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Mathematical Model of Yaw Angle Distributions for Bicycle Wheels
Zach McCormick, Nick McCormick, Justin Clark
Abstract
The purpose of this paper is to derive a probability distribution of yaw angles to be used for the design of
aerodynamic bicycle wheels. We derived a probability distribution for wind speed based on the commonly used
Rayleigh distribution, and then used that distribution to write an algorithm to solve for the probability of a given
range of yaw angles, given inputs such as rider speed and average wind speed. We then compared our results to
experimental results from the Ironman World Championships in Kona. Our model gave us results within 5% of these
experimental results. Our main finding was that yaw angles are generally much lower than the published estimates
of major bicycle wheel manufacturers.
Motivation
skills to work and create a mathematical model based
on climatic data.
This research project started while we were doing the
computational fluid dynamics optimization of the aero
shells for our upcoming Arsenal Wheel System. It is well
known that different rim profiles perform well at
different yaw angles. For example, wider rims tend to
perform better at higher yaw angles, while narrower
rims tend to perform better at lower yaw angles. In
order to optimize the aerodynamic design we needed to
know what yaw angles were most common in real
world racing conditions. What we quickly realized was
that there is no consensus on the matter within the bike
industry. Most wheel companies were using higher yaw
angles, but the (very limited) real world data showed
that lower yaw angles were more common. This
prompted us to try to figure it out for ourselves.
There were two ways we could do this: experimentally
or mathematically. The first option would have required
us to build a device to measure yaw angle, and then go
out and ride hundreds of miles in various locations to
collect data. There are a few major problems with this,
though. First, the amount of data we could realistically
collect in a reasonable time frame would be very
limited. Specifically, we would be limited to collecting
data in locations that we could drive to in a couple
hours, which would skew our data towards the wind
conditions in the region around us. The second major
problem is that we would have to build a device to
measure the yaw angle, and then we would need to
take it to a wind tunnel to properly calibrate it. This
whole process would take at least a few months, and it
would be quite expensive. With neither time nor money
to spare, we decided instead to put our mathematical
Wind Angle and Yaw Angle
In this paper we will use two seemingly similar terms,
wind angle and yaw angle, so we will go ahead and
define what we mean by these terms. Wind angle is the
simpler of the two. It is just the angle between the
direction of the wind and the direction of the rider.
We’ll define it such that a direct headwind corresponds
to a wind angle of 0°, and the angle increases as the
direction the wind is coming from moves clockwise
around the circle. So if the rider is heading north and
the wind is coming from the east then the wind angle is
90°. Note that the wind angle does not take into
account the speed of the wind or the speed of the rider.
It is just the angle between their directions. Also notice
that the exact direction of the rider and the exact
direction of the wind do not matter; only the relative
direction of the two matter. So riding south into a wind
coming from the south is exactly the same as riding
north into a wind coming from the north.
The yaw angle will require a little more math. We first
need to find the component of the wind that is
perpendicular to the rider. This is found by taking the
sine of the wind angle and multiplying it by the wind
speed w. This is the wsinθ in the formula below. Next,
we need to find the effective speed of the rider. If a
rider is traveling at 25 mph into a 5 mph headwind, the
effective speed is 30 mph. To find the effective speed of
the rider we need to find the component of the wind in
the direction the rider is travelling. This is found by
taking the cosine of the wind angle and multiplying it by
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2
weather reports and it is also what is used for the long
term average wind speeds that we’ll be using.
Fortunately, this variation in wind speed has been well
studied and can be accurately predicted with a simple
formula called the wind profile power law. This formula
relates the wind speed at a given height above the
ground to the wind speed at another height. Thus, this
formula allows you to convert readily available surface
wind speeds (measured at 10 meters above the ground)
to speeds more applicable to bicycle aerodynamics. The
wind profile power law is as follows:
𝑣(ℎ) = 𝑣(𝑠) ∙ (ℎ⁄𝑠)𝛼
the wind speed. This is the 𝑤𝑐𝑜𝑠𝜃 in the formula.
