Drag coefficient

The drag coefficient ${\displaystyle c_{\mathrm {d} }}$ is defined as

${\displaystyle c_{\mathrm {d} }={\dfrac {2F_{\mathrm {d} }}{\rho u^{2}A}}}$

where:

${\displaystyle F_{\mathrm {d} }}$ is the drag force, which is by definition the force component in the direction of the flow velocity,[6]

${\displaystyle \rho }$ is the mass density of the fluid,[7]

${\displaystyle u}$ is the flow speed of the object relative to the fluid,

${\displaystyle A}$ is the reference area.

The reference area depends on what type of drag coefficient is being measured. For automobiles and many other objects, the reference area is the projected frontal area of the vehicle. This may not necessarily be the cross-sectional area of the vehicle, depending on where the cross-section is taken. For example, for a sphere ${\displaystyle A=\pi r^{2}}$ (note this is not the surface area = ${\displaystyle 4\pi r^{2}}$).

For airfoils, the reference area is the nominal wing area. Since this tends to be large compared to the frontal area, the resulting drag coefficients tend to be low, much lower than for a car with the same drag, frontal area, and speed.

Airships and some bodies of revolution use the volumetric drag coefficient, in which the reference area is the square of the cube root of the airship volume (volume to the two-thirds power). Submerged streamlined bodies use the wetted surface area.

Two objects having the same reference area moving at the same speed through a fluid will experience a drag force proportional to their respective drag coefficients. Coefficients for unstreamlined objects can be 1 or more, for streamlined objects much less.

It has been demonstrated that drag coefficient ${\displaystyle c_{d}}$ is a function of Bejan number (${\displaystyle Be}$), Reynolds number (${\displaystyle Re}$) and the ratio between wet area ${\displaystyle A_{w}}$and front area ${\displaystyle A_{f}}$:[8]

${\displaystyle c_{d}=2{\frac {A_{w}}{A_{f}}}{\frac {Be}{Re_{L}^{2}}}}$

where ${\displaystyle Re_{L}}$is the Reynold Number related to fluid path length L.

Flow around a plate, showing stagnation. The force in the upper configuration is equal to
${\displaystyle F=\rho u^{2}A}$
and in the down configuration
${\displaystyle F_{d}={\tfrac {1}{2}}\rho u^{2}c_{d}A}$
.

The drag equation

${\displaystyle F_{d}={\tfrac {1}{2}}\rho u^{2}c_{d}A}$

is essentially a statement that the drag force on any object is proportional to the density of the fluid and proportional to the square of the relative flow speed between the object and the fluid.

C_d is not a constant but varies as a function of flow speed, flow direction, object position, object size, fluid density and fluid viscosity. Speed, kinematic viscosity and a characteristic length scale of the object are incorporated into a dimensionless quantity called the Reynolds number ${\displaystyle \scriptstyle Re}$. ${\displaystyle \scriptstyle C_{\mathrm {d} }}$ is thus a function of ${\displaystyle \scriptstyle Re}$. In a compressible flow, the speed of sound is relevant, and ${\displaystyle \scriptstyle C_{\mathrm {d} }}$ is also a function of Mach number ${\displaystyle \scriptstyle Ma}$.

For certain body shapes, the drag coefficient ${\displaystyle \scriptstyle C_{\mathrm {d} }}$ only depends on the Reynolds number ${\displaystyle \scriptstyle Re}$, Mach number ${\displaystyle \scriptstyle Ma}$ and the direction of the flow. For low Mach number ${\displaystyle \scriptstyle Ma}$, the drag coefficient is independent of Mach number. Also, the variation with Reynolds number ${\displaystyle \scriptstyle Re}$ within a practical range of interest is usually small, while for cars at highway speed and aircraft at cruising speed, the incoming flow direction is also more-or-less the same. Therefore, the drag coefficient ${\displaystyle \scriptstyle C_{\mathrm {d} }}$ can often be treated as a constant.[9]

