In this work we study a restricted nonlinear (RNL) model for plane Couette flow. The RNL model partitions the dynamics of the flow field into a streamwise averaged mean flow and streamwise varying perturbations about that mean. This model forms a second order closure of the Navier Stokes equations in which the statistical state dynamics of the flow is simulated through the evolution of a streamwise mean flow and an approximation of its covariance. The RNL system was previously shown to exhibit self-sustaining turbulence that closely resembles DNS of turbulence but has the computational advantage of being supported by a small number of streamwise modes. We discuss RNL turbulence first in terms of the behaviour of its underlying structures. In particular, the the roll and streak structures that are known to be critical in the self-sustaining process of wall-turbulence. We compare the RNL structures to those obtained from DNS by examining the temporal spectra of their streak and roll energies as well as the spectral densities of these structures at di fferent wall-normal positions. The results show close correspondence between the structure and spectra of the rolls and streaks as well as agreement between the mean velocity pro files obtained from RNL simulations and DNS. We then discuss the support of RNL turbulence in streamwise Fourier space and show that this support is naturally restricted to a small number of modes. Moreover, we demonstrate that a realistic RNL SSP is supported by further truncation to as few as one streamwise mode, which constitutes an analytically and computationally attractive reduced order model for studying wall-turbulence.
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