turbulenceteam | Reconnection of vortex tubes in Eulerian and Lagrangian coordinates @turbulenceteamms | Uploaded 6 years ago | Updated 3 hours ago
The left panel shows the dynamics of two vortex tubes in a three-dimensional periodic box. The centers of the vortex tubes approach each other and a reconnection occurs, where one half of the fist vortex tube disconnects from its other half and connects to a half of the second vortex tube. Note that one can also consider a vortex tube in a periodic box to be a vortex ring as its ends are connected to each other.
The right panel shows the same phenomenon in Lagrangian coordinates, i.e. from the point of view of Lagrangian particles (tracers), which have been mapped back to their point of origin for the purpose of this visualization.
Here the same dynamics looks very different. The vortex tubes don't move but a complex vortical structure emerges around the centers of the vortices; this is where the reconnection happens. The high intensity of the vorticity in this structure (indicated by the red color) might due to the topological change (which a reconnection represents).
When one transforms the Navier-Stockes to Lagrangian coordinates, one sees that the only term driving the dynamics stems from the viscosity of the fluid. Hence without viscosity the vorticity distribution in right panel would not change at all (although it is not clear what would happen in left panel). The changes one sees are induces by viscosity and its influence on the metric tensor of the Lagrangian coordinates.
The visualizations are based on simulation of the 3D Navier-Stokes equation in a periodic box and a simulation of Lagrangian particles (tracers) which allows it to view the dynamics in the Lagrangian coordinates.
[1] A. Daitche, Statistische und geometrische Eigenschaften turbulenter Strömungen, Master’s thesis, Institut for Theoretical Physics, University of Münster.
The left panel shows the dynamics of two vortex tubes in a three-dimensional periodic box. The centers of the vortex tubes approach each other and a reconnection occurs, where one half of the fist vortex tube disconnects from its other half and connects to a half of the second vortex tube. Note that one can also consider a vortex tube in a periodic box to be a vortex ring as its ends are connected to each other.
The right panel shows the same phenomenon in Lagrangian coordinates, i.e. from the point of view of Lagrangian particles (tracers), which have been mapped back to their point of origin for the purpose of this visualization.
Here the same dynamics looks very different. The vortex tubes don't move but a complex vortical structure emerges around the centers of the vortices; this is where the reconnection happens. The high intensity of the vorticity in this structure (indicated by the red color) might due to the topological change (which a reconnection represents).
When one transforms the Navier-Stockes to Lagrangian coordinates, one sees that the only term driving the dynamics stems from the viscosity of the fluid. Hence without viscosity the vorticity distribution in right panel would not change at all (although it is not clear what would happen in left panel). The changes one sees are induces by viscosity and its influence on the metric tensor of the Lagrangian coordinates.
The visualizations are based on simulation of the 3D Navier-Stokes equation in a periodic box and a simulation of Lagrangian particles (tracers) which allows it to view the dynamics in the Lagrangian coordinates.
[1] A. Daitche, Statistische und geometrische Eigenschaften turbulenter Strömungen, Master’s thesis, Institut for Theoretical Physics, University of Münster.
