Drag coefficient

In fluid dynamics, the drag coefficient (commonly denoted as: c_d, c_x or c_w) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment such as air or water. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a particular surface area.^[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 a lifting airfoil or hydrofoil also includes the effects of lift-induced drag.^[2][3] The drag coefficient of a complete structure such as an aircraft also includes the effects of interference drag.^[4][5]

Definition

The drag coefficient $c_\mathrm d\,$ is defined as:

$c_\mathrm d = \dfrac{2 F_\mathrm d}{\rho v^2 A}\, ,$

where:

$F_\mathrm d\,$ is the drag force, which is by definition the force component in the direction of the flow velocity,^[6]

$\rho\,$ is the mass density of the fluid,^[7]

$v\,$ is the speed of the object relative to the fluid, and

$A\,$ is the reference area.

The reference area depends on what type of drag coefficient is being measured. For automobiles and many other objects, the reference area is the projected frontal area of the vehicle. This may not necessarily be the cross sectional area of the vehicle, depending on where the cross section is taken. For example, for a sphere $A = \pi r^2\,$ (note this is not the surface area = $\!\ 4 \pi r^2$).

For airfoils, the reference area is the planform area. Since this tends to be a rather large area compared to the projected frontal area, the resulting drag coefficients tend to be low: much lower than for a car with the same drag and frontal area, and at the same speed.

Airships and some bodies of revolution use the volumetric drag coefficient, in which the reference area is the square of the cube root of the airship volume. Submerged streamlined bodies use the wetted surface area.

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 unstreamlined objects can be 1 or more, for streamlined objects much less.

Background

Flow around a plate, showing stagnation.

Main article: Drag equation

The drag equation:

$\vec F_\mathrm d = -\frac{\rho v^2 c_\mathrm d A}{2} \hat v$

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 speed between the object and the fluid.

C_d is not a constant but varies as a function of speed, flow direction, object position, object size, fluid density and fluid viscosity. Speed, kinematic viscosity and a characteristic length scale of the object are incorporated into a dimensionless quantity called the Reynolds number or $Re\, .$ $c_\mathrm d\,$ is thus a function of $Re\, .$ In compressible flow, the speed of sound is relevant and $c_\mathrm d\,$ is also a function of Mach number $Ma\, .$

For a certain body shape the drag coefficient $c_\mathrm d\,$ only depends on the Reynolds number $Re\, ,$ Mach number $Ma\,$ and the direction of the flow. For low Mach number $Ma\, ,$ the drag coefficient is independent of Mach number. Also the variation with Reynolds number $Re\,$ within a practical range of interest is usually small, while for cars at highway speed and aircraft at cruising speed the incoming flow direction is as well more-or-less the same. So the drag coefficient $c_\mathrm d\,$ can often be treated as a constant.^[8]

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 high form drag results in a broad wake. 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.^[9]

For other objects, such as small particles, one can no longer consider that the drag coefficient $c_\mathrm d\,$ is constant, but certainly is a function of Reynolds number.^[10][11][12] At a low Reynolds number, the flow 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. At very low Reynolds numbers, without flow separation, the drag force $F_\mathrm d\,$ is proportional to $v\,$ instead of $v^2\, ;$ for a sphere this is known as Stokes law. Reynolds number will be low for small objects, low velocities, and high viscosity fluids.^[9]

A $c_\mathrm d\,$ 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. Only considering the front size, the $c_\mathrm d\,$ of a real flat plate would be less than 1; except that there will be suction on the back side: a negative pressure (relative to ambient). The overall $c_\mathrm d\,$ of a real square flat plate perpendicular to the flow is often given as 1.17. Flow patterns and therefore $c_\mathrm d\,$ for some shapes can change with the Reynolds number and the roughness of the surfaces.

