3.1 The R and Q joint distributions

In this section, we compare the invariants computed from the filtered and unfiltered velocity fields. The results are presented in figure 4(a), which shows the joint probability density functions (p.d.f.s) of $R$ and $Q$ for all of the cases in table 2. For comparison, figure 4(b) includes the conditionally averaged velocity $\boldsymbol{v}$ from (2.5) for case F0.25. Similar vector fields are obtained for the other cases although they are omitted for the sake of brevity. Consistent with earlier findings (Borue & Orszag Reference Borue and Orszag1998; van der Bos et al. Reference van der Bos, Tao, Meneveau and Katz2002; Lüthi et al. Reference Lüthi, Ott, Berg and Mann2007), the iso-probability lines maintain a tear-drop shape for the filtered cases, and the strain production dominates over enstrophy production in strain-dominated regions also in the inertial scales. This result differs from the more symmetrical p.d.f.s obtained for the coarse-grained velocity reported in previous works (Chertkov et al. Reference Chertkov, Pumir and Shraiman1999; Naso & Pumir Reference Naso and Pumir2005; Naso et al. Reference Naso, Pumir and Chertkov2007; Pumir & Naso Reference Pumir and Naso2010) and computed through a stochastic tetrad model based on the evolution of four tracer points. Nevertheless, Lüthi et al. (Reference Lüthi, Ott, Berg and Mann2007) showed that the latter results are aliased and that the tear-drop shape is recovered when the velocity derivatives are computed using a larger number of tracers.

Figure 4.

(a) Joint p.d.f.s of $R$ and $Q$ . The iso-probability contours contain 90 % and 98 % of the data (that is, 90 % and 98 % of the data points are enclosed by each contour respectively), corresponding to $1.5\times 10^{-4}$ and $3.3\times 10^{-6}$ of the maximum of the p.d.f.s. The dashed lines are $R=0$ and $D=0$ . Symbols and colour are as in table 2. (b) Vector field $\boldsymbol{v}$ for F0.25. The solid lines contain 90 % and 98 % of the data. (c) Ratio of the magnitude of the conditionally averaged velocity deviation $\boldsymbol{v}_{std}$ and the mean $\boldsymbol{v}$ , conditioned on the $R$ – $Q$ plane. Results for the unfiltered case. Although the colourbar ranges from 0 to 10, values up to 100 are attained close to the Vieillefosse tail. (d) Magnitude of the conditionally averaged velocity deviation $\boldsymbol{v}_{std}h/u_{\unicode[STIX]{x1D70F}}$ . The red solid lines in (c) and (d) contain 90 % and 98 % of the data for the probability density function of $R$ and $Q$ .

The five filtered cases collapse well except for some small differences for large values of $Q$ . However, they differ from the unfiltered case along the horizontal axis where the iso-contours of the filtered cases tend to broaden, meaning that for a given level of $Q$ , the invariant $R$ is stronger than in the unfiltered case.

The fairly good overlap between the filtered cases also indicates that the normalization with $\widetilde{Q}^{\prime }$ is appropriate. Without the normalization, the values for the filtered cases are strongly reduced by several orders of magnitude with respect to the unfiltered case (not shown), since strong and intermittent events are mostly caused by the small-scale structure of turbulence (Batchelor & Townsend Reference Batchelor and Townsend1949; Jiménez Reference Jiménez2000). The good collapse for the filtered cases suggests that the dynamics on the $R$ – $Q$ plane is self-similar in the inertial range, although it will be shown in § 3.3 that this is not the case for its enstrophy and strain components.

It should be noted that the vectors in figure 4(b) are statistical representations of the conditional dynamics of the flow, and only represent the evolution of individual particles when their standard deviation is small with respect to the mean. Otherwise, they should be considered as small residuals of a more complex underlying evolution. This does not mean that they are irrelevant to the flow. They represent evolutionary trends, in the same sense as the bulk flow of a low-Mach-number fluid is a small residue of the much faster random motion of its molecules. To shed some light on this, figure 4(c) shows the ratio of the magnitudes of $\boldsymbol{v}$ and of the deviation vector

where the subscript std denotes standard deviation at each $R/{Q^{\prime }}^{3/2},Q/Q^{\prime }$ point. The average velocity field is relevant at those points where $|\boldsymbol{v}_{std}|$ is much smaller than $|\boldsymbol{v}|$ , with $|\cdot |$ the $L^{2}$ -norm. In most of the $R$ – $Q$ plane, $|\boldsymbol{v}_{std}|/|\boldsymbol{v}|=3$ –5, meaning that $\boldsymbol{v}$ is not highly representative of the trajectories of individual fluid particles but rather a weak trend of their motion. This is particularly pronounced along the Vieillefosse tail, where the ratio achieves values of up to $100$ . Similar results were reported by Lüthi et al. (Reference Lüthi, Holzner and Tsinober2009). This effect is caused by the small values attained by the mean rather than by the standard deviation, as seen in figure 4(d), which shows $|\boldsymbol{v}_{std}|$ without dividing by the mean. Figure 4(c) corresponds to the unfiltered case, and qualitatively similar values are obtained for the filtered cases (see the example in appendix D).

3.2 Conditional mean trajectories

The CMTs are shown in figure 5. It is important to remark that accurate calculation of CMTs is numerically challenging, and the inwards spiralling observed in many works is spurious. We have shown in § 2.2 that our numerics are good enough to recover closed CMTs when the full domain is considered. The results in figure 5 are physical and not the byproduct of numerical artefacts.

Figure 5.

The CMTs in the $R$ – $Q$ plane computed as shown in (2.5). The initial $R$ and $Q$ positions (marked by ▫) are the same for all of the cases, $(R/{Q_{s}^{\prime }}^{3/2},Q/Q_{s}^{\prime })=(0.1,0),(0.5,0),(0.75,0),(1,0),(1.25,0)$ . (a) Case F0; (b) case F0.1; (c) case F0.2; (d) case F0.25; (e) case F0.3; (f) case F0.4.

Lozano-Durán et al. (Reference Lozano-Durán, Holzner and Jiménez2015) showed that unfiltered CMTs, scaled as in (2.5), describe closed trajectories when the whole channel domain is considered (see figure 1), and the same result is found to be valid here for the filtered cases (not shown). When the p.d.f. is computed in a subdomain defined by two wall-parallel planes, as is done for the logarithmic region in this paper, the trajectories need not be closed any more. In this case, the equation for the conservation of probability of $J=J(R/{Q^{\prime }}^{3/2},Q/Q^{\prime })$ for a stationary state is

where

and $\boldsymbol{v}$ is as defined in (2.5). Here, $\unicode[STIX]{x1D713}_{b}$ and $\unicode[STIX]{x1D713}_{t}$ are the probability fluxes at the bottom plane ( $x_{2}^{+}=100$ ) and at the top plane ( $x_{2}/h=0.4$ ),

where $V_{b}$ , $V_{t}$ , $J_{b}$ and $J_{t}$ are functions of $(R/{Q^{\prime }}^{3/2},Q/Q^{\prime })$ . The terms $V_{b}$ and $V_{t}$ are the conditional wall-normal velocities on the $R/{Q^{\prime }}^{3/2}$ – $Q/Q^{\prime }$ plane at $x_{2}=x_{b}$ and $x_{2}=x_{t}$ respectively, $J_{b}$ and $J_{t}$ are the p.d.f.s at those same heights and $\unicode[STIX]{x1D6FC}$ is a scale factor equal to $1/(x_{t}-x_{b})$ . For $x_{b}$ and $x_{t}$ equal to the bottom and top walls, the fluxes (3.4) and (3.5) become zero and (3.2) is equivalent to the stationary Fokker–Planck equation used in previous works (van der Bos et al. Reference van der Bos, Tao, Meneveau and Katz2002; Chevillard et al. Reference Chevillard, Meneveau, Biferale and Toschi2008, Reference Chevillard, Lvque, Taddia, Meneveau, Yu and Rosales2011). Some guidelines for deriving equation (3.2) are provided in appendix C.

Figure 5(a) shows that, for the unfiltered case, the CMTs describe clockwise cycles around the origin in almost closed trajectories, consistent with the results of Lozano-Durán et al. (Reference Lozano-Durán, Holzner and Jiménez2015), who observed a probability flux of strong $R$ – $Q$ leaving the buffer layer and entering the outer zone, but which cancelled when both boundaries were considered.

