3.1. Drag modulation and stress balance

The fluid viscosity, turbulence, polymers and particle phase all contribute to the total stress within the channel. In order to assess their relative contributions, the mean total stress $\tau _{tot}$ can be expressed in terms of its constituents,

(3.1)\begin{align} &\overbrace{\dfrac{\beta}{Re} (1-\phi) \dfrac{\boldsymbol{\text{d}}{\langle u_f \rangle}}{\text{d}{y}}}^{\displaystyle\tau_{\mu}} \ \overbrace{-[(1-\phi) \langle u'_f v'_f \rangle + \phi \langle u'_p v'_p \rangle]}^{\displaystyle\tau_{Re}} \nonumber\\ &\quad + \underbrace{\dfrac{(1-\beta)}{Re}(1-\phi) \langle \boldsymbol{\mathsf{T}}_{xy} \rangle}_{\displaystyle\tau_{\beta}} + \underbrace{\phi \langle \sigma_{p,xy} \rangle}_{\displaystyle\tau_{\phi}} = \underbrace{\langle \tau_{w}\rangle(1-y)}_{\displaystyle\tau_{tot}}. \end{align}

From left to right, the components are the viscous stress $\tau _\mu$, the turbulent Reynolds stress $\tau _{Re}$, the polymer stress $\tau _\beta$ and the particle stress $\tau _\phi$. When integrated over $0<y<1$ and normalized by $0.5\langle \tau _{w} \rangle ^{{\text {W0}}}$ from the single-phase Newtonian case, the right-hand side of (3.1) reduces to $\langle \tau _{w} \rangle /\langle \tau _{w} \rangle ^{{\text {W0}}}$, i.e. the normalized mean wall shear stress (drag), and the left-hand side expresses the contribution of different stress constituents. Figure 3 reports the normalized total drag (height of the bars) and its constituents (layering of the bars) from the various flow conditions. While the primary focus of the present work is on dense concentration ($\varphi = 20\,\%$), for the purpose of comparison two dilute cases ($\varphi = 5\,\%$) from Esteghamatian & Zaki (Reference Esteghamatian and Zaki2019) are also included.

Figure 3.

Contributions of different stress components to the total drag, integrated between $0<y<1$ and normalized by $0.5\langle \tau _{w} \rangle ^{{\text {W0}}}$. Results are arranged to display the effect of (a) Weissenberg number, (b) solid volume fraction. The intensity of fluctuations of the total stress in time, measured by the standard deviation of $\langle \tau _{w} \rangle _{xz}/\langle \tau _{w} \rangle ^{{\text {W0}}}$, is shown by uncertainty bars. Case designations are listed in table 1.

We first focus on the effect of the Weissenberg number (figure 3a). In single-phase flows, with increasing viscoelasticity, the Reynolds shear stress markedly diminishes while the polymer stress is marginally enhanced. The net effect is the well-established reduction in the total drag. At $20\,\%$ particle concentration, increasing viscoelasticity results in the reduction of the Reynolds shear stress, a mild increase in the particle stress and significant enhancement of the polymer stress.

These combined effects give rise to a non-monotonic behaviour in the total stress: the drag is decreased with an increase in viscoelasticity up to the point where the Reynolds shear stress is nearly eradicated in case W5P20. Beyond $Wi=5$, further increasing $Wi$ only enhances the polymer and particle stresses and, in turn, the total drag.

The non-monotonic trend is not universal for all particle concentrations. Results from the earlier study at $5\,\%$ particle concentration (Esteghamatian & Zaki Reference Esteghamatian and Zaki2019) predicted smaller particle and polymer stresses and lower drag relative to the $20\,\%$ cases, and a consistent reduction in the total stress with increasing elasticity. The Reynolds stress, $\tau _{Re}$, was also eradicated in the dilute case, but only at the highest Weissenberg number (case W25P5) – an indication that turbulence inhibition in particle-laden viscoelastic flows is weaker at dilute conditions. A comparison of the drag makeup at both concentrations suggests that increasing the Weissenberg number beyond $Wi=25$ in the dilute case ($\varphi =5\,\%$) would enhance the polymer stress and potentially increase the total drag, although additional simulations at the lower concentration would be required to verify this outcome. Nonetheless, the general conclusion is that increasing $Wi$ in particle-laden flows reduces drag when the reduction in $\tau _{Re}$ is appreciable, and hence $\tau _{Re}$ must be finite.

The intensity of fluctuations of the total stress in time, measured by the standard deviation of $\langle \tau _{w} \rangle _{xz}/\langle \tau _{w} \rangle ^{{\text {W0}}}$, is shown by uncertainty bars in figure 3. Strong fluctuations in time are a signature of intermittency, or the alternation of the turbulence between a hibernating and an active state – an effect that intensifies at higher $Wi$. In cases where the turbulence is eradicated, the intermittency thus vanishes. At $\varphi = 20\,\%$, the intermittency is most pronounced at $Wi=3$. The results highlight that intermittency sets in and is most pronounced at lower $Wi$ for larger particle concentrations.

