3.1. Particle dynamics

We start by describing the dynamics of the dispersed phase under the effect of lift force models using a simple 1WC approach. This allows us to measure observables sampled by the particle that dictate the validity region of the lift models (e.g. $Sr$, $\varepsilon$ and ${\textit {Re}}_p$). As we will see, current lift models are bound to fail in the buffer layer.

Figure 2 shows the profiles of the normalized mean particle concentration for all particle sizes. When the lift force is not taken into account, the preferential accumulation of particles in the wall region is significant, which has been observed widely (Picano, Sardina & Casciola Reference Picano, Sardina and Casciola2009; Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012; Lee & Lee Reference Lee and Lee2015; Gao et al. Reference Gao, Samtaney and Richter2023; Gualtieri et al. Reference Gualtieri, Battista, Salvadore and Casciola2023), and termed as turbophoresis (Reeks Reference Reeks1983; Johnson, Bassenne & Moin Reference Johnson, Bassenne and Moin2020). However, the particles show less wall accumulation when subjected to lift forces, and this effect is more pronounced the larger the particle size. The Saffman model yields a much smaller peak, while the other two models feature smaller deviations, with the highest peak corresponding to the Mei model. When the particles are very small ($D^+=0.1$), the differences in the lift force models become almost indistinguishable, with all the models robustly showing the same reduction of wall accumulation. Yet the difference in accumulation in the absence of lift is clear.

Figure 2.

Local particle volume fraction $\phi$, normalized by the bulk value $\varPhi$, as a function of the wall-normal distance (1WC): (a) CL1, (b) CM1, (c) CS1. The insets show the normalized particle volume fraction versus the inner-scaled wall distance.

To qualitatively describe the lift force effect on the spatial localization of wall accumulation, figure 3 shows instantaneous snapshots of streamwise velocity fluctuations in a wall-parallel plane within the buffer layer ($y^+=12$) and in a plane of constant streamwise location ($x^+=540$), along with the corresponding particle positions for the case with largest particles, CL1. As expected, the particles show strong spatial localization and inhomogeneous distribution, with larger local density occurring in elongated clusters (Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012). Effects of the lift force model on the gross characteristics of the particle distribution are readily discernible. One notable manifestation is the difference in particle numbers in the wall-parallel ($x$–$z$) plane. Consistent with figure 2, the lift-free model yields the highest number of particles, while the Saffman model yields the lowest. Conversely, the Mei and CBP models fall somewhere between these two. This indicates the weakening accumulation of particles near the wall by the lift force, consistent with observations by Marchioli et al. (Reference Marchioli, Picciotto and Soldati2007) and Shin et al. (Reference Shin, Portela, Schaerer and Mangiavacchi2022). The other manifestation concerns small-scale particle clustering. At first glance, lift-free, Mei and CBP simulations show a more significant tendency to over-sample low-speed regions than the Saffman one. This tendency is checked in figure 4, showing the probability density functions (p.d.f.s) of streamwise fluid velocity fluctuations sampled by the particles within the $y^+ \le 12$ region. For larger particle sizes, particles under the Saffman lift force sample a less focused range of streamwise velocity fluctuations than other models for larger particle sizes. In particular, for $D^+=3$ and $1$, the CBP and Mei cases show approximately the same tendency to sample low-speed regions, differently than the Saffman model and comparable to the lift-free case for $D^+=3$. Preferential accumulation gets more focused with decreasing particle size, with the lift-free case producing the strongest preferential accumulation for $D^+=1$ and $0.1$.

Figure 3.

Inner-scaled instantaneous streamwise velocity fluctuation and the corresponding instantaneous snapshots of particle locations in the $x$–$z$ plane ($y^+=12$, left-hand column) and $y$–$z$ plane ($x^+=540$, right-hand column) (case CL1, 1WC). The wall-parallel plane shows all particles with inner-scaled wall normal position $Y_p^+ \le 12$. (a) Lift-free, (b) Saffman model, (c) Mei model, (d) CBP model.

Figure 4.

Probability density functions (p.d.f.s) of inner-scaled streamwise fluid velocity fluctuations conditioned at the particle location within the $y^+ \le 12$ region (1WC): (a) CL1, (b) CM1, (c) CS1.

