Editing Drag coefficient (section)

== Background ==
{{Main|Drag equation}}
[[File:Cd flat plate.svg|thumb|200px|Flow around a plate, showing stagnation. The force in the upper configuration is equal to<br /><math>F = \frac {1}{2} \rho u^2 A</math><br /> and in the lower configuration<br /><math>F_d = \tfrac12 \rho u^2 c_d A</math>]]

The drag equation

:<math>F_{\rm d} = \tfrac12 \rho u^2 c_{\rm d} A</math>

is essentially a statement that the [[drag (physics)|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. The factor of <math>1/2</math> comes from the [[dynamic pressure]] of the fluid, which is equal to the kinetic energy density.

The value of <math>c_\mathrm d</math> 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]] <math>\mathrm{Re}</math>. <math>c_\mathrm d</math> is thus a function of <math>\mathrm{Re}</math>. In a compressible flow, the speed of sound is relevant, and <math>c_\mathrm d</math> is also a function of [[Mach number]] <math>\mathrm{Ma}</math>.

For certain body shapes, the drag coefficient <math>c_\mathrm d</math> only depends on the Reynolds number <math>\mathrm{Re}</math>, Mach number <math>\mathrm{Ma}</math> and the direction of the flow. For low Mach number <math>\mathrm{Ma}</math>, the drag coefficient is independent of Mach number. Also, the variation with Reynolds number <math>\mathrm{Re}</math> 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 <math>c_\mathrm d</math> can often be treated as a constant.<ref>Clancy, L. J.: ''Aerodynamics''. Sections 4.15 and 5.4</ref>

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 (physics)|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 [[viscosity|viscosities]] contribute to larger Reynolds numbers.<ref name=Clancy_4.17>Clancy, L. J.: ''Aerodynamics''. Section 4.17</ref>

[[File:Drag coefficient on a sphere vs. Reynolds number - main trends.svg|thumb|Drag coefficient ''C''<sub>d</sub> 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:<br />
•2: attached flow ([[Stokes flow]]) and [[steady flow|steady]] [[flow separation|separated flow]],<br />
•3: separated unsteady flow, having a [[laminar flow]] [[boundary layer]] upstream of the separation, and producing a [[vortex street]],<br />
•4: separated unsteady flow with a laminar boundary layer at the upstream side, before flow separation, with downstream of the sphere a chaotic [[turbulence|turbulent]] wake,<br />
•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 <math>c_\mathrm d</math> is constant, but certainly is a function of Reynolds number.<ref>Clift R., Grace J. R., Weber M. E.: ''Bubbles, drops, and particles''. Academic Press NY (1978).</ref><ref>Briens C. L.: ''Powder Technology''. 67, 1991, 87-91.</ref><ref>Haider A., Levenspiel O.: ''Powder Technology''. 58, 1989, 63-70.</ref>
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 <math> F_\mathrm d</math> is proportional to <math>v</math> instead of <math>v^2</math>; for a sphere this is known as [[Stokes' law]]. The Reynolds number will be low for small objects, low velocities, and high viscosity fluids.<ref name="Clancy_4.17" />

A <math>c_\mathrm d</math> 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 <math>c_\mathrm d</math> of a real flat plate would be less than 1; except that there will be suction on the backside: a negative pressure (relative to ambient). The overall <math>c_\mathrm d</math> of a real square flat plate perpendicular to the flow is often given as 1.17.{{Citation needed|date=March 2013}} Flow patterns and therefore <math>c_\mathrm d</math> for some shapes can change with the Reynolds number and the roughness of the surfaces.
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