Figure 3(b) draws the focus to the effect of the solid volume fraction on different components of the total stress. In both Newtonian and viscoelastic conditions, the addition of particles increases the total drag. In the Newtonian cases, underlying the increase in total drag is a non-monotonic behaviour of the Reynolds stress: as the particle concentration is increased, $\tau _{Re}$ first increases (W0P5) and then reduces (W0P20) relative to the single-phase configuration. This trend is consistent with a previous study in Newtonian flows and similar conditions by Picano et al. (Reference Picano, Breugem and Brandt2015). The increase in the total drag at higher particle concentration is more pronounced for the viscoelastic conditions due to the significant increase of the particle and polymer stresses with the solid volume fraction. Also, the non-monotonic trend in the Reynolds stress reported for Newtonian carrier fluids disappears in the viscoelastic cases: the addition of particles monotonically reduces and ultimately eliminates $\tau _{Re}$.

The particle stress $\tau _\phi$ makes an important contribution to the total drag at $\varphi = 20\,\%$, and warrants consideration. Two relevant quantities are the solid volume fraction $\phi$ and the mean shear rate $\dot {\gamma } \equiv {\boldsymbol {\text {d}}{\langle u_f \rangle }}/{\text {d}{y}}$ (figure 4a,b). For $Wi\ge 5$, particles migrate away from the wall in all the particle-laden cases and the solid volume fraction reaches near $45\,\%$ at the channel centre (figure 4a) – an explanation of this migration trend is provided in § 3.4. In all cases, the $\phi$ profile has a local maximum near $y\approx d_p$, due to the asymmetric interaction of near-wall particles that are stabilized by the wall lubrication. For $Wi \geq 5$, a second local maximum near $y\approx 2d_p$ is indicative of a propensity for the particles to form a layered structure in the absence of a vertical mixing mechanism. The mean shear rate (figure 4b) exhibits a non-monotonic behaviour with increasing viscoelasticity. At the wall, it decreases to a minimum at $Wi=5$ and increases beyond that value. Away from the wall, $y \ge 0.2$, that trend is reversed and the mean shear rate is the highest for $Wi=5$ and decreases at higher $Wi$.

Figure 4.

Wall-normal profiles of (a) solid volume fraction; (b) mean shear rate $\dot {\gamma } \equiv {\boldsymbol {\text {d}}{\langle u_f \rangle }}/{\text {d}{y}}$ in particle-laden cases; (c) particle stress $\tau _\phi$; and polymer stress $\tau _\beta$ in (d) single-phase and (e) particle-laden conditions. Line types are listed in table 1.

The profile of $\tau _\phi$ (figure 4c) is influenced by viscoelasticity in three ways. (i) The polymers directly increase the stresslet contribution to the particle stress by altering the surface traction (Einarsson et al. Reference Einarsson, Yang and Shaqfeh2018). (ii) The mean shear rates are higher in the region $y\ge 0.2$. Since particles resist deformation under shear, the stresslet contribution to the particle stress is enhanced in that region. (iii) Due to the migration of particles towards the channel centre, the particle concentration and in turn the particle stress are larger at $y\geq 0.65$. A combination of all these effects leads to the larger particle stresses in the viscoelastic cases away from the wall. Near the wall, local peaks in $\tau _\phi$ coincide with the local maxima in the solid volume fraction. In addition, conditions with the stronger near-wall shear rates ($Wi<5$ cases) have larger peaks in the particle stress profiles.

The other important contribution to drag in the particle-laden viscoelastic cases is the polymer stress, especially when compared with the single-phase cases (figures 4d and 4e). The latter sustain small levels of $\tau _\beta$, with very weak variation in the wall-normal direction. In contrast, in presence of the particle phase, the polymer stress increases appreciably, is highest in the near-wall region where particle layering is observed, and decays to zero towards the channel centre. Evidently, the particles contribute to the deformation of polymers and, in turn, increasing the polymer stress. It is unexpected, however, that the polymer stress vanishes in the region where the particles are most accumulated ($0.8 < y < 1.0$). In this region, the mean flow is akin to a plug profile with negligible shear rate due to the high concentration of particles (see figure 4b for the shear rate). This observation underscores that the presence of particles and of a finite shear rate are two essential components for effective polymer deformation. The differences between the polymer conformation in the near-wall region and the channel centre are further examined in § 3.3.

