4.1 Segment length statistics

4.1.1 The mean segment length

The mean length of vectorline segments quantifies the scale of the local structure. Figure 7 shows for both vectors the dependence of the segment mean length on the wall-normal distance $y^{+}$ , where the superscript $+$ denotes normalization by the viscous unit. Meanwhile, the Taylor scale, which for boundary layer turbulence is defined as $\unicode[STIX]{x1D706}(y^{+})=\sqrt{10\unicode[STIX]{x1D708}k(y^{+})/\unicode[STIX]{x1D716}(y^{+})}$ , is also plotted for comparison. Here $k(y^{+})$ and $\unicode[STIX]{x1D716}(y^{+})$ are the averaged turbulent kinetic energy and the averaged energy dissipation conditioned at specified $y^{+}$ , respectively. For streamline segments, the mean length $\langle l_{u}\rangle ^{+}$ decreases quickly in the viscous sublayer ( $y^{+}<10$ ) and buffer layer ( $10<y^{+}<30$ ), and reaches its minimum in the log-law region at approximately $y^{+}\approx 80$ . Afterwards $\langle l_{u}\rangle ^{+}$ increases more and more rapidly until the edge of the boundary layer. For vorticity line segments in the viscous sublayer and part of the buffer layer, the mean length $\langle l_{\unicode[STIX]{x1D714}}\rangle ^{+}$ decreases slowly and then starts to increase in the entire log-law region and beyond.

Figure 7.

The variation of (a) $\langle l_{u}\rangle$ for the streamline segments and (b) $\langle l_{\unicode[STIX]{x1D714}}\rangle$ for the vorticity line segments, as a function of $y^{+}$ .

Physically, the mean length is dominated by the scale of the local flow structure and the perturbation strength, which can roughly be understood as the local turbulence intensity. On the one hand, the flow structure, for instance shown in figure 1, evolves to be larger from the wall boundary towards the outside, making the segments longer. On the other hand, stronger perturbation, as well as larger turbulence intensity, tends to disturb the flow field to generate more extremal points, making the segments shorter. Within the viscous sublayer, the turbulent intensity increases and then continues to decrease from the buffer region. Therefore the mean length of both the streamline segments and vorticity line segments decreases first and then increases. Overall, for vorticity line segments, the mean length $\langle l_{\unicode[STIX]{x1D714}}\rangle ^{+}$ is much smaller than $\langle l_{u}\rangle ^{+}$ . A sharp difference is that from the solid wall surface $\langle l_{u}\rangle ^{+}$ decreases largely till $y^{+}\approx 80$ , while $\langle l_{\unicode[STIX]{x1D714}}\rangle ^{+}$ decreases weakly in a much shorter range till $y^{+}=20$ , which may be an indication that the vorticity line segments are more influenced by the flow structure than turbulence perturbation.

4.1.2 Probability density function of the length distribution

More detailed information on the segment length can be viewed from its probability density function (p.d.f.). Work has been done in studying the scale of dissipation elements (Wang & Peters Reference Wang and Peters2006, Reference Wang and Peters2008). Physically, the segment length $l$ is determined by the generation and annihilation of extremal points. On the one hand, turbulent random motion disturbs the flow to generate new extremal points. On the other hand, viscous diffusion annihilates the closely located extremal points to smoothen the flow field. Besides these two abrupt effects, there is another drift velocity representing the continuous change of $l$ . In a general scenario, the evolution of $l$ can be modelled based on a Poisson random cutting–reconnection process. Considering one-dimensional space, each segment is bounded by two extremal points, and all the segments are connected with each other to fill the entire space. Once a newly generated extremal point is added into a segment, it is then cut into two smaller segments. Meanwhile, if two adjacent extremal points merge and then annihilate because of viscous diffusion, two smaller segments can connect to generate a large one. In total, the birth and death of segments are summarized as the following mechanism:

1. (i) generation by cutting ( $GC$ );

2. (ii) generation by reconnection ( $GR$ );

3. (iii) removal by cutting ( $RC$ );

4. (iv) removal by reconnection ( $RR$ ).

