3.1. Statistical moments of particle phase

Instantaneous particle velocity is decomposed into mean and fluctuating components, e.g. $u_{px} = U_{px} + u_{px}^{\prime }$, and the mean and r.m.s. of the fluctuating particle velocities are shown in figure 3 for Set 1 (see table 2). Here $y = R - r$, where $R$ is the pipe radius. The data are temporally averaged over $\Delta t^+ = 3600$ (which is equal to 1190 integral time units, i.e. ratio of pipe radius to bulk velocity) from the beginning of particle injection. As expected, the $St^+ = 0.1$ particles behaving like tracers do not show a shift from the fluid in their mean velocity nor r.m.s. profiles. Note that fluid statistics are shown by solid grey lines. As $St^+$ increases, deviation between the streamwise velocity of the particle ($U_{px}^{+}$) and the fluid ($U_{x}^{+}$) is minor, except for $St^+ = 100$ particles, where the $U_{px}^{+}$ profile is significantly flatter. This is more clearly observed in the inset of figure 3(a), where the difference between the particle and fluid velocity $\Delta U^{+} = U_{px}^{+} - U^{+}$ is plotted. This behaviour has also been observed by Mortimer, Njobuenwu & Fairweather (Reference Mortimer, Njobuenwu and Fairweather2019) previously in channel flows. Although the precise details of this mechanism remains unknown, $St^+ = 100$ particles lag the fluid in the bulk flow but retain their inertia as they approach the near-wall region and, consequently, have higher-than-fluid streamwise velocity for $y^{+} \lessapprox 10$.

Figure 3. Turbulent statistics for simulations in Set 1 (see table 2 for symbols). Mean particle velocities in the (a) streamwise and (b) radial directions; r.m.s. of particle velocity fluctuations in the (c) streamwise, (d) radial directions and (e) the particle Reynolds shear stresses. Note that fluid statistics are shown by solid grey lines. Also plotted in (d) with the same symbols as in table 2 but grey-coloured is analytical prediction of the r.m.s. of radial particle velocity fluctuation, obtained from the DNS determined eddy viscosity and r.m.s. profile for the fluid via (4.7) and (4.8a,b) from § 4.2.

The particle mean radial velocity ($U_{pr}^{+}$) profiles show a peak at approximately $y^+ = 36$, indicating that particle drift towards walls is a feature of wall turbulence (recall that radial velocity is positive when directed towards the wall and $U_r^{+}$ for the fluid is zero). When particle–wall collisions are modelled as elastic, there are competing effects contributing to the $U_{pr}^{+}$; and some of them are, particles bouncing against the wall, particle sluggishness and preferential sampling of fluid events by the particles. The $St^+ = 100$ particles are qualitatively observed to collide with the wall and flee the near-wall region, re-entraining into the pipe turbulent core, whereas $St^+ = 10$ particles that begin near the wall and are wall-fleeing often show high curvatures in their trajectory of motion in the buffer region and return to the viscous sublayer.

The results of an international benchmark effort in turbulent channels presented in Marchioli et al. (Reference Marchioli, Soldati, Kuerten, Arcen, Taniere, Goldensoph, Squires, Cargnelutti and Portela2008) reveal considerable scatter between different numerical schemes in the prediction of the r.m.s. velocity fluctuations of particles. Results in their benchmark effort however consistently found, compared with the fluid, an increase in the peak for streamwise particle velocity fluctuations for particle populations $St^+ = 5$ and $25$, which is similar to figure 3(c) for pipes. These velocity fluctuations near the wall are significantly enhanced for the largest particles as they are able to retain their velocity from the buffer region while entering the viscous sublayer. Furthermore, consistent with Mortimer et al. (Reference Mortimer, Njobuenwu and Fairweather2019), figure 3(c) shows that at the wall particle velocity fluctuations do not go to zero for large $St^+$ values. However, we recall a limitation in our PP-DNS study: particle radius effects are not incorporated into the particle–wall collision model. The particle-tracking algorithm follows a particle until cell face crossing occurs and in the case of a boundary, applies the particle–wall boundary condition, (2.6) and (2.7). Having recognised this limitation, the largest particles in core simulation sets of this study are $d_p^{+} = 1.341$, and so $r_p^{+} = 0.67$ would be the collision radius but the high particle r.m.s. profiles persist until $y^{+} = 10$. In addition, in Appendix B, figure 18 reports that the near-wall profiles are characterised by the $St^{+}$ number as this finding persists with higher-density $\rho _p / \rho _f= 10\ 000$ particles.

The $u_{px,rms}^{\prime +}$ profiles in the outer region approximately collapse for all particle populations. In figure 3(d) the $u_{pr,rms}^{\prime +}$ profiles in the radial direction monotonically decrease in the outer region with increasing $St^+$ number, and all except the largest $St^+$ particles collapse onto the fluid profile very close to the wall. Turning attention to the particle Reynolds shear stresses $\overline {u_{px}^{\prime }u_{pr}^{\prime }}^{+}$ in figure 3(e), we observe a slight gain in the peak of the profile for $St^{+} = 0.1,1$ and $10$ particles and then collapse onto the fluid Reynolds shear stress away from the wall. The situation is different for $St^{+} = 100$ particles, where there is a slight gain close to the wall but thereafter, dampening in the profile compared with fluid. This is highly likely to be driven by the substantial decrease in the radial velocity fluctuations observed in figure 3(d).

The profiles presented in figure 3 for Set 1 are supplemented by comparison with the corresponding profiles for the remaining simulation sets in Appendix B, figures 18 and 19. Inclusion of the Saffman–Mei lift force for the elastic simulations yields similar statistics ($U_{px}^{+},u_{px,rms}^{\prime +},u_{pr,rms}^{\prime +}$) for particles with $St^+ = 0.1,1 \ \textrm {and} \ 10$ but $St^+ = 100$ particles show a striking difference in their near-wall quantities. This is most likely because the Saffman–Mei lift force for the latter plays a crucial role by assisting in wall-fleeing, and providing one mechanism for lowering the residence time of the particles in the viscous sublayer. Note that the ‘wall-fleeing’ effect of larger $St^+$ particles is due to the lift force (see (2.4)) where the wall-normal shear and lower particle velocity compared with fluid constitute a lift force away from the wall. Comparison between the elastic and absorbing wall simulation sets (cf. figures 18 and 19) shows increased $U_{pr}^{+}$ for absorbing walls because of the consequent lower wall-fleeing. Again absorbing walls highlight the distinctive $St^+ = 100$ particles, and classifies these particles as ballistic rather than merely deviating from flow behaviour. The $St^+ = 100$ particles also show the most significant deviation for $U_{pr}^{+}$ profiles between simulation sets, which suggests the importance of the wall and Saffman–Mei lift force for heavier or larger particles in future modelling efforts. The peculiar characteristics of $St^+=100$ particles will become apparent later when we consider modelling aspects.

