1. Introduction
Particle-laden wall-bounded turbulence is widely observed in both natural phenomena and industrial processes. Examples of these particle-laden flows include the aeolian sandstorm in atmospheric surface layers (Liu et al. Reference Liu, He and Zheng2023), snowflakes in the atmosphere (Li et al. Reference Li, Abraham, Guala and Hong2021a ), the sediment ridges on the river bed (Scherer et al. Reference Scherer, Uhlmann, Kidanemariam and Krayer2022), the powder streams in continuous pharmaceutical manufacturing (Velez-Silva et al. Reference Velez-Silva, Drennen and Anderson2023) and fluidised bed combustion (Valizadeh et al. Reference Valizadeh, Skoglund, Forsberg, Lycksam and Öhman2024), to name just a few. The fluctuating, turbulent velocity field underlying these flows can notably affect the transportation of particles. Understanding the particle transport processes in wall-bounded flows is important for designing and optimising industrial applications involving particle transport and for understanding the mechanism behind natural two-phase phenomena.
Particle dispersion and transportation in turbulent flows over smooth walls have received extensive study. Inertial particles distribute non-uniformly in turbulent flows, appearing in the form of small-scale clustering and turbophoresis (Narayanan et al. Reference Narayanan, Lakehal, Botto and Soldati2003; Picano et al. Reference Picano, Sardina and Casciola2009; Toschi & Bodenschatz Reference Toschi and Bodenschatz2009; Balachandar & Eaton Reference Balachandar and Eaton2010; Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012a ). The non-uniform spatial distribution of particles is attributed to the finite inertia of particles which prevents them from following fluid trajectories and selectively filters turbulent structures (Milici et al. Reference Milici, De Marchis, Sardina and Napoli2014). The small-scale local clustering, occurring in both homogeneous and inhomogeneous flows, results from the combined effects of particle inertia and fine-scale turbulent fluctuations, causing particles to be less prone to aggregate in the vortical regions. Chen et al. (Reference Chen, Goto and Vassilicos2006) and Goto & Vassilicos (Reference Goto and Vassilicos2008) proposed that the small-scale cluster occurs as a result of the sweep-stick mechanism which causes inertial particles with relaxation times smaller than the integral timescale of the flow to aggregate at the acceleration stagnation points.
A considerable number of particles accumulating at the walls is a distinctive feature of wall-bounded flows, especially more pronounced for moderately inertial particles (Fessler et al. Reference Fessler, Kulick and Eaton1994; Kulick et al. Reference Kulick, Fessler and Eaton1994; Rouson & Eaton Reference Rouson and Eaton2001; Fong et al. Reference Fong, Amili and Coletti2019). By analysing particle motion equations, Reeks (Reference Reeks1983) proposed the concept of turbophoresis and suggested that inertial particles tend to aggregate in regions with low turbulent fluctuations. In wall-bounded turbulence, the gradient of turbulent intensity in the wall-normal direction leads to a characteristic particle flux towards the wall and particles achieve large concentrations at the wall. Meanwhile, a lot of works have focused on the mechanism of turbulent wall structures on particle transport in the last few decades. Marchioli & Soldati (Reference Marchioli and Soldati2002) and Soldati & Marchioli (Reference Soldati and Marchioli2009) observed that particle transport is primarily controlled by quasi-streamwise vortex and coherent flow motions (sweep/ejection events) in the wall-bounded turbulence. In the proximity of the buffer region, particles cannot effectively follow the long turbulent quasi-streamwise vortical structures due to their inertia and tend to aggregate outside the vortex structures. Subsequently, sweep events transfer the particles closer to the wall. However, the spanwise fluid motions near the wall diminish the potential for ejection events to transport particles away from the wall, ultimately leading to the near-wall accumulation of inertial particles. Particles that stay at the wall for long residence times tend to aggregate in low-velocity regions and form elongated particle streaks along the flow direction (Sardina et al. Reference Sardina, Picano, Schlatter, Brandt and Casciola2011, Reference Sardina, Schlatter, Brandt, Picano and Casciola2012a ).
Turbulent flows laden with particles in industrial processes and natural environments are often bounded by rough walls. Such rough surfaces are often highly complex and require multiple morphological parameters to characterise their properties, such as roughness height, frontal solidity, effective slope, plan solidity, skewness and solid volume fraction (Chung et al. Reference Chung, Hutchins, Schultz and Flack2021). These parameters govern the interaction between wall geometry and near-wall turbulent structures, thereby influencing flow separation, reattachment, wake formation, turbulence production and momentum transfer (Jiménez Reference Jiménez2004; Flack & Schultz Reference Flack and Schultz2014; MacDonald et al. Reference MacDonald, Chan, Chung, Hutchins and Ooi2016; Bruno et al. Reference Bruno, Leonardi and De Marchis2024). Compared with smooth-wall turbulent flow, rough walls can significantly alter the coherent structures of turbulence and the drag coefficients in wall turbulence (Jiménez Reference Jiménez2004; Chung et al. Reference Chung, Hutchins, Schultz and Flack2021). It has been revealed experimentally and numerically that rough walls reduce the streamwise length of near-wall turbulent structures (Orlandi et al. Reference Orlandi, Leonardi and Antonia2006; Volino et al. Reference Volino, Schultz and Flack2007; Chan et al. Reference Chan, MacDonald, Chung, Hutchins and Ooi2018), increase the inclination of quasi-streamwise vortex structures (Krogstad & Antonia Reference Krogstad and Antonia1994; Coceal et al. Reference Coceal, Dobre, Thomas and Belcher2007) and weaken the coherent structures of turbulence (MacDonald et al. Reference MacDonald, Chan, Chung, Hutchins and Ooi2016; Chan et al. Reference Chan, MacDonald, Chung, Hutchins and Ooi2018). In addition to altering the carrier phase, wall roughness significantly modifies the properties of the dispersed phase. Early works on particle–wall interactions have adopted a stochastic approach, neglecting the deterministic shape of the irregular wall structure (Sommerfeld & Huber Reference Sommerfeld and Huber1999; Squires & Simonin Reference Squires and Simonin2006; Konan et al. Reference Konan, Kannengieser and Simonin2009; Mando & Yin Reference Mando and Yin2012). Without details of wall asperities, the change of the turbulent flow field cannot be resolved and, thus, the realistic trajectory of rebounding particles cannot be captured.
Recent studies have modelled the actual geometric profiles of rough surfaces, considering the effect of these surfaces on coherent flow structures and particle dynamics. Milici et al. (Reference Milici, De Marchis, Sardina and Napoli2014) and De Marchis et al. (Reference De Marchis, Milici, Sardina and Napoli2016) investigated particle-laden turbulent flow in a channel bounded by irregular two-dimensional rough surfaces and found that wall roughness induces a reduction of particle accumulation in the near-wall region and enhances the streamwise transport velocities of particles. They attributed these changes to alterations in turbulent coherent structures and particle–wall interactions (collision processes and collision frequency), thus affecting the particle transport process; consequently, inertial particles tend to accumulate at the centre of the channel. In the turbulent boundary layer over a rough wall, where the roughness is represented by a series of staggered arranged hemispherical elements, Luo et al. (Reference Luo, Dai, Liu and Fan2019) similarly observed a more uniform distribution of particles along the wall-normal direction. They noted that the increased frequency of particle–wall collisions, the destruction of vortices by wall roughness, the strong shear layer generated at the top of the roughness elements, and the vortical structures disrupted by particle back-reaction, all contribute to the reduction in near-wall particle accumulation. Chan et al. (Reference Chan, Zahtila, Ooi and Philip2021) analysed the transport and deposition mechanisms of inertial particles in three-dimensional sinusoidal roughness turbulent pipe flows, while varying both the roughness size and the Stokes number of the particles. They observed that the transport of particles in turbulent rough-wall flow is closely related to particle inertia: small inertial particles are primarily controlled by turbulent structures, large inertial particles are predominantly influenced by particle–wall collisions and moderate inertial particles experience a combined effect of turbulent structures and wall collisions. Saccone et al. (Reference Saccone, De Marchis, Milici and Marchioli2023) examined the transport of rigid elongated particles in turbulent channel flows and revealed that roughness elements significantly modulate their distribution and orientation. However, in these studies, collisions between inertial particles and curved rough-wall surfaces can directly enhance the wall-normal rebound velocities of particles, thereby suppressing near-wall accumulation through a purely kinematic mechanism and promoting wall-normal particle transport. Consequently, the effects of particle–wall collisions and roughness-induced turbulence modifications remain intrinsically coupled and difficult to disentangle. In contrast, spanwise rod-roughened elements with flat surfaces do not significantly amplify the wall-normal particle velocities during collisions, thereby largely suppressing direct collision-induced particle redistribution. This configuration therefore provides an opportunity to isolate the influence of roughness-modified coherent turbulent structures on inertial-particle transport. The present study is aimed at clarifying how particle distribution and transport are altered in rough-wall turbulence when the direct kinematic effects of particle–wall collisions are minimised.