3.2. Flow structures and statistics

In this section, we examine the effect of particles on turbulent structures, mean flow statistics and Reynolds stresses. Figure 5 compares the instantaneous contours of $u_f - \langle u_f \rangle$, and the correlation of $u'_f$ in the spanwise direction,

(3.2)\begin{equation} R_{u'_fu'_f}(y, {\rm \Delta} z) = \dfrac{\langle u'_f(x,y,z,t)u'_f(x,y,z+{\rm \Delta} z,t)\rangle}{ \langle u'_f(x,y,z,t)u'_f(x,y,z,t)\rangle}. \end{equation}

In single phase, relative to the Newtonian streaks, viscoelasticity inflates these structures (see figures 5a and 5b). Addition of particles to a Newtonian flow also increases the width of the streamwise structures (figure 5c). The particle-laden viscoelastic case is markedly different: the classical turbulent streaks are nearly absent at $Wi =25$ (figure 5d), and the spanwise correlation length reduces to the order of a particle diameter.

Figure 5.

Instantaneous contours and isosurfaces of $u_f\text {--}\langle u_f\rangle$ and particle positions in $xz$ plane (top); and two-point correlation of $u'_f$ as a function of spanwise spacing (bottom). (a) Single-phase Newtonian (W0); (b) single-phase viscoelastic (W25); (c) particle-laden Newtonian (W0P20); (d) particle-laden viscoelastic (W25P20). The wall-normal location of the contours and correlations is $y u_\tau /\nu \approx 20$.

The mean fluid-velocity profiles and their deviation from Poiseuille flow are reported in figure 6. In single-phase flows, the mean profiles monotonically approach the Poiseuille condition with increasing elasticity. In particle-laden flows, a non-monotonic behaviour that is consistent with the change in drag is observed: the flow approaches the laminar condition with increasing elasticity up to $Wi=5$ at which the turbulence is completely eradicated. Beyond that level, the polymer stress dominates the momentum transfer in the wall-normal direction. As a result, the mean velocity profiles are flattened with further increase in elasticity.

Figure 6.

Mean streamwise fluid velocity (top); and its deviation from the laminar profile $u_{{L}}$ (bottom) in (a) single-phase and (b) 20 % particle concentration. Line types are listed in table 1.

The normal components of the Reynolds stress tensor are plotted in figure 7. In single-phase flows, the peak of $\langle u'_f u'_f \rangle$ shifts away from the wall with increasing elasticity due to the swelling of the streamwise streaks, and the stresses in the cross-flow plane are progressively attenuated at higher $Wi$. In particle-laden conditions, the co-presence of the solid phase and elasticity effectively suppresses the turbulent fluctuations. At $Wi\ge 5$, fluctuations are significantly suppressed in all three directions and become independent of elasticity. This observation reinforces our previous interpretation of flow perturbations (figure 5): they are not due to conventional turbulent eddies, but rather fluctuations due to the particles in shear.

Figure 7.

Wall-normal profiles of (a,d) streamwise, (b,e) wall-normal and (c,f) spanwise fluid velocity fluctuations. Line types are listed in table 1.

Quadrant analysis of the Reynolds shear stress is reported in figure 8. The second and fourth quadrants ($\langle u'_f v'_f \rangle _\text {Q2}$ and $\langle u'_f v'_f \rangle _\text {Q4}$) are associated with ejections and sweeps that are responsible for the turbulence production, while $\langle u'_f v'_f \rangle _\text {Q1}$ and $\langle u'_f v'_f \rangle _\text {Q3}$ correspond to interaction events that counter the wall-normal momentum flux (Wallace, Eckelmann & Brodkey Reference Wallace, Eckelmann and Brodkey1972). In single-phase flows, elasticity reduces the magnitude of $u_f'v_f'$ in all quadrants, while the largest drops are associated with $\langle u'_f v'_f \rangle _\text {Q2}$ and $\langle u'_f v'_f \rangle _\text {Q4}$. As a result, the net effect is a reduction in the turbulent shear stress and in turn the wall-normal momentum transfer.

Figure 8.

Quadrant analysis of the Reynolds shear stress, $\langle u'_f v'_f \rangle$, in (a) single-phase and (b) particle-laden conditions. Line types are listed in table 1.

The addition of particles to the Newtonian flow increases the contributions in $\langle u'_f v'_f \rangle _\text {Q1}$, $\langle u'_f v'_f \rangle _\text {Q3}$ and $\langle u'_f v'_f \rangle _\text {Q4}$, while weakening $\langle u'_f v'_f \rangle _\text {Q2}$ ejection events. These modifications are marginal, and their net effect is an $11\,\%$ reduction in the turbulent shear stress relative to the single-phase condition. At $Wi=\{1,3\}$ the turbulence is still active, and the addition of particles results in $\{15,31\}\%$ decrease in the turbulent shear stress mostly due to weakened $\langle u'_f v'_f \rangle _\text {Q2}$ ejection events. Beyond $Wi = 5$, a steep reduction in the contributions to $\langle u'_fv'_f\rangle$ is observed in all four quadrants. The nearly symmetric contributions of $\langle u'_f v'_f \rangle _\text {Q1}$ to $\langle u'_f v'_f \rangle _\text {Q4}$ highlights that the turbulence production is completely disrupted. Also, the remarkable drop in the peak values underlines the reduction in correlated motions in the $x$ and $y$ directions.