For the filtered cases (figures 5 b–5 f), the CMTs spiral outwards. Intuitively, this is caused by the unbalanced $R$ and $Q$ associated with the fluid leaving and entering the subdomain through the boundaries, and can be better understood by studying the fluxes $\unicode[STIX]{x1D713}_{b}$ and $\unicode[STIX]{x1D713}_{t}$ shown in figure 6(a,b) for case F0.25. At the bottom boundary, incoming fluxes are located in the first ( $R>0$ and $Q>0$ ) and third ( $R<0$ and $Q<0$ ) quadrants, whereas at the top one they concentrate at the centre of the $Q$ – $R$ plane. The resulting net effect $\unicode[STIX]{x1D713}_{b}+\unicode[STIX]{x1D713}_{t}$ (figure 6 c) is dominated by incoming flux of weak events into the logarithmic layer, and a secondary outflow of stronger events distributed in the second and fourth quadrants. Qualitatively similar results are obtained for other filter widths (not shown).

In conclusion, the outward spiralling is mainly due to weaker normalized $R$ and $Q$ transported from the outer region into the logarithmic layer, where they are amplified. Results from the remaining cases (not shown) reveal that the increasing spiralling with wider filter widths is caused by the influx from the lower boundary being damped by the filter, while fluxes across the upper boundary remain similar. If the effect of the viscous terms is considered to be negligible for the filtered cases, the outward spiralling of inertial CMTs may be attributed to the combined effect of self-amplification, pressure and interscale transfer. It should be noted that the outward spiralling also implies that the residual CMTs, i.e. those that added to the filtered cases result in the unfiltered one, must spiral inwards in order to recover the almost closed CMTs in figure 5(a). This may be caused by viscous and/or interscale transfer effects (among others), and a term-by-term analysis of the dynamic equations of $\widetilde{R}$ and $\widetilde{Q}$ (and of the residual counterparts) would be necessary to address this question in detail. This will be tackled in future studies.

Figure 6.

Probability fluxes from (3.4) and (3.5) at plane (a) $x_{2}^{+}=100$ ( $\unicode[STIX]{x1D713}_{b}$ ) and (b) $x_{2}/h=0.4$ ( $\unicode[STIX]{x1D713}_{t}$ ); (c) net flux through both boundaries ( $\unicode[STIX]{x1D713}_{b}+\unicode[STIX]{x1D713}_{t}$ ). Positive values are represented by hot colours and solid contours. Negative values are represented by cold colours and dashed lines. The contours are $\pm 10^{-1}$ , $\pm 10^{-2}$ and $\pm 10^{-3}$ of the maximum. The data are for F0.25.

3.3 Strain and enstrophy components

Despite the similarities found for the joint distributions of $R$ and $Q$ in the previous section, we show below that the decomposition of the invariants into their strain and enstrophy components leads to quite pronounced differences. Following Ooi et al. (Reference Ooi, Martín, Soria and Chong1999), $Q$ and $R$ are decomposed as

so that $Q=Q_{s}+Q_{\unicode[STIX]{x1D714}}$ and $R=R_{s}+R_{\unicode[STIX]{x1D714}}$ . It should be noted that this decomposition differs from that in Soria et al. (Reference Soria, Sondergaard, Cantwell, Chong and Perry1994), Blackburn et al. (Reference Blackburn, Mansour and Cantwell1996) and Davidson (Reference Davidson2004), who considered the invariants of the rate-of-strain, $\unicode[STIX]{x1D634}_{ij}$ , and rate-of-rotation, $\unicode[STIX]{x1D6FA}_{ij}$ , tensors. In those cases, $Q_{s}$ and $R_{s}$ coincide with the second and third invariants of $\unicode[STIX]{x1D634}_{ij}$ , and so does $Q_{\unicode[STIX]{x1D714}}$ with the second invariant of $\unicode[STIX]{x1D6FA}_{ij}$ . However, the third invariant of $\unicode[STIX]{x1D6FA}_{ij}$ is zero and $R_{\unicode[STIX]{x1D714}}$ is not.

Relations (3.6) and (3.7) show that $Q_{\unicode[STIX]{x1D714}}$ is proportional to the enstrophy density, whose intense values tend to concentrate in tube-like structures (Jiménez et al. Reference Jiménez, Wray, Saffman and Rogallo1993). The term $Q_{s}$ is proportional to the strain, which is proportional to the local rate of viscous dissipation of kinetic energy, $\unicode[STIX]{x1D700}=-4\unicode[STIX]{x1D708}Q_{s}$ , with high values organized in sheets or ribbons (Moisy & Jiménez Reference Moisy and Jiménez2004). The meaning of $R_{s}$ and $R_{\unicode[STIX]{x1D714}}$ is closely connected to the evolution equations for the strain and enstrophy densities, which for the filtered cases are (see Ooi et al. (Reference Ooi, Martín, Soria and Chong1999) for the original equations for the unfiltered case)

where $\unicode[STIX]{x1D70F}_{mk}=\widetilde{u}_{m}\widetilde{u}_{k}-\widetilde{u_{m}u_{k}}$ and $B_{ijk}=1/2(\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}_{ik}/\unicode[STIX]{x2202}x_{j}+\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}_{jk}/\unicode[STIX]{x2202}x_{i})$ are responsible for the interscale transfer of strain and enstrophy. The corresponding relations for the unfiltered case are recovered by taking $\unicode[STIX]{x1D70F}_{ik}=0$ . Relations (3.8) and (3.9) show that $R_{s}$ and $R_{\unicode[STIX]{x1D714}}$ are proportional to the strain self-amplification and enstrophy production respectively. We will use $Q_{s}^{\prime }(x_{2})$ to non-dimensionalize quantities related to the strain (such as $Q_{s}$ and $R_{s}$ ) and $Q_{\unicode[STIX]{x1D714}}^{\prime }(x_{2})$ for those related to the enstrophy (such as $Q_{\unicode[STIX]{x1D714}}$ and $R_{\unicode[STIX]{x1D714}}$ ). The material derivatives are computed for the normalized quantities as in (2.5).

Figure 7.

Joint p.d.f.s of (a,b) $R_{\unicode[STIX]{x1D714}}$ – $Q_{\unicode[STIX]{x1D714}}$ , (c,d) $R_{s}$ – $Q_{s}$ , (e,f) $Q_{s}$ – $Q_{\unicode[STIX]{x1D714}}$ and (g,h) $R_{s}$ – $R_{\unicode[STIX]{x1D714}}$ . Panels (a,c,e,g) include the conditionally averaged velocity $\boldsymbol{v}$ for case F0. Symbols and colours as in table 2. Panels (b,d,f,h) include the joint p.d.f. and $\boldsymbol{v}$ for F0.25. The contours contain 90 % and 98 % of the data. The red dot in (b) is $\boldsymbol{v}=\mathbf{0}$ . See the text for details about the normalization.

The joint p.d.f.s for several combinations of (3.6) and (3.7) are plotted in figure 7. Figure 7(a,c,e,g) shows the iso-probability contours for all of the cases in table 2, and the corresponding conditionally averaged velocity for the unfiltered case. For comparison, figure 7(b,d,f,h) shows the contours and velocities for F0.25. In all of the panels, the corresponding conditional velocity $\boldsymbol{v}$ reveals a cyclical behaviour of the CMTs that is consistent with previous literature, e.g. Ooi et al. (Reference Ooi, Martín, Soria and Chong1999), Lüthi et al. (Reference Lüthi, Holzner and Tsinober2009). Although not shown for the distributions in figure 7, $|\boldsymbol{v}_{std}|/|\boldsymbol{v}|$ attains values similar to those reported above in the $R$ – $Q$ plane (see examples in appendix D).

One of the most remarkable results is the lack of collapse of the p.d.f.s for the different filter widths. The distributions in the $Q_{\unicode[STIX]{x1D714}}$ – $R_{\unicode[STIX]{x1D714}}$ and $Q_{s}$ – $R_{s}$ planes lose their skewed shape, at least partially compared with the unfiltered case, and become more symmetric, especially for $\unicode[STIX]{x1D6E5}_{2}\geqslant 0.3h$ . On the contrary, the enstrophy and strain densities become increasingly anticorrelated as the filter width grows, and follow the relation $Q_{s}\approx -Q_{\unicode[STIX]{x1D714}}$ . The same result applies to the enstrophy production and strain self-amplification distributions which exhibit a strong anticorrelation, $R_{s}\approx -R_{\unicode[STIX]{x1D714}}$ , although in this case the trend saturates above $\unicode[STIX]{x1D6E5}_{2}=0.2h$ . It should be noted that $Q_{s}=-Q_{\unicode[STIX]{x1D714}}$ , i.e. $Q=0$ , represents a degenerate flow topology (Chong et al. Reference Chong, Perry and Cantwell1990) that can be associated with pure shear, and that $R_{s}=-R_{\unicode[STIX]{x1D714}}$ implies a linear relation between enstrophy production and strain self-amplification.

