Welcome to our first guest post! It is a simple experiment that was developed by Bruce R. Sutherland for a high school newsletter. However, it could also be used to demonstrate a range of different phenomena including that 2D turbulence tends toward larger scales over time whereas 3D turbulence decays to smaller scales. I plan on using this in my environmental fluid mechanics class and may use is as a quick demonstration in my undergraduate fluids class as an interesting phenomena (if I can find a spare 15 minutes in the schedule). Bruce Sutherland is a Professor in the Department of Mathematical and Statistical Sciences at the University of Alberta. His e-mail address is (Bruce dot Sutherland at ualberta.ca). His website is http://taylor.math.ualberta.ca/~bruce. The write up also includes general reflections on the subject of fluid mechanics, its applications, and why he decided to study the subject. I decided to leave the full text of the write up with Bruce’s reflections rather than try to edit it down to just the demonstration since it flows nicely as is.
The study of fluids is pervasive in such scientific disciplines as mathematics, physics, chemistry, engineering and medicine. To name but a few examples, Fluid dynamics researchers might examine methods for extracting oil from Alberta’s tar sands and transporting it through pipes, or they might study how medicine is distributed though the body’s cardiovascular system, or they may try to predict how the cold waters in the Equatorial Pacific will affect Canada’s weather during a La Nina event. Although the equations describing the motion of fluids were derived two centuries ago, exact solutions have been found for only a few special cases. The challenge of finding exact or approximate solutions have continually pushed the frontiers of mathematics, most recently through devising efficient and reliable computer codes, and through the development of new fields of mathematics, including chaos theory and pattern formation, about which many popular science books have recently been written.
Much of my work examines mixing and waves in fluids with varying density. Such fluids are said to be “stratified” because they act as if they are composed of slabs of fluid layered one on top of the other. Oceans, lakes and the atmosphere are stratified fluids. (Indeed, the stratosphere gets its name because its density decreases relatively rapidly with altitude.) The air in the room where you are sitting is a stratified fluid: hot, less dense air floats near the ceiling and cooler air is closer to the floor.
I have been drawn to study fluid dynamics not only because its applications are of such practical importance, but because of the intuition and breadth of knowledge required by the discipline. Furthermore, because of the remarkable growth in the speed and memory size of computers, many fundamental problems in fluid dynamics that were previously unsolvable can now be modelled numerically and studied in laboratory experiments using lasers and digital image processing. The following describes an experiment that can be done in the kitchen and which demonstrates some of the beauty and surprising complexity of stratified fluid motion. You will need the following:
- 9″ x 11″ glass baking dish (or similarly large glass dish)
- a 4 cup measurer or bowl of at least this volume
- 8″ x 8″ piece aluminum foil
- ½ cup sugar
- food coloring (two colors)
Figure 1: What you need to make a stratified fluid.
Figure 2: After filling the bottom half of the pan with sugar water, pour tap water, dyed red, into the aluminum boat. This will inhibit mixing and most of the red water will end up floating on top of the sugar water solution.
Figure 3: Put in a few drops of blue food coloring and watch the patch evolve into spirals.
Figure 4: Dragging a knife through the patch makes more complex patterns of vortices.
To make a stratified fluid, add 4 cups of water and 1=2 cup of sugar to the glass dish. (You may wish to put the dish on a white cloth to observe the fluid motions more easily later on.) Mix these together to form a strong sugar solution (the density of the solution should be about 1.1 gcm-3, compared with fresh water which has a density of about 1:0 gcm-3). Now measure out another four cups of water and add a light colored dye to it (four drops of red food coloring should be enough). We want to layer this dyed fresh water on top of the dense sugar water. The following is a crude but effective way to do this. Make a “boat” from the aluminum foil with a flat bottom and sides as high as the sugar water in the baking dish (about 1 cm). Float the boat on the sugar water and slowly pour in the dyed water. The boat will lower into the sugar water and the dyed water will eventually overflow spilling over the sugar water. You will notice that as the dyed water spills out, it floats over the sugar water. Continue to pour all four cups of the dyed water into the overflowing boat, pouring at a rate so that it takes about a minute to do this. When you are done, you should still be able to see some clear, undyed sugar water at the bottom of the dish.