3.3. Polymer conformation

Velocity gradients in the fluid phase, whether induced by the mean shear, turbulent fluctuations or particles, modify the polymer conformation away from equilibrium. As explained in detail by Hameduddin et al. (Reference Hameduddin, Meneveau, Zaki and Gayme2018), these modifications cannot be accurately interpreted with reference to isolated components of $\boldsymbol{\mathsf{c}}$. Instead, those authors introduced three rigorous measures to characterize the conformation tensor and its perturbations: (i) the logarithmic volume ratio, $\log (\det (\boldsymbol{\mathsf{c}})/\det (\bar {\boldsymbol{\mathsf{c}}}))$, which compares the volume of the instantaneous polymer conformation $\boldsymbol{\mathsf{c}}$ relative to a reference conformation tensor $\bar {\boldsymbol{\mathsf{c}}}$; (ii) the squared geodesic distance between $\boldsymbol{\mathsf{c}}$ and $\bar {\boldsymbol{\mathsf{c}}}$ on the manifold of positive-definite matrices; (iii) the anisotropy which is a function of (i) and (ii), and therefore is not considered in the present study. The squared geodesic distance between a pair of positive-definite matrices ${\boldsymbol{\mathsf{A}}}$ and ${\boldsymbol{\mathsf{B}}}$ is defined by

(3.3)\begin{equation} \textrm{d}^2({\boldsymbol{\mathsf{A}}},{\boldsymbol{\mathsf{B}}}) \equiv \text{tr}(\log^2({\boldsymbol{\mathsf{A}}}^{-1/2} \boldsymbol{\cdot} {\boldsymbol{\mathsf{B}}} \boldsymbol{\cdot} {\boldsymbol{\mathsf{A}}}^{-1/2})) . \end{equation}

Wall-normal profiles of the first two scalar measures are reported in figure 9, where the reference tensor was selected to be $\bar {\boldsymbol{\mathsf{c}}} = \boldsymbol{\mathsf{I}}$ in order to highlight the departure of the polymer from the isotropic equilibrium state. This choice of the reference state also affords a fair comparison across the different flow configurations. In both the single-phase and particle-laden cases, $\langle \log (\det (\boldsymbol{\mathsf{c}}))\rangle$ and $\langle \textrm {d}^2(\boldsymbol{\mathsf{I}},\boldsymbol{\mathsf{c}}) \rangle$ increase with $Wi$, with marginal difference between $Wi = 15$ and $Wi=25$. In single-phase flows, and at the lowest $Wi$, the profiles of both measures decay in the outer flow due to the fast polymer relaxation away from the peak turbulence production. At the higher $Wi$ cases, the profiles only have weak variations in the wall-normal direction. Due to the long polymers’ relaxation times in these cases, the perturbations in polymer conformations are effectively transported by turbulent motion across the channel, explaining the almost uniform profile of both measures in the bulk of the channel at $Wi=\{15,25\}$. In the particle-laden cases with $Wi \ge 5$, the wall-normal mixing is inhibited due to the suppression of turbulence (cf. figure 8b), yet both measures still exhibit small variations in most of the channel, except near the centre. The controlling factors in polymer perturbations in particle-laden cases with $Wi \ge 5$ are the mean shear rate and the particle concentration. From the wall to the centre, the polymer perturbation is enhanced with an increase in the number of particles, yet hindered by a decrease in the mean shear rate. These competing effects explain the weak variations in most of the channel. Nevertheless, near the channel centre where the mean shear decays to zero and the particles population is the highest, the significant drop in both measures of polymer deformation is a signature of a different particle–polymer interaction mechanism, which is further detailed in the following.

Figure 9.

Wall-normal profiles of (a,b) average logarithmic volume and (c,d) squared geodesic distance between the conformation and the identity tensors (cf. (3.3)). Line types are listed in table 1.