The results above reveal that a lot of information is hidden in the $R$ – $Q$ plane, and that decomposing the invariants into their strain and enstrophy contributions offers a more comprehensive view to study the dynamics of the flow. The resulting dynamics in the $R$ – $Q$ plane is obtained by adding the quantities $Q_{s}+Q_{\unicode[STIX]{x1D714}}$ and $R_{s}+R_{\unicode[STIX]{x1D714}}$ , which have similar magnitude but opposite signs most of time, making it difficult to predict the final shape of the $R$ – $Q$ iso-contour in figure 4(a) from those in figure 7(a). It is still intriguing how the tear-drop shape persists at different scales despite the changes undergone by the strain and enstrophy components of $R$ and $Q$ .

Figure 8.

(a) Ratios of the standard deviations of the terms defined by  (3.10) and (3.11), and those in (3.6) and (3.7), averaged along the log-layer and as a function of the filter width, $\unicode[STIX]{x1D6E5}_{2}$ : ▫, $\overline{Q_{s}^{\prime S}/Q_{s}^{\prime }}$ ; ◃, $\overline{Q_{\unicode[STIX]{x1D714}}^{\prime S}/Q_{\unicode[STIX]{x1D714}}^{\prime }}$ ; ▪, $\overline{R_{s}^{\prime S}/R_{s}^{\prime }}$ ; ◂, $\overline{R_{\unicode[STIX]{x1D714}}^{\prime S}/R_{\unicode[STIX]{x1D714}}^{\prime }}$ . (b) Skewness of the probability density functions of ▫, $R_{s}/{Q_{s}^{\prime }}^{3/2}$ and ▵, $R_{\unicode[STIX]{x1D714}}/{Q_{\unicode[STIX]{x1D714}}^{\prime }}^{3/2}$ as a function of the wall-normal filter width, $\unicode[STIX]{x1D6E5}_{2}$ . The skewnesses for the unfiltered cases are omitted and are roughly equal to $\pm 40$ .

The lack of collapse in the previous results may be explained by taking into account the increasing contribution of $\unicode[STIX]{x2202}\widetilde{u}_{1}/\unicode[STIX]{x2202}x_{2}$ with the filter width. We can write the relations for $\widetilde{Q}_{s}$ , $\widetilde{Q}_{\unicode[STIX]{x1D714}}$ , $\widetilde{R}_{s}$ and $\widetilde{R}_{\unicode[STIX]{x1D714}}$ in the limiting case in which the wall-normal derivative of $\widetilde{u}_{1}$ is the most important gradient,

The superscript $S$ is used to distinguish them from the regular definitions in (3.6) and (3.7). Relations (3.10) and (3.11) show that $\widetilde{Q}_{s}^{S}=-\widetilde{Q}_{\unicode[STIX]{x1D714}}^{S}$ and $\widetilde{R}_{s}^{S}=-\widetilde{R}_{\unicode[STIX]{x1D714}}^{S}$ , which is consistent with the trends observed in figure 7(e,g). In order to test whether the decomposed invariants are dominated by the contribution of $\unicode[STIX]{x2202}\widetilde{u}_{1}/\unicode[STIX]{x2202}x_{2}$ as the filter width increases, figure 8(a) shows the ratios of the standard deviations, $Q_{s}^{\prime S}/Q_{s}^{\prime }$ , $Q_{\unicode[STIX]{x1D714}}^{\prime S}/Q_{\unicode[STIX]{x1D714}}^{\prime }$ , $R_{s}^{\prime S}/R_{s}^{\prime }$ and $R_{\unicode[STIX]{x1D714}}^{\prime S}/R_{\unicode[STIX]{x1D714}}^{\prime }$ , averaged in $x_{2}$ along the log-layer, denoted by $\overline{(\cdot )}$ , and as a function of the filter width. The results suggest that the dynamics of the eddies is progressively controlled by $\unicode[STIX]{x2202}\widetilde{u}_{1}/\unicode[STIX]{x2202}x_{2}$ as their scale increases. It should be noted that $\unicode[STIX]{x2202}\widetilde{u}_{1}/\unicode[STIX]{x2202}x_{2}$ is the instantaneous gradient but is related to the mean shear, $S$ , by averaging in the homogeneous directions and in time. This is in agreement with Corrsin’s argument (Corrsin Reference Corrsin1958) whereby the dynamics of the eddies with sizes comparable or larger than the Corrsin scale, $l_{C}=(\unicode[STIX]{x1D700}/S^{3})^{1/2}$ , are dominated by the effect of the mean shear. It is also consistent with $l_{C}$ being on average ${\approx}0.08h$ in the range considered for the logarithmic layer, which is well below the filter widths used (see table 2). The key role of $S$ in the dynamics of wall-attached eddies in the logarithmic layer has also been highlighted in previous works (Jiménez Reference Jiménez2013; Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014b ). At this point, it is interesting to add that the trends shown above are much weaker if the filter is only performed in the homogeneous directions but not in $x_{2}$ , and the reader is referred to appendix E for more details and some examples.

Another conclusion from (3.10) and (3.11) is that the distributions $\widetilde{Q}_{s}$ – $\widetilde{R}_{s}$ and $\widetilde{Q}_{\unicode[STIX]{x1D714}}$ – $\widetilde{R}_{\unicode[STIX]{x1D714}}$ should become mirror images of each other as the filter width increases, since their variables may be interchanged as $\widetilde{Q}_{s}\rightarrow -\widetilde{Q}_{\unicode[STIX]{x1D714}}$ and $\widetilde{R}_{s}\rightarrow -\widetilde{R}_{\unicode[STIX]{x1D714}}$ . This is clearly visible in figure 7(a,c) for F0.4. Figure 8(b) shows that the skewness of $\widetilde{R}_{s}$ and $\widetilde{R}_{\unicode[STIX]{x1D714}}$ decreases, and justifies the increasingly symmetrical shape of the $\widetilde{Q}_{s}$ – $\widetilde{R}_{s}$ and $\widetilde{Q}_{\unicode[STIX]{x1D714}}$ – $\widetilde{R}_{\unicode[STIX]{x1D714}}$ distributions with the $\unicode[STIX]{x1D6E5}_{2}$ (figures 7(a) and 7(c) respectively). Exact zero average enstrophy production is not expected for any filter width, since averaging (3.8) yields

where $\langle \cdot \rangle$ denotes ensemble average. Assuming that the viscous effects are negligible at the inertial scales, (3.12) implies that the average filtered enstrophy production is not zero but is balanced by the interscale transfer of enstrophy density.

Figure 9.

Results computed for the velocity fluctuations. Joint p.d.f.s of (a) $R$ – $Q$ , (b) $Q_{s}$ – $Q_{\unicode[STIX]{x1D714}}$ , (c) $R_{s}$ – $Q_{s}$ and (d) $R_{\unicode[STIX]{x1D714}}$ – $Q_{\unicode[STIX]{x1D714}}$ . The iso-probability contours contain 90 % and 98 % of the data. Symbols and colours are as in table 3. For comparison, one more case is included: $\times$ , case F0.3. (e–h) Conditionally averaged velocity field computed as in (2.5) in the planes (e) $R$ – $Q$ , (f) $Q_{s}$ – $Q_{\unicode[STIX]{x1D714}}$ , (g) $R_{s}$ – $Q_{s}$ and (h) $R_{\unicode[STIX]{x1D714}}$ – $Q_{\unicode[STIX]{x1D714}}$ . The contours are 90 % and 98 % of the data of the corresponding p.d.f.s. The results are for case S0.25. See the text for details about the normalization.

