Invariant solutions of minimal large-scale structures in turbulent channel flow for up to 1000

Abstract

Understanding the origin of large-scale structures in high Reynolds number wall turbulence has been a central issue over a number of years. Recently, Rawat et al. (J. Fluid Mech., 2015, 782, p515) have computed invariant solutions for the large-scale structures in turbulent Couette flow at Re τ ≃ 128 using an over-damped LES with the Smagorinsky model to account for the effect of the surrounding small-scale motions. Here, we extend this approach to an order of magnitude higher Reynolds numbers in turbulent channel flow, towards the regime where the large-scale structures in the form of very-large-scale motions (long streaky motions) and large-scale motions (short vortical structures) energetically emerge. We demonstrate that a set of invariant solutions can be computed from simulations of the self-sustaining large-scale structures in the minimal unit (domain of size L x = 3.0h streamwise and L z = 1.5h spanwise) with midplane reflection symmetry at least up to Re τ ≃ 1000. By approximating the surrounding small scales with an artificially elevated Smagorinsky constant, a set of equilibrium states are found, labelled upper-and lower-branch according to their associated drag. It is shown that the upper-branch equilibrium state is a reasonable proxy for the spatial structure and the turbulent statistics of the self-sustaining large-scale structures.

Self-sustaining processes at all scales in wall-bounded turbulent shear flows

“…Probably the most important new finding is that large-scale [44,52] and moderate-scale [45] structures can self-sustain when smaller-scale motions are artificially quenched in over-damped large-eddy simulations of the turbulent flow. A continuum of self-sustaining motions therefore exists which is, furthermore, statistically and dynamically consistent with the features of Townsend's attached eddies, including an approximate self-similarity in the logarithmic region [35,45,50,64]. The existence of coherent self-sustained processes at large scale is also confirmed by the computation of invariant large-scale solutions of the filtered (fully nonlinear) Navier-Stokes equations [52,56,61,62,64].…”

“…A continuum of self-sustaining motions therefore exists which is, furthermore, statistically and dynamically consistent with the features of Townsend's attached eddies, including an approximate self-similarity in the logarithmic region [35,45,50,64]. The existence of coherent self-sustained processes at large scale is also confirmed by the computation of invariant large-scale solutions of the filtered (fully nonlinear) Navier-Stokes equations [52,56,61,62,64]. The ability of the moderate-and large-scale structures to self-sustain is supported by generalized linear stability analyses revealing that the non-normal amplification of coherent quasi-streamwise streaks from coherent quasi-streamwise vortices provides a robust mechanism for the extraction of kinetic energy from the turbulent mean flow at scales ranging from those of buffer-layer streaks to those of the large-scale and very-large-scale structures [26,[31][32][33][34][35][36].…”

A self-sustaining process model of inertial layer dynamics in high Reynolds number turbulent wall flows

“…In contrast to the recent work by Hwang et al [11,12,16] discussed in §1, this study captures only a single interacting UMZ/VF 'unit'. Nevertheless, based on the self-similar hierarchical rescaling of VWI states performed by Blackburn et al [38] and of 'resolvent modes' by Moarref et al [39], we speculate that the flow structures investigated here may constitute one member of a family of similar motions on a hierarchy-the varying spacing between VFs with distance from the wall will naturally generate a hierarchy of scales.…”

“…These structures, too, have been observed and studied in laboratory experiments and DNS. Moreover, Hwang et al [11,12] recently have employed over-damped large-eddy simulations strategically designed to isolate large-scale structures away from the wall for Re τ ≈ 100-1000. To date, however, quantitative theoriessystematically developed from the governing incompressible Navier-Stokes (NS) equation-that describe the essential nonlinear dynamics by which self-organized flow features in the inertial layer are maintained have not been developed to the extent that is true for their near-wall counterparts.…”

Vertically localised equilibrium solutions in large-eddy simulations of homogeneous shear flow

“…In particular, it is known that the large-scale motion is maintained in channels even when the small scales have been filtered in LES (Scovazzi et al 2001;Hwang & Cossu 2010). The key idea of extending the dynamical system approach to higher Reynolds numbers is to focus on the quasi-autonomous behaviour of the larger scales by modelling the effect of the smaller ones using a sub-grid (SG) model (Yasuda et al 2014;Rawat et al 2015;Hwang et al 2016;Sasaki et al 2016), where some LES steady states and periodic orbits are numerically obtained in wall-bounded flow and they are considered as a representation of large-scale motions. However, as mentioned above, the anisotropy and inhomogeneity of the length scales introduced by the wall makes the role of the eddy-viscosity term, which should be zero on the wall, on the invariant solutions ambiguous.…”