Restricted optimal paths to transition in a plane Couette flow

Abstract

To identify laminar/turbulent transition paths in plane Couette flow, a variational formulation incorporating a restricted nonlinear (RNL) system that retains a single streamwise Fourier mode, is used. Considering the flow geometry originally used by Monokrousos et al. (2011) and Duguet et al. (2013) and the same Reynolds numbers (Re), we show that initial perturbations obtained by RNL optimizations exhibit spatial localization. Two optimal states are found with comparable initial energy levels above which the flow structure evolves to turbulence. It is found that this level is twice that of the minimal threshold energy which has been obtained using the full nonlinear equations (Duguet et al. (2013)). Especially, the Re dependence of energy thresholds is studied within a RNL optimization framework for the first time, with evidence for a ORe−2.65 scaling close to the one found using the full Navier–Stokes equations (ORe−2.7). The first state is obtained for a short target time. It is symmetric with respect to the mid-plane y=0 and spanwise localized. For a long target time, the optimal appears to be localized in both spanwise and wall-normal directions. The mechanisms highlighted within the scope of nonlinear nonmodal theory (Kerswell (2018)): Orr mechanism, oblique wave interaction, lift-up, streak breakdown, localized pocket of turbulence and turbulence spreading, are also observed in the RNL simulations. Although greatly simplified, the RNL system provides a good approximation of these different fundamental mechanisms. The analysis gives then some insight into the potential of RNL optimizations for estimating Re scaling laws and routes to turbulence for shear flows.