Collapse of transitional wall turbulence captured using a rare events algorithm

Abstract

This text presents one of the first successful applications of a rare events method for the study of multistability in a turbulent flow without stochastic energy injection. The trajectories of collapse of turbulence in plane Couette flow, as well as their probability and rate of occurrence are systematically computed using \emph{Adaptive Multilevel Splitting} (AMS). The AMS computations are performed in a system of size $L_x\times L_z=24\times 18$ at Reynolds number $R=370$ with an acceleration by a factor $\mathcal{O}(10)$ with respect to DNS and in a system of size $L_x\times L_z=36\times 27$ at Reynolds number $R=377$ with an acceleration by a factor $\mathcal{O}(10^3)$. The AMS results are validated with a comparison to DNS in the system of size $L_x\times L_z=24\times 18$. Visualisations in both systems indicate that turbulence collapses because the self sustaining process of turbulence fails locally. The streamwise vortices decay first in streamwise elongated holes, leaving streamwise invariant streamwise velocity tubes that experience viscous decay. These holes then extend in the spanwise direction. The examination of more than a thousand of trajectories in the $(E_{c,x}=\int u_x^2/2\,{\rm d}^3\mathbf{x},E_{c,y-z}=\int (u_y^2/2+u_z^2/2)\,{\rm d}^3\mathbf{x})$ plane in the system of size $L_x\times L_z=24\times 18$ confirms the faster decay of streamwise vortices and shows concentration of trajectory. This hints at an instanton phenomenology in the large size limit. The computation of turning point states, beyond which laminarisation is certain, confirms the hole formation scenario and shows that it is more pronounced in larger systems. Finally, the examination of non reactive trajectories, where a hole opens then closes, indicates that hole opening and closing are distinct processes. Both the vortices and the streaks reform concomitantly when the laminar holes close.