3.1. Drag modification

Firstly, we study the effects of changing the Reynolds number on the drag of the system. For the global analysis of the drag modification by bubbles or particles, the friction coefficient, $c_{f,\varPhi }$, which evaluates the skin frictional drag of the flow system, has been extensively used in various flow geometries (Sanders et al. Reference Sanders, Winkel, Dowling, Perlin and Ceccio2006; Verschoof et al. Reference Verschoof, Van Der Veen, Sun and Lohse2016; Olivucci, Wise & Ricco Reference Olivucci, Wise and Ricco2021). In the TC system the friction coefficient is defined as (Lathrop, Fineberg & Swinney Reference Lathrop, Fineberg and Swinney1992; van Gils et al. Reference van Gils, Huisman, Bruggert, Sun and Lohse2011)

(3.1)\begin{equation} c_{f,\varPhi}= \frac{(1-\eta)^2}{{\rm \pi} \eta^2}\frac{G}{Re^2}. \end{equation}

Figure 2 shows the normalized friction coefficients of the system, $c_{f,\varPhi }/c_{f,\varPhi =0}$, where $c_{f,\varPhi =0}$ is the friction coefficient of the single-phase case. It is found that the drag modification increases with increasing $\varPhi$ for a given $Re$. In the investigated ranges, the maximum drag enhancement is found to be around $20\,\%$, which is observed in the spherical case at $Re=6.5\times 10^3$ and $\varPhi =10\,\%$ (figure 2b). Additionally, $c_{f,\varPhi }/c_{f,\varPhi =0}$ decreases as $Re$ increases, which has also been found and is referred to as the shear-thinning effect in the suspension of deformable particles (Adams, Frith & Stokes Reference Adams, Frith and Stokes2004) and the emulsions (Rosti & Takagi Reference Rosti and Takagi2021; Yi, Toschi & Sun Reference Yi, Toschi and Sun2021). Note that, the overall drag of the system ($G$ or $\tau$, which are not shown here) increases with increasing $Re$ and is consistent with the literature (Stickel & Powell Reference Stickel and Powell2005; Fall et al. Reference Fall, Lemaitre, Bertrand, Bonn and Ovarlez2010; Picano et al. Reference Picano, Breugem, Mitra and Brandt2013).

Figure 2. The normalized friction coefficient of the TC system, $c_{f,\varPhi }/c_{f,\varPhi =0}$, for (a) $\lambda =1/3$ (oblate), (b) $\lambda =1$ (spherical) and (c) $\lambda =3$ (prolate), where $c_{f,\varPhi =0}$ is the friction coefficient for the single-phase case. The accuracy of the experiments is indicated by the error bars, which are typically less than 1 %.

The modification of the friction coefficient is related to the changes of the wall stress, which can be decomposed into three parts when no external force and torque is applied on particles, i.e.

(3.2)\begin{equation} \tau_{w,\varPhi} = \tau_{v}+\tau_{T}+ \tau_{p}= \tau_{v}+\tau_{T_f}+\tau_{T_p} +\tau_{p}, \end{equation}

where $\tau _{v}$ the viscous stress of the fluid phase, $\tau _{p}$ the particle-induced stress and $\tau _{T}= \tau _{T_f}+\tau _{T_p}$ the turbulent Reynolds stress of the combined phase, $\tau _{T_f}$ and $\tau _{T_p}$ the turbulent Reynolds stress of fluid and particulate phase, respectively. Following the studies in Zhang & Prosperetti (Reference Zhang and Prosperetti2010) and Picano et al. (Reference Picano, Breugem and Brandt2015), the explicit expression of each term above in a Couette flow (Batchelor Reference Batchelor1970; Wang, Abbas & Climent Reference Wang, Abbas and Climent2017a) can be written as

(3.3)\begin{gather} \tau_{v} = \mu (1-\varPhi) \frac{\mathrm{d}U_f}{\mathrm{d}r}, \end{gather}

