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The natural scale separation in the restricted nonlinear (RNL) modelling approach is exploited to build upon recent studies, e.g., Wangsawijaya (2020), that have used scale separation to provide insight into mechanisms underlying secondary motions in turbulent flow over spanwise heterogeneous roughness. In the RNL decomposition the large-scale comprises the streamwise averaged mean and the small-scales are defined through a dynamical restriction that leads to computational tractability, while providing good agreement with salient flow features. In agreement with the experimental work, our results indicate that energy of the large-scales is amplified over the low roughness region due to the secondary flow. The small-scales are shown to play a dominant role in the Reynolds stresses responsible for generation of the secondary flow. Conditional averaging of the RNL mean field reveals stronger momentum pathways over low roughness regions experiencing downwash in instances that differ from the time-averaged trends. Further analysis of the large scale indicates that meandering of low speed streaks in the RNL flow is in response to secondary flow momentum mixing. 

In this work, we apply structured input-output analysis to study optimal perturbations and dominant flow patterns in transitional plane Couette-Poiseuille flow. The results demonstrate that this approach predicts the high structured gain of perturbations with wavelengths corresponding to the oblique turbulent bands observed in experiments. The inclination angles of these structures and their Reynolds number dependence are also consistent with previously observed trends. Reynolds number scalings of the maximally amplified structures for an intermediate laminar profile that is equally balanced between plane Couette and Poiseuille flow show an exponent that is at the midpoint of previously computed values for these two flows. However, the dependence of these scaling exponents on the shape of laminar flow as the relative contribution moves from predominately plane Couette to Poiseuille flow is not monotonic and our analysis indicates the emergence of different optimal perturbation structures through the parameter regime. Finally we adapt our approach to estimate the advection speeds of oblique turbulent bands in plane Couette flow and Poiseuille flow by computing their phase speed. The results show good agreement with prior predictions of the convection speeds of these structures from direct numerical simulations, which suggests that this framework has further potential in examining the dynamics of these structures.