Turbulence structure in the quasi-linear restricted nonlinear (RNL) model is analyzed and compared with DNS of turbulent Poiseuille flow at Reynolds number R=1650. The turbulence structure is obtained by POD analysis of the two components of the flow partition used in formulating the RNL model: the streamwise-mean flow and the associated perturbations. The dominant structures are found to be similar in RNL simulations and DNS despite the neglect of perturbation-perturbation nonlinearity in the RNL formulation. POD analysis of the streamwise-mean flow indicates that the dominant structure in both RNL and DNS is a coherent roll-streak structure in which the roll is collocated with the streak in a manner configured to reinforce the streak by the lift-up process. This mechanism of roll-streak maintenance accords with analytical predictions made using the second order statistical state dynamics (SSD) model, referred to as S3T, which shares with RNL the dynamical restriction of neglecting the perturbation-perturbation nonlinearity. POD analysis of perturbations from the streamwise-mean streak reveals that similar structures characterize these perturbations in both RNL and DNS. The perturbation to the low-speed streak POD are shown to have the form of oblique waves collocated with the streak that can be identified with optimally growing structures on the streak. Given that the mechanism sustaining turbulence in RNL has been analytically characterized, this close correspondence between the streamwise-mean and perturbation structures in RNL and DNS supports the conclusion that the self-sustaining mechanism in DNS is the same as that in RNL.
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