2. Theoretical formalism

To quantify the generation mechanism of particle streaks, we leverage the equation for the PDF of the position and velocity of a particle in three-dimensional space. The equations for the zeroth-order and first-order statistical moments can be obtained, i.e. the particle mass and momentum conservation laws, which can help the physical interpretation of the uneven distribution of particles in space.

We start with the PDF of the position and velocity of a particle, which is defined as

(2.1) \begin{align} f(\boldsymbol {x},\boldsymbol {v}; t) = \overline {\delta (\boldsymbol {x}_{\!p}(t)-\boldsymbol {x})\,\delta (\boldsymbol {v}_{\!p}(t)-\boldsymbol {v})}, \end{align}

where $\delta ({\cdot })$ is the Dirac delta function, and $\boldsymbol {x}_{\!p}$ and $\boldsymbol {v}_{\!p}$ are the position and velocity of the particle, respectively. The overline indicates ensemble average across multiple realisations of the system.

By differentiating (2.1) with time, one can obtain the kinetic equation for the PDF of particle position and velocity in the three-dimensional space (Marble Reference Marble1963; Buyevich Reference Buyevich1971; Reeks Reference Reeks1983; Wang & Ni Reference Wang and Ni1991; Zaichik Reference Zaichik1999; Sikovsky Reference Sikovsky2014; Johnson et al. Reference Johnson, Bassenne and Moin2020; Bragg et al. Reference Bragg, Richter and Wang2021a ) as

(2.2) \begin{align} \frac {\partial f}{\partial t} + \boldsymbol{\nabla} _{\!\boldsymbol {x}} \boldsymbol{\cdot }(\boldsymbol {v}f) + \boldsymbol{\nabla} _{\!\boldsymbol {v}} \boldsymbol{\cdot }( {\overline {\boldsymbol {a}}}|_{\boldsymbol {x,v}}\, f) = \dot {f}_c, \end{align}

where $\boldsymbol{\nabla} _{\!\boldsymbol {x}}=(\partial /\partial x, \partial /\partial y, \partial /\partial z)$ , $\boldsymbol{\nabla} _{\!\boldsymbol { v}}=(\partial /\partial v_x, \partial /\partial v_y, \partial /\partial v_z)$ , and $\overline {\boldsymbol{a}}|_{\boldsymbol {x,v}} = \overline {\boldsymbol{a}}_p(\boldsymbol {x}_{\!{p}}=\boldsymbol{x}, \boldsymbol {v}_{{p}}=\boldsymbol{v}, t)$ is the mean particle acceleration. Inter-particle collisions are not considered due to the one-way coupling assumption in the dilute regime, so the collision term is $\dot {f}_c=0$ .

The zeroth-order moment of the PDF is particle number density $C_n$ , which can be obtained by integrating $f$ over all possible particle velocities as

(2.3) \begin{align} C_n(\boldsymbol {x};t) = C_{n0}(t) \int _{\mathbb{R}^3} f(\boldsymbol {x},\boldsymbol {v}; t)\, \textrm {d}\boldsymbol {v}, \end{align}

where $C_{n0}(t) = \int _{{\varOmega }} C_n(\boldsymbol {x},t)\, \textrm {d}\boldsymbol {x}$ is the bulk particle number ( $\varOmega \subset \mathbb{R}^3$ is the domain of the flow).

Following Johnson et al. (Reference Johnson, Bassenne and Moin2020), by integrating (2.2) over the entire velocity space and multiplying by $C_{n0}$ , one can obtain the mass conservation law for the particle phase as

(2.4) \begin{align} \frac {\partial C_n}{\partial t} + \boldsymbol{\nabla} _{\!\boldsymbol {x}} \boldsymbol{\cdot }({\overline {\boldsymbol {v}}}|_{\boldsymbol {x}}\, C_n) = 0, \end{align}

in which, $\overline {\boldsymbol {v}}|_{\boldsymbol {x}} = \overline {\boldsymbol {v}}_p (\boldsymbol {x}_{\!p} = \boldsymbol {x}, t)$ .

By multiplying (2.2) with $\boldsymbol {v}$ and $C_{n0}$ , then integrating over the entire velocity space, one can obtain the momentum conservation law for the particle phase as

(2.5) \begin{align} \frac {\partial (C_n\, \overline {\boldsymbol {v}}|_{\boldsymbol {x}})}{\partial t} + \boldsymbol{\nabla} _{\!\boldsymbol {x}} \boldsymbol{\cdot }(\overline {\boldsymbol { v}\boldsymbol {v}}|_{\boldsymbol {x}}\, C_n) - \overline {\boldsymbol {a}}|_{\boldsymbol {x}}\, C_n = 0, \end{align}

where $\overline {\boldsymbol {v}\boldsymbol {v}}|_{\boldsymbol {x}}=\overline {\boldsymbol {v}_{\!p}(\boldsymbol {x}_{\!p}=\boldsymbol {x}, t)\,\boldsymbol { v}_p(\boldsymbol {x}_{\!p}=\boldsymbol {x}, t)}$ is the mean particle velocity covariance tensor.

Considering unidirectional particle distribution, let $f_i$ be the PDF of particle position and velocity in the $i$ th direction, which is defined by

(2.6) \begin{align} f_i({x_i},{v_i}; t) = \overline {\delta ({x}_{p,i}(t)-{x_i})\,\delta ({v}_{p,i}(t)-{v_i})}. \end{align}

Here $i=1,2,3$ , indicating the $x$ , $y$ or $z$ direction, respectively. Then the kinetic equation for the PDF in the $i$ th direction is

(2.7) \begin{align} \frac {\partial f_i}{\partial t} + \frac {\partial (v_i f_i)}{\partial x_i} + \frac {\partial \big(\overline {a_i}|_{x_i,v_i}\,f_i \big)}{\partial v_i} = 0. \end{align}

The mass and momentum conservation laws are

(2.8) \begin{align} \frac {\partial C_{n,i}}{\partial t} + \frac {\partial \big(\overline {v_i}|_{x_i}\,C_{n,i} \big)}{\partial x_i} = 0 \end{align}

and

(2.9) \begin{align} \frac {\partial (C_{n,i}\, \overline {v_i}|_{x_i})}{\partial t} + \frac {\partial }{\partial x_i} \left (\overline {v_i^2}|_{x_i}\,C_{n,i}\right )- \overline {a_i}|_{x_i}\, C_{n,i} = 0, \end{align}

respectively, where

(2.10) \begin{align} C_{n,i}(x_i;t) = C_{n0,i}(t) \int f_i(x_i,v_i;t)\,\text{d}v_i, \end{align}

which is the particle number density along the $i$ th direction.

Next, we derive a formal relationship for the particle number density in an arbitrary direction.

