Eigenspace-based reinforcement learning acceleration of statistical convergence in turbulent flows

Direct numerical simulation approaches play a critical role in advancing the understanding and accurate modeling of turbulent flow phenomena. However, the direct simulation of high-Reynolds-number turbulent flows is computationally intensive, requiring high spatial resolution and extended simulation times to achieve statistically reliable results. The strategy leverages deep reinforcement learning to apply dynamic, data-driven perturbations to the Reynolds stresses eigenspace. This allows an optimized control policy to guide the evolution of the Reynolds stresses tensor across all degrees of freedom (magnitude, anisotropy and orientation) within a physics-constrained framework. For the present numerical experiments, a reduced three-dimensional action space targeting magnitude and shape perturbations is utilized as an initial proof-of-concept. To that end, the framework is thoroughly described and its performance evaluated in the context of canonical turbulent channel flows at friction Reynolds numbers $R{e}_{\tau }=100$ and $R{e}_{\tau }=180$ . The trained agent successfully accelerates the convergence of key flow statistics, including the fluctuating velocity components and the underlying turbulent structure, i.e., anisotropy, particularly in the $R{e}_{\tau }=100$ training case. A comprehensive computational cost analysis demonstrates that the training overhead is rapidly amortized during unsupervised evaluation phases.Depending on the training duration and early-stopping criteria, the initial training overhead is recovered within 3 to 10 deployment runs under the executed conservative time steps, and within 2 to 5 runs under the projected CFL-optimal conditions. This yields a net computational saving when accelerating the convergence of the fluctuating velocity components to their targeted error tolerances when extrapolating from the $R{e}_{\tau }=100$ training case to the $R{e}_{\tau }=180$ deployment case. Furthermore, the evaluation provided critical insights into the complex nature of the control problem, identifying main performance trade-offs during training, the strong dependency on the agent's actuation frequency, and the fundamental challenges of policy generalization to a new flow regime. This work validates, therefore, the presented approach as a viable tool for targeting the convergence acceleration of specific flow statistics, and demonstrates a clear path toward more computationally efficient high-fidelity simulations of turbulence.