Bicycle Aerodynamics, or:How to lose the Tour de France.

Duncan Sutherland

ΣUMS

Duncan Sutherland Bicycle Aerodynamics

Background

1989 Tour de France final stage. Greg LeMond (right) beatLaurent Fignon (left) by 8 seconds. The closest margin in Tour deFrance history.

Duncan Sutherland Bicycle Aerodynamics

From Wikipedia: “ Fignon, who rode after LeMond, lost 58seconds during the stage, and although he became third in thestage, he lost the lead to LeMond. It was calculated afterwardsthat if Fignon had cut off his ponytail, he would have reduced hisdrag that much that he would have won the Tour.[McCann]”

Jolly good, but how is such a problem approached? Paper byThijis Defraeye, et al. “Aerodynamic study of different cyclistpositions: CFD analysis and full-scale wind-tunnel tests.” Journalof Biomechanics, v43, i7, pp 1262-1286 (2010).

Duncan Sutherland Bicycle Aerodynamics

Streamlined and bluff bodies

I Streamlined bodies: flow follows the contours of the body.Example: airfoils.

I Bluff bodies: flow around the body separates and vorticies areformed by rolling up of shear layers. Example: A brick.

I Cyclists are combinations of these types of body. Areas suchas the helmet may be quite streamlined, but areas such as thetorso may be bluff bodies.

(a) Airfoil (b) Brick

Duncan Sutherland Bicycle Aerodynamics

Resistance

I A cyclist’s performance is affected by the resistance, eg, drag,friction and gradients

I At the highest speeds 50kms/h and above, most of theresistance is aerodynamic drag, mostly accounted for by theposition of the cyclist’s body.

I Two sources of drag: form drag contributes more than 90% ofthe drag and viscous drag, caused by roughness on the surfaceof the cyclist.

Duncan Sutherland Bicycle Aerodynamics

Dimensional analysis and the drag equation

Dimensional analysis is way of deriving an equation for an unknownquantity without thinking too hard:

I Take variables of interest: drag FD, velocity u, density ρ, areaL2 and viscosity ν

I Writefa(FD, u, ρ, L

2, ν) = 0

and note that 0 is dimensionless, thus the function must bedimensionless.

I Rearranging into dimensionless groups

fb

(FDρL2u2

,uL

ν

)= 0

since only FD is unknown:

I FD = 12ρL

2u2fc(Re), Re = uLν and then define

CD = fc(Re). CD ≈ 0.23.

Duncan Sutherland Bicycle Aerodynamics

Fluid Dynamics

Fluid mechanics is basically concerned with solving theNavier-Stokes equation and the continuity equation

ρ

(∂ui∂t

+ uj∂ui∂xj

)= − ∂p

∂xi+∂Tij∂xi

∂ρ

∂t+ uj

∂ρ

∂xi+ ρ

∂uj∂xj

= 0

Simplifying assumptions: incompressibility, ∂ui∂xi

= 0; Newtonian

fluid, Tij = µ(∂ui∂xj

+∂uj∂xi

)These are complicated and we want to

use a computer. Problems are cost, stability and accuracy.

Duncan Sutherland Bicycle Aerodynamics

Reynolds Averaged Navier Stokes

For an incompressible, viscous, Newtonian fluid, we maydecompose velocity, ui into the time averaged part Ui and thefluctuating part, with zero mean, u:

ui = Ui + ui

pi = Pi + pi

Tij = Tij + τij

This gives the RANS equation:

ρ

(∂Ui∂t

+ Uj∂Ui∂xj

)= − ∂P

∂xi+

∂

∂xj(Tij − ρ〈uiuj〉)

Turbulence Closure Problem: use models of turbulence todetermine 〈uiuj〉.

Duncan Sutherland Bicycle Aerodynamics

Free pizza

Duncan Sutherland Bicycle Aerodynamics

Large Eddy Simulation

Low-pass filter applied to the Navier-Stokes equation.

¯ϕ(x, t) =

∫ ∞−∞

∫ ∞−∞

ϕ(r, t′)G(r− x, t− t′)dt′dr

∂ui∂t

+∂

∂xj(uiuj) = −1

ρ

∂p

∂xi+ 2ν

∂

∂xjSij −

∂τ rij∂xj

Sij is the rate of strain tensor. The residual stress tensorτ rij = Lij + Cij +Rij Lij represents interation of large scales, Cijcross scale interations and Rij is subscale stress.

(c) Direct sim-ulation

(d) LES sim-ulation ∆ =L/32

(e) LES sim-ulation ∆ =L/16

Duncan Sutherland Bicycle Aerodynamics

Finite Element Method

FEM is a means of approximating the solution to a PDE on amesh, using certain basis functions eg: peicewise polynomials. Themesh can cope with complicated geometries and the computationis relatively cheap.

Duncan Sutherland Bicycle Aerodynamics

Defraeye paper

I Comparison of the area drag L2CD, of different cyclistpositions.

I Wind tunnel experiments and a comparison of LES and RANScalculations based upon the cyclist’s position alone.

I Experimental setup of a bicycle was photographed, digitisedand the cyclist was smoothed into a mesh suitable for finiteelement computations.

Duncan Sutherland Bicycle Aerodynamics

Meshing

The mesh is generated by a 3D laser scanning and digitisingsystem. The bicycle, stand and other pieces of experimentalequipment were omitted from the computational domain.

(f) Computationaldomain

(g) Experimentalsetup

Duncan Sutherland Bicycle Aerodynamics

Comparison of simulation and experiment

Duncan Sutherland Bicycle Aerodynamics

Conclusion

Maurice Garin, disqualified in 1904 for catching a train.

Duncan Sutherland Bicycle Aerodynamics