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.