A theoretical framework of vorticity dynamics for two dimensional flows on fixed smooth surfaces

TL;DR

This paper develops a theoretical framework for vorticity dynamics in two-dimensional flows on fixed smooth surfaces, incorporating surface curvature effects into fundamental equations and relations.

Contribution

It extends classical vorticity dynamics to curved surfaces by deriving new governing equations and relations that include surface curvature effects.

Findings

Derived vorticity governing equation on curved surfaces

Extended Lagrange theorem and Caswell formula to surface flows

Proposed stream function and vorticity algorithm with pressure Poisson equation

Abstract

Two dimensional flows on fixed smooth surfaces have been studied in the point of view of vorticity dynamics. Firstly, the related deformation theory including kinematics and kinetics is developed. Secondly, some primary relations in vorticity dynamics have been extended to two dimensional flows on fixed smooth surface through which a theoretical framework of vorticity dynamics have been set up, mainly including governing equation of vorticity, Lagrange theorem on vorticity, Caswell formula on strain tensor and stream function & vorticity algorithm with pressure Possion equation for incompressible flows. The newly developed theory is characterized by the appearances of surface curvatures in some primary relations and governing equations.