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Drag coefficient

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In <a href="Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, the drag coefficient (commonly denoted as: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

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<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {d} }}</annotation>

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</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{x}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{x}}</annotation>

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</math><img src="./fb6e0397e797e2cde37718a8e2b2e0fad6252c8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.179ex; height:2.009ex;" alt="{\displaystyle c_{x}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\rm {w}}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{\rm {w}}}</annotation>

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</math><img src="./25b056e9682186d4051fddda8032aa344a8c69f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.426ex; height:2.009ex;" alt="{\displaystyle c_{\rm {w}}}" loading="lazy">) is a <a href="Dimensionless_quantity" title="Dimensionless quantity">dimensionless quantity</a> that is used to quantify the <a href="Drag_(physics)" title="Drag (physics)">drag</a> or resistance of an object in a fluid environment, such as air or water. It is used in the <a href="Drag_equation" title="Drag equation">drag equation</a> in which a lower drag coefficient indicates the object will have less <a href="Aerodynamics" title="Aerodynamics">aerodynamic</a> or <a href="Hydrodynamics" class="mw-redirect" title="Hydrodynamics">hydrodynamic</a> drag. The drag coefficient is always associated with a particular surface area.<a href="#cite_note-3">[3]</a>

The drag coefficient of any object comprises the effects of the two basic contributors to <a href="Fluid_dynamics" title="Fluid dynamics">fluid dynamic</a> drag: <a href="Skin_friction" class="mw-redirect" title="Skin friction">skin friction</a> and <a href="Form_drag" class="mw-redirect" title="Form drag">form drag</a>. The drag coefficient of a lifting <a href="Airfoil" title="Airfoil">airfoil</a> or <a href="Hydrofoil" title="Hydrofoil">hydrofoil</a> also includes the effects of <a href="Lift-induced_drag" title="Lift-induced drag">lift-induced drag</a>.<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> The drag coefficient of a complete structure such as an aircraft also includes the effects of interference drag.<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The drag coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

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<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {d} }}</annotation>

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</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy"> is defined as

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<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {d} }={\dfrac {2F_{\mathrm {d} }}{\rho u^{2}A}}}</annotation>

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where:

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<annotation encoding="application/x-tex">{\displaystyle F_{\mathrm {d} }}</annotation>

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</math><img src="./94c70082e398a95e8d658918edea80a198a2351d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.641ex; height:2.509ex;" alt="{\displaystyle F_{\mathrm {d} }}" loading="lazy"> is the <a href="Drag_(physics)" title="Drag (physics)">drag force</a>, which is by definition the force component in the direction of the <a href="Flow_velocity" title="Flow velocity">flow velocity</a>;<a href="#cite_note-9">[9]</a></li>

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is the <a href="Mass_density" class="mw-redirect" title="Mass density">mass density</a> of the fluid;<a href="#cite_note-10">[10]</a></li>

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> is the <a href="Flow_speed" class="mw-redirect" title="Flow speed">flow speed</a> of the object relative to the fluid;</li>

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is the reference <a href="Area" title="Area">area</a></li></ul>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\pi r^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle A=\pi r^{2}}</annotation>

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</math><img src="./33f7b7f93f93e7ba7bebb97efbe88e181ce332e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.276ex; height:2.676ex;" alt="{\displaystyle A=\pi r^{2}}" loading="lazy"> (note this is not the surface area = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi r^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle 4\pi r^{2}}</annotation>

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</math><img src="./b81fcce302776a01dc66fc186a1ce0a616b4d772.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.597ex; height:2.676ex;" alt="{\displaystyle 4\pi r^{2}}" loading="lazy">).

For <a href="Airfoil" title="Airfoil">airfoils</a>, 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.

<a href="Airship" title="Airship">Airships</a> and some <a href="Solid_of_revolution" title="Solid of revolution">bodies of revolution</a> use the volumetric drag coefficient, in which the reference area is the <a href="Square_(algebra)" title="Square (algebra)">square</a> of the <a href="Cube_root" title="Cube root">cube root</a> 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.

