Fig. 4: Probability distribution functions (PDF) for the fluid velocity and correlation functions for the fluid velocity and vorticity.

Reynolds numbers are: a \({\rm{Re}}=0\) and b \({\rm{Re}}=0.1\). Spatial correlation of the c fluid velocity 〈u(r) ⋅ u(r + r₀)〉 and d fluid vorticity \(\langle \left(\nabla \times {\bf{u}}({\bf{r}})\right)\cdot \left(\nabla \times {\bf{u}}({\bf{r}}+{{\bf{r}}}_{0})\right)\rangle \). Here r = (x, y) is the spatial position, and r₀ is the displacement vector. The vertical axis for (c) and (d) are normalized so the curves have intercepts equal to one. The gray curve shows the lane formation case, \({\rm{Re}}=0\), and the blue curve shows the turbulent-like case, \({\rm{Re}}=0.1\). In the case of lanes, the PDF (a) exhibits a clear bi-modal distribution as expected with formation of lanes, while for sustained turbulent-like motion, the PDF (b) exhibits a Gaussian distribution. For reference, the dashed curve provides the Gaussian distribution with the same mean and standard deviation.