Adding this to the rider speed, 𝑟, gives you the effective
rider speed, 𝑟 + 𝑤𝑐𝑜𝑠𝜃. Once we have the effective
rider speed and the perpendicular wind speed, all we
have to do is find the angle between the two and we
have the yaw angle. This is found by taking the inverse
tangent of the perpendicular wind speed divided by the
effective rider speed. The final equation is as follows:
𝜑 = tan−1(
𝑤𝑠𝑖𝑛𝜃
)
𝑟 + 𝑤𝑐𝑜𝑠𝜃
The Variation of Wind Speed with Height above the
Ground
Wind speed data is readily available for thousands of
locations across the globe. It is important to note,
though, that wind speed varies with height above the
ground. So the height at which it is measured needs to
be accounted for. The reason this occurs is that friction
between the air and the ground causes the wind speed
at ground level to be exactly zero. The higher you get
off the ground, the higher the wind speed gets. If you
have ever wondered why wind turbines are so tall, the
answer lies in this phenomenon; wind at 80 meters
above the ground is much stronger than wind at 10
meters above the ground. This is important for bicycle
aerodynamics because the international standard
height for measuring so-called surface wind speed is 10
meters above the ground, whereas you and your bike
sit between 0 and 1.5 meters above the ground. This
surface wind speed is usually what is reported in
where α is the Hellman exponent, which is determined
by the terrain; s is the height the wind speed is given for
(10 meters in our case), and h is the height you’re
interested in. Since we’re studying wheel aerodynamics,
we’ll use the height at the center of a wheel,
approximately 0.33 meters, for our h. The Hellman
exponent varies based on the type of environment
you’re riding in. In a desert or open grassland area, it is
around 0.14; in a wooded or suburban area it is closer
to 0.33, and in a city it can be 0.50 or more (Belfort
Instrument, 2012). Most race courses have a variety of
terrain, so the Hellman exponent must be estimated
based on the relative proportion of different types of a
terrain. For example, if a course features an equal mix
of forested and wide open grassland terrain, you’ll want
to use a Hellman exponent right in between 0.14 and
0.33, which would be around 0.23. Clearly the choice of
Hellman exponent is a bit subjective, so a degree of
error in the final result can certainly be attributed to it.
Let’s do some sample calculations using different
Hellman exponents. Using the above equation, we need
to multiply the surface wind speed by a factor of
(0.33⁄10)0.14 = 0.62 for an open landscape,
(0.33⁄10)0.33 = 0.30 for a wooded landscape, and
(0.33⁄10)0.50 = 0.18 for an urban environment. So the
wind speed at the level of the wheel is quite a bit lower
than the surface wind speed shown in weather data. As
we will see later, this has some very important
implications for bicycle wheel aerodynamics.
Typical Wind Speeds for Different Locations
Next, we need to find wind speed data for various
locations. This, along with rider speed, will allow us to
©2015 Catalyst Cycling LLC. All Rights Reserved.
3
create a probability distribution of yaw angles. To do
this we will use climatic data from WeatherSpark (the
data is sourced from NOAA, but WeatherSpark makes
the data easier to find). We’ll pick three popular
Ironman courses and convert the average wind speed
for those locations at 10 meters to an average wind
speed at 0.33 meters.
First, we’ll look at our hometown race of Ironman
Chattanooga. The daily mean surface wind speed
(measured at 10 meters above the ground) in
Chattanooga in September is 4.2 mph. In order to get
the actual wind speed at the height of the wheel, we
need to first choose an appropriate Hellman exponent,
and then use the wind profile power law. Ironman
Chattanooga is held in a mostly forested area, with a
little bit of urban riding mixed in, so we’ll use a Hellman
exponent of 0.35. Plugging this into our wind profile
power law yields (0.33⁄10)0.35 = 0.30. Multiplying this
factor by our average surface wind speed of 4.2 mph
leaves us with an average wind speed, as seen by the
wheel, of 0.30 × 4.2 𝑚𝑝ℎ = 1.26 𝑚𝑝ℎ.
above. A sample Rayleigh distribution for an average
wind speed of 4.5 m/s is provided below (Caleb
Engineering).
As you can see, the wind speed with the highest
probability, about 3.7 m/s, is a bit lower than the mean
wind speed. Wind speeds very close to zero are very
unlikely, as are wind gusts of over 14 m/s.
Probability Distribution for the Wind Angle
Doing the same thing for Ironman Arizona, we choose a
Hellman exponent of 0.25 to account for the mix of
urban and open desert riding. WeatherSpark shows an
average November wind speed for Phoenix of 5.1 mph.
Plugging this in yields an effective wind speed of 2.17
mph.
Finally, we’ll look at a course known to have very windy
conditions: Kona. The average wind speed in October at
the Kailua Kona airport is 8.1 mph. Since the course in
Kona is very wide open, we’ll use a very low Hellman
exponent of 0.14. This yields an effective wind speed of
5.02 mph, much higher than the other two courses.
