Drag coefficient

Drag Coefficient

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Drag coefficient

The drag coefficient (C_d, C_x or C_w) is a dimensionless quantity

Dimensionless quantity

In dimensional analysis, a dimensionless quantity is a quantity without any physical units and thus a pure number. Such a number is typically defined as a product or ratio of quantity which do have units, in such a way that all the units cancel out....
which 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

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 a fluid that it is moving through....
, 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....
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 the sub-discipline of fluid mechanics dealing with fluid flow — the natural science of fluids in motion....
drag: skin friction and form drag.

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The drag coefficient (C_d, C_x or C_w) is a dimensionless quantity

Dimensionless quantity

Drag equation

Aerodynamics

Fluid dynamics

In physics, fluid dynamics is the sub-discipline of fluid mechanics dealing with fluid flow — the natural science of fluids in motion....
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.An airfoil-shaped body moved through a fluid produces a force perpendicular to the motion called lift ....
or hydrofoil

Hydrofoil

A hydrofoil is a boat with wing-like airfoils mounted on struts below the hull . As the craft increases its speed the hydrofoils develop enough lift for the boat to become foilborne - i.e....
also includes the effects of induced drag. The drag coefficient of a complete structure such as an aircraft also includes the effects of interference drag.

The dimensionless drag coefficient is a ratio between two numbers in the same units of area. The reference area chosen for comparison depends on what type of drag coefficient is being measured. For airfoils

Airfoil

Chord (aircraft)

In reference to aircraft, chord refers to the distance between the leading edge and trailing edge of a wing, horizontal stabilizer or vertical stabilizer, measured in the direction of the normal airflow....
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

Airship

An airship or dirigible is a aerostat that can be steered and propelled through the air using rudders and propellers or other thrust. Unlike other aerodynamics aircraft such as fixed-wing aircraft and helicopters, which produce lift by moving a wing, or airfoil, through the air, aerostatic aircraft, such as airships and Balloon , stay...
and bodies of revolution

Solid of revolution

In mathematics, engineering, and manufacturing, a solid of revolution is a Shape 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 rough unstreamlined objects can be 1 or more, for streamlined objects much less.

explanation of terms on drag equation page

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 a fluid that it is moving through....
.

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

The drag equation is essentially a statement that the drag

Drag (physics)

The term drag is widely used in Physics and Engineering and is central to the field of fluid dynamics. "Drag" refers to forces that oppose the motion of a solid object through a fluid ....
force

Force

In physics, a force is that which can cause an object with mass to change its velocity. Force has both Euclidean_vector#Length of a vector and Direction , making it a Vector quantity....
on any object is proportional to the density of the fluid, and proportional to the square of the relative velocity

Velocity

In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
between the object and the fluid. The drag coefficient of an object varies depending on its orientation to the vector

Vector

Vector may refer to:...
representing the relative velocity between the object and the fluid.

C_d is not a constant but varies as a function of speed, object length, fluid density and fluid viscosity. These are generally combined to a dimensionless quantity called the Reynolds number

Reynolds number

In fluid mechanics and heat transfer, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscosity forces and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions....
or Re. C_d is thus a function of Re. In compressible flow, the speed of sound is relevant and C_d is also a function of Mach number

Mach number

Mach number is the speed of an object moving through air, or any fluid substance, divided by the speed of sound as it is in that substance. It is commonly used to represent an object's speed, when it is travelling at the speed of sound....
. Provided Re is the same and the flow is subsonic and steady, C_d is the same regardless of fluid or variation of other variables. Although Re and hence C_d can vary greatly, the variation within a practical range of interest is usually small, so that C_d is often treated as a constant. The drag coefficient can be considered constant for objects moving at high Re, e.g. a car at highway speed or an airplane at cruising speed. For other objects, such as small particles, one can no longer consider that the drag coefficient is 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. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal heat, moisture or momentum transfer to or from the surface....
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 turbulence immediately to the rear of a solid body caused by the flow of air or water around the body.In fluid dynamics, a wake is the region of separated flow downstream of a solid body moving relative to the fluid, caused by the flow of liquid around the body....
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

Reynolds number

In fluid mechanics and heat transfer, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscosity forces and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions....
of the flow around the body is high enough. Larger velocities, larger objects, and lower viscosities

Viscosity

Viscosity is a measure of the Drag of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness"....
contribute to larger Reynolds numbers.

