Dimensional Analysis and turbulent jets

This demonstration is a good way to illustrate / visualize a simple dimensional analysis example.

Equipment

1. Fish tank full of fresh water
2. A couple of syringes of different sizes (though you can get away with a single syringe)
3. Food coloring

Analysis

I do this demonstration in the reverse order compared to most of my demonstrations. That is, I do the analysis first and then demonstrate that the predicted result is observed. The basic premise is that, provided the Reynolds number is large enough, the flow of a round turbulent jet is not influenced by the fluid viscosity. Therefore, the radius of the jet (r) is a function of the source diameter (D), downstream distance (x), and the source velocity (U). That is,

r=r(D,x,U).

Writing out the parameter dimensions gives [r]=L, [D]=L, [x]=L, [U]=LT-1. There are four parameters and two independent dimensions leading to two non-dimensional groups. Taking x and U as the two repeating variables one gets

π1=r.xa.Ub                      and         π2=D.xc.Ud

solving for the exponents leads to a=-1, b=0, c=-1, and d=0. That is,

r/x=f(D/x).

In the limit of x>>D the term in brackets approaches zero and the function must approach a constant. Therefore, far from the nozzle outlet, the radius will increase linearly with distance and the radial spread rate will be independent of the source velocity. The demonstration illustrates these two results.

Demonstration

1. Fill the fish tank with fresh water. You may want to put a white background behind it for contrast. Taping some paper works fine.
2. Fill two syringes with water dyed with food coloring
3. Inject the two syringes vertically down into the fresh water to form two turbulent jets. Try to ensure that the exit velocities are different for the two jets.

If the demonstrations goes well the students should observe the classic conical shape for each jet with approximately the same radial growth rate. The problem with this demonstration is that, when the jets impinge on the base of the tank, they overturn and the entire tank is rapidly filled with dyed water. It is, therefore, hard to re-run the demonstration. It is also hard to do the demonstration smoothly. Below is an image of two jets. The one on the left has a much lower source velocity. They are both somewhat conical in shape (behind the head of the starting cap) though the radial growth rate differs. This is due to the source wobbling as it is hard to keep them still when trying to do two at once. Doing one at a time solves this problem (or building a mount for the syringes) but you don’t get the direct the comparison. It is also hard to get them vertical.

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 (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.