The aerodynamics of aero socks and fabrics - Part 2

In my previous post about aero socks, I explained how the ribs and grooves on the fabric of aero socks can induce boundary layer 'transition', from laminar to turbulent, which helps to keep the airflow around the ankle and calf 'attached' for longer, which then leads to a narrower wake of separated flow and therefore less drag.

I also explained an important aerodynamic parameter called the Reynolds number of the flow, which depends on the speed of the airflow and also the size of the object, and how that affects the behaviour of the flow.  

I showed the plot above to illustrate the factors that determine whether the fabric of the aero socks will cause boundary layer transition to occur at a lower Reynolds number, compared with what would happen naturally on a smooth surface.  This plot shows how the drag coefficient of a cylinder varies as a function of Reynolds number for different surface roughness heights.  "k/d" in the plot is the surface roughness height divided by cylinder diameter.  Surface roughness has a similar effect to aerodynamic fabric ribs in that it causes a the boundary layer to transition from laminar to turbulent earlier (i.e. at lower Reynolds numbers) than what would occur naturally on a smooth surface.

In this post, I'll dive deeper into these effects and discuss what else we can infer from plots like the one above.

Why size is important

The Reynolds number depends on the size of an object, in addition to the speed of the flow.  For a given object shape, for example a cylinder, a larger object will have a higher Reynolds number than a smaller object having the same shape, even if the airflow speed (i.e. the riding speed) is the same.  This is shown in the plot above; the ankle, calf and thigh lines (the red, green and blue lines respectively) are at different Reynolds numbers, different positions on the x-axis, despite the airflow speed being the same 50kph for all three.

This difference in Reynolds number for different body parts is, I think, one of the reasons why we see clothing manufacturers use different fabric textures on different parts of skinsuits.  Looking at the plot above, it's clear that a cylinder with a smaller diameter, such as an ankle, will have a lower Reynolds number, and therefore needs to have a larger surface roughness (or height of the fabric ribs), in order to trigger the boundary layer transition in the optimum way, to achieve the lowest drag coefficient (Cd) at that Reynolds number.  The red ankle line on the plot above would achieve it's lowest Cd at 50 kph by having a roughness of ~k/d=0.005.  For an 82 mm diameter cylinder, which is my ankle diameter, a k/d value of 0.005 corresponds to a roughness height of 0.41 mm.

On the other hand, for a thigh, the plot above shows that the lowest drag coefficient at 50 kph would instead be achieved with a much smaller roughness of about k/d=0.0003.  For an 175 mm diameter cylinder, which is my thigh diameter, a k/d value of 0.0003 corresponds to a roughness height of just 0.05 mm.  To put that in context, the width of a human hair is 0.05-0.10 mm.

So for the ankle, the optimal roughness is 0.41mm and for the thigh it's 0.05 mm. That's an order of magnitude difference in optimal roughness (fabric texture height) values for the ankle and thigh!  This wouldn't be obvious or intuitive to most people I think, and would probably be surprising even for most aerodynamicists too.  This could be why we don't see ribbed aero fabrics being used on the thigh area of shorts and tights, like the offerings shown above from Rule28, despite ribbed fabrics being used on the lower leg.  The natural texture of a standard lycra clothing fabric probably provides the optimal amount of roughness (0.05 mm) for the thigh area at speeds around 50kph.

Interestingly, I remember an Q&A reply provided by Rule28's founder, Sam Calder, in a TrainerRoad forum post. In that post he explained that aero fabrics don't work on the thigh because of the rotating nature of the thigh when pedalling, which makes it difficult to find a fabric texture that reduces drag.  The rotating element might indeed be an additional factor that complicates things, but I believe the difference in Reynolds number that the thigh experiences, compared with the ankle, is another big reason why aero fabrics don't 'work' on thighs.

Why speed is important too

The Reynolds number also depends on the airflow speed, however, in addition to the size.  This is important, because very few of us (sadly) cycle at 45-50 kph, which is the speed that cycling wind tunnel tests are often performed at.  45-50 kph is a suitable speed for professional and high level amateur time triallists.