For a streamlined body to achieve a low drag coefficient, the boundary layer around the body must remain attached to the surface of the body for as long as possible, causing the wake to be narrow. A high form drag results in a broad wake. The boundary layer will transition from laminar to turbulent if Reynolds number of the flow around the body is sufficiently great. Larger velocities, larger objects, and lower viscosities contribute to larger Reynolds numbers.[10]

Drag coefficient C_d for a sphere as a function of Reynolds number Re, as obtained from laboratory experiments. The dark line is for a sphere with a smooth surface, while the lighter line is for the case of a rough surface. The numbers along the line indicate several flow regimes and associated changes in the drag coefficient:
•2: attached flow (Stokes flow) and steady separated flow,
•3: separated unsteady flow, having a laminar flow boundary layer upstream of the separation, and producing a vortex street,
•4: separated unsteady flow with a laminar boundary layer at the upstream side, before flow separation, with downstream of the sphere a chaotic turbulent wake,
•5: post-critical separated flow, with a turbulent boundary layer.

For other objects, such as small particles, one can no longer consider that the drag coefficient ${\displaystyle \scriptstyle C_{\mathrm {d} }}$ is constant, but certainly is a function of Reynolds number.[11][12][13] At a low Reynolds number, the flow around the object does not transition to turbulent but remains laminar, even up to the point at which it separates from the surface of the object. At very low Reynolds numbers, without flow separation, the drag force ${\displaystyle \scriptstyle F_{\mathrm {d} }}$ is proportional to ${\displaystyle \scriptstyle v}$ instead of ${\displaystyle \scriptstyle v^{2}}$; for a sphere this is known as Stokes law. The Reynolds number will be low for small objects, low velocities, and high viscosity fluids.[10]

A ${\displaystyle \scriptstyle C_{\mathrm {d} }}$ equal to 1 would be obtained in a case where all of the fluid approaching the object is brought to rest, building up stagnation pressure over the whole front surface. The top figure shows a flat plate with the fluid coming from the right and stopping at the plate. The graph to the left of it shows equal pressure across the surface. In a real flat plate, the fluid must turn around the sides, and full stagnation pressure is found only at the center, dropping off toward the edges as in the lower figure and graph. Only considering the front side, the ${\displaystyle \scriptstyle C_{\mathrm {d} }}$ of a real flat plate would be less than 1; except that there will be suction on the back side: a negative pressure (relative to ambient). The overall ${\displaystyle \scriptstyle C_{\mathrm {d} }}$ of a real square flat plate perpendicular to the flow is often given as 1.17. Flow patterns and therefore ${\displaystyle \scriptstyle C_{\mathrm {d} }}$ for some shapes can change with the Reynolds number and the roughness of the surfaces.

As noted above, aircraft use their wing area as the reference area when computing ${\displaystyle c_{\mathrm {d} }}$, while automobiles (and many other objects) use frontal cross sectional area; thus, coefficients are not directly comparable between these classes of vehicles. In the aerospace industry, the drag coefficient is sometimes expressed in drag counts where 1 drag count = 0.0001 of a ${\displaystyle C_{d}}$.[45]

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

• L. J. Clancy (1975): Aerodynamics. Pitman Publishing Limited, London, ISBN 0-273-01120-0
• Abbott, Ira H., and Von Doenhoff, Albert E. (1959): Theory of Wing Sections. Dover Publications Inc., New York, Standard Book Number 486-60586-8
• Hoerner, Dr. Sighard F., Fluid-Dynamic Drag, Hoerner Fluid Dynamics, Bricktown New Jersey, 1965.
• Bluff Body: http://user.engineering.uiowa.edu/~me_160/lecture_notes/Bluff%20Body2.pdf
• Drag of Blunt Bodies and Streamlined Bodies: http://www.princeton.edu/~asmits/Bicycle_web/blunt.html
• Hucho, W.H., Janssen, L.J., Emmelmann, H.J. 6(1975): The optimization of body details-A method for reducing the aerodynamics drag. SAE 760185.