Ian Stokes edited k.tex  over 9 years ago

Commit id: 8f45426715bf0bfc5b70240891d008aca8426985

deletions | additions      

       

The first situation is the simplest and thus I will hash out this scenario before the others. The magnitude of the lift force on an object in flight is given by the density of the fluid the object is traveling through times the velocity of the object times the circulation, where circulation is the closed path integral of the fluid velocity around a wing's cross section. For a bird in flight, the circulation ( $\Gamma$ ) can be approximated as $\Gamma$ = $\ (C_{L}/2)(v_{o}A_{p})$ where$\ A_{p}$ is the planiform area of the wing: that is, the area of the wing projected onto the relevant xy plane. Therefore the magnitude of the lift force is given by (1/2)($\rho_{o}$)($\ A_{p})(v^2)$. The drag force follows the same relation, but$\ C_{L}$ is replaced by$\ C_{D}$. Choosing the xy plane to be parallel with the plane of the bird's flight in a standard, right handed cartesian coordinate system, we can set $\Sigma(F_{x})$ and $\Sigma(F_{z})$ both equal to zero as we are solving for the angle where the bird flies at a constant altitude. Solving for this angle, $\theta$, we find that $\tan$( $\theta$ ) =$\ C_{D}$/$\ C_{L}$. I will refer to this as the glide angle equation.   The solution to the second problem is beyond the scope of simple geometric arguments. To solve for the air flow over ocean swells, a good method is to introduce an ensatz, or in other words, a guess. As a first step, I approximated an ocean swell as a sinusoidal wave of amplitude$\ h$. From there I required that at a large height above the wave the wind would feel no effects from the wave, and would travel with$\ v_{x}$ = constant =$\ v_{o}$, the undisturbed wind speed. Initially Both$\ v_{x}$ and$\ v_{z}$ decay at equal rates with additional altitude$\ z$ and therefore obey the same exponential decay factor, $\ e^($\alpha$)($\ x$)$ Since at this point I do not have any intuition for what this factor is, I introduced the undetermined constant $\alpha$.Initially  I thought that due to radial accelleration along the trough, combined with the force of gravity pushing air down the backside of an arbitrary swell, that the maximum wind velocity would occur at the deepest point of the trough, trough. On the flipside the wind would be least on top of the wave as it has to fight gravity in traversing up the face. One may think the wave would block the wind, but I am considering waves  under assumption the constraint  that wavelength $\lambda$ >>$\ h$. It turns out, h$ to negate that factor.    An ancient creature, estimates have it that this majestic bird has been cruising the skies for at least thirty million years. Historical geologists would call this period of time the "Oligocene epoch" of the "Paleogene period," but to the rest of us these implications can be put in much simpler terms--Pelicans are dinosaurs! Perhaps it is to this that the Brown Pelican owes its aerial expertise. Thirty million years has allowed evolution to take its course, and over the generations the pelican has been able to develop the artform of "compression surfing."