Figure 5 shows the inner-scaled profiles of streamwise mean particle velocity $\langle u_p \rangle$ and local slip velocity $\langle u_s \rangle$. Compared with the lift-free particles with $D^+ \ge 1$, lift forces increase $\langle u_p \rangle$ in the viscous sublayer and buffer layer, while the differences between cases become negligible beyond $y^+ \sim 30$. Once more, particles under the Saffman lift force show the highest variation with respect to the lift-free case, yielding the highest $\langle u_p \rangle$; particles under the Mei lift force yield the lowest, and the ones under the CBP lift force fall in between. Consistently, for the smallest particle sizes, the overall modification of $\langle u_p \rangle$ by lift forces is quite small. This tendency of lift forces becoming less important with decreasing particle size is expected – as we will discuss later in more detail (§ 3.3), the relative importance of lift to drag forces close to the wall approximately scales with the inner-scaled particle diameter $D^+$.

Figure 5.

Inner-scaled streamwise mean particle velocity and slip velocity profiles (1WC): (a,d) CL1, (b,e) CM1, (c,f) CS1. The horizontal dashed lines denote $\langle u_s \rangle = 0$.

At the wall, while the fluid velocity must vanish, particles can feature a mean apparent slip where they flow faster than the fluid due to their inertia (Zhao, Marchioli & Andersson Reference Zhao, Marchioli and Andersson2012). This is shown clearly in the particle mean slip (figures 5d–f). At larger wall distances, the tendency of particles to over-sample slower-than-average fluid velocity (Kiger & Pan Reference Kiger and Pan2002; Baker & Coletti Reference Baker and Coletti2021) is also clearly reflected in the mean particle slip velocity. A similar trend can be observed in the corresponding streamwise mean slip velocity profiles; however, the difference in $\langle u_s \rangle$ is larger than $\langle u_p \rangle$, and this is attributed to the negative apparent slip velocity ($\langle u |_{\boldsymbol {x}=\boldsymbol {X}_p} \rangle - \langle u \rangle < 0$), which reflects preferential sampling of slower-than-average fluid. Interestingly, lift forces increase the apparent slip velocity near the wall. This can be understood from the sign of the lift force in this region, which points towards the wall due to the positive slip velocity. Particles sampling higher-momentum regions will be driven towards the wall by the lift force, and their inertia results in a higher mean slip.

The sign of $\langle u_s \rangle$ changes in approximately the same location of the buffer layer, irrespective of the particle size and lift force model. This mechanism seems relatively robust and has been observed in numerous other studies with inertial particles at different Stokes numbers and different flow Reynolds numbers (see e.g. Mortensen et al. Reference Mortensen, Andersson, Gillissen and Boersma2008; Zhao et al. Reference Zhao, Marchioli and Andersson2012; Wang & Richter Reference Wang and Richter2020; Gao et al. Reference Gao, Samtaney and Richter2023). It will be shown that this change in sign makes lift force modelling in wall turbulence extremely challenging (see discussions in § 3.3).

Figure 6 shows the $x$–$y$ component inner-scaled velocity covariances for the particle and fluid phases, respectively. The spanwise velocity variances are hardly affected by lift forces, and thus not shown. As expected, away from the wall where the mean shear is low, the particle statistics closely follow those of the lift-free cases. Near the wall, instead, the lift force enhances the streamwise and wall-normal particle velocity variances ($\langle u'^2_p \rangle$ and $\langle v'^2_p \rangle$) and Reynolds stress ($\langle u'_p v'_p \rangle$), even for the smallest particle size ($D^+=0.1$). Enhancement of wall-normal velocity fluctuations near the wall has indeed been reported in previous numerical and experimental studies (Fong, Amili & Coletti Reference Fong, Amili and Coletti2019; Costa et al. Reference Costa, Brandt and Picano2020a). Indeed, as also shown in Costa et al. (Reference Costa, Brandt and Picano2020a), this enhancement of velocity fluctuations near the wall cannot be reproduced when lift forces are not considered in the particle dynamics. It seems that lift force is very effective at correlating streamwise and wall-normal velocity fluctuations near the wall – under high shear rate, small variations in streamwise velocity naturally translate to changes in vertical acceleration through the lift forces, which in turn induce wall-normal velocity fluctuations. Streamwise velocity fluctuations will also be amplified naturally, since streamwise and wall-normal velocity fluctuations are correlated through the mean shear. As we will see in § 3.4, this amplification of near-wall velocity fluctuations by lift forces has major consequences on turbulent modulation and drag changes in the flow at high particle mass loading.

Figure 6.

Second-order moments of mean particle (solid lines) and fluid (dashed lines) velocity (1WC). (a–c) Inner-scaled streamwise and (d–f) wall-normal velocity variances, and (g–i) Reynolds shear stress profiles, where (a,d,g), (b,e,h) and (c,f,i) denote cases CL1, CM1 and CS1, respectively.