By rearranging (2.9), we can get

(2.11) \begin{align} \frac {\partial \ln C_{n,i}}{\partial x_i} = \underbrace {\frac {\overline {a_{i}}}{\overline {v^2_{p,i}}}}_{\textit{MPA term}} \quad \underbrace {- \frac {\partial \ln \overline {v^2_{p,i}}}{\partial x_i}}_{\textit{MPKEG term}} - \underbrace { \frac {1}{C_{n,i} \overline {v_{p,i}^2} }\frac {\partial \big(C_{n,i} \overline {v_{p,i}}\big)}{\partial t}}_{\textit{unsteady term}}, \end{align}

which depicts the contributions to the spatial gradient of mean particle number density by the mean particle acceleration (MPA) term, the mean particle kinetic energy gradient (MPKEG) term and the unsteady term.

If the unsteady term is much smaller than the MPA and MPKEG terms (see Appendix B), then we have

(2.12) \begin{align} \frac {\textrm {d}\ln C_{n,i}}{\textrm {d}x_i} = \underbrace {\frac {\overline {a_{i}}}{\overline {v^2_{p,i}}}}_{\textit{MPA}} \quad \underbrace {- \frac {\textrm { d}\ln \overline {v^2_{p,i}}}{\textrm {d}x_i}}_{\textit{MPKEG}}. \end{align}

It can be seen that the MPA term ${\overline {a_{i}}}/{\overline {v^2_{p,i}}}$ imposes an increasing trend of particle concentration along the direction of mean particle acceleration. If $\overline {v^2_{p,i}}$ is much smaller than $\overline {a_{i}}$ , then this term may play a prominent role in determining the particle concentration gradient. In contrast, it can be neglected if $\overline {v^2_{p,i}}$ is much larger than $\overline {a_{i}}$ . In addition, the MPKEG term $\textrm {d}\ln \overline {v^2_{p,i}}/\textrm {d}x_i$ always opposes the mean particle concentration gradient, which implies that particles are likely to accumulate in the regions of small $\overline {v^2_{p,i}}$ , e.g. the classical turbophoresis effect (Johnson et al. Reference Johnson, Bassenne and Moin2020; Bec & Vallée Reference Bec and Vallée2024), as their derivatives are opposite.

In particular, in the wall-normal ( $z$ ) or spanwise ( $y$ ) direction, the relationship (2.12) can be written as

(2.13) \begin{align} \frac {\textrm {d}\ln C_{n}(z)}{\textrm {d}z} = \frac {\overline {a_{z}}}{\overline {v^2_{p,z}}} {- \frac {\textrm { d}\ln \overline {v^2_{p,z}}}{\textrm {d}z}}, \quad \frac {\textrm {d}\ln C_{n}(y)}{\textrm {d}y} = \frac {\overline {a_{y}}}{ \overline {v^2_{p,y}}} {- \frac {\textrm { d}\ln \overline {v^2_{p,y}}}{\textrm {d}y}}. \end{align}

If the contribution by the mean particle acceleration term can be neglected (see figures 8 and 9 below), then

(2.14) \begin{align} \frac {\textrm {d}\ln C_{n}(z)}{\textrm {d}z} \approx - \frac {\textrm {d}\ln \overline {v^2_{p,z}}}{\textrm {d}z}, \quad \frac {\textrm {d}\ln C_{n}(y)}{\textrm {d}y} \approx - \frac {\textrm {d}\ln \overline {v^2_{p,y}}}{\textrm {d}y}, \end{align}

which indicates that a local maximum (minimum) particle number density $C_n$ corresponds to a local minimum (maximum) $\overline {v^2_{p}}$ .

Relations (2.13) will be adopted to reveal the physical mechanism that leads to uneven particle distribution and the role of the lift force in the wall-normal direction (near-wall particle accumulation) and spanwise direction (particle streaks), which can also be regarded as a relation between particle distribution and a generalised turbophoresis effect. Although the system may be homogeneous in the $i$ th direction (such as the spanwise direction of a channel flow), the MPKEG term $\textrm {d}\ln \overline {v^2_{p,i}}/\textrm {d}x_i$ is a function of $x_i$ , which can have local variations. Therefore, the generalised turbophoresis effect can also play a significant role in a homogeneous direction.

Moreover, a sufficiently long time average may eliminate the spanwise variations in particle statistics, as the flow and particle streaks can meander with equal probability at all spanwise locations over a very long time. As a result, the time-averaged particle concentration may be uniform in the spanwise direction. This phenomenon is actually owing to the ergodicity of the system and is dependent on the time-averaging window. However, we would like to stress that (2.12) is derived regardless of the size of the time-averaging window once the flow is statistically stationary. In other words, (2.12) and (2.13) always hold exactly, regardless of whether or not the spanwise variations of statistics are smoothed out.

3. Simulation methodologies

3.1. Flow dynamics simulation

The governing equations of the flow are the three-dimensional Navier–Stokes equations for an incompressible fluid, as well as the continuity equation, which are

(3.1) \begin{align} \frac {\partial \boldsymbol {u}_{\kern-1pt f}}{\partial t}+ (\boldsymbol {u}_{\kern-1pt f} \boldsymbol{\cdot }\boldsymbol{\nabla }) \boldsymbol {u}_{\kern-1pt f}= - \frac {1}{\rho _f} \boldsymbol{\nabla }p_f+ \nu _f\, {\nabla} ^2 \boldsymbol {u}_{\kern-1pt f}, \\[-28pt] \nonumber \end{align}

(3.2) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol {u}_{\kern-1pt f}=0, \\[0pt] \nonumber \end{align}

where $\boldsymbol {u}_f$ is the flow velocity, $p_f$ is the fluid pressure, $\rho _f$ is the fluid density, and $\nu _f$ is the fluid kinematic viscosity.

In this study, an open-source code, namely CP3d, is utilised for the simulations of particle-laden flows (Gong et al. Reference Gong, Wu, An, Zhang and Fu2023). Code CP3d is a three-dimensional DNS solver developed for particle-laden flows, employing an Euler–Lagrangian framework, and supporting one-way, two-way and particle-resolved/unresolved four-way coupling methods. The incompressible Navier–Stokes equations are spatially discretised using a fourth-order central finite-difference scheme on a staggered grid. Temporal advancement is implemented via the second-order Adams–Bashforth method. The pressure Poisson equation is solved efficiently using the fast Fourier transform, while viscous terms are treated with a fully implicit scheme to enhance numerical stability. The code has been well validated in the DNS of turbulent flows and particle-laden flows (Gong et al. Reference Gong, Wu, An, Zhang and Fu2023).

