Stratified Flow and Instability¶
By Elise Coates and Zoe Hallman | April 1st, 2024
Stratified Flow of Two Densities¶
Introduction¶
Stratified flow involves of difference densities of fluid interacting with eachother under gravity. On Earth this can be seen when fresh glaciel water meets the oceans salty water, or in the atmosphere.
In the ocean with stratified flow, disturbances can cause a standing wave in the heavier fluid to propogate, called a seiche. The period of this standing wave oscillation varies with the density differences can be calculated by $T = \frac{2L}{\sqrt{gh}}$ where $L$ is the length, $h$ is the height, and $g$ is gravity.
Procedure¶
Stratified flow is visualised in a narrow tank with a divider half way along, called a lock exchange. Fresh water is used as the lighter fluid, coloured with yellow dye, and the heavier fluid is made with varying amounts of salt, coloured with blue dye.
1500 ml of each of the fluids are poured into either side of the lock, and when they are reasonably settled to reduce turbulence, the barrier is removed. The fluids are then observed to move accross eacher other and the seiche period is recorded.
Results and Discussion¶
The following video is for salt water density of 1.007 $g/cm^3$ (15ml), the lowest of the tested densties. The videos for other densities can be found in the appendix.
Five different densities of salt water were tested yielding the following results:
| Amount of Salt [$g$] | Density of Salt Water for 1500ml [$g/cm^3$] | Density Difference to water ($0.998 g/cm^2$) [$g/cm^3$] | Seiche Period [s] |
|---|---|---|---|
| 17.3 | 1.007 | 0.009 | 54.4 |
| 23.1 | 1.009 | 0.011 | 51.2 |
| 34.6 | 1.015 | 0.017 | 40.4 |
| 46.2 | 1.021 | 0.023 | 34.4 |
| 53.1 | 1.025 | 0.027 | 30.2 |
The chosen amounts of salt in this experiment were relatively high, and in this high density difference regime the period of the seiche is found to increase linearly with decreasing density. Since the seiche wave dissipated relatively quickly, it was timed for only one half of the period, from one end of the tank to the other and then doubled. The timer was not started as soon as the barrier was lifted (middle to middle timing) to ensure that any turbulence or extra movement due to removing the barrier was negligable by the time it reached one end and reflected off of the tank.
Tips for next time:¶
- Start with smaller amounts of salt such as 5g rather than 15g and increase it by small incriments. This may give more interesting results that are comparable to theory
- Experiment with the more or less water, i.e changing the depth may also give insight. Keep the same amounts of both fluids though or else there is too much wave action that mixes the fluids.
Kelvin-Helmholtz Instability¶
Introduction¶
Stratified flow of three densities is used to demonstrate Kelvin-Helmholtz insability. This insability occurs when two different densities sheer against eachother, and periodic vortices are produced. This is most commonly observed in the atmopshere when clouds of different densities than air move across eachother at different speeds. It is also observed in the atmospheres of stars as well as Jupiter's famous red spot. Photo credit: https://en.wikipedia.org/wiki/Kelvin–Helmholtz_instability
Procedure¶
Following the previous experiment, the two fluids are allowed to settle after the barrier is released until the seiche wave and purterbations have dissipated. The barrier is placed again, and one side of the lock is mixed to get a medium density. Black dye is added in the mixing to easily visualize the layers. Three densisites are used here to increase the chances of seeing the instability on the top or bottom side of the black dye.
The barrier is removed and the black fluid moves between the yellow and blue layers. As it shears against the other layers, the characteristic bumps and curls of Kelvin-Helmholtz instability can be seen.
Results and Discussion¶
The periodic bumps of the Kelvin-Helmholtz instability can be seen as the black dye is about half way from the the middle to the wall. At the interface of the black dye and blue dye.