Motivated by the above questions, this work investigated the effects of spanwise rod-roughened elements on the dynamics of the dispersed phase in particle-laden turbulent channel flows, mainly focusing on the spatial distribution and transport mechanisms of spherical particles with different inertia. The remainder of this paper is organised as follows. In § 2, the governing equations for the carrier and dispersed phase are described, along with the detailed numerical methods. Section 3 presents the statistics of fluid and particles, focusing particularly on the mechanisms of particle transport. Conclusions are drawn in § 4.
2. Methodology
2.1. Eulerian fluid representation
The present study investigates incompressible Newtonian fluid flows in a turbulent channel with rough walls. The governing equations of the mass and momentum conservation are
\begin{equation} \begin{aligned} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u} &= 0, \\ \frac {\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{u} &= - \frac {1}{\rho _{\!f}} \boldsymbol{\nabla }\!p + \nu {\nabla} ^2 \boldsymbol{u} + \boldsymbol{f}_{\textit{IB}}, \end{aligned} \end{equation}
where
$\boldsymbol{u}$
,
$p$
,
$\rho _{\!f}$
, and
$\nu$
represent the velocity, pressure, density and dynamic viscosity of the fluid, respectively. The rough walls are represented using an immersed boundary method (IBM) based on the discrete direct-forcing approach (Huang & Sung Reference Huang and Sung2007). In this method, the roughness geometry is analytically prescribed and represented on a fixed staggered Cartesian grid through forcing points located immediately adjacent the immersed boundary inside the solid region (see figure 1
c). The spacing of the forcing points is identical to the local Eulerian grid spacing near the wall. The IBM introduces an additional forcing term
$\boldsymbol{f}_{\textit{IB}}$
into the momentum equation to enforce the no-slip and non-penetration boundary conditions on the rough surfaces. Following Huang & Sung (Reference Huang and Sung2007), the forcing term is evaluated as
\begin{equation} \boldsymbol{f}_{\textit{IB}} = \frac {\boldsymbol{U}^{\#}-\boldsymbol{u}^{\#}}{\Delta t}, \end{equation}
where
$\boldsymbol{u}^{\#}$
is the predicted velocity obtained without the immersed-boundary forcing by the forward Euler explicit scheme,
$\boldsymbol{U}^{\#}$
is the desired velocity at the forcing points reconstructed through second-order bilinear interpolation together with the prescribed wall velocity and
$\Delta t$
is the time increment step. The forcing term is subsequently added to the momentum equation such that the corrected velocity field satisfies the no-slip condition at the immersed boundary wall. Furthermore, no retraction distance was introduced when imposing the immersed-boundary forcing. Periodic boundary conditions are imposed in both the streamwise and spanwise directions. The diffusion and convection terms are treated using the Crank–Nicolson discretisation (Kim et al. Reference Kim, Baek and Sung2002).
(a) Channel with rough walls; (b) schematic diagram of the roughness elements; (c) distribution of the immersed-boundary forcing points (marked by red points) for the streamwise velocity component on the staggered Cartesian grid near the roughness elements (represented by blue lines).
Figure 1. Long description
Panel A: A three-dimensional representation of a channel with rough walls. The walls are depicted with a series of rectangular protrusions arranged in a staggered pattern. The coordinate axes x, y, and z are labeled. Panel B: A schematic diagram of the roughness elements. The diagram shows two rectangular blocks with dimensions labeled as k for height and w for width. The distance between the blocks is labeled as λT. The coordinate axes x and y are labeled. Panel C: A two-dimensional grid showing the distribution of immersed-boundary forcing points for the streamwise velocity component. The grid is marked with red points indicating the forcing points and blue lines representing the roughness elements. The coordinate axes x and y are labeled with units in centimeters.
As illustrated in figure 1(a), the roughness elements are periodically arranged in the streamwise direction while homogeneously applied along the spanwise direction and symmetrically distributed on the upper and lower walls. Here
$x$
,
$y$
and
$z$
denote the streamwise, wall-normal and spanwise directions of an inertial Cartesian frame, respectively. The height and width of the roughness elements are both denoted as
$k$
, and the streamwise spacing between the roughness elements is represented as
$w$
, as shown in figure 1(b). Therefore, the roughness element period is given by
$\lambda _T = k + w$
, and the ratio of spacing to height is
$w/k$
. It has been indicated that the ratio of roughness spacing to height is a crucial parameter for investigating the modulation of turbulent coherent structures and drag coefficients in such a rough-wall channel flow (Leonardi et al. 2003, 2004, 2007). In the current simulation cases, the height and width of the roughness elements are set to
$k = 0.1h$
(where
$h$
represents half the channel height). The ratio
$w/k$
is varied as
$1, 3, 5, 7, 9, 11, 99$
and
$199$
, aiming to explore how the dynamics of inertial spherical particles in turbulent channel flows are affected by roughness elements with the same height but different streamwise spacings, while focusing on the underlying mechanisms. According to Perry et al. (Reference Perry, Schofield and Joubert1969) and Bandyopadhyay (Reference Bandyopadhyay1987), a ribbed wall features k-type roughness if
$w/k \geqslant 3$
, but transitions to d-type roughness if
$w/k \leqslant 2$
. The rough-wall cases considered in the present study were designed to cover both flow regimes.
The turbulent channel flow is driven under a constant bulk flow-rate condition. For different roughness cases, the fluid viscosity
$\nu$
is adjusted accordingly to ensure that all simulations achieve the same friction Reynolds number. The friction Reynolds number based on the frictional velocity
$u_\tau$
and the half-height of the channel
$h$
is
$Re_\tau = 180$
, with
$u_\tau = \sqrt {-h/\rho _{\!f}\partial {P}/\partial x }$
and
$\partial {P}/\partial x$
is the average pressure gradient in the streamwise direction. The computational domain has dimensions of
$L_x \times L_y \times L_z = 10h \times 2h \times 4h$
. The uniform grid has a size of
$N_x \times N_y \times N_z = 512 \times 400 \times 128$
, with the corresponding grid spacing as
$ \Delta x^+ \times \Delta y^+ \times \Delta z^+ = 3.52 \times 0.9 \times 5.6$
. Specifically, for the case with the maximum ratio (
$w/k=199$
), the streamwise computational domain spans
$L_x = 20h$
with
$N_x = 1024$
grids, while maintaining all other settings identical to those mentioned earlier. The sufficiently high streamwise resolution is capable of avoiding spurious numerical oscillations near the leading edge of the roughness elements (Ashrafian et al. Reference Ashrafian, Andersson and Manhart2004; Krogstad et al. Reference Krogstad, Andersson, Bakken and Ashrafian2005). The wall units are defined with respect to the bottom reference plane of the rough-wall channel, where
$y^+=0$
corresponds to the lowest wall location. The roughness height is
$k=0.1h$
, corresponding to
$k^+\approx 18$
, such that the roughness crests are located approximately at
$y^+\approx 18$
. The grid resolution in the wall-normal direction is
$\Delta y^+=0.9$
, corresponding to approximately 20 grid points across the roughness height in the wall-normal direction. This wall-normal resolution of
$\Delta y^+=0.9$
is comparable to or finer than those commonly used in previous direct numerical simulation (DNS) studies of rough-wall turbulent channel flows (Leonardi et al. 2003, 2004; Bruno et al. Reference Bruno, Leonardi and De Marchis2024). The adopted wall-normal grid resolution is therefore sufficient to accurately resolve the near-wall turbulent flow structures and velocity gradients relevant to particle transport.
