The drag coefficient is a number that aerodynamicists use to model
all of the complex dependencies of drag on shape,
inclination, and some
flow conditions. This equation is simply a
rearrangement of the drag equation where we
solve for the drag coefficient in terms of the other variables.
The drag coefficient **Cd**
is equal to the drag **D** divided by the quantity: density **r** times
half the velocity **V** squared times the reference area **A**.

Cd = D / (A * .5 * r * V^2)

This equation gives us a way to determine a value for the drag coefficient. In a controlled environment (wind tunnel) we can set the velocity, density, and area and measure the drag produced. Through division we arrive at a value for the drag coefficient. As pointed out on the drag equation slide, the choice of reference area (wing area, frontal area, surface area, ...) will affect the actual numerical value of the drag coefficient that is calculated. When reporting drag coefficient values, it is important to specify the reference area that is used to determine the coefficient. We can predict the drag that will be produced under a different set of velocity, density (altitude), and area conditions using the drag equation.

The drag coefficient contains not only the complex dependencies of
object shape and inclination, but also the effects of air
viscosity and compressibility. To correctly use the drag
coefficient, we must be sure that the viscosity and compressibility
effects are the same between our measured case and the predicted
case. Otherwise, the prediction will be inaccurate. For very low
speeds (< 200 mph) the compressibility effects are negligible. At
higher speeds, it becomes important to match Mach numbers between the
two cases. **Mach number** is the ratio of the velocity to the
speed of sound. At supersonic speeds, shock waves will be present in
the flow field and we must be sure to account for the wave drag in
the drag coefficient. So it is completely incorrect to measure a drag
coefficient at some low speed (say 200 mph) and apply that drag
coefficient at twice the speed of sound (approximately 1,400 mph,
Mach = 2.0). It is even more important to match air viscosity
effects. The important matching parameter for viscosity is the
**Reynolds number** that expresses the ratio of inertial forces to
viscous forces. In our discussions on the sources of drag,
recall that skin friction drag depends directly on the viscous
interaction of the object and the flow. If the Reynolds number of the
experiment and flight are close, then we properly model the effects
of the viscous forces relative to the inertial forces. If they are
very different, we do not correctly model the physics of the real
problem and will predict an incorrect drag.

The drag coefficient equation will apply to any object if we properly
match flow conditions. If we are considering an aircraft, we can
think of the drag coefficient as being composed of two main components; a basic
drag coefficient which includes the effects of skin friction and shape (form),
and an additional drag coefficient related to the lift of the aircraft.
This additional source of drag is called the
induced drag
and it is produced at the wing tips due to aircraft lift. Because of pressure
differences above and below the wing, the air on the bottom of the wing is
drawn onto the top near the wing tips. This creates a swirling flow
which changes the effective angle of attack along the wing and "induces"
a drag on the wing. The induced drag coefficient **Cdi** is equal to
the square of the lift coefficient **Cl** divided by the quantity: **pi**
(3.14159) times the aspect ratio **AR** times an
efficiency factor **e**.

Cdi = (Cl^2) / (pi * AR * e)

The aspect ratio is the square of the
span divided by the wing area. For a
rectangular wing this reduces to the ratio of the span to the chord.
Long, slender, high aspect ratio wings have lower induced drag than
short, thick, low aspect ratio wings. Lifting line theory shows that
the optimum (lowest) induced drag occurs for an elliptic distribution
of lift from tip to tip. The efficiency factor (e) is equal to 1.0
for an elliptic distribution and is some value less than 1.0 for any
other lift distribution. (Typical e = .7) (The outstanding
aerodynamic performance of the British Spitfire of World War II is partially
attributable to its elliptic shaped wing which gave the aircraft a very low
amount of induced drag.)
The total
drag coefficient **Cd** is equal to the drag coefficient at zero lift **Cdo**
plus the induced drag coefficient **Cdi**.

**
Cd = Cdo + Cdi
**

The drag coefficient in this equation uses the wing area for the reference area. Otherwise, we could not add it to the square of the lift coefficient, which is also based on the wing area.

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- Beginner's Guide to Propulsion
- Beginner's Guide to Model Rockets
- Beginner's Guide to Kites
- Beginner's Guide to Aeronautics

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*byTom
Benson
Please send suggestions/corrections to: benson@grc.nasa.gov *