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Aerodynamic Coefficients: Lift

November 16, 2011
NACA 23012 XFOIL

I am transferring RacingTech to a New Race Car Technology Page.

This article can be seen in this Aerodynamics Article.

The technology of race cars is a vast topic involving many disciplines. As tempting as it may be to dive right in to multi element wings and Navier-Stokes it is important to have a solid grasp on the fundamental topics. One of these topics is Aerodynamic Coefficients.

Before talking about the properties of fluids, I would like to talk about the aerodynamic coefficients and later show the dependence on fluid properties. If you are reading this I will assume you know what lift is. You should be able to look at an airplane wing for example, and know that a wing with flaps engaged has more lift than a wing with the flaps up, and that a wing at a high angle of attack will create more lift than a wing with a low angle of attack. Let’s take that intuition and show why it works.

We know that the faster the airflow over a body the greater the force exerted on the body. We also know that the more dense the fluid, the greater the force exerted on the body. Try paddling an oar in the air and see how far you will travel. These two properties describe the dynamic pressure (*2):

q_\infty=\frac{1}{2} \rho V_\infty^{2}

where rho is the density of the fluid.

Intuition tells us that based on the equation, the higher the dynamic pressure, the greater the lift and drag. It is also apparent that velocity has a greater effect on the dynamic pressure than the density, due to the nonlinear relationship.

Another fact is the larger the oar, the greater the force we can put on the fluid. Last, think about what angle you want the oar to face relative to the direction it is traveling. To “slice” the water would be nearly useless, so we naturally angle the oar so that it is effective. All of these common sense facts combine to create the general lift equation (*2):

L=q_\infty C_L S_{ref}

L=\frac{1}{2} \rho C_L S_{ref} V_\infty^{2}

where CL is the coefficient of lift and S is the reference wing area.

This should all make sense. A higher coefficient of lift should produce more lift, but what controls the coefficient of lift? Well, the equations that go into finding the coefficient of lift are beyond the topic of this writing, but I will cover some ways to find the coefficient of lift.

The first way would be to buy the book “Theory of Wing Sections” by Abbott (Link below). My understanding is that even for those with access to expensive software and experience this book is a must have. Here is the famous airfoil, the NACA 23012 chart (*1), from the famous book:

NACA 23012 from “Theory of Wing Sections” by Abbott

Looking at the chart to the left, you can see the independent variable is the angle of attack, and the dependant variable is the coefficient of lift. These values were obtained experimentally. Take for example an angle of attack of 10 degrees, we see the coefficient of lift is near 1.25. Continuing to increase the angle of attack raises the coefficient of lift up to a point around 19 degrees where the coefficient of lift abruptly decreases. This as you may already know is called stall.

Another way, and the last I will cover, is using recommended software developed by MIT called XFOIL. Running XFOIL at the same Reynolds number, 8.8e6, as that in Abbott, I got the following results which are very similar (*4):

XFOIL NACA23012 Data

As you can see the values from this very quick run of XFOIL are very close to the data from Abbott. I could have put in better resolution in my alpha values, but this is just for demonstration.

At an angle of attack of 5 degrees the airflow is smooth. Increasing the angle of attack toward stall shows a still attached flow, but a coefficient of lift of 1.72. Increasing further to near the verge of stall shows a thicker boundary layer and a higher lift. Push beyond the stall angle, and flow separates just after the nose of the wing, and the lift begins to drop rapidly.

Skip to 1:20

EXAMPLE

Sidepodcast

Let’s put this to use. The highest lateral G loading is at Suzuka’s 130r, which is a 130m radius corner where F1 cars have reportedly pulled 5-6G’s. Here are some assumptions which will help simplify the solution:

1)      640Kg vehicle mass

2)      Coefficient of Friction is 2.5 (FSAE data, F1 is likely higher)

3)      Front wing contributes 35% of downforce (good guess)

4)      Standard Air Density 1.225Kg/m^3 (Standard value)

5)      Corner Radius 130m (Given. Assume circular race line)

6)      6 G’s pulled

7)      Planar Wing 1.5m X .25m (No curve, flip ups, endplates etc)

8)      Ignore Aspect Ratio (This is NOT negligible in real life)

9)      Ignore Ground Effect (This is NOT negligible in real life)

10)   Treat as a point, no weight transfer etc. Just a simplified system

Let’s see if the standard NACA 23012 will work, and if so, what angle of attack should it have?

F=m a (*3)

a=\frac{v^{2}}{r}=6*9.81=\frac{v^{2}}{130}

V=87.5 \frac{m}{s}

F=m a=640*6*9.81=37,670N

F=\mu N

37,670=2.5*N

{Downforce Required}=15068N * .35 = 5274N

L=\frac{1}{2} \rho C_L S_{ref} V_\infty^{2}

5274=\frac{1}{2}*1.225*C_L*1.5*.25*87.5^{2}

{Lift Coefficient Required}=3.0 > C_LMAX of NACA23012

Since the required lift coefficient is greater than that of the standard NACA 23012, a high lift device will be required. One example could be a NACA 65-118 profile at zero angle of attack, but with a modification (*1). This wing profile would have a dual slotted flap, which will be similar to a 3 element formula one wing. According to the chart I have in front of me,  with the wing at zero angle of attack the flap angle will need to be 65 degrees. On the other hand, with the wing at an angle of attack of 4 degrees, the flap angle only needs to be around 40 degrees. The chart I am using is on page 221 of Abbott’s “Theory of Wing Sections”.

Obviously this little thought experiment is highly simplified. For example, I didn’t factor in the vehicles weight into the Normal Force, which would have decreased the front wing’s required downforce to be about 3076N instead of 5274N. As a mere mortal I have no clue what the friction coefficient is of a F1 tire, but I can use what resources I have to make educated guesses like this, and then try to validate them. For example, the maximum speed an F1 car reaches on Suzuka (*5) is nearly the same speed I calculated the 130R corner is taken at, which makes sense since 130R is at the end of the long straight after the exit of Spoon.

SOURCES

In the interested of giving credit where credit is due, I will list others work that contributed to mine. These are all books I personally own and highly recommend.


(*1)

(*2)

(*3)

(*4) XFOIL

(*5) http://www.f1fanatic.co.uk/f1-information/going-to-a-race/suzuka-circuit-japan/

Sidepodcast

Recommended Reading

http://www.ae.illinois.edu/m-selig/ads.html

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