Calculating Wind Tunnel Speed for Aerodynamic Similarity in Automotive Design

[Kenny Lee]

The aerodynamic drag of a new sports car is to be predicted at a
speed of 100 km/hr at air temperature of 25oC. Automotive engineers
build a ¼ scale model of the car to test in a wind tunnel, where the air
temperature is 10oC. A drag balance is used to measure the drag, and
the moving belt is used to simulate the moving ground. Determine the
speed of the wind tunnel that the engineers must run in order to achieve
similarity between the model and prototype.

I would say something more... like maybe tell you what I've done and stuffs, but I'm just plain stuck. It's the kind of stuck that has you running up walls, and scratching brains out. Please, just some direction.

Thanks.

 

[mezarashi]

This has been sometime since I've done this in my fundamental engineering classes, and I am no longer in the discipline of civil or mechanical engineering. In anycase, as a guidance, this is what you should do to similitude.

We know that we can compare things if we find similar dimensions. For example, if we want two triangles to be congruent we must make the ratio between their sides equal. A similar thing happens in similitude, but through a much more formal derivation.

You'll find in fact that in the end, we must ensure the similarity of all the related numbers (Reynolds, Strouhal, Froude, Mach, and etc). For example if we knew that the experiment relied ONLY on the equivalent of the prototype's Reynolds number then we would have

$$\frac{\rho_1 V_1 L_1}{\mu_1} = \frac{\rho_2 V_2 L_2}{\mu_2}$$

and you can easily solve from there. Now, the question of which numbers are important is not always obvious and such we have the more formal approach using the modeling theorems and if you recall from class, the Buckinham Pi Theorem. We choose a number of "repeat variables" that we will scale with the other variables such that the fundamental units (length, temperature, mass) cancel out and we have a dimensionless term.

So to conclude, the strategy is:

1. Find the Pi terms.
2. Write the equivalent equation (e.g.)

$$\frac{\omega L}{V} = \phi ( \frac{D}{H}, \frac{\rho V L}{\mu})$$

3. Equivalate the model and prototype dimensions.

If you don't understand the Buckinham Pi part, this gets a bit confusing. I don't know if I can get a proper link on the web that teaches it.

 

[Kenny Lee]

I really appreciate the info. Yeah, we've been doing Pi theorem; but as usual, sort of understand what's written in the book, but when it comes to doing the problems... yeah, i just get lost.

Thanks again.