Furthermore, we observe the lift force gains importance with increasing $St^{+}$ in the near wall-region, and approaches in the mean sense $20\,\%$ of the drag force magnitude close to the wall for the largest particles studied. In Appendix B, we further quantify the relative importance of the lift force by comparing its magnitude to that of the drag force as a function of distance from the wall. The role of the lift force is not immediately obvious, as typically in wall flows it aids particles to migrate away from walls but Vreman (Reference Vreman2007) found inclusion of the lift force causes the near-wall particle concentration to increase by a factor $1.5$ or $2$, dependent on the particle size and overall particle loading, which is also transiently observed in this paper prior to strong concentration gradients being established (see discussion towards the end of Appendix B).

3.2. Time-evolving concentration and spatial distribution profile

The time-varying number fraction of inertial particles ($N_p / N_t$, where $N_p$ is the instantaneous number of particles residing in a region and $N_t$ is the total number of particles in the simulation) that reside within each of the turbulent regions, viscous sublayer, buffer, logarithmic and core, is presented in figure 4. This region-by-region analysis has been previously employed by Picano et al. (Reference Picano, Sardina and Casciola2009) where particles were modelled by drag force $\boldsymbol {F}_d$ alone and released on the centreline; their results for ${St^+ = 10}$ are also plotted in figure 4 by solid (grey) lines. In addition, reference is made to cross-sectional slices of instantaneous particle distributions in figure 5 plotted over pseudo-colours of fluid streamwise vorticity. Profiles of particle wall-normal number fraction and spatial preferential concentrations are jointly presented in this section as they are intimately connected (Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012); e.g. biased sampling of fluid events in turn leads to ejection mechanisms near the wall that control particle transfer. Figure 5 highlights the preferential concentration in particle-laden flows (e.g. Squires & Eaton Reference Squires and Eaton1991) as well as the diffusional deposition that preferentially occurs in streamwise oriented streaks (e.g. Narayanan et al.'s (Reference Narayanan, Lakehal, Botto and Soldati2003) observations in channel flows).

Figure 4. Time-evolving averaged local particle number ($N_p/N_t$) in regions: $(\textit {a})$ viscous sublayer, $(\textit {b})$ buffer, $(\textit {c})$ logarithmic and $(\textit {d})$ core. Solid grey line from Picano et al. (Reference Picano, Sardina and Casciola2009) for $St^+=10$, which injected particles on the centreline. See table 2 for symbols. Green lines are plotted to represent expected fraction in the case of uniform dispersion.

Figure 5. Streamwise slices of thickness $\approx R/12$ for instantaneous particle positions for (a) $t^+ = 36$, (b) $t^+ = 900$, (c) $t^+ = 1800$, (d) $t^+ = 2700$ and (e) $t^+ = 3600$ coloured by streamwise vorticity $\omega _x$. Results from simulation Set 1.

The key feature that we see in figures 4 and 5 for $St^+ = 0.1$ particles is very low levels of accumulation in the viscous sublayer with time. Even after $\approx 300$ integral time units ($\equiv U_b t / R$), wall-normal region-by-region particle number fraction shows only minor deviation due to accumulation in the viscous sublayer. These particles are able to trace flow structures of all time scales (see figure 5), thus remaining evenly distributed throughout the domain, both in terms of wall-normal distance and their local organisation with respect to flow structures.

Particles with $St^+ = 1$ accumulate at a moderate rate in the viscous sublayer suggesting that deposition cannot be neglected for this population. Accumulation of $St^+ = 1$ particles did not appreciably vary between simulation Sets 1–4 indicating that deposition modelling for this species does not require inclusion of $\boldsymbol {F}_l$, the lift force and particle–wall interactions did not have any significant effect. These particles are unable to trace high-wavenumber fluid motions and accumulate outside regions of high vorticity magnitude which can be observed in figure 5; particle–turbulence interactions highlighted by preferential concentration can be observed in all regions of the cross-section.

The situation is more complex for larger particles. For simulations in Set 1, $St^+ = 10$ and $St^+ = 100$ particles both wall-accumulate at a rate that is similar but orders of magnitude greater than the $St^+ = 0.1$ and $St^+ = 1$ species. To delineate the effects of $\boldsymbol {F}_l$ and wall-boundary conditions separately, let us first consider the absorbing wall case (so that particle ejection from the viscous sublayer effects are absent). For $St^+=10$ with absorbing walls, figure 4(a) shows an increased $N_p / N_t$ with $\boldsymbol {F}_l$ included (back diamonds) than without $\boldsymbol {F}_l$ (green diamonds). Note that $\boldsymbol {F}_l$ direction is towards (or away from) the wall when particle velocity is higher (or lower) than the fluid velocity. As the inertial $St^+=10$ particles approach the wall they retain their higher velocity and, hence, experience a wall-ward lift force increasing the $N_p / N_t$. On the other hand, $N_p / N_t$ distribution of $St^+=100$ does not seem to be significantly affected by $\boldsymbol {F}_l$. For elastic walls, in general, $N_p / N_t$ is lower because of particle rebound. Interestingly, for $St^+=10$ particles with elastic walls, although initially $\boldsymbol {F}_l$ helps to increase $N_p / N_t$ (similar to the absorbing wall case), as the time increases particles show longer residence times in the wall region. Next, they may be ejected by fluid motions or rebound against the wall or their now lower-than-fluid velocity directs $\boldsymbol {F}_l$ away from the wall. Therefore, over long time intervals particles with $\boldsymbol {F}_l$ included have a reduced $N_p / N_t$. This effect is accentuated for $St^+=100$ particles with elastic walls (compare red and blue pentagram symbols in figure 4(a).

Increased (decreased) $N_p / N_t$ within the near-wall region implies a decreased (increased) number fraction away from the wall. This is reflected for the buffer region shown in figure 4(b) especially in the case of $St^{+} = 100$ particles where the number fraction even increases relative to the initial distribution. This is consistent with the inertia-moderated regime described by Guha (Reference Guha1997) (see also figure 1), where deposition velocity decreases slightly with increasing $St^+$. Simulations in Set 3 reveal a distinctive phenomena; rather than wall-accumulate, particles converge to a steady state. Convergence is faster in $St^+ = 100$ particles and an order of magnitude decrease in viscous sublayer concentration compared with results from Set 1 suggests the importance of $\boldsymbol {F}_l$. Spatially, $St^+ = 10$ particles show major deviation from randomness, accumulating outside regions of intense vorticity (see again figure 5). However, $St^+ = 100$ particles with their significant inertia appear ballistic in nature, as they filter fluid events and remain randomly distributed. Note, it should be clear that $St^+ = 1$ and $St^+ = 100$ particles both remain randomly distributed but for totally different reasons.

