1. [20 pts.] We want to simplify Prandtl's lifting-line theory by assuming that a wing with elliptical loading, peak circulation $\Gamma_0$, and span $b$ can be represented by a single horseshoe vortex with constant circulation $\Gamma_0$ and span $b'$. (This is a crude but simple model that is useful for quick computations of interference effects.)
(a) [3 pts] Determine the span $b'$ of the simplified system such that the lift on the two wings is the same.
(b) [3 pts] Calculate the induced drag for the simplified system assuming that the downwash at the wing root can be used for the entire wing. What is the percentage error in the induced drag compared to the value for the elliptic wing using lifting-line theory?
(c) [2 pts] Let us consider the idea of modeling a wing-tail combination by using a single horseshoe vortex with constant circulation to model each of the two lifting surfaces. For this baseline case, the tail bound vortex is located a length $\ell = 3b'/8$ downstream of the wing bound vortex, where $b'$ is the wingspan as shown in the image. Determine the induced drag for the tail, $D_{i,t}$, as a function of $\Gamma_w$, $\Gamma_t$, $b'$, $V_\infty$ and $\rho_\infty$ (note that your answer might be in terms of some but not all these parameters). Simplify your final result. Assume the downwash can be approximated by the downwash at the center of the tail's bound vortex and that the span of the tail is $b'/4$. i.e.: the downwash on the tail should be evaluated at the point $(\ell = 3b'/8, 0, 0)$.
(d) [2 pts] Again, for this baseline case, determine the induced drag for the wing, $D_{i,w}$, as a function of $\Gamma_w$, $\Gamma_t$, $b'$, $V_\infty$ and $\rho_\infty$ (note that your answer might be in terms of some but not all of these parameters). Simplify your final result. Assume the downwash can be approximated by the downwash at the center of the wing's bound vortex i.e.: the downwash on the wing should be evaluated at the point $(0, 0, 0)$.
(e) [10 pts.] Use the simple theory to calculate the percentage change in the induced drag for the wing and the tail as a function of the separation distance $\ell$, as shown above when compared to the baseline case values found in part (c) and (d).
Write a computer program (using MATLAB/python/C++ etc. is fine) to compute and plot the percentage change in induced drag for the wing and tail as a function of the nondimensional streamwise distance $1 \le \ell/b' \le 10$. Discuss how the induced drag changes with the separation distance and whether the trends seem to make physical sense. (Hint: use the Biot-Savart law given in the class notes to compute the downwash velocity at the wing and tail root but be sure to account for the contributions of all the pieces of the vortex systems, keeping in mind that a straight-line vortex does not induce any motion on itself. Assume $\Gamma_w = 1$ and $\Gamma_t = 0.1$.