(3.4)\begin{gather}\tau_{T_f} ={-}\rho(1-\varPhi) \langle u^{\prime}_fv^{\prime}_f \rangle, \end{gather}

(3.5)\begin{gather}\tau_{T_p} ={-}\rho \varPhi \langle u^{\prime}_pv^{\prime}_p \rangle, \end{gather}

(3.6)\begin{gather}\tau_{p} = \varPhi \langle \sigma^p_{xy} \rangle, \end{gather}

where $\mu$ is the dynamic viscosity of the fluid, $U_f$ is the mean fluid velocity in the azimuthal direction, $u^{\prime }$ and $v^{\prime }$ are the velocity fluctuations in the azimuthal and radial directions with the subscripts $f$ and $p$ denoting the fluid and particulate phase, respectively, $\sigma ^p_{xy}$ is the general stress in the particulate phase, normal to the cylindrical surface and pointing in the radial direction. Here we note that the contribution to fluid viscous stress due to velocity in the axial direction is neglected since it is a minor effect.

At a fixed $Re$, the wall stress of the single-phase case, $\tau _{w,\varPhi =0} = \mu ({\mathrm {d}U}/{\mathrm {d}r})|_{r=r_i}$, is a constant, and the particle-induced stress ($\tau _{p}$) increases with increasing $\varPhi$. In addition, it has been reported that in channel flow (Picano et al. Reference Picano, Breugem and Brandt2015), for a fixed $Re$ in the current low particle volume fraction ranges, the viscous stress ($\tau _{v}$) weakly depends on $\varPhi$, and the turbulent Reynolds stress of the combined phase ($\tau _{T}$) increases with increasing $\varPhi$. Though there are differences between the channel flow and the TC flow, the result in the channel flow is instructive for the understanding of the current study. Considering the dependence above of the stress on $\varPhi$ at a fixed $Re$, the normalized friction coefficient increases with increasing $\varPhi$, which is consistent with the trends in figure 2. On the other hand, for a fixed low $\varPhi$ (as in this work), the turbulent Reynolds stress of the fluid phase in single-phase flow ($\tau _{T_f,\varPhi =0}$) increases greatly as $Re$ increases, which makes the contribution of particulate phase ($\tau _{T_p}$ and $\tau _{p}$) relatively insignificant and, therefore, results in the decreasing trends of normalized friction coefficients with increasing $Re$. Another feature of the normalized friction coefficients in figure 2 is that, as $Re$ increases, the differences between various $\varPhi$ decrease. Specifically, when $Re$ is small, the contributions of the particulate phase to the wall stress are non-negligible, hence, the normalized wall stress (friction coefficients) highly depends on $\varPhi$ through $\tau _{T_p}$ and $\tau _{p}$. However, when $Re$ is high enough and $\varPhi$ is low (as in this work), the turbulent Reynolds stress due to the fluid phase in the particle-laden flow and single-phase flow ($\tau _{T_f}$ and $\tau _{T_f,\varPhi =0}$) plays the dominant role while $\tau _{T_p}$ and $\tau _{p}$ become minor; therefore, the contributions from particles become less and less important as $Re$ increases. Hence, the effect of changing $\varPhi$ on the normalized friction coefficients becomes smaller at higher $Re$ in the current parameter regimes.

On the other side, taking the particle-laden flow as a continuous effective fluid, the interaction between the particles and the resulting particle distributions are related to the rheology of a particle-laden flow, which can be quantified by the effective viscosity, $\nu _{eff}$. In the Stokes regime ($Re\ll 1$) the rheology of dense granular suspensions has been extensively investigated, and the dependence of the effective viscosity on the particle volume fraction has been discussed (Guazzelli & Pouliquen Reference Guazzelli and Pouliquen2018). In the current semi-dilute regime ($\varPhi \leq 10\,\%$) the normalized effective viscosity of the granular flow in the Stokes regime, $\nu ^S_{eff}$, can be approximated using the Krieger & Dougherty (K-D) formula (Krieger & Dougherty Reference Krieger and Dougherty1959),

