3.1 Lagrangian stretching analysis

Ni et al. (Reference Ni, Ouellette and Voth2014) observed that not only prolate spheroids but also the fluid vorticity vector $2\unicode[STIX]{x1D734}$ aligned in the direction of Lagrangian stretching in homogenous isotropic turbulence. The crucial question that therefore arises is whether this finding is universally valid, e.g. also in anisotropic channel-flow turbulence in presence of substantial mean vorticity. By considering the alignment of the fluid rotation vector $\unicode[STIX]{x1D734}$ relative to the maximum eigenvector of the left Cauchy–Green strain tensor $\boldsymbol{e}_{L1}$ in figure 2(a), we can observe that a modest alignment $\langle (\boldsymbol{e}_{L1}\boldsymbol{\cdot }\unicode[STIX]{x1D734})^{2}\rangle \approx 0.50$ still exists, although the alignment has attenuated compared with $\langle (\boldsymbol{e}_{L1}\boldsymbol{\cdot }\unicode[STIX]{x1D734})^{2}\rangle \approx 0.60$ in homogeneous isotropic turbulence (Ni et al. Reference Ni, Ouellette and Voth2014). They argued that the stretching of fluid vorticity is the same as the stretching of a material line and tends to align the vorticity in the same direction as $\boldsymbol{e}_{L1}$ . They explained the observed plateau $\langle (\boldsymbol{e}_{L1}\boldsymbol{\cdot }\unicode[STIX]{x1D734})^{2}\rangle \approx 0.60$ as a result of a dynamic balance between vorticity stretching and viscous effects, the latter which tend to move $\unicode[STIX]{x1D734}$ and $\boldsymbol{e}_{L1}$ apart. Their conjecture is indeed supported by our results in figure 2(b) which reveal qualitatively different alignments in the near-wall region and at the centre of the channel. Statistics in figure 2(b) is computed by averaging the instantaneous data of alignment between fluid vorticity vector and Lagrangian stretching over the two homogeneous directions (i.e. the streamwise and spanwise) in each of the 100 bins from wall to wall and over the time window from $t^{+}=t_{0}^{+}+136$ to $t^{+}=t_{0}^{+}+144$ where the correlations of those quantities are in an almost statistically steady state. The fluid vorticity is preferentially aligned with the direction of strongest Lagrangian stretching $\boldsymbol{e}_{L1}$ in the nearly isotropic centre region, just as in homogenous isotropic turbulence. Ni et al. (Reference Ni, Ouellette and Voth2014) argued that the vorticity became preferentially aligned with $\boldsymbol{e}_{L1}$ simply due to angular momentum conservation. However, this preferred alignment gradually reduces as the wall is approached and tends almost to zero. The vanishing alignment is most pronounced inside of $z^{+}<30$ where viscous effects gradually become dominating and, apparently, orient the vorticity perpendicular to $\boldsymbol{e}_{L1}$ . The fluid rotation $\unicode[STIX]{x1D734}$ , i.e. half of the fluid vorticity, exhibits fundamental differences between the almost isotropic centre region and the near-wall regions. The velocity field in the vicinity of the channel walls is characterized not only by a substantial mean vorticity entirely in the spanwise direction, but also by highly anisotropic vorticity fluctuations (Andersson, Zhao & Variano Reference Andersson, Zhao and Variano2015). The leading component of the instantaneous rotation vector $\unicode[STIX]{x1D734}$ is therefore in the $y$ -direction and the results in figure 2(b) accordingly show that maximum Lagrangian stretching is in the ( $x,z$ )-plane.

Figure 2.

Alignment of the fluid rotation vector $\unicode[STIX]{x1D734}$ relative to the three eigenvectors $\boldsymbol{e}_{Li}$ of the left Cauchy–Green strain tensor. (a) Time evolution of $\langle (\boldsymbol{e}_{Li}\boldsymbol{\cdot }\unicode[STIX]{x1D734})^{2}\rangle$ ; (b) variation of $\langle (\boldsymbol{e}_{Li}\boldsymbol{\cdot }\unicode[STIX]{x1D734})^{2}\rangle$ from the wall to the channel centre at $z^{+}=180$ sampled in the almost statistically steady state from $t^{+}=t_{0}^{+}+136$ to $t^{+}=t_{0}^{+}+144$ . Notice that the vorticity vector is two times the rotation vector, i.e.  $2\unicode[STIX]{x1D734}$ .

