The three-dimensional structure of momentum transfer in turbulent channels

Abstract

The quadrant analysis of the intense tangential Reynolds stress in plane turbulent channels is generalized to three-dimensional structures (Qs), with special emphasis on the logarithmic and outer layers. Wall-detached Qs are background stress fluctuations. They are small and isotropically oriented, and their contributions to the mean stress cancel. Wall-attached Qs are larger, and carry most of the mean Reynolds stresses. They form a family of roughly self-similar objects that become increasingly complex away from the wall, resembling the vortex clusters in del Álamo et al. (J. Fluid Mech., vol. 561, 2006, pp. 329–358). Individual Qs have fractal dimensions of the order of $D= 2$, slightly fuller than the clusters. They can be described as ‘sponges of flakes’, while vortex clusters are ‘sponges of strings’. The number of attached Qs decays away from the wall, but the fraction of the stress that they carry is independent of their sizes. A substantial fraction of the stress resides in a few large objects extending beyond the centreline, reminiscent of the very large structures of several authors. The predominant logarithmic-layer structure is a side-by-side pair of a sweep (Q4) and an ejection (Q2), with an associated cluster, and shares dimensions and stresses with the conjectured attached eddies of Townsend (J. Fluid Mech., vol. 11, 1961, pp. 97–120). Those attached eddies tend to be aligned streamwise from each other, located near the side walls between the low- and high-velocity large-scale streaks, but that organization does not extend far enough to explain the very long structures in the centre of the channel.

Wall-based identification of coherent structures in wall-bounded turbulence

“…Indeed, the differences in the position and intensity of the peaks corresponding to (ρ v,pw ) min and (ρ v,pw ) max are consistent with the results of these authors, who obtained slightly different velocity structures when performing conditional averages for positive and negative pressure fluctuations. It is also interesting to note that the pressure fluctuations at the wall do not seem to be aware of the side-by-side organization of the v-velocity structures observed in previous works in the logarithmic and outer region [5]. In the pressure velocity correlation, this organization is only apparent in the near-wall region.…”

“…Also, by choosing σ = l c /3, approximately 67% of the area of (4) lies inside the circle of diameter l c defined by (5). As usual, the unfiltered or filtered correlations are normalized with the corresponding standard deviations of p w ,p w and v, to yield the correlation coefficients…”

“…Some of them are semi-empirical, based on observation of coherent structures [2] on canonical wall-bounded turbulent flows [3,4,5]. Some are time-periodic solutions of simpler wall-bounded flows [6], which can be related to the dynamics of fully turbulent flows [7].…”

A new identification method in sampled quadrant analysis for wall-bounded turbulence

“…11 Here, we must emphasize that the present identification method is different from the former ones used by Willmarth and Lu, 8 Lu and Willmarth, 9 Lozano-Durán, Flores, and Jimenez, 11 and Narasimha et al 24 First of all, our threshold R c is a case-determined value which depends on the Reynolds number and the wall distance, and it is not a free parameter. 8,9,11,24 Secondly, the present threshold R c singles out only the Q2 and Q4 motions in each wall parallel plane and these singled-out motions directly contribute to the total Reynolds stress in an average sense, while the former ones might identify the Q1 and Q3 events as well, as the "hole" size is not so large. Thirdly, although we can connect all the regions in three dimensions as done by Lozano-Durán, Flores, and Jimenez 11 (see Figure 4(c) as an example), the resultant three-dimensional structures are not the same as those objects described by Lozano-Durán, Flores, and Jimenez.…”

“…Thirdly, although we can connect all the regions in three dimensions as done by Lozano-Durán, Flores, and Jimenez 11 (see Figure 4(c) as an example), the resultant three-dimensional structures are not the same as those objects described by Lozano-Durán, Flores, and Jimenez. 11 In Lozano-Durán, Flores, and Jimenez, 11 the three-dimensional criterion, Equation (3), was used with the local mean u m and v m (which are computed over the object domain of all of its constituent points) as the quadrant classification quantities, while in the present work, criterion Equation (4) motions and the green ones indicate the selected Q2 motions. In the near wall region (Figure 4(a) with y + = 12.68), the identified objects are small and dense.…”