Drag coefficient c_d examples

General

In general, $c_\mathrm d\,$ is not an absolute constant for a given body shape. It varies with the speed of airflow (or more generally with Reynolds number). A smooth sphere, for example, has a $c_\mathrm d\,$ that varies from high values for laminar flow to 0.47 for turbulent flow.

Shapes

cd	Item
0.7	a typical bicycle plus cyclist[citation needed]
0.48	rough sphere (Re = $\!\ 10^6$)
0.1	smooth sphere ($\!\ Re = 10^6$)
0.001	laminar flat plate parallel to the flow ($\!\ Re < 10^5$)
0.005	turbulent flat plate parallel to the flow ($\!\ Re > 10^5$)
0.195	General Motors EV1 1996[citation needed]
0.24	Mercedes-Benz E-Class Coupé 2010[13]
0.25	3rd Generation Toyota Prius
0.295	bullet (not ogive, at subsonic velocity)
1.28	flat plate perpendicular to flow (3D)
1.0–1.1	skier
1.0–1.3	wires and cables
1.0–1.3	man (upright position)
1.1-1.3	ski jumper[14]
1.3–1.5	Empire State Building
1.8–2.0	Eiffel Tower
1.98–2.05	flat plate perpendicular to flow (2D)
2.1	a smooth brick[citation needed]

Aircraft

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

Aircraft^[15]

cd	Aircraft model
0.021	F-4 Phantom II (subsonic)
0.022	Learjet 24
0.024	Boeing 787[16]
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)

Main article: Automobile drag coefficient

See also

• Automotive aerodynamics
• Automobile drag coefficient
• Drag crisis
• Zero-lift drag coefficient

References

• Clancy, L. J. (1975): Aerodynamics. Pitman Publishing Limited, London, ISBN 0-273-01120-0
• Abbott, Ira H., and Von Doenhoff, Albert E. (1959): Theory of Wing Sections. Dover Publications Inc., New York, Standard Book Number 486-60586-8
• Hoerner, S. F. (1965): Fluid-Dynamic Drag. Hoerner Fluid Dynamics, Brick Town, N. J., USA

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". Wright.nasa.gov. 2010-03-25. http://wright.nasa.gov/airplane/drageq.html. Retrieved 2010-12-07. 
5. ^ Clancy, L. J.: Aerodynamics. Section 11.17
6. ^ See lift force and vortex induced vibration for a possible force components transverse to the flow direction.
7. ^ Note that for the Earth's atmosphere, the air density can be found using the barometric formula. Air is 1.293 kg/m³ at 0 °C and 1 atmosphere
8. ^ Clancy, L. J.: Aerodynamics. Sections 4.15 and 5.4
9. ^ ^{a b} Clancy, L. J.: Aerodynamics. Section 4.17
10. ^ Clift R., Grace J. R., Weber M. E.: Bubbles, drops, and particles. Academic Press NY (1978).
11. ^ Briens C. L.: Powder Technology. 67, 1991, 87-91.
12. ^ Haider A., Levenspiel O.: Powder Technology. 58, 1989, 63-70.
13. ^ Carter, Marc (2009-02-17). "Geneva 09' Preview: 2010 Mercedes-Benz E-Class Coupe Unveiled". The Torque Report. http://www.thetorquereport.com/2009/02/geneva_09_preview_2010_mercede.html#more. Retrieved 2010-12-07. 
14. ^ "Drag Coefficient". Engineeringtoolbox.com. http://www.engineeringtoolbox.com/drag-coefficient-d_627.html. Retrieved 2010-12-07. 
15. ^ "Ask Us - Drag Coefficient & Lifting Line Theory". Aerospaceweb.org. 2004-07-11. http://www.aerospaceweb.org/question/aerodynamics/q0184.shtml. Retrieved 2010-12-07. 
16. ^ "Boeing 787 Dreamliner : Analysis". Lissys.demon.co.uk. 2006-06-21. http://www.lissys.demon.co.uk/samp1/index.html. Retrieved 2010-12-07.