1 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 ©2015 Catalyst Cycling LLC. All Rights Reserved. 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: (ℎ) = () ∙ (ℎ⁄) the wind speed. This is the in the formula. Adding this to the rider speed, , gives you the effective rider speed, + . 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: = tan−1( ) + 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 ℎ = 1.26 ℎ. 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 () = − 2 ⁄42 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. () = 1 360 Probability Distribution of Yaw Angles We’ve already found an equation for the yaw angle based on the wind speed and the wind angle θ. As explained previously, the yaw angle is given by = tan−1( ©2015 Catalyst Cycling LLC. All Rights Reserved. ) + 4 where is the wind speed, is the rider speed, and θ is the wind angle. Using the change-of-variable technique on our probability density function for , along with our equation for ϕ we can find a p.d.f. that is independent of w: () = 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. () csc( − ) 22 −(() csc(−))2 42 2 × × () ( − ) 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.
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