3.2. Mechanism for near-wall accumulation

To understand the role of the lift force in altering the near-wall particle distributions, it is essential to examine three primary mechanisms responsible for wall-normal inertial particle transport. The first is the lift-induced migration, which is the direct consequence of the inertial lift force. The second is turbophoresis (Reeks Reference Reeks1983), causing particle migration from regions of higher to lower turbulence intensity. In wall-bounded flows, turbophoresis results in strong particle accumulation in the viscous sublayer, where turbulent fluctuations vanish. Note, however, that in most cases, the peak in the particle concentration appears not at the wall but at a distance of $O(D)$ from the wall (Marchioli & Soldati Reference Marchioli and Soldati2002; Costa et al. Reference Costa, Brandt and Picano2020a), owing to the wall–particle collisions as well as hydrodynamic wall–particle interactions (Goldman, Cox & Brenner Reference Goldman, Cox and Brenner1967; Vasseur & Cox Reference Vasseur and Cox1977; Zeng, Balachandar & Fischer Reference Zeng, Balachandar and Fischer2005). Finally, there is a biased sampling effect, due to the tendency of particles to sample ejection regions in the buffer layer featuring negative streamwise velocity and positive (i.e. repelling) wall-normal velocity fluctuations (Marchioli & Soldati Reference Marchioli and Soldati2002; Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012). As a result, biased sampling usually provides a net slow particle drift away from the wall. The interaction among the three mechanisms is sketched in figure 7 and will be elaborated below.

Figure 7.

Illustration of the interplay between different mechanisms responsible for the wall-normal particle transport.

Lift-induced migration is directed along $\boldsymbol {\omega }\times \boldsymbol {U}_s$, which in the case of a turbulent wall flow is predominantly in the wall-normal direction. According to figures 5(d–f), particles lead the liquid flow for $y^+\lesssim 10$, and lag behind the flow further away from the wall. Hence the lift-induced migration is towards the wall in the viscous sublayer, and acts cooperatively with the turbophoresis effect. The way these two mechanisms interact in the viscous sublayer may seem surprising at first glance, as figure 2 clearly indicates a suppression of the near-wall particle accumulation by lift force. The reasoning behind this suppression may be understood by noting the following two issues. First, the lift force is towards the channel centre within the buffer layer and beyond $y^+\approx 10$. The resulting outward migration, together with the strong turbulence intensity in the buffer layer, tends to entrain particles in this region outwards, leading to a net decrease in the concentration in the region $y^+\lesssim 10$ as revealed by figure 2. Second, the lift force influences the wall-normal particle transport by turbophoresis. This phenomenon is linked to the particle wall-normal velocity variance $\langle v'^2_p \rangle$ (Reeks Reference Reeks1983; Johnson et al. Reference Johnson, Bassenne and Moin2020). Indeed, in the limit of vanishing particle Reynolds number, the particle wall-normal momentum balance at steady state yields a turbophoresis pseudo-force proportional to $\mathrm {d}\langle v'^2_p \rangle /\mathrm {d}y$, which creates the migration of particles down gradients in particle wall-normal velocity variance (Sikovsky Reference Sikovsky2014; Johnson et al. Reference Johnson, Bassenne and Moin2020). Recall figures 6(d–f), where the variation of $\langle v'^2_p \rangle$ with the wall distance is depicted. Clearly, the lift force tends to increase the ‘apparent inertia’ of the particles, enhancing the deviation between the particle velocity variance and that of the fluid within the viscous sublayer. This leads to an attenuation in the corresponding turbophoresis pseudo-force. In particular, for the two cases where $D^+\geq 1$ (figures 6d,e), the prediction with the Saffman lift model yields $\mathrm {d}\langle v'^2_p \rangle /\mathrm {d}y<0$ in the viscous sublayer, indicating that the turbophoresis pseudo-force is outwards, driving the particles away from the wall. Consequently, the lift force competes with the turbophoresis pseudo-force in these two cases, leading to a highly flattened near-wall peak in the particle fraction, as seen in figures 2(a,b). In contrast, they cooperate in predictions using the Mei and the CBP models, as figure 2(a) shows. There, the lift force compensates the attenuation in turbophoresis, making the negative slopes in $\phi (\kern0.7pt y^+)$ from results using Mei and the CBP models closely follow the lift-free case (highlighted in the figure insets). Note, however, that the magnitude of the Mei lift (CBP lift) is proportional to ${\textit {Re}}_\omega ^{1/2}$ (${\textit {Re}}_\omega$), which approaches $D^+$ (${D^+}^2$) in the viscous sublayer. Consequently, for the two cases where $D^+\leq 1$, the lift-induced migration is weak and incapable of compensating the still-pronounced attenuation in turbophoresis revealed by figure 6(e,f). This is why in these two cases the slopes in $\phi (\kern0.7pt y^+)$ from results employing the Mei and CBP models are smaller in magnitude than in the lift-free case. Still, the difference in accumulation with respect to the lift-free case for smaller sizes is clear, due to near-wall particles residing for long times near the wall while being subjected to small but persistent lift forces.