Tips for next time¶
- Try out different densites to find the optimal combonation
- Try to get a faster shear to create the curls by varying the amounts of fluid used
Rayleigh-Taylor Instability¶
Introduction¶
The Rayleigh-Taylor instability is the result of a lighter fluid pushing against a heavier fluid. The dominant pressure gradient is hydrostatic and occurs when a lower-density fluid accelerates a higher-density fluid. This acceleration occurs because the pressure gradient, Δ𝑃 opposes the density gradient, Δ𝜌. Examples of such phenomenon include the mushroom clouds that form from a nuclear or volcanic reaction, when water is suspended above oil, mantle plumes, and when ablation occurs during inertial confinement fusion. A particularily niche application of the Rayleigh-Taylor instability is described by an article from Science: “Snapping shrimp, which produce most of the ambient noise in subtropical shallow waters throughout the world, produce their sound through collapse of a cavitation bubble, generated by a water jet formed from rapid claw closure; the bubble is destroyed through a Rayleigh-Taylor instability.” (Versluis et al. 2000)
The governing differential equation for this instability is as follows:
$$\frac{d}{dz}(\rho\frac{dw}{dz}) - \rho k^2 w = -wg \frac{k^2}{n^2}\frac{d\rho}{dz}$$
where $k$ is the wave number, $g$ is gravity, and $w$ is the vorticity. The eigenvalue solution for this equation is:
$$ n = \sqrt{gk\frac{\rho _2 - \rho _1}{\rho _2 + \rho _1}} $$
where $\rho _1$ in the demonstration’s case was the density of water, and $\rho _2$ was the density of the black dye.
The Rayleigh-Taylor has four dominant stages. When the dye is first introduced, the perturbation amplitudes are small compared to their wavelengths, and therefore, the equations of motion are linearized and there is exponential growth in instability. In this stage, a sinusoidal shape in briefly maintained. In the second stage, non-linear affects appear and the presence of the characteristic mushroom-shaped structures of the heavy fluid begin to develop and grow into the lighter fluid’s region. In this step, bubbles also form. The growth rate is now approximately constant with time. The third stage is where the bubbles and the mushroom-structures interact and non-linear mode coupling occurs. Mode coupling can be seen in the diagram included below. Two vortices rotate in oppposite directions at the edges of the dense fluid bulge. The two, opposite moving vortices are key in the evolution of instability as the initial higher-density bulge accelerates into the lower-density fluid and grows with the production of the bubbles and mushroom-structures. The mode coupling causes the small, higher-density fluid structures and bubbles to combine and form larger structures. The smaller wavelength structures that are saturated are consumed by larger, unsaturated structures. The final stage is turbulent mixing of the fluids. If the Reynold’s number is large enough, the mixing region is self-sustaining and turbulent.
Procedure¶
A small test-tube without volume lines was used to allow full visibility of the instability. Water was used as the lower density fluid, and black dye used as the heavier fluid. The composition of the black dye was unknown however typical black dye densities are around $1.1 g/cm^3$. A pipette was used to deposit the black dye on top of the water and to initiate the instablity.
The test-tube filled with water was held still and the black dye was pipetted on top of the water. This procedure was repeated multiple times with non-exact amounts of dye. The amount of dye pipetted each time was varied because only as much dye as was needed to produce the instability was used.
Discussion¶
The way the black dye was inserted was not the most ideal as it was not very controlled and the flow was not optimally smooth. With the pipette, it was hard to control the force with which the dye was put into the test-tube. The black dye was already accelerating before it reached the lower-density fluid, which disturbed the natural process of the black dye accelerating into the water due to gravity only. Ideally, the dye would have been inserted under the force of gravity, such as letting it drip into the test-tube down the sides, or gently siffoning it. The siffoning would still potentially produce an acceleration if it inserted the dye with too much force.
Through video analysis, it was evident that the introduction of a more dense fluid on top of a less dense fluid initiates a Rayleigh-Taylor instability. The main issue with producing the instability by pipetting a dense dye onto a realtively deep layer of water was that the acceleration of the dye out of the pipette interfered with the acceleration that was supposed to occur because of the pressure gradient, $\Delta P$, opposing the density gradient,$\Delta \rho$ between the fluids.
Tips for next time¶
- A suggestion for future demonstrations would be to develop a piece of apparatus that allowed one to have the two fluids, one above the other, with a removeable gate seperating them.
- Vary the size of the 'test tube' or container and see if the size of the mushroom clouds change as well.
References¶
- Versluis, M., Schmitz, B., von der Heydt, A. and Lohse, D. (2000) How snapping shrimp snap: Through cavitating bubbles. Science, 289:2114-2117
- https://www.sciencedirect.com/science/article/abs/pii/S157418181830106X#:~:text=The%20classical%20Rayleig Taylor%20instability,1%5D%2C%20%5B2%5D.
- https://en.wikipedia.org/wiki/Rayleigh–Taylor_instability
- http://www.scholarpedia.org/article/Rayleigh-Taylor_instability_and_mixing
Appendix¶
20 ml of salt ($1.009 g/cm^3$) lock exchange video:
30 ml of salt ($1.015 g/cm^3$) lock exchange video:
40 ml of salt ($1.021 g/cm^3$) lock exchange video: