Large-scale structures predicted by linear models of wall-bounded turbulence

Abstract

The objective of this article is to determine for which scales stochastic forcing of the linearized Navier-Stokes equations, recast as the resolvent operator, is sufficient to reproduce second-order statistics in turbulent channel flow. Our focus is on the large scales at a friction Reynolds number of Reτ = 2003. We consider a molecular resolvent operator, where only the kinematic viscosity appears, and an eddy resolvent operator, where the kinematic viscosity is augmented with an eddy viscosity profile. The molecular resolvent operator is able to identify the wall-normal height where the maximum energy of a structure is located, but it fails to predict the most energetic wave speed. It also overestimates the streamwise velocity component and underestimates the spatial support of the structures in the wall-normal direction. When the eddy resolvent operator identifies the most energetic wave speed, it also predicts the correct statistics for a given spatial scale. For spatial scales where this criterion is not met, the eddy viscosity overdamps the linear response. As a result, it predicts energetic wave speeds which are too low and velocity structures which are too energetic close to the wall. We conclude that eddy viscosity works best for structures which are most energetic in the wake region while its performance deteriorates for structures that are active in the log region.