Profiles of $\langle \textrm {d}^2(\langle \boldsymbol{\mathsf{c}} \rangle _{\log },\boldsymbol{\mathsf{c}}) \rangle$ are reported for the single-phase and particle-laden conditions in figure 10. This quantity is the average of the squared geodesic distance between the instantaneous and the log-Euclidean mean conformation, and is therefore a measure of the intensity of the perturbation from that mean at different $y$ locations. Note that since volumetric compressions and expansions of polymers with respect to $\langle \boldsymbol{\mathsf{c}} \rangle _{\log }$ are symmetric, the average value of $\log (\det (\boldsymbol{\mathsf{c}})/\det (\langle \boldsymbol{\mathsf{c}} \rangle _{\log }))$ is zero and is therefore not reported. In the single-phase flows, higher elasticity is associated with larger departures of the conformation from its mean value, and the profiles of $\langle \textrm {d}^2(\langle \boldsymbol{\mathsf{c}} \rangle _{\log },\boldsymbol{\mathsf{c}}) \rangle$ become nearly flat for $Wi \geq 5$. The recorded values are, as can be anticipated, smaller than $\langle \textrm {d}^2(\boldsymbol{\mathsf{I}}, \boldsymbol{\mathsf{c}}) \rangle$. Another notable difference is the appreciable near-wall drop in figure 10(a) relative to figure 9(c): although the polymers are highly deformed in the near-wall region, the fluctuations away from their log-Euclidean mean are relatively small. In the particle-laden cases, local maxima near the wall coincide with the peaks of the solid volume fraction profiles (cf. figure 4a). In the outer bulk flow, where the particle population is larger, the perturbations relative to the mean are enhanced at larger $Wi$. Nevertheless, in all cases the fluctuations decay significantly beyond $y \approx 0.8$, confirming the trends in the profiles of $\langle \log (\det (\boldsymbol{\mathsf{c}}))\rangle$ and $\langle \textrm {d}^2(\boldsymbol{\mathsf{c}}, \boldsymbol{\mathsf{I}}) \rangle$ in figures 9(b) and 9(d).

Figure 10.

Wall-normal profiles of the mean squared geodesic distance between instantaneous and the log-Euclidean mean conformation tensors in (a) single-phase and (b) particle-laden conditions. Line types are listed table 1.

We now direct the focus to select $y$ locations where we provide a further detailed account of the polymer deformation when $Wi=25$; the results for $Wi=15$ are qualitatively similar. Figure 11 shows the joint probability density function (JPDF) of $\langle \textrm {d}^2(\boldsymbol{\mathsf{I}}, \boldsymbol{\mathsf{c}}) \rangle$ and $\log (\det (\boldsymbol{\mathsf{c}}))$. At $y=0.15$, polymers depart appreciably from the isotropic state and, in the particle-laden condition, are more deformed and expanded compared with the single-phase counterpart (figure 11a). The figure also shows instantaneous realizations of the polymer conformation, visualized using ellipsoids whose axes are the eigenvectors of the tensor each scaled by the corresponding eigenvalues; the ellipsoid are also coloured by the logarithm of the volume of the tensor. These visualizations suggest dissimilar mechanisms for polymers’ deformation in the single-phase and particle-laden cases. In the former, polymers are most perturbed in the vicinity of the coherent turbulent structures, visualized by contours of $u_f\text {--}\langle u_f \rangle$. On the other hand, in the particle-laden condition, large polymer deformation are observed in the vicinity of particles. The JPDF of $\langle \textrm {d}^2(\boldsymbol{\mathsf{I}}, \boldsymbol{\mathsf{c}}) \rangle$ and $\log (\det (\boldsymbol{\mathsf{c}}))$ at the channel centre (figure 11b) shows appreciable departure of the polymer from equilibrium in the single-phase condition. In contrast, in the particle-laden case the polymer conformation is most akin to the isotropic equilibrium state.

Figure 11.

The JPDF of squared geodesic distance $\textrm {d}^2(\boldsymbol{\mathsf{I}},\boldsymbol{\mathsf{c}})$ and the logarithmic volume $\log (\det (\boldsymbol{\mathsf{c}}))$ at (a) $y=0.15$ and (b) $y = 1$. Flood (particle-laden) and line (single-phase) contours are plotted in the same range, and refer to the cases with the $Wi=25$ condition. Flow visualizations show snapshots of $u_f\text {--}\langle u_f \rangle$ contours in the single-phase condition and position of particles that are cut by the $x$–$z$ plane in the particle-laden case. The ellipsoids represent instantaneous realizations of the polymer conformation, are coloured by the logarithmic volume of $\boldsymbol{\mathsf{c}}$, and their axes are the eigenvectors of $\boldsymbol{\mathsf{c}}$ scaled by the corresponding eigenvalues.

The instantaneous visualizations at $y=1$ show that the polymer conformation is relatively small in volume in the two-phase case, in the confined regions between the crowd of particles. Scattered instances of deformed polymer conformation are still observable, although their presence is rare and their magnitude is small (note that the colourbars for $\log (\det (\boldsymbol{\mathsf{c}}))$ in the particle-laden condition have different ranges in figures 11a and 11b).

