3.1 Momentum and TKE budget analysis

In particulate turbulent flows, the particle-induced stress also contributes to the mean momentum transfer in the wall-normal direction, in addition to the viscous and Reynolds shear stresses. Based on the formulation proposed by Zhang & Prosperetti (Reference Zhang and Prosperetti2010), Picano et al. (Reference Picano, Breugem and Brandt2015), we can write the mean momentum balance in the channel as:

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{f}u_{\unicode[STIX]{x1D70F}}^{2}\left(1-\frac{y}{h}\right)=\unicode[STIX]{x1D707}_{f}(1-\unicode[STIX]{x1D6F7})\frac{\text{d}U_{f}}{\text{d}y}-\unicode[STIX]{x1D70C}_{f}[\unicode[STIX]{x1D6F7}\langle u_{p}^{\prime }v_{p}^{\prime }\rangle +(1-\unicode[STIX]{x1D6F7})\langle u_{f}^{\prime }v_{f}^{\prime }\rangle ]+\unicode[STIX]{x1D6F7}\langle \unicode[STIX]{x1D70E}_{xy}^{p}\rangle ,\end{eqnarray}$$

where the first term on the right-hand side is the viscous shear stress, denoted  $\unicode[STIX]{x1D70F}_{V}$ , the second and the third terms are the turbulent Reynolds shear stress of the combined phase, $\unicode[STIX]{x1D70F}_{T}$ , the sum of the separate contributions from the two phases and the fourth term the particle-induced stress  $\unicode[STIX]{x1D70F}_{P}$ . $\unicode[STIX]{x1D70E}_{xy}^{p}$ in the equation above indicates the general stress in the particle phase, normal to the streamwise plane and pointing in the wall-normal direction, $\unicode[STIX]{x1D6F7}$ the mean local solid volume fraction as a function of wall-normal distance  $y$ , $\unicode[STIX]{x1D70C}_{f}$ the density of the fluid, $U_{f}$ the mean fluid velocity and $u^{\prime }$ and $v^{\prime }$ the velocity fluctuations in the streamwise and wall-normal directions with the subscripts $f$ and $p$ denoting fluid and particle phases. Note that in the absence of particles ( $\unicode[STIX]{x1D6F7}=0$ ), this equation reduces to the classic momentum balance for single-phase turbulent channel flow (Pope Reference Pope2000).

Integrating (3.1) from 0 to $h$ and substituting $\unicode[STIX]{x1D70C}_{f}u_{\unicode[STIX]{x1D70F}}^{2}$ with the drag at the wall, $\unicode[STIX]{x1D70F}_{w}$ , leads to the contribution of each stress to $\unicode[STIX]{x1D70F}_{w}$ , which can be written in discrete form as:

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{w}=\frac{2}{h}\left[\sum \unicode[STIX]{x1D70F}_{V}+\sum \unicode[STIX]{x1D70F}_{T}+\sum \unicode[STIX]{x1D70F}_{P}\right].\end{eqnarray}$$

Each contribution times $2/h$ , normalized by the drag of the single-phase flow, $\unicode[STIX]{x1D70F}_{w_{\unicode[STIX]{x1D719}=0\,\%}}$ , is given in figure 1(a) for the different particle aspect ratios and the single-phase flow (denoted S-P) for a better comparison. An overall drag reduction is observed for the cases with ${\mathcal{A}}{\mathcal{R}}\leqslant 1/3$ , increasing up to 10 % for the smallest ${\mathcal{A}}{\mathcal{R}}=1/5$ . The largest increase with respect to the unladen case is obtained for spheres, whereas a small drag increase is still observed for ${\mathcal{A}}{\mathcal{R}}=1/2$ . Finally, the drag for the case with prolate particles is almost the same as for the single-phase flow. Considering the different contributions, we note the non-monotonic behaviour of the particle-induced stress with decreasing ${\mathcal{A}}{\mathcal{R}}$ as it reaches a minimum for ${\mathcal{A}}{\mathcal{R}}=1/3$ before slightly increasing on further decreasing the aspect ratio. The contribution of the viscous shear stress to the total drag is slightly varying among the different aspect ratio cases, while being smaller than the single-phase flow due to the presence of $1-\unicode[STIX]{x1D6F7}$ in $\unicode[STIX]{x1D70F}_{V}$ of the particulate cases. In addition, we see turbulence attenuation for all non-spherical cases, an effect increasing as the aspect ratio departs from sphericity. The level of turbulence activity is quantified by the turbulence Reynolds number $Re_{T}\equiv u_{T}h/\unicode[STIX]{x1D708}$ , with $u_{T}$ the turbulent friction velocity, calculated as the square root of the wall-normal derivative of the Reynolds stress profile for the combined phase, $\langle u_{c}^{\prime }v_{c}^{\prime }\rangle$ , at the centreline of the channel:

