2. Governing equations and numerical set-ups

The TC system consists of a viscous fluid confined in the gap between two coaxially rotating cylinders. The inner and outer cylinders rotate with angular velocities $\omega _i$ and $\omega _o$ , respectively. Parameters $r_i$ and $r_o$ represent the radii of the inner and outer cylinders, respectively. The gap width between the two cylinders is denoted as $d=r_o-r_i$ . The geometric control parameters of the TC system include aspect ratio $\varGamma = L/d$ and the gap ratio $\eta = r_i / r_o$ , where $L$ is the height of the TC system. In this study, the outer cylinder is fixed, and the inner cylinder rotates with a constant angular velocity $\omega _i$ .

The Euler–Lagrangian framework is used to investigate the inertial particles settling in turbulent TC flow. The fluid phase is described by the incompressible Navier–Stokes equations, while the particle phase is described by the dispersed point-particle. The coupling between the fluid and particle phases is achieved through the hydrodynamic force acting on the particles. The dimensionless governing equations for the fluid phase are the incompressible Navier–Stokes equations, given by

(2.1) \begin{align} \displaystyle {\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}=0,} \\[-28pt] \nonumber \end{align}

(2.2) \begin{align} \displaystyle {\frac {\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla }) \boldsymbol{u}=-\boldsymbol{\nabla }p+\frac {1}{{\textit{Re}}_i} {\nabla} ^{2} \boldsymbol{u} + \boldsymbol{f}_{\kern-2.5pt p},} \\[-28pt] \nonumber \end{align}

(2.3) \begin{align} \displaystyle {\boldsymbol{f}_{\kern-2.5pt p} = -\frac {\rho _p}{\rho _f} V_p \sum _{j=1}^{N_p}\left(\frac {{\rm d}\boldsymbol{v}}{{\rm d}t}-\frac {1}{\textit{Fr}^2}\boldsymbol{e}_z \right)\delta (\boldsymbol{x}-\boldsymbol{y}_{\!j}),}\\[6pt] \nonumber \end{align}

where ${\textit{Re}}_i = r_i \omega _i d/\nu$ is the Reynolds number based on the inner cylinder rotation, $\boldsymbol{u}$ and $\boldsymbol{v}$ represent the fluid and particle velocities, respectively, $\boldsymbol{f}_{\kern-1.5pt p}$ denotes the feedback force exerted by particles per unit mass of the fluid phase, $\rho _p$ and $\rho _f$ represent the particle and fluid densities, respectively, $V_{\!p}$ is the particle volume, $N_{\!p}$ is the number of particles, $\textit{Fr} = \omega _i \sqrt {r_i/g}$ is the Froude number, $\boldsymbol{e}_z$ is the unit vector in the axial direction and $\boldsymbol{y}_j$ denotes the instantaneous position of the $j$ th particle. The particle feedback force on the fluid is given by (2.3), where $\delta$ is the Dirac delta function and ${\rm d}\boldsymbol{v}/{\rm d}t$ is the acceleration of the particles. Equations (2.1)–(2.3) are normalised by the inner cylinder rotation velocity $r_i \omega _i$ and gap width $d$ .

The particles dispersed in the fluid phase are tracked using a Lagrangian point-particle approach with the Stokes drag, added mass, Saffman–Mei lift, buoyancy and gravity force (Gatignol Reference Gatignol1983; Maxey & Riley Reference Maxey and Riley1983; Maxey Reference Maxey1987; Tsai Reference Tsai2022). The momentum equation for the particle phase is given by