So far, we have shown that the particles appreciably alter the polymer deformation and, in turn, the associated stresses. We will now examine how these deformations take place at the particle scale. To this end, we first describe the particles’ microstructure by evaluating the particle-pair distribution function, $q(r,\psi ,\theta )$, which depends on the pair separation $r$, the polar angle $\psi$ relative to the positive $z$ axis, and the azimuthal angle $\theta$ measured anticlockwise from the positive $x$ axis,

(3.4)\begin{gather} q(r,\psi,\theta) = \langle \text{d}N(r,\psi,\theta) \rangle / \langle \text{d} N(r,\psi,\theta)^{R} \rangle, \end{gather}

(3.5)\begin{gather}\text{d}N(r,\psi,\theta) = \sum_{m=1}^{N_p} \delta(r-|{\boldsymbol{r}}_{m}|)\delta(\psi - \psi_{m})\delta(\theta - \theta_{m}), \end{gather}

(3.6)\begin{gather}{\boldsymbol{r}}_{m} = {\boldsymbol{P}}_{m} - {\boldsymbol{P}}_{ref}, \end{gather}

(3.7a,b)\begin{gather}\psi_{m} = \cos^{-1}\left(\dfrac{{\boldsymbol{r}}_m\boldsymbol{\cdot} \boldsymbol{e}_z}{|{\boldsymbol{r}}_{m}|} \right); \quad \theta_{m} = \arctan\left(\dfrac{{\boldsymbol{r}}_m\boldsymbol{\cdot} \boldsymbol{e}_y}{{\boldsymbol{r}}_m\boldsymbol{\cdot} \boldsymbol{e}_x}\right). \end{gather}

The number of neighbouring particles within each bin is denoted $\text {d}N$, and ${\boldsymbol {P}}$ refers to the particles’ position vector. Similar to our previous study (Esteghamatian & Zaki Reference Esteghamatian and Zaki2019), the particle-pair distribution function $q$ is normalized by $\textrm {d}N^R$ which corresponds to a random distribution of particles with a no-overlap condition and the case-specific wall-normal profile of the mean solid fraction, in order to eliminate bias in the interpretation of the results.

Results for reference particles located at $y=0.2$ are shown in figure 12(a). Contours of $q(r = 1.05d_p)$ are projected onto the particle surface. In terms of the particle-attached coordinates, $\tilde {{\boldsymbol {x}}}=\boldsymbol {x} - {\boldsymbol {P}}_{ref}$, the pairwise distribution function shows accumulations of neighbouring particles in the regions $\tilde {x}\tilde {y}<0$ and depletion $\tilde {x}\tilde {y}>0$. This microstructure is generally observed when finite-size particles are suspended in shear flows: particles at $\tilde {y}>0$ advect faster than the reference one and, as a result, are more likely to accumulate behind it. A similar explanation describes the asymmetry of $q$ with respect to the $\tilde {y}$ axis at $\tilde {y}<0$ locations, where the neighbouring particles advect more slowly than the reference one.

Figure 12.

Particle-pair distribution and polymer conformation for case W25P20. (a) Reference particle located at $y=0.2$ coloured by $q(r = 1.05d_p)$. Ellipsoids represent the particle-conditioned log-Euclidean mean conformation $\langle \boldsymbol{\mathsf{c}} \rangle _{\log }$ coloured by their logarithmic volume and plotted in the particle coordinates ($\tilde {\boldsymbol {x}}=\boldsymbol {x} - \boldsymbol {P}_{ref}$). Four regions within one diameter from the particle centre are designated $\Omega _1$–$\Omega _4$. (b) The JPDF of squared geodesic distance $\textrm {d}^2(\langle \boldsymbol{\mathsf{c}} \rangle _{\log },\boldsymbol{\mathsf{c}})$ and logarithmic volume ratio $\log (\det (\boldsymbol{\mathsf{c}})/\det (\langle \boldsymbol{\mathsf{c}} \rangle _{\log }))$ for points located at $y=0.15$ ($\tilde {y}=-0.05$). Line contours are unconditional; colours are points within one diameter of a particle centre and exclusively located in (left) $\Omega _1$ or $\Omega _3$ versus (right) $\Omega _2$ or $\Omega _4$.

Figure 12(a) also shows the conditionally averaged log-Euclidean polymer conformation $\langle \boldsymbol{\mathsf{c}} \rangle _{\log }$. In order to aid the discussion, four regions are identified within one diameter of the particle centre and are designated $\Omega _1$ to $\Omega _4$. Regions $\Omega _1$ and $\Omega _3$ correspond to $\tilde { x} \tilde { y} > 0$, while $\Omega _2$ and $\Omega _4$ correspond to $\tilde { x} \tilde { y} < 0$. The conformation tensors have larger volumes in $\Omega _1$ and $\Omega _3$, where polymers are effectively deformed by the presence of the reference particle and unconstrained by adjacent particles, as demonstrated by the particles microstructure. In the other two regions, where the neighbouring particles are stabilized by collision, polymers are overall less deformed and large deformations are only recorded in the immediate vicinity of the reference particle.

