Statistical structure of self-sustaining attached eddies in turbulent channel flow

Abstract

The linear growth of the spanwise correlation length scale with the distance from the wall in the logarithmic region of wall-bounded turbulent flows has been understood as a reflection of Townsend's attached eddies. Based on this observation, in the present study, we perform a numerical experiment, which simulates energy-containing motions only at a given spanwise length scale in the logarithmic region, using their self-sustaining nature found recently. The self-sustaining energy-containing motions at each of the spanwise length scales are found to be self-similar with respect to the given spanwise length. Furthermore, their statistical structures are consistent with those of the attached eddies given in the original theory, providing direct evidence on the existence of Townsend's attached eddies. It is shown that the single self-sustaining attached eddy is composed of two distinct elements, one of which is a long streaky motion reaching the near-wall region, and the other is a relatively short vortical structure carrying all the velocity components. For the given spanwise length λ z between λ + z = 100 and λ z ≃ 1.5h, the former is found to be self-similar along y ≃ 0.1λ z and λ x ≃ 10λ z , while the latter is self-similar along y ≃ 0.5 ∼ 0.7λ z and λ x ≃ 2 ∼ 3λ z . The scaling suggests that the smallest attached eddy would be a near-wall coherent motion given in the form of a streak and quasi-streamwise vortices aligned, whereas the largest one would be an outer motion with a very-largescale motion and large-scale motions aligned. The attached eddies in-between, the size of which is proportional to their distance from the wall, contribute to the logarithmic region and fill the space caused by the length scale separation. The scaling is also found to yield consistent behaviour with the emergence of k −1 x spectra in a number of previous studies. Finally, a further discussion is provided specially on the Townsend's inactive motion and several recent theoretical findings.

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

“…In this respect, extending the present approach to computation of the relative periodic orbits would be a fruitful path to follow towards more accurate modelling of the large-scale dynamics (Kawahara & Kida 2001;Willis et al 2013Willis et al , 2016. Finally, it should be mentioned that the self-sustaining energycontaining motions in wall-bounded turbulent flows at high Reynolds numbers appear in a self-similar form throughout the entire logarithmic region Hwang 2015), as originally hypothesized by Townsend (1976) (i.e. attached eddy hypothesis).…”

“…By doing so, all the structures smaller than the large-scale structures are removed, while their roles are modelled with the artificially elevated eddy viscosity. It is very important to note that the removal with an appropriate increase of C s does not significantly distort the statistics and the self-sustaining dynamics of the largescale structures themselves, as extensively discussed in the previous studies (Hwang & Cossu 2010b;Hwang 2015;Rawat et al 2015;Hwang & Bengana 2016). Therefore, the computed invariant solutions at the artificially elevated C s would conceptually represent the structures in the 'presence of the surrounding small scales', which is modelled by the eddy viscosity, while enabling us to compute the invariant solutions with the relatively low resolutions at high Reynolds numbers considered.…”

“…attached eddy hypothesis). Each of the energy-containing motions is typically characterized by its spanwise length scale Hwang 2015;Hwang & Bengana 2016), suggesting that the invariant solutions with different spanwise length scales may be linked to these selfsimilar motions. This also implies that a thorough investigation of the invariant solutions with different sets of the length scales needs to be carried out in order to clarify the relation between the 'real' and 'exact' coherent structures using a numerical experiment given in the present study.…”

“…The self-sustaining outer structure is composed of two structural elements, a long streak and short quasi-streamwise vortices which respectively correspond to the VLSM and the LSMs (Hwang 2015). The selfsustaining process was also found to be almost identical to that in the near-wall region: 1) the streaky structure (VLSM) is amplified by the vortical structures (LSMs) via the lift-up effect (e.g.…”

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

“…This issue was further investigated by a set of numerical experiments [50,54] based on LESs introduced to isolate the statistical and dynamical behaviours of the self-sustaining motions at each spanwise length scale λ z between λ + z = 100 and λ z = 1.5δ by combining two techniques, one of which explicitly filters out the motions wider than λ z [69] and the other damps out the motions smaller than λ z by elevating C s as described in §3. The statistical structures of the computed self-sustaining motions at each spanwise length scale were found to be approximately self-similar with respect to the given spanwise length λ z , as shown in figure 7 by their spectra.…”