The direct effect of the mean shear in the previous results may be removed by computing the invariants of the velocity fluctuations instead of those of the total velocity. Hence, six more cases were computed using the fluctuating velocities, and their parameters are summarized in table 3. The new cases are denoted by $S\unicode[STIX]{x1D6FE}$ , where $\unicode[STIX]{x1D6FE}$ is the wall-normal filter width, $\unicode[STIX]{x1D6E5}_{2}/h$ , with the same aspect ratios $\unicode[STIX]{x1D6E5}_{1}/\unicode[STIX]{x1D6E5}_{2}$ and $\unicode[STIX]{x1D6E5}_{3}/\unicode[STIX]{x1D6E5}_{2}$ as in F $\unicode[STIX]{x1D6FE}$ . Although not shown, the differences in the probability distributions in figure 7 computed with and without the mean shear are insignificant for the unfiltered case, meaning that the small scales are barely affected by $S$ .

Table 3.

Summary of cases computed for the velocity fluctuations. The parameters $\unicode[STIX]{x1D6E5}_{1}$ , $\unicode[STIX]{x1D6E5}_{2}$ and $\unicode[STIX]{x1D6E5}_{3}$ are the filter widths in the streamwise, wall-normal and spanwise directions respectively. The fluctuating velocity field is filtered according to (2.7). The cases are denoted by $S\unicode[STIX]{x1D6FE}$ , where $\unicode[STIX]{x1D6FE}$ is the wall-normal filter width, $\unicode[STIX]{x1D6E5}_{2}/h$ . The symbols and colours in the last columns are used to denote the different cases in the figures.

The results for $S\unicode[STIX]{x1D6FE}$ are shown in figure 9. One noteworthy difference is the change in the joint p.d.f.s of $\widetilde{R}$ and $\widetilde{Q}$ , which become more symmetric and lose a large portion of the Vieillefosse tail (figure 9 a). The strong correlation between $\widetilde{Q}_{s}$ and $\widetilde{Q}_{\unicode[STIX]{x1D714}}$ is also lost (figure 9 b), and the good collapse of the iso-probability contours reinforces the conclusion that the mean shear is responsible for the trends observed in figure 7. The distribution of $\widetilde{R}_{s}$ – $\widetilde{R}_{\unicode[STIX]{x1D714}}$ behaves in a similar manner (not shown). The joint p.d.f.s of $\widetilde{Q}_{s}$ – $\widetilde{R}_{s}$ and $\widetilde{Q}_{\unicode[STIX]{x1D714}}$ – $\widetilde{R}_{\unicode[STIX]{x1D714}}$ become more symmetric too (figure 9 c,d), and, contrary to the results observed for cases F $\unicode[STIX]{x1D6FE}$ , the contours collapse quite well for $\unicode[STIX]{x1D6E5}_{2}>0.1h$ . Case S0.1 lies in between S0 and S0.2, probably because it is an intermediate stage between the small and inertial scales, and was omitted from figure 9 for the sake of clarity.

Interestingly, the conditionally averaged velocities for $S\unicode[STIX]{x1D6FE}$ , computed from $\text{D}/\text{D}t$ based on the fluctuating filtered velocity, do not always rotate around one centre as in F $\unicode[STIX]{x1D6FE}$ (see figure 7). As an example, figure 9(e–h) contains the conditional velocity $\boldsymbol{v}$ for S0.25, and shows that the CMTs in the $R$ – $Q$ and $R_{\unicode[STIX]{x1D714}}$ – $Q_{\unicode[STIX]{x1D714}}$ planes may be classified into two families according to their clockwise/counterclockwise rotation. It is remarkable that the trajectories in the upper quadrant of the $R$ – $Q$ plane now cycle counterclockwise, in contrast to the result for the invariants of the total velocity gradient shown in figure 4(c,d). The difference comes mostly from the behaviour of $R_{\unicode[STIX]{x1D714}}$ and $Q_{\unicode[STIX]{x1D714}}$ , since $R_{s}$ and $Q_{s}$ remain similar to those observed in F $\unicode[STIX]{x1D6FE}$ (cf. figures 7 and 9). The fluctuating $R$ – $Q$ plane may be explained by noting that intense contraction of vorticity ( $R_{\unicode[STIX]{x1D714}}>0$ ) is now associated with increasing $Q_{\unicode[STIX]{x1D714}}$ (figure 9 h), which is responsible for the counterclockwise part in the first quadrant of figure 9(e). Moreover, the strongest $Q_{\unicode[STIX]{x1D714}}$ at negative $R_{\unicode[STIX]{x1D714}}$ (vortex stretching) is associated with decreasing $Q_{\unicode[STIX]{x1D714}}$ , which explains the counterclockwise part of the second quadrant in figure 9(e). This very interesting behaviour comes from the increase of fluctuating enstrophy ( $Q_{\unicode[STIX]{x1D714}}$ ) in the presence of contraction of vorticity ( $R_{\unicode[STIX]{x1D714}}>0$ ), in contrast to the usual decrease observed for the total velocity. It also implies that $R_{\unicode[STIX]{x1D714}}$ is not the dominant term for the budget of $Q_{\unicode[STIX]{x1D714}}$ (or $Q$ ) in these regions, and that the interactions between mean and fluctuating gradients at the inertial scales or the interscale transfer are presumably responsible for these trends. Double cycles similar to those shown in figure 9(e) and (h) for S0.25 appear in the remaining filtered cases too (not shown).

Figure 10.

Results computed for the velocity fluctuations. Joint p.d.f.s of (a) $R$ – $Q$ , (b) $Q_{s}$ – $Q_{\unicode[STIX]{x1D714}}$ , (c) $R_{s}$ – $Q_{s}$ and (d) $R_{\unicode[STIX]{x1D714}}$ – $Q_{\unicode[STIX]{x1D714}}$ . The iso-probability contours contain 90 % and 98 % of the data. Symbols and colours are as in table 3. For comparison, one more case is included, $\star$ , isotropic turbulence from the JHU turbulence database (Li et al. Reference Li, Perlman, Wan, Yang, Meneveau, Burns, C., S. and E.2008) filtered with $\unicode[STIX]{x1D6E5}_{1}=\unicode[STIX]{x1D6E5}_{2}=\unicode[STIX]{x1D6E5}_{3}=90\unicode[STIX]{x1D702}$ . See the text for details about the normalization.

Another interesting question is whether the results for the velocity fluctuations presented above resemble those of isotropic turbulence (HIT). The idea is tested in figure 10(a–d) and turns out not to be the case. The data used are HIT values from the JHU turbulence database (Li et al. Reference Li, Perlman, Wan, Yang, Meneveau, Burns, C., S. and E.2008) filtered using (2.7) with $\unicode[STIX]{x1D6E5}_{1}=\unicode[STIX]{x1D6E5}_{2}=\unicode[STIX]{x1D6E5}_{3}=90\unicode[STIX]{x1D702}$ , which is comparable to the filter widths from table 3, where $\unicode[STIX]{x1D6E5}_{2}=0.3h\approx 90\unicode[STIX]{x1D702}$ . Borue & Orszag (Reference Borue and Orszag1998) reported distributions consistent with ours from their analysis of HIT using a top-hat filter. The distributions for HIT are more skewed than those for the fluctuations in the channel, especially in the $R$ – $Q$ and $Q_{s}$ – $R_{s}$ maps. The enstrophy/enstrophy-production p.d.f. for HIT is the only one close to the results obtained for cases $S\unicode[STIX]{x1D6FE}$ . However, the trends in the $Q_{s}$ – $Q_{\unicode[STIX]{x1D714}}$ plane are opposite, and the CMTs from HIT resemble the clockwise cycling of figure 7 rather than the ones observed in figure 9(e,h). These results suggest that considering only fluctuating velocities removes the direct effect of the shear on the dynamics but there still remains the indirect effect, which is expected since the mean shear and the fluctuations are coupled by the nonlinear dynamics of the Navier–Stokes equations.

3.4 Orbital periods

The CMTs are useful to compute orbital periods, i.e. the time $\unicode[STIX]{x1D70F}$ elapsed to complete one full revolution around the stagnation point, $\boldsymbol{v}=\mathbf{0}$ . This can be done for all of the distributions in figure 7, and the results will be shown to be of the same order throughout this section. We focus on the periods in the $Q_{\unicode[STIX]{x1D714}}$ – $R_{\unicode[STIX]{x1D714}}$ plane, and some examples are discussed for other cases. Figure 11(a) shows the values of $\unicode[STIX]{x1D70F}$ as a function of the initial distance from the stagnation point, $r_{0}$ (see figure 7 b). Many initial conditions randomly distributed in each plane ( $Q_{\unicode[STIX]{x1D714}}$ – $R_{\unicode[STIX]{x1D714}}$ , $R$ – $Q$ , etc.) were used to compute $\unicode[STIX]{x1D70F}$ , and its average value is shown in figure 11(a).