2.2. Lagrangian particle dynamics
The mass of a single particle is
$m = 4 \pi a^3 \rho _{\!p} /3$
, where
$\rho _{\!p}$
is the density of the particle while
$a$
is the radius of the particle. A high particle-to-fluid density ratio was adopted to represent heavy solid particles suspended in a gas or other low-density carrier fluid, such as dust, sand, ash or small solid contaminants transported by air. This choice is commonly used in point-particle modelling to ensure that the particles have appreciable inertia while their diameters remain sufficiently small relative to the smallest turbulent length scale (Voth & Soldati Reference Voth and Soldati2017). Since the particle density is much larger than the fluid density, the contributions of buoyancy, pressure-gradient, added-mass and Basset history forces are expected to be negligible (Elghobashi & Truesdell Reference Elghobashi and Truesdell1992; Daitche Reference Daitche2015; van Hinsberg et al. Reference van Hinsberg, Clercx and Toschi2017; Li et al. Reference Li, Wei, Bu and Yu2021b
). The magnitude ratio of the Saffman lift force to the Stokes drag force is approximately proportional to the particle radius,
$a^+$
, and the square root of the local fluid vorticity,
$\lvert \mathbf{\varOmega } ^+ \lvert ^{0.5}$
(Costa et al. Reference Costa, Brandt and Picano2020). Our quantitative analysis, not shown here, indicates that the shear layers induced by the roughness elements have only a modest effect on this ratio compared with the smooth-wall case. Since all particle radii considered in the present simulations are on the order of
$0.1$
in wall units, the Saffman lift force remains much smaller than the Stokes drag force (Costa et al. Reference Costa, Brandt and Picano2020). Accordingly, the lift force is neglected in this study, while one should note that the effect of the wall-normal lift force may arise for the particles near the top plane of the roughness element. The particle Froude numbers
$Fr_{\!p} = U_b /g \tau _{\!p}$
, representing the ratio of drag force and gravity, are all greater than
$1$
, where
$U_b$
is the bulk velocity of the flow and
$\tau _{\!p}$
is the response time of the particle defined later. It means that the effect of gravity on the transport of particles is smaller compared with the effect of the bulk flow motion induced drag force (Sardina et al. Reference Sardina, Schlatter, Picano, Casciola, Brandt and Henningson2012b
; Milici et al. Reference Milici, De Marchis, Sardina and Napoli2014). Therefore, only the Stokes drag with a semi-empirical correction (Schiller & Maumann Reference Schiller and Maumann1933) is considered in the point-particle model in the present study. The governing equations of the particles are
\begin{equation} \begin{aligned} u_{\textit{p,i}} & = \frac {\mathrm{d} x_{\textit{p,i}}}{\mathrm{d} t}, \\[3pt] \frac {\mathrm{d} u_{\textit{p,i}}}{\mathrm{d} t} & = \frac {1}{\tau _{\!p}} (u_{\!f@\textit{p,i}}-u_{\textit{p,i}})\left(1+0.15Re_{\!p}^{0.687}\right), \end{aligned} \end{equation}
where
$x_{\textit{p,i}}$
is the location of the particle,
$u_{\textit{p,i}}$
is the velocity of the particle, while
$u_{\!f@\textit{p,i}}$
is the fluid velocity seen by the particle. Here
$\tau _{\!p} = 2a^2 \rho _{\!p} / 9 \rho \nu$
is the response time of the particle and the Stokes number of a particle is defined as
$St = \tau _{\!p} / \tau _\nu$
, where
$ \tau _\nu$
is viscous fluid time scale. The fluid velocity seen by the particle,
$u_{\!f@\textit{p,i}}$
, is obtained from the Eulerian velocity field using a second-order Lagrangian spatial interpolation scheme based on the surrounding Eulerian grid points. The particle volume fraction is sufficiently low such that both the particle feedback on the carrier flow and particle–particle interactions are negligible. Under these dilute conditions, the particle transport behaviour is primarily governed by the roughness-induced modification of the turbulent flow structures rather than by particle back-reaction (De Marchis & Milici Reference De Marchis and Milici2016). The collision between a particle and the wall is modelled as a fully elastic collision. Finite-size effect is taken into account by treating particles as spherical particles with a finite radius. A particle–wall collision is assumed to occur when the distance between the particle centre and the wall surface becomes smaller than the particle radius. Upon collision, the local wall-normal velocity component (perpendicular to the local wall surface) is reversed while the local tangential velocity components (parallel to the local wall surface) remain unchanged. No soft-sphere or penalty-force model of particle–wall interaction is employed in the present simulations. To ensure the validity of the point-particle approach, the grid resolution and particle sizes must be carefully assessed against the smallest turbulence length scale. In the present study, the local Kolmogorov length scale
$\eta ^+$
ranges from approximately
$1.5$
near the wall to
$3.8$
in the channel centre. The maximum dimensionless particle radius investigated is
$a^+=0.577$
, which remains strictly smaller than the minimum Kolmogorov scale, thereby justifying the sub-Kolmogorov particle assumption. An explicit second-order Adams–Bashforth scheme is employed to solve (2.3).The simulation parameters are provided in table 1. Particles are released randomly in the computational domain when the turbulent channel flow is fully developd (
$t^+ = 0$
). Subsequently, statistics are collected over a duration of
$T_{\textit{span}}^+ \gt 10\,000$
after the particle distribution reaches a steady state (see figure 6). The number of each type of particle is
$N_{\!p}=5 \times 10^5$
.
Parameters for inertial spherical particles.
Table 1. Long description
A table comparing parameters for inertial spherical particles. The table has five columns and seven rows. The columns are labeled St, ρp/ρf, a/h, and a+. The rows present values for different Stokes numbers (St) ranging from 1 to 200. Row 1: St, 1; ρp/ρf, 2700; a/h, 2.27 x 10-4; a+, 4.09 x 10-2. Row 2: St, 5; ρp/ρf, 2700; a/h, 5.07 x 10-4; a+, 9.13 x 10-2. Row 3: St, 10; ρp/ρf, 2700; a/h, 7.17 x 10-4; a+, 1.29 x 10-1. Row 4: St, 30; ρp/ρf, 2700; a/h, 1.24 x 10-3; a+, 2.24 x 10-1. Row 5: St, 50; ρp/ρf, 2700; a/h, 1.60 x 10-3; a+, 2.89 x 10-1. Row 6: St, 100; ρp/ρf, 2700; a/h, 2.27 x 10-3; a+, 4.08 x 10-1. Row 7: St, 200; ρp/ρf, 2700; a/h, 3.21 x 10-3; a+, 5.77 x 10-1.
3. Results and discussion
3.1. Fluid statistics and flow structures
As described in § 2.1, the roughness elements are periodically arranged in the streamwise direction while homogeneously applied along the spanwise direction. Therefore, by using the triple decomposition (Reynolds & Hussain Reference Reynolds and Hussain1972), an instantaneous physical quantity
$f(x, y, z)$
can be decomposed as
\begin{equation} f(x,y,z)=\left \langle f \right \rangle (x,y)+f^\prime (x,y,z,t) =\overline {f} (y)+f_r (x, y) + f^\prime (x,y,z,t). \end{equation}
Here,
$\langle f\rangle$
denotes the phase-averaged component, obtained by averaging in time, along the
$z$
direction, and further at the streamwise rough periods. The global-averaged component
$\overline {f}$
is calculated by averaging
$ \langle f\rangle$
along the
$x$
direction. We use
$f_r$
and
$f^\prime$
to denote the rough-induced and turbulent fluctuations, respectively. The former is the difference between
$\langle f \rangle$
and
$\overline {f}$
, equal to 0 in smooth-wall flows.
