Interaction of forced Orr-Sommerfeld and Squire modes in a low-order representation of turbulent channel flow

Abstract

A resolvent-based reduced-order representation is used to capture time-averaged second-order statistics in turbulent channel flow. The recently-proposed decomposition of the resolvent operator into two distinct families related to the Orr-Sommerfeld and Squire operators [K. Rosenberg and B. J. McKeon, Efficient representation of exact coherent states of the Navier-Stokes equations using resolvent analysis, Fluid Dynamics Research 51, 011401 (2019)] results in dramatic improvement in the ability to match all components of the energy spectra and the uv cospectrum. The success of the new representation relies on the ability of the Squire modes to compete with the vorticity generated by Orr-Sommerfeld modes, which is demonstrated by decomposing the statistics into contributions from each family. It is then shown that this competition can be used to infer a phase relationship between the two sets of modes. Additionally, the relative Reynolds number scalings for the two families of resolvent weights are derived for the universal classes of resolvent modes presented by Moarref et al. [R. Moarref, A. S. Sharma, J. A. Tropp, and B. J. McKeon, Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels, Journal of Fluid Mechanics 734, 275 (2013)]. These developments can be viewed as a starting point for further modeling efforts to quantify nonlinear interactions in wall-bounded turbulence.

Scaling of turbulence intensities up to Reτ=106 with a resolvent-based quasilinear approximation

“…mean velocity). This is the main difference from the other previous data-driven approaches utilising the linearized Navier-Stokes equations as the modelling template [24,[42][43][44][45][46][47]. Indeed, the QLA qualitatively preserves all the basic scaling behaviours in the velocity spectra at the level of DNS, except the one for energy cascade [22] originating from the self-interacting nonlinear terms modelled phenomenologically: i.e.…”

“…Here, we note that the emergence of such odd features with increasing N is partly due to the delta-correlated nature of the resolvent modes considered in (9). The assumption may be relaxed by allowing the amplitude of higher-order resolvent modes to vary as in [43,46] -the resolvent modes are orthogonal, thus the interpolation capability may well be utilized for the purpose of low-dimensional approximation for the velocity spectra. However, in such a case, the extrapolation/prediction capability of the present QLA, which relies only on the mean velocity given by Eqs.…”

Energy transfer in turbulent channel flows and implications for resolvent modelling

“…In Rosenberg & McKeon (2019), a componentwise analysis of the resolvent operator yielded two distinct families of modes which, when correctly weighted, destructively interfere to reduce bias towards the streamwise velocity component. The approach has been applied in McMullen, Rosenberg & McKeon (2020) to higher Reynolds number flows where the destructive interference is more pronounced due to stronger non-normality and higher mean shear.…”

Closing the loop: nonlinear Taylor vortex flow through the lens of resolvent analysis

“…For example, Moarref et al. (2014) and McMullen, Rosenberg & McKeon (2020) used convex optimization to compute reduced-order representations of turbulent statistics and Rigas, Sipp & Colonius (2021) studied solutions of the harmonic-balanced NSEs to identify optimal nonlinear mechanisms leading to boundary layer transition. Additionally Morra et al.…”