cyclopeon said:

Ah, okay; thank you. But may I point out that **that** (at Wiki) is Drag at High Velocity. *Which means approaching Mach 1 (ie approaching the speed of sound).* Cyclists do not travel at anywhere near such speeds, not even with a motor-assist. Wrong conditions, wrong equation.

At cycling speeds power will vary roughly as the square of the velocity. 20kph, 100W; 40kph, 400W. On flat roads. (Uphill, less than the square: it depends on steepness but that's a whole nother issue)

Nowhere does it mention Mach is the description of 'High Velocity'. It defines high velocity as having a Reynolds number greater than 1000. Reynolds number is a non-dimension number relating inertial forces to viscous forces. For a high Reynolds number (Re), the inertial (drag) forces dominate, and the viscous forces can be neglected.

The Reynolds number is given by the equation:

Re=rho*V*L/mu

where:

Re is the Reynolds number

rho is the density of the fluid (air in our example)

V is the relative velocity of the fluid to the object (or vice-versa)

mu is the dynamic viscosity of the fluid.

L is the charactheristic length of the object (the bike in our example)

for Air at nominal sea pressure level and room temperature (25C):

rho = 1.184 kg/m^3

mu = 1.85E-5 N*s/m^2

source: Fundamentals of Fluid Mechanics Fifth Edition, Munson,2006, Appendix B, pg 763.

Assume a velocity of 13m/s (approx 29mph or 48kph).

It is difficult to calculate the characteristic length, so lets see what minimum characteristic length is necessary for the Reynolds number to be greater than 1000.

Solving the Reynolds Number equation for L:

L = Re*mu/(rho*V)

L = (1000*1.85E-5)/(1.184*13)

L = 0.001m or 1mm.

We can see from the equation that Re increases with increasing L, so any characteristic length greater than 1mm will qualify as 'high velocity'. I don't know the exact value, but the characteristic length of a bicycle is clearly greater than 1mm. Therefore, in this example, a bicycle travelling through air at 29mph is 'high velocity'.

FYI, the variance of density and viscosity are small over the ranges found in cycle racing.

at 0C, rho = 1.292, mu=1.71E-5

at 40C, rho = 1.127, mu = 1.84E-5

at 3000m, rho = 0.9093, mu = 1.694E-5

None of these are significant enough to change the fact that a bicycle in air at 29mph is 'high velocity', no matter the altitude and temperature.

Power is a function of velocity cubed, not velocity squared.

If you want to talk more fluid dynamics, let me know.

EDIT: I just noticed that in the wiki example, they use a car at 100mph, far from mach, and they state that double speed need 8 times the power.