The effect of the microstructure on the polymer conformation in case W25P20 is quantified using the JPDF of the squared geodesic distance and logarithmic volume ratio in figure 12(b). The adopted reference conformation is $\bar {\boldsymbol{\mathsf{c}}} = \langle \boldsymbol{\mathsf{c}} \rangle _{\log }$, that is to say the polymer deformation is evaluated relative to the log-Euclidean mean conformation. The line contours in both panels show the unconditional JPDF at $y = 0.15$. Near symmetry of positive and negative values of the logarithmic volume ratio confirms that the log-Euclidean mean is representative of the ensemble. The colour contours display the JPDF conditional on the fluid points belonging to specific regions: the left panel for points in $\Omega _1$ and $\Omega _3$, and the right for points in $\Omega _2$ and $\Omega _4$. In both figures, the colour contours are skewed towards larger values of the logarithmic volume ratio, highlighting the role of particles in the deformation of the polymers. The shift of the peak in the JPDFs towards larger expansions and squared geodesic distances is more appreciable in $\Omega _1$ and $\Omega _3$, which is consistent with the interpretation of figure 12(a): polymers are more deformed in $\tilde {x}\tilde {y}>0$ regions, where the reference particle is not immediately constrained by immediate neighbours.

3.4. Particle dynamics: diffusion and migration

The diffusion of particles is examined by evaluating their mean square displacement in the wall-normal direction $\langle {\rm \Delta} y^2\rangle$ as a function of the separation time ${\rm \Delta} t$:

(3.8)\begin{equation} \langle {\rm \Delta} y^2\rangle \equiv \langle (P_y(t+{\rm \Delta} t) - P_y(t))^2\rangle, \end{equation}

where the average is performed over the ensemble of particles which at time $t$ are located in a wall-parallel bin centred at $y$. The extent of the bin in the wall-normal direction is $1.5d_p$, while in the $x$ and $z$ directions it spans the full domain. The mean square displacements as a function of ${\rm \Delta} t$ are reported in figure 13, and exhibit quadratic and linear growth at small and large separation times, respectively. The former is indicative of correlated motion driven by collisions, and the latter is due to uncorrelated diffusive motion. In viscoelastic conditions, the eradication of turbulence results in faster decorrelation of particles motion compared with the Newtonian flow. In addition, the mean square displacements are orders of magnitude smaller in the viscoelastic flow due to the weaker mixing of particles in the wall-normal direction.

Figure 13.

Particles’ mean squared displacement in the wall-normal direction (cf. (3.8)) normalized by the particle diameter squared $\langle {\rm \Delta} y^2 \rangle /d_p^2$, for cases (a) W0P20 and (b,c) W25P20. In panel (c), the ordinate is also normalized by the mean shear rate of the streamwise fluid velocity $\dot {\gamma }$. The colours indicate different $y$ locations of the reference particle at ${\rm \Delta} t = 0$.

In the Newtonian condition, the first bin for the computation of $\langle {\rm \Delta} y^2\rangle$ is located at $y^+\approx 35$. At higher wall-normal positions, and during the correlated regime $\langle {\rm \Delta} y^2\rangle \propto {\rm \Delta} t^2$, the mean squared displacements weakly decrease as the turbulence activity decreases. In the viscoelastic flow, $\langle {\rm \Delta} y^2\rangle$ is a stronger function of $y$ during both the correlated and diffusive regimes. In the absence of the turbulence, it is expected that the particles’ dynamics is driven by shear-induced self-diffusion. We examine this hypothesis by normalizing $\langle {\rm \Delta} y^2\rangle$ with the mean shear rate of the streamwise fluid velocity, $\dot {\gamma }$, in figure 13(c). The better collapse of the curves corresponding to different $y$ locations confirms that the shear-induced self-diffusion is a dominant mechanism for the dispersion of particles in the wall-normal direction.

In § 3.1, we have shown that the particles’ concentration increases away from the wall for $Wi \ge 5$ (cf. figure 4a). An explanation was sought by performing an additional simulation that probes the transient particles’ migration dynamics. The initial condition was a snapshot from the statistically stationary Newtonian particle-laden flow, where the particle concentration is almost flat in the majority of the channel. Viscoelasticity was then suddenly introduced, with $Wi=25$. The particles migrated towards the channel centre over approximately 400 time units, and the migration velocity progressively weakened. The equilibrium concentration profile for case W25P20 was established beyond $t=500$.

A particle in wall-bounded shear has an equilibrium position farther away from the wall when the carrier fluid is viscoelastic compared with Newtonian (e.g. Karnis & Mason (Reference Karnis and Mason1966); Leshansky et al. (Reference Leshansky, Bransky, Korin and Dinnar2007), and the Appendix). This lateral migration of isolated particles, or in the very dilute regime, is well documented and is due to an imbalance of elastic normal stresses (see D'Avino, Greco & Maffettone (Reference D'Avino, Greco and Maffettone2017), for a review). In the present concentrated suspension, interparticle collisions play an important role in shear-induced migration towards the channel centre. In order to elucidate the mechanism, the statistical measures introduced by Sundaram & Collins (Reference Sundaram and Collins1997) are tailored for our configuration. The propensity of particle interactions to affect a reference particle is determined by (i) the probability of the presence of a neighbouring particle and (ii) the intensity of their relative velocity. The first measure is quantified by the particle-pair distribution function, previously defined as $q$ in (3.4). Here the quantity is redefined with a different normalization,

