- p is peloton density -- equivalent to fluid density in kg m^3
- d is maximum width of the peloton -- equivalent to tube diameter in m
- v is mean peloton speed in m/s -- equivalent to mean fluid velocity along direction of flow
To flesh this out, the following are the notes I've included in the "Info" section of my Netlogo model:
Introduction
Riders in a peloton, or group of cyclists, are coupled by the energy
savings benefit of drafting. By drafting, riders in zones of reduced
air-pressure behind others expend less energy than those who are facing
the wind. By coupling, cyclists share and distribute energetic resources
among the group as a whole. In this sense, drafting may be thought of as
an attractive or cohesive force among riders.
In mass-start bicycle racing, pelotons may consist of up to 200 riders.
In such groups, cyclists attain high density in continuously deforming
peloton shapes. Density changes are generated by cyclists’ collective
power output. At very low power output, riders tend to leave a lot of
space between each other, and so density is comparatively low. As
outputs and speeds increase, riders tuck in as closely as possible to
others to obtain maximum drafting benefit, increasing density. At
intermediate outputs/speeds, the peloton shape is roughly square or
round and density is high. As cyclists’ outputs approach anaerobic
threshold, the peloton begins to stretch into a line.
A fluid dynamics approach to peloton behavior
Certain basic properties of fluids, such as viscosity (internal fluid
friction, or “thickness”), and its flow characteristics, like laminar
flow (streamlined flow in which particle paths do not cross) or
turbulent flow [1], appear to exhibit similarities to the “flow”
properties of a peloton. In measuring liquid fluid properties (gasses
are not considered here), liquid is conceptualized as flowing through a
tube [1]. The fluid is considered to fill the tube, and its internal
properties can be measured using parameters indicated by the dimensions
of the tube (like tube diameter and volume). The Reynolds number is a
dimensionless value that describes the transition between laminar and
turbulent flows [1]. Generally laminar flow occurs at Reynolds numbers
less than 2000; above 3000, turbulent flow is expected; between 2000 and
3000 the fluid flow is unstable, meaning flow may be laminar but easily
perturbed into turbulence [1].
Although a peloton displays properties similar to liquid flow generally,
a peloton is unlike a fluid flowing through a tube which constrains the
shape of the fluid. The diameter of a tube may vary along the length of
the tube, and this changes the “shape” of the fluid inside the tube, but
the tube is an external constraint on the shape of the fluid inside the
tube. Liquid by itself doesn’t generally flow in shapes of varying
diameters, and on an open flat surface, the liquid will simply spread
out to its minimum thickness.
By contrast, the shape of a peloton is continuously changing as a result
of oscillations in the collective power of the riders. True, the course
over which a peloton travels can vary in width and, if sufficiently
narrow, the course does constrain the shape of the peloton. However, we
can imagine a road of infinite width, and the peloton would still
self-organize into its ordinary expected shapes, changing continuously
and independently of the course width. As a result, if we want to find
equivalencies between peloton flow and fluid flow, we need to account
for this changing shape.
Constant and variable density
Another difference between peloton flow and liquid flow is that a
peloton exhibits significant changes in density, unlike liquid fluids
which are mostly incompressible [1].
In [2] a method used to calculate density showed that density falls as
the peloton stretches. However, another conception of peloton density is
that as long as the wheel spacing between cyclists is roughly constant,
changing peloton shape by stretching does not mean density decreases. In
other words, the peloton could be either stretched in a single-file
line, or in a compact round shape, but the mass of cyclists per volume
is still the same.
In the peloton fluid dynamical model proposed here, density can be set
as a constant parameter despite continuous peloton deformation. To be
sure, in conditions of low drafting such as up steep hills, or during
full-out sprints, spacing between riders increases and peloton density
falls. However, although low density is observed in periods of very low
and very high output, a peloton retains high-density over a broad range
of intermediate outputs, and density may be considered to be constant
over this range.
Constant density is useful for this peloton fluid model. As a result we
can maintain constant density, then apply an adjustment factor to the
length and width of the peloton according to cyclists’ coupled outputs
as a fraction of their mean maximum output. Both the length and width
measurements have considerable effect on viscosity value and the
Reynolds number, and so adjusting them appropriately is a key
consideration.
