Wall-model response to departure from equilibrium in turbulent flows over rough surfaces

Most turbulent boundary-layer flows in engineering and natural sciences are out of equilibrium. While direct numerical simulation and wall-resolved large-eddy simulation can accurately account for turbulence response under such conditions, lower-cost approaches like wall-modelled large-eddy simulation often assume equilibrium and struggle to reproduce non-equilibrium effects. The recent ‘Lagrangian relaxation-towards-equilibrium’ (LaRTE) wall model (Fowler et al. 2022 J. Fluid Mech. vol. 934, 137), formulated for smooth walls, applies equilibrium modelling only to the slow dynamics that are more likely to conform to the assumed flow state. In this work, we extend the LaRTE model to account for wall roughness (LaRTE-RW) and apply the new model to turbulent flow over heterogeneous roughness and in accelerating and decelerating flows over rough surfaces. We compare predictions from the new LaRTE-RW model with those from the standard log-law equilibrium wall model (EQWM) and with experimental data to elucidate the turbulence response mechanisms to non-equilibrium conditions. The extended model transitions seamlessly across smooth-wall and fully rough regimes and improves prediction of the skin-friction coefficient, especially in recovering trends at roughness transitions and in early stages of pressure-gradient-driven flow acceleration or deceleration. Results show that LaRTE-RW introduces response delays that are beneficial when EQWMs react too quickly to disturbances, but it is less effective in flows requiring rapid response, such as boundary layers subjected to accelerating–decelerating–accelerating free stream conditions. These findings emphasize the need for further model refinements that incorporate fast turbulent dynamics not currently captured by LaRTE-RW.