Influence of the turbulent closure for the prediction of the linear response of a flow bounded by a corrugated wall

We investigate the influence of the turbulent closure in Reynolds-Averaged Navier–Stokes (RANS) simulations for the prediction of the linear response of a turbulent boundary layer developping over a small-amplitude corrugated wall. Experimental studies by Hanratty and co-workers (Zilker et al. 1977; Abrams and Hanratty, 1985; Frederick and Hanratty, 1988) show a phase shift between the wall shear stress and the wall undulation, that depends on the wall wavenumber. Historically, this problem was studied by the means of linear forced response analyses using a mixing length model. It was shown that an ad-hoc correction is required to recover the experimental results (Thorsness et al. 1978; Charru et al. 2013). In this study, we ran Reynolds Averaged Navier–Stokes (RANS) computations using different types of turbulent closures. The results confirm the inadequacy of the Boussinesq assumption, leading to the failure of Eddy Viscosity Models (EVM) to properly recover the wall shear stress phase angle. Moreover, it is shown that a second moment closure performs better in capturing the effects of a small deformation of the wall. Additionally, a general strategy based on the modification of the balance of the closure coefficients of a $k-\omega$ model is found to be an effective approach to improve the performance of first order turbulence models. We establish corrections adapted to the $k-\omega$ model which can be seen as a pragmatic way to recover the expected behaviors.