3.2. Particle dynamics simulation

Particles are modelled by spherical volumeless points, and their dynamics are determined by

(3.3) \begin{align} \frac {\textrm {d}\boldsymbol {x}_{\!p}}{\textrm {d}t} = \boldsymbol {v}_{\!p}, \\[-28pt] \nonumber \end{align}

(3.4) \begin{align} \frac {\textrm {d}{\boldsymbol {v}_{p}}}{\textrm {d}t} = \boldsymbol {a}_{p} = \boldsymbol {a}_d + \boldsymbol {a}_l + \boldsymbol {a}_g, \\[0pt] \nonumber \end{align}

in which the terms on the right-hand side of (3.4) are the accelerations by fluid drag force ( $\boldsymbol { a}_d$ ), flow-shear-induced lift force ( $\boldsymbol {a}_l$ ) and gravitational force ( $\boldsymbol {a}_g$ ).

The drag-induced acceleration is modelled by $\boldsymbol {a}_d=(\boldsymbol {u}_f - \boldsymbol {v}_{\!p})/\tau _p$ , where $\boldsymbol { u}_f$ is the flow velocity seen by the particle. The drag-related time scale is $\tau _p=\tau _s/k_p$ , where $\tau _s={\rho _p d_p^2}/{(18\mu _f)}$ is the particle relaxation time based on the Stokes drag, $\rho _p$ is particle density, $d_p$ is particle diameter, $\mu _f$ is the dynamic viscosity of the fluid, and $k_p= 1+0.15\,\textit{Re}_p ^{0.687}$ is the finite particle Reynolds number correction coefficient (Schiller & Naumann Reference Schiller and Naumann1933). The particle Reynolds number is defined by $\textit{Re}_p={\rho _f\,| \boldsymbol {u}_f- \boldsymbol {v}_{\!p}|\, d_p}/{\mu _f}$ .

For the flow-shear-induced lift force, the most well-known model is the classical Saffman model (Saffman Reference Saffman1965, Reference Saffman1968) for the lift force acting on an isolated spherical particle in a linear uniform shear flow. However, the Saffman model is limited to small particle Reynolds numbers and is free of wall effects, the latter of which may be critical in our case. Therefore, we resort to the composite lift force model of Wang et al. (Reference Wang, Squires, Chen and McLaughlin1997), based on several existing models that account for wall effects. The details of the model are provided in Appendix A. The composite lift model as well as the Saffman model are validated against the PR-DNS result of Costa et al. (Reference Costa, Brandt and Picano2020a ) at friction Reynolds number 180. Figure 1 demonstrates good agreement of the composite lift model prediction with the PR-DNS data, while the Saffman model significantly underestimates the particle concentration near the wall, similar to the finding of Gao et al. (Reference Gao, Shi, Parsani and Costa2024).

Figure 1.

Predictions of the wall-normal particle concentration distribution in the turbulent channel flow at $\textit{Re}_{\tau } = 180$ using two different lift force models against the PR-DNS data of Costa et al. (Reference Costa, Brandt and Picano2020a ), in which $C_v$ is the particle volume fraction, and $C_{v0}$ is the bulk mean particle volume fraction.

The equations of particle dynamics (3.3) and (3.4) are numerically advanced by the second-order Adams–Bashforth scheme in time (Johnson & Meneveau Reference Johnson and Meneveau2018; Johnson et al. Reference Johnson, Bassenne and Moin2020; Gong et al. Reference Gong, Wu, An, Zhang and Fu2023). A trilinear interpolation scheme is adopted to calculate the fluid velocity at the particle centre (Johnson & Meneveau Reference Johnson and Meneveau2018; Johnson et al. Reference Johnson, Bassenne and Moin2020; Gong et al. Reference Gong, Wu, An, Zhang and Fu2023).

Table 1.

Particle parameters in the simulations, in which $St$ is the particle Stokes number, $d_p$ is the particle diameter, $d_p^+$ is the viscous-scaled particle diameter, $\rho _p/\rho _f$ is the particle-to-fluid density ratio, $Sv$ is the settling parameter, and $C_{v0}$ is the bulk mean particle volume fraction. The terms ‘D’, ‘DL’ and ‘DLG’ refer to cases that include only drag force, cases that include both drag force and lift force, and cases that include drag force, lift force and gravitational force, respectively.

Table 1. Long description

A table with 8 rows and 7 columns comparing particle parameters in simulations. The columns are labeled Case, St, d p (μm), d p+, ρ p/ρ f, S v, and C v0. The rows present different cases with varying values for each parameter. The cases are St5D, St5DL, St5DLG, St25D, St25DL, St25DLG, St100D, St100DL, and St100DLG. Each row lists specific values for the particle Stokes number, particle diameter, viscous-scaled particle diameter, particle-to-fluid density ratio, settling parameter, and bulk mean particle volume fraction.

3.3. Problem set-up

The flow configuration in the present study is a horizontal turbulent channel flow. The carrying fluid is air, with kinematic viscosity $\nu _f=1.5\times 10^{-5}$ m $^2$ s⁻¹ and density $\rho _f=1.3$ kg m⁻³. The friction Reynolds number of the turbulent channel flow is $Re_{\tau }=u_\tau H/\nu _f=200$ , where $H$ is the half-channel height, and $u_\tau$ is the friction velocity. The simulation domain size is $L_x\times L_y\times L_z=2\pi H\times \pi H\times 2 H$ , and the grid number is $N_x\times N_y\times N_z=128\times 96\times 192$ , where $x$ , $y$ and $z$ denote the streamwise, spanwise and wall-normal directions, respectively. Uniform grids are used in the streamwise and spanwise directions, with grid spacings $\Delta x^+=9.8$ and $\Delta y^+=6.5$ . Here, the plus superscript indicates the normalisation using the viscous units, i.e. $u_\tau$ and $\nu _f$ . In the wall-normal direction, the grid is stretched such that it is finer approaching the wall, and the vertical grid size is in the range $\Delta z^+=0.6{-} 3.45$ . In the simulations, the Courant–Friedrichs–Lewy (CFL) number is used to determine the time step for the flow advancement at first, then a fixed time step is utilised after the turbulent flow has reached a statistically stationary state, before introducing particles in the simulations. In this work, the CFL number is approximately $0.2$ , and the fixed flow time step is $\Delta t_f^+=0.06$ . The flow boundary conditions are periodic in the streamwise and spanwise directions, and no-slip at the top and bottom walls.

Three types of particles with different sizes are considered, as displayed in table 1. The particle Stokes number is defined by $St=\tau _p/\tau _f$ , in which $\tau _f = \nu _f/u_\tau ^2$ is the viscous time scale. The particle density is $\rho _p=2649$ kg m⁻³. The number of particles is $N_p=100\,000$ in each simulation case. Since the dispersed phase is dilute (see table 1), the one-way coupling method is incorporated in the simulations, in which the feedback force of particles to fluid flow and particle–particle collisions is neglected, and particles collide elastically with the walls. In addition, the gravitational force is in the negative $z$ direction, and quantified by the settling number ${S}{\textit{v}} = \tau _p g / u_\tau$ (where $g$ is the gravity acceleration). In the present study, particles are randomly placed in the channel at the initial instant. The initial particle velocity is set to be equal to the flow velocity at its centre. The particle boundary conditions are periodic in the streamwise and spanwise directions, and elastically rebound at the top and bottom walls. The Shannon entropy of particle distribution is monitored (Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012), and particle statistics are collected once the Shannon entropy reaches statistical stationarity, as shown in figure 2.