Although conceptually the model applies a constant density parameter in
a broad intermediate range of peloton speeds (the range in which the
majority of peloton behavior occurs), it does allow for changing
density. By adjusting the density slider between 9kg m^3, approximately
the minimum density at which cyclists remain coupled by drafting (i.e.
if cyclists are 1m behind and to the sides of others, or more, drafting
benefit drops off dramatically – here this corresponds to about 18
cyclists in a 100m^2 area).
Application of Peloton-Convergence-Ratio (PCR)
In [2,3] we demonstrated the “Peloton-convergence-ratio” (PCR), which
corresponds directly with the changing shape of the peloton. PCR is an
equation that accounts for cyclists’ reduced power requirements in
drafting positions, while also indicating cyclists output at any given
moment as a fraction of their maximum sustainable output (MSO).
As PCR increases (i.e. as cyclists approach MSO), the peloton stretches,
and conversely as PCR falls, the peloton resumes its compact formation.
In this peloton fluid model, I apply PCR to adjust the shape of the
peloton in direct proportion to the length which increases as PCR
increases; inversely to width, which decreases as PCR increases (i.e. as
the peloton stretches). When PCR = 1 for a peloton collectively, that
means the all following (drafting) riders are at their maximum even with
the benefit of drafting. This represents the extreme outputs for riders
in a maximally stretched formation. Before reaching this point, however,
we can think of riders adjusting their relative positions in a highly
turbulent manner, and that as speeds are adjusted incrementally higher,
each corresponding adjustment in position must happen faster than the
previous adjustment, and hence the Reynolds number is higher. When PCR >
1, riders de-couple. Although there is a temporary decrease in density
while cyclists remain in drafting zones, the globally coupled systems
breaks down.
Applying PCR to account for cyclists’ power output relative to their
maximums is important for a peloton model because these factors affect
the force and velocity parameters of the fluid viscosity and Reynolds
number equations, and scales them appropriately to typical peloton
speeds and power output values. See Notes 3-5.
MSO can be adjusted, which can be thought of as changing “how easy it
is” for the cyclist to travel at the given speed. A higher MSO, means
the given pace will be easier for him/her, and conversely for a lower
MSO.
Temperature-viscosity relationship
As temperature increases, viscosity falls, as the speed of interactions
between molecules reduces friction between them [4]. Increasing the
relative-passing speed is effectively like changing the temperature of
the peloton, resulting in decreasing viscosity, as expected [1].
However, here I don’t address whether the equations for the relationship
between temperature and viscosity [6] accord with the equations I’ve
developed that adjust the shape of the peloton according to increasing
speed, and hence the peloton viscosity and Reynolds number.
Things to try
So, with these considerations in mind, in
my Netlogo peloton fluid dynamics model we can plug some realistic values
into the model and see the resulting viscosity and Reynolds number, and
make predictions as to the values at which laminar flow transitions to
turbulent flow. Generally, we may predict that at appropriate
combinations of PCR, relative passing speed, and peloton shape,
turbulence sets in.
I used the following as constants: 100 cyclists 75kg each in a 10m * 10m grid * 1.5m (height of crouching cyclist on bike) yielding constant high density of 50kg m^3 (75kg * 100 / 150m^3) ; base peloton width of
10m -- equating to tube diameter; minimum peloton “side-area” of 12.4m^2, based on 5 cyclists on bikes 1.65m (tip of front-wheel to outside tip of
rear-wheel) * 1.5m (height of cyclist’s head/shoulder in crouched
position on bike) -- equating to parameter A in the viscosity equation
(area of the fluid boundary layer).
Starting with these values, we can make adjustments to mean-peloton speed and passing speed to generate
relative-passing speed. Generally, unless they are
“attacking” to escape the peloton, cyclists do not pass each
other much faster than between approximately 5km/h (1.39 m/s) and 10km/h (2.78 m/s) in normal peloton
movement. Regardless, inputting any values you can see how viscosity and Reynolds number change in
correlation to relative passing speed. Speeds in turn affect the
drafting rate, and PCR, which affect the shape of the peloton (length
and width).
You can adjust the MSO values for a drafting rider. As noted above, this
affects PCR, and the comparative “ease” of pace for the drafting
cyclist. It does not account for the front cyclist’s output, since most
of the riders in the peloton are assumed to be drafting.
