Forcing statistics in resolvent analysis: application in minimal turbulent Couette flow

An analysis of the statistics of the nonlinear terms in resolvent analysis is performed in this work for turbulent Couette flow at Reynolds number 400. Data from a direct numerical simulation of a minimal flow unit is used to compute the covariance matrix of the velocity. From the same data, we computed the nonlinear terms of the Navier-Stokes equations (treated as forcing), which allowed us to compute the covariance matrix of the forcing. The quantitative relation between the two covariances via the resolvent operator is confirmed here for the first time, accounting for relevant signal processing issues related to the windowing procedure for frequency-domain quantities. Such exact correspondence allowed the eduction of the most relevant force components for the dominant structures in this flow, which participate in the self-sustaining cycle of turbulence: (i) streamwise vortices and streaks, and (ii) spanwise-coherent fluctuations of spanwise velocity. The results show a dominance by a subset of the nonlinear terms for the prediction of the full statistics of streamwise vortices and streaks; a single term is seen to be dominant for spanwise motions. A relevant feature observed in these cases is that the forcing covariance is dominated by its first eigenfunction, showing that nonlinear terms also have a coherent structure at low frequencies in this flow. Different forcing components are also coherent between them, which leads to constructive and destructive interferences that greatly modify the flow response. These are key features of forcing 'colour' for the present flow.

Automated summary

This study analyzes the nonlinear terms in turbulent Couette flow at Reynolds number 400 and establishes a quantitative relationship between the covariance matrices of velocity and forcing through the resolvent operator. The dominance of certain nonlinear terms in predicting the statistical properties of streamwise vortices and streaks, as well as the coherent structure of forcing at low frequencies, is observed as key features of the flow. Constructive and destructive interferences between different forcing components greatly modify the flow response, illustrating the importance of forcing 'colour' in this context.