Calculating Drag Coefficient from Wind Tunnel Data

[zehkari]

Homework Statement

Using a wind tunnel to measure force and velocity at different drive %, we obtained some data for drag. We used one dimpled sphere and one smooth sphere. There are a couple of questions I am stuck on.

"The force you have measured is known as the drag force and can be calculated using the following equation:
$$F_D=\frac {1}{2}ρ.v^2.C_D.A$$
(f): Re-arrange this equation in the form $y=mx+c$, and state the formula for the gradient of the line."
(d) Explain whether the relationship between wind speed and force is linear or non-linear - and give your best estimate of the relationship (equation).
(g) Plot a second graph of force on the y-axis and wind speed squared on the x-axis, add a trend line and display this and the R2 value on the graph. Calculate the drag co-efficient for each sphere.

Homework Equations

$$F_D=\frac {1}{2}ρ.v^2.C_D.A$$
$$y=mx+c$$

The Attempt at a Solution

Attached are graph results of both the raw data obtained and a velocity$^2$.
I have little background knowledge as I am new to calculating drag.
But taking a guess,
(f) $F_D=\frac {1}{2}ρ.v^2.C_D.A$ re-arranged to $y=mx+c$ is: $$F_D=\frac {ρ.C_D.A}{2}v^2$$
Therefore, the gradient would be:$$m=\frac {ρ.C_D.A}{2}$$
(d) I can only see a non-linear relationship between wind speed and force from the raw data and then a linear relationship for velocity$^2$. Not quite sure where I can estimate a relationship with an equation.
(g) To calculate the Drag Co-efficient, you could re-arrange Drag Force equation to get: $$C_D=\frac {2F_D}{ρ.v^2.A}$$
For the Dimpled Sphere:
A(Cross sectional area) =$1.45×10^{-5}$
ρ(Fluid density/kgm$^{-3}$)=$1.225 kgm^{-3}$
So,$$C_D=m×\frac {2}{ρ.A}$$
ie,
$$C_D=0.0003×\frac {2}{1.225×1.45×10^{-5}}$$
=$33.78$

Then repeat for the smooth sphere.

Any help would be appreciated. Thank you for your time,
Zehkari.

 

[haruspex]

the gradient would be:

Yes, but you should have quoted that y is to represent force and x the square of velocity. The image is hard to read.

Not quite sure where I can estimate a relationship with an equation.

I believe they are asking you to write y=mx+c, but plugging in the numbers (and units) for m and c, and replacing x and y by the variables they represent.

Plot a second graph of force on the y-axis and wind speed squared on the x-axis

Your two graphs look the same shape even though the x axes are different. Looks to me like you plotted F against v² in both, whereas the first should have been F against v. Again, it is hard to read, so I'm not sure.

 

[zehkari]

Hello,
Thank you for the help.
Here is a link to a better quality file (apologies for the small picture before, should of checked):
https://imgur.com/a/EWfsy
The pattern of results unfortunately do not give a clear result. There is a slight curve for force against velocity and a more linear relationship for force against velocity squared. I have added a trend line for force vs velocity graph now as well to see a difference.

(f) Thank you for confirming and I will quote that y is to represent force and x the square of velocity.
(g) I think because its force against velocity, then the equation would be stating $f ∝ v^2$, ie, $y=x^2$.

Once again, thanks for the help.
Zehkari.

 

[haruspex]

There is a slight curve for force against velocity and a more linear relationship for force against velocity squared

Yes. There seem to be a couple of rogue datapoints in the middle of the dimpled data which mess it up.
You could make it a bit better by forcing the curves through the origin. You know that is a rock solid datapoint. A simple way to do that is to replicate it in the dataset, maybe ten copies of (0,0).

I will quote that y is to represent force and x the square of velocity.

No, I meant that the question in the image you attached said that, but you omitted to mention it in the post. It would not have mattered if the image had been clear.

the equation would be stating f∝v²

No, I think they want you to fill in the actual number, like F=2.17 v² kg/m, or whatever number you determine from the data. Note the units.