Non-equilibrium effects in turbulent boundary layers over riblets: DNS of step changes in surface texture

Abstract

We computationally study the response of zero-pressure-gradient (ZPG) turbulent boundary layers (TBLs) to streamwise step changes from a smooth wall to riblets (SM_RI), and vice versa (RI_SM). To quantify the departure from equilibrium due to the step changes, we conduct reference calculations of ZPG TBLs over an entirely smooth wall, and an entirely riblet-covered surface. To save the computational cost, we generate an optimal grid for an unstructured spectral-element code, consistent with the size of turbulent scales across the TBL. By the step change, the momentum thickness Reynolds number reaches $Re_{\theta_0} \simeq 680$ (friction Reynolds number $Re_{\tau_0} \simeq 283$), and by the domain outlet downstream of the step change, $Re_{\theta} \simeq 1000$ ($Re_{\tau} \simeq 400$). The TBL departure from equilibrium due to the step change, and its subsequent relaxation, recall previous studies on step changes in surface roughness. Downstream of the step change, growth of the internal equilibrium layer thickness $\delta_\text{IEL}$, hence recovery to equilibrium, follows two stages. Stage I corresponds to the recovery up to the buffer region ($y^+ \simeq 10$), which is slower during the RI_SM step change than the SM_RI counterpart. For the RI_SM cases during Stage I, $\delta_\text{IEL} \propto (x/k)^{0.6}$, and this stage is completed by $x \simeq 100k \simeq (5\delta_0 - 20\delta_0)$ downstream of the step change, where $k$ is the riblet height. Stage II recovery i.e.\ recovery of the outer region, is quite slow. Therefore, for drag-increasing riblets with $k^+ \ge 25$, $\delta_\text{IEL}$ does not reach the boundary layer thickness, even up to $50\delta_0$ downstream of the step change, owing to the advected frozen wake from upstream. As a result, skin-friction coefficient reaches to more than $90\%$ of its equilibrium counterpart, but does not reach its $100\%$.