(3.9)\begin{equation} q'(r,\psi,\theta) = \langle \text{d}N(r,\psi,\theta) \rangle / (r^2\,\text{d}\theta \, \text{d}\psi \,\text{d}r), \end{equation}

namely using the volume of the spherical bin, $r^2\,\text {d}\theta \,\text {d}\psi \,\text {d}r$. Unlike the previous definition, $q'(r,\psi ,\theta )$ is intentionally biased by the inhomogeneity of particles distribution in the wall-normal direction. As we will show shortly, this inhomogeneity plays an important role in the migration dynamics. The second measure is expressed by the approaching velocity between the reference and a neighbouring particle $m$,

(3.10)\begin{gather} {\rm \Delta} u^{n,-} (r,\psi,\theta)= \dfrac{ \displaystyle\sum_{m=1}^{N_p} \delta(r-|{\boldsymbol{r}}_{m}|)\delta(\psi - \psi_{m})\delta(\theta - \theta_{m}) {\rm \Delta} u_{m}^{n,-}}{\text{d}N(r,\psi,\theta)}, \end{gather}

(3.11)\begin{gather}{\rm \Delta} u_{m}^{n,-}= \max\left\{0, -(\boldsymbol{u}_{p,m} - \boldsymbol{u}_{p,{ref}})\boldsymbol{\cdot} \dfrac{{\boldsymbol{r}}_m}{|{\boldsymbol{r}}_m|}\right\}. \end{gather}

On average, $\langle {\rm \Delta} u^{n,-} \rangle (r,\psi ,\theta )$ describes the spatial variations in the approaching velocity of neighbouring particles in the vicinity of the reference one. Finally, the product of $q'$ and $\langle {\rm \Delta} u^{n,-} \rangle$, referred to as the collision kernel when $r=d_p$ (Sundaram & Collins Reference Sundaram and Collins1997), quantifies the intensity of collision events. Note that in addition to direct collisions, neighbouring particles also indirectly affect the motion of the reference one by displacement of the interstitial fluid between them. Hence, the observable $q' \times \langle {\rm \Delta} u^{n,-} \rangle$ will be examined even at $r>d_p$, with the knowledge that values at larger separation distances are less relevant to the dynamics of the reference particle.

These statistical measures are reported in figure 14 for case W25P20. The figure compares the initial transient of the simulation and a statistically stationary interval. During the former, the profile of solid volume fraction is similar to its initial condition adopted from the Newtonian flow, i.e. it increases away from the wall at a relatively weak rate compared with its statistically stationary state. As a result, $q'$ is slightly larger above the reference particle than below it during the transient, and this asymmetry becomes much more pronounced in the statistically stationary state (figure 14a). Due to the positive mean shear rate, neighbouring particles experience a finite approaching velocity only in regions $\Omega _2$ and $\Omega _4$ (cf. figure 12a for a definition). Also, the values of $\langle {\rm \Delta} u_p^{n,-} \rangle$ are larger in $\Omega _4$ due to the curvature of the mean streamwise velocity profile: the mean shear rate decreases away from the wall, giving rise to a stronger approaching velocity for neighbouring particles below the reference one (figure 14b). The combination of these two effects is shown in figure 14(c). While in the statistically stationary state $q' \times \langle {\rm \Delta} u_p^{n,-} \rangle$ is symmetric about the origin, it is larger in $\Omega _4$ during the initial transient. Hence, stronger collisions from below drive the reference particle towards the channel centre. In short, the curvature in mean streamwise velocity profile induces the migration of particles towards the channel centre, and this effect is eventually counterbalanced by the gradient of solid volume fraction in the wall-normal direction.

Figure 14.

Measures of microstructure for case W25P20 and particles located at $0.6 \le y \le 0.8$, averaged during (top) transient and (middle) statistically stationary intervals. Lines compare the statistics averaged over $1\le r/d_p \le 2$. (a) Particle-pair distribution function $q'(r,\psi ,\theta )$ (3.9), (b) mean approaching velocity $\langle {\rm \Delta} u_p^{n,-}(r,\psi ,\theta ) \rangle$ (3.10) and (c) intensity of particles collision measured by $q'\times \langle {\rm \Delta} u_p^{n,-} \rangle$. All quantities are averaged over the polar angle range of $- {{\rm \pi} }/{8} < \psi - {{\rm \pi} }/{2} < {{\rm \pi} }/{8}$.

The results presented herein showed that, while both the presence of particles and viscoelasticity can suppress and even eradicate turbulence at moderate particle concentrations and high elasticity, drag can increase appreciably. Both the particle and polymer stresses become important in these conditions. The large contribution to the polymer stress takes place in the high-shear region of the flow, where the polymers undergo appreciable deformation in the vicinity of the particles, and decays in the channel centre. The particle contribution to the stress shifts away from the wall at higher $Wi$ due to the change in the preferential migration of the particles.