(2.4) \begin{align} \rho _p V_p \frac {{\rm d}\boldsymbol{v}}{{\rm d}t} &= \rho _p V_p \boldsymbol{g} - C_D \frac {\pi d_p^2}{8} (\boldsymbol{v} - \boldsymbol{u}) |\boldsymbol{v} - \boldsymbol{u}| + \rho _f V_p C_M \left ( \frac {{\rm D}\boldsymbol{u}}{{\rm D}t} - \frac {{\rm D}\boldsymbol{v}}{{\rm D}t} \right )\notag \\ &\quad + \rho _f V_p \left ( \frac {{\rm D}\boldsymbol{u}}{{\rm D}t} - \boldsymbol{g} \right ) - C_L \rho _f V_p (\boldsymbol{v} - \boldsymbol{u}) \times \boldsymbol{\omega }, \end{align}

where the terms on the right-hand side represent the gravitational force, Stokes drag force, added mass force, pressure gradient force and lift force, respectively. The particle density is denoted as $\rho _p$ , $V_p = \pi d_p^3/6$ is the particle volume, $d_p$ denotes the particle diameter, $\boldsymbol{u}$ and $\boldsymbol{\omega }$ are the velocity and vorticity of the fluid at the particle position, respectively, $\boldsymbol{v}$ is the particle velocity, $\boldsymbol{g}$ is the gravitational acceleration, while $C_D$ , $C_M$ and $C_L$ represent the drag, added mass and lift coefficients, respectively. Using the characteristic velocity $r_i \omega _i$ , characteristic length $r_i$ and fluid density $\rho _f$ , (2.4) can be normalised and simplified as

(2.5) \begin{eqnarray} \frac {{\rm d}\boldsymbol{v}}{{\rm d}t} = \frac {C_D}{\textit{St}}(\boldsymbol{u}-\boldsymbol{v}) + (\beta +1)\frac {{\rm D}\boldsymbol{u}}{{\rm D}t} + \frac {\beta +1}{3}\ (\boldsymbol{u}-\boldsymbol{v}) \times \boldsymbol{\omega }+ \beta \frac {1}{\textit{Fr}^2}\boldsymbol{e}_z. \end{eqnarray}

Here, $\textit{St} = {d_p^*}^2 {\textit{Re}}_i / 12 (\beta +1)$ represents the particle Stokes number based on the inner cylinder rotation speed and $\beta = 2 (1-\rho ^*)/(1+2\rho ^*)$ is a parameter related to density ratio $\rho ^*$ . For the drag coefficient, the Shiller–Naumann relation $C_D = 1+0.15{\textit{Re}}_p^{0.687}$ is used, which is valid for particle Reynolds number ${\textit{Re}}_p = d_p|\boldsymbol{v}-\boldsymbol{u}|/\nu$ up to 1000 (Naumann & Schiller Reference Naumann and Schiller1935). The added mass coefficient is set to $C_M = 0.5$ and the lift coefficient is set to $C_L = 0.5$ (Spandan et al. Reference Spandan, Ostilla-Mónico, Verzicco and Lohse2016b ; Sugiyama et al. Reference Sugiyama, Calzavarini and Lohse2008).

Direct numerical simulations of the carrier phase are performed using a second-order-accurate finite-difference method in cylindrical coordinates $( \boldsymbol{e}_r, \boldsymbol{e}_{\theta }, \boldsymbol{e}_z)$ . The governing equations are numerically solved by using a fraction third-order Runge–Kutta scheme (Verzicco & Orlandi Reference Verzicco and Orlandi1996; Ostilla et al. Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013; Song, Shishkina & Zhu Reference Song, Shishkina and Zhu2024). The time step is chosen so as not to exceed the the Courant–Friedrichs–Lewy condition and particle relaxation time. A uniform grid is employed in the azimuthal and axial directions with grid spacing $r_i^+\Delta \theta = r_i\Delta \theta /\delta _{\nu ,i} = 6.64$ and $\Delta z^+ = \Delta z/\delta _{\nu ,i} = 4.22$ , and a non-uniform grid spacing using a clipped Chebyshev-type clustering method in the radial direction with $\Delta r^+ = \Delta r/\delta _{\nu ,i}$ ranging from 0.17 to 1.29 (single-phase inner viscous length scale $\delta _{\nu ,i} = \nu / u_{\tau ,i}$ ; Huisman et al. Reference Huisman, Scharnowski, Cierpka, Kähler, Lohse and Sun2013). We ensure 2–3 grid points within the viscous sublayer $(y^+\lt 1)$ . The velocity profiles normalised by the friction velocity can be found in § 1 of the supplementary material available at https://doi.org/10.1017/jfm.2025.10850. As a result, the numbers of grid points in the $\theta$ , $r$ and $z$ directions are $N_{\theta }=150$ , $N_r = 200$ and $N_z = 180$ , respectively.

