In
fluid dynamics, the
drag
coefficient (commonly denoted as
Cd,
Cx or
Cw) 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
Cd is defined as:
- C_d= \frac{F_d}{\tfrac{1}{2} \rho v^2 A},
where
- Fd is the drag
force, which is by definition the force component in the
direction of the flow velocity,
- ρ is the mass density of
the fluid, 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
- 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 frontal area of the vehicle (i.e., the
cross-sectional area when viewed from ahead). For example, for a
sphere
A =
π r2 (note
this is not the surface area =
4 π r2).
For
airfoils, the reference area is the
chord of the airfoil multiplied
with the length of span, which can be easily related to wing 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, 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.
The drag equation
- F_d= \tfrac{1}{2} \rho v^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
speed between the
object and the fluid.
Cd is not a constant but varies as a function
of speed, flow direction, object shape, 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.
Cd is thus a function of
Re. In
compressible flow, the speed of sound is relevant and
Cd is also a function of
Mach number Ma.
For a certain body shape the drag coefficient
Cd 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
Cd 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
form
drag results in a broad wake. The boundary layer will remain
attached longer if it is
laminar than if it
is
turbulent. 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.Clancy, L.J.,
Aerodynamics, Section 4.17
For other objects, such as small particles, one can no longer
consider that the drag coefficient
Cd 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
Fd is proportional to
v instead of
v2; for a sphere this is known as
Stokes law. Reynolds number will be low for small
objects, low velocities, and high viscosity fluids.
A
Cd 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
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
d
of a real square flat plate perpendicular to the flow 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.
Drag coefficient Cd examples
General
In general, C
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
d that varies from about 0.1 for laminar (slow) flow to
0.47 for turbulent (faster) flow.
Shapes [26073]
| Cd |
Item |
| 0.9 |
a typical bicycle plus cyclist |
| 0.4 |
rough sphere (Re =
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) |
| 0.25 |
lowest of production cars (e.g., Toyota Prius) |
| 0.295 |
bullet (not ogive, at subsonic
velocity) |
| 1.0–1.3 |
man (upright position) |
| 1.28 |
flat plate perpendicular to flow (3D) |
| 1.0–1.1 |
skier |
| 1.0–1.3 |
wires and cables |
| 1.1-1.3 |
ski jumper |
| 1.3–1.5 |
Empire State Building |
| 1.8–2.0 |
Eiffel Tower |
| 1.98–2.0 |
flat plate perpendicular to flow (2D) |
| 2.1 |
a smooth brick |
| >2.5 |
spacecraft in LEO |
Aircraft
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.
Automobile
Notes
- McCormick, Barnes W. (1979), Aerodynamics, Aeronautics, and
Flight Mechanics, p.24, John Wiley & Sons, Inc., New York
ISBN 0-471-03032-5
- Clancy, L.J., Aerodynamics, Section 5.18
- Abbott, Ira
H., and Von Doenhoff, Albert E., Theory of Wing
Sections, Sections 1.2 and 1.3
- NASA’s Modern Drag Equation
- Clancy, L.J., Aerodynamics, Section 11.17
- See lift force
and vortex induced vibration for a
possible force components transverse to the flow direction.
- Clancy, L.J., Aerodynamics, Sections 4.15 and 5.4
- Clift R., Grace J.R., Weber M.E., "Bubbles, drops, and
particles", Academic Press NY (1978).
- Briens C.L., Powder Technology, 67, 1991, 87-91.
- Haider A., Levenspiel O., Powder Technology, 58, 1989,
63-70.
-
http://www.engineeringtoolbox.com/drag-coefficient-d_627.html
-
http://www.aerospaceweb.org/question/aerodynamics/q0184.shtml
- http://www.lissys.demon.co.uk/samp1/index.html
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
See also
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