Equivalence of dipole sheet and vortex sheet both of variable strength density

Abstract

강도가 일정치 않은 dipole sheet와 vortex sheet의 등가성을 고찰한다. 한 개 곡면 위에 분포된 dipole 에 의하여 유도된 속도가 그 곡면 위에 놓인 많은 ring vortex 에 의하여 유도된 속도들의 중첩과 같음을 보이고 이 중첩을 vorticity 분포에 의한 유도속도임을 나타내는 두 적분 즉 개곡면의 윤곽선을 따르면서 하는 선적분과 개 곡면 위에서의 면적적분의 합으로 변형한다. 이와 같은 결과의 유도는 개 곡면을 덮는 grid 의 cell 을 생각하여 인접 cell 에서의 dipole 강도변화에 대한 해석을 통하여 이루어지며 한편 순수한 수학적 수식조작에 의해서도 동일한 결과가 얻어짐을 보인다.

The equivalence of dipole sheet and vortex sheet is considered in this paper. The velocity induced by dipoles distributed on an open surface is shown, by the use of vector differentiation identities, to be superposition of velocities induced by a large number of ring vortices placed on the surface. This expression is then changed to the sum of two terms; one a line integral along the boundary of the open surface and the other a surface integral on the surface, both integrals carrying the implication of velocity induced by vorticity. The derivation of this expression is through the analysis of dipole strength variation across the cells of grid which covers the open surface. An alternative derivation of equivalence through pure mathematical manipulation is also presented.

The equivalence of dipole sheet and vortex sheet is considered in this paper. The velocity induced by dipoles distributed on an open surface is shown, by the use of vector differentiation identities, to be superposition of velocities induced by a large number of ring vortices placed on the surface. This expression is then changed to the sum of two terms; one a line integral along the boundary of the open surface and the other a surface integral on the surface, both integrals carrying the implication of velocity induced by vorticity. The derivation of this expression is through the analysis of dipole strength variation across the cells of grid which covers the open surface. An alternative derivation of equivalence through pure mathematical manipulation is also presented.