Nonlinear Physics Lab


 

Re = 70

Re = 237

Figure 1: Curvature fields in the Kolmogorov flow for Re = 70, where the flow is a hexagonal vortex lattice, and Re = 237, where the flow is spatiotemporally chaotic. Blue corresponds to low curvature, while red is high curvature. Elliptic points are marked with circles, while hyperbolic points are marked with crosses.

 

Re = 70

Re = 237

Figure 2: Trajectories of the critical points for the same two Reynolds numbers. Elliptic point trajectories are shown in blue and hyperbolic point trajectories are in red. At the smaller Reynolds number, the critical points are pinned to their lattice positions, while in the spatiotemporally chaotic state they can wander.

 

Animations

Re = 70

Re = 237

Figure 3: Animations of the motion of the critical points superimposed on the curvature fields (top row) and the vorticity fields (bottom row).

 

Figure 4: The mean number of critical points per frame as a function of Reynolds number. The number drops as the driving is increased, showing a coarsening of the pattern.

The number of critical points can change in three ways: they can enter or leave the measurement volume (the central 10 cm x 10 cm of the flow cell), they can appear or disappear singly (part of which may be due to experimental difficulties in locating them), or they can be created or annihilated in hyperbolic/elliptic pairs. Below are the rates for these processes as a function of the driving:

Figure 5: The rates at which the critical points enter and leave the measurement area, as a function of Reynolds number.

Figure 6: The rates at which the critical points appear and disappear singly within the measurement area.

Figure 7: The rates at which the critical points are created and annihilated in pairs.

We have also computed the probability density functions for the separation of the critical points along the two coordinate axes, where the y coordinate runs parallel to the Kolmogorov Flow's shear bands and the x coordinate is transverse.

Figure 8: PDFs of the separation distance r (scaled by the shear band wavelength λ) between two critical points for Re = 70 and Re = 237. The peaks in the Re = 70 data clearly show its lattice structure. For the Re = 237 data, where the flow is spatiotemporally chaotic, the forcing structure is lost in the direction parallel to the shear bands, but is still retained in the transverse direction.

Finally, we have investigated the movement of these points by computing their mean-squared displacement.

Figure 9: Evolution of the magnitude of the vector mean-squared separation of the critical points in all directions. As the Reynolds number is increased, the critical points become unpinned from their lattice sites and begin to wander, collapsing onto a power law with an exponent of 1.4.

 

Figure 10: Evolution of the mean-squared separation of the critical points in the direction transverse to the shear bands. Even at high Reynolds number, it is still difficult for the critical points to travel across the forced shear bands. The mean-squared displacement in this direction is an order of magnitude smaller than in the parallel direction.