Collecting all the above contributions, together with the drift velocity, one obtains the following equation for the p.d.f. of the segment length $l$ as (Wang & Peters Reference Wang and Peters2006, Reference Wang and Peters2008):

(4.1) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}P(l,t)}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}(v(l)P(l,t))}{\unicode[STIX]{x2202}l} & = & \displaystyle \int _{0}^{\infty }2\unicode[STIX]{x1D706}P(l+z)\,\text{d}z-\unicode[STIX]{x1D706}lP(l,t)\nonumber\\ \displaystyle & & \displaystyle \times \,\int _{0}^{l}2\unicode[STIX]{x1D707}\frac{z}{l}P(l-z)P(z)\,\text{d}z-2\unicode[STIX]{x1D707}P(l,t).\end{eqnarray}$$

Here $v(l)$ is the drift velocity at scale $l$ , $\unicode[STIX]{x1D706}$ is the cutting frequency per unit length, and $\unicode[STIX]{x1D707}$ is the disappearance rate of extreme points, which can be determined as $\unicode[STIX]{x1D707}=4D(\unicode[STIX]{x2202}P(x,t)/\unicode[STIX]{x2202}x)|_{x=0}$ , where $D$ is the fluid viscosity. The four terms on the right-hand side represent $GC$ , $RC$ , $GR$ and $RR$ , respectively. In the non-dimensional form, the p.d.f. of $\tilde{l}=l/\langle l\rangle$ , i.e. the length scale normalized by its mean $\langle l\rangle$ , becomes

(4.2) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}P(\tilde{l},t)}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{l}}(\tilde{v}(\tilde{l})P(\tilde{l},t)) & = & \displaystyle \unicode[STIX]{x1D6EC}\left[\int _{0}^{\infty }2P(\tilde{l}+\tilde{z},t)\,\text{d}\tilde{z}-\tilde{l}P(\tilde{l},t)\right]\nonumber\\ \displaystyle & & \displaystyle \left.+\,8\frac{\unicode[STIX]{x2202}P(\tilde{l},t)}{\unicode[STIX]{x2202}\tilde{l}}\right|_{\tilde{l}=0}\left[\int _{0}^{\tilde{l}}\frac{\tilde{z}}{\tilde{l}}P(\tilde{l}-\tilde{z},t)P(\tilde{z},t)\,\text{d}\tilde{z}-P(\tilde{l},t)\right].\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x1D6EC}$ is an eigenvalue, which can be determined from the p.d.f.’s normalization condition. At the statistically stationary state, there is a dynamic balance under the action of all these mechanisms and the p.d.f. equation (4.2) has a unique time-independent solution. Results from different problems such as the dissipation element scale in passive scalar turbulence (Wang & Peters Reference Wang and Peters2006), the scale distribution in Lagrangian turbulence (Wang Reference Wang2014) and reactive turbulence (Wang, Chakraborty & Zhang Reference Wang, Chakraborty and Zhang2013) justify that such a solution is applicable with good generality.

Figure 8 shows the p.d.f. of streamline segment length $\tilde{l}_{u}$ at different wall-normal distances, in both linear–linear and log–linear coordinates. Compared with the stationary solution of (4.2), the agreement is satisfactory, which implies that the aforementioned smoothing mechanism, the random perturbation mechanism and the continuous drift are generally valid in describing the streamline segment behaviour. It is noteworthy that in the log–linear plot, the p.d.f.’s exponential tail can be explained by a Poisson-like process at the large scale in deriving (4.2).

Figure 8.

P.d.f. of $\tilde{l}_{u}=l_{u}/\langle l_{u}\rangle$ compared with the solution of (4.2) in (a) linear–linear coordinates and (b) log–linear coordinates.

For vorticity line segments, the length p.d.f. is shown in figure 9, in both linear–linear and log–linear coordinates. Differently from the $\tilde{l}_{u}$ case, the $\text{p.d.f.}(\tilde{l}_{\unicode[STIX]{x1D714}})$ at different $y^{+}$ is not universal. At larger $y^{+}$ (for instance, when $y^{+}>30$ ) the numerical result matches the solution of (4.2); however, when closer to the wall, the numerical result deviates more from the equation solution. A tentative explanation of such an outcome is that the strong shear stress near the solid wall may invalidate the smoothing and random perturbation mechanisms in deriving (4.2).