Congratulations! You have made a stratified fluid!
At this point it is easy to see an astounding property of stratified fluids: add a single drop of dark colored dye (such as green or blue) to the center of the dish and observe what happens. If you do this within a few minutes of making the stratified fluid, it is likely you will see the dye stretched out into a spiral-like vortex looking not unlike a nebula in various Star Trek movies. A coherent slowly swirling vortex such as this typically does not occur in unstratified, homogeneous fluids. To see this, just add a drop of dye to fresh water in a bowl or another baking dish. You will likely find that the dye in this case gets pulled into filaments of ever finer structure in a motion that is typically chaotic and progressively less predictable.
To emphasize further the difference between the behaviors of stratified and unstratified fluids, take a knife and slowly drag it through both fluids creating a wake about 1 mm wide. In both cases, you will see small-scale turbulence in the knife’s wake. But what happens over time as the turbulence decays? In the unstratified fluid, mixing occurs near the wake and the resulting motion dies down after a minute or so. In the stratified fluid, from the small-scale mixing emerge large, slow-moving vortices which grow in size as they combine with other vortices and which continue to evolve for many minutes. The collapse and decay of turbulence in a stratified fluid involves many complicated processes that are the subject of active research today. How might a mathematician approach this problem? The first step is to write the exact equations of fluid motion appropriate to this problem. Although it requires an understanding of calculus to make any sense, the equations describing the motion of sugar water are given below (with an English translation of their meaning in parentheses below them):
Figure 5: But if you leave the fluid undisturbed for a long time, it will form one large spiral. This spiral took 5 minutes to form.
Here ρ is the density, the velocity, and p the pressure, all three of which are functions of space and time. The constant g is the acceleration due to gravity and ν is the kinematic viscosity (which is a measure of friction within a fluid). The symbols D/Dt , ∇, and ∇2 are convenient notations involving derivatives which are used to describe infinitesimal changes in time and space. Similar equations also exist describing how the density changes in time. The equations describing the motion of unstratified, fresh water are the same as those for stratified water but with the ρgz term removed. Although, with experience, it is a relatively simple matter to write down the equations, at present they cannot be solved to describe the turbulent motions in the above experiment. Indeed, they may never be solved; one would be hard pressed to think of a function that could encompass such complexity of evolution in time and space. Nonetheless, mathematical progress has been made. From experiments it was realized that unstratified (homogeneous) turbulence exhibits a special kind of symmetry, which today we describe as being fractal: a close up view of turbulence looks almost identical to turbulence seen from farther away.
For example, the turbulent plume formed by pouring cream in your coffee is similar in many respects to the turbulent plume from a chimney or an exploding volcano. Using scaling theory, scientists have been able to estimate how quickly energy is dissipated and how fast pollutants are mixed in turbulence. Stratified turbulence is much more difficult to model in this way, however. Mathematically, this is due to the presence of the ρgz term in the equations of motion for stratified fluids, which represents buoyancy forces acting vertically to carry relatively heavy fluid downward and light fluid upward. Physically it means that fluid, loosely speaking, “prefers” to move horizontally when it is stratified. You can see this in the experiments. When the knife is dragged through the stratified fluid, vertical motions are suppressed in its turbulent wake and only horizontal motions persist. Effectively, the motion evolves from one which is three dimensional (moving horizontally and vertically), to one that can be thought of as two dimensional (moving strictly horizontally). Although the transition from three dimensional to two dimensional motion is not yet well understood, scaling theory can be applied to “two dimensional turbulence” to predict that large scale, slowly evolving vortices should develop, as observed. Progress is being made in understanding turbulence with the aid of computer models. To this end, scientists are reformulating equations, like that above, into a form that can be calculated numerically. Such methods can only approximate the exact solution because one must ultimately impose a restriction on the smallest sizes of motion that can be resolved by the computer. Computers with greater speed and memory are providing ever more accurate solutions which are only now capable of reproducing some of the observations in laboratory experiments. It is not unreasonable to hope that with improvement in computers and laboratory measurements, further mathematical breakthroughs are just around the corner.
An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (firstname.lastname@example.org). I also welcome comments (through the comments section or via email) on improving the demonstrations.