Given the discussion above, it is worth examining in more detail the link between the chosen lift model and the corresponding modulation in the turbophoresis pseudo-force, particularly to what extent this modulation relates to the lift force magnitude. Combining the turbophoresis pseudo-force (figures 6d–f) and lift force as shown in figure 8, a monotonic increase in the suppression of the turbophoresis pseudo-force with increasing lift is observed at $D^+ \leq 1$, while no such trend is evident at $D^+=3$. In the latter case, the suppression of the turbophoresis pseudo-force is most pronounced in the prediction using the Saffman model, even though the magnitude of the corresponding lift force is only half that of the prediction using the CBP model for $y^+\lesssim 5$. Hence the differences in peak concentration are directly linked not to the lift force magnitudes, but rather to the modulation in the turbophoresis pseudo-force induced by these models. In the present work, the turbophoresis pseudo-force is always suppressed by the presence of lift, leading to a decrease in the peak concentration. Hence the precise relationship between the extent of this suppression and the lift force magnitude remains unclear, making it challenging to directly connect the chosen lift models with the changes in the predicted concentration profiles. Finally, it would be worth investigating these dynamics at lower Stokes numbers, as both turbophoresis and preferential sweeping will be less pronounced, and the nature of the interaction between lift and turbulence may change drastically. Given the increased issues with lift force modelling at lower Stokes number, discussed in the next subsection, addressing this question should be the object of a future, dedicated study.

Figure 8.

The inner-scaled lift-induced acceleration (i.e. normalized by $u^3_{\tau }/\nu$) profiles along the wall-normal direction (1WC): (a) CL1, (b) CM1, (c) CS1. The horizontal dashed lines denote $\langle F_{l,y} \rangle = 0$.

To gain insights into biased sampling of sweeps/ejection events under lift forces, we carried out a quadrant analysis for the fluid flow experienced by the particles. In the $u^{\prime }\unicode{x2013} v^{\prime }$ plane, with positive $v^{\prime }$ directed away from the wall, the flow experienced by the particle is categorized into four types of events: first quadrant events (Q1), characterized by outward motion of high-speed fluid, with $u^{\prime } > 0$ and $v^{\prime } > 0$; second quadrant events (Q2), characterized by outward motion of low-speed fluid, with $u^{\prime } < 0$ and $v^{\prime } > 0$, which are usually called ejections; third quadrant events (Q3), characterized by inward motion of low-speed fluid, with $u^{\prime } < 0$ and $v^{\prime } < 0$; and finally, fourth quadrant events (Q4), which represent motions of high-speed fluid towards the wall, with $u^{\prime } > 0$ and $v^{\prime } < 0$, and are usually called sweeps. Figure 9 shows the joint p.d.f.s in the $u^{\prime }\unicode{x2013} v^{\prime }$ plane of the fluid fluctuation seen by ascending ($v_p >0$) and descending ($v_p <0$) particles within the $y^+ \le 12$ region. The first observation is that a larger proportion of particles tends to move outwards (ascending), irrespective of the presence of lift force. The proportion of ascending particles without lift force decreases as the particle size decreases, whereas particles subject to lift force exhibit an inverse trend. Second, the lift-free ascending particles are highly likely to sample ejections (Q2), while the lift-free descending ones are prone to sample sweeps (Q4) for particles with $D^+ \geq 1$. This agrees with experimental observations in channel flow laden with finite-size particles (Kiger & Pan Reference Kiger and Pan2002; Yu, Vinkovic & Buffat Reference Yu, Vinkovic and Buffat2016; Baker & Coletti Reference Baker and Coletti2021). However, the particles under the effect of lift forces tend not to over-sample sweeps, irrespective of the wall-normal particle velocity sign, which indicates that the wall-normal velocity of the particles is less correlated with the fluid one. This also confirms that inertial particles under the effect of lift forces are not prone to slowly drive towards the wall in low-speed regions, because the competing wall-repelling effect of lift forces seems to dominate their dynamics. In close inspection of the high-intensity region of the joint p.d.f.s on the $u'$-axis, we note that particles under the Saffman lift force tend to sample a wider range of negative $u'$ regions, compared with particles under the Mei and CBP lift forces. This aligns with the p.d.f.s of $u'$ as shown in figure 4. Third, irrespective of the presence of the lift force or wall-normal particle velocity sign, the smallest particles ($D^+=0.1$) over-sample ejection (Q2) and inward motions of low-speed fluid regions (Q3). Sweep events, bringing high-momentum fluid towards the wall, are experienced only weakly by particles. Overall, the percentage of particles sampling Q2 events only slightly surpasses that for Q3, suggesting that biased sampling may play a minor role in driving high-inertia particles ($St^+=50$) to depart from the inner wall region. This observation aligns with numerical results from Marchioli & Soldati (Reference Marchioli and Soldati2002), where biased sampling is found to be more efficient for transferring particles with smaller inertia.

