Vortex dynamics on a 2D curved surface using discrete exterior calculus

Abstract

We present examples of vortex dynamics on a curved 2D surface using simulations based on the approach of discrete exterior calculus (DEC) developed by Mohamed et al. [8]. The DEC method allows a coordinate independent implementation and preserves vector calculus identities discretely. In addition, it has been demonstrated to exhibit good conservation of secondary flow quantities such as kinetic energy (for inviscid flows) and discretely satisfies vorticity conservation both locally and globally. These structure-preserving properties of DEC make it a suitable tool to investigate vortex and vorticity dynamics on 2D curved surfaces. In particular, we focus on the flow past a stationary cylinder (a canonical problem in fluid dynamics) embedded on a sphere (positive Gaussian curvature) or another cylinder (zero Gaussian curvature). We compute quantities of dynamical importance such as the drag coefficient, lift coefficient and Strouhal number. Another example presented is the evolution of initially randomly distributed vortices on a unit sphere in to a quadrupolar vortical structure similar to reported in Dritschel et al. [4].