Figure 2(a) shows profiles of the global-averaged streamwise velocity normalised by the frictional velocity
$u_\tau$
for the smooth and rough cases. The presence of rough walls makes the velocity smaller than that of smooth-wall flow. In the near-wall region, the negative velocity indicates there are intense recirculation flows for
$1 \leqslant w/k \leqslant 11$
. But near the channel centre, the profiles still satisfy the logarithmic law with a slope of around 0.4. In these regions, the results for
$w/k=1$
and
$w/k=199$
are closer to that of the smooth wall, while those for
$w/k=7, 9, 11$
are inversely. The enlarged view shows that the profile for
$w/k=9$
exhibits the greatest deviation from that of the smooth-wall case. The downward shift of
$\overline {u}^+$
suggests the effect of drag increase by the rod-roughened wall. The friction drag coefficient
$C_{\!f}^*$
and form drag coefficient
$C_{\!p}^*$
are defined as
\begin{equation} C_{\!f}^*=\lambda _T^{-1}\int _{0}^{\lambda _T} \left \langle \tau _w^* \right \rangle \,{\mathrm{d}} x, \end{equation}
\begin{equation} C_{\!p}^* = {\lambda _T ^{ - 1}}\int _0^{k} \langle {\Delta p^*}\rangle\, {\mathrm{d}}y. \end{equation}
where
$\tau _w$
is the viscous shear stress along the
$x$
direction at the wall and
$\Delta p$
represents the pressure difference between the windward and leeward surfaces at the same wall-normal position. From figure 2(b), the drag coefficients vary drastically with
$w/k$
. The form and total drag coefficients increase first and then decrease, reaching the maximum around
$w/k=9$
. But
$C_{\!f}^*$
shows the exact opposite trend. For fully rough flows (
$w/k=5,7, 9,11$
), the viscous drag plays a negligible role compared with the form drag. Significantly, the total drag coefficients for rough walls are always larger than that for the smooth wall.
(a) Global-averaged streamwise velocity for the smooth and rough walls. The dashed and dash-dotted lines represent the logarithmic law
$\overline {u}^+=2.5 \ln y^+ +5.5$
and the height of roughness elements, respectively. An enlarged view of the local region is shown in the lower-right corner. (b) Friction, form (or pressure) and total drag coefficients versus
$w/k$
. The dotted line is the friction drag coefficient for the smooth-wall flow.
Figure 2. Long description
Panel A: A line graph shows the global-averaged streamwise velocity for smooth and rough walls. The x-axis is labeled y+ and the y-axis is labeled u+. Multiple lines represent different w/k values, with a legend indicating these values. The dashed and dash-dotted lines represent the logarithmic law and the height of roughness elements, respectively. An enlarged view of a local region is shown in the lower-right corner. Panel B: Another line graph displays friction, form (or pressure), and total drag coefficients versus w/k. The x-axis is labeled w/k and the y-axis is labeled with the coefficients. Different lines represent Cf, Cp, and Cf + Cp, with a legend indicating these coefficients. The dotted line represents the friction drag coefficient for the smooth-wall flow.
Figure 3 displays the turbulent velocity fluctuations and turbulent kinetic energy. The results for the smooth and rough walls coincide near the channel centre, but there are noteworthy differences in the buffer region (
$5\lt y^+\lt 30$
). Near the crest plane, the streamwise turbulence intensity and turbulent kinetic energy both increase with
$w/k$
and gradually approach the results of smooth-wall flow. The spanwise and wall-normal intensities reach the maximum at
$w/k=99$
, which is larger than that for the smooth case. Overall, the results for
$w/k=199$
are closest to the smooth-wall case, while those for
$w/k=1$
are the opposite. In wall-bounded turbulent flows, the wall accumulation of inertial particles is closely related to turbulence inhomogeneity and the associated turbophoretic mechanism (Caporaloni et al. Reference Caporaloni, Tampieri, Trombetti and Vittori1975; Reeks Reference Reeks1983), according to which particles preferentially migrate toward regions of lower turbulent kinetic energy. Since the turbulent kinetic energy is strongly modified by the rough walls (figure 3
d), especially the appearance of local minima near the crest plane, the wall-normal transport and accumulation of particles are also expected to differ significantly from those in smooth-wall turbulence.
Global-averaged (a–c) turbulent velocity fluctuations and (d) turbulent kinetic energy.
The roughness elements disrupt the uniformity of turbulent flow in the
$x$
direction. The phase-averaged streamlines and streamwise velocity are shown in figure 4. The separation of mean flows occurs at the trailing edge of the crest, forming a recirculation region in the cavity (Leonardi et al. 2003, 2007). When the roughness spacing is minimum (
$w/k=1$
), a stable separation vortex exists within the recirculation zone, and the streamline and streamwise velocity above the crest plane exhibit minor variation along the streamwise direction. It indicates that in the case of closely packed roughness elements, the rough-wall channel flow resembles turbulent flows over smooth walls positioned on the crest plane. Such behaviour is indicative of a d-type rough-wall flow pattern. For d-type roughness, a stable recirculation vortex is well extended and occupies almost the entire cavity between adjacent ribs, while the mean outer flow skims over the cavity rather than penetrating into it (Perry et al. Reference Perry, Schofield and Joubert1969; Bandyopadhyay Reference Bandyopadhyay1987; Mahmoodi-Jezeh & Wang Reference Mahmoodi-Jezeh and Wang2022). As
$w/k$
increases, the vortex core of a single separation vortex gradually moves closer to the windward surface, eventually evolving into two separation vortices of different sizes. At the same time, the streamlines separating above the roughness elements gradually approach the bottom wall until reattachment occurs. In addition, the variations of the streamwise velocity become more pronounced along the flow direction. The mean flow patterns in figure 4(b–f) are typical in k-type rough-wall flows. For k-type roughness, unstable vortices are generated by the ribs and shed upwards, forming roughness-sublayer structures, while the flow inside the cavities interacts strongly with the outer flow (Perry et al. Reference Perry, Schofield and Joubert1969; Shafi & Antonia Reference Shafi and Antonia1997; Leonardi et al. Reference Leonardi, Orlandi and Antonia2007). A comparison of figures 4(e) and 4(f) indicates that, when the spacing between roughness elements is sufficiently large, the separation vortices are already fully developed, whereas the separation and reattachment locations remain nearly unchanged. Moreover, the streamline patterns near the roughness elements exhibit only minor variations. Because the roughness-affected region occupies a smaller fraction of the computational domain in the case with
$w/k=199$
, the corresponding statistical results are closer to those of the smooth-wall case, as shown in figures 2 and 3.
Phase-averaged streamlines and streamwise velocity
$\langle u\rangle ^+$
for the rough cases: (a)
$w/k=1$
, (b)
$w/k=5$
, (c)
$w/k=9$
, (d)
$w/k=11$
, (e)
$w/k=99$
and (f)
$w/k=199$
.
Figure 4. Long description
Panel A: A contour plot shows phase-averaged streamlines and streamwise velocity for a rough case. The x-axis is labeled x/k, ranging from 2 to 12, and the y-axis is labeled y/h, ranging from 0.1 to 0.5. The plot features a color gradient from blue to red, indicating different velocity magnitudes, with contour lines showing streamline patterns. Panel B: Another contour plot with similar axes and labels, displaying distinct streamline patterns and velocity distributions. Panel C: A contour plot with the same axes and labels, showing different streamline and velocity patterns compared to Panels A and B. Panel D: A contour plot with identical axes and labels, featuring unique streamline and velocity distributions. Panel E: A zoomed-in contour plot with the x-axis labeled x/h, ranging from 4.5 to 6.5, and the y-axis labeled y/h, ranging from 0.1 to 0.5. It shows detailed streamline patterns and velocity distributions. Panel F: Another zoomed-in contour plot with the x-axis labeled x/h, ranging from 9.5 to 11.5, and the y-axis labeled y/h, ranging from 0.1 to 0.5. It displays distinct streamline patterns and velocity distributions.
Instantaneous iso-surfaces of
$Q^+ =0.01$
coloured by streamwise vorticity in the lower half channel for smooth-wall and rough-wall turbulent flows. Red and blue colours indicate positive and negative streamwise vorticity, respectively. (a) Smooth-wall turbulence. (b–g) Rough-wall turbulence with
$w/k=1$
,
$w/k=5$
,
$w/k=9$
,
$w/k=11$
,
$w/k=99$
and
$w/k=199$
, respectively.