(3.7)\begin{equation} \frac{\nu^S_{eff}}{{\nu_f}} =\left(1-\frac{\varPhi}{\varPhi_m}\right)^{-[\eta]\varPhi_m}, \end{equation}

where $\nu _f$ is the viscosity of the fluid, $\varPhi _m$ is the maximum packing particle fraction ($\varPhi _m = 0.65$ is used here as done in Stickel & Powell Reference Stickel and Powell2005) and $[\eta ]={\frac 52}$ for rigid spheres. In this work, $\nu _{eff}$ was calculated using the same method as in previous works (Bakhuis et al. Reference Bakhuis, Ezeta, Bullee, Marin, Lohse, Sun and Huisman2021; Yi et al. Reference Yi, Toschi and Sun2021), and the details can be found in Appendix B. As shown in figure 3, we compare the experimental results with the K-D formula. Though the model captures the increasing trend of the effective viscosity, the $Re$-dependence of the effective viscosity is missing, indicating that the relation in the Stokes regime is no longer valid in the current situation. The disparity found between the experimental data and the model can be understood since (3.7) was obtained based on the assumptions that the flow is inertialess and the particles distribute uniformly in the Stokes regime. However, for the current turbulent flow ($Re\sim 10^4$) and the suspended finite-size particles, the particle inertia is non-negligible and can be measured by the Stokes number, which is far beyond unity as shown in the later section. Moreover, as has been reported in previous numerical studies (Picano et al. Reference Picano, Breugem and Brandt2015; Ardekani & Brandt Reference Ardekani and Brandt2019), the particles show collective effects, which are also found in our experiments and will be discussed later. Note that when $Re=6.5\times 10^3$, the flow could be in the classical regime with laminar boundary layers and a turbulent bulk with Taylor vortex (Grossmann et al. Reference Grossmann, Lohse and Sun2016), which may account for the tendency for the effective viscosity in this Re case to increase faster than the model and other experimental data sets. Using the effective viscosity, the shear-thinning effect of the particle-laden flow is also examined, which can be well described by the Herschel–Bulkley model (Herschel & Bulkley Reference Herschel and Bulkley1926), the details of which can be found in Appendix B.

Figure 3. The non-dimensional effective viscosity, $\nu _{eff}/\nu _f$, as a function of the particle volume fraction, $\varPhi$, for (a) $\lambda =1/3$ (oblate), (b) $\lambda =1$ (spherical) and (c) $\lambda =3$ (prolate), where $\nu _f$ is the kinematic viscosity of the solution.

3.2. Effects of the particle aspect ratio

Next, we investigate the effects of changing the particle aspect ratio on the drag of the system. Particles with three kinds of aspect ratio are used: $\lambda =1/3$ (oblate), $\lambda =1$ (spherical) and $\lambda =3$ (prolate). The results are already given in figures 2 and 3. It is found that the particles could increase the drag of the system regardless of their aspect ratio, which is attributed mostly to the hydrodynamics interactions and rarely to the frictional contact between the particles in the current low $\varPhi$ range (Guazzelli & Pouliquen Reference Guazzelli and Pouliquen2018). The shear-thinning effect and the increasing trend of the drag modification with increasing $\varPhi$ are found to be ubiquitous in all cases.