Figure 9.

P.d.f. of $\tilde{l}_{\unicode[STIX]{x1D714}}=l_{\unicode[STIX]{x1D714}}/\langle l_{\unicode[STIX]{x1D714}}\rangle$ compared with the solution of (4.2) in (a) linear–linear coordinates and (b) log–linear coordinates.

4.2 Joint p.d.f. between the characteristic parameters

Besides the segment length p.d.f., the joint p.d.f. between the characteristic parameters is more informative. Figure 10 shows such joint p.d.f.s for streamline segments corresponding to grid points at four different wall-normal distances, i.e. $y^{+}=2.6$ , $y^{+}=20$ , $y^{+}=80$ and $y^{+}=275$ , which according to the mean velocity profile in figure 2 represent the typical samples in the viscous sublayer, buffer layer, log-law region and outer layer, respectively. Here $\unicode[STIX]{x0394}u$ is non-dimensionalized by the speed of sound. Each joint p.d.f. has two branches, one with positive $\unicode[STIX]{x0394}u$ and the other with negative $\unicode[STIX]{x0394}u$ . Interestingly, in the viscous sublayer, these two branches are almost symmetrical. When moving to the buffer layer, the positive branch becomes clearly larger, whereas in the log-law region and outer layer such asymmetry weakens again and finally almost disappears.

Figure 10.

Joint p.d.f. of $l_{u}$ and $\unicode[STIX]{x0394}u$ at different wall-normal distances: (a) $y^{+}=2.6$ (in the viscous sublayer); (b) $y^{+}=20$ (in the buffer layer); (c) $y^{+}=80$ (in the log-law region); and (d) $y^{+}=275$ (in the outer layer).

Because of the monotonic change of $u$ along the streamline segments, $\unicode[STIX]{x0394}u/l_{u}$ can be interpreted as a kind of averaged velocity derivative. Therefore the asymmetry of the joint p.d.f. implies the unequal contributions from the positive and negative velocity derivative parts in the flow field, which can be approximately quantified by the skewness of the velocity derivative. For the present boundary layer flow, streamline segments are most probably aligned along the streamwise direction. Therefore it is meaningful to investigate the skewness of the streamwise velocity component derivative, i.e.

(4.3) $$\begin{eqnarray}S=\langle (\text{d}u_{1}/\text{d}x)^{3}\rangle /\langle (\text{d}u_{1}/\text{d}x)^{2}\rangle ^{3/2}.\end{eqnarray}$$

As demonstrated in figure 11, in most of the boundary layer region, the skewness is negative. Around $y^{+}=20$ ,  $S$ reaches its minimum and then increases. A similar behaviour has been observed by Vreman & Kuerten (Reference Vreman and Kuerten2014). As $y^{+}$ varies, the change of skewness in figure 11 conforms with the change of the joint p.d.f. asymmetry.

Figure 11.

Dependence of the skewness of the streamwise velocity component derivative on the wall-normal distance. In coincidence with the asymmetry of the joint p.d.f.s in figure 10, the skewness reaches a minimum in the buffer layer.

Physically, the negative skewness of the velocity derivative represents the average rate of vorticity production through vortex stretching (Taylor Reference Taylor1938; Monin & Yaglom Reference Monin and Yaglom2007). Vreman & Kuerten (Reference Vreman and Kuerten2014) believed that the negative peak is caused by local coherent structures. In the context of streamline segment analysis, the negative skewness can be interpreted from the joint p.d.f. of $l_{u}$ and $\unicode[STIX]{x0394}u$ . As has been extensively studied (Wang Reference Wang2010; Boschung et al. Reference Boschung, Schaefer, Peters and Meneveau2014), the asymmetry of the joint p.d.f. is one of the most important results from streamline segment statistics in isotropic turbulence. Considering an individual streamline segment, the velocity magnitude $u$ along the vector direction will increase monotonically if the segment is positive, or decrease if the segment is negative. Therefore, when evolving with time, positive segments with increasing $u$ tend to be elongated, while negative segments tend to be compressed because of decreasing $u$ along the vector direction. Consequently, on average, the cube of the negative derivative becomes larger, while the cube of the positive derivative becomes smaller, resulting in negative  $S$ .