A key finding in this section is the identification of relative importance of the lift force $\boldsymbol {F}_l$ and particle–wall interaction type for non-dimensional $St^+$ number. The effect of $\boldsymbol {F}_l$ is significant for $St^+ = 10$ particles and dramatic for $St^+ = 100$ particles when particle–wall interaction is elastic. The effect of $\boldsymbol {F}_l$ is less important when particle–wall interaction is absorbing type, and is unimportant for $St^+ = 0.1$ and $St^+ = 1$ particles. As such, figure 4 is useful in decision-making on modelling of particle dynamics and particle–wall interactions. In addition, in Appendix B, we investigate the relevance of density ratio $\rho _p/\rho _f$ and whether $St^{+}$ number alone is sufficient to characterise particle wall-accumulation by comparing velocity and particle concentration for Sets 3 and 5. There, it is reported that although the velocity statistics are similar, the density ratio may influence the ratio of Saffman–Mei lift force to drag force acting on the particle (because increased $\rho _p / \rho _f$ for some $St^{+}$ implies reduced $d_p$ and, hence, reduced $F_l$), and this in turns modifies the particle wall-normal concentration.

3.3. Deposition velocity

Figures 6(a), 6(b), 6(c) and 6(d) corresponding to $St^+=0.1,1,10$ and $100$ show the time series of $V_{dep}^+$ revealing varying levels of agreement between Sets $1$–$4$ for different Stokes numbers. Simulations with absorbing walls (green and black symbols) quickly attain a steady deposition rate. The slight exception is the $St^+ = 10$ particles in Set 2, where $V_{dep}^+$ gradually increases with time before becoming approximately constant. The presence of elastic walls results in either a gradual increase in the deposition velocity when only the drag force $\boldsymbol {F}_d$ is included, or with the inclusion of the Saffman–Mei lift force $\boldsymbol {F}_l$, a fast equilibrium concentration is achieved by $t^+ \approx 600$ and, thereafter, fluctuations occur in particles departing/re-accumulating in the near-wall region.

Figure 6. Deposition velocity time series for Sets 1–4: (a) $St^+ = 0.1$; (b) $St^+ = 1$; (c) $St^+ = 10$; and (d) $St^+ = 100$. Note that results from simulation Set 3, in which particles each time-interval may resuspend into the flow after wall-accumulating yielding ‘negative’ deposition velocity are marked by absolute valued filled circles. For symbols, see table 2.

These evolving $V_{dep}^+$ profiles in Sets 1 and 3 that feature elastically rebounding walls can be attributed to evolving wall-normal concentration gradients. As particle number fraction increases in the viscous sublayer (reported in § 3.2), there is a growing role of ejection mechanisms by vortical motions (e.g. Marchioli, Picciotto & Soldati Reference Marchioli, Picciotto and Soldati2003). In addition, the importance of the lift force $\boldsymbol {F}_l$ becomes clear as this additional mechanism assists particles to flee the viscous sublayer. In particular, for rebounding walls, time-evolving deposition velocity changes by several orders of magnitude for $St^+ = 10$ and $100$ particles when lift force $\boldsymbol {F}_l$ is included. The findings in this section indicate that the analytical modelling that prominently features absorbing walls and steady deposition rates (Guha Reference Guha1997; Young & Leeming Reference Young and Leeming1997) is on good footing, but the situation involving rebounding walls is more complex, where near-wall fleeing mechanisms are of vital importance.

The large variation in $V_{dep}^+$ for changing $St^+$ can be better illustrated by taking the average of $V_{dep}^+$ across time. These average values of $V_{dep}^+$ as a function of $St^+$ is shown in figure 1 for different Sets. The symbols denote an averaging time from $t^+=0$ to $720$, which corresponds to a time based on bulk flow, $t U_b ( = R \,(t U_\tau ^2/\nu ) (U_b R/\nu )(\nu ^2/(U_\tau ^2 R^2))=R t^+ Re_b/Re_{\tau }^2 \approx 0.165 R t^+) \approx 119 R$ that is based on our case where $Re_b=5357$ and ${Re_\tau = 180}$. These distances are not unusual for typical experiments; for example, the length of deposition pipe in Liu & Agarwal (Reference Liu and Agarwal1974) is $\approx 160R$. The slight right-shifted symbols in figure 1 correspond to long average up to $t^+=3600$. It is clear that the DNS results for the absorbing wall with one or two particle forces (Sets $2$ and $4$) are in closest agreement with the experimental data of Liu & Agarwal (Reference Liu and Agarwal1974). As discussed in the previous paragraph, reflecting walls show a slightly different behaviour especially for the large $St^+$ cases where particle bouncing or ejection from the wall is common, thus leading to a reduced $V_{dep}^+$.

Recall that deposition velocity is simply the normalised particle flux $J$ at the wall. Knowledge of $J$ as function of $r$, on the other hand, would not only enable us to estimate concentration $c$ at the wall (i.e. $V_{dep}^+$) but also the full distribution of $c$ with $r$ as well as its variation in time. In the next section we address several aspects of this modelling that will provide further insight into particle migration and later also allow us to estimate $c(r;t)$.

4.1. Determining turbophoretic and diffusive coefficients with the DNS database

To determine the turbophoretic and diffusive coefficients $D_1$ and $D_2$, respectively, the particle fluxes are measured across imagined cylindrical surfaces in the domain. The radii of these cylinders vary from a minimum of $r^{+} = 4$ to a maximum of $r^{+} = 178$ with an increment of $\Delta r^{+} = 6$. The expression for particle flux is given by

(4.3)\begin{equation} J = \frac{n_p}{A\Delta t}, \end{equation}

where $n_p$ is the number of particles that crosses the surface, $A$ is the area of the cylindrical surface and $\Delta t$ is the time interval over which subsequent snapshots of particle interval are recorded. Note that $J$ is positive in the positive $r$ direction, i.e. when particles migrate towards the wall. Here the concentration $c(r)$ is evaluated by $c(r) = N_{p,l} / ({\rm \pi} L_z (r_{o}^2 - r_{i}^2))$ where $N_{p,l}$ is the local number of particles in a thin annulus with length, $L_z$ (the pipe length), with outer and inner radius $r_o$ and $r_i$, the thickness of which approach each other in the limit. The concentration is evaluated at twice as many points as the flux is measured, for a typical calculation, $r_o^+ - r_i^+ \approx 3$, and the concentration gradient is obtained through finite differencing. To obtain $D_1$ and $D_2$, (1.2) is arranged, such that

(4.4)\begin{equation} \frac{J}{c(r)} = D_1(r) - D_2(r) \frac{1}{c(r)} \frac{\partial c(r)}{\partial r}.\end{equation}

Determination of $D_1$ and $D_2$ follows a statistical approach through the simultaneous observations of particle flux $J(r)$, concentration $c$ and concentration slope $\partial c/\partial r$. To evaluate $J$, the radial positions of individual particles are recorded at sequential snapshots in time. Particles whose linearly interpolated trajectory cross a surface are determined as either wall-directed or wall-fleeing, and a net particle number flux is determined. Now that we know $J$, $c$ and $\partial c / \partial r$, we linearly regress (4.4) through ensembles of the known terms to determine the two remaining unknowns $D_1$ and $D_2$. One thousand equally spaced instances are collected spanning $3600$ viscous time units; measured instances are graphed, and a line of regression drawn.

