The aerodynamics of aero socks and fabrics - Part 1

 

Aero socks, like the Rule 28 socks shown in the picture to the left, are a popular clothing choice for time triallists and racers seeking and advantage.

In fact, aero socks are often mentioned as being one of the the best value bang-for-your-buck upgrades, considering the performance advantage they provide for their relatively modest price.

In this blog post, I'll explain why aero socks work.

During my 30-year career as a professional aerodynamicist, I've worked on many aircraft R&D projects that involve the same aerodynamic phenomena that apply to sock and leg aerodynamics.  I'm conscious that the majority of readers won't be familiar with many of the aerodynamics concepts I'll talk about, so I'll start with a basic explanation.  More knowledgeable readers might want to skip the early paragraphs.

Bluff body aerodynamics

The legs of cyclists, and cyclist's bodies in general, are what an aerodynamicist would call bluff bodies.  A bluff body is an object that will typically have a wide or irregular shape, and the nature of that shape means that the air cannot flow smoothly around it.  A streamlined body, on the other hand, is shaped so that the air can flow smoothly around it from the front all the way to the back.  An aeroplane wing or a dolphin are examples of streamlined bodies.  Unlike a bluff body, a streamlined body will have much lower drag.

The flow over a bluff body like a cyclist's leg will tend to be smooth only at the front of it, as shown in the right-hand picture above.  Towards the back, often at or close to the widest part, the air is unable to continue flowing smoothly, and it 'detaches' or 'separates' from the surface, as shown above.  In the region of separated flow there tends to be large eddies and a low pressure region, which 'sucks' the object backwards, contributing to the majority of the object's drag.  A cyclist's leg is similar to a cylinder, or a tapered cylinder to be more precise, where the thigh has a larger diameter than the calf and the ankle.  Clearly, the cross section of a leg is not exactly circular, as it is for a cylinder, but for the purposes of explaining leg aerodynamics and aero socks, the cylinder analogy works well.  There have been plenty of studies concerning the flow around cylinders, so we can use cylinder aerodynamic data to understand how aero socks work.

Cylinder aerodynamics

Before getting into the aerodynamics of cylinders, it's important to first explain that the drag coefficient for an object, denoted by the abbreviation "Cd", is in general not a fixed value.  Instead Cd is dependent on the flow conditions like the speed, air temperature and the size of the object.

For cyclists, whose frontal area can be easily adjusted by changing the torso angle and arm position, it's often more convenient to use the drag area parameter, "CdA", which is the drag coefficient multiplied by the frontal area.  Cyclists and time triallists often talk about their CdA as if the CdA value is a constant value for a given setup, but it's not really true.  To be fair, over the range of relevant cycling speeds, the changes in CdA are likely to be fairly small, so for practical purposes, considering CdA to be a fixed value is a reasonable simplification.

Changes in CdA occur because the change of a parameter called the Reynolds Number (Re). The Mach number will also affect CdA, but because we cycle at a small fraction of the speed of sound (which is 1230 kph) we can ignore that dependency of CdA on Mach number and focus only on the Reynolds number dependency.  Reynolds number describes the ratio between the inertial properties of the flow and the viscous properties of the flow.  This won't mean much to many people, so it's more helpful to explain what things change the Reynolds number:

• Doubling the speed will double the Reynolds number.  Riding at 40 kph means your Re number is twice as large as if you are cycling at 20 kph. 
• Doubling the size of the object, even if it has the same shape, will double the Reynolds number.  If an ankle has half the diameter of a thigh, the flow around the ankle will have a Reynolds number that's half the Reynolds number of the flow around the thigh.
• The air density and temperature will also affect the Reynolds number.  Increasing altitude will result in a lower Reynolds number, although there isn't a linear relationship like there is with the first two dependencies, speed and size.

The reason for explaining Reynolds number is because the drag of a cylinder-like object, such as a leg, is highly dependent on the Reynolds number.  The plot to the left shows the drag coefficient of a smooth cylinder as a function of Reynolds number.  Note that the Reynolds number dependency is plotted on the x-axis using a logarithmic scale, so it covers a very wide range of flow conditions.

I've annotated the plot to show the region (in blue) that's relevant for cyclist's legs, covering the 10-60 kph speed range.  Across this speed range, you can see that the drag coefficient is very similar for a smooth cylinder, and the Cd is typically a value around 1.2 for the whole blue range.  However, you will notice that at Reynolds numbers that are slightly higher than the blue region, at about 300,000-400,000, the drag coefficient curve reduces significantly.  This point, where the drag coefficient drops substantially is called the 'critical Reynolds number', and it describes a point where the flow around the cylinder behaves very differently.

At Reynolds numbers below the critical Re number, the flow around a cylinder looks like the flow shown in the top sketch on the left, having a wide wake, often with regular vortex shedding occurring from the cylinder and those vortices are transported downstream in wake.  This is where the drag coefficient is around 1.2.

Once the Reynolds number is larger than the critical Reynolds number, at about 300,000-400,000, the wake becomes much smaller, as shown by the bottom sketch on the left.  A narrower wake causes a smaller low-pressure region at the back, hence less drag.

