Constant-Strength Source Distribution

Consider a source distribution along the x axis as shown in Fig. 10.5. It is assumed that the source strength per length is constant such that o(x) = a — const. The influence of this distribution at a point P is an integral of the influences of the point elements (described in the previous section) along the segment Xi~*x2.

The integral for the velocity potential (Eq. (10.11)) appears in Appendix B, Eq. (B.5), (note In r2 = 2 In r is used in the derivation) and in terms of the

• p

U. z)

Returning to x—z variables these equations become

z -*i

Ф = 4jt ~ln “Xl)>2 + ~(x~*2),n К* – xz)2 + z2]

Of particular interest is the case when the point P is on the element (usually at the center). In this case z = 0± and the potential becomes

Ф(х, 0±) = [(x – Xi) ln (x – x,)2 – (x – x2) ln (x – x2)2] (10.22)

and at the center of the element it becomes

ф(£іті-г’0±)-й(Іг-^1п(і1?;і)2 (Щ22л)

The x component of the velocity at z = 0 becomes

“(j:’0±)’5lni(f^)i (ШЗ>

which is zero at the panel center and infinite at the panel edges.

For evaluating the w component of the velocity, it is important to distinguish between the conditions when the panel is approached from its upper or from its lower side. For the case of P being above the panel 0, = 0, while в2 = л. These conditions are reversed on the lower side and therefore

w(x, 0±) = ±^

This is the same result obtained in Section 3.14 for the source distribution.