The observed alteration in the alignment of the fluid rotation vector $\unicode[STIX]{x1D734}$ with the eigenvectors $\boldsymbol{e}_{Li}$ of the left Cauchy–Green strain tensor from the channel centre to the channel walls in figure 2(b) could explain the weaker alignment $\langle (\boldsymbol{e}_{L1}\boldsymbol{\cdot }\unicode[STIX]{x1D734})^{2}\rangle \approx 0.50$ observed in the channel flow in figure 2(a) as compared with that found in homogenous isotropic turbulence by Ni et al. (Reference Ni, Ouellette and Voth2014). Indeed, $\langle (\boldsymbol{e}_{L1}\boldsymbol{\cdot }\unicode[STIX]{x1D734})^{2}\rangle \approx 0.57$ at the centre where the fluid vorticity is almost isotropic.

3.2 Orientation correlations

To uncover the mechanism responsible for the interesting orientational behaviour of inertia-free spheroids in near-wall turbulence, as illustrated in figure 1, we examined the correlation between the particle orientation vector $\boldsymbol{p}$ and the three eigenvectors $\boldsymbol{e}_{Li}$ of the left Cauchy–Green strain tensor for spheroids with aspect ratios $\unicode[STIX]{x1D706}$ ranging from 50 to 0.01. The statistics shown in figure 3 were obtained by averaging 55 000 Lagrangian particle trajectories over the whole channel. As the elongation of rod-like particles increased to $\unicode[STIX]{x1D706}=50$ , their orientation vector $\boldsymbol{p}$ approached the direction of strongest Lagrangian stretching $\boldsymbol{e}_{L1}$ , similarly to what was observed by Ni et al. (Reference Ni, Ouellette and Voth2014) for prolate spheroids with $\unicode[STIX]{x1D706}=20$ suspended in homogeneous isotropic turbulence. This finding suggests that rod-like particles preferentially orient their symmetry axis in the direction of the strongest Lagrangian stretching independently of their actual trajectories in the anisotropic channel flow. While Ni et al. (Reference Ni, Ouellette and Voth2014) reported that the orientation dynamics for rods with aspect ratio larger than approximately 10 was insensitive to aspect ratio, the preferential alignment of the long rods in figure 3(a) was stronger than for the shorter rod-like particles in figure 3(b). By gradually decreasing the aspect ratio to unity (a–c), the alignment gradually attenuates.

Figure 3.

Time evolution of the alignment of the orientation vector $\boldsymbol{p}$ of spheroidal particles with aspect ratio $\unicode[STIX]{x1D706}$ relative to the three eigenvectors $\boldsymbol{e}_{Li}$ of the left Cauchy–Green strain tensor. Here, $\boldsymbol{e}_{L1}$ and $\boldsymbol{e}_{L3}$ are the maximum (extensional) and minimum (compressional) eigenvectors, respectively. The alignment $\langle (\boldsymbol{e}_{Li}\boldsymbol{\cdot }\boldsymbol{p})^{2}\rangle$ of two randomly oriented vectors is $1/3$ . (a)  $\unicode[STIX]{x1D706}=50$ , (b)  $\unicode[STIX]{x1D706}=3$ , (c)  $\unicode[STIX]{x1D706}=1.5$ , (d)  $\unicode[STIX]{x1D706}=0.8$ , (e)  $\unicode[STIX]{x1D706}=0.33$ and (f)  $\unicode[STIX]{x1D706}=0.01$ .

While the present alignment statistics in figure 3(a) showed striking similarities with the results in figure 1(a) of Ni et al. (Reference Ni, Ouellette and Voth2014), the orientation of disk-like particles with aspect ratios ranging from 0.01 to 0.8 evolved almost oppositely in figure 3(d–f). The oblate spheroids tended to align their symmetry axis perpendicular to the direction of strongest Lagrangian stretching, i.e.  $\langle (\boldsymbol{e}_{L1}\boldsymbol{\cdot }\boldsymbol{p})^{2}\rangle$ decayed and eventually approached zero for the flattest disks, whereas $\boldsymbol{p}$ instead aligned with the minimum eigenvector $\boldsymbol{e}_{L3}$ associated with Lagrangian compression. Similar observations for oblate tracer spheroids in homogeneous isotropic turbulence were recently reported by Hunt et al. (Reference Hunt, Tierney, Kramel and Voth2015). When the shape of the spheroid is close to a sphere, as shown in figure 3(c,d), the alignments are approaching to be random but weak preferential alignment still exists.