The periods collapse reasonably well for all of the filter widths when normalized by $\overline{\widetilde{Q}^{\prime }}^{1/2}$ (where the bar represents average along the logarithmic layer) and do not vary much with $r_{0}$ , i.e. weak and strong events show similar characteristic time scales. This is the consequence of the larger velocities sampled by the CMTs as $r_{0}$ increases, which compensates for the longer paths travelled, leading to $\unicode[STIX]{x1D70F}\approx$ constant.

For the unfiltered case, $\unicode[STIX]{x1D70F}$ decreases by 30 % by the time $r_{0}$ has tripled, but it always remains above the filtered cases. The dependence of $\unicode[STIX]{x1D70F}$ with $r_{0}$ suggests that weak small-scale events take a longer time to complete one dynamic cycle, although the underlying physical meaning is unclear. On average, $\unicode[STIX]{x1D70F}\overline{Q^{\prime }}^{1/2}\approx 17$ for the unfiltered case, and $\unicode[STIX]{x1D70F}\overline{\widetilde{Q}^{\prime }}^{1/2}\approx 10$ for the filtered ones. Results of the same order but slightly larger are obtained in the $Q_{s}$ – $R_{s}$ plane, and one example is included in figure 11(a).

A Kolmogorov-scale normalization, $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}\sim \overline{Q^{\prime }}^{-1/2}$ , where $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ is the Kolmogorov time scale, is the natural choice for the unfiltered case since $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ is the typical decorrelation time scale during Lagrangian evolution (Meneveau Reference Meneveau2011). The fact that such a normalization works with $\widetilde{Q}^{\prime }$ for the filtered cases suggests that the dominating eddies in the filtered flow are those with characteristic size $\unicode[STIX]{x1D6E5}_{2}$ and lifetimes proportional to their local eddy-turnover time $\overline{\widetilde{Q}^{\prime }}^{-1/2}$ .

Figure 11(a) also includes the orbital periods in the $R$ – $Q$ plane for case F0.2, and the trend is similar to those computed for other joint distributions. However, our experience shows that the conditionally averaged velocity on the $R$ – $Q$ plane changes direction very rapidly close to the Vieillefosse tail, especially for large filter widths, which poses some numerical issues and makes the CMTs follow wrong paths. For instance, it was not possible to complete a single orbit for case F0.4 for large $r_{0}$ (figure 5 d). The $Q_{\unicode[STIX]{x1D714}}$ – $R_{\unicode[STIX]{x1D714}}$ and $Q_{s}$ – $R_{s}$ planes are more reliable than the $R$ – $Q$ space to compute periods for large filter widths.

The last period computed corresponds to the CMTs from case S0.25 in the $R$ – $Q$ plane restricted to $Q>0$ (see figure 9 e). The results are included in figure 11(a) and remain within the scatter of previous orbital times. Many other periods can be computed and show different degrees of agreement with those shown before. These are not discussed here since we do not pretend to perform an exhaustive analysis of all of the possibilities, but just remark that they are of the same order.

Figure 11.

(a) Orbital periods of the CMTs, $\unicode[STIX]{x1D70F}$ , as a function of the distance of their initial condition from the stagnation point of $\boldsymbol{v}$ in the corresponding plane,  $r_{0}$ : ▪, periods computed in the $R$ – $Q$ plane for case F0.2; ▾, periods computed in the $Q_{s}$ – $R_{s}$ plane for case F0.2; *, periods computed in the $R$ – $Q$ plane for CMTs in the regions $Q>0$ and case S0.25. The rest of the symbols are as in table 2 and correspond to periods extracted from the $Q_{\unicode[STIX]{x1D714}}$ – $R_{\unicode[STIX]{x1D714}}$ plane and filter widths $\unicode[STIX]{x1D6E5}_{2}=0,0.1,0.2,0.25,0.3$  and  $0.4$ . (b) Orbital periods of the CMTs in the $Q_{\unicode[STIX]{x1D714}}$ – $R_{\unicode[STIX]{x1D714}}$ plane averaged in $r_{0}$ as a function of the wall-normal filter width, $\unicode[STIX]{x1D6E5}_{2}$ . The dashed line is $\unicode[STIX]{x1D70F}u_{\unicode[STIX]{x1D70F}}/h=8\unicode[STIX]{x1D6E5}_{2}/h+0.08$ .

The value $\unicode[STIX]{x1D70F}\overline{\widetilde{Q}^{\prime }}^{1/2}\approx 10$ extracted from figure 11(a) for cases F $\unicode[STIX]{x1D6FE}$ implies that the absolute orbital period increases with $\unicode[STIX]{x1D6E5}_{2}$ , since the magnitude of $\overline{Q^{\prime }}^{1/2}$ decreases with the filter width. Figure 11(b) shows that the relation between orbital periods and filter widths follows approximately the linear trend $\unicode[STIX]{x1D70F}u_{\unicode[STIX]{x1D70F}}/h\approx 8\unicode[STIX]{x1D6E5}_{2}/h+0.08$ when the time is normalized by the eddy-turnover time, $h/u_{\unicode[STIX]{x1D70F}}$ . The first and last points, $\unicode[STIX]{x1D6E5}_{2}=0$ and $\unicode[STIX]{x1D6E5}_{2}=0.4$ , were excluded from the previous fitting, the former for being dominated by the viscous effects, and the latter for exceeding or being at the edge of the usual range considered for the logarithmic layer (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). The dependence of $\overline{\widetilde{Q}^{\prime }}^{1/2}$ with $\unicode[STIX]{x1D6E5}_{2}$ can be estimated analytically for a known velocity gradient spectrum and, for the range of filter widths considered here, turns out to be almost linear, which explains the trend in figure 11(b). The constant factor based on the filter width and friction velocity, $\unicode[STIX]{x1D6E5}_{2}/u_{\unicode[STIX]{x1D70F}}$ , is a suitable normalization factor for the orbital period.

To close this section we compare our periods with those in the literature. The orbital periods discussed above for the unfiltered case, $\unicode[STIX]{x1D70F}\overline{Q^{\prime }}^{1/2}\approx 17$ , correspond to $\unicode[STIX]{x1D70F}^{+}=280$ and $\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}=26$ in wall and Kolmogorov units respectively, with $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}=\overline{(\unicode[STIX]{x1D708}/\unicode[STIX]{x1D700})^{1/2}}$ . The value obtained by Atkinson et al. (Reference Atkinson, Chumakov, Bermejo-Moreno and Soria2012) in a turbulent boundary layer flow at a comparable Reynolds number is $\unicode[STIX]{x1D70F}^{+}=658$ , which exceeds ours, although in their case the periods are computed for the logarithmic and wake region. Elsinga & Marusic (Reference Elsinga and Marusic2010) reported a smaller value, $\unicode[STIX]{x1D70F}^{+}=470$ , which is still larger than ours, although they used experimental data filtered over $50$ wall units in each direction. For isotropic turbulence, Martín et al. (Reference Martín, Ooi, Chong and Soria1998), Ooi et al. (Reference Ooi, Martín, Soria and Chong1999) and Lüthi et al. (Reference Lüthi, Holzner and Tsinober2009) obtained $\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}\approx 40$ and $\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}\approx 30$ respectively, which are not so far from the value of $26$ obtained here. Nevertheless, some differences are expected since we compute the Lagrangian time derivative of the normalized invariants, as in (2.5), and not of the invariants themselves, as is the case in previous works. It is difficult to find orbital periods for the filtered invariants in the literature, although values of the same order as those shown in figure 11(b) were obtained for the bursting periods, $T_{b}$ , of minimal log-layer channels by Flores & Jiménez (Reference Flores and Jiménez2010), $T_{b}u_{\unicode[STIX]{x1D70F}}/h\approx 6x_{2}/h$ , if we take $\unicode[STIX]{x1D6E5}_{2}=x_{2}$ . However, it is not simple to establish a link between orbital and bursting periods, and it is unclear whether they are related or not. A linear relation between lifetime and scale has also been observed in previous works (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006; LeHew, Guala & McKeon Reference Lehew, Guala and Mckeon2013; Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014b ), and is explained in the context of self-similar log-layer eddies with lifetimes proportional to their size. Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014b ) found $Tu_{\unicode[STIX]{x1D70F}}/h\approx \unicode[STIX]{x1D6E5}_{2}/h$ , with $T$ the lifetime of wall-attached eddies and $\unicode[STIX]{x1D6E5}_{2}$ their size, which will be considered to be equivalent to the filter width. However, these lifetimes are much shorter than the orbital periods reported here, $\unicode[STIX]{x1D70F}u_{\unicode[STIX]{x1D70F}}/h\approx 8\unicode[STIX]{x1D6E5}_{2}/h+0.08$ . This discrepancy may be related to the ratio $|\boldsymbol{v}_{std}|/|\boldsymbol{v}|$ shown in figure 4(b). The average $|\boldsymbol{v}_{std}|/|\boldsymbol{v}|$ along a CMT is $6$ – $12$ for all filter widths, and its inverse value can be interpreted as the fraction of time the fluid particles travel in the ‘correct’ direction to complete a cycle instead of drifting in other directions. This may responsible for the factor of eight between orbital periods and lifetimes of individual eddies mentioned above.