Figure 5 presents the instantaneous vortical structures in smooth-wall and rough-wall turbulent flows, identified by the Q-criterion (Chong et al. Reference Chong, Perry and Cantwell1990). In the smooth-wall case, the vortical structures, particularly those in the near-wall region, are predominantly aligned in the streamwise direction and extend away from the wall at relatively small inclination angles, as shown in figure 5(a). For the smallest-spacing case (
$w/k=1$
), the vortical structures above the crest plane exhibit characteristics similar to those observed in the smooth-wall case. However, the high-speed fluid swept downwards by the quasi-streamwise vortices induces strong shear near the crest plane, leading to the formation of numerous small-scale spanwise vortices. As the ratio
$w/k$
increases, the number of vortical structures in rough-wall turbulence increases markedly, and their morphology becomes more complex and disordered, characterised by larger streamwise inclination angles and stronger three-dimensionality. For the largest-spacing case (
$w/k=199$
), the vortical structures far upstream and downstream of the roughness elements resemble those observed in the smooth-wall case. In contrast, near the roughness elements, the vortical structures exhibit larger streamwise inclination angles, and vortex shedding occurs from the roughness crests. Recently, Ge et al. (Reference Ge, Xiong, Xu and Wang2026) also observed the strong shear layers triggered by the rib crests in their experimental study of turbulent ribbed-duct flow. Therefore, the roughness elements primarily influence the distribution of vortical structures through two primary mechanisms. First, the quasi-streamwise vortices are directly perturbed by the roughness elements, resulting in more irregular and isotropic vortical structures (Leonardi et al. Reference Leonardi, Orlandi, Djenidi and Antonia2004; Bruno et al. Reference Bruno, Leonardi and De Marchis2024). Second, spanwise vortices, generated by strong shear layers, are shed from the roughness crests and subsequently develop downstream with larger inclination angles (Ikeda & Durbin Reference Ikeda and Durbin2007; Ge et al. Reference Ge, Xiong, Xu and Wang2026). The morphological changes in the quasi-streamwise vortices indicate that the roughness elements affect the near-wall self-sustaining process and modify the classical coherent structures characteristic in smooth-wall turbulence. Given the important role of these coherent structures in particle transport and accumulation (Marchioli & Soldati Reference Marchioli and Soldati2002; Jie et al. Reference Jie, Cui, Xu and Zhao2022), rough-wall turbulence is expected to exhibit distinct particle distribution patterns and transport mechanisms.
3.2. Particle distribution and velocity statistics
Numerous particles are released randomly into fully developed turbulent flows. Under the transport of turbulence, inertial particles tend to preferentially accumulate in specific regions. Here, we employ the Shannon entropy to characterise the particle accumulation in the normal direction. It is defined as (Picano et al. Reference Picano, Sardina and Casciola2009; Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012a )
\begin{equation} S = \zeta /{\zeta _{\textit{max}}}.\end{equation}
The entropy
$\zeta$
is
\begin{equation} \zeta =-\sum ^{N_{yp}}_{i=1}\frac {N_{\textit{p,i}}}{N_{\!p}}\ln {\frac {N_{\textit{p,i}}}{N_{\!p}}}. \end{equation}
Here,
$N_{yp}$
is the number of bins distributed uniformly in the normal direction,
$N_{\!p}$
is the total number of particles,
$N_{\textit{p, i}}$
is the average number of particles in the
$i$
th bin and the maximum entropy is
$\zeta _{\textit{max}}=\ln {N_{yp}}$
. A smaller Shannon entropy corresponds to a higher degree of segregation. When all particles accumulate within a specific grid layer, the Shannon entropy reaches its minimum value of 0. Conversely, when particles are uniformly distributed along the normal direction,
$S$
attains its maximum value of 1. The wall-normal direction is uniformly divided into
$N_{yp}=400$
bins, corresponding to a bin width of approximately
$\Delta y^+=0.9$
. Instantaneous particle data are collected over a sufficiently long time window after the particle distribution reaches a statistically stationary state. The statistical convergence of the particle distribution is assessed using the Shannon entropy, and all particle statistics are averaged after the Shannon entropy reaches a statistically stationary value.
Shannon entropy of mean particle concentration versus time for each particle population: (a) smooth-wall turbulence; (b–f) rough-wall turbulence with
$w/k=1$
, 5, 9, 11 and 99, respectively.
Figure 6. Long description
Six line graphs depict the Shannon entropy of mean particle concentration over time for different particle populations in smooth-wall and rough-wall turbulence. Panel A: Line graph showing Shannon entropy (S) versus time (t+) for smooth-wall turbulence. The x-axis represents time (t+) and the y-axis represents Shannon entropy (S). Different lines represent various particle populations with distinct Stokes numbers (St). Panel B: Line graph showing Shannon entropy (S) versus time (t+) for rough-wall turbulence with a specific roughness height. The x-axis represents time (t+) and the y-axis represents Shannon entropy (S). Different lines represent various particle populations with distinct Stokes numbers (St). Panel C: Line graph showing Shannon entropy (S) versus time (t+) for rough-wall turbulence with a different roughness height. The x-axis represents time (t+) and the y-axis represents Shannon entropy (S). Different lines represent various particle populations with distinct Stokes numbers (St). Panel D: Line graph showing Shannon entropy (S) versus time (t+) for rough-wall turbulence with another roughness height. The x-axis represents time (t+) and the y-axis represents Shannon entropy (S). Different lines represent various particle populations with distinct Stokes numbers (St). Panel E: Line graph showing Shannon entropy (S) versus time (t+) for rough-wall turbulence with yet another roughness height. The x-axis represents time (t+) and the y-axis represents Shannon entropy (S). Different lines represent various particle populations with distinct Stokes numbers (St). Panel F: Line graph showing Shannon entropy (S) versus time (t+) for rough-wall turbulence with the highest roughness height. The x-axis represents time (t+) and the y-axis represents Shannon entropy (S). Different lines represent various particle populations with distinct Stokes numbers (St).
Figure 6 presents the time evolution of the Shannon entropy for inertial particles in the smooth and rough walls. At time
$t^+=0$
, Shannon entropy
$S \approx 1$
shows that particles of each type are almost uniformly distributed in the computational domain. Subsequently, the phenomenon of particle accumulation is demonstrated by the declining Shannon entropy. Finally, the convergence of Shannon entropy indicates that the particle distribution reaches a eventual steady state. It is observed that inertial particles attain such state more rapidly in rough-wall flows, displaying a reduced tendency for preferential accumulation. For the smooth-wall case (figure 6
a), the lightest particles (
$St=1$
) have the largest Shannon entropy and the most uniform distribution in the wall-normal direction, whereas particles with
$St=30$
exhibit the opposite behaviour. This is consistent with the results reported by previous studies (Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012a
; Bernardini Reference Bernardini2014; Motoori et al. Reference Motoori, Wong and Goto2022). However, for the rough-wall cases, the influence of particle inertia on the degree of segregation is more complex. The results presented below for the particles are all within a statistically steady state. To understand the smaller particle entropy observed at
$St=5$
and
$10$
in figure 6, we evaluated the characteristic time scale of the separation vortices between roughness elements. This characteristic time is defined based on the fluid angular velocity at the core of the phase-averaged separation vortex, i.e.
$T_e^+=1/\omega ^+$
. If two separation vortices exist within the cavity, the larger one near the windward face of the roughness element is selected. Results show that
$T_e^+$
ranges from
$4.8$
to
$9.3$
across different
$w/k$
cases. The relaxation times of particles with
$St=5$
and
$10$
are comparable to these characteristic time scales, yielding a local effective Stokes number of
$O(1)$
, which promotes the preferential accumulation of these particles near the roughness elements, thereby reducing the overall particle entropy.
Instantaneous distribution of particles with
$St=30$
in the
$x{-}y$
plane: (a) smooth-wall turbulence; (b–f) rough-wall turbulence with
$w/k=1$
, 5, 9, 11 and 99, respectively. The colour contours in the background show the instantaneous streamwise velocity
$u^+$
.
