Lie derivatives of fundamental surface quantities in incompressible viscous flows

Abstract

Lie derivative is an important concept in differential geometry. From the perspective of theoretical fluid dynamics, the present paper evaluates and interprets the Lie derivatives of the fundamental surface physical quantities (including skin friction, surface vorticity, and surface pressure) with respect to a characteristic velocity field in near-wall incompressible viscous flows. It is found that the Lie derivatives are directly associated with the boundary enstrophy flux, an orthogonal pair of skin friction and surface vorticity, and an orthogonal pair of surface enstrophy gradient and its conjugate vector, while components of the Lie derivatives in skin-friction-surface-vorticity orthogonal frame are related to four on-wall coupling scalar quantities (associated with the skin friction divergence and the surface vorticity divergence). The derived theoretical results are first evaluated in a laminar oblique Hiemenz flow and a turbulent channel flow. Then, features of the Lie derivatives are explored in a typical skin friction structure generated by a complex separated flow over a hill model. The present exposition provides a unique perspective of the Lie derivatives to the boundary vorticity dynamics and near-wall flow physics.