(3.3) $$\begin{eqnarray}u_{T}=\sqrt{\text{d}\langle u_{c}^{\prime }v_{c}^{\prime }\rangle /\text{d}(y/h)}|_{y/h=1}.\end{eqnarray}$$

Figure 1.

(a) The contribution of the viscous, $\unicode[STIX]{x1D70F}_{V}$ , Reynolds, $\unicode[STIX]{x1D70F}_{T}$ , and particle-induced stress, $\unicode[STIX]{x1D70F}_{P}$ , for different aspect ratios, normalized by the drag of the single-phase flow. Each contribution is depicted separately, versus ${\mathcal{A}}{\mathcal{R}}$ in the insets of panel (a) with S-P referring to the single-phase flow. (b) Turbulence Reynolds number $Re_{T}$ and friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ , normalized by the values for the single-phase flow case. (c) Production, dissipation and interphase injection of TKE, normalized with $(u_{\unicode[STIX]{x1D70F}}^{0})^{4}/\unicode[STIX]{x1D708}$ of the single-phase flow. The production is not depicted in the vicinity of the wall for clarity.

$Re_{T}$ , normalized by that of the single-phase flow, $Re_{T}^{0}\approx 174$ , is depicted in figure 1(b) for different ${\mathcal{A}}{\mathcal{R}}$ , together with the ratio $Re_{\unicode[STIX]{x1D70F}}/Re_{\unicode[STIX]{x1D70F}}^{0}=Re_{\unicode[STIX]{x1D70F}}/180$ . The results indicate that the turbulence activity decreases faster than the friction Reynolds number when decreasing the aspect ratio of oblate particles. This is due to the almost unchanged contribution of viscous shear stress ( $\sum \unicode[STIX]{x1D70F}_{V}$ ) and slight increase of particle-induced stress when decreasing the aspect ratio below ${\mathcal{A}}{\mathcal{R}}=1/3$ , meaning that a stronger turbulence attenuation is required in order to further reduce the drag.

We next consider the turbulent kinetic energy (TKE) budget to quantify the solid-phase contribution to production and dissipation of TKE of the fluid phase. Considering the Navier–Stokes (NS) equations with the IBM force $(\,\boldsymbol{f})$ , employed to impose no-slip at the particle surface, the TKE budget can be written for a fully developed turbulent channel flow as:

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle -\frac{\unicode[STIX]{x2202}{\mathcal{T}}_{j}}{\unicode[STIX]{x2202}x_{j}}-\unicode[STIX]{x1D716}+{\mathcal{P}}+{\mathcal{I}}=0, & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{P}}=-\langle u_{i}^{\prime }u_{j}^{\prime }\rangle \frac{\unicode[STIX]{x2202}U_{i}}{\unicode[STIX]{x2202}x_{j}}, & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D716}=\unicode[STIX]{x1D708}\left\langle \frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{j}}\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{j}}\right\rangle , & \displaystyle\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{T}}_{j}=\frac{1}{2}\langle u_{i}^{\prime }u_{i}^{\prime }u_{j}^{\prime }\rangle +\frac{1}{\unicode[STIX]{x1D70C}}\langle u_{j}^{\prime }p^{\prime }\rangle -\frac{1}{2}\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}\langle u_{i}^{\prime }u_{i}^{\prime }\rangle }{\unicode[STIX]{x2202}x_{j}}, & \displaystyle\end{eqnarray}$$