As a caution, note that although the above is the conventional definition for the drag coefficient, there are other definitions that one may encounter in the literature. The reason for this is that the conventional definition makes the most sense when one is in the Newton regime, such as what happens at high Reynolds number, where it makes sense to scale the drag to the momentum flux into the frontal area of the object. But, there are other flow regimes. In particular at very low Reynolds number, it is more natural to write the drag force as being proportional to a drag coefficient multiplied by the speed of the object (rather than the square of the speed of the object). An example of such a regime is the study of the mobility of aerosol particulates, such as smoke particles. This leads to a different formal definition of the "drag coefficient," of course.

<div class="mw-heading mw-heading2"><h2 id="Cauchy_momentum_equation">Cauchy momentum equation</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Cauchy_momentum_equation#Nondimensionalisation" title="Cauchy momentum equation">Cauchy momentum equation § Nondimensionalisation</a></div>

In the non dimensional form of the Cauchy momentum equation, the skin drag coefficient or skin friction coefficient is referred to the transversal area (the area normal to the drag force, so the coefficient is locally defined as:

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<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {d} }={\dfrac {\tau }{q}}={\dfrac {2\tau }{\rho u^{2}}}}</annotation>

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where:

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> is the local <a href="Shear_stress" title="Shear stress">shear stress</a>, which is by definition the stress component in the direction of the local <a href="Flow_velocity" title="Flow velocity">flow velocity</a>;<a href="#cite_note-11">[11]</a></li>

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> is the local <a href="Dynamic_pressure" title="Dynamic pressure">dynamic pressure</a> of the fluid</li>

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is the local <a href="Mass_density" class="mw-redirect" title="Mass density">mass density</a> of the fluid;<a href="#cite_note-12">[12]</a></li>

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> is the local <a href="Flow_speed" class="mw-redirect" title="Flow speed">flow speed</a> of the fluid</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Background">Background</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Drag_equation" title="Drag equation">Drag equation</a></div>

The drag equation

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<annotation encoding="application/x-tex">{\displaystyle F_{\rm {d}}={\tfrac {1}{2}}\rho u^{2}c_{\rm {d}}A}</annotation>

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</math><img src="./e0f33b9969caaa89780acd5209eaaa39e570f959.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.879ex; height:3.509ex;" alt="{\displaystyle F_{\rm {d}}={\tfrac {1}{2}}\rho u^{2}c_{\rm {d}}A}" loading="lazy"></dd></dl>

is essentially a statement that the <a href="Drag_(physics)" title="Drag (physics)">drag</a> <a href="Force" title="Force">force</a> on any object is proportional to the density of the fluid and proportional to the square of the relative <a href="Flow_speed" class="mw-redirect" title="Flow speed">flow speed</a> between the object and the fluid. The factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2}">

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</math><img src="./e308a3a46b7fdce07cc09dcab9e8d8f73e37d935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 1/2}" loading="lazy"> comes from the <a href="Dynamic_pressure" title="Dynamic pressure">dynamic pressure</a> of the fluid, which is equal to the kinetic energy density.

The value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

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</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy"> is not a constant but varies as a function of flow speed, flow direction, object position, object size, fluid density and fluid <a href="Viscosity" title="Viscosity">viscosity</a>. Speed, <a href="Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a> and a characteristic <a href="Length_scale" title="Length scale">length scale</a> of the object are incorporated into a dimensionless quantity called the <a href="Reynolds_number" title="Reynolds number">Reynolds number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} }</annotation>

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</math><img src="./ce9a978f729af1a34028e60d45f484f0908e4be7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Re} }" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

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</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy"> is thus a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} }">

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</math><img src="./ce9a978f729af1a34028e60d45f484f0908e4be7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Re} }" loading="lazy">. In a compressible flow, the speed of sound is relevant, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

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For certain body shapes, the drag coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Ma} }</annotation>

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</math><img src="./5900aa8d40d03d189fa5a5c253f7ad95114d39b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \mathrm {Ma} }" loading="lazy">, the drag coefficient is independent of Mach number. Also, the variation with Reynolds number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} }">

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</math><img src="./ce9a978f729af1a34028e60d45f484f0908e4be7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Re} }" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

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For a streamlined body to achieve a low drag coefficient, the <a href="Boundary_layer" title="Boundary layer">boundary layer</a> around the body must remain attached to the surface of the body for as long as possible, causing the <a href="Wake_(physics)" title="Wake (physics)">wake</a> 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 <a href="Viscosity" title="Viscosity">viscosities</a> contribute to larger Reynolds numbers.<a href="#cite_note-Clancy_4.17-14">[14]</a>