The temporal averaging for all statistical quantities reported in this study was performed over time window $\Delta t^+ \approx 3300$ once the flow and particles reach statistical stationarity. In viscous units, the characteristic lifetime scale of an individual flow streak is $T_{\kern-1pt \textit{fs}}^+ \approx O(1000)$ (Jiménez & Moin Reference Jiménez and Moin1991; Bae & Lee Reference Bae and Lee2021), while the large-eddy turnover period defined as $T_{e}=H/u_{\tau }$ is $T_{e}^+ \approx 200$ in the viscous units. Thus our employed averaging window ( $\Delta t^+ \approx 3300$ ) spans approximately more than 10 times the large-eddy turnover period, and is comparable to the formation time scale of particle streaks, which will be shown later. This duration is sufficiently long to average over temporal fluctuations, thereby ensuring the convergence of statistical budgets and rendering the unsteady term in (2.11) numerically negligible (Appendix B). However, as discussed in Appendix C, an averaging window $\Delta t^+ \approx 3300$ represents a finite-time averaging relative to the infinite time-averaging limit where streak meandering eventually homogenises the spanwise profiles.

Figure 2.

Temporal evolution of the Shannon entropy of all cases.

4. Particle distributions

4.1. Wall-normal distributions

First, we focus on the distributions of particles along the wall-normal direction, with the simulations neglecting the gravitational force. The effect of the gravitational force will be examined afterwards.

Figure 3.

Wall-normal profiles of particle concentration, where $C_n$ is the local mean particle number density, and $C_{n0}$ is the bulk mean particle number density. The dashed lines with hollow markers are the cases without lift force, and the solid lines are with lift force.

Figure 3. Long description

Panel: A line graph depicting wall-normal profiles of particle concentration. The x-axis represents the dimensionless wall-normal distance z plus, ranging from 10^0 to 10^2. The y-axis represents the normalized particle concentration C n over C n 0, ranging from 10^-2 to 10^2. The graph includes six data series: St 5 without lift (red dashed line with square markers), St 25 without lift (green dashed line with triangle markers), St 100 without lift (blue dashed line with circle markers), St 5 with lift (red solid line), St 25 with lift (green solid line), and St 100 with lift (blue solid line). The dashed lines with hollow markers represent cases without lift force, while the solid lines represent cases with lift force. The data series show a general trend of decreasing particle concentration with increasing wall-normal distance.

Figure 3 shows the profiles of mean particle number density, which is the mean particle number in a unit volume. The averaging is taken in horizontal slabs and over time. One can observe that the wall-normal particle distributions are highly non-uniform, with particles tending to have higher concentrations near the wall. If the lift force is neglected, then the preferential accumulation of particles in the near-wall region is significant, and most of the particles are located in the viscous sublayer, which is well known as the (classical) turbophoresis effect (Caporaloni et al. Reference Caporaloni, Tampieri, Trombetti and Vittori1975; Reeks Reference Reeks1983; Kallio & Reeks Reference Kallio and Reeks1989; McLaughlin Reference McLaughlin1989; Marchioli & Soldati Reference Marchioli and Soldati2002; Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012; Johnson et al. Reference Johnson, Bassenne and Moin2020; Wang et al. Reference Wang, Zhou and Zheng2024).

In contrast, when the lift force is included in the simulations, the near-wall preferential accumulation is significantly weakened, and the profiles become much flatter. This has been observed in some studies. For example, considering particles in an upward channel flow with gravity, Marchioli et al. (Reference Marchioli, Picciotto and Soldati2007) discovered that the shear-induced lift force can weaken the near-wall accumulation of particles, similar to the present finding. Shin et al. (Reference Shin, Portela, Schaerer and Mangiavacchi2022) found weakening of the near-wall accumulation of particles at a single and small $\textit{St}$ by including the Saffman lift in their simulations of particle suspension with a low particle-to-fluid ratio. Gao et al. (Reference Gao, Shi, Parsani and Costa2024) and Li et al. (Reference Li, Wang and Zheng2024) also reported similar findings.

4.2. Horizontal distributions

Figure 4 displays the particle distributions and streamwise velocity fluctuations in a horizontal $x{-}y$ plane at $z^+\lt 10$ . If the lift force is neglected and only the drag force is included in the simulations, then one can observe clear particle streaks in the near-wall region, as in other studies (Marchioli & Soldati Reference Marchioli and Soldati2002; Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012; Jie et al. Reference Jie, Cui, Xu and Zhao2022; Rousta et al. Reference Rousta, Lessani and Ahmadi2023; Zhang et al. Reference Zhang, Cui and Zheng2023). However, if the shear-induced lift force is included, then it is seen that near-wall particles are more uniformly distributed, and the strength of particle streaks is evidently weakened. This is also similar to the observations of Gao et al. (Reference Gao, Shi, Parsani and Costa2024) and Ruan & Xiao (Reference Ruan and Xiao2024).

Figure 4.

Particle distributions in a horizontal slab of $z^+\lt 10$ and the contours of streamwise velocity fluctuations at $z^+=10$ : (a–c) without lift force, (d–f) with lift force.

Figure 4. Long description

Panel A: A heat map showing particle distributions and streamwise velocity fluctuations without lift force. The x-axis represents the horizontal distance (x/H) ranging from 0 to 6, and the y-axis represents the vertical distance (y/H) ranging from 0 to 3. The color scale ranges from -2 to 2, with blue indicating lower values and red indicating higher values. The particles are distributed throughout the slab, with noticeable clusters and patterns. Panel B: A heat map showing particle distributions and streamwise velocity fluctuations without lift force at a different time scale. The axes and color scale are the same as in Panel A. The distribution of particles and velocity fluctuations shows more pronounced patterns compared to Panel A. Panel C: A heat map showing particle distributions and streamwise velocity fluctuations without lift force at an even higher time scale. The axes and color scale remain consistent. The patterns of particle distribution and velocity fluctuations are more defined and structured. Panel D: A heat map showing particle distributions and streamwise velocity fluctuations with lift force. The axes and color scale are the same as in the previous panels. The inclusion of lift force results in a different distribution of particles and velocity fluctuations. Panel E: A heat map showing particle distributions and streamwise velocity fluctuations with lift force at a different time scale. The axes and color scale remain consistent. The patterns of particle distribution and velocity fluctuations are distinct from those in Panel B. Panel F: A heat map showing particle distributions and streamwise velocity fluctuations with lift force at an even higher time scale. The axes and color scale are the same as in the previous panels. The patterns of particle distribution and velocity fluctuations are more pronounced and structured compared to Panel E.