It is remarkable, however, that particles with different aspect ratios could affect the drag of the system to different degrees, even for the same $Re$ or $\varPhi$. As shown in figure 2, for a given $\varPhi$, the $c_{f,\varPhi }/c_{f,\varPhi =0}$ of the oblate case decrease much more rapidly than that of the spherical and prolate cases, indicating that the suspended oblate particles result in a more pronounced shear-thinning effect, which can also be verified by the effective viscosity (see Appendix B). Moreover, for a given $Re$ and $\varPhi$, the suspensions of the spherical, prolate and oblate particles result in the maximum, moderate and minimal drag, respectively. In the investigated ranges, the largest discrepancy of $c_{f,\varPhi }/c_{f,\varPhi =0}$ between suspensions of particles with different aspect ratios is $4\,\%$, which occurs at $\varPhi =10\,\%$ and $Re=1.3\times 10^4$ in the spherical and oblate cases. Given the impressive accuracy of the drag measurements (less than 1 %) and the relatively low $\varPhi$, this disparity in the drag modification is quite noticeable. In addition, the difference of $c_{f,\varPhi }/c_{f,\varPhi =0}$ increases with increasing $\varPhi$, which is due to the increasing importance of particle-induced stress at higher $\varPhi$ (Ardekani & Brandt Reference Ardekani and Brandt2019). Therefore, it is reasonable to speculate that, as $\varPhi$ further increases, the differences of the drag modification between different aspect ratios would become larger. Note that this result is different from the particle-laden channel flow in the previous study (Ardekani & Brandt Reference Ardekani and Brandt2019), where they found that only the spherical particles increase the drag and the other two types of particles reduce the drag.

The dependence of the drag on the particle aspect ratio provides an experimental clue to the understanding of the mechanisms of bubbly drag reduction. Flow with dispersed bubbles can, under certain conditions, result in significant drag reduction (van den Berg et al. Reference van den Berg, Luther, Lathrop and Lohse2005, Reference van den Berg, van Gils, Lathrop and Lohse2007; Ezeta et al. Reference Ezeta, Bakhuis, Huisman, Sun and Lohse2019). However, bubbles are deformable, making it impossible to fix the bubble shape and, therefore, to isolate the effects of the bubble shape. van Gils et al. (Reference van Gils, Guzman, Sun and Lohse2013) and Verschoof et al. (Reference Verschoof, Van Der Veen, Sun and Lohse2016) have reported that the bubble deformation is crucial for achieving significant drag reduction, but the contributions of (i) the deformation process and (ii) the ultimate bubble shape after the deformation remain unknown. Obviously, the rigid particle suspensions are significantly different from bubbly flow, including the slip conditions at the surface, the resistance of the dispersed phase to straining motion, the polydispersity, the presence/absence of contaminants (surfactants), etc, and, therefore, a one-to-one comparison between these two types of flow is also unrealistic. However, the one that should be emphasized is that, as done in this work, the shape effect of rigid spheroidal finite-size particles on the overall drag provides a relatively effortless implement to study the shape effects of bubbles during the deformation process. Notwithstanding how simplistic the rigid particles are compared with bubbles, the kinematic motion of these particles (rigid particles and bubbles) is somehow similar, making the study of the rigid particles a reasonable choice to provide a reference case for bubbly flow studies. In this work the rigid spheroidal particles are used to study their shape effect on drag. Due to the frictional contact between solid surfaces, rigid particles dispersed in the flows result in the drag enhancement (Guazzelli & Pouliquen Reference Guazzelli and Pouliquen2018), which is contrary to the drag reduction in the bubbly flow. However, the fact that the particle aspect ratio affects the drag modification in the current particle-laden flow, also hints that the bubble shape could affect the drag of the bubbly flow. Given that the shape effect on the overall drag is relatively small (compared with the drag modulation amplitude in bubbly flow, which is around 40 % at $\varPhi = 4\,\%$ Verschoof et al. Reference Verschoof, Van Der Veen, Sun and Lohse2016), one would expect that the bubbles can also modulate the turbulence through other mechanisms next to the change of their shapes. Therefore, the unique features of bubbly flow relative to rigid particle-laden flow are worthy of attention for future studies.

