On the use of eddy viscosity in resolvent analysis of turbulent channel flow

Abstract

The predictions of resolvent analysis for turbulent channel flow are evaluated for a friction Reynolds number of Retau = 550. In addition to the standard resolvent operator with kinematic viscosity, a resolvent operator augmented with the Cess eddy viscosity is considered. Adding eddy viscosity significantly alters the low-rank behavior of the resolvent. Regardless of the wave speed selected, the eddy resolvent is low-rank for spanwise wavelengths of ly+ = 80 and ly/h = 3.5 in comparison to the standard resolvent whose low-rank behavior depends on the wave speed. The leading eddy modes have higher projections onto the leading mode from spectral proper orthogonal decomposition in comparison to standard resolvent modes. Neither analysis, however, reliably predicts the most energetic wave speed. The standard resolvent tends to overestimate it while the eddy resolvent underestimates it. When the most energetic wave speed is underestimated, the eddy modes are energetic too close to the wall. The eddy resolvent does, however, correctly identify the most energetic wave speed and mode shapes for structures associated with the near-wall cycle or that are most energetic at z/h = 0.5. These structures are likely to be correctly predicted for any friction Reynolds number due to the scaling of the Cess eddy viscosity profile. Finally, it is shown that the accuracy of eddy predictions relies on the right balance between positive and negative energy transfers. Even though eddy viscosity primarily adds dissipation, its wall-normal gradient injects energy in the near-wall region, resulting in mode shapes that are attached to the wall. For some scales the predicted positive energy transfer is too strong thus biasing structures towards the wall. The ability of the Cess eddy viscosity profile to model both positive and negative energy transfers suggests that it could be optimized for individual scales.