where the first three terms in (3.4) are respectively responsible for the spatial redistribution, viscous dissipation and production of TKE ( $0.5\langle u_{i}^{\prime }u_{i}^{\prime }\rangle$ ) (Mansour, Kim & Moin Reference Mansour, Kim and Moin1988), while the fourth term is the interphase energy injection, ${\mathcal{I}}=\langle u_{j}^{\prime }f_{j}\rangle$ (Tanaka & Teramoto Reference Tanaka and Teramoto2015).

The last three terms in (3.4) are calculated for the fluid phase, considering only the points located outside of the volume occupied by the particles (phase-ensemble average), and are depicted for different aspect ratios in figure 1(c). The data indicate a strong reduction of ${\mathcal{P}}$ in the near-wall region with decreasing ${\mathcal{A}}{\mathcal{R}}$ for oblate particles. A significant reduction is also observed for prolate particles, while a slight increase with respect to the single-phase flow is seen for spheres. The dissipation term $\unicode[STIX]{x1D716}$ is almost the same as in the single-phase flow for non-spherical particles, except at the wall, where its value reduces with ${\mathcal{A}}{\mathcal{R}}$ . For spherical particles, however, the dissipation is considerably larger throughout the channel. The energy injected by the fluid–solid interactions, i.e. the interphase energy injection at the particle surface, interestingly follows the same trend as the contribution of particle-induced stresses in the momentum balance ( $\unicode[STIX]{x1D70F}_{P}$ in figure 1 a): it reaches a minimum at ${\mathcal{A}}{\mathcal{R}}=1/3$ , before increasing again for lower aspect ratios. The energy injected by the particles first decreases when decreasing ${\mathcal{A}}{\mathcal{R}}$ , owing to the reduced rotation rate discussed below; however, it increases again for the smallest aspect ratio considered ( ${\mathcal{A}}{\mathcal{R}}=1/5$ ) as the particle surface area increases. This can be explained by the fact that a large value of $f$ indicates a force that acts on the fluid and also on the surface of the particle, meaning that in the absence of particle rigidity, a large deformation of the particle surface would have been expected. Indeed the particle rigidity opposes this deformation and instead increases the particle-induced stress.

3.2 Turbulence modulation

Fluid velocity fluctuations of the fluid phase are depicted in figure 2(a) for the different particle aspect ratios under investigations. The velocity fluctuations are reduced compared to the single-phase flow for non-spherical particles in the cross-flow directions. The effect is observed to increase as the aspect ratio of oblate particles decreases. Spherical particles, on the other hand, increase the cross-flow velocity fluctuations. The strongest reduction of $u^{\prime }$ is observed for prolates, followed by spheres, despite the slight increase in the core region. Interestingly, $u^{\prime }$ is observed to increase with reducing ${\mathcal{A}}{\mathcal{R}}$ in the oblate cases. The same behaviour is observed in the literature for other drag-reducing flows, e.g. polymeric flows, when reduced spanwise and wall-normal velocity fluctuations are observed (Dubief et al. Reference Dubief, White, Terrapon, Shaqfeh, Moin and Lele2004). Figure 2(b) shows the Reynolds shear stress of the fluid phase for all studied cases. A significant turbulence attenuation is observed for non-spherical particles, as opposed to the turbulence enhancement for spherical particles. The Reynolds shear stress $-u_{f}^{\prime }\,v_{f}^{\prime }$ decreases monotonically with decreasing ${\mathcal{A}}{\mathcal{R}}$ in the oblate cases. Interestingly, the $-u_{f}^{\prime }\,v_{f}^{\prime }$ pertaining to prolates with ${\mathcal{A}}{\mathcal{R}}=3$ approximately matches the case of oblates with ${\mathcal{A}}{\mathcal{R}}=1/3$ .

Figure 2.

