Gradient-free learning of a closed-loop wall controller for turbulent drag reduction

Closed-loop wall control learnt by multi-agent reinforcement learning can lower skin-friction drag in turbulent channels, but these gradient-based policies are trained on small periodic boxes and exhibit reduced performance when carried over to a larger domain.

We recently showed that such policies are also prone to saturated bang-bang actuations that collapse into standing streamwise waves whose scale is set by the computational box rather than by the near-wall cycle, and proposed architectural fixes that avoid these degeneracies.

Here, we employ Evolution Strategy (ES) to optimise a recurrent closed-loop controller directly on a large turbulent channel at $\mathit{Re}_{\tau}\simeq180$, evaluating policy performance over full flow episodes using an energy-aware criterion and processing candidate policies in parallel.

To our knowledge, this is the first application of an evolution strategy to the control of a turbulent flow.

The ES controller reduces the skin friction by about $26\%$, exceeding the gradient-based multi-agent controller of Cavallazzi et al.

(2026), GRU-MARL, trained on a minimal box ($17\%$), and marginally exceeding classic opposition control (OC, $22\%$).

A wall-normal decomposition of the friction, Reynolds-stress profiles and anisotropy invariants show that the ES and opposition-controlled flows follow separate trajectories through the buffer layer, reaching comparable drag reduction by different reorganisations of the near-wall turbulence.

In particular, the ES actuation correlates predominantly with the streamwise velocity fluctuations rather than with the wall-normal velocity that classical OC targets.