3.5 Alignment of the vorticity and the rate-of-strain tensor

In order to gain a better insight into the dynamics at different scales, we analyse the alignment of the vorticity vector, $\unicode[STIX]{x1D74E}$ , with the eigenvectors of the rate-of-strain tensor, $\unicode[STIX]{x1D706}_{1}$ , $\unicode[STIX]{x1D706}_{2}$ and $\unicode[STIX]{x1D706}_{3}$ , whose associated eigenvalues are $\unicode[STIX]{x1D706}_{1}>\unicode[STIX]{x1D706}_{2}>\unicode[STIX]{x1D706}_{3}$ (note that in this case the subscript refers to the sorting of the eigenvalues, in contrast to the convection used for other quantities, $\unicode[STIX]{x1D634}_{ij}$ , $u_{i},\ldots$ , where the index denotes the spatial direction $x_{i}$ ). This alignment is of interest since the enstrophy production may be expressed as (Betchov Reference Betchov1956)

where $p_{i}=\unicode[STIX]{x1D706}_{i}\cos ^{2}(\unicode[STIX]{x1D74E},\unicode[STIX]{x1D706}_{i})$ and $\unicode[STIX]{x1D714}$ is the $L^{2}$ -norm of $\unicode[STIX]{x1D74E}$ . An equivalent expression applies to the filtered cases.

Since the trace of $\unicode[STIX]{x1D634}_{ij}$ must be zero due to incompressibility, it is satisfied that $\unicode[STIX]{x1D706}_{1}>0$ and $\unicode[STIX]{x1D706}_{3}<0$ . The former eigenvalue represents stretching in the direction of $\unicode[STIX]{x1D706}_{1}$ , whereas the latter is a contraction of vorticity along $\unicode[STIX]{x1D706}_{3}$ . The second eigenvalue, $\unicode[STIX]{x1D706}_{2}$ , takes both positive and negative values, and it is well known that, for the small scales, $\unicode[STIX]{x1D74E}$ aligns preferentially with $\unicode[STIX]{x1D706}_{2}$ in isotropic turbulence (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987), free shear flows (Mullin & Dahm Reference Mullin and Dahm2006) and turbulent channels (Blackburn et al. Reference Blackburn, Mansour and Cantwell1996). Similar results hold for the filtered cases, as shown in figure 12(a). However, the alignment of $\unicode[STIX]{x1D74E}$ and $\unicode[STIX]{x1D706}_{2}$ intensifies for larger scales, and, as a result, the angles of $\unicode[STIX]{x1D706}_{1}$ and $\unicode[STIX]{x1D706}_{3}$ with $\unicode[STIX]{x1D74E}$ tend to zero, i.e. perpendicular to $\unicode[STIX]{x1D74E}$ (not shown).

Figure 12.

Results computed for the total velocities. (a) The p.d.f.s of the cosine of the angle of the vorticity, $\unicode[STIX]{x1D74E}$ , and the second eigenvector of the rate-of-strain tensor, $\unicode[STIX]{x1D706}_{2}$ . (b,c) The p.d.f.s of the eigenvalues of the rate-of-strain tensors (b) $\unicode[STIX]{x1D706}_{1}$ and $\unicode[STIX]{x1D706}_{3}$ , and (c) $\unicode[STIX]{x1D706}_{2}$ . For (a), (b) and (c) the symbols and colours are as in table 2. (d) Relative contributions to the enstrophy production of $p_{1}=\unicode[STIX]{x1D706}_{1}\cos ^{2}(\unicode[STIX]{x1D74E},\unicode[STIX]{x1D706}_{1})$ , $p_{2}=\unicode[STIX]{x1D706}_{2}\cos ^{2}(\unicode[STIX]{x1D74E},\unicode[STIX]{x1D706}_{2})$ and $p_{3}=\unicode[STIX]{x1D706}_{3}\cos ^{2}(\unicode[STIX]{x1D74E},\unicode[STIX]{x1D706}_{3})$ as a function of the wall-normal filter width, $\unicode[STIX]{x1D6E5}_{2}$ . The top panel shows the ratio of the means: ▫, $\langle p_{2}\rangle /\langle p_{1}\rangle$ ; ▵, $\langle p_{3}\rangle /\langle p_{1}\rangle$ . The bottom panel shows the ratio of the standard deviations: ▫, $p_{2}^{\prime }/p_{1}^{\prime }$ ; ▿, $p_{3}^{\prime }/p_{1}^{\prime }$ .

Figure 12(b,c) shows the p.d.f.s of $\unicode[STIX]{x1D706}_{i}$ scaled by the corresponding ${Q_{s}^{\prime }}^{1/2}$ of each case. The distribution for the unfiltered $\unicode[STIX]{x1D706}_{2}$ is skewed towards positive values, as already reported in previous works (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Vincent & Meneguzzi Reference Vincent and Meneguzzi1991; Blackburn et al. Reference Blackburn, Mansour and Cantwell1996). The p.d.f.s of $\unicode[STIX]{x1D706}_{1}$ do not collapse for the different filter widths, nor do they for $\unicode[STIX]{x1D706}_{2}$ and $\unicode[STIX]{x1D706}_{3}$ , although the distributions become more symmetric as $\unicode[STIX]{x1D6E5}_{2}$ increases, consistent with the results for the skewness of $R_{\unicode[STIX]{x1D714}}$ in figure 8(b).

The preferential alignment of $\unicode[STIX]{x1D74E}$ with $\unicode[STIX]{x1D706}_{2}$ is predicted by angular momentum conservation in the restricted Euler model (Cantwell Reference Cantwell1992), and was explained from a kinematic point of view in Jiménez (Reference Jiménez1992). Such an alignment, or at least part of it, may also be expected in the context of eddies controlled by the mean shear, as proposed in § 3.3. In fact, in the very simple scenario in which $\unicode[STIX]{x2202}u_{1}/\unicode[STIX]{x2202}x_{2}$ is the most important term, the flow behaves like a pure shear, and $\unicode[STIX]{x1D74E}$ and $\unicode[STIX]{x1D706}_{2}$ align (Tennekes & Lumley Reference Tennekes and Lumley1972). In this case, the associated $\unicode[STIX]{x1D706}_{2}$ tends to zero, which is a consequence of the oversimplifications made and could be solved by retaining higher-order terms.

The preferential alignment of $\unicode[STIX]{x1D74E}$ with $\unicode[STIX]{x1D706}_{2}$ does not imply that most of the contribution to the enstrophy production is due to $\unicode[STIX]{x1D714}^{2}p_{2}$ (Tsinober, Shtilman & Vaisburd Reference Tsinober, Shtilman and Vaisburd1997). Figure 12(d) shows the relative importance of $p_{1}$ , $p_{2}$ and $p_{3}$ in terms of its mean $\langle \cdot \rangle$ (top panel) and its standard deviation $(\cdot )^{\prime }$ (bottom panel) for different filter widths. For the unfiltered case, the contributions of $\langle p_{2}\rangle$ and $\langle p_{3}\rangle$ to the mean enstrophy production/destruction are, in magnitude, roughly one-half of the contribution of $\langle p_{1}\rangle$ , which turns out to be the most important term (Vincent & Meneguzzi Reference Vincent and Meneguzzi1994; Tsinober Reference Tsinober1998). When filtering, the ratios $\langle p_{2}\rangle /\langle p_{1}\rangle$ and $\langle p_{3}\rangle /\langle p_{1}\rangle$ approach $\pm 1$ respectively. This implies that the small-scale vortices are mostly dominated by vortex stretching, but the large-scale vorticity is equally influenced by the three terms from (3.13). From a geometric point of view, the results above suggest that the tube-like structures are favoured at the small scales but that sheet-like objects dominate at larger ones, as in the geometrical analysis of turbulent structures in Moisy & Jiménez (Reference Moisy and Jiménez2004). The scale dependence of the ratios is even more pronounced for the standard deviations of $p_{i}$ , and $p_{2}^{\prime }/p_{1}^{\prime }$ increases steadily, attaining values of up to three times those of $p_{3}^{\prime }/p_{1}^{\prime }$ .