It is noted that the Shannon entropy
$S$
only provides the global feature of particle distribution in the wall-normal direction. Figure 7 further shows the spatial distribution of particles with
$St=30$
in cross-sectional view. The black dots represent the particles located within a spanwise layer of thickness 20 wall units. For the smooth-wall case, the majority of particles accumulate in the near-wall region, while a small number of particles are inhomogeneously distributed in the core region. The presence of rough elements not only alters the instantaneous velocity field and turbulent structure of the fluid but also significantly influences the spatial distribution of inertial particles. In comparison with the smooth-wall case, the wall accumulation of particles has weakened, and the number of particles near the channel centre has increased in the rough-wall case. In the near-wall region, inertial particles are affected by the separation vortex (figure 4) and quasi-spanwise turbulent vortices (figure 5), and thus tend to accumulate in low-speed regions around the roughness elements. Away from the wall, inertial particles are driven by centrifugal effects induced by instantaneous turbulent vortical structures (figure 5), preferentially accumulating in low-vorticity regions outside the vortex cores and thereby forming distinct clustering. These results qualitatively demonstrate that the wall accumulation of inertial particles in turbulent channel flows with rod-roughened walls has weakened, leading to a more uniform spatial distribution.
Normalised phase-averaged concentration of particles with
$St = 30$
in the near-wall region for the rough cases: (a)
$w/k=1$
, (b)
$w/k=5$
, (c)
$w/k=9$
, (d)
$w/k=11$
and (e)
$w/k=99$
.
To focus on the spatial distribution of inertial particles near the roughness elements, figure 8 presents the phase-averaged concentration of particles with
$St = 30$
in the near-wall region of rough-wall turbulence. The mean concentration is normalised by the bulk concentration,
$C_0$
, defined as the ratio of the total number of particles to the domain volume. For d-type roughness (
$w/k = 1$
), inertial particles accumulate near the leeward side and the crest plane, while remaining depleted on the windward side. Above the crest plane, the particle concentration decreases with increasing wall-normal distance but exhibits little variation in the streamwise direction, resembling the distribution characteristics of inertial particles in smooth-wall turbulence. For k-type roughness (
$w/k \geqslant 3$
), the particle distribution near the roughness elements changes more markedly. As the roughness spacing increases, the preferential accumulation of particles shifts from the leeward side to the windward side. When the roughness spacing is sufficiently large, as shown in figure 8(e), particles mainly accumulate near the windward side, whereas almost no particles are observed near the leeward side. In rough-wall turbulence, the near-wall particle distribution is closely associated with the mean flow structures, i.e. the separation vortices. Particles are initially entrained by the upstream flow into the cavity and are subsequently strongly influenced by the separation vortices. Owing to the finite inertia, the particles cannot fully follow the curved vortex streamlines and are therefore prone to colliding with the windward side of the roughness elements. After collision, the particles move against the streamwise flow. Their momentum is subsequently dissipated by the low-velocity fluid near the leeward side of the roughness elements, causing them to remain in this region. However, as the roughness spacing increases, reattachment regions form between ribs, as shown in figure 4(d–f), making it difficult for particles to cross these regions and reach the leeward side. Consequently, inertial particles are transported back toward the windward side by the flow, leading to preferential accumulation in this region.
Mean normalised particle concentration
$\overline {C/C_0}$
as a function of wall distance
$y^+$
: (a) smooth-wall turbulence; (b–f) rough-wall turbulence with
$w/k=1$
, 5, 9, 11 and 99, respectively.
Figure 9 quantitatively presents the concentration distribution of inertial particles along the wall-normal direction in the smooth-wall and rough-wall flows. The particle number per unit volume
$C$
is normalised by the bulk concentration
$C_0$
. For the smooth-wall case, inertial particles preferentially accumulate in the wall region and the particle concentration
$\overline {C/C_0}$
attains its maximum near the wall. The distribution of the lightest particles (
$St=1$
) is relatively uniform, with a ratio of maximum concentration to minimum concentration less than 10. Particles with the intermediate Stokes number (
$10\leqslant St \leqslant 50$
) exhibit significant wall accumulation. These particles display a concentration surpassing 100 near the wall, which is approximately 1000 times higher than that in the channel centre. The wall accumulation of heavy particles (
$St=$
100, 200) begins to weaken, with a ratio in the order of hundreds between the maximum and minimum concentration. The same effect of particle inertia on wall accumulation in is also observed by Sardina et al. (Reference Sardina, Schlatter, Brandt, Picano and Casciola2012a
) and Marchioli & Soldati (Reference Marchioli and Soldati2002). However, as figure 9(b–f) shows, regardless of
$w/k$
, the distribution of inertial particles in the rough-wall case is significantly more uniform than that in the smooth-wall case. Specifically, the particle concentration near the wall is decreased, while that near the channel centre is increased. Milici et al. (Reference Milici, De Marchis, Sardina and Napoli2014), Luo et al. (Reference Luo, Dai, Liu and Fan2019) and Chan et al. (Reference Chan, Zahtila, Ooi and Philip2021) also reported analogous phenomena in rough-wall turbulent flows with different rough forms. In the region below the crest plane of square-shaped rough elements, particles with different inertia exhibit distinct concentration differences. Overall, with the increase in roughness spacing, the concentration difference in this region gradually becomes larger, and the inertial effects continue to strengthen. Above the crest plane, the particle distribution is relatively uniform and the difference between inertial particles is subtle.
The concentration distribution and velocity profile (not depicted here) of inertial particles are influenced by the presence of rough elements. Consequently, one anticipates a corresponding alteration in the efficiency of wall turbulence to transport the particle phase. The bulk velocity
$U_{b,p}$
introduced by Chan et al. (Reference Chan, Zahtila, Ooi and Philip2021) is adopted to represent the transport velocity of the particle phase. It is defined as the number average and time average of the streamwise velocities of all particles:
\begin{equation} U_{b,p}=\frac {1}{T_n N_{\!p}} \sum _{j=1}^{T_n} \sum _{i=1}^{N_{\!p}}u_{\!p}^{i,j}. \end{equation}
Here,
$u_{\!p}^{i,j}$
is the streamwise velocity of the
$i$
th particle in the
$j$
th time sample,
$N_{\!p}$
is particle numbers and
$T_n$
is the number of time samples in the statistical steady state. Figure 10(a) shows the particle bulk velocity normalised by
$U_b$
. In the smooth-wall case,
$U_{b,p}^*$
initially decreases and subsequently increases with particle inertia, attaining the maximum of 0.93 and minimum of 0.13 at
$St=$
1 and 30, respectively. The observed trend aligns with the degree of particle accumulation near the wall. Particles with small inertia easily follow the fluid and have a uniform spatial distribution, resulting in their bulk velocity closely matching that of the fluid. Particles with moderate inertia tend to accumulate near the wall, where the fluid velocity is very small. As a result, their bulk velocity is significantly lower than
$U_b$
. In rough-wall cases, the differences between particles with different inertia are not pronounced, as
$U_{b,p}^*$
spans from 0.45 to 0.95. This is mainly attributed to the more uniform distribution of inertial particles in the wall-normal direction in rough-wall flows, especially in regions above the crest plane of roughness elements. With the exception of particles with
$St=1$
, the ratio
$U_{b,p}/U_b$
for medium and large inertia particles in rough walls is greater than that in the smooth wall. The reduction in velocity for small inertia particles is primarily due to their tendency to accumulate near the windward and leeward sides of roughness elements, particularly in the near-wall regions. For particles with medium and large inertia, their distribution in rough walls is more uniform, making the bulk velocity of particle phase closer to that of the fluid phase. This is qualitatively consistent with the results in turbulent flows over curved rough walls (Milici et al. Reference Milici, De Marchis, Sardina and Napoli2014; Chan et al. Reference Chan, Zahtila, Ooi and Philip2021). Considering that
$U_{b,p}^*$
in smooth-wall turbulence increases with particle inertia after decreasing, while the inertial effect on this velocity is weaker in rough-wall turbulence, it can be speculated that the mean velocity in the smooth wall would exceed the rough-wall results when particle inertia is sufficiently high.