For other objects, such as small particles, one can no longer consider that the drag coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

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</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy"> is constant, but certainly is a function of Reynolds number.<a href="#cite_note-15">[15]</a><a href="#cite_note-16">[16]</a><a href="#cite_note-17">[17]</a>

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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mathrm {d} }}">

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{\mathrm {d} }}</annotation>

</semantics>

</math><img src="./94c70082e398a95e8d658918edea80a198a2351d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.641ex; height:2.509ex;" alt="{\displaystyle F_{\mathrm {d} }}" loading="lazy"> is proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v^{2}}</annotation>

</semantics>

</math><img src="./4faa98a21ac8133ab466999288849492be28b3d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.182ex; height:2.676ex;" alt="{\displaystyle v^{2}}" loading="lazy">; for a sphere this is known as <a href="Stokes'_law" title="Stokes' law">Stokes' law</a>. The Reynolds number will be low for small objects, low velocities, and high viscosity fluids.<a href="#cite_note-Clancy_4.17-14">[14]</a>

A <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {d} }}</annotation>

</semantics>

</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy"> equal to 1 would be obtained in a case where all of the fluid approaching the object is brought to rest, building up <a href="Stagnation_pressure" title="Stagnation pressure">stagnation pressure</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {d} }}</annotation>

</semantics>

</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {d} }}</annotation>

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</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy"> of a real square flat plate perpendicular to the flow is often given as 1.17. Flow patterns and therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {d} }}</annotation>

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</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy"> for some shapes can change with the Reynolds number and the roughness of the surfaces.

<div class="mw-heading mw-heading2"><h2 id="Drag_coefficient_examples">Drag coefficient examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="General">General</h3></div>

In general, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {d} }}</annotation>

</semantics>

</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy"> is not an absolute constant for a given body shape. It varies with the speed of airflow (or more generally with <a href="Reynolds_number" title="Reynolds number">Reynolds number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} }</annotation>

</semantics>

</math><img src="./ce9a978f729af1a34028e60d45f484f0908e4be7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Re} }" loading="lazy">). A smooth sphere, for example, has a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {d} }}</annotation>

</semantics>

</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy"> that varies from high values for <a href="Laminar_flow" title="Laminar flow">laminar flow</a> to 0.47 for <a href="Turbulent_flow" class="mw-redirect" title="Turbulent flow">turbulent flow</a>. Although the drag coefficient decreases with increasing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} }</annotation>

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</math><img src="./ce9a978f729af1a34028e60d45f484f0908e4be7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Re} }" loading="lazy">, the drag force increases.

<th>cd

</th>

</th></tr>

<tr>

<td>Laminar flat plate parallel to the flow (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} <10^{6}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} <10^{6}}</annotation>

</semantics>

</math><img src="./1ab689e51d7bff6f27e54575111b4cdbd579f44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.221ex; height:2.676ex;" alt="{\displaystyle \mathrm {Re} <10^{6}}" loading="lazy">)

</td></tr>

<tr>

<td>Turbulent flat plate parallel to the flow (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} >10^{6}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} >10^{6}}</annotation>

</semantics>

</math><img src="./de943b72cf16c99df06f751cf5d8b6cec4536965.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.221ex; height:2.676ex;" alt="{\displaystyle \mathrm {Re} >10^{6}}" loading="lazy">)

</td></tr>

<tr>

<td>Smooth sphere (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} =10^{6}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} =10^{6}}</annotation>

</semantics>

</math><img src="./f711b13e6ef453646fcf08d1a81ba62c7459f48d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.221ex; height:2.676ex;" alt="{\displaystyle \mathrm {Re} =10^{6}}" loading="lazy">)

</td></tr>

<tr>

<td>Rough sphere (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} =10^{6}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} =10^{6}}</annotation>

</semantics>

</math><img src="./f711b13e6ef453646fcf08d1a81ba62c7459f48d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.221ex; height:2.676ex;" alt="{\displaystyle \mathrm {Re} =10^{6}}" loading="lazy">)

</td></tr>

<tr>

<td>0.81

</td>

<td>Triangular trapeze (45°)

