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Top Speed: Speedskating Wind Drag Coefficient

c. P. Baum, 1997-98, pjbemail@gte.net

This analysis continues the Speedskating Santa Barbara top speed analysis by exploring the wind drag coefficient in some detail. My previous analysis took into account the velocity dependence of the drag force but assumed that the drag coefficient remained constant while the skater varied his other parameters. This page will explore the improvment which could come from properly exploiting the drag coefficient itself. (See alsorecent drag-reduction items) Here I will identify three items of importance:
1: The skater's cross-sectional area facing the wind (the "in your face" force).
2: The aspect ratio of the skater or his degree of being streamlined (be a fish, not a brick).
3: Skating "in the dip" or the "get off my back" effect to exploit the drag minimum.
First I will review some basic aerodynamics then present my interpretation in the Summary at the end.

Much of the material on this page was inspired by the Princeton University Aerodynamics website and most of the figures on this page are mirrored from the Princeton site:


[Princeton U.: Aerodynamics of Bicycles , Drag of Blunt Bodies and Streamlined Bodies]
Starting with the basics, I Quote from Princeton: " When the drag is dominated by viscous drag, we say the body is streamlined, and when it is dominated by pressure drag, we say the body is bluff. Whether the flow is viscous-drag dominated or pressure-drag dominated depends entirely on the shape of the body. A streamlined body looks like a fish, or an airfoil at small angles of attack, whereas a bluff body looks like a brick, a cylinder, or an airfoil at large angles of attack. For streamlined bodies, frictional drag is the dominant source of air resistance. For a bluff body, the dominant source of drag is pressure drag. For a given frontal area and velocity, a streamlined body will always have a lower resistance than a bluff body. For example, the drag of a cylinder of diameter 'D' can be ten times larger than a streamlined shape with the same thickness (see figure 1). Cylinders and spheres are considered bluff bodies because at large Reynolds numbers the drag is dominated by the pressure losses in the wake. "


So I would consider a skater standing up to be BLUFF while a skater bending down so that his head, shoulders, and torso all point straight ahead would be more (but not fully) STREAMLINED.

Figure 1. Drag coefficients of blunt and streamlined bodies.

The drag coefficient and Reynolds Numbers formulas on the right side of Figure 1 are from the Team Arian website: [Team Ariane Aerodynamics Formulas] and will be used shortly.
Note from figure one that the drag coefficient is about constant for the circular and elliptical shapes until we get near the dip where the stick skater appears. Then the drag coefficient drops sharply at a particular Reynolds Number of about 5*10^5 to 10^6. Where is the skater in terms of Reynolds Number? Suppose his speed is 30 mph = 44 feet per sec = 14-15 meters per second. From the Ariane handy formula for Re on Figure 1, the skater's dimension : t ~ Re/( 70000*v) yields
t ~(5*10^5 - 10^6)/((70000)*(14-15)) so, finally,
t ~ 0.5 - 1.0 meters
or about 1.5 - 3 feet. If a skater looked like a circular or elliptical cylinder and were travelling at 30mph, he would be at the minimum drag ("in the dip") at 1x to 2x his size.

Figure 2. Drag coefficient as a function of Reynolds number for smooth circular cylinders and smooth spheres.
The variation of the drag coefficient with Reynolds number is shown in figure 2, and the corresponding flow patterns are shown in figure 3. : We see that as the Reynolds number increases the variation in the drag coefficient (based on cross-sectional area) decreases, and over a large range in Reynolds number it is nearly constant. I have placed a stick skater in the dip in the drag coefficient (the minimum).
Figure 3. Flow patterns for flow over a cylinder:
(A) Reynolds number = 0.2; (B) 12; (C) 120;
(D) 30,000; (E) 500,000.
Patterns correspond to the points marked on figure 2.
I have placed a stick skater's head and arms on (A) to establish the direction pointing into the wind..

Following Princeton's analysis again At a Reynolds number between 10^5 and 10^6, the drag coefficient takes a sudden dip. The size of the wake decreases, indicating that the boundary layer separation on the cylinder or sphere occurs further along the surface than before. What has happened? The phenomenon is related to the differences between laminar and turbulent boundary layer. The boundary layer and its interaction with the local pressure gradient plays a major role in affecting the flow over a cylinder. In particular, near the shoulder, the pressure gradient changes from being negative (decreasing pressure) to positive (increasing pressure). The force due to pressure differences changes sign from being an accelerating force to being a retarding force. In response, the flow slows down. However, the fluid in the boundary layer has already given up some momentum because of viscous losses and viscous friction, and it does not have enough momentum to overcome the retarding force. Some fluid near the wall actually reverses direction, and the flow separates.


It follows that, if the boundary layer of a sphere can be made turbulent at a lower Reynolds number, then the drag should also go down at that Reynolds number. This is the case, as we can show by using a trip wire. A trip wire is simply a wire located on the front face of the sphere and it introduces a large disturbance into the boundary layer. This disturbance causes an early transition to turbulence, and its effect on the size of the wake, and the total drag is quite dramatic, as shown in figure 4 .

Figure 4. Flow over a sphere: (a) Reynolds number = 15,000;
(b) Reynolds number = 30,000, with trip wire.


Summary

Lets summarize the three points I listed at the beginning:

1: The skater's cross-sectional area facing the wind.
 The skater's cross-sectional area should be minimized if it does not hinder the drag reduction mechanism in point 3. The area should probably always be minimized at low and moderate speeds. This will reduce the "in your face" force.
2: The aspect ratio of the skater or his degree of being streamlined.
The aspect ratio seems to me to be more important than the cross-sectional area and it is important at all speeds. This means bending down so that the head and torso all point straight into the wind. It can be augmented by extending the skater's apparent length in the direction of travel with aerodynamic "shrouds" to make him more like a fish.
3: Skating "in the dip" or the "get off my back" effect to exploit the drag minimum.
This point takes advantage of the dip in drag force when the laminar air boundary layer on the skater's back is shed at the onset of turbulence. It exploits the drop in drag force (by possibly a factor of 2 or more) which occurs if your backside generates a turbulent layer and not a smooth or laminar one. In order to operate in the dip at skating's top speed it may require an increase in the skater's size to get the Reynold's number up a factor ~2. I do not recommend increasing your area by standing up as this makes you less streamlined and hurts your aspect ratio.

I would conclude that at top speed a skater's drag would either be unchanged or actually decreased with a modest increase in his frontal area. If the drag is unchanged it would give you some room to employ streamlining shrouds. If the drag drops you will want to increase your area for it own sake.


Figure 4 here also showed that one can play tricks on your frontside to induce turbulence on your rearside. It appears that the US Speedskating Team used the trip wire effect at the 1998 Winter Olympics in Nagano, Japan whereas the Dutch used a vortex generator effect. Details Here Further investigation is sorely needed.
I close this page with some preliminary ideas for making the skater more streamlined. The model on the left is the "jet" model and on the right is the "stealth" model. The top row shows the skater in a side view. The bottom row shows a front view. Consider these body shrouds as an aerodynamic "helmet" on the front connnected to an aerodynamic "jersey" on the rear. I think the legs would have to have two separate aerodynamic shrouds. Maybe some of you creative types can come up with better designs.
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