To further quantify particle streaks, the Voronoï diagram (Monchaux et al. 2010, Reference Monchaux, Bourgoin and Cartellier2012) is leveraged. The Voronoï diagram is a decomposition of space into independent cells associated with each particle, and a Voronoï cell is defined as the set of spatial points that are closer to a particle than to any other particles. After a plane is divided into Voronoï cells according to particle position, the normalised Voronoï cell area is defined as the ratio of the cell area $A$ and the average cell area $\overline {A}$ , i.e. $\eta =A/\overline {A}$ . By comparing the PDF of the normalised Voronoï cell area $\eta$ and the random Poisson process, clusters and voids of Voronoï cells can be identified (Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2010).

Figure 5.

Voronoï diagrams in a near-wall horizontal slab of $z^+\lt 10$ : (a) $St=5$ without lift force; (b) $St=25$ without lift force; (c) $St=100$ without lift force; (d) $St=5$ with lift force; (e) $St=25$ with lift force; (f) $St=100$ with lift force.

Figure 5 shows representative Voronoï diagrams from the present DNS simulations in a near-wall $x{-}y$ plane. The horizontal slab for the Voronoï analysis is selected to be well within $z^+\lt 10$ , where a majority of particles reside. The cells of clusters and voids are coloured black and white, respectively. Figures 5(a–c) display the Voronoï diagrams in the cases without the lift force, from which it can be seen that the particles with the three Stokes numbers exhibit obvious streak-like clusters in the horizontal plane. Figures 5(d–f) depict the Voronoï diagrams in the cases with the lift force. It can be seen that the spatially local clustering distribution of particles is much weaker, and the areas of the cluster cells are much larger. These results indicate that the lift force not only weakens the preferential accumulation of particles in the wall-normal direction, but also significantly reduces the spatially local clustering and streaky structure of particles horizontally.

Figure 6.

Wall-normal profiles of the standard deviation of the normalised Voronoï cell area $\overline {\sigma _\eta }$ . The dashed lines with hollow markers are the cases without lift force, and the solid lines with filled markers are with lift force. Here, $\sigma _{\textit{RPP}}$ corresponds to the random Poisson process.

Figure 7.

The ADFs of particle distributions in the horizontal slab of $z^+\lt 10$ : (a–c) without lift force, (b–f) with lift force.

Figure 7. Long description

Panel A: A heat map showing particle distributions without lift force at a Stokes number of 5. The x-axis is labeled x/Dp and ranges from 0 to 15. The y-axis is labeled y/Dp and ranges from 0 to 15. The color scale ranges from blue to red, indicating values from 1.0 to 4.0. The distribution shows a gradient from blue to red, with higher values concentrated towards the upper right. Panel B: A heat map showing particle distributions without lift force at a Stokes number of 25. The x-axis is labeled x/Dp and ranges from 0 to 15. The y-axis is labeled y/Dp and ranges from 0 to 15. The color scale ranges from blue to red, indicating values from 1.0 to 4.0. The distribution shows a more uniform gradient from blue to red, with higher values concentrated towards the upper right. Panel C: A heat map showing particle distributions without lift force at a Stokes number of 100. The x-axis is labeled x/Dp and ranges from 0 to 15. The y-axis is labeled y/Dp and ranges from 0 to 15. The color scale ranges from blue to red, indicating values from 1.0 to 4.0. The distribution shows a more uniform gradient from blue to red, with higher values concentrated towards the upper right. Panel D: A heat map showing particle distributions with lift force at a Stokes number of 5. The x-axis is labeled x/Dp and ranges from 0 to 15. The y-axis is labeled y/Dp and ranges from 0 to 15. The color scale ranges from blue to red, indicating values from 1.0 to 4.0. The distribution shows a gradient from blue to red, with higher values concentrated towards the upper right. Panel E: A heat map showing particle distributions with lift force at a Stokes number of 25. The x-axis is labeled x/Dp and ranges from 0 to 15. The y-axis is labeled y/Dp and ranges from 0 to 15. The color scale ranges from blue to red, indicating values from 1.0 to 4.0. The distribution shows a more uniform gradient from blue to red, with higher values concentrated towards the upper right. Panel F: A heat map showing particle distributions with lift force at a Stokes number of 100. The x-axis is labeled x/Dp and ranges from 0 to 15. The y-axis is labeled y/Dp and ranges from 0 to 15. The color scale ranges from blue to red, indicating values from 1.0 to 4.0. The distribution shows a more uniform gradient from blue to red, with higher values concentrated towards the upper right.

To quantify the degree of local clustering of particles, the standard deviation of the normalised Voronoï cell area (Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2010), i.e. ${\overline {{\sigma }_{\eta }}}=(\overline {{\eta }^2}-1)^{1/2}$ , is utilised as a measure. The standard deviation of the random Poisson process is $\sigma _{\textit{RPP}}\approx 0.53$ . If $\overline {{\sigma }_\eta } \gt \sigma _{\textit{RPP}}$ , then it indicates that there exist particle clusters. The larger $\overline {\sigma _\eta }$ , the stronger the local clustering of particles. Figure 6 displays the wall-normal profiles of $\overline {\sigma _\eta }$ in the near-wall region, where most of the particles reside. If the drag force alone is included, then one can see that $\overline {\sigma _\eta }$ decreases with the wall-normal height, implying a stronger clustering degree approaching the wall. Generally, particles with higher inertia (larger $St$ ) exhibit stronger spatial clustering in this case. If the lift force is included, then it is seen that $\overline {\sigma _\eta }$ is significantly smaller than without the lift force, quantitatively demonstrating evidence that the lift force weakens near-wall particle streaks. Moreover, it does not greatly vary with the wall-normal height, and is only slightly larger than the standard deviation of the random Poisson process.

In addition, the angular distribution function (ADF) is employed as another measure to quantitatively characterise particle clustering and reveal its anisotropic features (Gualtieri et al. Reference Gualtieri, Picano and Casciola2009, Gualtieri et al. Reference Gualtieri, Picano, Sardina and Casciola2013). The ADF can capture both angular orientation and clustering intensity across different directions, making it suitable for analysing particle alignment and microstructure. In a horizontal $x{-}y$ slab, the ADF is calculated by

(4.1) \begin{align} \textit{ADF}(r,\theta ) = \frac {\sum _{i=1}^{n_p} \delta N_i(r,\theta )}{r\, \delta r\, \delta \theta } \Big/ \frac {N}{L_x L_y}, \end{align}

where $\delta N_i(r, \theta )$ are the numbers of particle pairs within the relative distance interval $(r, r + \delta r)$ and relative angle interval $(\theta , \theta + \delta \theta )$ , and $n_p$ is the total number of particle pairs. Thus a higher value larger than the ADF indicates stronger particle clustering.