Probability Distribution of Wind Speeds
Wind speed fluctuates constantly, so an average wind
speed only tells part of the story. What we really need is
a probability distribution of wind speeds. For this, we
will use the Rayleigh distribution, commonly used in
wind power models (Caretto, 2010). The probability
density function (p.d.f.) for this distribution is given by
𝑓(𝑣) =
𝜋𝑣 −𝜋𝑣 2 ⁄4𝜇2
𝑒
2µ2
where v is the speed of the wind at any given moment
in time and µ is the average wind speed, as calculated
Now that we have our probability distributions for wind
speed, we need a probability distribution for the wind
angle. Fortunately, this is easy, as every angle is equally
likely. On a specific course on a specific day this might
be a bit of a simplification, but over the long run it is a
very accurate approximation. The law of large numbers
ensures that any days that may be spent riding with a
near-constant crosswind will be counteracted over time
by other days with a near-constant headwind or
tailwind, so that over the life of the wheel it will see a
distribution of wind angles that is extremely close to a
uniform distribution. Assuming this uniform distribution
of wind angles, the p.d.f. for the wind angle is given by
the following equation.
𝑝(𝜃) =
1
360
Probability Distribution of Yaw Angles
We’ve already found an equation for the yaw angle
based on the wind speed 𝑣 and the wind angle θ. As
explained previously, the yaw angle is given by
𝜑 = tan−1(
©2015 Catalyst Cycling LLC. All Rights Reserved.
𝑤𝑠𝑖𝑛𝜃
)
𝑟 + 𝑤𝑐𝑜𝑠𝜃
4
where 𝑤 is the wind speed, 𝑟 is the rider speed, and θ
is the wind angle.
Using the change-of-variable technique on our
probability density function for 𝑤, along with our
equation for ϕ we can find a p.d.f. that is independent
of w:
𝑦(𝜑) =
that the yaw angle will be below 2° and a 98.9% chance
that the yaw angle will be below 6°. Yaw angles above
8° are extremely unlikely. For this race, you would want
a wheelset optimized for 0-2° of yaw.
𝜋𝑟𝑠𝑖𝑛(𝜑) csc(𝜃 − 𝜑)
2𝜇2
−𝜋(𝑟𝑠𝑖𝑛(𝜑) csc(𝜃−𝜑))2
4𝜇2
𝑒
2
×
× 𝑟𝑠𝑖𝑛(𝜃)𝑐𝑠𝑐 (𝜑 − 𝜃)
This function still depends on θ, though, which is not a
constant. So we need to find a way to incorporate our
p.d.f. for θ into this p.d.f. for ϕ. In theory, this can be
solved analytically by multiplying y(ϕ) by p(θ), and then
integrating with respect to θ. Unfortunately, the
resulting integral cannot be solved by hand easily, and
neither MATLAB nor Wolfram Alpha can compute a
symbolic solution for it. Thus, we will have to solve for it
numerically using MATLAB.
With Ironman Arizona, the yaw angles are a bit higher
than Chattanooga, but they are still quite low, with
98.6% of the time spent below 10°. For this race, the
ideal wheelset would be optimized for 2-4° of yaw.
Results for IMCHOO, IMAZ, and Kona
Now that we have our yaw angle p.d.f., we can use it to
predict yaw angles at various races. For these
calculations we used a rider speed of 25 mph, along
with the previously calculated average wind speeds for
each location. When looking at the charts, keep in mind
that the probabilities are rounded to the nearest tenth
of a percent. So when you see a 0.0% probability, it isn’t
really zero, but it is less than 0.05%, so it gets rounded
to 0.0%. That is also the reason why the probabilities
don’t quite add up to 100%.
As would be expected, yaw angles are much higher in
Kona than in Chattanooga or Arizona. But even in a
place as windy as Kona, you’re still spending a lot more
time below 10° (72.1%) than above it (27.9%). For Kona,
you would be best off with a wheelset optimized for 410° of yaw.
Agreement with Real-World Data
Looking at the results for Ironman Chattanooga, we can
see it is dominated by very low yaw angles. Our results
indicate that at any given point there is a 61.0% chance
A few days before the 2013 Ironman World
Championships, Mavic set up Lars Finager with a wind
vane on the front of his bike and had him ride the entire
112 mile course to experimentally determine the
distribution of yaw angles at the famed Kona course
(Mavic, 2013). According to Mavic, “wind conditions this
day were similar to those usually met at that time on
this race,” so their results should compare quite closely
with our results. Here is the distribution they measured.
©2015 Catalyst Cycling LLC. All Rights Reserved.