Figure 9.

Joint p.d.f.s of the streamwise and wall-normal fluid fluctuating velocities seen by ascending ($v_p >0$) and descending ($v_p <0$) particles within the $y^+ \le 12$ region (1WC): (a) CL1, (b) CM1, (c) CS1. The numbers in the images denote the proportion of ascending particles.

3.3. Emerging challenges in lift force modelling

The aforementioned results allow us to address several issues concerning the effects of lift force in turbulent wall flows, which may have been overlooked previously in PP-DNS. Ideally, such discussions would be based on results from PR-DNS. However, in this instance, we will extrapolate from PP-DNS results, in conjunction with the CBP model, as an analogue to PR-DNS outcomes. This approach is justified by findings from Costa et al. (Reference Costa, Brandt and Picano2020a,Reference Costa, Brandt and Picanob), which demonstrate that PP-DNS are capable of satisfactorily reproducing the corresponding PR-DNS results for the most challenging case considered in this study, with $D^+=3$.

Let us first address the circumstances under which the effects of lift force could be considered negligible for large inertial particles, and thereby disregarded in PP-DNS. Given that biased sampling does not play a key role in the wall-normal transport of inertial particles, a criterion based on the ratio of lift to the turbophoresis pseudo-force seems sufficient to describe the relative significance of the lift force. In this context, the dimensionless particle size $D^+$ seems a promising candidate, for the following two reasons. First, ${D^+}^2$ approaches the shear Reynolds number ${\textit {Re}}_\omega$ in the viscous sublayer; the latter directly measures the magnitude of the shear lift irrespective of the choice of lift models (Saffman Reference Saffman1965; McLaughlin Reference McLaughlin1991; Mei Reference Mei1992; Shi & Rzehak Reference Shi and Rzehak2019; Costa et al. Reference Costa, Brandt and Picano2020b). Second, the turbophoresis effects seem to decay with increasing $D^+$. This is clearly revealed by comparing the slopes of $\langle v'^2_p \rangle (\kern0.7pt y^+)$ in figures 6(d–f), which correspond to results with increasing $D^+$. Gluing these two aspects together, it seems that the force ratio of lift to turbophoresis approximately scales as ${D^+}^k$ with $k>0$. Although it is not feasible to determine the exact value of $k$, the results summarized in § 3.1 made it clear that by no means should the lift force be neglected for $D^+\geq 1$. On the other hand, it is reasonable for the lift force to be neglected for $D^+\leq 0.1$, except for the under-prediction of near-wall accumulation discussed above (recall figure 2).

We should now address a major issue in commonly applied lift force models for PP-DNS of turbulent wall flows. As outlined in § 2.1, the shear lift models originating from the pioneering work of Saffman (Reference Saffman1965) are applicable rigorously only in the double limits ${\textit {Re}}_p\to 0$ and ${\textit {Re}}_\omega \to 0$. In the viscous sublayer, ${\textit {Re}}_p\approx D^+(\langle u_s\rangle /u_\tau )\approx D^+$ for $D^+\geq 1$ according to figure 5, whereas ${\textit {Re}}_\omega \approx {D^+}^2$. Consequently, neither of the two limits is satisfied for $D^+\geq 1$. In addition to the double limits above, there is a constraint concerning the dimensionless shear rate $Sr$. This is made clear by noting that $Sr={\textit {Re}}_\omega ^{1/2}\,\varepsilon$ and hence $Sr\leq O(1)$ even in the strong shear limit $\varepsilon \geq 1$ considered in Saffman (Reference Saffman1965). This is why in the wall-bounded situation, the ‘outer-region’ lift solutions (Asmolov Reference Asmolov1990; McLaughlin Reference McLaughlin1993; Takemura et al. Reference Takemura, Magnaudet and Dimitrakopoulos2009) following the methodology of Saffman (Reference Saffman1965) match the corresponding ‘inner-region’ solutions (Cherukat & McLaughlin Reference Cherukat and McLaughlin1994; Magnaudet, Takagi & Legendre Reference Magnaudet, Takagi and Legendre2003; Shi & Rzehak Reference Shi and Rzehak2020) only for $Sr=O(1)$. For particle-laden channel flows with small inertial particles (Mortensen et al. Reference Mortensen, Andersson, Gillissen and Boersma2008; Zhao et al. Reference Zhao, Marchioli and Andersson2012; Wang & Richter Reference Wang and Richter2020; Gao et al. Reference Gao, Samtaney and Richter2023), this constraint is more seriously violated than that for the double limits above, as the slip velocity reverses at $y^+\approx O(10)$ (see e.g. figures 5(d–f) in the present work), making $Sr\to \infty$ for particles within the buffer layer. This issue is highlighted in figure 10, where we show the p.d.f.s of the inverse of $Sr$ (horizontal axis) as functions of the wall-normal distance $y^+$ (vertical axis). Apparently, a large portion of particles experience a strong shear where $Sr\geq 1$, in particular for the two cases where $D^+\geq 1$. Finally, it is worth noting that the probability of a particle experiencing very high values of $Sr$ near the wall will increase with decreasing values of $St^+$ as, for the same local shear rate, particles will experience an ever smaller slip velocity. This exacerbates the violation of the $Sr=O(1)$ constraint, meaning that current lift force models are even less suitable for turbulent channel flow laden with particles at lower Stokes numbers.