To reduce computational cost, we exploit a rotational symmetry of $n_{sym}=6$ , meaning that the computational domain spans only $1/6$ of the full azimuthal direction. The aspect ratio is set to $\varGamma =L/d=4$ and radius ratio $\eta =0.833$ . These parameters have been validated for simulation accuracy of two-phase TC flow in Spandan et al. (Reference Spandan, Ostilla-Mónico, Verzicco and Lohse2016b ). The no-slip boundary condition is applied at the inner and outer cylinder walls, and the periodic boundary condition is applied in the azimuthal and axial directions for the carrier phase and dispersed phase. The complete elastic collisions have been used to calculate the particle–wall collisions. The fluid information at the particle location and the feedback force (2.3) extrapolated from the particle location to grid nodes are both calculated by a tri-linear scheme. In order to be consistent with Sugiyama et al. (Reference Sugiyama, Calzavarini and Lohse2008) and Spandan et al. (Reference Spandan, Ostilla-Mónico, Verzicco and Lohse2016b ), we also fix the mean axial flow flux to be zero at each $z$ position. In this study, the particle volume fraction is fixed at $6 \times 10^{-5}$ , allowing particle–particle interactions to be neglected, while the Reynolds number of the carrier phase is maintained at ${\textit{Re}}_i = 3500$ . The particle Stokes numbers $\textit{St}$ range from 0.088 to 0.876, and the Froude number $\textit{Fr}$ varies from 0.16 to 2.56. For comparison, the cases with $\textit{Fr}^{-1} = 0$ are also considered, where the gravitational effects are neglected. For convenience, $\textit{Fr}^{-1}$ will be used to represent the magnitude of the effective gravity in the following sections. The details of the particle parameters are summarised in table 1.

Table 1.

Details of the settling particle parameters and results for the numerical simulations. Density ratio $\rho ^* = \rho _p/\rho _f$ , defined as the ratio of particle to fluid density, is varied in the range $50 \leqslant \rho ^* \leqslant 500$ . Stokes number $\textit{St} = {d_p^*}^2 {\textit{Re}}_i / 12 (\beta +1)$ is the particle bulk Stokes number. For all cases, the dimensionless particle diameter $d_p^* = d_p / d$ and volume fraction $\phi _v = N_p V_p/V_f$ are fixed at 0.003 and $6 \times 10^{-5}$ , respectively. The particle mass loading ratio $\phi _m = N_pm_p/m_f$ ranges from $3.1\times 10^{-3}$ to $3.1\times 10^{-2}$ . For the case of $\textit{Fr}^{-1} = 0$ , the gravitational effects can be neglected. For the single phase, the dimensionless angular velocity flux $\textit{Nu}_{\omega }=7.96$ , which differs from the result of Spandan et al. (Reference Spandan, Ostilla-Mónico, Verzicco and Lohse2016b ) by about 1.4 %.

3. Angular velocity flux and drag modulation

In TC flow, the torque needed to drive the cylinders $T$ and the angular velocity flux $J^{\omega }$ are related by $T = 2\pi L \rho J^{\omega }$ (Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007), and $J^{\omega }$ is conserved in the radial direction. The definition of $J^{\omega }$ is given in § 4. Parameter $\textit{Nu}_{\omega }$ represents the dimensionless angular velocity flux, where it is defined by $\textit{Nu}_{\omega }=J^{\omega }/J_{\textit{lam}}$ with normalisation by the flux in laminar flow $J_{\textit{lam}}=2\nu r_i^2\ r_o^2\ \omega _i/(r_o^2-r_i^2)$ . The net percentage drag reduction can be defined as follows (Spandan et al. Reference Spandan, Ostilla-Mónico, Verzicco and Lohse2016b ):

