# Drag coefficient

In fluid dynamics, the drag coefficient (commonly denoted as: ${\displaystyle \scriptstyle C_{\mathrm {d} }\,}$, ${\displaystyle \scriptstyle C_{\mathrm {x} }\,}$ or ${\displaystyle \scriptstyle C_{\mathrm {w} }\,}$) 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 in which 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 ${\displaystyle c_{\mathrm {d} }\,}$ is defined as

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

where:

${\displaystyle F_{\mathrm {d} }\,}$ is the drag force, which is by definition the force component in the direction of the flow velocity,[6]
${\displaystyle \rho \,}$ is the mass density of the fluid,[7]
${\displaystyle u\,}$ is the flow speed of the object relative to the fluid,
${\displaystyle A\,}$ 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 ${\displaystyle A=\pi r^{2}\,}$ (note this is not the surface area = ${\displaystyle \!\ 4\pi r^{2}}$).

For airfoils, the reference area is the nominal wing area. Since this tends to be large compared to the frontal area, the resulting drag coefficients tend to be low, much lower than for a car with the same drag, frontal area, and 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 (volume to the two-thirds power). 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.

It has been demonstrated that drag coefficient ${\displaystyle c_{d}}$ is a function of Bejan number (${\displaystyle Be}$), Reynolds number (${\displaystyle Re}$) and the ratio between wet area ${\displaystyle A_{w}}$and front area ${\displaystyle A_{f}}$ [8] :

${\displaystyle c_{d}={\frac {A_{w}}{A_{f}}}{\frac {Be}{Re_{L}^{2}}}}$

where ${\displaystyle Re_{L}}$is the Reynold Number related to fluid path length L.

## Background

The drag equation

${\displaystyle F_{d}\,=\,{\tfrac {1}{2}}\,\rho \,u^{2}\,c_{d}\,A}$

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 flow speed between the object and the fluid.

Cd is not a constant but varies as a function of flow 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 ${\displaystyle \scriptstyle Re}$. ${\displaystyle \scriptstyle C_{\mathrm {d} }\,}$ is thus a function of ${\displaystyle \scriptstyle Re}$. In a compressible flow, the speed of sound is relevant, and ${\displaystyle \scriptstyle C_{\mathrm {d} }\,}$ is also a function of Mach number ${\displaystyle \scriptstyle Ma}$.

For certain body shapes, the drag coefficient ${\displaystyle \scriptstyle C_{\mathrm {d} }\,}$ only depends on the Reynolds number ${\displaystyle \scriptstyle Re\,}$, Mach number ${\displaystyle \scriptstyle Ma\,}$ and the direction of the flow. For low Mach number ${\displaystyle \scriptstyle Ma}$, the drag coefficient is independent of Mach number. Also, the variation with Reynolds number ${\displaystyle \scriptstyle Re\,}$ 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 also more-or-less the same. Therefore, the drag coefficient ${\displaystyle \scriptstyle C_{\mathrm {d} }\,}$ can often be treated as a constant.[9]

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 if Reynolds number of the flow around the body is sufficiently great. Larger velocities, larger objects, and lower viscosities contribute to larger Reynolds numbers.[10]

For other objects, such as small particles, one can no longer consider that the drag coefficient ${\displaystyle \scriptstyle C_{\mathrm {d} }\,}$ is constant, but certainly is a function of Reynolds number.[11][12][13] 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 ${\displaystyle \scriptstyle F_{\mathrm {d} }\,}$ is proportional to ${\displaystyle \scriptstyle v\,}$ instead of ${\displaystyle \scriptstyle v^{2}\,}$; for a sphere this is known as Stokes law. The Reynolds number will be low for small objects, low velocities, and high viscosity fluids.[10]

A ${\displaystyle \scriptstyle C_{\mathrm {d} }\,}$ 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 side, the ${\displaystyle \scriptstyle C_{\mathrm {d} }\,}$ 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 ${\displaystyle \scriptstyle C_{\mathrm {d} }\,}$ of a real square flat plate perpendicular to the flow is often given as 1.17. Flow patterns and therefore ${\displaystyle \scriptstyle C_{\mathrm {d} }\,}$ for some shapes can change with the Reynolds number and the roughness of the surfaces.

