Birds & Insects - Natural Flyers
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Lift :





                To understand the exact mechanism of lift in case of a flying animal, it is easier to first treat the aerodynamics in a relatively simpler manner. Studying the movement of an object through a fluid is equivalent to studying the flow of the fluid about a stationary object, in terms of fluid dynamics. This is the Lagrangian viewpoint, as opposed to the Eulerian viewpoint.



Let a cylinder with circular cross section (of radius b) be immersed in a fluid flowing with a constant velocity u
. Now, if the cylinder starts to rotate (with spin s rad/sec), it can be shown that the cylinder experiences a lift! And, the lift per unit length of the cylinder is given by L = r G u.

Here,
G ( = 2 p b Vr ) is the circulation,
Vr ( = 2 p b s) is the rotational speed of the cylinder,
r is the fluid density.


http://www.nasa.gov/centers/glenn/home/index.html

A rotating cylinder immersed in a steady fluid flow (Ref: http://www.nasa.gov/centers/glenn/home/index.html )




The coefficient of lift is defined as

lift coefficient

with A being the area of the surface over which the fluid flows, and
r being the fluid density.



The Mathematics (our tool to comprehend nature's intricacies) :

From the condition of continuity in a fluid, flowing with velocity u, . u = 0.

With this condition satisfied, the flow can be described by a streamfunction, Ψ,  such that the streamlines are lines along which Ψ is constant. The radial velocity(ur)  and angular velocity (uΘ)  are then given by (in polar co-ordinates)

velocity components


If the flow is irrotational, or the vorticity is zero, then  ∇ X u = 0.

Another function, Φ, better known as the velocity potential, can be derived from this equation, such that,

velocity components


A complex function, w(z), can be defined as w =  Φ + i ψ. w(z) is called the complex potential.

Now, if a constant flow, with a clockwise (hence, negative) circulation, -Γ, is considered around a stationary cylinder, the resulting complex potential looks like

complex potential

Hence, taking the imaginary part of w to get
ψ, and then differentiating it yields uΘ.

angular velocity

The value for
uΘ at r = b can be determined from this equation. It may also be noted that  ur= 0 at r=b.

The fact that this expression for the angular velocity at the surface of the cylinder gives rise to a pressure, is obtained from Bernoulli’s equation, which states that,

Bernoulli's equation


where p is the pressure,
r is the density, and h is the given height for the fluid.

  Hence, considering that
ur= 0 at r=b, and that there is no horizontal pressure due to symmetry, the vertical component of the pressure can be integrated to get the lift per unit length of the cylinder :

lift


Now, it needs to be noted that, the above expression for lift is valid not just for cylinders with circular cross sections, but for all other cross sections as well!

The following applet illustrates this point. You may change  the input option to "Shape/Angle", and then vary the option just underneath the movie to "airfoil", "cylinder", etc. To enjoy more of the fun, play with all the other options!



No wonder, then, that animals with wings, the cross-sections of which resemble air foils, can sail through air!



          
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 PHYS 645, University of Alaska Fairbanks
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