Drag coefficient

Encyclopedia

In fluid dynamics

, the

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.

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

. The drag coefficient of a complete structure such as an aircraft also includes the effects of interference drag.

where: is the drag force

, which is by definition the force component in the direction of the flow velocity, is the mass density of the fluid, is the speed

of the object relative to the fluid, and 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 (note this is not the surface area = ).

For airfoil

s, 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.

Airship

s 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.

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.

. Speed, kinematic viscosity and a characteristic length scale

of the object are incorporated into a dimensionless quantity called the Reynolds number or is thus a function of In compressible flow, the speed of sound is relevant and is also a function of Mach number

For a certain body shape the drag coefficient only depends on the Reynolds number Mach number and the direction of the flow. For low Mach number the drag coefficient is independent of Mach number. Also the variation with Reynolds number 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 can often be treated as a constant.

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

contribute to larger Reynolds numbers.

For other objects, such as small particles, one can no longer consider that the drag coefficient is constant, but certainly is a function of Reynolds number.

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 is proportional to instead of for a sphere this is known as Stokes law

. Reynolds number will be low for small objects, low velocities, and high viscosity fluids.

A 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 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 of a real square flat plate perpendicular to the flow is often given as 1.17. Flow patterns and therefore for some shapes can change with the Reynolds number and the roughness of the surfaces.

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, the

**drag coefficient**(commonly denoted as:*c*,_{d}*c*or_{x}*c*) is a dimensionless quantity_{w}Dimensionless quantity

In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...

that is used to quantify the drag

Drag (physics)

In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

or resistance of an object in a fluid environment such as air or water. It is used in the drag equation

Drag equation

In fluid dynamics, the drag equation is a practical formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid....

, where a lower drag coefficient indicates the object will have less aerodynamic

Aerodynamics

Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...

or hydrodynamic drag. The drag coefficient is always associated with a particular surface area.

The drag coefficient of any object comprises the effects of the two basic contributors to fluid dynamic

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

drag: skin friction and form drag. The drag coefficient of a lifting airfoil

Airfoil

An airfoil or aerofoil is the shape of a wing or blade or sail as seen in cross-section....

or hydrofoil

Hydrofoil

A hydrofoil is a foil which operates in water. They are similar in appearance and purpose to airfoils.Hydrofoils can be artificial, such as the rudder or keel on a boat, the diving planes on a submarine, a surfboard fin, or occur naturally, as with fish fins, the flippers of aquatic mammals, the...

also includes the effects of lift-induced drag

Lift-induced drag

In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is a drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings or a lifting body redirecting air to cause lift and also in cars...

. The drag coefficient of a complete structure such as an aircraft also includes the effects of interference drag.

## Definition

The drag coefficient is defined as:where: is the drag force

Drag (physics)

In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

, which is by definition the force component in the direction of the flow velocity, is the mass density of the fluid, is the speed

Speed

In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...

of the object relative to the fluid, and is the reference area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

.

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 (note this is not the surface area = ).

For airfoil

Airfoil

An airfoil or aerofoil is the shape of a wing or blade or sail as seen in cross-section....

s, the reference area is the planform

Planform

In aviation, a planform is the shape and layout of a fixed-wing aircraft's fuselage and wing. Of all the myriad planforms used, they can typically be grouped into those used for low-speed flight, found on general aviation aircraft, and those used for high-speed flight, found on many military...

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.

Airship

Airship

An airship or dirigible is a type of aerostat or "lighter-than-air aircraft" that can be steered and propelled through the air using rudders and propellers or other thrust mechanisms...

s and some bodies of revolution

Solid of revolution

In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line that lies on the same plane....

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

The drag equation:is essentially a statement that the drag

Drag (physics)

In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

force

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

on any object is proportional to the density of the fluid and proportional to the square of the relative speed

Speed

In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...

between the object and the fluid.