## Drag coefficient cd examples

### General

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

cd Item[14]
0.001Laminar flat plate parallel to the flow (${\displaystyle \!\ Re<10^{6}}$)
0.005Turbulent flat plate parallel to the flow (${\displaystyle \!\ Re>10^{6}}$)
0.0512Ecorunner V, 2015 [15]
0.07Nuna 3
0.075Pac-car
0.076Milan SL (one of the fastest practical velomobiles) [16]
0.1Smooth sphere (${\displaystyle \!\ Re=10^{6}}$)
0.47Smooth sphere (${\displaystyle \!\ Re=10^{5}}$)
0.81 Triangular trapeze (45°)
0.9-1.7 Trapeze with triangular basis (45°)
0.14Fiat Turbina 1954[17]
0.15Schlörwagen 1939 [18]
0.18 Mercedes-Benz T80 1939
0.186-0.189Volkswagen XL1 2014
0.19Alfa Romeo B.A.T. 7 1954[19]
0.19General Motors EV1 1996[20]
0.212Tatra 77A 1935
0.23Tesla Model 3[21], Audi A4 B9 (2016)[22]
0.23Alfa Romeo Giulia Advanced Efficiency 2016[23]
0.24Tesla Model S[24]
0.24Hyundai Ioniq [25]
0.24Toyota Prius (4th Generation)[26]
0.25Toyota Prius (3rd Generation)
0.26BMW i8
0.26Nissan GT-R (2011-2014)
0.26Toyota Prius (2nd Generation)[27]
0.27BMW E39 5 Series (1995-2003, Germany)[28]
0.27Mercedes-Benz CLS-Class Type C257[29]
0.27Nissan GT-R (2007-2010)
0.27Chrysler 200 (2015-2017)
0.281959 Alfa Romeo Giulietta Sprint Speciale[30]
0.281969 Dodge Charger Daytona and 1970 Plymouth Superbird
0.281986 Opel Omega saloon.[31]
0.28Mercedes-Benz CLA-Class Type C 117.[32]
0.29Mazda3 (2007) [33], Nissan 350Z (Track and Grand Touring models) [34]
0.295Bullet (not ogive, at subsonic velocity)
0.3Saab 92 (1949), Audi 100 C3 (1982)
0.31Maserati Ghibli (2013) [35], 0.29 after restyle
0.324Ford Focus Mk2/2.5 (2004-2011, Europe)
0.33BMW E30 3 Series (1984-1993, Germany)
0.35Maserati Quattroporte V (M139, 2003–2012)
0.36Citroen CX (1974-1991, France), Tesla Semi (2017, United States of America)
0.37Ford Transit Custom Mk8 (2013, Turkey)
0.48Rough sphere (${\displaystyle \!\ Re=\!\ 10^{6}}$),
Volkswagen Beetle[36][37]
0.58Jeep Wrangler TJ (1997-2005)[38]
0.75A typical model rocket[39]
1.0Coffee filter, face-up[40]
1.0road bicycle plus cyclist, touring position[41]
1.0–1.1Skier
1.0–1.3Wires and cables
1.0–1.3Man (upright position)
1.1-1.3Ski jumper[42]
1.2Usain Bolt[43]
1.28Flat plate perpendicular to flow (3D) [44]
1.3–1.5Empire State Building
1.4Formula One car
1.8–2.0Eiffel Tower
1.98–2.05Flat plate perpendicular to flow (2D)

## Aircraft

As noted above, aircraft use their wing area as the reference area when computing ${\displaystyle c_{\mathrm {d} }\,}$, while automobiles (and many other objects) use frontal cross sectional area; thus, coefficients are not directly comparable between these classes of vehicles. In the aerospace industry, the drag coefficient is sometimes expressed in drag counts where 1 drag count = 0.0001 of a ${\displaystyle C_{d}}$.[45]

cd Drag CountAircraft type[46]
0.021 210F-4 Phantom II (subsonic)
0.022 220Learjet 24
0.024 240Boeing 787[47]
0.0265 265Airbus A380[48]
0.027 270Cessna 172/182
0.027 270Cessna 310
0.031 310Boeing 747
0.044 440F-4 Phantom II (supersonic)
0.048 480F-104 Starfighter

