by Bruce Finwall

Have you ever wondered how the wind affects the performance of a Derby racer? You may have seen some interesting results on a gusty race day, when a big ''sled'' somehow beat the ''sure winner''. Let's examine mathematically how this works.

Once a Derby racer gets up to speed it's estimated that aerodynamic drag accounts for over 80% of the total drag an the car. So, as cars approach the finish line, it is common to see the more aerodynamic cars seem to pull away. Since air drag is a highly nonlinear function of apparent wind velocity, and since a Derby has a larger lateral area than frontal area, it appears that there are aerodynamic effects that work to increase the drag forces of crosswinds and even "quartering'' tailwinds. This article describes calculations for the, effects.

The results are interesting:

- The most beneficial wind is not a direct tailwind but a ''quartering'' tailwind of about 30 degrees from the rear.
- The most detrimental wind is not a direct headwind, but rather a ''quartering'' headwind blowing at about 45 degrees to the racer's path.
- Crosswinds are not harmless. A pure crosswind (at 90 degrees to the direction of travel) actually slows a racer down almost as much as a direct headwind.
- ''Marginal'' tailwinds (i.e. winds Making an angle of about 100 degrees to 140 degrees to the direction of the racer) can be either beneficial or detrimental depending on there speed, the speed of the racer, and the design of the racer.

All of these effects are due to the fact that the lateral area of a Derby racer is about 4 to 7 times the size of the direct frontal area of the racer. The racer is simply a bigger ''Sail'' I then viewed from the side. It should be noted that energy consumed by wind drag is a nonlinear function of both apparent wind speed and actual racer speed.

So, a new area that you may want to consider when designing
your next Derby racer is the wind pattern of your local track (or
maybe Akron or Ft. Wayne). It is probable that the best design in
a wind tunnel is not the best when it comes to a Derby track
where there are wind conditions. So get out to your favorite
track with a wind sock to start working on your new design.
(Note: see related stories on "quartering" tailwind
racer and "quartering" head-wind racer.) *see below
- Online ED*

__MATHEMATICS__

- A = angle of actual wind vector (v) relative to the direction of travel of Derby racer (u)
- B = angle of apparent wind vector (w) relative to direction of travel of Derby racer
- r = density of air
- Cd = drag coefficient of Derby racer
- u = velocity vector of Derby racer (relative to ground)
- v = velocity vector of wind (relative to ground)
- w = apparent wind vector (windspeed relative to Derby racer) (w = u + v, in vector analysis)
- Al = lateral area of Derby racer
- Af = frontal area of Derby racer
- At = total area of Derby racer
- Fw = total aerodynamic force on Derby racer
- Dw = aerodynamic drag force under windy conditions
- Do = aerodynamic drag force under windless conditions

The basic approach is to do a vector addition of the wind speed and the Derby speed to obtain an apparent wind speed and direction (see figure 1, and then calculate the drag force produced by this apparent wind acting on the projected area of the car exposed to the apparent wind. The results are shown in figure 2 as a ratio of drag forces on the Derby in windy conditions to the drag under windless conditions. These ratios are given as a function of wind angle (A) and the ratio of the wind speed to the racer speed (v/u). Drag ratios less than 1.0 mean that the wind is helping to speed up the racer; drag ratios greater than 1.0 indicate an increase in drag.

The apparent velocity of the wind relative to the Derby (w) is the vector sum of the actual wind velocity (v) and the velocity of the ground relative to the racer under windless conditions (u):

w = u + v (in vector addition)

This vector addition is shown in figure 1. By the Pythagorean theorem:

which can be rearranged to:

The total aerodynamic drag force exerted on the racer is given by the following expression:

This total drag force has components in both parallel and perpendicular to the direction of travel. But only the parallel component has any influence on the racer's speed, because the perpendicular component is absorbed in tire-pavement skid resistance. (supposedly, in sufficiently windy and slick conditions the perpendicular component of aerodynamic drag could push the racer off of the road. That possibility is ignored in this analysis.) The component of drag parallel to the direction of Derby, travel is:

Substituting Equation 2 into Equation 3 gives:

By contrast, the drag force under windless conditions is given b3, the sir.simpler expression:

Dividing Equation 4 by Equation 5 gives the ratio of the drag under windy conditions to that under windless conditions:

This constitutes the primary expression for the drag-related effects of crosswinds. The task is to evaluate the right-hand side of Equation 6 in terms of quantities we already know, such as racer speed (u), wind speed (v), the angle between them (A), and the area terms and the drag coefficients.

But immediately there is a problem we have no data on the effective area and the drag coefficient of the racer in crosswinds. To my knowledge no one has ever actually measured the drag coefficient of a Derby racer except in conditions where the apparent wind is head on. This could be a topic for future research.

To proceed with the analysis, I will assume that the race car behave aerodynamically like a box having frontal area Af and lateral area Al. Inclusion of lateral area is of surprising importance. As the apparent wind vector shifts from directly head-on, the large lateral area becomes exposed to the apparent wind, like a sail, thereby increasing the drag force. The projected area exposed to the apparent wind is given by the expression:

[7] At = Af * [cosB] + Al*|sinB]

It turns out that only the ratio of lateral area to frontal area is im- portant in evaluating Equation 6.

The drag coefficient also drops out of Equation 6; that is, the important point about the drag coefficient is not its actual value, but rather its variation with apparent wind direction. For this analysis, and for simplification, drag coefficient is assumed to be ''invariant'' with apparent wind direction. To calculate the angle B, the following trig identities were used:

[8] cosB = (u + v.cosA) /w

[9] sinB = (v * sinA) /w

Substituting Equations 7, 8, and 9 into Equation 6, and simplifying, gives the following expression for drag force as a function of wind speed/ racer speed ratio (v/u), wind angle (A), and lateral to frontal area ratio:

A plot for this expression is shown in figure 2, with wind angle along the horizontal axis, and the wind speed/derby speed ratio treated as a parameter. 5

THEORY

THE ''QUARTERING'' TAILWIND RACER by Bruce Finwall

In designing a car to run primarily in ''quartering tailwind conditions, there are several ideas that a designer may want to consider.

- The ideal design would probably have a ''cusped'' tail (looking from a top view) which would work a more effective ''sail'' :

- The side view would probably have a LONG, FLAT rear:

- The frontal area might be TALL and THIN:

- The tail sections might be concaved:

Note: unfortunately many of these ideas don't allow much room for the driver.

THEORY

THE ''QUARTERING'' HEADWIND RACER by Bruce Finwall

The ''pencil nosed'' designs that ran well in 1986 are good examples of a car designed to perform in ''quartering'' headwind conditions.

- Reducing the lateral area is the key. Using a SMALL, LOW, LONG nose is one way to achieve a smaller lateral area; this could be a ''pencil nose'' or "wedge'' type lateral section.

- A side profile based off of a standard aerofoil section would have a lateral area of about 1030 square inches. It would be possible to get the lateral area down to the 700 square inch range using one of the above mentioned techniques.

- Also, a ROUND nose design could help to reduce the Cd (see Equation 4). (Also, see BIG AND ROUND VS. SMALL AND FLAT, DT JAN/FEB 87)

Note: I hope that you readers enjoyed this ''Food for Thought''. Ed