Optimal, Data-Driven Wall Models for Efficient Large Eddy Simulations of Metastable von Kármán Flows

The von K\'arm\'an turbulent swirling flow exhibits intriguing large-scale metastable dynamics, including low-frequency state switching. The study of state switching demands long-duration high-fidelity simulations at high Reynolds numbers that capture the flow generated by the impellers. Blade-resolved Large Eddy Simulations (LES) are computationally prohibitive, limiting access to these slow dynamics. Here, we develop a model for the action of the impellers on the flow using experimental data from Particle Image Velocimetry (PIV) and torque measurements of the von K\'arm\'an flow. The impeller-region velocity is parametrized via B-splines and coupled to the LES through momentum forcing. An initial set of B-spline coefficients is inferred using the Optimizing a DIscrete Loss (ODIL) framework constrained by the Reynolds-Averaged Navier-Stokes (RANS) equations, PIV measurements in the optically accessible portion of the device, and impeller torque measurements. The coefficients are then refined by the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), which minimizes the discrepancy between the LES time-averaged velocity and torque and their experimental counterparts. Using the data-driven impeller model, we perform long-duration LES of the von K\'arm\'an flow. We find that the simulation reproduces the mean flow in the bulk and displays metastable state-switching dynamics. We further show that these metastable states are not axisymmetric and consist of an alternating four-cell flow pattern that slowly rotates around the axis of the cylindrical vessel. The proposed approach provides a practical and computationally efficient route to investigating large-scale dynamics in impeller-driven turbulent flows.