## Blunt and streamlined body flows

### Concept

Drag, in the context of fluid dynamics, refers to forces that act on a solid object in the direction of the relative flow velocity (note that the diagram below shows the drag in the opposite direction to the flow). The aerodynamic forces on a body come primarily from differences in pressure and viscous shearing stresses. Thereby, the drag force on a body could be divided into two components, namely frictional drag (viscous drag) and pressure drag (form drag). The net drag force could be decomposed as follows:

{\displaystyle {\begin{aligned}c_{\mathrm {d} }&={\dfrac {2F_{\mathrm {d} }}{\rho v^{2}A}}\ \\&=c_{\mathrm {p} }+c_{\mathrm {f} }\\&=\underbrace {{\dfrac {1}{\rho v^{2}A}}\ \displaystyle \int \limits _{S}\mathrm {d} A\,(p-p_{o})\left({\hat {\mathbf {n} }}\cdot {\hat {\mathbf {i} }}\right)} _{c_{\mathrm {p} }}+\underbrace {{\dfrac {1}{\rho v^{2}A}}\ \displaystyle \int \limits _{S}\mathrm {d} A\,\left({\hat {\mathbf {t} }}\cdot {\hat {\mathbf {i} }}\right)T_{w}} _{c_{\mathrm {f} }}\end{aligned}}}

where:

${\displaystyle c_{\mathrm {p} }\,}$ is the pressure drag coefficient,
${\displaystyle c_{\mathrm {f} }\,}$ is the friction drag coefficient,
${\displaystyle {\hat {\mathbf {t} }}}$ = Tangential direction to the surface with area dA,
${\displaystyle {\hat {\mathbf {n} }}}$ = Normal direction to the surface with area dA,
${\displaystyle T_{\mathrm {w} }\,}$ is the shear Stress acting on the surface dA,
${\displaystyle p_{\mathrm {o} }\,}$ is the pressure far away from the surface dA,
${\displaystyle p\,}$ is pressure at surface dA,
${\displaystyle {\hat {\mathbf {i} }}}$ is the unit vector in direction of free stream flow

Therefore, when the drag is dominated by a frictional component, the body is called a streamlined body; whereas in the case of dominant pressure drag, the body is called a blunt body. Thus, the shape of the body and the angle of attack determine the type of drag. For example, an airfoil is considered as a body with a small angle of attack by the fluid flowing across it. This means that it has attached boundary layers, which produce much less pressure drag.

The wake produced is very small and drag is dominated by the friction component. Therefore, such a body (here an airfoil) is described as streamlined, whereas for bodies with fluid flow at high angles of attack, boundary layer separation takes place. This mainly occurs due to adverse pressure gradients at the top and rear parts of an airfoil.

Due to this, wake formation takes place, which consequently leads to eddy formation and pressure loss due to pressure drag. In such situations, the airfoil is stalled and has higher pressure drag than friction drag. In this case, the body is described as a blunt body.

A streamlined body looks like a fish (Tuna), Oropesa, etc. or an airfoil with small angle of attack, whereas a blunt body looks like a brick, a cylinder or an airfoil with high angle of attack. For a given frontal area and velocity, a streamlined body will have lower resistance than a blunt body. Cylinders and spheres are taken as blunt bodies because the drag is dominated by the pressure component in the wake region at high Reynolds number.

To reduce this drag, either the flow separation could be reduced or the surface area in contact with the fluid could be reduced (to reduce friction drag). This reduction is necessary in devices like cars, bicycle, etc. to avoid vibration and noise production.

#### Practical example

The aerodynamic design of cars has evolved from the 1920s to the end of the 20th century. This change in design from a blunt body to a more streamlined body reduced the drag coefficient from about 0.95 to 0.30.

Time history of cars' aerodynamic drag in comparison to change in geometry of streamlined bodies (blunt to streamline).

## Notes

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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. Liversage, P., and Trancossi, M. (2018). Analysis of triangular sharkskin profiles according to second law, Modelling, Measurement and Control B. 87(3), 188-196. http://www.iieta.org/sites/default/files/Journals/MMC/MMC_B/87.03_11.pdf
9. Clancy, L. J.: Aerodynamics. Sections 4.15 and 5.4
10. Clancy, L. J.: Aerodynamics. Section 4.17
11. Clift R., Grace J. R., Weber M. E.: Bubbles, drops, and particles. Academic Press NY (1978).
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13. Haider A., Levenspiel O.: Powder Technology. 58, 1989, 63-70.
14. Shapes
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