A new velocity decomposition for viscous flows: Lighthill's equivalent-source method revisited

A convenient decomposition for vector fields is presented that falls within the vast class of potential-vorticity decompositions of the type upsilon = del phi + w, which includes the classical Helmholtz and Clebsch decomposit ions. The distinguishing feature of the present decomposition is that the r otational velocity contribution iv (obtained by a line-integration along a path normal to the boundary) vanishes in much of the irrotational region. A s a consequence, in the irrotational region one obtains upsilon = del phi. The mathematical formulation of the problem and the application to the anal ysis of incompressible viscous flows around solid bodies are discussed. For flows in which the vortical region is concentrated in a thin layer surroun ding the surface of the body and the wake mid-surface, the relationship bet ween the present formulation and the classical Lighthill's equivalent-sourc e approach is addressed. In addition, an exact extension of the Lighthill f ormulation is presented. The equivalence of the two approaches is establish ed. Numerical results for the reconstruction of the two-dimensional velocit y field (from a prescribed vorticity field) are presented to illustrate the differences between various approaches. (C) 1999 Elsevier Science S.A. All rights reserved.