The precise set of parameters governing transition to turbulence in wall-bounded shear flows remains an open question; many theoretical bounds have been obtained, but there is not yet a consensus between these bounds and experimental or simulation results. In this work, we focus on a method to provide a provable Reynolds-number-dependent bound on the amplitude of perturbations a flow can sustain while maintaining the laminar state. Our analysis relies on an input-output approach that partitions the dynamics into a feedback interconnection of the linear and nonlinear dynamics (i.e., a Luré system that represents the nonlinearity as static feedback). We then construct quadratic constraints of the nonlinear term that is restricted by system physics to be energy-conserving (lossless) and to have bounded input-output energy. Computing the region of attraction of the laminar state (set of safe perturbations) and permissible perturbation amplitude are then reformulated as linear matrix inequalities, which allows more computationally efficient solutions than prevailing nonlinear approaches based on the sum of squares programming. The proposed framework can also be used for energy method computations and linear stability analysis. We apply our approach to low-dimensional nonlinear shear flow models for a range of Reynolds numbers. The results from our analytically derived bounds are consistent with the bounds identified through exhaustive simulations. However, they have the added benefit of being achieved at a much lower computational cost and providing a provable guarantee that a certain level of perturbation is permissible.

This paper employs an input–output based approach to analyse the convective velocities and transport of fluctuations in turbulent channel flows. The convective velocity for a fluctuating quantity associated with streamwise–spanwise wavelength pairs at each wall-normal location is obtained through the maximization of the power spectral density associated with the linearized Navier–Stokes equations with a turbulent mean profile and delta-correlated Gaussian forcing. We first demonstrate that the mean convective velocities computed in this manner agree well with those reported previously in the literature. We then exploit the analytical framework to probe the underlying mechanisms contributing to the local convective velocity at different wall-normal locations by isolating the contributions of each streamwise–spanwise wavelength pair (flow scale). The resulting analysis suggests that the behaviour of the convective velocity in the near-wall region is influenced by large-scale structures further away from the wall. These structures resemble Townsend’s attached eddies in the cross-plane, yet show incomplete similarity in the streamwise direction. We then investigate the role of each linear term in the momentum equation to isolate the contribution of the pressure, mean shear, and viscous effects to the deviation of the convective velocity from the mean at each flow scale. Our analysis highlights the role of the viscous effects, particularly in regards to large channel spanning structures whose influence extends to the near-wall region. The results of this work suggest the promise of an input–output approach for analysing convective velocity across a range of flow scales using only the mean velocity profile.

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.

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.