Exact coherent states of attached eddies in channel flow

Abstract

A new set of exact coherent states in the form of a travelling wave is reported in plane channel flow. They are continued over a range in $Re$ from approximately $2600$ up to $30\,000$, an order of magnitude higher than those discovered in the transitional regime. This particular type of exact coherent states is found to be gradually more localised in the near-wall region on increasing the Reynolds number. As larger spanwise sizes $L_{z}^{+}$ are considered, these exact coherent states appear via a saddle-node bifurcation with a spanwise size of $L_{z}^{+}\simeq 50$ and their phase speed is found to be $c^{+}\simeq 11$ at all the Reynolds numbers considered. Computation of the eigenspectra shows that the time scale of the exact coherent states is given by $h/U_{cl}$ in channel flow at all Reynolds numbers, and it becomes equivalent to the viscous inner time scale for the exact coherent states in the limit of $Re\rightarrow \infty$. The exact coherent states at several different spanwise sizes are further continued to a higher Reynolds number, $Re=55\,000$, using the eddy-viscosity approach (Hwang & Cossu, Phys. Rev. Lett., vol. 105, 2010, 044505). It is found that the continued exact coherent states at different sizes are self-similar at the given Reynolds number. These observations suggest that, on increasing Reynolds number, new sets of self-sustaining coherent structures are born in the near-wall region. Near this onset, these structures scale in inner units, forming the near-wall self-sustaining structures. With further increase of Reynolds number, the structures that emerged at lower Reynolds numbers subsequently evolve into the self-sustaining structures in the logarithmic region at different length scales, forming a hierarchy of self-similar coherent structures as hypothesised by Townsend (i.e. attached eddy hypothesis). Finally, the energetics of turbulent flow is discussed for a consistent extension of these dynamical systems notions to high Reynolds numbers.

The Simulation of Vortex Structures Induced by Different Local Vibrations at the Wall in a Flat-Plate Laminar Boundary Layer

“…However, the main affected area is in the vicinity of the wall, which is the region near the wall where hydrodynamic instability occurs actively. Results of the vortex distribution agree with the research results of Shinneeb et al [12].…”

“…At t = 95, the amplitude of vortex structures rose rapidly to nearly 1.0, and the scales of the high-speed fluid and low-speed fluid were smaller than those in Figure 5c in the stream-wise direction; however, there were mainly high-speed stripes near the wall, and the characteristics of the high-speed fluid occupied a larger area than the low-speed fluid, similar to Figure 5c. Results of the vortex structure stripe agree with the results of Wall et al [12].…”

Self-similar hierarchies and attached eddies

“…Flores and Jiménez [11] and Hwang and Bengana [12] have identified self-sustaining turbulent solutions to the Navier-Stokes equations in channel flow simulations with domains restricted to be proportional to the distance from the wall, while recent work by Yang et al [13] identifies families of attached exact coherent structures. The statistics of these flows are consistent with the properties of attached eddies, and thus appear to provide evidence that the latter are multiscale and self-sustaining within the minimal unit construction introduced by Jiménez and Moin [14].…”

Shear stress-driven flow: the state space of near-wall turbulence as

“…The stability boundary of the laminar flow, which separates initial conditions that relaminarise from those that become fully turbulent, is referred to as the edge (Skufca et al 2006;Schneider et al 2008Schneider et al , 2007 and plays a fundamental role in structuring the state space of turbulence. The computation of invariant solutions of the Navier-Stokes equations has allowed for a simplified analysis of a number of physical processes, including an equilibrium selfsustaining process (Waleffe 1998), the self-similarity of equilibria localised in the wallnormal direction (Eckhardt & Zammert 2018) and the high-Re inner-scaling of wallattached equilibria (Yang et al 2019).…”