas. The fact the the temporal spectrum agrees with the spatial
spectrum is another test of the imaging technique.
A number of key assumptions underly the ‘derivation’ of Kolmogorov’s law
from the theory of interacting propagating waves[4]. They are 1) the random
phase approximation and 2) a closure hypothesis. In its simplest form the closure
hypothesis says that the average of the product of action of two modes is the product of
the averages [this can be called a stosszahl ansatz]. The abiding question of turbulence is
the extent to which these assumptions are violated.
These issues will be addressed in a microgravity environment where ripples will
be studied on the surface of a positioned fluid sphere that is about 12cm in
diameter. The sphere will be held in place and excited by acoustic forces. Light will be
brought in with an optical fiber to the center of the sphere. From that point light will diffuse
out to the surface where it will be imaged.
On the surface of a sphere the wavenumber k is replaced with l/R where l is an
integer and R is the radius of the sphere. For a cascade starting at 1Hz [l
about 8], the turbulent regime in k should extend over more than 1.5 decades.
The advantages of microgravity are that the forces of capillarity dominate the
motion over a wider ranger of wave numbers than on the ground. This is desirable
as the nonlinear properties of capillary waves are much stronger than those of gravity
waves.
At these longer wavelengths capillary waves also suffer less dissipation and so
the effects of turbulent motion are more accessible. In low g it is possible to levitate a
large drop of fluid. In this arrangement capillary waves do not scatter off of boundaries but
run around the drop and scatter only off other waves which is precisely the effect that we
seek. This can be called turbulence without boundaries.
Wright, W., Putterman, S., Diffusing Light Photography of Containerless Ripple Turbulence, Proceedings of the Fifth Microgravity Fluid Physics and Transport Phenomena Conference, NASA Glenn Research Center, Cleveland, OH, CP-2000-210470, pp. 800-819, August 9, 2000.