(a) Fluid velocity fluctuations $u^{\prime }$ , $v^{\prime }$ and $w^{\prime }$ in the streamwise, ——, wall-normal, – – –, and spanwise, $\cdots \cdots$ , directions and (b) Reynolds shear stress, scaled in outer units for different aspect ratios. The velocity fluctuations for the cases with ${\mathcal{A}}{\mathcal{R}}=1/2$ and $1/4$ are not shown in (a) to improve clarity.

Figure 3.

(a) Reynolds shear stress, conditionally averaged outside of the particles, for each quadrant of $u_{f}^{\prime }\,v_{f}^{\prime }$ plane for different aspect ratios. (b,c)  $Q_{2}$ and $Q_{4}$ events, conditionally averaged over the fluid inside the spheroidal shells, 5 % larger than the particles. The dashed lines in (b,c) indicate the average over the shells, while the overall averages inside the fluid phase are indicated with the solid lines.

The Reynolds shear stresses are then conditionally averaged over each quadrant of the $u^{\prime }-v^{\prime }$ plane, with $Q_{1}$ to $Q_{4}$ referring to the four quadrants. $Q_{2}$ (ejections: $u_{f}^{\prime }<0$ , $v_{f}^{\prime }>0$ ) and $Q_{4}$ (sweeps: $u_{f}^{\prime }>0$ , $v_{f}^{\prime }<0$ ) events result in turbulence production, while $Q_{1}$ and $Q_{3}$ are responsible for damping. The results of this analysis are depicted in figure 3(a), where it is observed that the presence of oblate particles reduces the value of $u^{\prime }v^{\prime }$ over all four quadrants, with a far more pronounced dampening of the ejection events. Spherical particles, on the other hand, increase the strength of sweeps close to the wall and reduce the contribution of $Q_{3}$ events, without altering the others noticeably. Prolate particles exhibit an effect similar to oblates, but with lower dampening of the ejections. Similar analysis is performed, using only the fluid inside the spheroidal shells, 5 % larger than the particles. $Q_{2}$ and $Q_{4}$ events, conditionally averaged over these shells, are depicted in figures 3(b) and 3(c) (dashed lines), respectively and compared against the obtained results in figure 3(a). Figure 3(b) shows that the ejection events are strongly attenuated in the vicinity of the particles compared to the single-phase flow and also compared to the results in figure 3(a). The attenuation is observed to be stronger for non-spherical particles. Interestingly, the results in figure 3(c) indicate a region near the wall where the sweeps are strong close to the particles before decreasing below the results of figure 3(a) in the middle of the channel. The peak is stronger for spherical and prolate particles and it decreases with the aspect ratio of oblates. It should be noted that similar results are obtained by considering smaller spheroidal shells, 2.5 % larger than the particles.

Figure 4.

Contours of two-point spatial correlation of the streamwise velocity fluctuations as a function of the spanwise spacing for different wall-normal distances: (a) single-phase flow, (b)  ${\mathcal{A}}{\mathcal{R}}=1$ , (c)  ${\mathcal{A}}{\mathcal{R}}=1/5$ and (d)  ${\mathcal{A}}{\mathcal{R}}=3$ . The inner length scale $\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$ , used here, is calculated from the single-phase flow case to allow for a direct comparison. The solid and dashed lines correspond to positive and negative values, ranging from 0.8 to $-0.2$ with a step of 0.2.

To gain a better understanding of the turbulence structures in the wall region, the two-point spatial correlation of the streamwise velocity fluctuations as a function of the spanwise spacing is computed for different wall-normal distances:

(3.8) $$\begin{eqnarray}R_{uu}(y,\unicode[STIX]{x0394}z)=\frac{\langle u^{\prime }(x,y,z,t)u^{\prime }(x,y,z+\unicode[STIX]{x0394}z,t)\rangle }{{u^{\prime }}^{2}}.\end{eqnarray}$$

