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.

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.

Definition

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

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

where:

$F_{\mathrm {d} }\,$ is the drag force, which is by definition the force component in the direction of the flow velocity,

$\rho \,$ is the mass density of the fluid,

$v\,$ 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 blunt objects can be 1 or more, for streamlined objects much less.

Background

Concept

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

Flow across an airfoil showing the relative impact of drag force to the direction of motion of fluid over the body. This drag force gets divided into frictional drag and pressure drag. The same airfoil is considered as a streamlined body if friction drag (viscous drag) dominates pressure drag and is considered as bluff body when pressure drag (form drag) dominates friction drag.

$c_{\mathrm {d} }={\dfrac {2F_{\mathrm {d} }}{\rho v^{2}A}}\ =c_{\mathrm {p} }+c_{\mathrm {f} }=\underbrace {{\dfrac {1}{\rho v^{2}A}}\ \textstyle \int \limits _{S}(p-p_{o}).{\hat {n}}.{\hat {i}}dA} _{c_{\mathrm {p} }}+\underbrace {{\dfrac {1}{\rho v^{2}A}}\ \textstyle \int \limits _{S}T_{w}.{\hat {t}}.{\hat {i}}dA} _{c_{\mathrm {f} }}$

where:

$c_{\mathrm {p} }\,$ is the pressure drag coefficient,

$c_{\mathrm {f} }\,$ is the friction drag coefficient,

${\hat {t}}$ is the tangential direction to the surface with area dA,

${\hat {n}}$ is the normal direction to the surface with area dA,

$T_{\mathrm {w} }\,$ is the shear Stress acting on the surface dA,

$p_{\mathrm {o} }\,$ is the pressure far away from the surface dA,

$p\,$ is pressure at surface dA,

${\hat {i}}$ is the unit vector in direction normal to the surface dA, forming a unit vector $d{\hat {A}}$

Therefore, when the drag is dominated by frictional component, body is called as streamlined body where as in case of dominant pressure drag, the body is called bluff body.Therefore, shape of the body and the angle of attack determines the type of a drag. For e.g.; Airfoil is considered as body with small angle of attacks by fluid flowing across it. This means that it has have attached boundary layers which produce very less pressure drag.

Trade-off relationship between pressure drag and friction drag

The wake produced is very small and drag is dominated by friction component. Therefore, such body (here airfoil) is described as streamlined whereas for bodies with fluid flow at high angle of attack; boundary layer separation takes place. This mainly occurs due to adverse pressure gradient at top and rear parts of an airfoil. Due to this, wake formation takes place which consequently leads to eddies formation and henceforth, pressure losses due to pressure drag. In such situations, the airfoil is termed as being stalled and has higher pressure drag than friction drag. In this case, the body (here airfoil) is described as a bluff body. A streamlined body looks like a fish ( Tuna, Oropesa ,etc.) or an airfoil with small angle of attacks whereas a bluff body looks like a brick, a cylinder or an airfoil with high angle of attack. For a given frontal area and velocity, streamlined body will have lower resistance than a bluff body. Cylinders and spheres are taken as bluff bodies because the drag is dominated by pressure component in the wake region at high Reynolds number.

So as to reduce this drag; flow separation could be reduced (to reduce pressure drag) or surface area in contact with the fluid could be reduced (to reduce friction drag). This reduction is necessary in devices used like cars, bicycle, etc. to avoid considering them as bluff bodies because less the streamlining, more is the vibration and noise produced.

Practical example

Example that could explain the concept of bluff and streamlined flows with the years of revolution in design is the history of aerodynamic drag of cars. Cars have seen the difference in the aerodynamic design from 1920s to the end of 20th century. This change in design from a bluff body to a more streamlined body reduced the drag coefficient from about 0.95 to 0.15. This endorses the fact that better the streamlined design, less is the drag coefficient.

Time history of Aerodynamic drag of cars in comparison with change in geometry of streamlined bodies (bluff to streamline).

Time history of Aerodynamic drag of cars in comparison with change in geometry of streamlined bodies (bluff to streamline).

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 $Re$). 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
0.001	laminar flat plate parallel to the flow ($\!\ Re<10^{5}$)
0.005	turbulent flat plate parallel to the flow ($\!\ Re>10^{5}$)
0.075	Pac-car
0.1	smooth sphere ($\!\ Re=10^{6}$)
0.186	Schlörwagen 1939
0.186-0.189	Volkswagen XL1 2014
0.19	General Motors EV1 1996
0.23	Mercedes-Benz CLA-Class Type C 117. With exception of forthcoming CDI180, which will have 0.22
0.25	Toyota Prius (3rd Generation)
0.295	bullet (not ogive, at subsonic velocity)
0.3	Audi 100 C3 (1982)
0.48	rough sphere ($\!\ Re=\!\ 10^{6}$), Volkswagen Beetle
0.75	a typical model rocket
.8-.9	coffee filter, face-up
1.0	road bicycle plus cyclist, touring position
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
1.28	flat plate perpendicular to flow (3D)
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

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. In the aerospace industry the drag coefficient is sometimes expressed in drag counts where 1 drag count = 0.0001 of a $C_{d}$.

Aircraft

cd	Aircraft type
0.021	F-4 Phantom II (subsonic)
0.022	Learjet 24
0.024	Boeing 787
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)

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
• Bluff Body: http://www.engineering.uiowa.edu/
• Hucho, W.H., Janssen, L.J., Emmelmann, H.J. 6(1975): The optimization of body details-A method for reducing the aerodynamics drag. SAE 760185.