Drag coefficient

Drag Coefficient (commonly denoted as: c_d, c_x, or c_w) is a dimensionless quantity used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It indicates how aerodynamic or hydrodynamic a body is. A lower drag coefficient corresponds to lower aerodynamic drag for a given shape.

Definition

The drag coefficient c_d is defined as:

c_d = (2 * F_d) / (ρ * u² * A)

where:

• F_d is the drag force.
• ρ is the mass density of the fluid.
• u is the flow velocity relative to the fluid.
• A is the reference area (e.g., frontal area for cars, wing area for aircraft).

Key Points

• The reference area depends on the object and context.
• Airfoils use wing area; cars use projected frontal area.
• For streamlined bodies (e.g., fish, aircraft), c_d is typically lower.
• For bluff bodies (e.g., brick, sphere), c_d is higher due to flow separation and pressure drag.

Cauchy Momentum Equation

In terms of local shear stress τ and local dynamic pressure q:

c_d = τ / q = (2 * τ) / (ρ * u²)

where:

• τ is the local shear stress.
• q is the dynamic pressure (q = 0.5 * ρ * u²).

Drag Equation

The general drag force formula:

F_d = 0.5 * ρ * u² * c_d * A

Dependence on Reynolds Number

The drag coefficient is influenced by the Reynolds number (Re):

• Low Re: laminar flow, drag dominated by viscous forces.
• High Re: turbulent flow, drag dominated by pressure forces.
• For a sphere: c_d drops sharply at the critical Reynolds number.

Drag Coefficient Examples

General Shapes

Shape	c_d
Smooth sphere (Re = 10⁶)	0.1
Rough sphere (Re = 10⁶)	0.47
Flat plate perpendicular to flow (3D)	1.28
Empire State Building	1.3–1.5
Eiffel Tower	1.8–2.0
Long flat plate perpendicular to flow (2D)	1.98–2.05

Aircraft

Aircraft Type	c_d	Drag Count
F-4 Phantom II (subsonic)	0.021	210
Learjet 24	0.022	220
Boeing 787	0.024	240
Airbus A380	0.0265	265
Cessna 172/182	0.027	270
Boeing 747	0.031	310
F-104 Starfighter	0.048	480

Blunt and Streamlined Body Flows

• Streamlined bodies: Flow remains attached longer; friction drag dominates.
• Blunt bodies: Flow separates early; pressure drag dominates.

Boundary layer behavior is critical:

• Laminar flow = lower drag
• Turbulent flow = higher drag but more stable separation.

Drag Crisis

At critical Reynolds numbers, c_d can drop dramatically due to a transition to turbulent boundary layer flow (e.g., golf ball dimples reduce c_d).

See Also

• Automotive aerodynamics
• Ballistic coefficient
• Drag crisis
• Zero-lift drag coefficient

References

1. Clancy, L.J. (1975). Aerodynamics. ISBN 0-273-01120-0.
2. Abbott, Ira H., and Von Doenhoff, Albert E. (1959). Theory of Wing Sections.
3. Hoerner, Dr. Sighard F., Fluid-Dynamic Drag.
4. EngineeringToolbox.com - Drag Coefficient resources.
5. NASA - Shape Effects on Drag.