*C*is not a constant but varies as a function of speed, flow direction, object position, object size, fluid density and fluid viscosity_{d}Viscosity

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

. Speed, kinematic viscosity and a characteristic length scale

Length scale

In physics, length scale is a particular length or distance determined with the precision of one order of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other and are said to decouple...

of the object are incorporated into a dimensionless quantity called the Reynolds number or is thus a function of In compressible flow, the speed of sound is relevant and is also a function of Mach number

Mach number

Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...

For a certain body shape the drag coefficient only depends on the Reynolds number Mach number and the direction of the flow. For low Mach number the drag coefficient is independent of Mach number. Also the variation with Reynolds number 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 can often be treated as a constant.

For a streamlined body to achieve a low drag coefficient the boundary layer

Boundary layer

In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

around the body must remain attached to the surface of the body for as long as possible, causing the wake

Wake

A wake is the region of recirculating flow immediately behind a moving or stationary solid body, caused by the flow of surrounding fluid around the body.-Fluid dynamics:...

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 viscositiesViscosity

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

contribute to larger Reynolds numbers.

For other objects, such as small particles, one can no longer consider that the drag coefficient is constant, but certainly is a function of Reynolds number.

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 is proportional to instead of for a sphere this is known as Stokes law

Stokes law

Stokes' law can refer to:*Stokes' law for friction force*Stokes' law law describing attenuation of sound in Newtonian liquidsFor integration, see Stokes' theorem....

. Reynolds number will be low for small objects, low velocities, and high viscosity fluids.

A 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

Stagnation pressure

In fluid dynamics, stagnation pressure is the static pressure at a stagnation point in a fluid flow.At a stagnation point the fluid velocity is zero and all kinetic energy has been converted into pressure energy . Stagnation pressure is equal to the sum of the free-stream dynamic pressure and...

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 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 of a real square flat plate perpendicular to the flow is often given as 1.17. Flow patterns and therefore for some shapes can change with the Reynolds number and the roughness of the surfaces.

### General

In general, 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 that varies from high values for laminar flow to 0.47 for turbulent flow. c_{d} |
Item |
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0.7 | a typical bicycle Bicycle A bicycle, also known as a bike, pushbike or cycle, is a human-powered, pedal-driven, single-track vehicle, having two wheels attached to a frame, one behind the other. A person who rides a bicycle is called a cyclist, or bicyclist.... plus cyclist |

0.48 | rough sphere (Re = ) |

0.1 | smooth sphere () |

0.001 | laminar flat plate parallel to the flow () |

0.005 | turbulent flat plate parallel to the flow () |

0.195 | General Motors EV1 1996 |

0.24 | Mercedes-Benz E-Class Coupé 2010 |

0.25 | 3rd Generation Toyota Prius Toyota Prius (XW30) The Toyota Prius is a mid-size hatchback that has been produced by Toyota. Toyota debuted the third generation Prius at the January 2009 North American International Auto Show, and sales began in Japan on May 18, 2009. Replacing the XW20 series, the XW30 represents the third generation of the... |

0.295 | bullet (not ogive Ogive An ogive is the roundly tapered end of a two-dimensional or three-dimensional object.-Applied physical science and engineering:In ballistics or aerodynamics, an ogive is a pointed, curved surface mainly used to form the approximately streamlined nose of a bullet or other projectile.The traditional... , at subsonic velocity) |

1.28 | flat plate perpendicular to flow (3D) |

1.0–1.1 | skier Skiing Skiing is a recreational activity using skis as equipment for traveling over snow. Skis are used in conjunction with boots that connect to the ski with use of a binding.... |

1.0–1.3 | wires and cables |

1.0–1.3 | man (upright position) |

1.1-1.3 | ski jumper |

1.3–1.5 | Empire State Building Empire State Building The Empire State Building is a 102-story landmark skyscraper and American cultural icon in New York City at the intersection of Fifth Avenue and West 34th Street. It has a roof height of 1,250 feet , and with its antenna spire included, it stands a total of 1,454 ft high. Its name is derived... |

