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.
• $q={\frac {1}{2}}\rho u^{2}$ = 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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_d}
Smooth sphere (Re = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^6} )	0.1
Rough sphere (Re = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_d} can drop dramatically due to a transition to turbulent boundary layer flow (e.g., golf ball dimples reduce Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.