From the joint p.d.f., the conditional mean $\unicode[STIX]{x0394}u$ can be defined as $\langle \unicode[STIX]{x0394}u\rangle =\int \!\unicode[STIX]{x0394}uP(\unicode[STIX]{x0394}u,l_{u})\,\text{d}\unicode[STIX]{x0394}u$ . To differentiate the positive and negative branches, $\langle |\unicode[STIX]{x0394}u|\rangle =\int \!|\unicode[STIX]{x0394}u|P(\unicode[STIX]{x0394}u,l_{u})\,\text{d}\unicode[STIX]{x0394}u$ are calculated separately for positive segments and negative segments. The results are shown in figure 12. In the log-law region and the outer layer, $\langle |\unicode[STIX]{x0394}u|\rangle$ from the positive and negative segments are almost identical for $l_{u}>20$ , with a scaling exponent approximately 0.8. Differently, in the viscous and buffer sublayers, the positive and negative segments have different conditional mean $\langle |\unicode[STIX]{x0394}u|\rangle$ and there is no clear scaling behaviour. Especially in the smaller $l_{u}$ range, $\langle |\unicode[STIX]{x0394}u|\rangle$ from the positive segments is smaller, while in the larger $l_{u}$ range, $\langle |\unicode[STIX]{x0394}u|\rangle$ from the negative segments is smaller.

Figure 12.

The conditional mean of $|\unicode[STIX]{x0394}u|$ , calculated separately for the positive and negative segments, at different wall-normal distances corresponding to figure 10: (a) $y^{+}=2.6$ (in the viscous sublayer); (b) $y^{+}=20$ (in the buffer layer); (c) $y^{+}=80$ (in the log-law region); and (d) $y^{+}=275$ (in the outer layer).

For vorticity line segments, the joint p.d.f.s between $l_{\unicode[STIX]{x1D714}}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}$ at different $y^{+}$ are shown in figure 13. Here $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}$ is non-dimensionalized by the speed of sound and the boundary layer thickness. Compared with figure 10, a clear difference is that the positive $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}$ and negative $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}$ branches are symmetric throughout the boundary layer. The reason is that, for vorticity line segments, the aforementioned kinematic mechanism, leading on average to longer positive streamline segments and shorter negative streamline segments, does not exist (Wang Reference Wang2012). The positive $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}$ and negative $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}$ branches are well separated in the viscous sublayer, and then become closer when $y^{+}$ increases. Moreover, at larger $y^{+}$ ,  $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}$ decreases because of smaller velocity gradient and thus vorticity.

Figure 13.

Joint p.d.f. of $l_{\unicode[STIX]{x1D714}}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}$ at different wall-normal distances: (a) $y^{+}=2.6$ (in the viscous sublayer); (b) $y^{+}=20$ (in the buffer layer); (c) $y^{+}=80$ (in the log-law region); and (d) $y^{+}=275$ (in the outer layer).

Figure 14 plots the conditional mean of $|\unicode[STIX]{x0394}\unicode[STIX]{x1D714}|$ as a function of $l_{\unicode[STIX]{x1D714}}$ , which is almost identical for the positive and negative segments. In different boundary layer regions, no obvious scaling can be observed. Especially in the viscous sublayer and buffer layer, $\langle |\unicode[STIX]{x0394}\unicode[STIX]{x1D714}|\rangle$ levels off and then drops down at larger $l_{\unicode[STIX]{x1D714}}$ .

Figure 14.

The conditional mean of $|\unicode[STIX]{x0394}\unicode[STIX]{x1D714}|$ , calculated separately for the positive and negative vorticity line segments, at different wall-normal distances corresponding to figure 10: (a) $y^{+}=2.6$ (in the viscous sublayer); (b) $y^{+}=20$ (in the buffer layer); (c)  $y^{+}=80$ (in the log-law region); and (d) $y^{+}=275$ (in the outer layer).