</td></tr>

<tr>

<td>0.9-1.7

</td>

<td>Trapeze with triangular basis (45°)

</td></tr>

<tr>

<td>Bullet (not <a href="Ogive" title="Ogive">ogive</a>, at subsonic velocity)

</td></tr>

<tr>

<td><a href="Skier" class="mw-redirect" title="Skier">Skier</a>

</td></tr>

<tr>

<td>Wires and cables

</td></tr>

<tr>

<td>Adult human (upright position)

</td></tr>

<tr>

<td>Ski jumper<a href="#cite_note-tool-19">[19]</a>

</td></tr>

<tr>

<td>Flat plate perpendicular to flow (3D)<a href="#cite_note-20">[20]</a>

</td></tr>

<tr>

<td><a href="Empire_State_Building" title="Empire State Building">Empire State Building</a>

</td></tr>

<tr>

<td><a href="Eiffel_Tower" title="Eiffel Tower">Eiffel Tower</a>

</td></tr>

<tr>

<td>Long flat plate perpendicular to flow (2D)

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Aircraft">Aircraft</h3></div>

As noted above, aircraft use their wing area as the reference area when computing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {d} }}</annotation>

</semantics>

</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy">, while automobiles (and many other objects) use projected frontal 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 <a href="Drag_count" title="Drag count">drag count</a> = 0.0001 of a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\mathrm {d} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {d} }}</annotation>

</semantics>

</math><img src="./9dc7e69baef1b4c3e6ace4295b31f31ca98dcb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {d} }}" loading="lazy">.<a href="#cite_note-21">[21]</a>

<th>cd

</th>

<th>Drag Count</th>

<th>Aircraft type<a href="#cite_note-22">[22]</a>

</th></tr>

<tr>

<td>0.021

</td>

<td><a href="F-4_Phantom_II" class="mw-redirect" title="F-4 Phantom II">F-4 Phantom II</a> (subsonic)

</td></tr>

<tr>

<td>0.022

</td>

<td><a href="Learjet_24" title="Learjet 24">Learjet 24</a>

</td></tr>

<tr>

<td>0.024

</td>

<td><a href="Boeing_787" class="mw-redirect" title="Boeing 787">Boeing 787</a><a href="#cite_note-23">[23]</a>

</td></tr>

<tr>

<td>0.0265

</td>

<td><a href="Airbus_A380" title="Airbus A380">Airbus A380</a><a href="#cite_note-24">[24]</a>

</td></tr>

<tr>

<td>0.027

</td>

<td><a href="Cessna_172" title="Cessna 172">Cessna 172</a>/<a href="Cessna_182" class="mw-redirect" title="Cessna 182">182</a>

</td></tr>

<tr>

<td>0.027

</td>

<td><a href="Cessna_310" title="Cessna 310">Cessna 310</a>

</td></tr>

<tr>

<td>0.031

</td>

<td><a href="Boeing_747" title="Boeing 747">Boeing 747</a>

</td></tr>

<tr>

<td>0.044

</td>

<td><a href="McDonnell_Douglas_F-4_Phantom_II" title="McDonnell Douglas F-4 Phantom II">F-4 Phantom II</a> (supersonic)

</td></tr>

<tr>

<td>0.048

</td>

<td><a href="F-104_Starfighter" class="mw-redirect" title="F-104 Starfighter">F-104 Starfighter</a>

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Automobile">Automobile</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Automobile_drag_coefficient" title="Automobile drag coefficient">Automobile drag coefficient</a></div>

<div class="mw-heading mw-heading2"><h2 id="Blunt_and_streamlined_body_flows">Blunt and streamlined body flows</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Concept">Concept</h3></div>

The force between a fluid and a body, when there is relative motion, can only be transmitted by normal pressure and tangential friction stresses. So, for the whole body, the drag part of the force, which is in-line with the approaching fluid motion, is composed of frictional drag (viscous drag) and pressure drag (form drag). The total drag and component drag forces can be related as follows:

<mtr>

<mtd>

<msub>

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</msub>

</mtd>

<mtd>

<mfrac>

<mrow>

<msub>

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</mrow>

<mrow>

<msup>

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</msup>

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</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

