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Birds have a very unique way of flying that sets them apart from the techniques used by humans in their aircraft. In this case, the wings serve a dual purpose, generating both lift and thrust, even when the bird is stationary.

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Concept

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Concept

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If you study the video of a pigeon flying from a lateral perspective, you can observe how it advances and retracts its wings.

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During the forward phase, the bird manages to generate lift, while during the backward phase, it seeks propulsion.

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Concept

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If you study the video of a pigeon flying from a frontal perspective, you can observe how it extends and retracts its wings.

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During the forward phase, the bird extends its wings for the first time to generate lift, while during the backward phase, it extends them for the second time to propel itself forward.

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Description

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To model the wing, we need to estimate the wing span ($L$), the width the wing width ($\Delta$), and the wing height ($\delta$) of the wing in order to calculate the surface that generates lift ($S_w$) and the total object profile ($S_p$). An article with data for migratory birds can be found in [1]:

Note: In this case, wing areas and spans are provided, so the width can be estimated as $S_w/L$. Similarly, the wing height can be estimated from the profile area divided by the span $S_p/L$, although in this case, we are not considering that the profile includes the bird's body section.

[1] "Field Estimates of Body Drag Coefficient on the basis of dives in passerine Birds," Anders Hedenström, Felix Liechti, The Journal of Experimental Biology, 204, 1167-1175 (2001).

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Image

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When we compare different types of wings, we notice that raptors tend to have shorter and broader wings, whereas migratory birds have longer and narrower ones. Therefore, it makes sense to define the wing factor ($\gamma_w$) as the relationship between the wing span ($L$) and the wing width ($\Delta$):

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Selected parameter

Calculations

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Equation

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The surface that generates lift ($S_w$) can be estimated using the wing span ($L$) and the wing width ($\Delta$) as follows:

$ S_w = L \Delta $

Conditions

Variables

$S_w$

Surface that generates lift

$m^2$

$L$

Wing span

$m$

$\Delta$

Wing width

$m$

Relation

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Equation

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The total object profile ($S_p$) can be estimated using the wing span ($L$) and the wing height ($\delta$) as follows:

$ S_p = L \delta $

Conditions

Variables

$S_p$

Total object profile

$m^2$

$\delta$

Wing height

$m$

$L$

Wing span

$m$

Relation

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Equation

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The wing factor ($\gamma_w$) is defined as the relationship between the wing span ($L$) and the wing width ($\Delta$):

$ \gamma_w =\displaystyle\frac{ L }{ \Delta }$

Conditions

Variables

$\gamma_w$

Wing factor

$-$

$L$

Wing span

$m$

$\Delta$

Wing width

$m$

Relation

This factor tends to be larger in migratory birds and smaller in raptors.

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Equation

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By analogy to the wing factor ($\gamma_w$), we can define the wing profile factor ($\gamma_p$). This relates the wing span ($L$) to the wing height ($\delta$) as follows:

$ \gamma_p =\displaystyle\frac{ L }{ \delta }$

Conditions

Variables

$\delta$

Wing height

$m$

$\gamma_p$

Wing profile factor

$-$

$L$

Wing span

$m$

Relation

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Equation

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Just as the power of flight ($P$) is related to the density ($\rho$), the total object profile ($S_p$), the coefficient of resistance ($C_W$), the body mass ($m$), the gravitational Acceleration ($g$), the proportionality constant coefficient sustainability ($c$), the surface that generates lift ($S_w$), and the speed with respect to the medium ($v$) through

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we can express the power in terms of the wing factor ($\gamma_w$) and the wing profile factor ($\gamma_p$) as

$ P_w =\displaystyle\frac{1}{2} \rho L ^2 C_w v ^3\displaystyle\frac{1}{ \gamma_p }+\displaystyle\frac{2 m ^2 g ^2}{ c ^2 L ^2 \rho } \gamma_w \displaystyle\frac{1}{ v }$

Conditions

Variables

Relation

Development

As the power of flight ($P$) is related to the density ($\rho$), the total object profile ($S_p$), the coefficient of resistance ($C_W$), the body mass ($m$), the gravitational Acceleration ($g$), the proportionality constant coefficient sustainability ($c$), the surface that generates lift ($S_w$), and the speed with respect to the medium ($v$) through

$ P =\displaystyle\frac{1}{2} \rho S_p C_W v ^3 + \displaystyle\frac{2 m ^2 g ^2}{ c ^2 S_w \rho }\displaystyle\frac{1}{ v }$

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with the definitions of the surface that generates lift ($S_w$) in terms of the wing width ($\Delta$)

$ S_w = L \Delta $

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and the wing factor ($\gamma_w$)

$ \gamma_w =\displaystyle\frac{ L }{ \Delta }$

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along with wing Profile ($S_p$) in relation to the wing height ($\delta$)

$ S_p = L \delta $

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and the wing profile factor ($\gamma_p$)

$ \gamma_p =\displaystyle\frac{ L }{ \delta }$

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finally, as

$ P_w =\displaystyle\frac{1}{2} \rho L ^2 C_w v ^3\displaystyle\frac{1}{ \gamma_p }+\displaystyle\frac{2 m ^2 g ^2}{ c ^2 L ^2 \rho } \gamma_w \displaystyle\frac{1}{ v }$

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