Initially at $t^{+}=0$ , the plots in figure 3 show the instantaneous alignments of the spheroids’ symmetry axis relative to the eigenvectors $\boldsymbol{e}_{i}$ of the Eulerian strain-rate tensor. Large aspect ratio rods are more aligned with the eigenvectors $\boldsymbol{e}_{1}$ (0.4) and $\boldsymbol{e}_{2}$ (0.4) than with the direction of least strain $\boldsymbol{e}_{3}$ (0.2). These alignments are slightly different from the observation in HIT by Ni et al. (Reference Ni, Ouellette and Voth2014) where the alignments relative to $\boldsymbol{e}_{i}$ ( $i=1$ , 2, and 3) are 0.4, 0.44 and 0.16, respectively. The modest deviation can be due to the different turbulent flow configurations and Reynolds numbers. As the aspect ratio decreases, the shorter rods preferentially align with $\boldsymbol{e}_{2}$ both for $\unicode[STIX]{x1D706}=3$ and 1.5. In contrast, disks initially exhibit a preferred alignment with the direction of least shear $\boldsymbol{e}_{3}$ and this preference strengthens with increasing flatness of the disks (i.e. as $\unicode[STIX]{x1D706}$ decreases). However, much stronger alignments of the spheroids with the eigenvectors $\boldsymbol{e}_{Li}$ of left Cauchy–Green tensor can be observed when $\boldsymbol{e}_{Li}$ is computed after a time interval $t^{+}-t_{0}^{+}$ of the order 100, slightly dependent on the particle’s asphericity $\unicode[STIX]{x1D6EC}$ . The Kolmogorov time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ varies from the wall to the centre with a channel-average of approximately $8.15\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}^{2}$ at the present Reynolds number. The time interval required for the alignment to saturate therefore amounts to approximately 12 $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , which compares well with $t^{+}-t_{0}^{+}\approx 10\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ in HIT (Ni et al. Reference Ni, Ouellette and Voth2014).

At first sight (2.1) may suggest that fluid rotation and shear contribute equally to the particle rotation for highly aspherical spheroids ( $|\unicode[STIX]{x1D6EC}|\approx 1$ ). However, whereas $\unicode[STIX]{x1D74E}^{\prime }$ measures rotation in the particle frame, rotation of inertia-free spheroids can alternatively be expressed by means of the rate of change of the particle orientation unit vector $\boldsymbol{p}$ in the inertial frame:

see e.g. Leal & Hinch (Reference Leal and Hinch1972), Parsa et al. (Reference Parsa, Calzavarini, Toschi and Voth2012) and Ni et al. (Reference Ni, Kramel, Ouellette and Voth2015). The rotation rate tensor $\unicode[STIX]{x1D619}_{ij}$ is related to the fluid rotation vector $\unicode[STIX]{x1D6FA}_{i}$ as $\unicode[STIX]{x1D619}_{ij}=\unicode[STIX]{x1D716}_{ijk}\unicode[STIX]{x1D6FA}_{k}$ . Here, in contrast to (2.1), all quantities refer to the frame of reference in which the flow field is computed by means of DNS. Now, according to (3.1), the role played by the fluid rotation reduces if the spheroids align themselves along the vorticity vector $2\unicode[STIX]{x1D734}$ . This may explain the transition from vorticity-dominated particle rotation for near spheres ( $\unicode[STIX]{x1D6EC}\approx 0$ ) to shear-dominated particle rotation for highly aspherical particles ( $|\unicode[STIX]{x1D6EC}|\approx 1$ ). According to above equation, the rate of change of particle orientation responds immediately to the flow field and particle orientation. However, the particle orientation $p_{i}$ does not respond instantaneously to the local flow field but evolves gradually towards alignment with the preferred Lagrangian stretching direction, as seen in figure 3. The actual particle orientation is therefore influenced by the history of the fluid velocity gradients along the Lagrangian trajectory.