4.3 The segment spatial structure

We denote the Cartesian coordinate differences between the segment starting point $s$ and ending point $e$ as $\unicode[STIX]{x0394}x=x_{e}-x_{s}$ , $\unicode[STIX]{x0394}y=y_{e}-y_{s}$ and $\unicode[STIX]{x0394}z=z_{e}-z_{s}$ . In isotropic turbulence without any preference of spatial orientation of the segments, $\unicode[STIX]{x0394}x$ , $\unicode[STIX]{x0394}y$ and $\unicode[STIX]{x0394}z$ need to be uncorrelated. Therefore, the anisotropy of the turbulent boundary layer can be characterized via the segment orientation statistics.

For streamline segments, the correlations between $\unicode[STIX]{x0394}x$ , $\unicode[STIX]{x0394}y$ and $\unicode[STIX]{x0394}z$ , or between $\unicode[STIX]{x0394}x^{+}$ , $\unicode[STIX]{x0394}y^{+}$ and $\unicode[STIX]{x0394}z^{+}$ in the normalized form, are shown in figure 15. The main findings are summarized as follows.

1. (i) Because of the dominant velocity component along the streamwise direction, segments tend to align with $x$ . Therefore $\unicode[STIX]{x0394}x^{+}$ is very probably positive and the magnitude of $\unicode[STIX]{x0394}x^{+}$ is much larger than that of $\unicode[STIX]{x0394}y^{+}$ and $\unicode[STIX]{x0394}z^{+}$ , as shown in the joint p.d.f.s between $\unicode[STIX]{x0394}x^{+}$ and $\unicode[STIX]{x0394}y^{+}$ , and between $\unicode[STIX]{x0394}x^{+}$ and $\unicode[STIX]{x0394}z^{+}$ . As $y^{+}$ increases, on average $\unicode[STIX]{x0394}x^{+}$ first decreases and then increases, which can be explained from the variation of the mean segment length, as shown in figure 7.

2. (ii) For the joint p.d.f.s between $\unicode[STIX]{x0394}x^{+}$ and $\unicode[STIX]{x0394}y^{+}$ , in the near vicinity of the wall, e.g. $y^{+}=2.6$ , because of the boundary confinement, the streamline segments are forced to bend up, and thus $\unicode[STIX]{x0394}y^{+}$ is probably positive. When moving to the buffer layer at $y^{+}=20$ , the positive $\unicode[STIX]{x0394}y^{+}$ part becomes larger. Such an observation indicates that the sweep events in the ‘bursting cycle’ (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967) predominate the production and regeneration of near-wall turbulence, which is consistent with quadrant analysis (Wallace Reference Wallace2016). From the buffer layer to the outer layer, the segments then gradually bend down with negative $\unicode[STIX]{x0394}y^{+}$ . Especially at large $y^{+}$ as $y^{+}=275$ , the negative $\unicode[STIX]{x0394}y^{+}$ part can be clearly observed.

3. (iii) The joint p.d.f.s between $\unicode[STIX]{x0394}y^{+}$ and $\unicode[STIX]{x0394}z^{+}$ demonstrate that at $y^{+}=2.6$ on average the magnitude of $\unicode[STIX]{x0394}z^{+}$ is much larger than that of $\unicode[STIX]{x0394}y^{+}$ , which means that the streamline segments are more likely to swing along the spanwise direction than the wall-normal direction. A possible explanation is that the internal shear layers (or vorticity sheets) dominate the dynamic behaviour of the near-wall streaks (Schoppa & Hussain Reference Schoppa and Hussain2002). With the increase of $y^{+}$ , the magnitude of $\unicode[STIX]{x0394}z^{+}$ shrinks gradually and $\unicode[STIX]{x0394}y^{+}$ and $\unicode[STIX]{x0394}z^{+}$ are comparable. Such a tendency agrees with the change of the turbulence kinetic energy of each velocity component. In principle, $\unicode[STIX]{x0394}y^{+}$ and $\unicode[STIX]{x0394}z^{+}$ are closely related to the velocity fluctuations along the $y$ and $z$ directions, or equivalently to $\langle u_{2}^{\prime 2}\rangle$ and $\langle u_{3}^{\prime 2}\rangle$ , respectively. For $y^{+}$ as small as $y^{+}=2.6$ , $\langle u_{3}^{\prime 2}\rangle$ is much larger than $\langle u_{2}^{\prime 2}\rangle$ , while for larger $y^{+}$ , $\langle u_{3}^{\prime 2}\rangle$ and $\langle u_{2}^{\prime 2}\rangle$ become closer.

