Here’s some light bedtime reading for all those interested in the aerodynamics of cycling.
These are just summary’s of the chapters so please be sure to click on the chapters titles to be taken to the full chapter over at eFluids
1. Introduction: Aerodynamics and Friction Losses
AERODYNAMICS have preoccupied bicycle designers since the early part of this century. The most advanced bicycles today are deployed in track racing. The recently unveiled SB II, or Superbike II, has a lightweight carbon-fiber frame. It also has a range of aerodynamic design elements. Similar features are incorporated into bicycles for some road-racing events in which Lance Armstrong competes.
As the bicycle and its rider move along the road, the air exerts a force that increases sharply with speed. The force is due to friction between the air and the exposed surfaces of the rider and bicycle. At high speed, this drag force can be the most importance source of resistance, and with a wind blowing, it can also lead to significant side forces.
2. Adding Wind Speed and Directions
When cycling in still air, you feel a wind that is caused by your own motion, and you feel it blowing directly in your face at a speed equal to your own velocity. The wind you feel in the moving reference frame that is moving with the cyclist is known as the “relative wind.” If instead of cycling, you are standing still by the side of the road, you would feel no wind. The wind relative to a stationary observer is known as the “absolute wind,” which in still air would obviously be zero.
When cycling in a head wind, so that the wind blows in the direction directly opposite to your own motion, the magnitude of the relative wind would be equal to the sum of your own speed and the absolute wind speed. For a tail wind, the magnitude of relative wind would be equal to your own speed minus the absolute wind speed.
3. Characteristics of Fluids
The principal difference in the mechanical behavior of fluids compared to solids is that when a shear stress is applied to a fluid it experiences a continuing and permanent distortion. Fluids offer no permanent resistance to shearing, and they have elastic properties only under direct compression: in contrast to solids which have all three elastic moduli, fluids possess a bulk modulus only.
Thus, a fluid can be defined unambiguously as a material that deforms continuously and permanently under the application of a shearing stress, no matter how small.
4. Pressure
Pressure is a stress. It is a scalar given by the magnitude of the force per unit area. In a gas, it is the force per unit area exerted by the change of momentum of the molecules impinging on the surface. We know from Newton’s second law that a net resultant force will cause a change of momentum in a body, and that the rate of change of momentum is equal to the applied force. It is a vector relationship, so that even if the magnitude of the momentum is unchanged, a change in the direction of motion requires a resultant force.
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4.1 Pressure: direction of action
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4.2 Pressure: transmission through a fluid
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4.3 Compressibility in Fluids
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4.4 Pressure and Lift
5. Continuity Equation
When a fluid is in motion, it must move in such a way that mass is conserved. To see how mass conservation places restrictions on the velocity field, consider the steady flow of fluid through a duct (that is, the inlet and outlet flows do not vary with time). The inflow and outflow are one-dimensional, so that the velocity V and density \rho are constant over the area
6. Bernoulli’s Equation
The Bernoulli equation states that,
where
- points 1 and 2 lie on a streamline,
- the fluid has constant density,
- the flow is steady, and
- there is no friction.
7. Streamlines and Streamtubes
A streamline is a line that is tangential to the instantaneous velocity direction (velocity is a vector, and it has a magnitude and a direction). To visualize this in a flow, we could imagine the motion of a small marked element of fluid. For example, we could mark a drop of water with fluorescent dye and illuminate it using a laser so that it fluoresces.
8. Flows With Friction
If there are no shear stresses present, there is no fluid deformation, and the behavior of a fluid is described by the bulk modulus relating the pressure and the compression strain. In the presence of a shear stress, however, the shear angle will grow indefinitely if the shear stress is maintained. The shear stress is not related to the magnitude of the shear angle, as in solids, but to the rate at which the angle is changing. For many fluids, the relationship is linear, so that
where the coefficient of proportionality is called the dynamic viscosity of the fluid, or simply the fluid viscosity, and it is a material property of the fluid. Fluids which obey this linear relationship between stress and strain rate are called Newtonian fluids. Most common fluids are Newtonian, including air and water over very wide ranges of pressures and temperatures.
9. Transition and Turbulence
This section was adapted from The Engine and the Atmosphere: An Introduction to Engineering by Z. Warhaft, Cambridge University Press, 1997.
