The Poincare Transformation [4] 



It is well known that the velocity field induced by a vortex ring is 

 identical to that of a doublet sheet on a surface capping the ring. The 

 Poincare transformation, derived below, enables more general relationships 

 between fields of vorticity and irrotational distributions to be obtained. 

 Since the entire disturbance flow field is induced by the vorticity alone, 

 the velocity field of the equivalent irrotational distributions would be 

 identical to that induced by the vorticity in the regions exterior to the 

 vorticity domain. For the case where the vorticity lies in the boundary 

 layer and wake of a body, the vorticity induces not only an irrotational 

 field exterior to its outer boundary T, but also an irrotational field 

 within the body. Because of the nonslip condition, the induced velocity 

 within the body must be zero. Thus, in contrast with the distributions 

 previously considered, the Poincare transformation offers the possibility of 

 matching the boundary conditions on both the interior and exterior bound- 

 aries of BLW. 



Let us suppose that vorticity co = curl v is present in a domain D, 

 bounded by a closed surface S, and denote the domain exterior to D by E. 

 Here v denotes the velocity vector of the fluid flow. We shall distinguish 

 between a fixed point P(x,y,z) at which induced velocities are calculated, 

 and variable points of integration, Q{E,,T],C,) . The position vector from P 

 to Q is r and has the magnitude Tp^. 



The velocity induced by a vorticity distribution can be expressed 

 either by means of the Biot-Savart Law, 



1_ f^ 



^ dx (51) 



PQ 



or in terms of the vector potnetial. 



CO 



J^PQ 



Vp = 1^ Vp X I f^ dx (52) 



20 



