Iowa Sailing Club

Car in a Crosswind

an article by David Norwood

Several years ago, a debate raged among the members of the University of Iowa Sailing Club as to whether or not you should slow down if you were towing a big boat in a heavy crosswind. (Actually, everybody agreed that slowing down was safer. The disagreement was over why it was a good idea, so the debate was really more about WHY, not WHETHER. But I digress.) One position was that you could actually decrease the SIDEWAYS force by reducing your FORWARD speed.....WOW!! Let's see if this makes sense.

 First, a diagram of the situation:

The car + thing feel a wind that is partly due to the speed of motion and partly due to the crosswind. The net wind (or apparent wind) is off at an angle from the direction of motion. The speed of the crosswind will be called v_x and the driving speed (and thus the speed of the wind from driving) will be called v_y. So the magnitude of the apparent wind is , and the angle between the apparent wind and the direction of travel is found from . The drag force will be in the same direction as the apparent wind, and, for large enough wind speeds, quadratic in the apparent wind speed: F~v². What's large enough?

ASIDE: We got Reynold's number.... It is found empirically (with some theoretical support) that if you plot drag force as a function of something called the "Reynold's number" (named after Chico Reynold's), it increases linearly when "Reynold's number" is less than about 1000 and quadratically when "Reynold's number" is more than about 10,000. Sounds pretty big, pretty hard to get to. But Reynolds number doesn't just depend on wind speed, it also depends on the size of the object being blown and the viscosity of the fluid blowing. Specifically, the Reynold's number of a system is given by (1) where v is the speed of the wind, D is the size of the object, and n is something called the kinematic viscosity (basically, just a measure of how thick the fluid is). So, how about this towed thing. It's "size" is roughly 2 meters or so. The kinematic viscosity of air is about 0.15 cm2-s-1 (trust me). And the apparent wind speed will be about 63 mph (=28 m/s), using a driving speed of 60 mph and a 20 mph crosswind. (Incidentally, the direction of the apparent wind will be about 18º off straight ahead). With these numbers, "Reynold's number" for the towed thing will be about 4 million. We're so far in the F~v2 range, you could fry eggs on us. (But transfer of thermal energy in fluid flow is governed by the Prandtl number, and we're not going there).

So, the force varies as the square of the apparent wind speed, and we have

(2)

where c is the drag constant. Part of this drag force will be in the sideways (x) direction, and will tend to knock over the towed thing. This would be bad. The sideways component of the force is given by:

(3)

You can already see that something is up because the sideways force, F_x, depends on the forward speed, v_y. (Remember that .) The smallest this sideways force can be is when you're stopped, because then v_y=0 and F_x,min=c v_x². We can avoid the complication of the drag constant c by considering the ratio of the force F_x to it's minimum value, F_x,min. That looks like:

(4)

Here it is explicitly. The sideways, knock-me-over force is definitely affected by the motion perpendicular to that direction. In the range where the speed of driving is much larger than the speed of the crosswind (v_y>>v_x), the square of the ratio of speeds in (4) is much larger than 1 and (4) can be approximated as:

(5)

So for large driving speeds, the driving speed has a direct, linear effect on the sideways force. In other words, the sideways force depends strongly on the forward speed. On the other hand if the driving speed is small enough that v_y<<v_x (but still large enough that the wind force is still quadratic in the wind speed), The sideways force looks like:

. (6)

In this regime, the sideways force changes only slowly as the forward speed changes.

It is exactly this sort of odd, unfriendly, nonlinear behavior that makes fluid dynamics such a tough field to study.