Figure 7 left and right panels respectively show $D_1$ and $D_2$ (with symbols as in table 2). In general, value of $D_1$ increases with increasing $St^+$, which is consistent with the increased role of turbophoresis in particle deposition with increasing $St^{+}$. The values of $D_2$ on the other hand show only mild dependency on $St^{+}$, again consistent with our notion of turbulent diffusive action. Before further discussion, we report some analytical estimates for $D_1$ and $D_2$ from the properties of the underlying turbulent fluid. Subsequently, we compare the analytical estimates with DNS in figure 7.

Figure 7. Empirically determined coefficients $D_1$ (left panel) and $D_2$ (right panel), corresponding respectively to turbophoretic drift and gradient-driven diffusivity. (a,c,e,g) Solid grey lines, $\widetilde {D_1}$ from the analytical study of Guha (Reference Guha1997), i.e. (4.9); blue solid and dashed lines, (4.6) (with ${u}^{\prime 2}_{pr, rms}$ from DNS) and (4.6) using (4.7) to estimate ${u}^{\prime 2}_{pr, rms}$ (with $u_{r, rms}^{\prime 2}$ from DNS). (b,d, f,h) Solid grey lines, $\tilde {D}_2/10$ from Guha's (Reference Guha1997) estimates, where $\widetilde {D_2} = \nu _t$; dash-dotted grey lines, $\tilde {D}_2/10$ from (4.10) with the $\nu _t$ expression from Guha (Reference Guha1997); and dashed lines, $\tilde {D}_2/10$ from (4.10) with $\nu _t$ from present DNS. All grey lines are divided by a numerical value of $10$ to be shown on the same vertical axis limit. Blue lines indicate $\tilde {D}_2$ from (4.11), whereas dark/light blue are where $f(St)$ is taken for elastic/absorbing walls. Inset included for $St^{+} = 0.1$ and $1$ to aide judgement of agreement between the models and present data.

4.2. Analytical estimation of turbophoretic and diffusive coefficients, and comparisons with DNS results

Theoretical efforts to determine the coefficients in (4.4) (e.g. Caporaloni et al. Reference Caporaloni, Tampieri, Trombetti and Vittori1975; Guha Reference Guha1997; Zhao & Wu Reference Zhao and Wu2006) have broadly suggested that the expression for wall-directed flux can be written as

(4.5)\begin{equation} J = \widetilde{D_1} c - \widetilde{D_2}\frac{\partial c}{\partial r},\end{equation}

where $\widetilde {D_2}$ is the particle eddy diffusivity, and $\widetilde {D_1}$ is the turbophoretic velocity. Note that (1.2) and (4.5) are similar, however, in (1.2) the coefficients ($D_1,D_2$) are evaluated from a DNS whereas in (4.5), which follows the work of others, the coefficients ($\widetilde {D_1},\widetilde {D_2}$) are connected to the underlying turbulent velocity field. Additional terms that can be included in (4.5) (but do not form a part of this study) are, for example, the effect of Brownian diffusion, gravitational settling or electrostatic force.

A model for turbophoresis, $\widetilde {D_1}$ was first given by Caporaloni et al. (Reference Caporaloni, Tampieri, Trombetti and Vittori1975) as (also see (1.1))

(4.6)\begin{equation} \widetilde{D_1} ={-} \tau_p \frac{d {u}^{\prime 2}_{pr, rms}}{d{r}}. \end{equation}

As the particle properties are difficult to model, the interest is to relate them to the underlying fluid properties. The first step is to estimate the parameter, $\mathfrak {R}$, which is defined as the ratio of mean square of the particle velocity fluctuations to the mean square of fluid velocity fluctuations:

(4.7)\begin{equation} \mathfrak{R} = \frac{u^{\prime 2}_{pr, rms}}{u_{r, rms}^{\prime 2}}.\end{equation}

If we know $\mathfrak {R}$, and given that estimates of $u_{r, rms}^{\prime 2}$ are easier to obtain, we can find $u_{pr, rms}^{\prime 2}$ and hence $\widetilde {D_1}$. An experimental correlation for the $\mathfrak {R}$ parameter given by Binder & Hanratty (Reference Binder and Hanratty1991) (which is inspired by the original works of Friedlander Reference Friedlander1957) is

(4.8a,b)\begin{equation} \mathfrak{R} = \frac{1}{1 + 0.7(\tau_p / \tau_L)} \quad \text{and} \quad \tau_L = \frac{\nu_t}{u_{r, rms}^{\prime 2}}, \end{equation}

where $\tau _L$ is the Lagrangian time scale for the fluid (see Johansen Reference Johansen1991) and $\nu _t$ is the eddy viscosity (defined in the usual manner as $\nu _t \equiv -\overline {u_r^{\prime }u_x^{\prime }}/(\textrm {d} U_x/\textrm {d} r)$). It can be observed that in the limit of small $\tau _p$ that represents the case where particles faithfully follow the fluid eddies, $\mathfrak {R}$ is indeed equal to unity. This completes one empirical formulation for $\widetilde {D_1}$.

Guha (Reference Guha1997), on the other hand, starts with Eulerian equations for particle continuity and momentum equation, and then uses the Reynolds decomposition for mean and turbulence; after neglecting a number of terms mostly related with triple correlations, they arrive at an equation for the mean wall-normal particle velocity, identified with $\widetilde {D_1}$. The resulting ODE is

(4.9)\begin{equation} \widetilde{D_1} \frac{{\rm d} \widetilde{D_1}}{{\rm d} r} + \frac{\widetilde{D_1}}{\tau_p} ={-}\frac{{\rm d} u_{pr,rms}^{\prime 2}}{{\rm d} r}.\end{equation}

Note the similarity of this with (4.6), except that the particle acceleration term is absent in (4.6). For $u_{pr,rms}^{\prime 2}$, Guha (Reference Guha1997) uses (4.7) and (4.8a,b) along with the analytically fitted experimental correlations for $\nu _t$ and $u_{r,rms}^{\prime 2}$ from Davies (Reference Davies1966) and Kallio & Reeks (Reference Kallio and Reeks1989), respectively.