Contours of the one-dimensional autocorrelation of the streamwise velocity fluctuations as a function of the spanwise spacing are given in figure 4 for different aspect ratios. It is well known that the autocorrelations of the streamwise velocity along the spanwise direction in a single-phase turbulent flow (figure 4 a), indicate a negative local minimum value in the near-wall region around $\unicode[STIX]{x0394}z^{+}\approx 60$ (Pope Reference Pope2000). This value indicates half of the spacing between the near-wall streamwise low-speed streaks, characteristic of wall-bounded turbulence. Results in figure 4 indicate a clear increase in the spacing of these structures for the oblate particles with ${\mathcal{A}}{\mathcal{R}}=1/5$ (figure 4 c), a characteristic observed in drag-reducing flows (Dubief et al. Reference Dubief, White, Terrapon, Shaqfeh, Moin and Lele2004); however the low-speed streaks are observed to become less correlated for the flow with prolate particles (figure 4 d) and to a lesser extent with spherical ones (figure 4 b).

3.3 Particle dynamics

Preferential sampling, particle rotation and orientation, especially close to the wall, can lead to a strong modulation of the turbulence. Here we discuss the link between the solid-phase averaged data and the drag modifications and turbulence activity reported in the previous sections.

Figure 5.

Profiles of solid-phase averaged data versus $y/h$ ; (a) mean local volume fraction  $\unicode[STIX]{x1D6F7}$ ; (b) cosine of the mean particle inclination angle  $\unicode[STIX]{x1D703}$ , measured with respect to the wall; (c) magnitude of angular velocity vector $\unicode[STIX]{x1D734}_{p}$ ; (d) particle velocity fluctuations $\boldsymbol{u}_{p}^{\prime }$ ; (e) average of spanwise angular velocity and (f) wall-normal fluctuations of angular velocity. Angular and translational velocities are normalized with $U_{b}/h$ and $U_{b}$ respectively.

The mean local volume fraction, $\unicode[STIX]{x1D6F7}$ , is depicted versus the wall-normal distance $y/h$ in figure 5(a). Interestingly, a peak close to the wall is only observed for the spherical particles. This peak corresponds to a layer of particles that form due to the wall–particle interactions (Picano et al. Reference Picano, Breugem and Brandt2015) and is partially responsible for the increase of drag in this case (Costa et al. Reference Costa, Picano, Brandt and Breugem2016). Non-spherical particles are observed to avoid the wall; the local volume fraction profiles for disc-like particles seem to converge for ${\mathcal{A}}{\mathcal{R}}\leqslant 1/3$ . To better understand this, we report in figure 5(b) the average particle orientation throughout the channel. The orientation is measured as the cosine of the inclination angle  $\unicode[STIX]{x1D703}$ , which is the smallest angle between the spheroid axis of symmetry and the normal to the wall. The data clearly show the strong tendency for oblate and prolate particles to have their largest semi-axis parallel to the wall. The alignment of prolate particles with the wall in its vicinity is also observed in the studies of Eshghinejadfard et al. (Reference Eshghinejadfard, Hosseini and Thévenin2017), Wang (Reference Wang2017), similar to rigid fibres in the work of Do-Quang et al. (Reference Do-Quang, Amberg, Brethouwer and Johansson2014). An arbitrary symmetry axis is chosen for spheres to demonstrate the spheres’ random orientations. The magnitude of the angular velocity vector  $\unicode[STIX]{x1D734}_{p}$ , normalized by $U_{b}/h$ , is depicted for the different aspect ratios in figure 5(c): the angular velocities drastically decrease for oblate particles. Prolate particles, similarly to spheres, display a non-negligible angular velocity; this large value is mainly due to the wall-normal component of the angular velocity ${\unicode[STIX]{x1D6FA}^{\prime }}_{p}^{y}$ , depicted in figure 5(f) for different aspect ratios. Prolate particles are observed to have the largest wall-normal angular velocity in the vicinity of the wall, while this value is significantly smaller for oblate particles and decreasing with their aspect ratio. Spherical particles also experience noticeably smaller wall-normal angular velocity close to the wall, however they have the largest in the middle of the channel. Conversely, in the case of spherical particles, the major contribution to the angular velocity is due to the mean spanwise component $\unicode[STIX]{x1D6FA}_{p}^{z}$ , as a direct consequence of the mean shear. The mean spanwise angular velocities for different aspect ratios are given in figure 5(e). Larger angular velocities result in a larger particle-induced stress as the rotation of particles larger than the smallest scale of the flow opposes stretch and deformation. To better distinguish the effect of mean flow and fluctuations on the dynamics of particles, we have simulated the behaviour of single particles kept at a fixed distance from the wall ( $y/h=0.1$ ) in the laminar flow defined by the turbulent profile. These additional runs (not shown in the figure) show that a sphere tends to a constant angular velocity in the spanwise direction, an oblate particle with ${\mathcal{A}}{\mathcal{R}}=1/2$ exhibits a periodic spanwise angular velocity, with the majority of time spent parallel to the wall and, more interestingly, the motion of particles with ${\mathcal{A}}{\mathcal{R}}=1/3$ , $1/4$ , $1/5$ and 3 converges to zero angular velocity with the broad side approximately parallel to the wall (slightly tilted). The tendency of non-spherical particles to remain parallel to the wall is therefore a mean flow effect and quenches effectively the velocity fluctuations, resulting in turbulence attenuation. This mechanism is more effective when the particles are larger than the turbulence structures. The role of the particle size can be deduced from the results in figure 5: local volume fraction, particle rotation and orientation attain similar values for oblates with ${\mathcal{A}}{\mathcal{R}}\leqslant 1/3$ , however the turbulence attenuation increases for broader oblates. This can also be concluded from figure 5(d), where the magnitude of the particle velocity fluctuations, normalized by $U_{b}$ is shown to decrease with ${\mathcal{A}}{\mathcal{R}}$ . The local velocity of the solid phase are computed, taking into account the rigid body motion of the particles, $\boldsymbol{u}_{p}+\unicode[STIX]{x1D734}_{p}\times (\boldsymbol{x}-\boldsymbol{x}_{c})$ , with $\boldsymbol{x}$ and $\boldsymbol{x}_{c}$ denoting an arbitrary point inside the particle and the particle centre respectively.

