the rayleigh-taylor instability by: paul canepa and mike cromer team leftovers
TRANSCRIPT
The Rayleigh-Taylor Instability
By: Paul Canepa and Mike CromerTeam Leftovers
The Rayleigh-Taylor Instability
Outline
● Introduction● Experiment● Data● Theory● Model● Data Analysis● Interface● Conclusion
The Rayleigh-Taylor Instability
Introduction
The Rayleigh-Taylor instability occurs when a light fluid is accelerated into a
heavy fluid. The acceleration causes perturbations at the interface, which are the
cause of the instability. The study of liquid layer dynamics is important in many
applications, for example, coating non-uniformities, flotation and filtration, even
using the motion induced by instabilities to provide rational models for predicting
continental drift and volcanic activity.
Our goal is to develop a model which will give us the wavelength of the most
unstable mode and reconstruct the early onset of perturbations of the interface. The
model will depend heavily on two ideas: the fluid-fluid interface and the energy in
the fluids. We take a dynamical approach in order to be able to determine the
wavelength which dominates the time evolution of the velocity field. Finally, we
compare our model predictions with the experimental results to assess the validity,
as well as attempt to recreate the evolution of the interface over time.
The Rayleigh-Taylor Instability
Experiment
Setup:1) We filled a rectangular box first with a thin film of the heavy fluid (molasses),
then poured the lighter fluid (water) on top, completely filling the box, and then allowed
time for the two fluids to separate.
2) We filled half of a rectangular box with corn syrup then poured silicon oil on top
until the box was completely filled.
3) Poured a thin layer of silicon oil on a piece of glass then flipped it over.
Procedure:We flipped the box over so as to have the lighter fluid accelerated into the heavier
fluid by gravity. For (1) we took pictures from the top to see the bubbles that formed;
for (2) we used the pixel camera to capture the interfacial motion; for (3) we used the
high-speed camera to capture the interfacial motion.
The Rayleigh-Taylor Instability
Data
Data collection*:Our goal is to find the wavelength of
the most unstable mode. For (1) we
measured the distance between centers of
neighboring drops; for (2) we measured the
distance between neighboring spikes; for
(3) we measured the distance between peaks
of drops. For (3) we also measured the
height of the drops, to compare with our
interface reconstruction.
*All data collected from the experiment can be found
found on the wiki.
The Rayleigh-Taylor Instability
Theory
Note: The following theory and model proposal follow an approach due to Chang and Bankoff
(reference on wiki).
We first assume that the fluid is incompressible and the motion is
irrotational, and that the fluid motion in the horizontal direction is sinusoidal. We
can then assume that the velocity potentials have the form:
The Rayleigh-Taylor Instability
Theory
We will consider two systems. First, we assume infinitesimally thin
layers of fluid (i.e. y --> 0), for which the velocity field becomes:
Second, we will look at infinite layers of fluid, for which the velocity
field is:
The Rayleigh-Taylor Instability
Theory
In order to follow a particle we must find the particle path slopes:
Now, let the initial position of the particle be given by , then
integrating the above, and noting that at the interface, we
find the equation of the interface:
The Rayleigh-Taylor Instability
Model
Now that we have an expression for how a particle moves along the interface,
all we need to do is find q. In order to do this we use an energy balance over one
wavelength. For this system there are several energies which must be considered –
kinetic, potential and surface. However, we believe that viscosity affects the rate of
deformation of the fluid.
The Rayleigh-Taylor Instability
Model 1
First we consider infinitesimally thin layers of fluid, say height: . After
computing the integrals we arrive at the following equation for q, F & K are on the
wiki:
The Rayleigh-Taylor Instability
Model 1
Now we consider q small:
The Rayleigh-Taylor Instability
Model 1
Now we look at what dominates the time evolution of the interface:
The Rayleigh-Taylor Instability
Model 1
In an attempt to find q, we consider the next order equation:
We are currently unsure of the appropriate initial conditions, but for now we will
go with:
The Rayleigh-Taylor Instability
Model 2
For the infinite layers we first let h represent the height of the fluid. Computing the
integrals results in:
The Rayleigh-Taylor Instability
Model 2
Once again we consider small q:
The Rayleigh-Taylor Instability
Model 2
To the leading order, q satisfies the equation:
Now, to account for an infinite height of fluid:
The Rayleigh-Taylor Instability
Model 2
There are two unknowns in the previous equation, k & q; in order to deal with
this problem we let q assume a particular form:
n = n(k) is the growth rate, which satisfies the equation:
Our goal now is to find the wavenumber, k, associated with maximum growth,
i.e.:
The Rayleigh-Taylor Instability
Data Analysis
Experiment vs Thin Layer Theory
Experiment
Average measured wavelength:
Silicon Oil – Air:
12.17 mm
Molasses – Water:
15.66 mm
Model
Most unstable wavelength:
Silicon Oil – Air:
13.23 mm
Molasses – Water:
7.11 mm
Corn Syrup – Silicon Oil:
10.08 mm
~ 3.78 Dynes/cm
The Rayleigh-Taylor Instability
Data Analysis
Experiment vs Infinite Layer Theory
Experiment
Average measured wavelength:
Corn Syrup - Silicon Oil:
4.85 mm
Model
Most unstable wavelength:
Corn Syrup - Silicon Oil:
9.74 mm
Molasses – Water:
7.14 mm
Silicon Oil – Air:
8.14 mm
~ 4.84 Dynes/cm
The Rayleigh-Taylor Instability
Interface 1
Molasses–Water Interface
The Rayleigh-Taylor Instability
Interface 2
Corn syrup–Silicon oil Interface
The Rayleigh-Taylor Instability
Interface 3
Silicon oil–Air Interface
The Rayleigh-Taylor Instability
Conclusion
There is most likely a lot of error in our liquid-liquid data due to the difficulty
of finding good experimental methods and the inability to determine which spikes
were on the same line. Our thin layer model appears to be a good approach; it
compares very well with the good data that we have. The infinite layer model
may be good, as can be seen by comparing to thin layer predictions (except for
silicon oil-air). Also, for the liquid-liquid system, the interface takes on odd
shapes, ones that could only be described by using a nonlinear ode for q – for
future work we could either use an experimental wavelength or the thin layer
approximation to use for k, then solve the original ode for q numerically.