The velocity
$U_{b,p}$
indicates the degree of proximity between the particle bulk velocity and fluid bulk velocity. However, it does not consider the alterations in drag coefficients caused by roughness elements (figure 2). The particle bulk velocity normalised by the frictional velocity
$u_\tau$
can more accurately reflect the effect of roughness elements on particle transport efficiency. The bulk velocity
$U_{b,p}^+$
of inertial particles is shown in figure 10(b). In general, in comparison with the smooth wall, the existence of rough walls still reduces the bulk velocity of particles with low Stokes number, and increases it for particles with medium and high Stokes number. Nevertheless, the significant increase in the drag coefficient restricted the bulk velocity
$U_{b,p}^+$
in the rough walls. In some rough-wall cases, the bulk velocity of particles with
$St=5$
or
$St=200$
is even lower than that in the smooth-wall case. Generally, the spanwise-uniform square roughness elements make the bulk velocity of particles with medium inertia larger than that of the fluid phase, but their obstruction to fluid flow reduces the degree of improvement in particle transport efficiency. However, it is noteworthy that for cases with large
$w/k$
, the rough walls can enhance the transport efficiency for most types of particles. As observed in figure 10(b), the
$U_{b,p}^+$
values for large
$w/k$
cases generally exceed those in the smooth wall across the range of Stokes numbers except the case with
$St=1$
. This indicates that wider roughness spacing can effectively promote particle transport efficiency, outweighing the drag penalty. To the best of the authors’ knowledge, such an enhancement of particle transport across a wide range of inertia, attributed solely to the modification of near-wall coherent structures by rough walls, has not been reported in previous studies.
Particle bulk velocity normalised by (a) the outer scale (
$U_b$
) and (b) the inner scale (
$u_\tau$
) in the smooth and rough walls.
3.3. Mechanisms of particle transport
The results presented in the preceding section confirm that the presence of roughness elements modifies the spatial distribution characteristics of inertial particles, consistent with previous studies (Milici et al. Reference Milici, De Marchis, Sardina and Napoli2014; Luo et al. Reference Luo, Dai, Liu and Fan2019; Chan et al. Reference Chan, Zahtila, Ooi and Philip2021). This alteration reduces particle accumulation near the wall and induces a more uniform distribution in the wall-normal direction. This section focuses on the mechanisms of particle transport in rod-roughed walls.
In turbulent flows over rough walls, the reduction of particle accumulation near the wall is mainly attributed to alterations in flow structures and collisions between particles and the rough walls (Kussin & Sommerfeld Reference Kussin and Sommerfeld2002; Vreman Reference Vreman2007; Milici et al. Reference Milici, De Marchis, Sardina and Napoli2014; Luo et al. Reference Luo, Dai, Liu and Fan2019; Chan et al. Reference Chan, Zahtila, Ooi and Philip2021). The addition of rough elements increases the frequency of particle–wall collisions and modifies the behaviour of particles. Collisions between inertial particles and curved rough elements often increase the absolute value of the velocity in the wall-normal direction, making particles more prone to moving away from the wall afterward (Luo et al. Reference Luo, Dai, Liu and Fan2019; Chan et al. Reference Chan, Zahtila, Ooi and Philip2021). Hence, in turbulent flow over curved rough walls, the wall collision directly contribute to the elimination of preferential wall accumulation. The rod-roughed elements examined in this study have their normal directions aligned parallel to the coordinate axis. In such case, the absolute value of
$v_{\!p}$
remains unchanged before and after the collision. Figure 11 shows the probability density functions (PDFs) of the wall-normal particle velocity and fluid velocity in the buffer layer (
$5 \leqslant y^+ \leqslant 30$
). The larger the inertia of particles, the poorer their ability to follow the fluid, leading to a more pronounced discrepancy between the PDFs of
$v_{\!p}$
and
$v_{\!f@\textit{p}}$
. It is evident that in rough-wall turbulence, at significant wall-normal velocities, the probability density of inertial particles, particularly those with medium and large inertia, is consistently lower than that of the fluid. This indicates that the wall collision does not increase the wall-normal velocity of particles. Instead, the particles continue to approach and move away from the wall under the transport of the fluid.
Probability density functions (PDFs) of the wall-normal particle velocity (
$v_{\!p}$
) and fluid velocity sampled by the particle (
$v_{\!f@\textit{p}}$
) in the buffer layer: (a) smooth-wall turbulence; (b–f) rough-wall turbulence with
$w/k=1$
, 5, 9, 11 and 99, respectively.
PDFs of the streamwise particle velocity (
$u_{\!p}$
) and fluid velocity sampled by the particle (
$u_{\!f@\textit{p}}$
) in the buffer layer: (a) smooth-wall turbulence; (b–f) rough-wall turbulence with
$w/k=1$
, 5, 9, 11 and 99, respectively.
Figure 12 depicts the PDFs of the streamwise particle velocity and fluid velocity in the buffer layer. In smooth-wall turbulence, the results of particles and the local fluid are highly consistent, and both
$u_{\!p}$
and
$u_{\textit{p}@\textit{f}}$
are always greater than 0. In rough-wall cases, the distribution of the streamwise velocity for small inertial particles with
$St=1$
overlaps with that of the local fluid, with the probability peaking near 0. Nevertheless, notable distinctions exist in the distribution of particles with
$St=$
30 and 100 when compared with the results for the local fluid. The probability of these particles possessing significantly negative streamwise velocity is considerably higher than that of the fluid in rough-wall turbulence. Evidently, this phenomenon arises from collisions between inertial particles and the windward surfaces of rough elements. Hence, wall collisions primarily modify the distribution of particle streamwise velocity in the near-wall region rather than that of particle wall-normal velocity.
(a) Reynolds shear stress
$-\overline {u^\prime v^\prime }^+$
versus wall-normal coordinate
$y^+$
. (b) The contribution of Q2 events to Reynolds shear stress. (c) The contribution of Q4 events to Reynolds shear stress. (d) The ratio of the contributions of Q2 and Q4 events to Reynolds shear stress.
Joint probability density functions of wall-normal particle velocity and particle-sampled fluid velocity in the buffer layer: (a–c)
$w/k=1$
, (d–f)
$w/k=9$
, (g–i)
$w/k=99$
; (a,d,g)
$St=1$
, (b,e,h)
$St=30$
, (c, f,i)
$St=200$
. The solid lines represent the results of smooth-wall turbulence.
The PDFs of the wall-normal and streamwise velocities indicate that particle–rough-wall collisions do not contribute to the reduction of particle accumulation near the wall. Thus, modifications to the near-wall flow structures in rough-wall turbulence, including both modified canonical coherent structures and newly generated roughness-induced flow structures, may directly promote a more uniform particle distribution. Marchioli & Soldati (Reference Marchioli and Soldati2002) suggested that ejections (Q2 events) and sweeps (Q4 events) dominate the transport mechanism of inertial particles in smooth-wall turbulence. When particles are uniformly distributed, Q4 events are effective in driving particles toward the wall, resulting in an increasing numbers of particles near the wall. Due to the spanwise motion of streamwise vortices, these particles are trapped in the low-speed streaks for a long time, with some being subsequently entrained into the outer flows by the Q2 events. This phenomenon causes the eventual accumulation of particles in the near-wall region in smooth-wall turbulence (Marchioli & Soldati Reference Marchioli and Soldati2002; Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012a
). As the presence of roughness elements significantly affects the canonical coherent structures, further investigation is needed into the behaviour of Q2 and Q4 events in rough-wall turbulent flows. The Reynolds shear stress is primarily generated by Q2 and Q4 events. The contribution of Q2 events is defined as the product of the Reynolds shear stress generated by the Q2 event and the probability of its occurrence, denoted as
$-\overline {u^\prime v^\prime _{Q2}}^+$
. Likewise, the contribution of Q4 events is denoted as
$-\overline {u^\prime v^\prime _{Q4}}^+$
. As depicted in figure 13(a–c), the distribution of
$-\overline {u^\prime v^\prime _{Q2}}^+$
and
$-\overline {u^\prime v^\prime _{Q4}}^+$
exhibit shows the same trend as that of
$-\overline {u^\prime v^\prime }^+$
. They both increase with
$w/k$
in the near-wall region, reaching peak values at
$w/k = 99$
. Furthermore, in the regions away from the wall, the results of different cases nearly overlap. The ratio of
$-\overline {u^\prime v^\prime _{Q2}}^+$
and
$-\overline {u^\prime v^\prime _{Q4}}^+$
, as illustrated in figure 13(d), reflects the relative strengths of ejections and sweeps. In smooth-wall turbulence, Q4 events contribute more to Reynolds shear stress than Q2 events in the region
$y^+\lt 15$
, while the opposite is observed farther away from the wall. This is consistent with the results of previous studies (Wallace 2016). In rough-wall flows, this ratio decreases as
$w/k$
increases and remains consistently larger than that of the smooth-wall case. This decline in the ratio signifies an augmented relative strength of Q2 and Q4 events in the near-wall region. Considering that Q2 and Q4 events dominate particle fluxes towards and away from the wall, an increase in their relative strength suggests a reduction in wall accumulation by coherent structures. Therefore, the alterations in ejections and sweeps in rough-wall turbulence diminish the wall accumulation of inertial particles.
