Self-similar hierarchies and attached eddies

It is demonstrated that time-evolving coherent structures with features consistent with Townsend's attached eddies and the developments associated with the reconstruction of flow statistics using the attached eddy hypothesis can be obtained from analysis of the (linear) resolvent operator associated with the Navier-Stokes equations. A discrete number of members of a single self-similar hierarchy, chosen by assuming a constant ratio of critical layer heights between neighboring members with overlapping wall-normal footprints, produces a geometrically self-similar, fractal-like structure of the spatial distribution of the velocity field and an associated signature of uniform momentum zones. The range of convection velocities associated with the hierarchy gives rise to time evolution of the velocity. The scaling of the streamwise wave number on a hierarchy can be reconciled with Townsend's distance-from-the-wall scaling for self-similar, attached eddies, at least conceptually, by considering coherent spatial structures in the velocity field and thus enforcing an exact equivalence between compared features. It has been shown previously that both the linear and nonlinear terms in the resolvent framework have self-similar representations in the logarithmic overlap region of the turbulent mean velocity profile. The conditions under which self-sustaining self-similar behavior may be obtained from self-similar hierarchical members in this region are elaborated and some similarities with other results and theories associated with self-similarity in this region are identified.

Figure 1

Self-similar structure on a resolvent hierarchy in channel flow at ${\text{Re}}_{\tau }=15000$: $\mathcal{S}\left({\mathbf{\lambda }}_r\right)$, with ${\mathbf{\lambda }}_r={\mathbf{\lambda }}_5=(29,123)$ and $c_r=c_5=U\left(y_{c,5}\right)$ defined as the largest scale on the hierarchy. $y_{c,1}=0.4$, $y_{c,m}=A^{1-m}{y}_{c,1}$, and $\left|A\right|=2$. Here, all modes have the same peak amplitude, $0.1U\left(1\right)$. (a)–(d) Isocontours of instantaneous streamwise fluctuation on an $(x,y)$ plane, $u(x,y,z_0=0,t_0)$ for $z=0$ symmetric modes. $y_{cm}$ marked by black dashed lines and green dashed boxes in (a) identify the regions extracted in (b)–(d). (e) Three-dimensional isosurfaces of negative $u(x,y,z,t_0)$ for $z=0$ antisymmetric modes. (f), (g) Aggregated structures in a single period of the largest scale, color coded by the appropriate scaling height $y_{cm}$. Red, blue, and yellow: $m=1,2,3$, respectively. (h) Instantaneous streamwise velocity on an $(x,y)$ plane, $U(x,y,z_0,t_0)$, for the fluctuations in (a)–(d) added to the turbulent mean shown in (i). Note the aspect ratios in this figure.