Linearization of the Tangency Condition

The tangency condition reads V. nobstacie = 0. Consider a small element of the profile surface (dx, dz) = tdl where t is the tangent unit vector and dl the length of the element. A normal vector to the profile is ndl = (dz, – dx). The above condition can be written

(U + U, w). (dz, – dx)obstacle = ((U + u)dz – wdx) obstacle = 0 (3.18)

Dividing by dx (dx = 0) and applying the condition on the obstacle

(U + u(x, z(x))) – w(x, z(x)) = 0

Using the fact that u, w, z and || are all of order O(d + c + a), the above expression can be simplified and expanded to first order as

dz dz d w d e 2

U – w(x, 0) + u(x, 0) – z(x) (x, 0) + O +- + a = 0 (3.20)

dx dx dz c c

Note that the third and fourth terms are already second order terms and can be included in the remainder. Finally, keeping only the leading terms results in the new form of the tangency condition

dz

w(x, 0) = U (x)

dx

To be more specific, one can formulate the tangency condition in terms of the profile shape and incidence as

w(x, 0±) = U (f ±(x) – a^

This operation is called the transfer of the tangency condition from the profile surface to the x-axis. Note that, as regard the mathematical treatment of the tangency condition, the profile is replaced by a slit along the x-axis with an upper (0+) and a lower (0-) edge.

Note also that the tangency condition is expressed only in terms of w.