1.8–2.0 | Eiffel Tower Eiffel Tower The Eiffel Tower is a puddle iron lattice tower located on the Champ de Mars in Paris. Built in 1889, it has become both a global icon of France and one of the most recognizable structures in the world... |

1.98–2.05 | flat plate perpendicular to flow (2D) |

2.1 | a smooth brick |

### Aircraft

As noted above, aircraft use wing area as the reference area when computing while automobiles (and many other objects) use frontal cross sectional area; thus, coefficients are**not**directly comparable between these classes of vehicles. c_{d} | Aircraft model |
---|---|

0.021 | F-4 Phantom II F-4 Phantom II The McDonnell Douglas F-4 Phantom II is a tandem two-seat, twin-engined, all-weather, long-range supersonic jet interceptor fighter/fighter-bomber originally developed for the United States Navy by McDonnell Aircraft. It first entered service in 1960 with the U.S. Navy. Proving highly adaptable,... (subsonic) |

0.022 | Learjet 24 Learjet 24 |-See also:-References:* Taylor, John W. R. Jane's All The World's Aircraft 1976–77. London:Jane's Yearbooks, 1976. ISBN 0-354-00538-3.-External links:**... |

0.024 | Boeing 787 Boeing 787 The Boeing 787 Dreamliner is a long-range, mid-size wide-body, twin-engine jet airliner developed by Boeing Commercial Airplanes. It seats 210 to 290 passengers, depending on the variant. Boeing states that it is the company's most fuel-efficient airliner and the world's first major airliner to use... |

0.027 | Cessna 172 Cessna 172 The Cessna 172 Skyhawk is a four-seat, single-engine, high-wing fixed-wing aircraft. First flown in 1955 and still in production, more Cessna 172s have been built than any other aircraft.-Design and development:... /182 Cessna 182 The Cessna 182 Skylane is an American four-seat, single-engine, light airplane, built by Cessna of Wichita, Kansas. It has the option of adding two child seats, installed in the baggage area.... |

0.027 | Cessna 310 Cessna 310 The Cessna 310 is an American six-seat, low-wing, twin-engined monoplane that was produced by Cessna between 1954 and 1980. It was the first twin-engined aircraft that Cessna put into production after World War II.-Development:... |

0.031 | Boeing 747 Boeing 747 The Boeing 747 is a wide-body commercial airliner and cargo transport, often referred to by its original nickname, Jumbo Jet, or Queen of the Skies. It is among the world's most recognizable aircraft, and was the first wide-body ever produced... |

0.044 | F-4 Phantom II (supersonic) |

0.048 | F-104 Starfighter F-104 Starfighter The Lockheed F-104 Starfighter is a single-engine, high-performance, supersonic interceptor aircraft originally developed for the United States Air Force by Lockheed. One of the Century Series of aircraft, it served with the USAF from 1958 until 1969, and continued with Air National Guard units... |

0.095 | X-15 (Not confirmed) |

## See also

- Automotive aerodynamicsAutomotive aerodynamicsAutomotive aerodynamics is the study of the aerodynamics of road vehicles. The main concerns of automotive aerodynamics are reducing drag , reducing wind noise, minimizing noise emission, and preventing undesired lift forces and other causes of aerodynamic instability at high speeds...
- Automobile drag coefficientAutomobile drag coefficientThe drag coefficient is a common metric in automotive design pertaining to aerodynamic effects. As aerodynamic drag increases as the square of speed, a low value is preferable to a high one...
- Drag crisisDrag crisisIn fluid dynamics, drag crisis is a phenomenon in which drag coefficient drops off suddenly as Reynolds number increases. This has been well studied for round bodies like spheres and cylinders. The drag coefficient of a sphere will change rapidly from about 0.5 to 0.2 at a Reynolds number in the...
- Zero-lift drag coefficient