Variational construction of tubular and toroidal streamsurfaces for flow visualization

TL;DR

This paper introduces a variational finite-element method to construct approximate streamsurfaces in 3D flows, overcoming previous Fourier series limitations, and demonstrates its effectiveness across various complex flow domains.

Contribution

It presents a novel variational approach using finite-element methods to compute approximate first integrals for flow visualization, applicable to arbitrary geometries and boundary conditions.

Findings

Effective visualization of vortical regions in complex 3D flows.

Successful application to various flow domains including spheres, cylinders, and convection.

Extraction of momentum barriers in Rayleigh-Bénard convection.

Abstract

Approximate streamsurfaces of a 3D velocity field have recently been constructed as isosurfaces of the closest first integral of the velocity field. Such approximate streamsurfaces enable effective and efficient visualization of vortical regions in 3D flows. Here we propose a variational construction of these approximate streamsurfaces to remove the limitation of Fourier series representation of the first integral in earlier work. Specifically, we use finite-element methods to solve a partial-differential equation that describes the best approximate first integral for a given velocity field. We use several examples to demonstrate the power of our approach for 3D flows in domains with arbitrary geometries and boundary conditions. These include generalized axisymmetric flows in the domains of a sphere (spherical vortex), a cylinder (cylindrical vortex), and a hollow cylinder (Taylor-Couette flow) as benchmark studies for various computational domains, non-integrable periodic flows (ABC and Euler flows), and Rayleigh-B\'enard convection flows. We also illustrate the use of the variational construction in extracting momentum barriers in Rayleigh-B\'enard convection.