The present alignment statistics was gathered along arbitrarily chosen particle trajectories, along which a particle experiences an almost isotropic vorticity field in the channel centre (see Andersson et al. Reference Andersson, Zhao and Variano2015) but highly anisotropic turbulence near the channel walls. The preferential alignment results in figure 3 are nevertheless strikingly similar with the corresponding results in homogenous isotropic turbulence reported by Ni et al. (Reference Ni, Ouellette and Voth2014, Reference Ni, Kramel, Ouellette and Voth2015) and Hunt et al. (Reference Hunt, Tierney, Kramel and Voth2015). We are therefore inclined to infer that the observed tendency of rods to align with, and disks to orient normal to, the strongest Lagrangian stretching direction $\boldsymbol{e}_{L1}$ is robust and universal since the same trends are found in isotropic as well as in highly anisotropic turbulence. The time evolution of these alignment correlations indicates the dependence of the particles’ orientational behaviour on the history of experienced fluid stretching in a Lagrangian view. By revisiting (2.1), the tracer particle rotation in the particle frame only depends on instantaneous local fluid vorticity and strain rate. However, the particle orientation does not respond instantaneously to the local flow field but gradually evolves towards alignment with the preferred Lagrangian stretching direction. This process might cause nearby particles to orient and rotate differently (in the laboratory frame) in spite of almost identical local flow conditions.

After having discovered the major alteration in the alignment $\langle (\boldsymbol{e}_{L1}\boldsymbol{\cdot }\unicode[STIX]{x1D734})^{2}\rangle$ (figure 2 b) between the strongest Lagrangian stretching direction $\boldsymbol{e}_{L1}$ and the fluid rotation vector $\unicode[STIX]{x1D734}$ across the channel, we finally examined how the alignment $\langle (\boldsymbol{e}_{L1}\boldsymbol{\cdot }\boldsymbol{p})^{2}\rangle$ between $\boldsymbol{e}_{L1}$ and the particle orientation $\boldsymbol{p}$ varied with $z^{+}$ in figure 4. As one could anticipate from the data in figure 3, the major axis of a tracer spheroid with high asphericity $|\unicode[STIX]{x1D6EC}|$ is either aligned with $\boldsymbol{e}_{L1}$ (long rods with $\unicode[STIX]{x1D706}\gg 1$ in figure 4 a) or oriented normal to $\boldsymbol{e}_{L1}$ (thin disks with $\unicode[STIX]{x1D706}\ll 1$ in figure 4 f). These preferred particle orientations remain almost constant all over the entire channel and confirm the earlier results in figure 3. However, the results in figure 4(b–e) show that the actual particle shape parameterized by the aspect ratio $\unicode[STIX]{x1D706}$ also plays an important role. The $\unicode[STIX]{x1D706}=3$ spheroids, for instance, were mostly aligned with $\boldsymbol{e}_{L1}$ in the centre region, but oriented perpendicular to $\boldsymbol{e}_{L1}$ in the vicinity of the channel walls. This changeover of the preferred direction of the orientation vector $\boldsymbol{p}$ of the short rods in figure 4(b) corresponds qualitatively with the changeover of the orientation of the fluid rotation vector $\unicode[STIX]{x1D734}$ in figure 2(b). With reduced asphericity, as shown in figure 4(c,d), the alignment becomes almost independent of location.

Figure 4.

Variation of the alignment of the orientation vector $\boldsymbol{p}$ of spheroidal particles with aspect ratio $\unicode[STIX]{x1D706}$ relative to the three eigenvectors $\boldsymbol{e}_{Li}$ of the left Cauchy–Green strain tensor from the channel wall to the channel centre at $z^{+}=180$ sampled in the almost statistically steady state from $t^{+}=t_{0}^{+}+136$ to $t^{+}=t_{0}^{+}+144$ . The alignment $\langle (\boldsymbol{e}_{Li}\boldsymbol{\cdot }\boldsymbol{p})^{2}\rangle$ of two randomly oriented vectors is $1/3$ . (a)  $\unicode[STIX]{x1D706}=50$ , (b)  $\unicode[STIX]{x1D706}=3$ , (c)  $\unicode[STIX]{x1D706}=1.5$ , (d)  $\unicode[STIX]{x1D706}=0.8$ , (e)  $\unicode[STIX]{x1D706}=0.33$ and (f)  $\unicode[STIX]{x1D706}=0.01$ .