As a quick aside, another reason why wind tunnel operators prefer to test at the higher end of the speed spectrum is because the drag measuring equipment in the wind tunnel, called the balance, will be more precise at higher speeds.  Higher speeds produce significantly higher forces.   The drag at 50 kph is approximately 2.8 times more than the drag at 30 kph, for a given drag coefficient.  The wind tunnel balance will normally have a certain force precision, in terms of Newtons of force.  Therefore, if testing at 30 kph instead of 50 kph, then the drag force is 2.8 times less, so the precision of the data collection will be 2.8 times worse.  The guys from Specialized made a similar comment in an interview in this BikeRumour interview.

If the riding speed is slower, the Reynolds number will be proportionately lower. The plot below shows the Reynolds numbers for the ankle, calf and thigh for 40 kph, instead of 50 kph shown previously. 40 kph is closer to the speed that I would average for a 10 mile time trial.

Note that at 40 kph, the optimal roughness for the ankle is now a k/c of 0.007, which corresponds to a roughness height of 0.57 mm.  For 50 kph the optimal roughness was a smaller k/c of 0.005 (0.41mm).  Therefore, slower speeds need a larger surface roughness, or more prominent aero fabrics ribs and grooves, to achieve the lowest drag.  If the optimal aero fabric for 50 kph is used at 40 kph, there is a chance that the fabric texture is insufficient to cause boundary layer transition.

At speeds even than 40 kph, the effect is even more significant.  A speed of 25 kph is too slow for anybody riding a time trial or triathlon, no matter how unfit they are, but it's an appropriate average speed for many off-road racing scenarios.  The plot below shows how the Reynolds numbers for the ankle, calf and thigh move to the left on the x-axis when considering 25 kph.

There are a couple of really interesting observations to make from this 25 kph plot, compared withe previous ones:

1) Aero socks probably won't work at 25 kph: The k/d value that gave the lowest ankle drag at 50 kph (i.e. k/d=0.005) is completely ineffective at reducing the drag at 25 kph, according to the plot.  In fact, even the optimal k/d for 40 kph (0.007), is ineffective too.  At 25 kph, the drag coefficient for those k/d lines is at the upper 1.2 value, which is the drag coefficient for cylinder when it experiences fully laminar flow and laminar separation.  Those 0.005 and 0.007 k/d roughness values are not able to 'trip' the boundary layer, to cause the early transition to turbulent flow.  At 25 kph, a k/c of 0.02 is needed instead, which is 1.64 mm, which is larger than what the UCI now allows.

2) Aero fabric might work elsewhere though: The k/d value that provides the optimum drag for the thigh, however (the blue line below), is now 0.004.  For a thigh diameter of 175 mm, this requires a roughness height (k) of 0.7 mm.  Without that roughness, for k/d=0, the plot below shows that the thigh would experience laminar separation and has a Cd of 1.2.  For an optimum roughness height of 0.7mm (k/d=0.004), the drag coefficient would be half that, 0.6.  

What this shows it that the effectiveness of roughness, and therefore the effectiveness of aero fabrics, depends on the size of the object and the speed.

At 50 kph or 40 kph an aero fabric is not needed on the thigh area, and may be counter-productive, whereas at 25 kph it might actually be needed to improve the flow around the thigh and reduce the drag.  This is something that so far, I haven't seen any clothing manufacturers investigate or utilise.

There are a few assumptions behind what I've explained in this post, particularly the use of the roughened cylinder analogy to explain how textured aero fabrics work on the legs and arms.  Still, I think there is a good chance that aero fabrics used in unconventional places, like the thigh, could produce clothing that performs very well at lower speeds associated with off-road events like gravel, mountain bike and cyclocross races.  There is a big market there, and a potential opportunity for a clothing manufacturer to produce something that performs well for those events.

As a final remark, there may be people that read these last few paragraphs and think to themselves that aerodynamic improvements have little benefit at slow speeds.  This is a common misperception, and a wrote a blog post recently (see here) that shows that aerodynamic improvements have a surprisingly similar benefit at slower off-road speeds compared with their benefits at faster road and time trial speeds.