The modified canonical coherent structures exhibit a reduced ability to promote near-wall particle accumulation, whereas the contribution of roughness-induced flow structures to particle transport remains non-negligible. To further elucidate the combined effects of these flow structures on particle transport in rough-wall turbulence, figure 14 displays the joint probability density functions (JPDFs) of wall-normal particle velocity
$v_{\!p}^+$
and particle-sampled fluid velocity
$v_{\!f@\textit{p}}^+$
in the buffer layer. The sampled instantaneous fluid velocities reflect the wall-normal transport effects associated with both the modified canonical coherent structures and the roughness-induced flow structures. In rough-wall turbulence, the lightest particles with
$St=1$
(figures 14
a, 14
d and 14
g) exhibit a slightly diminished ability to comply with the fluid. Nevertheless, as the roughness parameter
$w/k$
increases, their behaviour gradually converges towards that observed in smooth-wall turbulence. Hence, these small inertial particles display a uniform spatial distribution in both smooth-wall and rough-wall flows. For particles with moderate and large inertia, the JPDF distribution in rough-wall cases is more scattered than that in the smooth-wall case, signifying a weakened correlation between
$v_{\!p}^+$
and
$v_{\!f@\textit{p}}^+$
. Significantly, in rough-wall cases, the distribution in the third quadrant is distinctly further from the diagonal, leaning towards horizontal or vertical lines. In cases with
$w/k=$
1 and 9, the distribution tends to be more horizontal, suggesting that particles move more slowly than the local fluid in approaching motions. As the roughness spacing increases (
$w/k=99$
), the influence of wall collisions leads to a distribution more inclined toward the vertical direction. A similar but less-accentuated trend is observed in the first quadrant. Therefore, the existence of rough elements substantially diminishes the capacity of flow structures to transfer particles towards the wall. Consequently, the wall accumulation of these particles decreases in rough-wall turbulence.
4. Concluding remarks
The transport and distribution of inertial particles in turbulent channel flows over rough walls with spanwise rod-roughened elements have been investigated using DNS at a friction Reynolds number of
$Re_\tau = 180$
. The effects of roughness topology have been systematically explored by varying the streamwise spacing-to-height ratio.
The introduction of rod-roughened elements significantly modifies flow structures and the resulting drag coefficient. Both form drag and total drag coefficients exhibit a non-monotonic dependence on roughness spacings and reach a maximum at approximately
$w/k=9$
. The fluid flow separates at the leeward vertex and reattaches at the windward vertex of the roughness element. Flow reattachment also occurs between roughness elements when the roughness spacing was sufficiently large. As the spacing-to-height ratio increases, a single stable separation vortex gradually evolves into two isolated separation vortices of different sizes, while the influence of roughness elements on the surrounding flow continues to intensify. Roughness elements also modify the organisation of turbulent structures. Quasi-streamwise vortices are directly disrupted by the roughness elements, leading to more irregular and three-dimensional vortical structures. Meanwhile, spanwise vortices generated by strong shear layers are shed from the roughness crests and evolve downstream with larger inclination angles.
Regarding the dispersed phase, the rough walls effectively suppress the turbophoretic accumulation of inertial particles in the near-wall region and lead to a more uniform spatial distribution compared to the smooth-wall case. This behaviour differs from flows over curved or hemispherical roughness elements where wall-normal rebounding upon collision dominates particle distribution. In the case of rod elements, collisions do not directly enhance the wall-normal velocity of particles. Instead, the alteration of near-wall flow structures plays a crucial role. Instead, the alteration of near-wall flow structures plays a crucial role. Roughness elements significantly weaken the correlation between the motion of inertial particles and the local fluid, particularly in the wall-normal direction. This decorrelation reduces the effectiveness of turbulent structures, including both modified canonical coherent structures and newly generated roughness-induced flow structures, in transporting particles deep into the viscous sublayer. Consequently, their ability to trap particles in the near-wall region is weakened, leading to the observed reduction in preferential accumulation. Furthermore, roughness elements significantly enhance the streamwise transport of particles, characterised by an increased mean bulk velocity. This enhancement suggests that the dispersed phase is transported more effectively in rough-wall flows, although the accompanying rise in flow drag limits the overall transport efficiency.
These findings highlight the critical role of surface topology in particle-laden rough-wall turbulence. The results suggest that different roughness geometries, such as spanwise rods versus curved elements, can fundamentally alter the relative importance of turbulence modulation and particle–wall collision mechanisms in controlling particle transport. In the present study, the spanwise rod-roughened configuration serves as a simplified canonical model that minimises direct collision-induced wall-normal particle redistribution, thereby allowing the isolated influence of roughness-modified turbulent structures to be examined. Future work should systematically investigate the effects of roughness geometry and topology, including circular, curved and fully three-dimensional irregular roughness elements, in order to establish a more general physical understanding of inertial-particle transport in rough-wall turbulence. In particular, clarifying how roughness shape modifies the balance between turbulence-induced transport and collision-induced redistribution remains an important open question. In addition, investigating the influence of particle shape anisotropy and higher Reynolds numbers on transport efficiency represents another direction for future study.
Funding
Authors acknowledge the support from the National Science Foundation of China (grant numbers 12472224, 12532011 and 12388101). This work is also funded by State Key Laboratory of Advanced Space Propulsion (No. 2025ASP0101).
Declaration of interests
The authors report no conflict of interest.
Appendix A. Validation of the solver
To validate the present DNS solver for rough-wall flows, the simulation results are compared with data from Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003, Reference Leonardi, Orlandi, Djenidi and Antonia2004) at a streamwise spacing-to-height ratio
$w/k = 7$
(other parameters are listed in § 2.1). Figure 15 shows the comparison of the two-point correlation
$R_{vv}$
at
$y^+ \approx 10$
in the streamwise direction and the mean drag coefficient
$\langle C_{\!f} \rangle$
, confirming the accuracy of the current solver for rough-wall flow simulations.
Two-points correlation
$R_{vv}$
at
$y \approx 0.22h$
in streamwise direction (a) and mean drag coefficient
$\langle C_{\!f} \rangle$
(b) for rough walls with streamwise spacing-to-height ratio
$w/k=7$
. The solid line represents the present results, and the circle represents the data from Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003, Reference Leonardi, Orlandi, Djenidi and Antonia2004).
u¯+=2.5lny++5.5
w/k
⟨u⟩+
w/k=1
w/k=5
w/k=9
w/k=11
w/k=99
w/k=199
Q+=0.01
w/k=1
w/k=5
w/k=9
w/k=11
w/k=99
w/k=199
w/k=1
St=30
x−y
w/k=1
u+
St=30
w/k=1
w/k=5
w/k=9
w/k=11
w/k=99
C/C0¯
y+
w/k=1
Ub
uτ
vp
vf@p
w/k=1
up
uf@p
w/k=1
−u′v′¯+
y+
w/k=1
w/k=9
w/k=99
St=1
St=30
St=200
Rvv
y≈0.22h
⟨Cf⟩
w/k=7
