# Dimensional analysis, conservation of volume, and the filling box model

The filling box model has been around for decades and is a simple model for the stratification that develops in enclosures when there is a localized source of buoyancy that forms a turbulent plume. There is a lot to the model but the prediction of the first front requires only knowing the flow rate in a plume and conservation of volume. The plume flow rate can be established from dimensional analysis after which the first front movement can be calculated from conservation of volume.

The filling box

The original filling box model (Baines, W. D. & Turner, J. S. Turbulent buoyant convection from a source in a confined region J. Fluid Mech., 1969, 37, 51-80) looks at a single round turbulent plume ascending or descending in an enclosure of constant cross sectional area. The following discussion focuses on dense plumes as they are easier to create in the lab, but the model works for buoyant plumes as well. When the plume reaches the base of the enclosure it spreads out and forms a dense layer. Over time more dense fluid is added to this layer from the plume and it thickens. Assuming that the ambient fluid in the enclosure is fairly quiescent then the density of the layer inhibits any mixing across the layer. Therefore, the only fluid entering the layer is from the plume. See the figure for a schematic of this flow.

Equipment

1. Fish tank filled with fresh water
2. Salt
3. Food coloring
4. Tubing with some mechanism for holding it steady at the top of the tank.
5. A method for delivering a constant flow rate of dense fluid into the tank (A constant head tank works but it may be hard to bring into the class room.)
6. A valve for turning the flow of dense water on and off.
7. Ruler
8. Stop watch

Image of the experimental setup with the constant head tank filled with dyed salt water, the tubing with valve to control the flow and the glass tank full of fresh water.

Below is an image of a simple constant head tank with a reservoir on the right, a pump to drive water to the left of the weir wall and an outlet at the base of the left hand side. The pump moves more fluid than is required for the experiment so that there is a return flow over the weir. Provided the height from the experiment to the top of the weir wall is large compared to the head over the weir, the total head is very close to constant. The head over the weir will only vary by a millimeter or two over the course of the experiment as the reservoir drains, the pump head increases and the pump flow rate decreases. This setup can be used for hours with the reservoir being topped up to make sure the pump intake remains flooded.

The constant head tank operating with the dyed salt water flowing over the weir wall back into the reservoir.

Demonstration

1. Mix up a dense salt solution in the constant head tank. The denser the better.
2. Fix the tubing to the outlet on the constant head tank and place the other end of the tubing at the water surface in the fish tank
3. Turn on the supply of dense water. A salt plume should form under the outlet that should fall to the base of the fish tank.
4. Measure the thickness of the dense salt layer as a function of time. The time interval will depend on the plume buoyancy flux and the tank dimensions. The larger the tank the larger the time step. The timing starts when the plume first touches the base of the fish tank.
5. Plot the layer depth as a function of time.
6. See analysis for later steps

Here is an image of the demonstration in progress. As you can see the interface can be a bit wavy early on but eventually settles down.

Plume volume flux dimensional analysis

For a fully turbulent plume the volume flux (Q) is independent of the Reynolds number and is only a function of the distance (h) from the source (provided the source is small) and the buoyancy flux of the plume. The buoyancy flux is the source volume flux multiplied by the reduced gravity of the source fluid

F=Q0g0

Where g’=g(ρap)/ρa, ρa is the ambient fluid density, ρp is the plume source fluid density, and the subscript ‘0’ means a source condition. Writing the dimensions of each of the parameters leads to

[h]=L,      [Q]=L3T-1,      and      [F]=L4T-3

There are three parameters, two independent dimensions, and, therefore, one non-dimensional group. This can be written as:

Π =QhaFb

Writing in terms of dimensions leads to

1= L3T-1LaL4bT-3b.

Writing out equations for the powers of L and T results in

0=3+a+4b      and      0=-1-3b

Solving and substituting back into the original expression for Π leads to

Q=CF1/3h5/3

where C is an unknown constant that can be determined experimentally.

Conservation of volume analysis

Returning to the filling box model we can draw a control volume around the dense layer and write conservation of volume

dV/dt+∑ Qout-∑Qin=0

where V is the volume of dense fluid in the lower dense layer. Denoting the tank cross sectional area as AT the volume of fluid in the dense layer is V=AT(H-h). There is no flow rate out of the dense layer and the flow rate in is the flow rate in the plume at the height of the interface between the dense layer and the ambient fluid above. Therefore, conservation of volume can be reduced to

dV/dt=-ATdh/dt= CF1/3h5/3.

Separating variables and integrating leads to

H-2/3-h-2/3=(2CF1/3/3AT)t

Plotting h-2/3 versus t should result in a relatively straight line.

The demonstration is nice for a number of reasons. It is a little more complex than a simple draining flow problem and you can throw in a little dimensional analysis for the plume and talk about data presentation with the final plot of h-2/3 versus t. That said, it is quite time consuming and can be a bit messy. When the plume is first turned on the dense lower layer can be very wavy. I usually use this in my graduate environmental fluid mechanics class.

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.