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<msub>

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</mtr>

<mtr>

<mtd>

<mrow>

<mfrac>

<mrow>

<msup>

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</msup>

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</mfrac>

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</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mover>

</mrow>

</mover>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

<mover>

</mrow>

</mover>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c_{\mathrm {d} }&={\dfrac {2F_{\mathrm {d} }}{\rho v^{2}A}}\\&=c_{\mathrm {p} }+c_{\mathrm {f} }\\&=\underbrace {{\dfrac {2}{\rho v^{2}A}}\displaystyle \int _{S}\mathrm {d} S(p-p_{o})\left({\hat {\mathbf {n} }}\cdot {\hat {\mathbf {i} }}\right)} _{c_{\mathrm {p} }}+\underbrace {{\dfrac {2}{\rho v^{2}A}}\displaystyle \int _{S}\mathrm {d} S\left({\hat {\mathbf {t} }}\cdot {\hat {\mathbf {i} }}\right)T_{\rm {w}}} _{c_{\mathrm {f} }}\end{aligned}}}</annotation>

</semantics>

</math>

where:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is the planform area of the body,</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is the wet surface of the body,</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {p} }}</annotation>

</semantics>

</math><img src="./3892da19d1c67f5c41f04824e608eeeb7f89e99c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.153ex; height:2.343ex;" alt="{\displaystyle c_{\mathrm {p} }}" loading="lazy"> is the <a href="Pressure" title="Pressure">pressure</a> drag coefficient,</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{\mathrm {f} }}</annotation>

</semantics>

</math><img src="./022b43175cf0d4b459b67a845b75bd9cc5a4db00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.851ex; height:2.009ex;" alt="{\displaystyle c_{\mathrm {f} }}" loading="lazy"> is the <a href="Friction" title="Friction">friction</a> drag coefficient,</li>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {t} }}}</annotation>

</semantics>

</math><img src="./230b09600348e313329f8e8050abc8681e4d0d99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.676ex;" alt="{\displaystyle {\hat {\mathbf {t} }}}" loading="lazy"> is the unit vector in the direction of the shear stress acting on the body surface dS,</li>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation>

</semantics>

</math><img src="./aae87b164ba005e99b51066c46d1eacc7f56564a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}" loading="lazy"> is the unit vector in the direction perpendicular to the body surface dS, pointing from the fluid to the solid,</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {w} }}</annotation>

</semantics>

</math><img src="./5d9903630b148bfaaa2406d9414cc4a498f365e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.776ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {w} }}" loading="lazy"> magnitude of the <a href="Shear_stress" title="Shear stress">shear stress</a> acting on the body surface dS,</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{\mathrm {o} }}</annotation>

</semantics>

</math><img src="./3fc079cce4a5c36c6880c5c722e5720c652a5e07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{\mathrm {o} }}" loading="lazy"> is the pressure far away from the body (note that this constant does not affect the final result),</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is pressure at surface dS,</li>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {i} }}}</annotation>

</semantics>

</math><img src="./c67b997d3ff51d850c47e3172ceabe2402937bff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathbf {i} }}}" loading="lazy"> is the unit vector in direction of free stream flow</li></ul>

Therefore, when the drag is dominated by a frictional component, the body is called a streamlined body; whereas in the case of dominant pressure drag, the body is called a blunt or bluff body. Thus, the shape of the body and the angle of attack determine the type of drag. For example, an airfoil is considered as a body with a small angle of attack by the fluid flowing across it. This means that it has attached <a href="Boundary_layer" title="Boundary layer">boundary layers</a>, which produce much less pressure drag.

The <a href="Wake_(physics)" title="Wake (physics)">wake</a> produced is very small and drag is dominated by the friction component. Therefore, such a body (here an airfoil) is described as streamlined, whereas for bodies with fluid flow at high angles of attack, boundary layer separation takes place. This mainly occurs due to adverse <a href="Pressure_gradient" title="Pressure gradient">pressure gradients</a> at the top and rear parts of an <a href="Airfoil" title="Airfoil">airfoil</a>.

Due to this, wake formation takes place, which consequently leads to eddy formation and pressure loss due to pressure drag. In such situations, the airfoil is <a href="Stall_(flight)" class="mw-redirect" title="Stall (flight)">stalled</a> and has higher pressure drag than friction drag. In this case, the body is described as a blunt body.