Figure 7 shows the ADFs of particle distributions in the horizontal plane at $z^+ \lt 10$ . If the lift force is not included, then enormous ADF values along the $x$ direction are observed, indicating strong streamwise clusterings, and formation of streamwise-elongated streaks with a width of several particle diameters and a length exceeding tens of particle diameters. Similar to figure 6, it is found that the clustering effect gradually intensifies with the particle Stokes number. If the shear-induced lift is included, then the ADFs become much smaller and more uniform across all distances and angles, indicating that the lift force can significantly weaken local particle clustering, and suppress particle streaks. It should be noted that the radial stripes in the ADFs shown in figures 7(a–c) arise from stable particle cluster orientation over the finite averaging period, and would be smoothed out by streak meandering in the long-time limit. Nevertheless, it does reflect the instantaneous or finite-time preferential distribution of particles.

5. Theoretical analyses

We employ (2.13) to analyse the mechanism for the reduced near-wall concentration. In figures 8(a–c), the profiles of the wall-normal gradient of mean particle number density and the sum of the right-hand-side terms in (2.13) are compared. It is shown that they are in excellent agreement, indicating the correctness of (2.13) whether the lift force is included or not. Johnson et al. (Reference Johnson, Bassenne and Moin2020) successfully demonstrated this for cases without lift force, while here we provide further evidence for cases that include lift force. Moreover, the near-wall magnitude of the mean particle number density gradient is significantly reduced when the lift force is included, and this reduction is more pronounced at moderate and large $St$ , consistent with figure 3.

Figure 8.

Wall-normal profiles of the mean particle number density gradient and the right-hand-side terms in (2.12).

Figures 8(d–f) further elucidate the role of different terms in the right-hand side of (2.13). In principle, it is seen that the wall-normal distribution of particle concentration is prominently determined by the counteraction between the MPKEG term (turbophoresis) and the MPA term from particle drag (biased sampling). In contrast, the direct contribution from the lift force is negligible. As particle inertia ( $St$ ) increases, the relative importance of the drag force term rapidly decreases. At a moderate or high $St$ , the turbophoresis effect predominantly governs the wall-normal particle distribution, which is also consistent with Johnson et al. (Reference Johnson, Bassenne and Moin2020). Furthermore, one can see that although the lift force introduces a negligible influence, it introduces a significant reduction in the magnitude of the MPKEG term (turbophoresis). Therefore, the indirect influence of the lift force on the turbophoresis effect is the main mechanism for the reduction of near-wall particle concentration.

Figure 9.

Profiles of the spanwise gradient of the mean particle number density and the contributions of the right-hand-side terms in (2.13).

Figure 9. Long description

Panel A: A line graph shows the spanwise gradient of the mean particle number density and contributions of various terms for St = 5 without lift. The x-axis is labeled y+ and ranges from 0 to 600. The y-axis is labeled d ln C_n(y)/d(y/H) and ranges from -10 to 10. Different line styles and symbols represent different terms. Panel B: A line graph shows the same variables for St = 25 without lift. The axes and line styles are similar to Panel A. Panel C: A line graph shows the same variables for St = 100 without lift. The axes and line styles are similar to Panel A. Panel D: A line graph shows the same variables for St = 5 with lift. The axes and line styles are similar to Panel A. Panel E: A line graph shows the same variables for St = 25 with lift. The axes and line styles are similar to Panel A. Panel F: A line graph shows the same variables for St = 100 with lift. The axes and line styles are similar to Panel A.

Figure 10.

Temporal evolution of the normalised standard deviation of the Voronoï cells for particles within $z^+\lt 10$ . The dashed lines indicate the approximate time for particles to reach a quasi-stationary state.

Similar to the wall-normal distribution of particle concentration, we also employ (2.13) to elucidate the mechanism that leads to the inhomogeneous streaky distribution of particles and the degradation of particle streaks by the lift force. Figure 9 shows the profiles of the spanwise gradient of mean particle number density, i.e. $\textrm {d} \ln C_n/\textrm {d}y$ , as well as the right-hand-side terms in (2.13). It is noted that the particle number density and other quantities are averaged in streamwise, wall-normal and time, thus they are functions of $y$ alone. Therefore, although the spanwise direction is homogeneous, the quantities shown in figure 9 can vary with $y$ . One can see that if the lift force is excluded, then the spanwise gradient of the mean particle number density has significant variations in the spanwise direction, as displayed in figures 9(a–c), which manifest the streaky structures of particles. In contrast, if the lift force is included, then the spanwise gradient of the mean particle number density becomes much more uniform, as in figures 9(d–f). This method reveals the weakening effect of the lift force on the streaky particle clustering. Figure 9 also shows the MPA and MPKEG terms and their sum. It can be seen that the mean particle concentration gradient and right-hand-side term of (2.13) excellently coincide with each other, confirming the correctness of (2.13) again. Moreover, the MPA contributions by the drag and lift forces are negligible, while the MPKEG term dominates and determines the spanwise distributions of particles, which is similar to the wall-normal particle distribution. Therefore, the primary mechanism for the weakening of particle streaks is also the indirect influence of the lift force on the generalised turbophoresis effect, other than the direct effects of the drag force (biased sampling) or the lift force. This finding cannot be revealed from the analyses of Voronoï diagrams and ADFs.

As we stated before, a sufficiently long time average may eliminate the spanwise variation of particle distribution (particle streaks), as the flow and particle streaks can meander to equal probability at all spanwise locations. As a result, the time-averaged particle concentration may be uniform in the spanwise direction. This is demonstrated in Appendix C. One can clearly see that the degree of particle streaking is decreased by increasing the time-averaging window, which also implies that the streaks are not unphysically locked in their spanwise locations over a long time. Meanwhile, (2.13) always holds regardless of the time-averaging window. Therefore, the spanwise peaks shown in figure 9 reflect the instantaneous or finite-time preferential distribution of particles induced by coherent structures. The significant attenuation of these peaks by the lift force remains a valid physical observation. It demonstrates that the lift force reduces the tendency of particles to form persistent spanwise-ordered structures. The finite-time averaging reveals such features, and the sensitivity to the lift force itself supports the conclusion that the lift force alters the clustering dynamics.

Considering the response of particle streaks to flow variations, it is informative to compare their respective time scales. First, we evaluate the time scale for particles to aggregate from an initially uniform distribution into stable, distinct streaks, quantified by the standard deviation $\overline {\sigma _\eta }$ of the normalised Voronoï cells for particles within $z^+ \lt 10$ , as presented in figure 10. It can be seen that the degree of near-wall particle streaking reaches a quasi-stationary state after a characteristic time scale $T_{ps}^+ \approx 2000 {-} 3500$ , which is larger for particles with a larger Stokes number. In comparison, the characteristic lifetime scale of near-wall flow streaks is $T_{\kern-1pt \textit{fs}}^+ \approx O(1000)$ (Jiménez & Moin Reference Jiménez and Moin1991; Bae & Lee Reference Bae and Lee2021). Therefore, $T_{ps}^+$ and $T_{\kern-1pt \textit{fs}}^+$ are of the same order of magnitude ( $O(10^3)$ ), indicating that the formation time scale of particle streaks is commensurate with the lifetime and evolution time scale of the driving flow structures ( $T_{ps}^+ \sim T_{\kern-1pt \textit{fs}}^+$ ). This finding is physically consistent, because particle aggregation occurs on a time scale comparable to the evolution of flow streaks, and particles have sufficient time to respond to quasi-streamwise vortices dynamically, and organise into distinct streaky patterns.

