Mitigating ill-conditioning of Reynolds-averaged Navier–Stokes equations for experimental data-driven turbulence closures

Experiments are a fundamental source of high-fidelity data in turbulence, providing reliable targets for modeling. As Direct Numerical Simulations (DNS), they can capture flow dynamics without an underlying model but are less restrictive than DNS regarding flow geometry and Reynolds number limits. In this work, we conducted an experimental campaign to generate quality data on the turbulent quantities in square duct flows. The recent literature has revealed the ill-conditioned nature of the Reynolds-averaged Navier–Stokes (RANS) equations. These studies suggest that other target quantities besides the Reynolds stress tensor (RST) can compose closures that lead to more accurate solutions. We investigated statistical quantities derived from these experiments to feed the RANS equations, testing various methods and comparing the different closure strategies. As anticipated, using the measured RST directly in the RANS equations led to significant error propagation in the predicted average velocity fields due to the equations' ill-conditioned nature. Conversely, other approaches successfully yielded accurate propagated mean velocity fields. Notably, implicit treatment of the linear components of the RST and Reynolds force vector provided precise results. By extending the range of tested Reynolds numbers beyond those achievable by DNS, this study underscores the critical need for careful selection of turbulent quantities in experimental data-driven turbulence closures and highlights the importance of exploring new methods for closing the RANS equations.