For comparison of $\widetilde {D_1}$ with DNS, we first write the ODE (4.9) in finite difference form and we then iteratively solve. The necessary boundary conditions for the solution are no flux through the pipe wall and at the centreline. The solution to (4.9) is plotted in figure 7 (left panel) using solid grey lines. The more basic expression arising from (4.6) is plotted in ‘blue’ solid lines with ${u}^{\prime 2}_{pr, rms}$ from DNS, whereas ‘blue’ dashed lines correspond to (4.6) but now using (4.7) to estimate ${u}^{\prime 2}_{pr, rms}$ (with $u_{r, rms}^{\prime 2}$ from DNS). Significant differences are only apparent for $St^+ = 100$ particles.

The empirical findings obtained from DNS and our flux-splitting method (using (4.4)) strongly corroborate the analytically obtained turbophoretic velocity for smaller particles of $St^+ = 0.1$ and $St^+ = 1$. This comparison validates the analytical treatments via (4.9) and (4.6) that have been used for many decades in modelling particle-laden flows. The empirical and analytical profiles for $St^+ = 10$ are in modest agreement but there is a significant gap between the profiles for $St^+ = 100$ particles. These results suggest that the expression for $D_1$ from (4.6) that comes basically from (1.1) is essentially in agreement with the DNS, except of course for $St^+ = 100$ particles. Possible causes for the anomalous behaviour of $St^+ = 100$ particles are discussed in § 5, but briefly, they are related to the fact that Lagrangian auto-correlation based integral time scale for $St^+ = 100$ particles in the radial direction is much longer compared with the corresponding time scale for smaller $St^+$ particles.

Furthermore, it is intriguing to compare the $D_1$ profiles from the DNS with absorbing and elastically rebounding boundary conditions. Physically, this corresponds to a pipe with and without an adhesive surface. Particles of $St^+ = 0.1,1$ and $10$ show little difference between their turbophoretic velocity with these boundary conditions, suggesting that the wall collision effects are not important. However, for the case of $St^+ = 100$ particles (cf. figure 7 bottom left panel), there is a fairly dramatic reduction in the magnitude of $D_1$ for elastically rebounding particles. We now turn to the discussion of $D_2$.

It appears that unlike $\widetilde {D_1}$, there is relatively less understanding with regard to modelling $\widetilde {D_2}$. For example, Guha (Reference Guha1997) simply takes the particle eddy diffusivity $\widetilde {D_2}$ to be equal to the turbulent eddy viscosity $\nu _t$. Zhao & Wu (Reference Zhao and Wu2006), however, incorporate the effect of finite particle relaxation time and use (a relation attributed by them to Hinze Reference Hinze1975)

(4.10)\begin{equation} \frac{\widetilde{D_2}}{\nu_t} = \left(1 + \frac{\tau_p}{\tau_L} \right)^{{-}1}. \end{equation}

The DNS and the analytical estimates (in grey lines) for $D_2$ are compared in figure 7 (right panel). Here the analytical estimates of $D_2$ are divided by a factor of $10$ so that they can be shown in figures with same vertical axes. Solid grey lines are from Reference GuhaGuha's (Reference Guha1997) estimates $\widetilde {D_2} = \nu _t$, whereas dash-dotted and dashed grey lines come from (4.10) with the $\nu _t$ expression from Guha (Reference Guha1997) and with $\nu _t$ from present DNS, respectively. It is clear that $\widetilde {D_2} = \nu _t$ is not appropriate, and even though (4.10) furnishes a slightly better comparison with the DNS (for larger $St^+$) in terms of magnitude, they are still off, and more importantly the peak location of $D_2$ in the analytical estimates is not at the correct wall-normal location.

To delve a little deeper into this issue of $D_2$, note that the form (4.10) can be obtained by starting from (1.1), where $D_2 \sim v^2 \tau _p \sim u_{pr,rms}^{\prime 2} \tau _p$. Using (4.7) and (4.8a,b) leads to $D_2 \sim \mathfrak {R} u_{r,rms}^{\prime 2} \tau _p$, and if one takes (in absence of any realistic data) the eddy viscosity $\nu _t \sim u_{r,rms}^{\prime 2} \tau _p$, it will result in $D_2 \sim \nu _t \mathfrak {R}$, which is the functional form of (4.10). Note that the least realistic assumption here seems to be $u_{r,rms}^{\prime 2} \tau _p \sim \nu _t$, where one relates the turbulent flow property $\nu _t$ directly to the particle time-scale $\tau _p$. We, therefore, revert to the basic expression: $D_2 \sim u_{pr,rms}^{\prime 2} \tau _p$, or $D_2^+ (= D_2/\nu ) \sim u_{pr,rms}^{\prime 2 +}\, St^+$ and, more generally,

(4.11)\begin{equation} D_2^+= u_{pr,rms}^{\prime2 +} f(St^+),\end{equation}

where $f(St^+)$ is a function of $St^+$. It is evident from (4.11) that, for a fixed $St^+$ particle, the form of wall-normal distribution of $D_2^+$ is completely specified by $u_{pr,rms}^{\prime 2 +}$ except for a constant $f(St^+)$, and this seems appropriate when we compare $D_2^{+}$ data in figure 7 (right panel) and $u_{pr,rms}^{\prime 2 +}$ profiles in figures 3(d) and figures 18 and 19. Furthermore, by examining the DNS data in figure 7 (right panel) it is obvious that both the absorbing-wall sets as well as the elastic ones behave similarly, and the variation is only between absorbing and elastic cases. Therefore, using our DNS database we find that $f(St^+)$ follows approximately a power law with $St^+$. Specifically, for absorbing walls, $f(St^+) \approx 0.62 {St^+}^{2/3}$, whereas for elastic walls $f(St^+) \approx 0.7 {St^+}^{3/4}$. The light/dark blue solid lines in figure 7 (right panel) correspond to (4.11) with these $f(St^+)$ for absorbing/elastic walls. At present these power-law variations are merely empirical. The simplistic (4.11) is by no means perfect, but it does capture the peak position in $D_2^+$ and the wall-normal variation of lower $St^+$ cases reasonably well. The $St^+=100$ case is problematic, as has been the case in modelling $D_1^+$ also, and it is likely due to similar reasons.

Even if we have perfect models for $D_1$ and $D_2$, it is not clear whether the time-dependent concentration $c(t)$ can be predicted with the phenomenological model (4.2). If $c(t)$ from (4.2) compares well with the DNS, then we can use the reduced (4.2) (compared with the full Navier–Stokes, continuity and particle equations) to shed light on the key features of particle migration.