The apparent alignment between $\boldsymbol{p}$ and $\unicode[STIX]{x1D734}$ for short rod-like spheroids arose because both vectors independently tended to align with the strongest Lagrangian stretching direction $\boldsymbol{e}_{L1}$ in the channel centre. Long rods (flat disks) consistently aligned with (normal to) the strongest Lagrangian stretching direction all across the anisotropic channel flow, similarly as in homogeneous isotropic turbulence. It is noteworthy, however, that the long rods with $\unicode[STIX]{x1D706}\gg 1$ preferentially oriented along $\boldsymbol{e}_{L1}$ even though the fluid vorticity $2\unicode[STIX]{x1D734}$ aligned normal to $\boldsymbol{e}_{L1}$ . Challabotla et al. (Reference Challabotla, Zhao and Andersson2015b ) explored the preferential orientation of rod- and disk-like particles in near-wall turbulence. The reduction of particle rotation near the wall with increasing asphericity could be attributed to the strong preferential orientation of rods and disks, which makes the observed rotation in the viscous sublayer qualitatively consistent with spheroids in Jeffery orbits (Jeffery Reference Jeffery1922) in simple shear flow, in which rods (disks) rarely tumble (spin) when they align in streamwise (wall-normal) direction. The reasoning was incomplete, however, due to lack of understanding of why such preferential orientations in either streamwise or wall-normal directions exist. Now the strong alignment of spheroids relative to Lagrangian stretching or compression in the wall region with increasing departure from sphericity is evident in figure 4. This finding, together with previous observations, completes the understanding of orientational and rotational dynamics of tracer spheroids in near-wall turbulence. Furthermore, if triaxial ellipsoids are considered (e.g. Chevillard & Meneveau Reference Chevillard and Meneveau2013), it is likely that the strongest alignment of one of the particle’s principle axes with Lagrangian stretching or compression will determine the orientation and rotation of non-axisymmetric ellipsoids.

From the results in figure 2 we could only infer that the maximum stretching direction $\boldsymbol{e}_{L1}$ is in the plane of the mean shear, i.e. the ( $x,z$ )-plane. Now we have learned from figure 4(a) that the long rods are aligned with $\boldsymbol{e}_{L1}$ also in the near-wall region. Since several earlier studies, e.g. Mortensen et al. (Reference Mortensen, Andersson, Gillissen and Boersma2008a ,Reference Mortensen, Andersson, Gillissen and Boersma b ) and Lundell et al. (Reference Lundell, Söderberg and Alfredsson2011), have shown that long rods preferentially align themselves in the mean-flow direction, we can conclude that the strongest Lagrangian stretching direction is in the streamwise direction. The maximum (extensional) eigenvector $\boldsymbol{e}_{L1}$ of the left Cauchy–Green strain tensor is therefore perpendicular to the mean vorticity and parallel with the channel walls. Similarly, we can learn from disk alignments and conclude that the strongest Lagrangian compression is in the wall-normal direction.

The eigenvectors of the left Cauchy–Green tensor indicate the direction of stretching at time $t_{0}+T$ , i.e. after stretching (Parsa et al. Reference Parsa, Guasto, Kishore, Ouellette, Gollub and Voth2011). However, the tensor depends on the entire history of a particle’s motion. It should therefore be acknowledged that spheroids located in a particular bin ( $z$ -location) at $t^{+}-t_{0}^{+}=144$ have followed trajectories through other locations (different $z$ -values) during the time window $T^{+}=144$ . Moreover, different spheroids within the same bin might have followed rather different trajectories and thus have sampled different $z^{+}$ -regions. The spheroids studied herein are inertia free and translate passively along with the fluid. Fluid velocity fluctuations in the $z$ -direction are not very large and with zero mean. Some fluid particles are fluctuating towards the wall and others away from the wall. Fluid particles (and tracers) are thus primarily translating in the $x$ -direction, but with modest wall-normal excursions superimposed. Since the data in figures 2(b) and 4 represent averages over many particles, the inevitable $z$ -excursions will tend to smooth the results. The rather distinct variations of the alignments seen in figures 2(b) and 4 suggest that this smearing effect is modest, at least for the present non-inertial particles. This suggestion is further supported by the close correspondence of the alignments at the channel centre ( $z^{+}=180$ ) with the alignment reported in HIT by Ni et al. (Reference Ni, Ouellette and Voth2014). This is not surprising since the time window $T^{+}=144$ over which the integration of the Lagrangian trajectories is performed is long compared with the Kolmogorov time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ (representative for the small-scale turbulence) but shorter than the large-eddy turnover time $h/u_{\unicode[STIX]{x1D70F}}$ .