In this problem you will analyze NACA 4-digit airfoil camber lines numerically. To model the
flow, use N vortices distributed evenly on the camber line, as shown below. The strengths of these
vortices, Γj, 1 ≤ j ≤ N, are the unknowns. Note that to satisfy the Kutta condition, there is no
vortex at the trailing edge.
α
V
00
vortex strength Γ
camberline, ze(x)
ΓΝ
Γ
control point i
c/N
The equations are flow tangency enforced at N control points, which are also on the camber line,
half-way in between the vortices. The normal vector for enforcing flow tangency is perpendicular
to the camber line.
In a NACA 4-digit airfoil MPXX of chord length c, the camber line equation is
Zc(x)
C
m
[2]
[(1-2p)+2p-
cp
if x < p
m =
M
100'
p=
otherwise
P
10
a) Write down an expression for the flow tangency equation enforced at control point i, which
is at . Denote by ; the location of vortex j. Your equation will be in terms of z, Vo,
a, all of the Tj, the node locations Tj, and the camber line slope, dzc/dx. Do not assume a
or dze/dx are small.
b) Rewrite the flow tangency equations in part (a) as a linear system of equations,
АГ = F,
where A is an N × N matrix, F = [Γ1, ..., ΓΝ], and F is an N × 1 vector. Each row of this
system corresponds to a flow tangency equation at one control point. Give expressions for
the entries of the matrix, Aij, and the right-hand-side vector, Fi.