6. Physical mechanisms

From the viewpoint of turbulent coherent structures, it has been well recognised that particle transport in near-wall turbulence is prominently governed by quasi-streamwise vortices and their induced sweeping/ejection events (Brooke et al. Reference Brooke, Kontomaris, Hanratty and McLaughlin1992; Marchioli & Soldati Reference Marchioli and Soldati2002; Berk & Coletti Reference Berk and Coletti2020; Motoori et al. Reference Motoori, Wong and Goto2022; Huang et al. Reference Huang, Jie, Cui, Xu, Andersson and Zhao2025). For example, through detailed analysis of DNS data, Marchioli & Soldati (Reference Marchioli and Soldati2002) found that particles can be effectively transported downwards to the near-wall region by the sweeps, and form particle streaks in low-speed velocity structures, while the ejections are less effective at lifting particles due to the influence of offspring vortices very near the wall.

Figure 11.

Instantaneous particle distributions and flow structures in a $y{-}z$ slab with streamwise thickness $\Delta x^+ \approx 6.5$ : (a) without lift force, (b) with lift force. The green lines are the isolines of the square of the spanwise fluid velocity $v^{2+}_f=0.2$ , the grey lines are those of $v^{2+}_f=0.018$ , the colour contours are the $Q$ criterion, and the arrows represent flow velocity vectors in the plane. The $St=100$ particles are shown, among which the red and blue colours denote upward and downward moving ones, respectively.

Figure 11. Long description

Panel A: A heat map showing particle distributions and flow structures in a fluid slab without lift force. The x-axis is labeled y+ and ranges from 400 to 600. The y-axis is labeled z+ and ranges from 0 to 100. Green lines represent isolines of the square of the spanwise fluid velocity, grey lines represent isolines of another variable, color contours indicate a specific criterion, and arrows represent flow velocity vectors in the plane. Red and blue particles denote upward and downward moving particles, respectively. Panel B: A heat map showing particle distributions and flow structures in a fluid slab with lift force. The x-axis is labeled y+ and ranges from 300 to 500. The y-axis is labeled z+ and ranges from 0 to 100. Green lines represent isolines of the square of the spanwise fluid velocity, grey lines represent isolines of another variable, color contours indicate a specific criterion, and arrows represent flow velocity vectors in the plane. Red and blue particles denote upward and downward moving particles, respectively.

Figure 11 shows typical instantaneous distributions of particles, vortical structures via the $Q$ criterion (Hunt et al. Reference Hunt, Wray and Moin1988), and isolines of squared spanwise flow velocity in a $y{-}z$ plane. If only the drag force is considered, i.e. in figure 11(a), then it is evident that most of the particles accumulate between near-wall counter-rotating quasi-streamwise vortices within the viscous sublayer, confirming the observations in previous studies. In figure 11(a), the regions with high and low magnitudes of the squared spanwise flow velocity are presented with green and grey isolines, respectively. It can be seen that near-wall particles are well accumulated within the regions with low spanwise flow velocity, and out of those with high spanwise velocity. This is consistent with (2.14) that a local maximum (minimum) particle concentration in spanwise corresponds to a local minimum (maximum) $\overline {v^2_p}$ . In simple terms, this phenomenon happens because particles in the viscous sublayer are pushed away from areas with strong spanwise velocity to areas with weaker spanwise velocity by the drag force. Therefore, the generation mechanism of particle streaks can be directly attributed to the generalised turbophoresis effect, which is also consistent with the centrifugal mechanism of preferential concentration of particles (Maxey Reference Maxey1987) that inertial particles prefer to be swept out of vortices. Figure 12(a) provides a simplified illustration of the mechanism when the lift force is absent, in which particles are prominently driven by the quasi-streamwise vortices, and accumulated in the low-flow-velocity streaks where the spanwise flow is weak.

Figure 12.

Schematics of particle distribution and turbulent structures. The circles represent streamwise vortices, and PSV and NSV indicate positive streamwise vortex and negative streamwise vortex, respectively. The blue and red regions with arrows indicate ejection and sweep turbulent motions, respectively. The grey arrows represent spanwise migration of particles. The purple arrows indicate the vertical elevation effect due to the lift force.

Figure 11(b) displays typical instantaneous distributions of particles, vortical structures, and regions with high and low spanwise flow velocity, when the lift force is included. It can be seen that the particles are elevated and more uniformly distributed in space, which is consistent with the preceding results. In particular, particles can be well located within quasi-streamwise vortices. The elevation of particles to a higher location should be attributed to the shear-induced lift force. We will present a hypothetical analysis as follows. Considering particles within the viscous sublayer, they will undergo spanwise migration due to the spanwise flow induced by quasi-streamwise vortices. As illustrated in figure 12(b), the combination of the spanwise migration of particles and the quasi-streamwise vortices will induce a localised upward lift to elevate the particles. For example, the particles beneath a negative streamwise vortex (NSV) undergo negative spanwise migration. Since ${L_z} \propto {\omega }_{f,x}({v}_{p,y}-{u}_{f,y})$ and $u_{f,y} \approx 0$ , one can determine a positive lift force for these particles. This is also true for the particles beneath a positive streamwise vortex (PSV), as the migration is in the positive spanwise direction. Therefore, it is suspected that the localised upward lift elevates the spanwise-migrating particles to a higher location. Since the net vertical particle flux is zero, and the upward- and downward-moving particles are cycling and balanced, an increasing number of spanwise migrating particles are ultimately elevated until they reach an equilibrium state.

