Identifying the length scales of the dominant coherent structures in channel flow: a resolvent analysis approach

Reduced order modeling approaches attempt to mitigate the costs of direct numerical simulations (DNS) by summarizing the flow with its dominant coherent motions or length scales. We propose identifying these length scales for the turbulent channel at Re_\tau = 186 using a modified resolvent analysis method that identifies the most amplified secondary perturbations that grow about a transiently evolving streak. Traditional resolvent analysis uses the equations of motion linearized about a mean profile to compute the maximally amplified Fourier modes of a chosen frequency. Though this method can correctly identify near-wall streaks as preferentially amplified structures, it does not account for transient growth, a key process during which a streak, which initially grows under the non-normal dynamics of the shear flow, then interacts nonlinearly with other length scales and breaks down. We propose using wavelet-based resolvent analysis---an extension developed to account for transient growth dynamics---to first compute a transiently growing ``optimal" streak; we then re-apply the method to the Navier-Stokes equations linearized about this streak, and compute the amplification of secondary modes in an attempt to identify the length scales that are most involved in the streak breakdown process. We compare these modes to those found from dissipation spectra computed from DNS data to tune restricted nonlinear turbulence models---modified numerical simulations that consider a subset of the resolved length scales.