Figure 10.

The p.d.f.s of $u_s/(\omega _z D)$ along the wall-normal direction (1WC DNS with the CBP model): (a) CL1, (b) CM1, (c) CS1. The vertical dashed lines denote $u_s/(\omega _z D) = \pm 1$.

Additionally, the discussion above focuses on the lift force arising from the shear in the ambient flow. In wall-bounded flows, as considered in this work, the lift force may deviate significantly from its unbounded counterpart if the particle is close to the wall. In brief, the presence of the wall leads to a repulsive transverse force in the absence of the ambient shear (Vasseur & Cox Reference Vasseur and Cox1977; Zeng et al. Reference Zeng, Balachandar and Fischer2005), and in the presence of an ambient shear, a suppression of the shear-induced lift (McLaughlin Reference McLaughlin1993; Magnaudet et al. Reference Magnaudet, Takagi and Legendre2003; Takemura et al. Reference Takemura, Magnaudet and Dimitrakopoulos2009). Although analytical solutions for the combined lift force are achievable in some asymptotic limits (see e.g. Wang et al. (Reference Wang, Squires, Chen and McLaughlin1997) and, more recently, Shi & Rzehak (Reference Shi and Rzehak2020) for an overview of these solutions), there is so far no satisfactory way to glue these solutions together to achieve a general treatment of the wall effect (Ekanayake et al. Reference Ekanayake, Berry, Stickland, Dunstan, Muir, Dower and Harvie2020; Ekanayake, Berry & Harvie Reference Ekanayake, Berry and Harvie2021). Here, instead of trying to make any further progress in achieving such a general solution, it is more feasible to discuss under which condition this intricate wall effect may be neglected. Within the $O(D)$-depth layer close to the wall, wall–particle collisions govern the wall-normal position of the peak particle concentration. Hence the hydrodynamic wall–particle interaction may play a significant role only beyond $y^+\approx D^+$. On the other hand, it is known that (McLaughlin Reference McLaughlin1993) the amount of reduction of the shear-induced lift due to the wall decreases as $(\ell /\ell _\omega )^{-5/3}$, and reduces to only $20\,\%$ of its magnitude in the limit $\ell /\ell _\omega \to 0$ at $\ell /\ell _\omega \approx 3$. Hence the wall effect may be considered negligible if the position of the peak particle concentration, which is of $O(D)$ from the wall, is larger than approximately $3\ell _\omega$, which yields approximately $D^+\geq 3$. This finding, together with the previous one where (shear-induced) lift force is non-negligible for $D^+\geq 1$, indicates that such a wall effect can be disregarded except in the overlap regime where $1\leq D^+ \leq 3$.