3.3. The collective effects of particles

To reveal the physical mechanism of the drag modification, we look into the particle distribution by performing optical measurements. It should be noted that, even at the minimal particle volume fraction ($\varPhi =2\,\%$), the particles cause violent light scattering and cannot be accurately detected by algorithms. Therefore, we start with a smaller volume fraction ($\varPhi =0.5\,\%$, the number of particles ${\simeq }1700$) to reduce the intensity of the light scattering, so that the particles in the images can be detected using the ellipse detection algorithms (Lu et al. Reference Lu, Xia, Shao and Fu2020). One may question as to whether the particle distribution obtained at this low volume fraction could qualitatively represent that at higher volume fractions since the particle dynamics change greatly with increasing $\varPhi$. To assuage this misgiving, we perform torque measurements at $\varPhi =0.5\,\%$ and the results have been shown in figures 2, 3, 8–10. It can be seen that the quantities related to the drag of the system at $\varPhi =0.5\,\%$ (including $c_f$, $\nu _{eff}$ and $G$) show similar trends to that at higher volume fractions, hinting that the particles behave in similar ways at the low and high volume fractions in the current parameter regimes. Here, as depicted in figure 4(a,c), we define the region where the plumes are ejected from the inner (outer) boundary layer to the bulk as the ejected (injected) region, and the region between them is defined as the vortex-centre region. The spherical particle positions which are obtained from different frames when $\varPhi =0.5\,\%$ at $Re=6.5\times 10^3$ and $Re=1.3\times 10^4$ are given in figures 4(a) and 4(b), respectively. When $Re=6.5\times 10^3$, it is found that the particles distribute non-uniformly, and the pattern of the particle distributions reminisces about the Taylor vortices in TC flow (Grossmann et al. Reference Grossmann, Lohse and Sun2016). The particle distributions show a lower clustering near the inner wall in the ejected region, while in the injected region the lower clustering emerges near the outer wall. However, when $Re$ is increased to $1.3\times 10^4$, the particles distribute nearly homogeneously in the entire system (figure 4b).

Figure 4. The spherical particle positions which are obtained from different frames for $\varPhi =0.5\,\%$ at (a) $Re=6.5\times 10^3$ and (b) $Re=1.3\times 10^4$. The value of the colour bar denotes the number of times that the particles appear at the position denoted by the points in the graph. The green, magenta and yellow rectangles represent the injected region, vortex-centre region and the ejected region, respectively, and the width of which is around $1.7d_e$. Note that most of the points in the graph are corresponding to values of $1{\sim}2$ in the colour bar. (c) Sketch of the flow structures and particle positions (dots) at $Re=6.5\times 10^3$. The injected and ejected plumes are represented by the red and blue arrows, respectively. Note that the particle positions in the sketch are not experimental data and are for demonstration only.

To quantitatively evaluate the collective effects of the particles, we choose three typical regions of the flow structures (i.e. the injected region, ejected region and the vortex-centre region) and calculate the probability density function (PDF) of the particle radial positions. As shown in figure 5, the PDF is consistent with the pattern of the particle distributions in figure 4. When $Re=6.5\times 10^3$, the PDF has the minimal value near the inner wall in the ejected region, indicating the emergence of the lower clustering region. While in the injected region, the PDF peaks near the inner wall (i.e. highly clustering region) and approaches the lowest value near the outer wall. Not surprisingly, as shown in figure 5(b), the PDF curves tend to overlap with each other when $Re$ increases, indicating that the distribution of the particles becomes less non-uniform in the entire system at higher $Re$. Note that, the inner wall peak appears in the ejected region at higher $Re$ (figure 5b), which is opposite to the behaviour found at low $Re$ (figure 5a), and this might result from the slight shift of flow structure positions since the positions of the Taylor vortex could be not exactly the same at low and high $Re$. The difference of the particle distributions at different $Re$ can be interpreted from the perspective of the evolution of flow structures and the finite-size effect of the particles. On the one hand, though the TC flow displays flow structure at high $Re$ (Huisman et al. Reference Huisman, Van Der Veen, Sun and Lohse2014), the Taylor vortices become turbulent and the velocity fluctuations increase as $Re$ increases (Grossmann et al. Reference Grossmann, Lohse and Sun2016), resulting in the more vigorous particle motion. On the other hand, the finite-size effect of the particle can be characterized by the particle Reynolds number. Since the slip velocity of the particles is not accessible in the current experiments, the particle Reynolds number is estimated using the bulk velocity of the flow $u_b$, i.e. $Re_p=u_bd_e/\nu$, where $u_b$ can be estimated as $u_b\simeq 0.4\cdot \omega _ir_i$ in the present Reynolds number regime (Grossmann et al. Reference Grossmann, Lohse and Sun2016). Considering that the flow Reynolds number is defined by (2.1), one can obtain the particle Reynolds number as

