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

Abstract

A convenient decomposition for vector fields is presented that falls within the vast class of potential-vorticity decompositions of the type υ = V ¯ ϕ + w , which includes the classical Helmholtz and Clebsch decompositions. The distinguishing feature of the present decomposition is that the rotational velocity contribution w (obtained by a line-integration along a path normal to the boundary) vanishes in much of the irrotational region. As a consequence, in the irrotational region one obtains υ = V ¯ ϕ . The mathematical formulation of the problem and the application to the analysis of incompressible viscous flows around solid bodies are discussed. For flows in which the vortical region is concentrated in a thin layer surrounding the surface of the body and the wake mid-surface, the relationship between the present formulation and the classical Lighthill's equivalent-source approach is addressed. In addition, an exact extension of the Lighthill formulation is presented. The equivalence of the two approaches is established. Numerical results for the reconstruction of the two-dimensional velocity field (from a prescribed vorticity field) are presented to illustrate the differences between various approaches.