Revisiting the validity of eddy viscosity models for predicting airflow over water waves

In this study, we revisit the validity of eddy viscosity models for predicting wave-induced airflow disturbances over ocean surface waves. We first derive a turbulence curvilinear model for the phase-averaged Navier–Stokes equations, extending the work of Cao, Deng & Shen (2020 J. Fluid Mech. 901, A27), by incorporating turbulence stress terms previously neglected in the linearised viscous curvilinear model. To verify our formulation, we perform a priori tests by numerically solving the model using mean wind and turbulence stress profiles from large-eddy simulations (LES) of airflow over waves across various wave ages. Results show that including turbulence stress terms improves wave-induced airflow predictions compared with the previous viscous curvilinear model. We further show that using a standard mixing-length eddy viscosity yields inaccurate predictions at certain wave ages, as it fails to capture wave-induced turbulence, which fundamentally differs from mean shear-driven turbulence. The LES data show that accurate representations of wave-induced stresses require a complex-valued eddy viscosity. The maximum magnitude of this eddy viscosity scales as $\sim \!u_\tau \zeta _{\textit{inner}}$ , where $u_\tau$ is the friction velocity and $\zeta _{\textit{inner}}$ is the inner-layer thickness, the height at which the eddy-turnover time matches the wave advection time scale. This scaling aligns with the prediction by Belcher & Hunt (1993 J. Fluid Mech. 251, 109–148). Overall, the findings demonstrate that traditional eddy viscosity models are inadequate for capturing wave-induced turbulence. More sophisticated turbulence models are essential for the accurate prediction of airflow disturbances and form drag in wind–wave interaction models.