Results
As relative-passing speed is increased, viscosity falls and Reynolds
number increases. When PCR > 1, viscosity is 0, and Reynolds number is
negative.
Overall the results seem to accord
reasonably well with what
the physics literature says is the Reynolds number range at which to
expect turbulence; i.e. above 3000, with an unstable region between
2000-3000 [1]. It also accords roughly with what we can expect in terms
of phase transitions according to PCR, where the peloton is stretched at
higher PCR. In terms of viscosity, the value is shown in N*m/s^2 (where
0.001 Nm/s^2 = .001 centipoise), and in the turbulent region, peloton
viscosity is roughly equivalent to high viscosity fluids, like syrup or
molasses [5].
However, the results indicate that the unstable region (Reynolds number
2000-3000) does not occur until the passing speed is about 3.65m/s, or
slightly more than 13km/h. Intuitively, it seems that turbulence occurs
at lower passing speeds than this. This suggests some of my initial
assumptions are inaccurate, or some adjustments need to be made in the
model overall -- or, indeed, that true turbulence as it would occur in analogous systems, does not emerge until passing speeds are > than 13km/h. I am inclined toward the two former options, however, and would want to check things quite a lot before I conclude the last option.
However, because the values are not extraordinarily outside the expected
range, the model is promising in terms of demonstrating viscosity and
the transitional regimes of laminar and turbulent flow.
Conclusions and model limitations
The model represents the passing speed of one segment of the peloton –
essentially one single-file line on one perimeter of the peloton. If the
standard viscosity model applies, we should assume that the speeds of
each line or layer (shear flow) diminishes to zero as they approach the
opposite side of the peloton. This of course does not occur in pelotons.
Further, aside from a simplified single point-in-time change in speed
which is assumed to represent an average among all passing cyclists, the
model does not indicate other differences in speeds among cyclists
within the peloton. Nor does it indicate trajectory changes, which
together with a range of speed differentials among riders, might indicate
the presence of vortices and eddies.
In addition, there is no attempt here to draw parallels to substantially
more complex Navier-Stokes equations of fluid dynamics [6].
As discussed, there is a relationship between temperature and viscosity
and Reynolds number. By increasing the passing speed relative to the
mean-peloton-speed (i.e. increase in relative-passing-speed), the
process is similar to increasing the temperature of the fluid. This
results in decreasing viscosity, which accords with what is expected
[4]. However, here I don’t address whether the equations I’ve developed to adjust the shape of the peloton according to increasing speed, and hence the peloton viscosity and Reynolds number, align with established equations for the
relationship between temperature and viscosity [4].
Also, the application of PCR, used to adjust the shape of the peloton, may in
principle be flawed. Nonetheless, this is an attempt to account for the
changing shape of the peloton, in contrast to other fluids that conform
to the shape of their containers.
While the proposed model is idealized, the key is that a
peloton exhibits relative changes in position and speed among globally
coupled cyclists, which, it seems, are amenable to a fluid dynamics
analysis. The proposed model identifies some of the key parameters which
are parallel to the standard fluid viscosity and Reynolds number, and
indicates, at least, that it is possible to model a peloton as a fluid.
Overall, I well expect there are flaws in the model. These flaws may be
conceptual and/or smaller ones that can be fixed without destroying the
the basic principles.
NOTES
1. Peloton density p is based on:
-
p = mass/vol. Mass = 75kg (rider plus bicycle), multiplied by number
of cyclists;
-
Volume = 150m^3 based on a selected area 10m x
10m (a standard road
width) x 1.5m (~height of slightly crouched cyclist on
bicycle).
-
Here, assuming 70 cyclists in a 100m^2 area, density is 50kg*m^3 (1
cyclist for every 70cm of space laterally for ~14 across, and 1 for
every 185cm lengthwise, for ~5 lengthwise).
While a peloton may be considered to be two-dimensional, to keep the
fluid model consistent with actual fluids analysed as 3-dimensional,
I’ve used a 3-dimensional density value for a peloton.