4.3. Time-dependent solution of the phenomenological concentration equation (4.2) and comparisons with the DNS

For the numerical solution procedure of (4.2) the physical pipe domain is represented as a 1D function of the distance from the wall, $y^+ = R^+ - r^+$; then ($D_1, D_2$) coefficients determined in § 4.1 are substituted into (4.2) and time marched. Details of the solution procedure are described in Appendix C. The simulations for all $St^{+}$ are initialised with a uniform concentration, i.e. $c/c_b = 1$ at $t = 0$. (Recall that $c_b$ is the bulk concentration at any instant in time.) The time evolution of $c(r)$ are presented in figures 8(a) to 8(d) for $St^+=0.1$, $1$, $10$ and $100$, respectively. Here we restrict our focus to simulation Set 1 (drag force only, elastic walls), but additional results are presented in Appendix C.

Figure 8. Predictive capacity for the simplified model against the observable from the DNS, corresponding to simulation Set 1: (a) $St^{+}=0.1$, (b) $St^{+}=1$, (c) $St^{+}=10$ and (d) $St^{+}=100$. Solid blue line indicates model prediction and light blue shading corresponds to the region bounding the 95$\%$ confidence interval for determined $D_1$ and $D_2$. The five different time instances correspond to $t^{+} = 0\unicode{x2013} 3600$ with intervals of ${\Delta t^{+} = 720}$.

In general, the major qualitative features observed from the DNS (in symbols) are well captured in this model (solid blue lines), particularly the negligible and low wall accumulation at lower $St^+$ values and dramatic increase in wall accumulation for particles with significant inertia. The shaded (light-blue) regions around the model profiles (in solid blue lines) are uncertainty estimates based on the regressed values of $D_1$ and $D_2$, and further details of the error estimation are presented in Appendix C1. Favourable agreement between our time-marched solution and the time-varying DNS concentration profiles allows us to probe further into the contributing factors to particle migration at the phenomenological level.

As shown in (4.2), $C_1$ and $C_2$ represent two turbophoretic terms whereas $G_1$ and $G_2$ the diffusive terms. Terms $C_2$ and $G_2$ show the effects of radial variations in $D_1$ and $D_2$, respectively. These terms are shown in figures 9(a) and 9(b) for $St^+=1$ and $10$, respectively. The terms are evaluated adjacent to the wall until $y^+=3$, and they are interpreted in terms of accumulation–depletion depending on the flux term signs. Further, the split contributions to the flux divergence are non-dimensionalised by the pipe radius $R$, deposition velocity $V_{dep}$ determined in § 3.3 and bulk concentration $c_b$.

Figure 9. Decomposition of contribution to normalised divergence of fluxes in numerical solution adjacent to the wall for (a) $St^+ = 1$ and (b) $St^+ = 10$. Black terms correspond to convective contributions and blue terms to diffusive contributions, and solid lines correspond to $C_1,G_1$ with dashed lines corresponding to $C_2,G_2$.

We observe that the relative magnitudes of the convective and diffusive contributions for $St^+ = 1$ and $St^+ = 10$ are similar but the magnitude of each term is larger for the more inertial particles suggesting a $St^{+}$ dependence. Higher magnitudes are expected as $St^+ = 10$ particles set up sharper concentration gradients close to the wall and are more likely to enter the viscous sublayer. This analysis suggests that close to the wall, the signs of the individual convection–diffusion mechanisms (and so their role) are similar for $St^+ = 1$ and $St^+ = 10$ particles, but their magnitude grows with $St^{+}$. This near-wall mechanism acts in conjunction with the high-turbophoretic force (via $D_1$) at $y^+ = 15$ which is much larger for $St^+ = 10$ than $St^+ = 1$ particles. Interestingly, the main balance is between the turbophoretic term $C_2$ (and a smaller effect from $C_1$ and $G_1$) that is driving accumulation and the diffusion term $G_2$ depleting the particles concentration. As such, it is the gradients in coefficients $D_1$ and $D_2$ that dominate the net particle migration. This implies that a constant turbophoretic velocity or diffusive coefficient will not capture the particle migration physics appropriately. Away from the wall we observe (but not shown here) that the turbophoretic components $C_1$ and $C_2$ are both negative, which implies that they are responsible for depletion of concentration in that region. Furthermore, the sum of all the fluxes at these locations are also negative, which implies a net reduction in concentration, and this is expected because most particles migrate from these regions towards the wall. This is, of course, in contrast to the near-wall region (see figure 9) where the total flux convergence is positive, suggesting an increase in particle concentration.

5.1. Lagrangian velocity correlation

Lagrangian statistics in homogeneous isotropic turbulence have been studied extensively (e.g. Yeung & Pope Reference Yeung and Pope1989), where particles are statistically identical, regardless of their initial position, but the case of wall turbulence is more complicated. Particles in wall flows have statistical dependency on their wall-normal distance. Lagrangian trajectories from the Set 1 DNS in this study are visualised in figure 10(a) (i.e. top panel), where it is apparent that the motion of inertial particles diverges from tracers. Particles are tagged and tracked once they reside within a small interval of the target heights of $y^+ = 5$, 30 and 118. Those tagged from the core, buffer and viscous regions (shown in figure 10(a) in the left, middle and right panels, respectively) show markedly different pathlines between $St^+ = 0.1$ and $St^+ = 100$ (identified by their symbols). In the near-wall region, there is a dramatic reduction in the pathline curvature of $St^+ = 100$ particles.

Figure 10. (a) Trajectories, coloured by streamwise distance traversed, of particles tracked after entering the turbulent core (left panel), buffer (middle panel) and viscous regions (right panel). Particles of different $St^+$ are identified by their respective symbols as indicated in table 2. (b) Wall-normal trajectories of inertial particles, starting from $y^{+} = 5$. Initial position are marked by black dots and final positions by red or blue dots. Trajectories plotted for a duration of $t^{+} = 360$, at which point, blue and red dots indicate particles that are respectively nearer to or further away from the wall than $y^{+} = 30$.

We make the additional observation that particle motion in the buffer region exhibits high curvature, which is due to the high fluid streamline curvature in this region (e.g. Perven, Philip & Klewicki Reference Perven, Philip and Klewicki2021). Particles with low $St^+$ enter this region and with even probability may journey to the pipe center or return to the wall. However, particles with higher $St^+$ are unable to follow these curvy motions and are flung wall-wards resulting in a reduced population that journeys to the pipe centre.

