The Rayleigh-Taylor Instability By: Paul Canepa and Mike Cromer Team Leftovers

The Rayleigh-Taylor Instability

By: Paul Canepa and Mike CromerTeam Leftovers


Outline

● Introduction● Experiment● Data● Theory● Model● Data Analysis● Interface● Conclusion


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.


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.


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.


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:


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:


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:


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.


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:


Model 1

Now we consider q small:


Model 1

Now we look at what dominates the time evolution of the interface:


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:


Model 2

For the infinite layers we first let h represent the height of the fluid. Computing the

integrals results in:


Model 2

Once again we consider small q:


Model 2

To the leading order, q satisfies the equation:

Now, to account for an infinite height of fluid:


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.:


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:


13.23 mm

Molasses – Water:

7.11 mm

Corn Syrup – Silicon Oil:

10.08 mm

~ 3.78 Dynes/cm


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


8.14 mm

~ 4.84 Dynes/cm


Interface 1

Molasses–Water Interface


Interface 2

Corn syrup–Silicon oil Interface


Interface 3

Silicon oil–Air Interface


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.

Documents

The Rayleigh-Taylor Instability By: Paul Canepa and Mike Cromer Team Leftovers