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Exercises

  1. Consider the integral

    $\displaystyle I_n(\phi) = \int_0^\pi \frac{\cos(n\,\phi')\,d\phi'}{\cos\phi'-\cos\phi},
$

    where $ n$ is a non-negative integer. This integral is defined by its Cauchy principal value

    $\displaystyle I_n(\phi)=\lim_{\epsilon\rightarrow 0}\left[ \int_0^{\phi-\epsilo...
..._{\phi+\epsilon}^\pi \frac{\cos(n\,\phi')\,d\phi'}{\cos\phi'-\cos\phi}\right].
$

    As was demonstrated in Section 9.9,

    $\displaystyle I_0 = 0.
$

    Show that

    $\displaystyle I_1=\pi,
$

    and

    $\displaystyle I_{n+1} + I_{n-1} = 2\,\cos\phi\,I_n,
$

    and, hence, that

    $\displaystyle I_n = \pi\,\frac{\sin(n\,\phi)}{\sin\phi}.
$

  2. Suppose that an airfoil of negligible thickness, and wingspan $ b$ , has a width whose $ z$ variation is expressed parametrically as

    $\displaystyle c(\phi) = \sum_{\nu=1,3,5,\cdots}c_\nu\,\sin(\nu\,\phi),
$

    for $ 0\leq \phi\leq \pi$ , where

    $\displaystyle z = -\frac{b}{2}\,\cos\phi.
$

    Show that the air circulation about the airfoil takes the form

    $\displaystyle {\mit\Gamma}(\phi) = \sum_{\nu=1,3,5,\cdots} {\mit\Gamma}_\nu\,\sin(\nu\,\phi),
$

    where $ {\mit\Gamma}_\nu = \pi\,V\,\alpha\,c_\nu$ . Here, $ \alpha $ is the angle of attack (which is assumed to be small). Demonstrate that the downwash velocity at the trailing edge of the airfoil is

    $\displaystyle w(\phi)= \sum_{\nu=1,3,5,\cdots} \frac{\nu\,{\mit\Gamma}_\nu}{2\,b}\,\frac{\sin(\nu\,\phi)}{\sin\phi}.
$

    Hence, show that the lift and induced drag acting on the airfoil take the values

    $\displaystyle L$ $\displaystyle = \frac{\pi}{4}\,\rho\,V\,b\,{\mit\Gamma}_1,$    
    $\displaystyle D$ $\displaystyle =\frac{\pi}{8}\,\rho\,\sum_{\nu=1,3,5,\cdots} \nu\,{\mit\Gamma}_\nu^{\,2},$    

    respectively. Demonstrate that the drag to lift ratio can be written

    $\displaystyle \frac{D}{L} = \frac{2}{A}\,\alpha\left(1+\sum_{\nu=3,5,7,\cdots}\frac{\nu\,c_\nu^{\,2}}{c_1^{\,2}}\right),
$

    where $ A$ is the aspect ratio. Hence, deduce that the airfoil shape (in the $ x$ -$ y$ ) plane that minimizes this ratio (at fixed aspect ratio) is an ellipse (i.e., such that $ c_\nu=0$ for $ \nu>1$ ).

  3. Consider a plane that flies with a constant angle of attack, and whose thrust is adjusted such that it cancels the induced drag. The plane is effectively subject to two forces. First, its weight, $ {\bf W} = -W\,{\bf e}_y$ , and second its lift $ {\bf L} = -k\,v\,v_y\,{\bf e}_x + k\,v\,v_x\,{\bf e}_y$ . Here, $ x$ and $ y$ are horizontal and vertical coordinates, respectively, $ {\bf v}$ is the plane's instantaneous velocity, and $ k$ is a positive constant. Note that the lift is directed at right angles to the plane's instantaneous direction of motion, and has a magnitude proportional to the square of its airspeed. Demonstrate that the plane's equations of motion can be written

    $\displaystyle \frac{dv_x}{dt}$ $\displaystyle = - \frac{v\,v_y}{h},$    
    $\displaystyle \frac{dv_y}{dt}$ $\displaystyle = \frac{v\,v_x}{h} - g,$    

    where $ h=k\,g/W$ is a positive constant with the dimensions of length. Show that

    $\displaystyle \frac{1}{2}\,v^{\,2} + g\,y = {\cal E},
$

    where $ {\cal E}$ is a constant. Suppose that $ v_x=\sqrt{g\,h}\,(1+u)$ and $ v_y=\sqrt{g\,h}\,w$ , where $ \vert u\vert$ , $ \vert w\vert\ll 1$ . Demonstrate that, to first order in perturbed quantities,

    $\displaystyle \frac{du}{dt}$ $\displaystyle \simeq - \sqrt{\frac{g}{h}}\,w,$    
    $\displaystyle \frac{dw}{dt}$ $\displaystyle \simeq 2\sqrt{\frac{g}{h}}\,u.$    

    Hence, deduce that if the plane is flying horizontally at some speed $ v_0$ , and is subject to a small perturbation, then its altitude oscillates sinusoidally at the angular frequency $ \omega= \sqrt{2}\,g/v_0$ . This type of oscillation is known as a phugoid oscillation.

next up previous
Next: Incompressible Viscous Flow Up: Incompressible Aerodynamics Previous: Simple Flight Problems
Richard Fitzpatrick 2016-03-31