To further illustrate visually the particle movement, in figure 10(b) (i.e. bottom panel) we present wall-normal trajectories of particles (for $St^+=0.1, 1, 10$ and $100$ from left to right panels, respectively). Particles with $St^+=100$ tend to show motions compared with other $St^+$ with less curvature and diminished total pathline distance. These visual features of particle paths can be quantified by using Lagrangian correlations. In general, Lagrangian correlations can be defined by

(5.1)\begin{equation} \rho_{ij}(\tau;y_0) = \frac{\langle v_{pi}^{\prime}(t_0;y_0) v_{pj}^{\prime}(t_0 + \tau;y_0)\rangle}{\langle v_{pi}^{\prime 2}(t_0;y_0) \rangle^{1/2} \langle v_{pj}^{\prime 2}(t_0 + \tau;y_0)\rangle^{1/2}}, \end{equation}

where $i,j$ represent the three coordinate directions, $t_0$ and $y_0$ are the initial time and wall-normal location of the particles, and the fluctuating particle velocity $v_{pi}^{\prime }(t_0 + \tau,y_0) = v_{pi}(t_0 + \tau,y_0) - \langle v_{pi}(t_0 + \tau,y_0)\rangle$ is the particle velocity fluctuation relative to the Lagrangian average over the total averaging time.

In figure 11 the panels in three columns (from left to right) show Lagrangian correlation functions where the particle populations commence their motion at the wall normal heights of $y^+_0=3$, $30$ and $118$, whereas the three rows (top to bottom) correspond to autocorrelation of particle velocity in the streamwise, radial and azimuthal directions $\rho _{xx}$, $\rho _{rr}$ and $\rho _{\theta \theta }$. Of particular interest for understanding the one-dimensional model discussed in § 4.3 is the radial component $\rho _{rr}$. Here, the profiles of the particle populations $St^+ = 0.1$ and $St^+ = 1$ show little difference, which is expected. Increasing in size, the $St^+ = 10$ particle population remains correlated for appreciably longer; but more notable is the order of magnitude increase in correlation period for the $St^+ = 100$ particles, as well as their flatter profiles in the neighbourhood of $\tau ^+=0$. This would suggest that an Eulerian gradient transport model for particle transport is less suitable for $St^{+} = 100$ particles, and this can be further quantified by evaluating several timescales associated with the Lagrangian correlations.

Figure 11. Lagrangian auto-correlation of streamwise ($\rho _{xx}$), wall-normal ($\rho _{rr}$) and azimuthal ($\rho _{\theta \theta }$) particle fluctuating velocity. For symbols see table 2.

In order to quantify the time interval over which particle velocity is correlated with itself, the Lagrangian integral scale associated to each velocity component is estimated according to

(5.2)\begin{equation} T_{ii} = \int^{\tau_c}_0 \rho_{ii} (\tau,y_0)\,{\rm d} \tau, \end{equation}

where $\tau _c$ is the time lag at which the auto-correlation first crosses 0.1 (e.g. Stelzenmuller et al. Reference Stelzenmuller, Polanco, Vignal, Vinkovic and Mordant2017). This definition is chosen because the usual definition of the integration over an infinite time interval cannot be applied to all profiles, as some profiles cross zero and others do not. Results of $T_{ii}$ for the correlations presented in figure 11 are shown in figure 12. Note that $St^+=100$ data are markedly different to other $St^+$ cases. Of particular interest are values of timescale in the radial direction $T_{rr}$ (middle panel), wherein $St^+=100$ has $T_{rr}^+ = T_{rr}U_\tau (U_\tau /\nu ) \approx 100$. Note that the pipe flow is for $Re_\tau =R (U_\tau /\nu ) = 180$, which implies that $T_{rr}U_\tau \approx 0.55R$, i.e. $St^+=100$ particle motions are correlated over about half $R$. This is perhaps not surprising for these particle which skip most of the small-scale fluid motions. These large correlation times also point towards the difficultly (we encountered in § 4) in modelling $St^+=100$ particles using gradient hypothesis (e.g. Corrsin Reference Corrsin1974).

Figure 12. Lagrangian integral scales ($T_{ii}$) corresponding to data in figure 11: (a) $T_{xx}^+$, (b) $T_{rr}^+$ and (c) $T_{\theta \theta }^+$. For symbols see table 2.

Notwithstanding the fact that timescale discussion sheds light on the modelling efforts, the physical movement of the particles is governed by the turbulent motions specific to wall flows. These coherent turbulent motions are discussed in the following section focusing on different wall regions, which can also help to clarify some observations, such as in the buffer region the empirically determined $D_1$ reaching a peak value, the dramatic difference for $St^+=100$ data and the effect of elastic/absorbing wall conditions.

5.2. Coherent motions and flow topology

We spatially correlate the particle wall-ward velocity in the buffer region with the turbulent coherent ‘sweep’ and ‘ejection’ motions (e.g. Willmarth & Lu Reference Willmarth and Lu1972; Brodkey, Wallace & Eckelmann Reference Brodkey, Wallace and Eckelmann1974; Robinson Reference Robinson1991). These turbulent coherent motions are made up of strongly coherent ($Q2$ and $Q4$) events and additionally interaction-type ($Q1$ and $Q3$) events which remain correlated over significantly less time. Classification into quadrants for turbulent events constitutes signage of turbulent fluctuations $u^{\prime }$ and $v^{\prime }$ where $Q2$ ejections are low-speed streaks ($u^{\prime } < 0$) moving away from the wall ($v^{\prime } > 0$), and $Q4$ sweeps are high-speed streaks moving towards the wall. Note that in pipe flows the wall-normal velocity $v = -u_{r}$. Particle transfer has previously been found to be controlled by strongly coherent sweeps and ejections which entrain particles (e.g. Marchioli & Soldati Reference Marchioli and Soldati2002). Visualisation of coherent motions (as line contours) with entrained particles (as dots) from simulation Set 1 are plotted in figures 13 and 14 for $Q2$ and $Q4$ events, respectively. These plots answer the question: if a particle is wall-directed (or wall-fleeing), how likely is it that they are entrained within a strongly coherent turbulent event? Particle points within the $Q2$/$Q4$ events indicate that the particles are correlated to, and hence likely transported by, these events. Strongly coherent $Q2$ and $Q4$ motions are able to transfer $St^+ = 1$ particles towards the wall, but also capable are $Q1$ and $Q3$ events. Inertial $St^+ = 10$ particles particles, however, require strongly coherent motions to be transferred towards the wall as the momentum of these particles is sufficient to filter $Q3$ events. Coherent motions are still able to transfer $St^+ = 100$ particles wall-ward but this mechanism is less effective.

Figure 13. Instantaneous overlay at $y^+ = 30$ between coherent $Q2$ ejections ($u_x^{\prime } < 0,u_r^{\prime } < 0$) and particles: (a) $St^+ = 0.1$, (b) $St^+ = 1$ (c) $St^+ = 10$ and (d) $St^+ = 100$. Contours levels for $Q2$ range from $(u_x^{\prime } < 0) \times (u_r^{\prime } < 0) = 0.001$–$5.0$. Particles plotted with instantaneously wall-fleeing velocity ($u_{pr} < 0$) and residing at $y^+ = [29,31]$.

