Structured input–output analysis of transitional wall-bounded flows

Abstract

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.

Structured input-output analysis of stably stratified plane Couette flow

“…In this paper, we have extended the structured input-output analysis (SIOA) originally developed for unstratified wall-bounded shear flows (Liu & Gayme 2021) to stratified plane Couette flow (PCF). We first apply SIOA to characterize highly amplified flow structures in the intermittent regimes of stratified PCF.…”

“…We need to define the spatio-temporal frequency response H S ∇ (y; k x , k z , ω) of stratified PCF that will form the basis of the structured input-output response. We use the superscript S to distinguish this operator from its counterpart for unstratified wallbounded shear flow (Liu & Gayme 2021). We employ the standard transformation to express the velocity field dynamics in (2.1) in terms of a formulation using the wall-normal velocity v and the wall-normal vorticity ω y := ∂ z u − ∂ x w (Schmid & Henningson 2012).…”

“…In this work, we extend the structured input-output analysis (SIOA) originally developed for unstratified PCF (Liu & Gayme 2021) to stratified PCF. Prior application of the SIOA approach to unstratified transitional wall-bounded shear flows (Liu & Gayme 2021) demonstrated that including the componentwise structure of nonlinearity uncovers a wider range of known key flow features identified through nonlinear analysis, experiments, and DNS, but not captured through traditional (unstructured) input-output approaches.…”

“…In this work, we extend the structured input-output analysis (SIOA) originally developed for unstratified PCF (Liu & Gayme 2021) to stratified PCF. Prior application of the SIOA approach to unstratified transitional wall-bounded shear flows (Liu & Gayme 2021) demonstrated that including the componentwise structure of nonlinearity uncovers a wider range of known key flow features identified through nonlinear analysis, experiments, and DNS, but not captured through traditional (unstructured) input-output approaches. Here, SIOA for stratified PCF includes the effect of nonlinearity in the momentum and density equations (under the Boussinesq approximation) within a computationally tractable linear framework through a feedback interconnection between the linearized dynamics and a structured forcing that is explicitly constrained to preserve the componentwise structure of the nonlinearity.…”

Structured input–output analysis of stably stratified plane Couette flow

“…In this work, we extend the structured input-output analysis (SIOA) originally developed for unstratified PCF (Liu & Gayme 2021) to stratified PCF. Prior application of the SIOA approach to unstratified transitional wall-bounded shear flows (Liu & Gayme 2021) demonstrated that including the componentwise structure of the nonlinearity uncovers a wider range of known key flow features identified through nonlinear analysis, experiments and DNS, but not captured through traditional (unstructured) input-output approaches.…”