The results above suggest that the dynamics of the flow differs at different scales. However, it was noticed in § 3.3 that this may be caused by the effect of the mean shear. For this reason, the calculations were repeated for cases $S\unicode[STIX]{x1D6FE}$ , where only the velocity fluctuations are considered, and the results are shown in figure 13. The fluctuating vorticity also aligns predominantly with the second eigenvector of the fluctuating rate-of-strain tensor (figure 13 a), implying that the alignment shown in figure 12(a) is not entirely caused by the effect of the mean shear, and the distributions of the three eigenvalues become more symmetric, especially for $\unicode[STIX]{x1D6E5}_{2}>0.1h$ (figures 13 b,c). The main difference compared with the results computed for the total velocities is the improved collapse of the p.d.f.s for both the alignments and the eigenvalues. Figure 13(d) is equivalent to figure 12(d), but the ratios remain roughly constant and independent of the filter width, suggesting that vortices defined through the fluctuating velocities are better candidates than the total vorticity to study the multiscale dynamics of the flow. This scale-independent behaviour also suggests that these large-scale vortices are geometrically self-similar as opposed to those described above for the total velocities, although this should be tested in more detail with a geometrical analysis of the structures, which is out of the scope of the present paper. From the results above, we can conclude that once the direct effect of the mean shear is removed, the multiscale dynamics of the enstrophy production becomes roughly self-similar, consistent with the results shown before for the joint distributions of $Q_{s}$ , $Q_{\unicode[STIX]{x1D714}}$ , $R_{s}$ and $R_{\unicode[STIX]{x1D714}}$ associated with the fluctuating velocity.

Figure 13.

The same as figure 13 but for the fluctuating velocities (cases $S\unicode[STIX]{x1D6FE}$ in table 3). For (a), (b) and (c) the symbols and colours are as in table 3.

3.6 The energy cascade in terms of vortex stretching

In the previous section we have studied the alignment of the vorticity and the eigenvectors of the rate-of-strain tensor at the same scale. Here, we analyse the alignment of vorticity and strain at different scales, i.e. the vorticity associated with the filtered velocity at scale $\unicode[STIX]{x1D6E5}^{\unicode[STIX]{x1D714}}$ , whose quantities will be denoted by $\widetilde{(\cdot )}$ , and the rate of strain associated with the filtered velocity at a scale $\unicode[STIX]{x1D6E5}^{s}$ , represented by $\widehat{(\cdot )}$ . This study is motivated by the classical energy cascade in terms of vortex stretching, where the strain at a given scale stretches the vortices at a smaller one and induces higher velocities by the conservation of angular momentum. This scenario provides a mechanism for the interscale energy transfer required by the energy cascade, but is presumably in contradiction to previous studies (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; She, Jackson & Orszag Reference She, Jackson and Orszag1991; Vincent & Meneguzzi Reference Vincent and Meneguzzi1994) and with § 3.5, where the vorticity aligns most probably with the intermediate strain eigenvector. This may be caused by the fact that $\unicode[STIX]{x1D74E}$ and $\unicode[STIX]{x1D634}_{ij}$ are both studied at the same scale, and it has been noted before that the alignment of vorticity with the intermediate strain eigenvector decreases when the local strain induced by the vortices is eliminated (Jiménez Reference Jiménez1992; Hamlington, Schumacher & Dahm Reference Hamlington, Schumacher and Dahm2008). This suggests that such an alignment could change for vorticity fields and strain tensors each calculated at a different scale.

Figure 14.

The p.d.f.s of the cosine of the angle of the vorticity, $\widetilde{\unicode[STIX]{x1D74E}}$ , filtered with $\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}$ , and the eigenvectors of the rate-of-strain tensor: ——, $\widehat{\unicode[STIX]{x1D706}}_{1}$ ; – – –, $\widehat{\unicode[STIX]{x1D706}}_{2}$ ;– ⋅ – ⋅ –, $\widehat{\unicode[STIX]{x1D706}}_{3}$ ; filtered with $\unicode[STIX]{x1D6E5}_{2}^{s}$ , for (a) $\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}=0.05h$ and $\unicode[STIX]{x1D6E5}_{2}^{s}=0.55h$ , (b) $\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}=0.55h$ and $\unicode[STIX]{x1D6E5}_{2}^{s}=0.05h$ . The results are computed for the total velocity.

The previous idea was tested in Leung, Swaminathan & Davidson (Reference Leung, Swaminathan and Davidson2012) for isotropic turbulence and a few scales. Here, we extend the study to the logarithmic layer of a turbulent channel and a wider range of scales. We will denote the streamwise, wall-normal and spanwise filter widths by $\unicode[STIX]{x1D6E5}_{1}^{j}$ , $\unicode[STIX]{x1D6E5}_{2}^{j}$ and $\unicode[STIX]{x1D6E5}_{3}^{j}$ , with $j=\unicode[STIX]{x1D714},s$ , and ratios as described in § 2.3. The interest of this analysis is further motivated by figure 14(a), which shows that the roles of $\widehat{\unicode[STIX]{x1D706}}_{1}$ and $\widehat{\unicode[STIX]{x1D706}}_{2}$ change completely for $\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}=0.05h$ and $\unicode[STIX]{x1D6E5}_{2}^{s}=0.55h$ compared with those reported for the vorticity and strain at the same scale. In this particular case, the smaller-scale vorticity aligns predominantly with the most extensional eigenvector and is stretched by the larger-scale strain. On the other hand, figure 14(b) shows that the p.d.f.s of $\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{i})$ follow a uniform distribution with no preferential alignment when computed for $\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}=0.55h$ and $\unicode[STIX]{x1D6E5}_{2}^{s}=0.05h$ , i.e. large-scale vorticity versus smaller-scale strain.

It is difficult to infer the causal relation between scales from instantaneous flow fields without more detailed time-resolved information on the flow. However, it is reasonable to suppose that, on average, the causality is from larger scales to smaller ones, since the characteristic times of the former are longer than those of the latter. Thus, for $\unicode[STIX]{x1D6E5}_{2}^{s}>\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}$ , we will assume that the vorticity is stretched/compressed by the larger-scale strain, and vice versa for $\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}>\unicode[STIX]{x1D6E5}_{2}^{s}$ .

To perform a more systematic analysis of the dominant alignment of $\widetilde{\unicode[STIX]{x1D74E}}$ and $\widehat{\unicode[STIX]{x1D706}}_{i}$ , we expand the number of filters previously used in order to sample the scale space in more detail. The new wall-normal filter widths range from $\unicode[STIX]{x1D6E5}_{2}^{j}=0.05h$ to $\unicode[STIX]{x1D6E5}_{2}^{j}=0.55h$ in increments of $0.05h$ with $j=\unicode[STIX]{x1D714},s$ , and all possible combinations of $(\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}},\unicode[STIX]{x1D6E5}_{2}^{s})$ are considered, which yields a total number of 121 cases. Alignments are measured by the ratio of probabilities of $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{i})|>0.9$ , with $i=1,2,3$ , for each $(\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}},\unicode[STIX]{x1D6E5}_{2}^{s})$ pair,

where $P$ stands for probability and $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{2})|>0.9$ is used as a reference value; $\cos (\unicode[STIX]{x1D6FC})=0.9$ corresponds to an angle close to $\unicode[STIX]{x03C0}/7$ radians or $25^{\circ }$ , and $r_{i}>1$ implies a dominant alignment of $\widetilde{\unicode[STIX]{x1D74E}}$ and $\widehat{\unicode[STIX]{x1D706}}_{i}$ . It was tested that values of the cosine equal to 0.7 and 0.8 do not change the discussion below.