Figure 14. Instantaneous overlay at $y^+ = 30$ between coherent $Q4$ sweeps ($u_x^{\prime } > 0,u_r^{\prime } > 0$) and particles: (a) $St^+ = 0.1$, (b) $St^+ = 1$, (c) $St^+ = 10$ and (d) $St^+ = 100$. Contours levels range from $(u_x^{\prime } > 0) \times (u_r^{\prime } > 0) = 0.001$–$5.0$. Particles plotted with instantaneously wall-ward velocity ($u_{pr} > 0$) and residing at $y^+ = [29,31]$.

To quantify the connection between particles and $Q2$/$Q4$ events illustrated in figures 13 and 14, we calculate probability of particle radial velocity direction ($u_{pr}$) conditioned on $Q2$ (where fluid velocity $u_x^{\prime } < 0,u_r^{\prime } < 0$) as well as on $Q4$ ($u_x^{\prime } > 0,u_r^{\prime } > 0$). (Recall that wall-normal velocity $v=-u_r$.) These conditional probabilities (as percentages) for varying $St^+$ are presented in figures 15(a) and 15(b) for $Q2$ and $Q4$ event, respectively. The statistics are collected over the entire time of the simulation within the wall-normal range of $y^+ = [25,35]$. The large conditional probability in figure 15(a) for all particles except $St^+=100$ suggest that indeed the coherent ejection motion is responsible for upward particle velocity (and wall-fleeing effect of particles). Furthermore, figure 15(a) also shows that modelling of the wall as either elastic or absorbing has little effect on these statistics when we consider the $St^+ = 0.1,1$ and $10$ particles, but a dramatic effect on the sluggish $St^+ = 100$ particles. This suggests a memory effect for $St^+ = 100$ pertaining to their approach towards $y^+ = [25,35]$ from either the wall, or from the pipe center. For the $St^+ = 100$ Stokes-drag-only cases, $Pr(u_{pr} < 0 | Q2) = 0.71$ for elastic walls and decreases to $Pr(u_{pr} < 0 | Q2) = 0.53$ for absorbing walls, both of which are smaller than for the probabilities for smaller $St^+$ suggesting these particles retain their attained inertia and can coast across strong vortical regions. To re-emphasise the point of conducting four distinct simulation sets, with toggled inclusion of the Saffman–Mei lift force $\boldsymbol {F}_l$, and wall-boundary type, we note that in the buffer region, for $St^+ = 0.1,1 \ \textrm {and} \ 10$ particles, there is negligible ($<1\,\%$) dependence of the conditional statistics on $\boldsymbol {F}_l$. Our findings are consistent with the general finding in wall flows where the Saffman–Mei lift force $\boldsymbol {F}_l$, is most prominent close to walls and decays approaching the centreline (e.g. Wang et al. Reference Wang, Squires, Chen and McLaughlin1997). For the $St^+ = 100$ particles, however, our simulations show that in the case of absorbing walls, inclusion of the Saffman–Mei lift force yields a moderate ($9\,\%$) reduction in the $Pr(u_{pr} < 0 | Q2)$ statistic between Set 2 and Set 4. In the case of elastic walls, we again see a reduction of $9\,\%$ in $Pr(u_{pr} > 0 | Q4)$ between Set 1 and Set 3. For sweep events, figure 15(b) shows characteristics similar to ejection events in figure 15(a). It seems, however, that sweep events (compared with ejection) events are more efficient in bringing even larger $St^+$ particles to the wall.

Figure 15. Probability of $(\textit {a})$ particle fleeing given it is sampling a Q2 event ($Pr(u_{pr} < 0 | Q2)$) or $(\textit {b})$ directed towards ($Pr(u_{pr} > 0 | Q4)$) pipe walls given sampling of a Q4 event, conditioned on the strongly coherent event types: $Q2 (u_x^{\prime } < 0,u_r^{\prime } < 0)$ and $Q4 (u_x^{\prime } > 0,u_r^{\prime } > 0)$. Statistics collected at $y^+ = [25,35]$ from simulation Set 1. For symbols see table 2.

Although the preceding coherent motions analysis sufficiently explains particle transfer mechanisms, study of local organisation of particles is needed to give insight into biased sampling of fluid events that lead to wall-ward transfer. We employ the flow classification of Chong, Perry & Cantwell (Reference Chong, Perry and Cantwell1990) that maps flow topology with respect to invariants of the velocity gradient tensor, $\boldsymbol {A} = \boldsymbol {\nabla } \boldsymbol {u}$. The invariants are $P = A_{ii}=0$ (owing to continuity), $Q = -(1/2)A_{ij}A_{ji}$ and $R = -(1/3)A_{ij}A_{jk}A_{ki}$. For data presented on a $Q$–$R$ plot, large positive and negative $Q$ indicate vorticity and strain dominated regions, respectively; in addition, for $Q<0$ a positive $R$ suggests bi-axial strain or sheet-like structures (e.g. Davidson Reference Davidson2015). In figure 16, we plot four decades of the joint probability density function (PDFs) for fluid events (in contour lines) separated by wall-normal region in the pipe (in three columns of figure 16) and then overlay the events sampled by inertial particles from Set 1 (shown by coloured scatter plots). In the $Q$–$R$ plane, there is little deviation between the conditional PDFs of particles able to trace flow events, $St^+ = 0.1$ and the fluid. There is a slight ‘down-shift’ in the sampling of $Q$ events for particles of comparable time scale to the small fluid scales, $St^+ = 1$, when they reside in the buffer and logarithmic regions (cf. figures 16e and 16f). This is suggestive of particles avoiding vortical regions in preference for strain-dominated regions. Another feature that we observe is a dramatic reduction for $St^+ = 10$ and $100$ events to sample more extreme events in $Q$–$R$ space. See also particle-laden channel DNS by Rouson & Eaton (Reference Rouson and Eaton2001). This is more prominent in the viscous sublayer than in the buffer/logarithmic regions. The consequences, first, are that once particles reach the viscous sublayer, they organise outside of the streamwise vortices, and prefer regions of low streamwise velocity rather than the high-speed streaks and, second, particles that do rebound from the wall must pass through the vortex cores present in the buffer region if they are to continue towards the centreline of the pipe. As they approach the vortex cores, however, they are likely to be flung back towards the wall region.

Figure 16. Joint PDF of $Q$–$R$, conditionally sampled by the fluid (contour lines) and particles (coloured scatter points). Solid grey line corresponds to the discriminant, $D = \frac {27}{4}R^2 + Q^3 = 0$. Left-to-right columns, viscous sublayer, buffer region and logarithmic layer. Top-to-bottom rows, $St^+ = 0.1,1,10$ and $100$.