So what is it about the increase in Reynolds number that causes this difference in the pattern of the separated flow and the size of the wake?  Well, the Reynolds number determines whether the air moving right next to the cylinder surface, called the boundary layer, is a laminar boundary layer or a turbulent boundary layer.  At higher Reynolds numbers, the boundary layer naturally becomes turbulent before the point where the flow separates.  This is important because turbulent boundary layers are much more resistant to flow separation than laminar boundary layers.  Therefore, at higher Reynolds numbers, the turbulent boundary layer resists flow separation at the widest point of the cylinder and instead the flow separates only at the very back of the cylinder, causing a narrow wake and a low drag coefficient.

So, to summarise:

• The drag of a cylinder depends of the size of its wake.
• The size of the wake depends on whether the boundary layer is laminar or turbulent at the widest part of the cylinder.
• The Reynolds number of the flow determines whether the boundary layer is laminar or turbulent.
• Hence the Reynolds number determines the drag of the cylinder.

However, the Reynolds number is not the only thing that determines whether the boundary layer is laminar or turbulent, as I'll explain in the next section.

Boundary layer transition tripping

As explained in the previous section, at higher Reynolds numbers the boundary layer will naturally transition from a laminar boundary layer to a turbulent boundary layer before the point of flow separation, and it's the turbulent boundary layer that enables the flow to resist separation at the widest part of the cylinder.

However, the boundary layer can also be 'forced' to transition from laminar to turbulent at Reynolds numbers below the critical Reynolds number.  This intervention to force the boundary layer to become turbulent is often called 'tripping' the boundary layer.  There are various ways to trip a boundary layer, but most methods involve some kind of protuberance, like a bump, a wedge or a band of roughness, that disturbs the laminar boundary layer and causes transition to turbulent boundary layer.

Hence, at low Reynolds numbers, below the critical Reynolds number, the only way to reduce the drag coefficient of a cylinder is to trip the boundary layer.  This is what the ridges and surface texture of aero socks do, and how they are able to reduce the drag of a cyclist's lower leg.

The plot above is similar to the one shown earlier, in the Cylinder Aerodynamics section, except that instead of showing just a single curve for a perfectly smooth cylinder, the plot shows several curves for cylinders with different levels of surface roughness.

The level of roughness is defined as "k/d", which is the roughness height divided by the cylinder diameter.  The perfectly smooth cylinder is the one with k/d=0, which you can see has a critical Reynolds number of about 300,000, as discussed earlier, above which the drag coefficient drops abruptly.  The other curves are for progressively rougher cylinders.  For example, the curve with triangular symbols is for a k/d of 4/10^3 (=0.004), which is equivalent to 0.4 mm roughness  on a 10 cm diameter cylinder.  0.4 mm roughness is about the same roughness as 40-grit sandpaper, which is a coarse sandpaper you'd use for DIY jobs.

For this k/d=0.004 example, you can see that the critical Reynold number is much lower, because the roughness is tripping the boundary layer at lower Reynolds numbers.  As a results, at a Reynolds number of 100,000, this rough cylinder has a lower drag coefficient, about 0.7, than the smooth cylinder has (which ahs a Cd of 1.2 at Re=100,000).  To non-aerodynamicists this might seems counter-intuitive, that adding surface roughness reduces the drag of the cylinder, but it's true, and it's all related to the state of the boundary layer.

This is how aero socks work.  The ridges in the fabric of an aero sock act like the roughness elements in this example, reducing the critical Reynolds number and therefore the leg drag at the Reynolds numbers that cyclists are operating at.

What trip height for what speed?

As a final word, it's worth mentioning that the transition trip height that's required, to give the lowest drag, depends on the Reynolds number.  Hence the trip height (which means the height of the ridges in the fabric), depends on the rider speed and also the size of the body part it's applied to.

The plot above shows the Reynolds numbers for a 50 kph speed.  This is the kind of speed that a high level time trialist would achieve and is approximately equivalent to a 20 minute time for a 10-mile time trial.  I've annotated the plot to show what the Reynolds numbers would be for an ankle, calf and thigh.  This is rather approximate and is based on my own ankle calf and thigh circumference values (26, 38 and 55 cm) to get an approximate equivalent cylinder diameters.  This is admittedly rather crude, because as mentioned previously, the leg doesn't have a circular cross-section. However, it's just to illustrate a point.

You can see that for the ankle, where (UCI-legal) aero socks are working, the best drag is  achieved with roughness height of about k/d=0.005.  For an 83 mm diameter cylinder, k/d=0.005 is a roughness height of 0.42 mm.  This is consistent with the fabric patterns used for aero socks, which have ridges and grooves that are about half a millimetre to one millimetre in depth.  There isn't a direct equivalence here, however, because aero sock fabrics use grooves and ridges, rather than distributed roughness, so it's likely that larger ridge height would be needed to trip the boundary layer in a way that's similar to how roughness behaves.  

Nevertheless, it's reassuring to see that plots of drag data for roughened cylinders is consistent with the fabrics that have been selected by manufacturers of aero socks.

In my next blog post, which I'll write in the coming weeks, I'll discuss this plot further and what other things it may reveal and imply.