## how wrong that porsche enthusiast was…

**Less practically,**

Downforce for a car body is calculated by

D = 1/2 ( Area (in meters^2) *angle of attack) * lift coefficient * air density (in kg/m^3) * volume ^2

lift coefficient is calculated the same way as drag coefficient, but instead of calculating for the frontal area, you calculate for the planform area, which is for a car much like a top down view, but would include any outward facing area, (like if you have a curve in your cars body, it is the area of the slice of the sphere that curve represents, not its top down length * width, fortunately planform areas are readily available for most cars)

Coeff of lift is given by C= [2 * Lift force] /[fluid density * true airspeed^2 *planform area]

Lift force is where it actually gets tricky(lol). See, Lift in this instance is derived from the Kutta-Joukowski theorem(it could be done other ways, i like this one because it is used by computer programs(NASA’s specifically) which usually means it is best), which approximates an airfoil as a cylinder moving in a fluid using a conformal map, which would be in this case the Joukowski airfoil. Fortunately, nasa has an animation of this, somehow.

http://www.grc.nasa.gov/WWW/k-12/airplane/cyl.html

But the java is broken for me, so I’ll explain.

The lift force can be gotten, eventually(after a v long formal derivation), from the density of the air,times the speed of the air on the slower side of the airfoil times the difference in speed with the faster side, times the section length, presumably in meters. For a car the chord length would be constant(unless you remove a bumper), whereas an aircraft wing usually has different lengths along it, getting thinner towards the outer part of the wing.

true airspeed is the next, and I think last thing we don’t know. It is not simply the speed, because

where T is the static air temperature in K, T0 is the standard temperature at sea level (288.15K, don’t ask why)

Tt is just temperature in K, M is speed in Mach so if you’re going 150 mph, thats 150/768 Machs or a Mach number of .1953125. You should use meters per second here though, 340.29 meters per second for mach, though its dimensionless, but since you’ve read this far, why the fuck not.

qc is actually Dynamic pressure. the ratio of static pressure to dynamic pressure is known to be, with gamma equaling 1.4

and

and so

Now I think we should have everything we need to compute downforce over a car body, given a speed, the angle of the car(probably in radians for these equations), the density of air is like 1.2ish kg/m^3

CRITICALLY,

Using these equations you have to remember that we aren’t looking for lift but downforce, so if the Coefficient of lift is positive, that merely means that we’ve fed it not the lift force, but the downforce, which are computed in exactly the same way( density * min airspeed * airspeed delta * chord length). Aerodynamic downforce is a fairly subjective topic and so most things you will find will put fluid and continuum mechanics in terms of either actual fluids or lift, because of their use in aeronautics, i was surprised to find a downforce equation at all.

**Practically, **

This is a lot of math, even for me, who is used to doing this. In one of my college exams for essentially applied fluid and continuum mechanics it was open book and take home, but it took literally days to finish, working all the time. Other students would be like yeah take home sounds nice, but no, no. Not at 4 in the morning on your seventh redbull.

To save yourself 28 hours and perhaps 6 redbulls, all you really need to know is how a change will effect the outcome.

For instance, chord length. Lets say someone was to reduce the length of a wing, or perhaps their car body in some way. Lets say removing their bumper, just for instance.

c = chord length, the length of the aerodynamic planform

chord length multiplies into L, the lift force or in our case, the Lift down force, assuming the vehicle is angled and adjusted in terms of the venturi principle to prevent lift(mostly being smooth on the bottom, and low at the front-not necessarily venturi ‘tunnels’),

eg:

(the following picture and equation basically means fast air, low pressure under constriction, in this case by the cross section; low pressure sucks things down in our case, why 80s f1 cars were so low, the h means pressure, delta h is the change in pressure head, lower is less pressure, like a waterfall, higher water has more pressure)

which race cars usually are. Unmodified street cars will default to lift actually(or eventually, in terms of speed, at 800 miles per hour there will be enough air moving under a toyota camry to rip it off the ground), but we are examining relative downforce here.

coefficient of planform lift is

which means a smaller L means a smaller Cl,

Downforce is given by, where F = Cl

so a larger lift coeff, means a larger D

or a shorter c means a smaller L, which means a smaller Cl, which is the same as F, which means a smaller D, for downforce(nominally a paradox, smaller lift coefficient and lift force means smaller downforce). So, reducing cord length will reduce downforce, in this example, where planform area per cord length is relatively slight, compared to an airplane wing.

To see how changes happen over different speeds or other numbers ,you could put this all into excel, or program an applet ( lol so 2002, crazy NASA).

consider other resulting changes, like in this case, no bumper means planform flows eventually moving around bluff objects(wheels, engine) and will create turbulent vortexes and slow down the speed of the air under the car, raising the under car pressure and lowering downforce still(and creating alternating eddys depending on the mach number lol), so our model is not perfect because it is depending on us to account for all changes in some way, which is why computer models are used to calculate things, but just knowing how changes interact can get you a lot further than most people, especially in racing where most people go by what they feel, not science.

to really get the full picture you would need to do about five of these, accounting for each aspect of change across changing variables(of different aero elements-perhaps speed for racing), and then if you have a spoiler or front wing, you need to do the same for that, so computer modelling is the way to go for anything quite serious.

another thing to consider is that fluid and continuum mechanics are very much a work in progress, 2 of the (1,000,000 dollar reward) millenium prize problems are basically fluid mechanics questions, looking to prove things on a very core level(i worked on one of them). for this instance that doesn’t mean much though, we aren’t ever going to get accurate enough to notice without a wind tunnel and an f1 team, but maybe this will help you pick out spoilers or suspension settings (like rear and front ride height).