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:

\[ c_d = \frac{2F_d}{\rho u^2 A} \]

where:

• \( F_d \) is the drag force.
• \( \rho \) 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 \( \tau \) and local dynamic pressure \( q \):

\[ c_d = \frac{\tau}{q} = \frac{2\tau}{\rho u^2} \]

where:

• \( \tau \) = local shear stress.
• \( q \) = \( \frac{1}{2} \rho u^2 \), the dynamic pressure.

Drag Equation

The general drag force formula:

\[ F_d = \frac{1}{2} \rho u^2 c_d A \]

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 = \( 10^6 \))	0.1
Rough sphere (Re = \( 10^6 \))	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.

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