In addition to the ambient shear, particle rotation is known to produce a lift force as well (Rubinow & Keller Reference Rubinow and Keller1961). This lift contribution is often neglected in prior PP-DNS and not considered in the present work. Nevertheless, it might be worth discussing its relevant importance in the following. Assuming that the particle inertia is small enough for the particle to always stay in the torque-free state, particle rotation leads to a lift contribution $C_L^\varOmega =0.5\,Sr$ in the double limits considered by Saffman (Reference Saffman1965). The ratio between this spin-induced lift contribution and that induced by ambient shear is proportional to ${\textit {Re}}_\omega ^{1/2}$, indicating that $C_L^\varOmega$ is merely a component of the second-order lift contribution from the ambient shear. This is confirmed by the recent work of Candelier et al. (Reference Candelier, Mehaddi, Mehlig and Magnaudet2023), where second-order lift contributions (with respect to ${\textit {Re}}_\omega ^{1/2}$) were obtained using matched asymptotic expansions. In addition to the spin-induced contribution $C_L^\varOmega =0.5Sr$, they also obtained a second-order contribution from the ambient shear, $C_L^{'(2)}=-0.505\,Sr$, which counterbalances the spin-induced contribution. In other words, the effect of particle rotation on the lift is negligible in the torque-free state. This conclusion is valid irrespective of the presence of the wall (Cherukat & McLaughlin Reference Cherukat and McLaughlin1994; Magnaudet et al. Reference Magnaudet, Takagi and Legendre2003; Shi & Rzehak Reference Shi and Rzehak2020). However, large-inertia particles with a large density ratio as considered in this work might not always stay torque-free. Specifically, for particles with density ratio $\rho _p/\rho \geq 100$, the relaxation time in response to torque is comparable with that to slip (see e.g. Bagchi & Balachandar Reference Bagchi and Balachandar2002). This finding, together with the pronounced slip in the viscous sublayer revealed by figures 5(a,b), indicates that particles with $D^+\geq 1$ may experience a strong angular acceleration and possibly a non-negligible lift contribution from forced rotation. Investigating the effects of rotation by accounting for the particles’ conservation of angular momentum would therefore be interesting but is outside the scope of the present work due to the higher complexity associated with drag and lift closures for angular momentum.

Finally, it is worth noting that while the CBP lift force model used in Costa et al. (Reference Costa, Brandt and Picano2020a,Reference Costa, Brandt and Picanob) appears to provide reasonable predictions in some cases, it is not valid universally. Specifically, it does not align with any of the inner-region solutions (Shi & Rzehak Reference Shi and Rzehak2020), which are applicable for particles extremely close to the wall (e.g. within the viscous sublayer), nor does it approach the unbounded solution (Saffman Reference Saffman1965; McLaughlin Reference McLaughlin1991) for particles located within the outer layer of the channel flow. The agreement between the PP-DNS utilizing this lift force model and PR-DNS suggests that the model offers a satisfactory blending of results in the two limits for the specific cases explored in Costa et al. (Reference Costa, Brandt and Picano2020a) (see table 1). A preliminary attempt to establish a more general lift force model can be found in the work of Ekanayake et al. (Reference Ekanayake, Berry, Stickland, Dunstan, Muir, Dower and Harvie2020, Reference Ekanayake, Berry and Harvie2021), where a lift model satisfying the above conditions was proposed in the limit of small shear Reynolds numbers, i.e. ${\textit {Re}}_\omega \ll 1$. However, the applicability of this model is limited to $D^+\ll 1$ because in the viscous layer, ${\textit {Re}}_\omega \approx (u_\tau ^2/\nu ) D^2/\nu =(D^+)^2$. While this provides some insights, its practicality may be limited since the lift force no longer plays a pivotal role for $D^+<1$, as demonstrated here. Hence further work extending Ekanayake et al. (Reference Ekanayake, Berry, Stickland, Dunstan, Muir, Dower and Harvie2020, Reference Ekanayake, Berry and Harvie2021) to finite ${\textit {Re}}_\omega$ (hence $D^+$) would be significant.

3.4. Lift-induced turbulence modulation

Finally in this section, we use 2WC point-particle DNS to illustrate that lift force models can have tremendous consequences in basic integral quantities such as the overall drag. To achieve this, we use the same parameters as in table 1, which fixes the total mass fraction and particle Stokes number while varying the particle diameter. We should reiterate that high-fidelity 2WC point-particle DNS are highly challenging and are currently being investigated actively. Still, the simple particle-in-cell 2WC DNS employed here suffice to illustrate the impact of lift force model choice on important turbulence metrics. In what follows, we illustrate the effects of lift force on the overall drag, and analyse the streamwise momentum fluxes that contribute to it.

Figure 11 shows the friction coefficient for the 2WC DNS, defined as $C_f=2 u^2_{\tau }/U^2_b$. While all small-inertia particles have a drag-increasing effect, the difference between lift force models is highly amplified as the particle size increases. At $D^+=3$, in particular, particles under the Saffman lift force show a drag reduction compared to the other particle sizes, almost reaching an overall drag decrease. Instead, the other lift models show a monotonic drag increase.

Figure 11.

The wall friction coefficient $C_f$ versus the particle diameter $D^+$ (2WC). The horizontal dashed line denotes the mean $C_f$ of the particle-free case.