For vorticity line segments, the main results are presented in figure 16 with the following findings.

(iv)

The joint p.d.f.s between $\unicode[STIX]{x0394}x^{+}$ and $\unicode[STIX]{x0394}y^{+}$ suggest that the magnitudes of $\unicode[STIX]{x0394}x^{+}$ and $\unicode[STIX]{x0394}y^{+}$ are comparable. At different $y^{+}$ , the positive $\unicode[STIX]{x0394}y^{+}$ part and the negative $\unicode[STIX]{x0394}y^{+}$ part are almost symmetric. When $y^{+}$ increases, on average the magnitudes of $\unicode[STIX]{x0394}y^{+}$ and $\unicode[STIX]{x0394}x^{+}$ increase, because of the larger turbulent structure. At $y^{+}=2.6$ ,  $\unicode[STIX]{x0394}x^{+}$ and $\unicode[STIX]{x0394}y^{+}$ are negatively correlated, while moving away from the wall such correlation becomes positive and $\unicode[STIX]{x0394}x^{+}$ on average is marginally larger than $\unicode[STIX]{x0394}y^{+}$ . The two peaks of the joint p.d.f. may be closely related to the hairpin vortex structure and the orientation of the maximum strain rate (Head & Bandyopadhyay Reference Head and Bandyopadhyay1981). Further quantitative investigation is needed to better understand the segment statistics.

(v)

From the joint p.d.f.s between $\unicode[STIX]{x0394}x^{+}$ and $\unicode[STIX]{x0394}z^{+}$ and between $\unicode[STIX]{x0394}z^{+}$ and $\unicode[STIX]{x0394}y^{+}$ , we see that at $y^{+}=2.6$ , because of the large shear strain rate and thus the vorticity magnitude, the segments are typically oriented in the $z$ direction and $\unicode[STIX]{x0394}z^{+}$ is very probably negative. When $y^{+}$ increases, the positive $\unicode[STIX]{x0394}z^{+}$ part becomes larger because of the stronger perturbation from turbulent eddies, while the negative part remains almost invariant. Overall the negative part is larger. In the log-law region and the outer layer, $\unicode[STIX]{x0394}x^{+}$ , $\unicode[STIX]{x0394}y^{+}$ and $\unicode[STIX]{x0394}z^{+}$ are all comparable in magnitude.

(vi)

The joint p.d.f.s between $\unicode[STIX]{x0394}y^{+}$ and $\unicode[STIX]{x0394}z^{+}$ are symmetric across the abscissa axis, which means that vorticity line segments have no preference to bend upwards or downwards. When $y^{+}$ increases, the positive and negative $\unicode[STIX]{x0394}y^{+}$ branches become less separated because the flow is more isotropic.

Figure 15.

For the streamline segments, the joint p.d.f. between the spatial coordinates ( $\unicode[STIX]{x0394}x$ , $\unicode[STIX]{x0394}y$ and $\unicode[STIX]{x0394}z$ ) at (a) $y^{+}=2.6$ , (b) $y^{+}=20$ , (c) $y^{+}=80$ and (d)  $y^{+}=275$ .

Figure 16.

For the vorticity line segments, the joint p.d.f. between the spatial coordinates ( $\unicode[STIX]{x0394}x$ , $\unicode[STIX]{x0394}y$ and $\unicode[STIX]{x0394}z$ ) at (a) $y^{+}=2.6$ , (b) $y^{+}=20$ , (c) $y^{+}=80$ and (d)  $y^{+}=275$ .

Figure 17.

Resolution independence check for the p.d.f. of $\tilde{l}_{u}=l_{u}/\langle l_{u}\rangle$ .