A streamlined body looks like a fish (<a href="Tuna" title="Tuna">tuna</a>), <a href="Oropesa_(minesweeping)" title="Oropesa (minesweeping)">Oropesa</a>, etc. or an airfoil with small angle of attack, whereas a blunt body looks like a brick, a cylinder or an airfoil with high angle of attack. For a given frontal area and velocity, a streamlined body will have lower resistance than a blunt body. Cylinders and spheres are taken as blunt bodies because the drag is dominated by the pressure component in the wake region at high <a href="Reynolds_number" title="Reynolds number">Reynolds number</a>.

To reduce this drag, either the flow separation could be reduced or the surface area in contact with the fluid could be reduced (to reduce friction drag). This reduction is necessary in devices like cars, bicycle, etc. to avoid vibration and noise production.

<ul><li><a href="Automotive_aerodynamics" title="Automotive aerodynamics">Automotive aerodynamics</a></li>

<li><a href="Automobile_drag_coefficient" title="Automobile drag coefficient">Automobile drag coefficient</a></li>

<li><a href="Ballistic_coefficient" title="Ballistic coefficient">Ballistic coefficient</a></li>

<li><a href="Drag_crisis" title="Drag crisis">Drag crisis</a></li>

<li><a href="Zero-lift_drag_coefficient" title="Zero-lift drag coefficient">Zero-lift drag coefficient</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFBaker1983" class="citation book cs1">Baker, W.E. (1983). <a rel="nofollow" class="external text" href="https://www.elsevier.com/books/explosion-hazards-and-evaluation/baker/978-0-444-42094-7">Explosion Hazards and Evaluation, Volume 5</a>. Elsevier Science. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-444-59988-9</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFAARØNÆS2014" class="citation book cs1">AARØNÆS, ANTON STADE (2014). <a rel="nofollow" class="external text" href="http://publications.lib.chalmers.se/records/fulltext/199998/199998.pdf">Dynamic response of pipe rack steel structures to explosion loads</a> (PDF). CHALMERS UNIVERSITY OF TECHNOLOGY.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFMcCormick1979" class="citation book cs1">McCormick, Barnes W. (1979). Aerodynamics, Aeronautics, and Flight Mechanics. New York: John Wiley & Sons, Inc. p. 24. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-03032-5</bdi>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFClancy1975" class="citation book cs1">Clancy, L. J. (1975). "5.18". Aerodynamics. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-15837-1</bdi>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="Ira_H._Abbott" class="mw-redirect" title="Ira H. Abbott">Abbott, Ira H.</a>, and Von Doenhoff, Albert E.: Theory of Wing Sections. Sections 1.2 and 1.3

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://wright.nasa.gov/airplane/drageq.html">"Modern Drag Equation"</a>. Wright.nasa.gov. 2010-03-25. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110302020853/http://wright.nasa.gov/airplane/drageq.html">Archived</a> from the original on 2011-03-02. Retrieved 2010-12-07.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> Clancy, L. J.: Aerodynamics. Section 11.17

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFHoerner1965" class="citation book cs1">Hoerner, Sighard F. (1965). <a rel="nofollow" class="external text" href="https://archive.org/details/FluidDynamicDragHoerner1965">Fluid-Dynamic Drag : Practical Information on Aerodynamic Drag and Hydrodynamic Resistance</a> (2 ed.). p. 3–17.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> See <a href="Lift_force" class="mw-redirect" title="Lift force">lift force</a> and <a href="Vortex_induced_vibration" class="mw-redirect" title="Vortex induced vibration">vortex induced vibration</a> for a possible force components transverse to the flow direction

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> Note that for the <a href="Earth's_atmosphere" class="mw-redirect" title="Earth's atmosphere">Earth's atmosphere</a>, the air density can be found using the <a href="Barometric_formula" title="Barometric formula">barometric formula</a>. Air is 1.293 kg/m3 at 0 °C (32 °F) and 1 <a href="Atmosphere_(unit)" class="mw-redirect" title="Atmosphere (unit)">atmosphere</a>.