To better understand and quantify this difference, we follow the analysis in Costa et al. (Reference Costa, Brandt and Picano2021) and investigate the stress budget in the 2WC limit of vanishing volume fraction, while keeping the mass fraction finite:

(3.1)\begin{equation} \tau = \rho u_\tau^2\left(1-\frac{y}{h}\right) \approx \underbrace{\left(\mu\,\frac{\mathrm{d}\langle u\rangle}{\mathrm{d}y} - \rho\langle u^\prime v^\prime\rangle\right)}_{\tau_\nu + \tau_{T,f}} - \underbrace{\langle \psi\rangle \langle u_p^\prime v_p^\prime\rangle}_{\tau_{T,p}}, \end{equation}

where $\langle \psi \rangle = \rho _p\langle \phi \rangle$ is the mean local mass fraction. The terms on the right-hand-side of this equation are the viscous stress, fluid Reynolds stress and particle Reynolds stress (hereafter $\tau _\nu$, $\tau _{T,f}$ and $\tau _{T,p}$). This equation boils down to the stress budget in the single-phase limit ($\tau _\nu$ and $\tau _{T,f}$), corrected by an additional turbulent particle momentum flux $\tau _{T,p}$ that involves the local mass fraction and the particle counterpart of the Reynolds shear stress. This term relates to the transfer of streamwise momentum by correlated particle velocity fluctuations, and shows the importance of high particle mass fraction near the wall when combined with correlated streamwise–wall-normal particle velocity fluctuations. We have seen in § 3.1 that while the former decreases in the presence of lift force, the latter may be increased by it.

The profiles of stress terms and the corresponding contributions of each momentum transfer mechanism to the overall friction coefficient, normalized by the overall friction coefficient of the particle-free case, are shown in figure 12. In the left-hand plots, the normalized shear stress profiles are shown, while on the right-hand side we show the so-called FIK (Fukagata–Iwamoto–Kasagi) identity employed to a particle suspension (Fukagata, Iwamoto & Kasagi Reference Fukagata, Iwamoto and Kasagi2002; Yu et al. Reference Yu, Xia, Guo and Lin2021). In the case of a turbulent plane channel flow, this identity decomposes the mean friction coefficient ($C_f$) into laminar and turbulent contributions. Here, the latter can be further decomposed into a contribution due to the fluid phase and one due to the particles. Each contribution is given by the weighted integral $C_{f,i} = (1/h^2)\int _0^h 6 (h-y) \tau _{i}\,\mathrm {d} y/(\rho U_b^2)$, with $\tau _i$ being one of the stress mechanisms described above. Note that the residual of this stress budget is virtually zero, which validates the approximation in (3.1).

Figure 12.

(a,c,e) Budget of streamwise momentum and (b,d,f) contribution of each stress contribution $\tau _i$ to the mean wall friction coefficient $C_f=2\tau _w/(\rho U_b^2)$, i.e. $(1/h^2)\int _0^h 6 (h-y) \tau _i \,\mathrm {d}y/(\rho U_b^2)$ normalized by that of the particle-free case: (a,b) CL1, (c,d) CM1, (e,f) CS1. Different line colours in (a,c,e) denote different cases, which match the legend in (c).

As anticipated from figure 11, all 2WC DNS show increased drag compared to the unladen flow. Yet, remarkably, different lift models show qualitatively different turbulence modulation for the largest particle size ($D^+=3$). Particles under the Saffman lift force have a major net turbulence attenuation effect, which results in an almost drag-reducing flow with respect to the unladen case. Conversely, particles under the Mei and CBP lift forces have a net drag increase, caused mostly by an increase in the 2WC stress term $\psi \langle u'_pv_p'\rangle$. This results from the amplification of particle Reynolds shear stresses $\langle u'_pv'_p\rangle$ discussed in § 3.1, which compensate for the decrease in local mass fraction $\psi$. Naturally, as the particle size decreases, the differences between lift models become less pronounced but still noticeable at all particle sizes, with particles under the Saffman lift force consistently showing the highest fluid turbulence attenuation. For the smallest particle size $D^+=0.1$, the two turbulent stress (fluid and particle) terms still show noticeable but in practice negligible differences (recall figure 11).

Hence it appears that the most important dynamical effects of lift forces in 2WC conditions can be seen as finite-size effects that directly modify single-point moments of particle velocity for $D^+\gtrsim 1$ (recall figure 6). However, the lift force still affects near-wall turbulence modulation when particles are relatively small, as they strongly affect the mean particle concentration (recall figure 2), directly linked to turbulence modulation via the 2WC stress term.