Neural Network-Based Closure Model of the Ensemble-Averaging Dynamics of Turbulent Puffs in Transitional Pipe Flow

The subcritical transition to turbulence in pipe flow is closely related to localized turbulent patterns called puffs. Individual puffs have chaotic dynamics as their trajectories wander around exact coherent states (ECS) of the Navier-Stokes equations in the phase space. Nevertheless, they share and maintain a well-defined characteristic spatial structure which resembles that of localized relative periodic orbits (RPOs) of Navier-Stokes equations (NSE). Such similarity indicates the feasibility of investigating puff dynamics from a statistical perspective to capture the common features of their time evolution. In this work, we derive the equations of the ensemble-average puff profile out of a simplified dynamical model (Barkley 2016). As the ensemble of puffs becomes large enough, the average profile stops fluctuating and approximates a stable equilibrium, which is due to the existence of unclosed terms in the average equations. We next put forward a closure model by training a neural network with an initial guess based on the eddy viscosity hypothesis. The model indicates that the unclosed terms strengthen the turbulent diffusion while decreasing the Reynolds number to stabilize the structure of the mean puff profile. Finally, we associate our model with previous conclusions on the differences between the dynamics of puffs and RPOs in actual pipe flow, which motivates our next step to investigate the mean evolution of real puffs governed by NSE.