(3.8)\begin{equation} Re_p=0.4{\frac{d_e}{d}}Re. \end{equation}

Hence, $Re_p$ ranges from $2.6\times 10^2$ to $10^3$ in the experiments, and the finite-size effects of the particle become more pronounced at higher $Re$.

Figure 5. The PDF of the particles position for $\lambda =1$ (spherical) at (a) $Re=6.5\times 10^3$ and (b) $Re=1.3\times 10^4$. The two red dashed vertical lines in each figure indicate the position of an equivalent radius of the particles (i.e. $r_e=0.5d_e$) away from the walls.

Furthermore, the particle inertia can also be measured by the particle Stokes number, $St=\tau _p/\tau _\eta$, where $\tau _p$ and $\tau _\eta$ are the inertia response time of the particles and the Kolmogorov time scale of the flow, respectively. For spherical particles suspended in TC flow, the Stokes number is reduced to

(3.9)\begin{equation} St={\frac{\rho_pd_p^2}{18\rho_f[(r_o^2-r_i^2)r_id]^{1/2}}(GRe)^{1/2}}, \end{equation}

i.e. the $St$ (or, the particle inertia) increases with increasing $G$ and $Re$. For non-spherical particles, $St_{non}=c\cdot St$, where $c=f(\lambda )$ is a constant only depending on $\lambda$ (Voth & Soldati Reference Voth and Soldati2017). Since the suspension is dilute here ($\varPhi =0.5\,\%$), the non-dimensional torque $G$ can be approximated to that of the single-phase case, $G_{sp}$, giving the $St$ ranging from $10$ to $60$. Therefore, in the current $Re_p$ and $St$ ranges, the suspended particles could induce unsteady wakes and exhibit inertia clustering (Toschi & Bodenschatz Reference Toschi and Bodenschatz2009; Mathai, Lohse & Sun Reference Mathai, Lohse and Sun2020). Specifically, for the case of $Re=6.5\times 10^3$ in figure 4(a) (correspondingly, $Re_p=2.6\times 10^2$ and $St=10$), the particles are principally driven by the strong plumes released from the boundary layer, which are depicted by the arrows in figure 4(c). While for the case of $Re=1.3\times 10^4$ in figure 4(b) (correspondingly, $Re_p=5.2\times 10^2$ and $St=25$), the particles with the increased inertia can escape more easily from the flow structures and distribute more uniformly.

Since the particle distributions are inhomogeneous in the axial direction of the system, we use the averaged PDF, which is obtained by calculating the arithmetic average value of the three typical regions, to quantitatively represent the collective effects of particles in the entire system. As shown in figure 6(a–c), for a given $\lambda$, the trends of the averaged PDF curves remain when the $Re$ increases from $6.4\times 10^3$ to $Re=1.3\times 10^4$, suggesting that the increasing turbulent intensity has negligible effects on the averaged particle migration in the radial direction of the system in the current situation.

Figure 6. The averaged PDF of the particles position (a,d) for $\lambda =1/3$ (oblate), (b,e) for $\lambda =1$ (spherical), (c,f) for $\lambda =3$ (prolate). Plots (a–c) are obtained at $\varPhi =0.5\,\%$ by calculating the arithmetic average value of the three typical regions in figure 4, while (d–f) are obtained at $\varPhi =2\,\%$ by manually detecting the particle positions. The two red dashed vertical lines in each figure are the same as in figure 5.

