Calculating Aerodynamic Shape
Country: United States
Date: March 2009
I am designing an experiment using a wind tunnel that my school
owns. I've made four small towers out of wood that have the same height and
base width - a cylinder, a rectangular prism, a triangular prism, and a
pyramid. I want to figure out which shape is the most aerodynamic. I will
put the towers one at a time into the wind tunnel, perpendicular to the
ground, and exert the same amount of horizontal wind force on each of them.
I need to know how I can calculate which shape is the most aerodynamic. How
can I measure each tower's air resistance against the wind?
That will be a good experiment.
What you need to determine is the area that the wind is blowing perpendicular to
on the tower and where the centroid of that area is. From the shapes you have
chosen I think you are making the area of the tower you see (from the direction
of the wind) equal and the centroid of that area would be at mid-height. This
keeps the comparisons more simple.
When the wind blows it tries to topple the tower and if the tower is fixed at the
bottom it creates what is called a 'moment' in the column, which is the twisting
the column experiences from the force of the wind. This wind force also deflects
the tower. So if you can either measure the deflection at the top of the tower or
measure the 'strain' at the tower base you can relate that to the wind force. The
'strain' is a lengthening of the windward side of the tower as it bends backward in
the wind. It is measured with a stain gage you cement onto the tower. When you
know the strain you can determine the stress in the tower from the equation E =
stress / strain with E being the modulus of elasticity for the material your tower
is made from. Probably, having a deflection gage and measuring the deflection at
the top of the tower would be the easier measurement.
Assuming you have chosen to measure deflection for each tower you now have data to
make a comparison of how the shape affects the wind load. You have to use a formula
for deflection of a cantilever (your tower is a cantilever fixed at the ground) and
from that determine the wind loading. That formula for deflection = w x L**4 / 8 E I.
It would be best to check a civil engineering text to get all the terms correct. w is
the wind load and is the unknown you are determining. With the same wind test speed it
will vary from tower to tower because of the different tower shapes and will give you
the information you want. L is the height of the tower; E is the modulus of elasticity
and is dependent on what you build the tower from; I is the moment of inertia of the
tower and is determined from the cross-section area of the tower about an axis
perpendicular to the wind. Based on the tower shapes you have chosen the I's will
be different and have to be taken into account because they will affect the strength
of the tower. You have to be consistent in your use of units throughout the
This all sounds pretty complicated but you will learn a lot doing the experiment and
calculations. There are factors or coefficients that have been determined from
tests like you are doing that compare how these shapes react to the wind; after you
have made you own determinations you might want to see how they compare to available
I am not an aerodynamic engineer, and I know you have to keep things simple. So here
are some "simple minded" suggestions.
1. Set up the experiment as you describe, and start the wind tunnel at a very slow
RPM which I assume you can control. Be careful because the objects have different
symmetry with respect to the direction of air flow. That is, the rectangular prism
and triangular prism are going to behave differently depending upon their orientation.
Record the RPM that is air speed at which each tower falls. You can alter different
air flows by putting weights in the tops of the block you are measuring.
Admittedly, this is pretty crude.
2. The following setup should give you better results. Attach the base of each
object to a loosely coiled spring (You will have to find one, maybe from a lab supply
vendor). I also assume you have a viewing port in the wind tunnel. Behind each object,
mount a piece of graph paper glued to a board, or alternatively have some way to mark
off etch marks of the background. In front of the viewing port mount a digital
camera -- this does not have to be fancy. Then start the air flow and photograph
the assembly as a function of wind speed. The object will tilt a certain angle (x)
as a function of wind speed. If I have done my trig correctly tan(x) will measure
the deflection of the blocks. The smaller the value of tan(x) , or "x" the more
aerodynamically "smooth" the object. You could also check the effect of the axial
symmetry. You could also do a relative comparison vs. the cylinder since that should
not show any axial differences like the other shapes.
Also note that you can reverse the configuration and use a hanging spring, but you
may have to put a ballast on the blocks to prevent them from "waving in the breeze".
Interesting experiment. It is not possible to calculate which of your
shapes has the least air resistance, without getting into fiendishly
difficult and involved equations that are likely beyond high school
level math, and even then the results will not be particularly
accurate. Nowadays, this type of a problem is solved using Finite
Element Analysis (FEA) software and even then it will take several
hours of time on a fast computer to come up with an accurate answer.
Millions of calculations are needed! Of course I suspect, you already
know the answer... the cylinder will have the least air resistance.
You can make measurements, however. You will need to measure the force
on the object as the air blast tries to push it "downwind". One method
might be to mount the object under test so it is restrained from
moving backwards (as a result of airflow impinging on it) with springs
holding it at the top and bottom. Thus, the drag caused when the air
flows, causes the springs to stretch a little. The amount that the
springs stretch, indicates how much drag is occurring. The test object
that stretches the springs the least, has the least air resistance.
this sounds like a great experiment! Your teachers might have told you that the
first step in science is defining the problem, and I think that is the first step
that is needed here. I suggest you start by defining what variables you can control,
measure, or calculate. For example, you can control the wind speed of the wind
tunnel and the design of the towers. You may be able to measure forces that the
towers exert on whatever surface they are resting on, but you cannot control it
directly. The force on the tower is a dependent variable, while the wind speed is
an independent variable (if you have not heard these terms, you can look them up
on-line). The force exerted on the tower by the wind would have to be calculated
from some other measurements, as it cannot be directly measured. My point here is
that you have to be careful in thinking about what you can actually control or
measure and what you cannot. A term like "most aerodynamic" is vague, and vague
terms are things we want to avoid in science.
So with this in mind, what would you like to test? You mentioned forces, so I am
assuming you have the ability to measure forces. Perhaps you could put your
pressure sensors under the posts of the towers and measure the values compared
with different wind speeds. You did not mention sizes of your equipment, but the
type of measuring apparatus definitely depends on the sizes involved. Alternatively,
if the towers are very flexible, you might be able to use a camera to capture them
bending in the wind. In both cases, you could then estimate/calculate how much force
is applied by the wind. You might also compare the cross-sectional area of each tower
to the force applied and draw some conclusions there.
Hope this helps,
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Update: June 2012