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> See <a href="Lift_force" class="mw-redirect" title="Lift force">lift force</a> and <a href="Vortex_induced_vibration" class="mw-redirect" title="Vortex induced vibration">vortex induced vibration</a> for a possible force components transverse to the flow direction

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> Note that for the <a href="Earth's_atmosphere" class="mw-redirect" title="Earth's atmosphere">Earth's atmosphere</a>, the air density can be found using the <a href="Barometric_formula" title="Barometric formula">barometric formula</a>. Air is 1.293 kg/m3 at 0 °C (32 °F) and 1 <a href="Atmosphere_(unit)" class="mw-redirect" title="Atmosphere (unit)">atmosphere</a>.

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> Clancy, L. J.: Aerodynamics. Sections 4.15 and 5.4

</li>

<li id="cite_note-Clancy_4.17-14">^ <a href="#cite_ref-Clancy_4.17_14-0">a</a> <a href="#cite_ref-Clancy_4.17_14-1">b</a> Clancy, L. J.: Aerodynamics. Section 4.17

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> Clift R., Grace J. R., Weber M. E.: Bubbles, drops, and particles. Academic Press NY (1978).

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> Briens C. L.: Powder Technology. 67, 1991, 87-91.

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> Haider A., Levenspiel O.: Powder Technology. 58, 1989, 63-70.

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070715171817/http://aerodyn.org/Drag/tables.html">Shapes</a>

</li>

<li id="cite_note-tool-19"><a href="#cite_ref-tool_19-0">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.engineeringtoolbox.com/drag-coefficient-d_627.html">"Drag Coefficient"</a>. Engineeringtoolbox.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20101204050919/http://www.engineeringtoolbox.com/drag-coefficient-d_627.html">Archived</a> from the original on 2010-12-04. Retrieved 2010-12-07.</cite>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.grc.nasa.gov/WWW/k-12/airplane/shaped.html">"Shape Effects on Drag"</a>. NASA. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130216192122/http://www.grc.nasa.gov/WWW/K-12/airplane/shaped.html">Archived</a> from the original on 2013-02-16. Retrieved 2013-03-11.</cite>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> Basha, W. A. and Ghaly, W. S., "Drag Prediction in Transitional Flow over Airfoils," Journal of Aircraft, Vol. 44, 2007, p. 824–32.

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.aerospaceweb.org/question/aerodynamics/q0184.shtml">"Ask Us – Drag Coefficient & Lifting Line Theory"</a>. Aerospaceweb.org. 2004-07-11. Retrieved 2010-12-07.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.lissys.demon.co.uk/samp1/index.html">"Boeing 787 Dreamliner : Analysis"</a>. Lissys.demon.co.uk. 2006-06-21. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100813094712/http://www.lissys.demon.co.uk/samp1/index.html">Archived</a> from the original on 2010-08-13. Retrieved 2010-12-07.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.dept.aoe.vt.edu/~mason/Mason_f/A380Dean.pdf">"Airbus A380"</a> (PDF). 2005-05-02. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150923213641/http://www.dept.aoe.vt.edu/~mason/Mason_f/A380Dean.pdf">Archived</a> (PDF) from the original on 2015-09-23. Retrieved 2014-10-06.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><a href="L._J._Clancy" class="mw-redirect" title="L. J. Clancy">L. J. Clancy</a> (1975): Aerodynamics. Pitman Publishing Limited, London, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-273-01120-0</bdi></li>

<li>Abbott, Ira H., and Von Doenhoff, Albert E. (1959): Theory of Wing Sections. Dover Publications Inc., New York, Standard Book Number 486-60586-8</li>

<li>Hoerner, Dr. Sighard F., Fluid-Dynamic Drag, Hoerner Fluid Dynamics, Bricktown New Jersey, 1965.</li>

<li>Bluff Body: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170809151428/http://user.engineering.uiowa.edu/~me_160/lecture_notes/Bluff%20Body2.pdf">http://user.engineering.uiowa.edu/~me_160/lecture_notes/Bluff%20Body2.pdf</a></li>

<li>Drag of Blunt Bodies and Streamlined Bodies: <a rel="nofollow" class="external free" href="http://www.princeton.edu/~asmits/Bicycle_web/blunt.html">http://www.princeton.edu/~asmits/Bicycle_web/blunt.html</a></li>

<li>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.</li></ul></div><div><div class="zim-footer">

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