2. Relative-passing-speed = (passing-speed) - (mean-peloton-speed)
-
equivalent to delta x (distance displacement) in the standard
viscosity equation
3. Side-area A = 12.4 + 12.4 * (14 ^ peloton-convergence-ratio), based
on:
-
~1.5m (height of crouched cyclist) * 1.65m (length of bike
from
tip of front wheel to outside tip of rear wheel), multiplied by the
number of
cyclists comprising the length of the single-file
perimeter line, here 5.
-
Number of cyclists in single-file (5), is determined by 10m length
(10mX10m grid) / 1.65m (bike length) + 0.20m (approx spacing between
wheels).
-
14 is the number of cyclists laterally who can fit in a 10m space,
given shoulder width of 0.50m, and 0.20 spacing between cyclists
side-to-side, for 10m/0.7m
-
exponent PCR, where PCR is <1, means that A increases
proportionately to increasing PCR; i.e. as PCR increases and riders
approach their maximum sustainable capacities, the peloton naturally
stretches; Where PCR > 1, it means riders become de-coupled.
4. lateral-distance = 10 - 9.5 * (Peloton-convergence-ratio);
-
10 is 10m, or the maximum width of the peloton and maximum density,
in this illustration;
-
since one rider at shoulder width is ~0.5, the minimum width of the
peloton is one-rider wide, or 0.5, so this is set so when PCR = 1,
peloton cannot be less than 0.5m wide, and so the lateral distance
diminishes proportionately to PCR, starting at 9.5m.
-
In order to correspond with PCR=1 and for the Reynolds number to
show negative values if PCR > 1, I’ve set this so that when there is
0 lateral distance, it means the peloton is stretched to a
single-file line (I could have set this so that PCR = 1 when the
peloton was 0.5m wide (one rider wide), but then Reynolds number
stays positive until lateral distance is 0).
-
passing is relative-passing-speed, which equals passing-speed -
mean-peloton-speed
5. PCR = [Pfront - (Pfront * (D/100)))] / Pmso
where:
-
Pfront is the power output of the front rider at the given speed
(the
same power output that would be required by the drafting (following)
rider
to maintain the speed set by the front rider,if the following rider
was
not drafting - hence “required output”;
-
D is the percentage of energy saved by drafting, here estimated as
1%
per mile/hour (see computer code for conversion);
-
Pmso is the maximal sustainable output of the drafting cyclist.
PCR is incorporated into this fluid dynamical model because a cyclist’s
capacity to pass others corresponds to the current peloton speed, the
power required for the given speed, and the fraction that this power
output is of his/her maximum capacity (MSO). The required output to
travel the given speed is reduced by drafting benefit, which is
accounted for by PCR. The closer the cyclist is to PCR = 1, the closer
his/her passing capacity, or relative passing speed, is to zero. When
relative passing speed is zero, the peloton may be said to be in
streamlined, or laminar flow. Only when the relative passing speed
reaches a certain threshold does peloton turbulence occur. Generally,
physics literature indicates turbulence in fluids occurs around Reynold
number 3000, although when between 2000 and 3000, the system is unstable
and could be perturbed into turbulence [1].
6. Average-effective-pedal-force, based on:
-
rider power-output / pedal velocity of 1.78 m/s, where vp = cadence *
crank-length * 2pi / 60 / 1000 using 100rpm, 170mm cranks;
see
analyticcycling.com for further information on
pedal-velocity and relationship to power
References
[a] Serway,R. 1996. 4th ed.
Physics for Scientists & Engineers with Modern Physics. Saunders College Publishing, Philadelphia
[1] Serway,Raymond and Jerry Faughn. 1989 2nd. ed.
College Physics.
Saunders College Publishing, Philadelphia
[2] Trenchard, H., Richardson, A., Ratamero, E., Perc, M. 2014.
Collective behavior and the identification of phases in bicycle
pelotons.
Physica A 405 (2014) 92-103.
[3] Trenchard, H. 2013. Peloton phase oscillations.
Chaos Solitons &
Fractals. 56 (2013) 194–201(56):194-201.
[4]
http://en.wikipedia.org/wiki/Temperature_dependence_of_liquid_viscosity
[5]
http://www.research-equipment.com/viscosity%20chart.html
[6]
http://catdir.loc.gov/catdir/samples/cam031/00053008.pdf