Actually, it is known that mean flow shear and streamwise particle slip can produce a negative mean vertical lift force towards the wall (Gao et al. Reference Gao, Shi, Parsani and Costa2024). The question is how significant the localised upward lift is, especially compared to the mean downward lift. To verify the proposed hypothesis, we conduct a conditional averaging analysis using the simulation data of the drag-only case St100D. First, the particles within $z^+\lt 10$ satisfying the condition of a net positive vertical force and strong spanwise migration,

(6.1) \begin{align} L_{z,m}+L_{z,l}+D_z\gt 0 \quad \text{and} \quad {v_{p,y}^2\gt 4\overline {v_{p,y}^2}}, \end{align}

are identified, where $L_{z,m}$ and $L_{z,l}$ are the mean and localised vertical lift forces, respectively, and $D_z$ is the vertical drag force. Then the corresponding $y{-}z$ flow fields in a slab of $\Delta x^+ \approx 10$ are used to obtain the conditional-averaged flow field around these particles. In addition, $v_{p,y}\lt 0$ and $v_{p,y}\gt 0$ particles are analysed separately. Figure 13 shows the conditionally averaged flow fields as well as the mean wall-normal height of these particles, denoted by a representative particle. It is seen that the particles subjected to a net positive vertical force are indeed at the bottom edges of an NSV or PSV, and are located within strong spanwise flows, which provides direct support for the proposed mechanism. Furthermore, it is noted that nine three-dimensional flow field snapshots are utilised for the conditional averaging, in which 1084 $v_{p,y}\lt 0$ particles and 1643 $v_{p,y}\gt 0$ particles are totally identified. Although the number of particles satisfying the condition is small, we mention that the above hypothetical analysis is purely based on the result of a drag-only case. Once a lift force is introduced, more and more particles will be elevated until they reach an equilibrium state, as the transport of particles is a cycling process.

Figure 13.

The conditionally averaged flow fields in the $y{-}z$ plane for particles with a net positive vertical force and strong spanwise migration at $z^+\lt 10$ in the St100D case: (a) $v_{p,y}\lt 0$ particles, (b) $v_{p,y}\gt 0$ particles. The mean heights of these particles are denoted by the representative particles in the figure.

Figure 13. Long description

Panel A: A heat map showing the flow field for particles with a net negative vertical force. The x-axis is labeled Δy+ and ranges from -60 to 60. The y-axis is labeled z+ and ranges from 0 to 60. The color scale on the right ranges from -100 to 100, with blue indicating lower values and red indicating higher values. Vector arrows indicate the direction and magnitude of the flow. There is a noticeable concentration of blue in the center and red regions on the sides. Panel B: A heat map showing the flow field for particles with a net positive vertical force. The x-axis is labeled Δy+ and ranges from -60 to 60. The y-axis is labeled z+ and ranges from 0 to 60. The color scale on the right ranges from -100 to 100, with blue indicating lower values and red indicating higher values. Vector arrows indicate the direction and magnitude of the flow. There is a noticeable concentration of red in the center and blue regions on the sides.

Figure 14.

The averaged flow fields in the $y{-}z$ plane conditionally sampled at the particles within $z^+\lt 100$ : (a) the St100D case, (b) the St100DL case. The mean heights of the particles are denoted by the representative particles in the figure.

Once particles are lifted to a higher location well within the quasi-streamwise vortices ( $z^+ \sim O(10)$ ) (Kim et al. Reference Kim, Moin and Moser1987), the spanwise accumulation of particles (particle streaks) will be weakened, as the variation of spanwise velocity within quasi-streamwise vortices is much weaker than that at the vortex bottom edge ( $z^+ \sim O(1)$ ). To verify this, we further examine the averaged flow fields around the particles at $z^+ \lt 100$ . Figure 14(a) shows that if only the drag force is included, then most particles reside within $z^+\lt 10$ and in the middle of two counter-rotating quasi-streamwise vortices, which is depicted by a representative particle informing their mean height. If the lift force is included, as seen in figure 14(b), then particles are elevated to mean height $20 \lt z^+ \lt 30$ , which is well within the near-wall quasi-streamwise vortices, as shown in figure 13. In contrast, the averaged flow field is quite uniform, and no evident vortices can be observed. The variation of the spanwise flow velocity at the mean particle height is much weaker in the St100DL case than in the St100D case, so that the generalised turbophoresis effect is much weaker, and particle streaks are degraded. Due to the very different conditionally averaged flow fields in figure 14, one may expect that the mean drag forces experienced by the particles are also quite different in the two cases. This can be confirmed in figures 8(e) and 8(f), in which the near-wall drag-induced MPA terms are enlarged.

Therefore, the physical mechanism for how the lift force degrades particle streaks can be clearly understood by (2.13) alongside the turbulent coherent structures and particle motions. As particles move to higher locations away from the viscous sublayer while within quasi-streamwise vortices, as a result of the excessive localised lift force caused by the combination of the quasi-streamwise vortices and spanwise migration of particles, the spanwise particle velocities are significantly decreased, which degrades particle streaks due to the much weakened spanwise turbophoresis force.

7. Effect of gravitational force

Several recent studies have highlighted the non-negligible role of the gravitational force in particle distribution, even when the settling number $Sv$ is very small (Lee & Lee Reference Lee and Lee2019; Bragg et al. Reference Bragg, Richter and Wang2021b ; Wang et al. Reference Wang, Li and Zheng2021). Figure 15(a) shows the wall-normal particle concentration profiles in the lower half-channel. It can be seen that the peak concentrations of particles with different $St$ all increase when the gravitational force (in the negative $z$ direction) is included in the simulations, which can be understood as the gravitational force drags particles towards the wall. Thus the gravitational force can counteract the upward localised lift force generated by the spanwise migration of particles and quasi-streamwise vortices. As a result, it can be anticipated that the degradation of particle streaks by the lift force will be relaxed, or particle streaks will be strengthened by the gravitational force. Figures 15(b–d) confirm this by showing the profiles of the spanwise gradient of particle concentration at the bottom wall, whose fluctuations are obviously increased compared to figures 9(d–f). Figure 16(a) shows the wall-normal particle concentration profiles in the upper half-channel. The gravitational force is found to evidently reduce the particle concentration near the upper wall. Figures 16(b–d) show that the spanwise fluctuations of particle distributions are not significantly altered compared to figures 9(d–f). This may be because the gravitational force has the same sign as the localised lift force near the upper wall.

Figure 15.

(a) Wall-normal profiles of the mean particle concentration in the lower half-channel. (b–d) Profiles of the spanwise gradient of mean particle number density at the bottom wall.

Figure 15. Long description

Panel A: A line graph shows wall-normal profiles of the mean particle concentration in the lower half-channel. The x-axis represents the dimensionless wall-normal distance (z+) on a logarithmic scale, and the y-axis represents the normalized particle concentration (Cn/Cn0) on a logarithmic scale. Different lines represent various Stokes numbers (St) with and without lift and gravity. Panel B: A line graph shows the spanwise gradient of mean particle number density at the bottom wall for St = 5 with lift and gravity. The x-axis represents the dimensionless spanwise distance (y+) and the y-axis represents the normalized gradient (d ln Cn(y)/d(y/H)). Different lines represent various components of the gradient. Panel C: A similar line graph for St = 25 with lift and gravity. Panel D: A similar line graph for St = 100 with lift and gravity.

Figure 16.

(a) Wall-normal profiles of the mean particle concentration in the upper half-channel. (b–d) Profiles of the spanwise gradient of mean particle number density at the upper wall.