Modeling Nonlinear Interactions in a Rough-Wall Turbulent Boundary Layer*

We model a wall with small-amplitude sinusoidal roughness as a small perturbation to a turbulent boundary layer. By altering the linearized boundary condition of the Navier-Stokes equations for a single spatial wavenumber, the roughness introduces a stationary, periodic velocity perturbation that interacts non-linearly with pairs of triadically-consistent convecting modes which make up the fluctuating velocity field. This study models these processes in a low-order fashion by approximating the stationary and convecting modes as the most-amplified resolvent modes for a given wavenumber and frequency. The model’s predictions are found to qualitatively reproduce the effects of two distinct sinusoidal roughnesses on the power spectrum of a turbulent boundary layer. Implications to more general roughness are discussed.