Investigation of unsteady secondary flows and large-scale turbulence in heterogeneous turbulent boundary layers

Following the findings in \cite{wangsawijaya2020}, we re-examine the turbulent boundary layers developing over surfaces with spanwise heterogeneous roughness of various roughness wavelengths $0.32 \leq S/\overline{\delta} \leq 3.63$,where$S$isthewidthoftheroughnessstripsand$\overline{\delta}$ is the spanwise-averaged boundary-layer thickness. The heterogeneous cases induce counter-rotating secondary flows, and these are compared to the large-scale turbulent structures that occur naturally over the smooth wall. Both appear as meandering elongated high- and low-momentum streaks in the instantaneous flow field. Results suggest that the secondary flows might be spanwise-locked turbulent structures, with $S\mathrm{/}\bar{\delta }$ governing the strength of the turbulent structures and possibly the efficacy of the surface in locking the structures in place (most effective when $S/\overline{\delta} \approx 1$). Conditional averages of the fluctuating velocity fields of both spanwise heterogeneous and smooth wall cases result in structures that are strongly reminiscent of the streak-vortex instability model. Secondary flows and large-scale structures coexist in the limits where either $S\mathrm{/}\bar{\delta }\gg 1$ or $S\mathrm{/}\bar{\delta }\ll 1$, where the secondary flows scale on $\delta$ or $S$, respectively. When $S\mathrm{/}\bar{\delta }\gg 1$, the secondary flows are locked about the roughness transition, while relatively unaltered large-scale structures occur further from the transition. In the case where $S\mathrm{/}\bar{\delta }\ll 1$, $S$-scaled secondary flows are confined close to the surface, coexisting with unaltered larger scale turbulent structures that penetrate much deeper into the layer.