Nevertheless, for different $\lambda$, the averaged PDF curves show distinct differences regardless of the $Re$. In other words, particles with different $\lambda$ show different collective behaviours near the walls and thereby are expected to affect the boundary layer to different degrees. Indeed, the formation of particle layers has been reported in the particle-laden channel flow in the previous simulation works (Costa et al. Reference Costa, Picano, Brandt and Breugem2016; Ardekani et al. Reference Ardekani, Costa, Breugem, Picano and Brandt2017; Ardekani & Brandt Reference Ardekani and Brandt2019). Ardekani & Brandt (Reference Ardekani and Brandt2019) have shown that the spherical particles form a particle layer, which could enhance the near-wall Reynolds stress. While for the cases of the oblate and prolate particles, the near-wall Reynolds stress is attenuated due to the absence of the particle layer. In the present work in TC flow, as indicated in figures 5 and 6(b), the spherical particles preferentially cluster and form a particle layer near the inner wall, where the boundary layer exists. Therefore, the maximum drag modification is observed in the flow laden with spherical particles. However, the near-wall clustering phenomenon disappears in the oblate case (figure 6a), of which the PDF peaks in the bulk and has the lowest value near the walls. The averaged PDF curves indicate that the oblate particles prefer to cluster in the bulk and thereby have a smaller effect on the boundary layer, accounting for the observations of the minimal drag modification in figures 2 and 3.

However, one more question remains since so far the analyses of particle distribution are based on the optical measurements at $\varPhi =0.5\,\%$: will the particle preferential clustering persist at even higher $\varPhi$? To bridge the gap of volume fractions between the torque measurements ($\varPhi \geq 2\,\%$) and optical measurements ($\varPhi =0.5\,\%$), we further conduct optical measurements at higher volume fractions. Since the domain is illuminated from the back, the light intensity decreases greatly at higher volume fractions, making it difficult to distinguish the particles from the background fluid. At higher volume fractions, the feasible method to detect the particles is manual detection, which is of relatively low accuracy but provides reliable information. The averaged PDF curves of particle distribution at $\varPhi =2\,\%$ are shown in figure 6(d–f). For each case, 100 frames are used and the total number of detected particles is more than 3000. For even higher volume fractions ($\varPhi \geq 4\,\%$), the enormous amount of unfocused particles make the particles inside the focal plane invisible; therefore, no useful data can be obtained.

One can see in figure 6(d–f) that the spherical particles still preferentially cluster near the inner wall and, therefore, form a particle layer, which is consistent with the result obtained at $\varPhi =0.5\,\%$. Additionally, at $\varPhi =2\,\%$, the distribution of spherical particles becomes more non-uniform in the radial direction than that at $\varPhi =0.5\,\%$, which has also been reported in Fornari et al. (Reference Fornari, Formenti, Picano and Brandt2016), hinting that the spherical particles tend to move toward the walls as $\varPhi$ increases. This growing non-uniform distribution possibly results from the stronger shear-induced particle–particle interactions at higher $\varPhi$, and could partially account for the increasing drag differences compared with other particle shapes as $\varPhi$ increases. Moreover, as $\varPhi$ increases, the particles might relaminarize the bulk flow and, thus, suppress the Reynolds stress (Fornari et al. Reference Fornari, Formenti, Picano and Brandt2016). This Reynolds stress suppression could therefore yield a stronger $Re$-dependence of the drag at higher $\varPhi$, which is verified by the stronger shear-thinning effects of the normalized friction coefficient (figure 2) and the normalized effective viscosity (figure 10). On the other hand, for oblate and prolate cases, it is clear that most of the particles distribute in the bulk region. The near-wall clustering phenomenon disappears and, therefore, the particle layer is absent, which is similar to that of $\varPhi =0.5\,\%$ and accounts for their similar overall drag.

So far, the optical measurements performed at $\varPhi =0.5\,\%$ and $\varPhi =2\,\%$ lead to the same conclusion that the larger drag modification is connected to the near-wall particle clustering in the current system. Though the result and conclusion of particle preferential clustering might not be directly extrapolated further to higher volume fractions, one reasonable speculation would be that it will play a significant role in the drag modulation at higher $\varPhi$. Of course, further studies are needed to confirm it in future work.