A tool to determine airfoil shape using rim cross section

Just a small tool to see how the airfoil shape of a wheel is modified when the distance from the ground to an horizontal cut plane is changed. As the wind is constrained to flow parallel to the ground, this airfoil shape is what the wind sees depending on the distance to the ground. Bontrager commented some time ago that they tried to improve the performance of their airfoil shape for h=0 (h is the distance from the hub axle to the horizontal cut plane) because the aspect ratio is the lowest and this was the worst case scenario. I haven't got neither CFD nor wind tunnel data to back this up but, as you will see in the following graphics, chord length and the shape of the airfoil changes notably when h is modified. To illustrate how this tool works, I will use the following rim cross section:

NACA 0030 (3.333:1) symmetric airfoil with an overlaid 20mm tire. This could be assimilated to a 70-75mm rim with tall blades and curved brake walls. 27mm at widest point

The following plot shows the tire leading side for some values of h:

Wheel radius, 0.34m. Yellow, h=0.1m. Green, h=0.21m. Blue, h=0.26m. Black, h=0.32m. As you can see, when h increases, the widest point approaches to the leading edge of the airfoil. When h>wheel radius-chord length, tire leading and rim leading sides form an unique airfoil

Finally, in the following graphic, the relation between aspect ratio and h for the interval with separated tire leading and rim leading sides airfoils is shown :

It goes from 3.333:1 to 8.535:1. Average is 4.069:1

That's all for today. Greetings

A yaw-independent quantification of aerodynamic performance

The relation between yaw and aerodynamic drag makes it difficult to discern which component is really faster. Wheel manufacturers usually use sentences like "X wheel is Y seconds faster than Z wheel  over 40km" but how do they calculate that time gain if wind and bike speed is variable as well as drag as a function of yaw? The answer is using a weighting function that multiplies the drag function obtained in the wind tunnel by a factor to obtain an unique drag value.

Until now, the weighting functions that every manufacturer use haven't been known but Mavic showed his during the presentation of the Mavic CXR80. In the previous post, I worked that CDF function to obtain a normal distribution that can be used as a weighting function. Next step is to consider the drag function obtained in the wind tunnel. Typically, aerodynamic drag is considered to vary linearly between measurement points. Another aspect that has to be considered is that aparent wind speed has been variable for that weighting law but, as the fluctuations of the Reynolds numbers are small, CdA can be considered independent for a given yaw.

Let's consider a drag function with n divisions, the first abscise is called x0 and the last one, xn. The CdA for every abscise is called di, with i going from 0 to n. The weighting fuction is assumed to be a null-mean normal distribution as Mavic has showed. Now, we can define the weighed yaw-independent CdA as:

erf, error function. sigma, standard deviation of the weighting function. The leading term (before the summation) takes into account that the drag function isn't defined in the whole range of possible yaw values

Let's make an example. Consider the following drag function taken from Slowtwitch's Cervélo P4 in the Tunnel article:

As you can see, the Cervélo P4 and the Trek Speed Concept are nearly tied. Considering that wind tunnel speed was set to 30mph, approximated CdA values are:

Using Mavic's weighting function with a standard deviation of 10.3388º, the terms of the summation over each interval are the following ones:

The leading term is equal to 1.056, the closer this term is to the unit, the surer we are of the conclusion. Finally, the yaw-independent CdA values of both bikes are:

The conclusion is: "A riderless Cervélo P4 "ridden" in typical conditions will produce 100*(0.003868-0.039685)/0.03868=2.598% less drag than a Trek Speed Concept with a confidence of the 100/1.056=94.697%". As the correlation between wind tunnel data and real world data has been already proved, this method, when coupled with pedaling-rider wind tunnel data, can answer the question "Which setup is fastest?" without any doubt. 

That's all for today. Thanks for reading!

About yaw

The subject of yaw is of vital importance when trying to design aerodynamic components. Typical values of yaw seen during cycling are intimately relationed with the speed of the cyclist and, therefore, the target client of these components

Dan Connelly developed some time ago a probability function for yaw motivated by some conflicts between ZIPP and HED about typical yaw values. This probability function used a Maxwellian distribution to describe wind speed and assumed that all the headings and wind directions were equally probable. The relation between wind speed at the height of a meteorological station and at bike level was obtained using the Hellman formula. I haven't checked all the process to obtain this function but Dan is a clever guy so I don't expect any mistakes to be found. My only concern was the calculation of v0, the standard deviation of the velocity components of the Maxwellian distribution. Dan assumed, maybe using a conclusion that is valid for normal distributions, that the peak of the probability function was equal to the average value of the probability function for speeds. Hellman formula works for average values of wind speeds so a correct relation between v0 and the average wind speed should be used. For this Maxwellian distribution for speeds, the average is given by the following limit:

 Working it, we get:

Now we can calculate v0 as a function of the average wind speed at bike level that is obtained using the Hellman formula. The following two graphics show the influence of this modification in the wind speed and yaw probability functions that Dan showed.

As you can see, considering that the average wind speed is equal to v0, there is an underestimation of the presence of low wind speeds and yaw values.

Mavic showed a probability function of yaw determined through experimental studies during the development of the new CXR80 wheel. There isn't many details about how that function was obtained (wind speeds, bike speeds, etc) but I have tried to replicate it using the yaw probabilty function that Dan developed. First, I fitted a normal distribution to Mavic's plot. The plot misleads a little bit because it shows the CDF between every tick - 1.25º and + 1.25º but I have calculated the standard deviation using the CDF of an arbitrary division of the x scale. The normal distribution and the CDF of yaw obtained from Mavic's data are the following ones:

Explanation: If you draw a vertical line taking as origin any abscise (let's call it Y), the ordinate of the intersection between that line and the function multiplied by 200 represents the percentage of time that you ride in the interval [-Y,Y]

That's all for today. Greetings