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. is no longer constant but varies with velocity, and is proportional to instead of . Reynolds number will be low for small objects, low velocities, and high viscosity fluids.

A C_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

Stagnation pressure

In fluid dynamics, stagnation pressure is the pressure at a stagnation point in a fluid flow, where the kinetic energy is converted into pressure energy....
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 C_d 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 C_d of a real square flat plate is often given as 1.17. Flow patterns and therefore C_d for some shapes can change with the Reynolds number and the roughness of the surfaces.

More C_dA examples

Skydiver's C_dA (in m²) (at 300 m)
Terminal velocity	Mass
Terminal velocity	60 kg Kilogram The kilogram or kilogrammeThe spelling kilogram is used by the International Committee for Weights and Measures and the U.S....	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 C_dA from the mass of the diver and equipment and terminal velocity. Skydiver C_dA examples are in both ft² and m² units.

C_d examples

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

In general, C_d is not an absolute constant, for a given shape body. It varies with the speed of airflow (or more generally with Reynolds number). A smooth sphere, for example, has a C_d that varies from about 0.47 for laminar (slow) flow to 0.1 for separated (faster) flow.

Aircraft
C_d	Aircraft model
0.021	F-4 Phantom II F-4 Phantom II The McDonnell Douglas F-4 Phantom II is a two-seat, twin-engined, all-weather, long-range supersonic interceptor jet fighter/fighter-bomber originally developed for the United States Navy by McDonnell Aircraft.... (subsonic)
0.022	Learjet 24 Learjet 24 The Learjet 24 is an United States six to eight seat twin-engined, high speed business jet. Manufactured by Learjet as the successor to the Learjet 23....
0.024	Boeing 787 Boeing 787 The Boeing 787 Dreamliner is a mid-sized, Wide-body aircraft, twinjet jet airliner currently under development by Boeing Commercial Airplanes....
0.027	Cessna 172 Cessna 172 The Cessna 172 Skyhawk is a four-seat, single-engine, high-wing fixed-wing aircraft.More Cessna 172s have been built than List of most produced aircraft.... /182 Cessna 182 The Cessna 182, marketed under the name Skylane, is a four-seat, single-engine, light Fixed-wing aircraft. It has the option of adding two child seats, installed in the baggage area....
0.027	Cessna 310 Cessna 310 The Cessna 310 was the first twin-engine aircraft design from Cessna to enter production after World War II....
0.031	Boeing 747 Boeing 747 The Boeing 747 is a wide-body aircraft commercial airliner, often referred to by the nickname "Jumbo Jet". It is among the world's most recognizable aircraft, and was the first widebody ever produced....
0.044	F-4 Phantom II (supersonic)
0.048	F-104 Starfighter F-104 Starfighter The Lockheed F-104 Starfighter was an United States single-engined, high-performance, supersonic interceptor aircraft that served with the United States Air Force from 1958 until 1967....
0.095	X-15 (Not confirmed)

Other shapes
C_d	Item
2.1	a smooth brick
0.9	a typical bicycle Bicycle The bicycle, bike, or cycle is a pedal-driven, human-powered transport with two bicycle wheel attached to a bicycle frame, one behind the other.... plus cyclist
0.4	rough sphere (Re Reynolds number In fluid mechanics and heat transfer, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscosity forces and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions.... = 10⁶)
0.1	smooth sphere (Re = 10⁶)
0.001	laminar flat plate parallel to the flow (Re = 10⁶)
0.005	turbulent flat plate parallel to the flow (Re = 10⁶)
0.295	bullet (not ogive Ogive An ogive is a curved shape, figure, or feature.... , at subsonic velocity)
1.0-1.3	man (upright position)
1.28	flat plate perpendicular to flow
1.0-1.1	skier Skiing Snow skiing is a group of sports using skis as primary equipment. Skis are used in conjunction with ski boots that connect to the ski with use of a ski bindings....
1.0-1.3	wires and cables
1.3-1.5	Empire State Building Empire State Building The Empire State Building is a 102-story Art Deco skyscraper in New York City at the intersection of Fifth Avenue and West 34th Street. Its name is derived from the List of U.S....
1.8-2.0	Eiffel Tower Eiffel Tower The Eiffel Tower is an Puddle iron tower built on the Champ de Mars beside the Seine River in Paris. The tower has become a global Cultural icon of France and is one of the most recognizable structures in the world....

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