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We employ a recently introduced structured input–output analysis (SIOA) approach to analyse streamwise and spanwise wavelengths of flow structures in stably stratified plane Couette flow. In the low-Reynolds-number ( $Re$ ) low-bulk Richardson number ( $Ri_b$ ) spatially intermittent regime, we demonstrate that SIOA predicts high amplification associated with wavelengths corresponding to the characteristic oblique turbulent bands in this regime. SIOA also identifies quasi-horizontal flow structures resembling the turbulent–laminar layers commonly observed in the high- $Re$ high- $Ri_b$ intermittent regime. An SIOA across a range of $Ri_b$ and $Re$ values suggests that the classical Miles–Howard stability criterion ( $Ri_b\leq 1/4$ ) is associated with a change in the most amplified flow structures when the Prandtl number is close to one ( $Pr\approx 1$ ). However, for $Pr\ll 1$ , the most amplified flow structures are determined by the product $PrRi_b$ . For $Pr\gg 1$ , SIOA identifies another quasi-horizontal flow structure that we show is principally associated with density perturbations. We further demonstrate the dominance of this density-associated flow structure in the high $Pr$ limit by constructing analytical scaling arguments for the amplification in terms of $Re$ and $Pr$ under the assumptions of unstratified flow (with $Ri_b=0$ ) and streamwise invariance. 

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. 

Input–output analysis of transitional channel flows has proven to be a valuable analytical tool for identifying important flow structures and energetic motions. The traditional approach abstracts the nonlinear terms as forcing that is unstructured, in the sense that this forcing is not directly tied to the underlying nonlinearity in the dynamics. This paper instead employs a structured-singular-value-based approach that preserves certain input–output properties of the nonlinear forcing function in an effort to recover the larger range of key flow features identified through nonlinear analysis, experiments and direct numerical simulation (DNS) of transitional channel flows. Application of this method to transitional plane Couette and plane Poiseuille flows leads to not only the identification of the streamwise coherent structures predicted through traditional input–output approaches, but also the characterization of the oblique flow structures as those requiring the least energy to induce transition, in agreement with DNS studies, and nonlinear optimal perturbation analysis. The proposed approach also captures the recently observed oblique turbulent bands that have been linked to transition in experiments and DNS with very large channel size. The ability to identify the larger amplification of the streamwise varying structures predicted from DNS and nonlinear analysis in both flow regimes suggests that the structured approach allows one to maintain the nonlinear effects associated with weakening of the lift-up mechanism, which is known to dominate the linear operator. Capturing this key nonlinear effect enables the prediction of a wider range of known transitional flow structures within the analytical input–output modelling paradigm.