Figure 6.

(a) Probability density function (PDF) of streamwise fluid velocity fluctuation, surrounding particles close to the wall ( $0.1<y/h<0.2$ ) and (b) the difference between the particle and the fluid mean velocity, normalized by the bulk velocity  $U_{b}$ .

Finally, we examine the probability density function (PDF) of the streamwise fluid velocity fluctuations in the vicinity of the particles and in a region close to the wall ( $0.1<y/h<0.2$ ), see figure 6(a). The fluid velocity fluctuations relative to each particle are the average on the spheroidal shell 5 % larger than the particle itself. The data indicate a clear tendency for particles to sample high-speed regions close to the wall. This observed preferential sampling is weakest for spheres, followed by prolates and more and more evident when decreasing the aspect ratio of oblate particles. Figure 6(b) shows the difference in the mean particle $U_{p}$ and fluid $U_{f}$ velocity: the difference in the region $y/h<0.2$ is consistent with the PDF reported in figure 6(a). It should be noted that the larger mean velocity of the particles, compared to the fluid phase, does not necessarily indicate a large slip velocity as particles are shown to have a tendency to sample the high-speed regions close to the wall. The larger mean velocity of the particles close to the wall is consistent with the results of Picano et al. (Reference Picano, Breugem and Brandt2015) for spherical particles and Do-Quang et al. (Reference Do-Quang, Amberg, Brethouwer and Johansson2014), Ardekani et al. (Reference Ardekani, Costa, Breugem, Picano and Brandt2017), Eshghinejadfard et al. (Reference Eshghinejadfard, Hosseini and Thévenin2017) for non-spherical ones. Wang, Abbas & Climent (Reference Wang, Abbas and Climent2018) recently reported an accumulation of spherical particles in the low-speed near-wall streaks of channel flow. This can be due to the smaller particles in viscous units ( $D^{+}\approx 10$ ), considered in their study, compared to the more inertial ones here ( $D^{+}=20$ ).