5
wind is blowing at a near constant crosswind the entire
time. If the wind angle varies randomly, like it would in
reality, the highest probability you can achieve for 1020° is around 30%, and that would require surface wind
speeds of roughly 15-40 mph (depending on terrain,
and hence the Hellman exponent). In fact, 0-10° will
always be more prevalent than 10-20°, which in turn
will always be more prevalent than 20-30° and so on.
This last observation is a simple consequence of the way
the probability distribution works, and is independent
of any variables such as wind speed, rider speed, or
Hellman exponent.
The three bars in the middle, representing 0-10° of yaw,
account for 70% of the total. Our model, adjusted to
match the height above the ground of the wind vane
used by Mavic, gives 66.1% for the 0-10° range. Looking
at 0-4°, our model gives 29.9%, compared to Mavic’s
32%. This is a very tight agreement, especially given
how many assumptions we had to make in our model.
One real world example isn’t enough to truly validate
our model, but it does indicate that we are at least on
the right track.
Comparison with Industry Assumptions
Let’s take a look at what some other wheel companies
have to say about real world yaw angles. According to
Zipp, “roughly 50% of real world riding occurs with
effective wind angles between 10 and 20 degrees (Zipp
Speed Weaponry).” Unsurprisingly, they also claim that
“ABLC allows the minimum drag to occur between 10
and 20 degrees.” (ABLC stands for Aerodynamic
Boundary Layer Control, which is what they call their
patented dimple technology.)
FLO Cycling quotes an even higher proportion of time
spent at high yaw angles (FLO Cycling). According to
them, “we spend roughly 80% of our time racing
between 10 and 20 degrees of yaw.” They use a formula
to calculate the weighted average drag savings of their
wheels. Naturally, their formula weights yaw angles
from 10°-20° much more heavily than lower yaw angles.
Zipp claims 50% and FLO claims 80%, but how do those
numbers compare with our model? Well for
Chattanooga, we get 0.1%; for Arizona, we get 1.4%,
and even for our extreme example of Kona we still only
get 24.6%. So what kind of conditions would it take to
get a probability of 80%, or even 50%, for yaw angles
between 10° and 20°? Well, it turns out neither of them
are even theoretically possible, unless you assume the
What this Means for Bicycle Wheel Aerodynamics
The main conclusion that can be drawn then is that
wheels should be designed for much lower yaw angles
than those that are currently being used. This means
using narrower tires and narrower rims, as well as less
bulbous rim shapes, among other changes.
Works Cited
Belfort Instrument. (2012, 8 19). Height of Wind
Measurements Above Ground. Retrieved
from Belfort Instrument:
http://belfortinstrument.com/heightwind-measurements-ground/
Caleb Engineering. (n.d.). How Much Wind.
Retrieved from Caleb Engineering:
http://www.calebengineering.com/howmuch-wind.html
Caretto, D. L. (2010). Use of Probability
Distribution Functions for Wind. Retrieved
from California State University
Northridge:
http://www.csun.edu/~lcaretto/me483/
probability.doc
FLO Cycling. (n.d.). Aerodynamics- Net Drag
Reduction Value. Retrieved from FLO
Cycling:
http://www.flocycling.com/aero.php
Mavic. (2013, October 16). Yaw Angle
Measurements in Real Conditions on Kona
Ironman Course. Retrieved from
Engineers Talk:
http://www.engineerstalk.mavic.com/ya
©2015 Catalyst Cycling LLC. All Rights Reserved.
6
w-angle-measurement-in-real-conditionson-kona-ironman-course/
WeatherSpark. (n.d.). About. Retrieved from
WeatherSpark:
https://weatherspark.com/about
WeatherSpark. (n.d.). Average Weather for
Chattanooga, TN, USA. Retrieved from
WeatherSpark:
https://weatherspark.com/averages/298
97/Chattanooga-Tennessee-United-States
WeatherSpark. (n.d.). Average Weather for
Kailua-Kona, HI, USA. Retrieved from
WeatherSpark:
https://weatherspark.com/averages/331
18/Kailua-Kona-Hawaii-United-States
WeatherSpark. (n.d.). Average Weather for
Phoenix, AZ, USA. Retrieved from
WeatherSpark:
https://weatherspark.com/averages/312
59/Phoenix-Arizona-United-States
Zipp Speed Weaponry. (n.d.). ABLC. Retrieved
from Zipp:
http://zipp.com/technologies/aerodyna
mics/ablc.php
©2015 Catalyst Cycling LLC. All Rights Reserved.