How many times a day do we turn on a faucet? Do it now. First very slowly, and you will see glassy, orderly flow. If there is no wind or other disturbance, nothing will change. This is called laminar flow. A photo taken now will be identical to one taken half an hour later. Such a flow is deterministic; information about its future behavior is completely determined by specification of the flow at an earlier time.
10. Separation
An important observation is that fluid flow is always irreversible. It is irreversible because fluids have viscosity, and whenever velocity gradients appear in the flow there will be friction and energy dissipation due to viscous stresses. The flow can be reversible only if there are no velocity gradients anywhere, since that is the only condition under which there is no friction. However, when frictional effects are small, a flow may be approximately reversible. For instance, in the flow through a large duct where the boundary layers are very thin, viscous effects are confined to a rather small region, and the fluid friction may sometimes be neglected. This is not the case for most practical flows. In most pipe and duct flows, for example, the velocity gradients extend over the entire cross-section and frictional stresses are all important. Even if the boundary layers are thin to begin with, they can thicken rapidly under some circumstances, and the flow can separate,
11. Drag of Blunt and Streamlined Bodies
A body moving through a fluid experiences a drag force, which is usually divided into two components: frictional drag, and pressure drag. Frictional drag comes from friction between the fluid and the surfaces over which it is flowing. This friction is associated with the development of boundary layers, and it scales with Reynolds number as we have seen above. Pressure drag comes from the eddying motions that are set up in the fluid by the passage of the body. This drag is associated with the formation of a wake, which can be readily seen behind a passing boat, and it is usually less sensitive to Reynolds number than the frictional drag.
12. Drafting
Another interesting phenomenon associated with cars and cyclists is the phenomenon of drafting or “tailgating.” It is well-known that the air resistance of a car following another in close proximity is reduced, and the leading car acts as a shield for the trailing car. The resulting flowfield is sketched in figure 12.1. The data shown in figure 12.2 indicate a significant decrease in drag coefficient for the trailing car when the separation is less than about one car length. Interestingly, the drag coefficient of the leading car is decreased by an even greater margin, suggesting that under race conditions both cars will travel faster in tandem than they could by themselves.
Figure 12.1-Left: Schematic description of the aerodynamic interaction during drafting. From Race Car Aerodynamics, J. Katz, Robert Bentley Publishers, 1995.
13. Golf Balls, Cricket Balls, and Tennis Balls
Tripping the boundary layer to reduce drag on spheres is widely used in sports. For example, it is the reason why golf balls are dimpled. The dimples act like a very effective trip wire, and the reduction in drag due to the delayed separation allows the ball to travel further for the same effort. A driver shot in golf can easily make a golf ball carry 250 yards, but the same shot using a smooth ball will only carry about 100 yards. Similarly, a tennis ball has a textured surface with a convoluted seam, much like a baseball. Figure 13.1 shows how effective different degrees of roughness can be in reducing the drag on a sphere.
14. Lift and Stall
We can see the role played by friction drag (sometimes called viscous drag) and pressure drag (sometimes called form drag or profile drag) by considering an airfoil at different angles of attack.
At small angles of attack, the boundary layers on the top and bottom surface experience only mild pressure gradients, and they remain attached along almost the entire chord length. The wake is very small, and the drag is dominated by the viscous friction inside the boundary layers. However, as the angle of attack increases, the pressure gradients on the airfoil increase in magnitude. In particular, the adverse pressure gradient on the top rear portion of the airfoil may become sufficiently strong to produce a separated flow.
15. Useful Links
- A good place to start is The Engineer and the Bicycle.
- Check out Yahoo Cycling.
- For a discussion of the science of bicycling, see Exploratorium Cycling.
- For a discussion of the aerodynamics of bicycling, visit Exploratorium Cycling Aerodynamics.
- For a particularly good site on aerodynamics, see Articles by Rainer Pivit.
- For more on aerodynamics, see M. S. Cramer’s wonderful site.
- For more on aerodynamics, visit Fit Werx’s site: lots of useful information.
- For a focus on aerodynamic handlebars, check out Lucas Pereira’s page.
- Here is a short article on aerodynamics HERE.
- Another short article on aerodynamics HERE.
- For a video on bicycle design, see Bicycle Design.
- The aerodynamics of the Cervelo P3SL
- Here is some scholarly work on the aerodynamic drag of mountain bike tyres.
- Some other interesting links in the area of fluid dynamics can be found HERE.
- For general links to pages on fluid flow, check out eFluids Links to Links page.
