Vortex boundaries as barriers to diffusive vorticity transport in two-dimensional flows

We put forward the idea of defining vortex boundaries in planar flows as closed material barriers to the diffusive transport of vorticity. Such diffusive vortex boundaries minimize the leakage of vorticity from the fluid mass they enclose when compared to other nearby material curves. Building on recent results on passive diffusion barriers, we develop an algorithm for the automated identification of such structures from general, two-dimensional unsteady flow data. As examples, we identify vortex boundaries as vorticity diffusion barriers in two flows: an explicitly known laminar flow and a numerically generated turbulent Navier-Stokes flow.

Figure 1

Family of limit cycles of Eq. (5). Each material curve has the same pointwise vorticity transport density ${\mathcal{T}}_0$. The outermost member of this family serves as the diffusive vortex boundary.

Figure 2

Streamlines of Eq. (14) with $\nu =0.001$ at $t=0$ overlaid on the initial vorticity field.

Figure 3

Barriers to vorticity transport (blue) superimposed on the $ln\left[{\text{DBS}}_{t_0}^{t_1}\left({\mathbf{x}}_0\right)\right]$ field inside a vortex array of Eq. (14). The analysis was performed for $t_0=0$ and $t_1=1$ (left) and $t_1=5$ (right). In both cases, we set the kinematic viscosity $\nu =0.001$.

Figure 4

Lagrangian and Eulerian vortex identification methods on a decaying turbulence simulation with integration time $t_1=50$. (a) Diffusive vortex boundaries (red) and material diffusion barriers (yellow) superimposed with the $ln\left[{\mathrm{DBS}}_0^{50}\left({\mathbf{x}}_0\right)\right]$ field. (b) Diffusive vortex boundaries (red) and black-hole eddies (yellow) overlaid on the $\text{FTLE}\left({\mathbf{x}}_0\right)$ field. (c) Diffusive vortex boundaries (red) and LAVD vortices (yellow) superimposed with the $\text{LAVD}\left({\mathbf{x}}_0\right)$ field. (d) Diffusive vortex boundaries (red) overlaid on the negative Okubo-Weiss parameter field. All panels show the entire computational domain $[0,2\pi ]\times [0,2\pi ]$.

Figure 5

Final position at time $t_1=50$ of advected diffusive vortex boundaries (red) and black-hole eddies (yellow) overlaid on the advected position of a uniform grid of $500\times 500$ tracers color coded with their ${\mathrm{DBS}}_0^{50}\left({\mathbf{x}}_0\right)$ value. Close-ups: Tangential filamentation of the diffusive vortex boundaries in contrast to the unstretched black-hole eddies.

Figure 6

Left: Extracted diffusive vortex boundary (red) overlaid on level sets of the Okubo-Weiss (OW) parameter. Right: Advected image of the diffusive vortex boundary and a set of tracers lying initially on the black level set of the OW parameter superimposed with the final position of an originally uniform grid of points color coded with their vorticity value.

Figure 7

Evolution of $|\widehat{\omega }({\mathbf{x}}_0,t)|$, the vorticity norm in Lagrangian coordinates. Left: Colored contours of $|\widehat{\omega }({\mathbf{x}}_0,0)|$ with the diffusive vortex boundary extracted from our algorithm (red) and OW level set (black). Right: Contours of $|\widehat{\omega }({\mathbf{x}}_0,50)|$ with the same red and black curves at $t_1=50$.