In this problem you will model non$-$lifting flow over a unit$-$chord NACA $0020$ airfoil using discrete

sources placed on the chordline, as shown below.

The unknowns are the strengths of the $N$ sources, $Q_j,1jN,$ whose coordinates are vec$(x{)}_j.$ The

equations are flow tangency enforced at the $N$ control points, vec$(x{)}_i^{cp}.$ By superposition, the velocity

at control point $i$ is

vec$(v{)}_i={\displaystyle \underset{j=1}{\overset{N}{}}}vec(v{)}_{ij}+U_{}$hat$(i)$

where $vec(v{)}_{ij}$ is the velocity induced by source $j$ on control point $i.$ Flow tangency requires that

vec$(v{)}_i*vec(n{)}_i=0,$ where vec$(n{)}_i$ is the normal vector $($perpendicular to the geometry$)$ at control point $i.$ The

geometry of the airfoil surface is

$y=+-[0\mathrm{.}2969\sqrt[\phantom{2}]{x}-0\mathrm{.}1260x-0\mathrm{.}3516x^2+0\mathrm{.}2843x^3-0\mathrm{.}1015x^4]$

a$)$ Derive an expression for $vec(v{)}_{ij}$ in terms of $Q_j,$ the coordinates of control point $i$ : vec$(x{)}_i^{cp}=$

$(x_i^{cp},y_i^{cp}),$ and the coordinates of node $j$:vec$(x{)}_j=(x_j,y_j).$

b$)$ Determine $A_{ij}$ in the expression

$vec(v{)}_{ij}*vec(n{)}_i=A_{ij}{Q}_j,(no$ implied sums$)$

Your answer should be in terms of vec$(x{)}_i^{cp},$vec$(x{)}_j,$ and vec$(n{)}_i.$ Give a formula for vec$(n{)}_i$ in terms of the

geometry and its slope.

c$)$ Show that the flow tangency condition applied at the $N$ control points gives a linear system

of equations for the unknown source strengths,

$AQ=b$

where $A$ is an $N$ by $N$ matrix with entries $A_{ij}($from the previous part$),$ Q is a vector of $N$

unknown source strengths, and $b$ is a right$-$hand$-$side vector $(N$ entries$)-$ give an expression

for the entries of $b.$