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Drag coefficient

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The drag coefficient (Cd, Cx or Cw) is a dimensionless quantity that describes how streamlined an object is. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic drag. [1] The drag coefficient of any object comprises the effects of the two basic contributors to fluid dynamic drag: skin friction and form drag. The drag coefficient of an airfoil also includes the effects of induced drag.[2][3] The drag coefficient of a complete aircraft also includes the effects of interference drag.[4][5]

Coefficient of drag is a dimensionless number, meaning it is a ratio between two numbers in the same units; in this case the units are units of area. The reference area chosen for comparison depends on what type of drag coefficient is being measured. For airfoils, the reference area is the square of the chord of the airfoil, which can be easily related to wing area. Since this tends to be a rather large area, the resulting drag coefficients tend to be low. For automobiles and many other objects, the reference area is the frontal area of the vehicle (i.e., the cross-sectional area when viewed from ahead); this area tends to be small, giving a higher drag coefficient than an airfoil with the same drag. Airships and bodies of revolution use the volumetric drag coefficient, in which the reference area is the square of the cube root of the airship volume.

Two objects having the same reference area moving at the same speed through a fluid will experience a drag force proportional to their respective drag coefficients. Coefficients for rough unstreamlined objects can be 1 or more, for streamlined objects much less.

\mathbf{F}_d= {1 \over 2} \rho \mathbf{v}^2 C_d A     explanation of terms on drag equation page.
Flow around a plate, showing stagnation.

A is the reference area, usually projected frontal area. For example, for a sphere A = πr2, (i.e., not the surface area.)

The drag equation is essentially a statement that the drag force on any object is proportional to the density of the fluid, and proportional to the square of the relative velocity between the object and the fluid. The drag coefficient of an object varies depending on its orientation to the vector representing the relative velocity between the object and the fluid. The drag coefficient does not vary with fluid density or relative velocity, providing the relative velocity is small compared with the speed of sound in the fluid, and providing the Reynolds number of the flow does not vary significantly.[6]

For a streamlined body to achieve a low drag coefficient the boundary layer around the body must remain attached to the surface of the body for as long as possible, causing the wake to be narrow. A broad wake results in high form drag. The boundary layer will remain attached longer if it is turbulent than if it is laminar. The boundary layer will transition from laminar to turbulent providing the Reynolds number of the flow around the body is high enough. Larger velocities, larger objects, and lower viscosities contribute to larger Reynolds numbers.[7]

At a low Reynolds number, the boundary layer around the object does not transition to turbulent but remains laminar, even up to the point at which it separates from the surface of the object. Cd is no longer constant but varies with velocity, and Fd is proportional to v instead of v2. Reynolds number will be low for small objects, low velocities, and high viscosity fluids.[8]

A Cd equal to 1 would be obtained in a case where all of the fluid approaching the object is brought to rest, building up stagnation pressure over the whole front surface. The top figure shows a flat plate with the fluid coming from the right and stopping at the plate. The graph to the left of it shows equal pressure across the surface. In a real flat plate the fluid must turn around the sides, and full stagnation pressure is found only at the center, dropping off toward the edges as in the lower figure and graph. The Cd of a real flat plate would be less than 1, except that there will be a negative pressure (relative to ambient) on the back surface. The overall Cd of a real square flat plate is often given as 1.17. Flow patterns and therefore Cd for some shapes can change with the Reynolds number and the roughness of the surfaces.

Contents

[edit] More CdA examples

Skydiver's CdA (in m²) (at 300 m)
Terminal
velocity		 Mass
60 kg 70 kg 80 kg 90 kg 100 kg
45 m/s 0.487 0.569 0.650 0.731 0.812
50 m/s 0.395 0.461 0.526 0.592 0.658
55 m/s 0.326 0.381 0.435 0.489 0.544
60 m/s 0.274 0.320 0.365 0.411 0.457
65 m/s 0.234 0.272 0.311 0.350 0.389
70 m/s 0.201 0.235 0.269 0.302 0.336
75 m/s 0.175 0.205 0.234 0.263 0.292

This value is extremely useful as either the area or drag coefficient alone are not enough to be used in any equation. Sometimes it is not possible to get either value, but it might be possible to deduce it. For a skydiver example below, it is possible to deduce CdA from the mass of the diver and equipment and terminal velocity. Skydiver CdA examples are in both  ft² and m² units.

[edit] Cd examples

As noted above, aircraft use wing area as the reference area when computing Cd, while automobiles (and many other objects) use frontal cross sectional area; thus, coefficients are not directly comparable between these classes of vehicles.

Aircraft[9]
Cd Aircraft model
0.021 F-4 Phantom II (subsonic)
0.022 Learjet 24
0.024 Boeing 787 [10]
0.027 Cessna 172/182
0.027 Cessna 310
0.031 Boeing 747
0.044 F-4 Phantom II (supersonic)
0.048 F-104 Starfighter
0.095 X-15 (Not confirmed)
Other shapes[11]
Cd Item
0.001 laminar flat plate parallel to the flow (Re = 106)
0.005 turbulent flat plate parallel to the flow (Re = 106)
0.1 smooth sphere (Re = 106)
0.11 Aptera Typ-1
0.18 Mercedes-Benz T80
0.19 GM EV1
0.25 Audi A2 1.2TDI, Honda Insight
0.26 1989 Opel Calibra, 2008 Toyota Prius
0.29 1996 Audi A8, 2004 Honda Accord
0.295 bullet
0.4 rough sphere (Re = 106)
0.57 2003 Hummer H2
0.9 a typical bicycle plus cyclist
1.0-1.3 man (upright position)
1.0-1.1 skier
1.0-1.3 wires and cables
1.28 flat plate perpendicular to flow
1.3-1.5 Empire State Building
1.8-2.0 Eiffel Tower
2.1 a smooth brick

Image:14ilf1l.gif

[edit] See also

[edit] References

  • Clancy, L.J. (1975), Aerodynamics, Pitman Publishing Limited, London ISBN 0 273 01120 0
  • Abbott, Ira H., and Von Doenhoff, Albert E., Theory of Wing Sections, Dover Publications Inc., New York, Standard Book Number 486-60586-8

[edit] Notes

  1. ^ McCormick, Barnes W. (1979), Aerodynamics, Aeronautics, and Flight Mechanics, p.24, John Wiley & Sons, Inc., New York ISBN 0-471-03032-5
  2. ^ Clancy, L.J., Aerodynamics, Section 5.18
  3. ^ Abbott, Ira H., and Von Doenhoff, Albert E., Theory of Wing Sections, Sections 1.2 and 1.3
  4. ^ NASA’s Modern Drag Equation
  5. ^ Clancy, L.J., Aerodynamics, Section 11.17
  6. ^ Clancy, L.J., Aerodynamics, Sections 4.15 and 5.4
  7. ^ Clancy, L.J., Aerodynamics, Section 4.17
  8. ^ Clancy, L.J., Aerodynamics, Section 4.17
  9. ^ http://www.aerospaceweb.org/question/aerodynamics/q0184.shtml
  10. ^ http://www.lissys.demon.co.uk/samp1/index.html
  11. ^ http://aerodyn.org/Drag/tables.html

[edit] External links


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