Results for $r_{i}$ are shown in figure 15, where low values of $r_{1}$ and $r_{3}$ appear along the diagonal $\unicode[STIX]{x1D6E5}_{2}^{s}=\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}$ , in agreement with the dominant alignment of $\widetilde{\unicode[STIX]{x1D74E}}$ and $\widetilde{\unicode[STIX]{x1D706}}_{2}$ discussed in § 3.5. For $\unicode[STIX]{x1D6E5}_{2}^{s}>\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}$ (upper diagonal), the alignment of $\widetilde{\unicode[STIX]{x1D74E}}$ and $\widehat{\unicode[STIX]{x1D706}}_{1}$ increases with the distance from the diagonal, consistent with the energy-cascade framework described above. This is especially the case for low values of $\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}$ , around $0.05-0.10h$ , and $\unicode[STIX]{x1D6E5}_{2}^{s}>0.2h$ , where $r_{1}\approx 1.6$ . For $\unicode[STIX]{x1D6E5}_{2}^{s}<\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}$ (lower diagonal), the alignment of $\unicode[STIX]{x1D74E}$ and $\widehat{\unicode[STIX]{x1D706}}_{3}$ increases (compression of the flow field by large-scale vorticity), although the effect is weaker than the one observed in the upper diagonal for $r_{1}$ . It is important to remark that the probability distributions flatten in this region, as shown in figure 14(b), and the peaks at $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{i})|=1$ are less pronounced than those found in the upper diagonal. It was checked that the kurtosis coefficients of the p.d.f.s in the lower diagonal are closer to the theoretical value of a perfectly flat distribution than those in the upper part (not shown). This suggests that the alignment of $\widetilde{\unicode[STIX]{x1D74E}}$ and $\widehat{\unicode[STIX]{x1D706}}_{3}$ is less relevant than that of $\widetilde{\unicode[STIX]{x1D74E}}$ and $\widehat{\unicode[STIX]{x1D706}}_{1}$ for $\unicode[STIX]{x1D6E5}_{2}^{s}>\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}$ .

Figure 15.

(a) Ratio of the probability of $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{1})|>0.9$ and $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{2})|>0.9$ as a function of the vorticity and strain wall-normal filter widths, $\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}$ and $\unicode[STIX]{x1D6E5}_{2}^{s}$ respectively. Values equal to one imply equal probability of finding $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{1})|>0.9$ and $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{2})|>0.9$ . (b) The same as (a) but for the ratio of the probability of $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{3})|>0.9$ and $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{2})|>0.9$ . The dashed line is $\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D6E5}_{2}^{s}$ . The results are computed for the total velocity.

The $\widetilde{\unicode[STIX]{x1D74E}}$ – $\widehat{\unicode[STIX]{x1D706}}_{i}$ alignments presented in figure 15 are of little value if their contribution to the enstrophy production of $\widetilde{\unicode[STIX]{x1D714}}^{2}$ is negligible. The dynamical equation for the average enstrophy from (3.12) can be rewritten neglecting the viscous term as

where $\widetilde{\unicode[STIX]{x1D634}}_{ij}$ has been decomposed into its contribution from scale $\unicode[STIX]{x1D6E5}_{2}^{s}$ and a residual term such that $\widetilde{\unicode[STIX]{x1D634}}_{ij}=\widehat{\unicode[STIX]{x1D634}}_{ij}+\unicode[STIX]{x1D634}_{ij}^{r}$ . Relation (3.15) shows that the enstrophy production may be expressed as the interaction of $\widetilde{\unicode[STIX]{x1D714}}_{i}$ and $\widehat{\unicode[STIX]{x1D634}}_{ij}$ plus a residual, and that the sum of both is balanced by the interscale transfer term on the right-hand side. It should be noted that the $\widetilde{\unicode[STIX]{x1D74E}}$ – $\widehat{\unicode[STIX]{x1D706}}_{i}$ alignment calculated above is directly related to the enstrophy production $\widetilde{\unicode[STIX]{x1D714}}_{i}\widetilde{\unicode[STIX]{x1D714}}_{j}\widehat{\unicode[STIX]{x1D634}}_{ij}$ . The importance of this term is quantified in figure 16(a), which shows the ratio

for all possible combinations of filter widths. By definition, the diagonal elements must be $1$ since $s_{ij}^{r}=0$ and $\langle \widetilde{\unicode[STIX]{x1D714}}_{i}\widetilde{\unicode[STIX]{x1D714}}_{j}\widehat{\unicode[STIX]{x1D634}}_{ij}\rangle >0$ . Interestingly, the data reveal that $r^{p}$ reaches values close to 1 for $\unicode[STIX]{x1D6E5}_{2}^{s}>\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}$ , and the contribution of $\langle \widetilde{\unicode[STIX]{x1D714}}_{i}\widetilde{\unicode[STIX]{x1D714}}_{j}\widehat{\unicode[STIX]{x1D634}}_{ij}\rangle$ is large enough to support the idea of a non-negligible role of vortex stretching in the energy cascade. Surprisingly, $r^{p}$ is also large in magnitude but negative for scales in the far lower part. The term (3.15) appears in the dynamic equation for $\widehat{\unicode[STIX]{x1D634}}_{ij}\widehat{\unicode[STIX]{x1D634}}_{ij}$ , acting as a source but with opposite sign. This suggests an interesting connection between large-scale vorticity and small-scale strain, closing the self-sustained cascade process. However, we have shown above that in this case there is no preferential alignment between the vorticity and the strain eigenframe (figure 14 b) and the physical relevance of this result is unclear. A simplified sketch of the different scenarios is shown in figure 17.

Figure 16.

Ratio of the average enstrophy productions, $r^{p}$ , as a function of the wall-normal filter width for the vorticity, $\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}$ , denoted by $\widetilde{(\cdot )}$ , and the strain, $\unicode[STIX]{x1D6E5}_{2}^{s}$ , denoted by $\widehat{(\cdot )}$ . The results are computed from (a) the total velocity and (b) the fluctuating velocity.

Figure 17.

Sketches of the vorticity–strain interaction: (a) at the same scale; (b) strain scale larger than vorticity scale; (c) strain scale smaller than vorticity scale.

The results shown so far were calculated for the total velocity, and following the structure of previous sections, they were repeated for the fluctuations. Most of the conclusions discussed for the total velocity apply to the fluctuating vorticity and rate-of-strain tensor. The most remarkable difference is found in the ratio $r^{p}$ in figure 16(b). The contribution of $\langle \widetilde{\unicode[STIX]{x1D714}}_{i}\widetilde{\unicode[STIX]{x1D714}}_{j}\widehat{\unicode[STIX]{x1D634}}_{ij}\rangle$ decays with the distance from the diagonal and is around 20–30 % of $\langle \widetilde{\unicode[STIX]{x1D714}}_{i}\widetilde{\unicode[STIX]{x1D714}}_{j}\widetilde{\unicode[STIX]{x1D634}}_{ij}\rangle$ at those places where the alignment of $\widetilde{\unicode[STIX]{x1D74E}}$ and $\widehat{\unicode[STIX]{x1D706}}_{1}$ is of the same order as that of $\widetilde{\unicode[STIX]{x1D74E}}$ and $\widehat{\unicode[STIX]{x1D706}}_{2}$ (figure 18 a). Figure 18 also shows that $r_{1}$ and $r_{3}$ attain values below those obtained for the total velocity, which may be connected to the counterclockwise behaviour in the $R_{\unicode[STIX]{x1D714}}$ – $Q_{\unicode[STIX]{x1D714}}$ plane in figure 9(h) and discussed in § 3.3.

Figure 18.

(a) Ratio of the probability of $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{1})|>0.9$ and $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{2})|>0.9$ as a function of the vorticity and strain wall-normal filter widths, $\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}$ and $\unicode[STIX]{x1D6E5}_{2}^{s}$ respectively. Values equal to one imply equal probability of finding $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{1})|>0.9$ and $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{2})|>0.9$ . (b) The same as (a) but for the ratio of the probability of $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{3})|>0.9$ and $|\!\cos (\widetilde{\unicode[STIX]{x1D74E}},\widehat{\unicode[STIX]{x1D706}}_{2})|>0.9$ . The dashed line is $\unicode[STIX]{x1D6E5}_{2}^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D6E5}_{2}^{s}$ . The results are computed for the fluctuating velocity.