Charaoteristias of Ship Boundary Layers 



IX. INTEGRAL EQUATION FOR A VORTEX SHEET FOR IRROTA- 

 TIONAL FLOW ABOUT A THREE-DIMENSIONAL FORM 



A three-dimensional form bounded by a surface S is im- 

 mersed in a uniform_stream of velocity U in the positive x-direc- 

 tion, of unit vector i. We shall suppose that the fluid is inviscid 

 and incompressible. Let us assume that the disturbance of the flow 

 due to the body may be represented by a vortex sheet of strength 

 y = ya where a is a unit vector tangent to the surface S such that 

 the fluid within the body is at rest. 



In crossing S in the direction of its outward normal, 

 designated by the unit vector n, there is a discontinuity in the tan- 

 gential component of the velocity of the fluid, of magnitude y, in 

 the direction with unit vector 



s = a X n. (40) 



By continuity, since the fluid oii_the interior side of S is at rest, 

 the velocity components in the o and n directions at the exterior 

 side of S must also vanish, and hence the velocity at the exterior 

 side of S is given by 



u=YcrXn = \'Xn. (41) 



Since, a priori , the mutually orthogonal directions of the 



streamlines, s", and of the vortex lines, a, are unknown, it is 



necessary to introduce a set of orthogonal, curvilinear coordinate 



lines on S, ^ = const, and r\ = consto _Denote unit vectors in the 



direcUons_of increasing i and r\ by e and e , with sense such 



that e X e„ = n. Put 

 I 2 



"Y=ey ^-ey, u=eu + e v . (42) 



Y ,Y, 2''2' 12 



Then, by (41), we have 



u = Vg* V = - Y| . (43) 



An integral equation for the vorticity vector y can be derived 

 from the condition that the contributions to the velocity on the interior 

 side of a point P of S must sum to zero. This gives 



in which the integral, obtained from the Biot-Savart law, represents 



465 



