(79j) Intermittent Hibernation of Newtonian and Viscoelastic Turbulence: Insight Into the Maximum Drag Reduction Asymptote
Maximum drag reduction (MDR), the asymptotic upper limit of reduction in turbulent friction drag by polymer additives, is the most important unsolved problem in viscoelastic turbulence. Recent studies of turbulence in minimal flow units have identified time intervals, denoted "hibernating turbulence" showing key features of MDR in both Newtonian and viscoelastic flow. The present study provides a comprehensive examination of this turbulence hibernation phenomenon in the minimal channel geometry, and discusses its impact on the turbulent dynamics and drag reduction. Similarities between hibernating turbulence and MDR are established in terms of both flow statistics (mean and fluctuating components of velocity) and flow structure (weak vortices and nearly streamwise-invariant kinematics). Hibernation occurs more frequently at high levels of viscoelasticity, leading to flows that increasingly resemble MDR. Viscoelasticity facilitates the occurrence of hibernation by suppressing the conventional "active" turbulence, but has little influence on hibernation itself. At low Weissenberg number Wi, the average duration of active turbulence intervals is constant, but above a transition value of Wi, the duration decreases dramatically, and accordingly, the frequency of hibernation intervals increases. This observation can be explained by a model that posits that the lifetime of an active turbulence interval is the time that it takes for the turbulence to stretch polymer molecules to a certain threshold value -- once the molecules exceed this threshold, they exert a large enough stress on the flow to suppress the active turbulence. This model predicts an explicit form for the duration as a function of Wi and the simulation results match this prediction very closely. The transition point where hibernation frequency becomes substantially increased coincides with the point where qualitative changes are observed in overall flow statistics. Probability density functions of important variables reveal a much higher level of intermittency in the turbulent dynamics after this transition. It is further confirmed that hibernating turbulence is a Newtonian structure during which polymer extension is small. Based on these results, a framework is proposed explaining key transitions in viscoelastic turbulence, especially the convergence toward MDR.