low-order analysis of effect of crosswind on riding speed

Back in July 2009 I did a series of first-order calculations on the effect of various parameters on riding speed. Calculating speed from power is difficult to do explicitly, but to first order the calculations become straightforward. First-order analysis is where much of the intuition is, anyway. One of these calculations was the effect of wind resistance on riding speed. Then in November 2009 I extended that analysis to the effect of wind speed on riding speed.

To my dismay, I found an error in that result. I'd even rationalized the wrong result with incorrect arguments. I had to track the consequences of that error through the following two blog posts. I think I fixed everything. The corrected result was:

d s / d s_w = 2f / [ 2f + 1 ‒ s_w / s ]
where s is the rider speed, s_w is the tailwind speed, and f is the initial fraction of retarding force due to wind resistance. There appears to be a singularity issue for strong tail-winds (see the denominator) but then if s = s_w, f = 0, so further analysis is needed in this case.

However, this only covers a wind in the same direction as the rider. The wind may also have a cross component.

First, a digression... in a cross-wind it's tempting to think that there should be no effect on speed. After all, if the rider is going north, and wind molecules are striking him from the east, then they are imparting on him a momentum component to the west, which induces a force to the west. But his wheels keep him moving north, and instead of the wind inducing an acceleration to the west, he leans his center-of-mass to the east, using gravity to balance the wind force. I've found myself falling into the trap of this reasoning.

The error is that the force of wind isn't just proportional to the wind's momentum relative to the rider (and the perpendicular momentum is unaffected by rider speed) but additionally by the rate at which the air molecules are striking the rider. And this is what is affected by rider speed in a cross-wind.

Consider a stationary rider with cross-section A (including his bike) is being hit by a wind from the side coming at speed s_w. The volume of air striking the rider per unit time = s_w A. Now the rider starts riding forward with speed s. The relative speed of the wind is now sqrt(s² + s_w²) > s_w. The result is he is struck by an increasing volume of air per unit time relative to when he was stationary, assuming the relative cross-sectional area is direction-independent (not an excellent approximation, but without it the analysis gets too complicated for me to avoid making errors, something I'm likely to do anyway, as demonstrated already here). Relative to the rider, the wind has a speed in the headwind direction of −s, and a speed in the perpendicular direction of s_w. The perpendicular momentum imparted goes into bike lean, and is thus canceled out without power penalty. But the momentum in the riding direction is what must be overcome with leg muscles, so yields a drag force. The drag force is thus proportional to s sqrt(s² + s_w²). This yields a power component proportional to s² sqrt(s² + s_w²).

So if I define f_w as the coefficient relating the square of relative wind speed to force magnitude, typically approximated by:

f_w = ½ ρ C_D A,
where ρ is the air mass-density, C_D is the coefficient of wind drag, and A is the effective cross-sectional area of the rider with his bike (assumed angle-independent), then the component of power due to wind resistance p_w is:
p_w = f_w s² sqrt(s² + s_w²).   (cross-wind)

This is fairly close to the analysis for a wind along the direction of rider motion, which is, using the convention that a positive wind is in the direction of rider motion (tail-wind):

p_w = f_w s (s − s_w)².   (tail-wind)

Back to the cross-wind: what I'm mostly interested in isn't the effect of wind on power, but more the effect of wind on speed. This is most readily considered under the assumption of constant power: dp = 0. For that, you need to add in the power due to climbing and rolling resistance, which is typically modeled as a weight-proportional force independent of speed. There's also drive-train losses, but the common (inaccurate) assumption is these are proportional to power, and thus unchanged under a constant power assumption.

First, there's wind resistance power. The differential of that is:

d p_w = f_w d [ s² sqrt(s² + s_w²) ].   (cross-wind)

Initially I made the mistake of assuming this difference was zero. In other words, I assumed constant wind resistance power But if the wind changes rider speed, then rolling power (which is mass-proportional) is also changed. It's total power, not just wind resistance power, which is fixed. The differential of rolling power is:

d p_m = f_m d s

where f_m is a coefficient for rolling power (which includes rolling resistance and road grade).

To solve ds versus ds_{, I set:}

d p = d p_w + d p_m = 0

I'll spare the details, but the end result is, written in terms of f which I define to the the fraction of propulsive power going into wind resistance (after removing drivetrain losses but not rolling resistance):

d s / d s_w = −f_w s² s_w / [ f_w s (3 s² + 2 s_w²) + f_m sqrt(s² + s_w²)] .

For small cross-wind speeds the rate of loss of speed with respect to cross-wind speed is proportional to the cross-wind speed, which implies to lowest order the speed loss from cross-winds is proportional to the square of the cross-wind.

The important point is while there is a first-order dependence of speed on head/tail winds, for cross-winds the dependence is second-order. It is therefore tempting to break the wind down into components: a head-wind component and a tail-wind component, then assuming each component is small, calculate a net dependence of rider speed on wind speed in each direction by summing. The problem with this is second-order terms for the headwind were already discarded in the first-order analysis there, so the net second-order wind dependence will be incorrect if just the cross-wind component is included: a proper first-order analysis is that the cross-wind is irrelevant. So to capture the cross-wind dependence, the tail/headwind analysis should be extended to second order. But if one doesn't care about about second-order errors unless the wind is close to a pure cross, then this approach might still be justified.