Re: Drafting in Bicycle Races

[John Denker <jsd@MONMOUTH.COM>]

At 05:54 AM 9/15/99 -0400, Ed Schweber wrote:
>    In bicycle races one rider will ride closely on the tail of the rider in
> front of him so that he will be riding in the "air resistance shadow" of the
> front rider.
>
>     I know of no reason based on basic physical laws of why this should
> impose an extra burden on the front rider except that it seems to violate my
> sense that there should be no free lunch.

Sometimes there is a mostly-free lunch, as discussed below.

>    So my question is: Does the front rider in any way need a greater power
> output to maintain the same speed by virtue of having someone drafting
> behind him?

** Executive summary:  probably a little bit, but not in proportion to the
benefit that the follwer gets.

** Explanation:

1) Obviously, if the riders are too far apart, then drafting is not effective.

2) If you imagine that each rider is fully streamlined, then drafting would
not be effective.  Each rider would separately have 100% pressure recovery,
as illustrated in
  http://www.monmouth.com/~jsd/how/htm/4forces.html#fig_recovery

3) Therefore we must consider the non-streamlined case.  The physics of
this case is captured by considering a "convoy" of flat plates, moving
perpendicular to their flat faces, closely following each other single file
as shown here:

                                 |  |  |  |  |  |
                                 |  |  |  |  |  |
                                 |  |  |  |  |  |
                                A| B| C| D| E| F| G
        motion   <---------      |  |  |  |  |  |
                                 |  |  |  |  |  |
                                 |  |  |  |  |  |
                                 |  |  |  |  |  |
                                 1  2  3  4  5  6

Behind the rearmost plate (point G) there will be a wake, as suggested in
  http://www.monmouth.com/~jsd/how/htm/4forces.html#fig_bluff

In front of the foremost plate (point A) there will be the full stagnation
pressure:
  P(A) = ambient + half rho vee squared
while between the plates the pressure will be closer to the static pressure
  P(B) ... P(F) = ambient
and a flat plate has very little pressure recovery in its wake (point G):
  P{G) = ambient + a little bit

(See http://www.monmouth.com/~jsd/how/htm/airfoils.html#sec_static_stagnation
for a discussion of the terminology of static pressure versus stagnation
pressure.)

Of course each plate will have some friction drag at its edge, but this
scales like the amount of edge, and does *not* scale like the frontal area
of the plate.

(See http://www.monmouth.com/~jsd/how/htm/4forces.html#fig_drag_venn for a
discussion of the difference between pressure drag and friction drag.)

Summary:
  *) Compared to riding alone, plate #1 gets a slightly worse ride, because
the folowers steal some of the pressure recovery that #1 might have gotten.
  *) Compared to riding alone, the intermediate plates #2 ... #5 get a much
better ride, because they don't have stagnation pressure on their fronts
like #1 does.
  *) Plates #6 gets the best ride of all:  low pressure ahead, and
presumably a little bit of pressure-recovery behind.

Note that the intermediate plates *do* get a free benefit:  They get
considerably less pressure drag (compared to riding alone) with essentially
no cost imposed on the first or last plate.  Explanation:  the convoy is
just a more efficient shape than a bunch of separate plates.

======================================

Partially-baked idea:  You could probably demonstrate this in the
classroom.  In my crystal ball I see a convoy of picnic-plates (the
relatively rigid styrofoam kind) harnessed with kite-strings to a
weight.... Dropping from high balcony....  Measuring the terminal velocity
as a function of the number of plates, and as a function of the spacing
between plates........