(3.1) \begin{gather} {\rm DR}=\frac {\left \langle \textit{Nu}_\omega \right \rangle _s-\left \langle \textit{Nu}_\omega \right \rangle _t}{\left \langle \textit{Nu}_\omega \right \rangle _s} \times 100\,\%=\frac {\big \langle C_f\big \rangle _s-\big \langle C_f\big \rangle _t}{\big \langle C_f\big \rangle _s} \times 100\,\%, \end{gather}

where $\left \langle \cdots \right \rangle _s$ and $\left \langle \cdots \right \rangle _t$ correspond to averaging for single-phase and two-phase systems, respectively, and $C_f = 2\tau _{w}/\rho u_{\theta ,i}^2$ represents the friction factor, where $\tau _{w}$ and $u_{t,i}$ are the averaged wall shear stress and inner cylinder rotation velocity, respectively.

Figure 1.

( $a$ ) Nusselt number $\textit{Nu}_{\omega }$ for two-phase TC flows under varying particle Stokes numbers ( $\textit{St}$ ) and inverse Froude numbers ( $\textit{Fr}^{-1}$ ), compared with single-phase (SP) flow (horizontal dashed line) at ${\textit{Re}}_i = 3500$ . ( $b$ ) The net percentage drag reduction (DR) for the two-phase TC flows, illustrating the influence of particle $\textit{St}$ and $\textit{Fr}^{-1}$ .

We first present results for the influence of the settling particles on the drag reduction in TC flow. Nusselt number $\textit{Nu}_\omega$ for two-phase TC flows is presented in figure 1(a), illustrating the influence of particle Stokes numbers ( $\textit{St}$ ) and inverse Froude numbers ( $\textit{Fr}^{-1}$ ) on the flow angular velocity transport and the corresponding drag reduction in figure 1(b). Here, $\textit{Nu}_\omega$ is obtained after statistical stationarity is reached, by averaging over at least 40 full cylinder rotations, using no fewer than $2.5\times 10^{4}$ snapshots, the same for all subsequent turbulent statistics. With varying strength of particle gravitational settling, $\textit{Nu}_\omega$ exhibits non-monotonic variation. As $\textit{Fr}^{-1}$ increases, there is more than 20 % drag reduction, where the amount of drag reduction reaches a maximum value at certain optimal $\textit{Fr}^{-1}$ . For heavy particles ( $\textit{St} = 0.375, 0.876$ ), $\textit{Nu}_{\omega }$ increases markedly after the optimal point leading to drag enhancement in the TC flow for large $\textit{Fr}^{-1}$ limit. The observed behaviour leads to the question as to what leads to the drag reduction and the subsequent enhancement in drag.

Figure 2.

Dimensionless contributions of angular velocity flux for varying particle $\textit{St}$ and $\textit{Fr}^{-1}$ , compared with single-phase flow (dashed line). ( $a$ , $b$ ) The advection terms and viscous diffusion terms for $\textit{St} = 0.088$ ; ( $c$ , $d$ ) the advection terms and viscous diffusion terms for $\textit{St} = 0.876$ .

4. Decomposition of flux contribution

To investigate the mechanism of drag modulation, we decompose the angular velocity flux into various parts. For the two-phase TC system, the angular velocity flux $J^{\omega }$ must also account for the feedback force from the particles. Following the approach proposed in Su et al. (Reference Su, Yi, Zhao, Wang, Xu, Wang and Sun2024), the angular velocity flux $J^{\omega }$ for two-phase TC flow can be decomposed into three parts where $J_A = r^3\langle u_r \omega \rangle _{\theta ,z,t}$ , $J_V = -r^3\nu \left \langle \partial _r \omega \right \rangle _{\theta ,z,t}$ and $J_P = -\int _{r_i}^{r} \langle r^2f_{\theta } \rangle _{\theta ,z,t}$ represent the contributions from the advection, viscous diffusion and particle feedback to the angular velocity flux, respectively. Here, $\left \langle \cdots \right \rangle _{\theta ,z,t}$ denotes averaging over time and in the $\theta ,z$ direction and $f_{\theta }$ represents the azimuthal components of (2.3).

The contributions of $J_A$ , $J_V$ normalised by the corresponding values can be obtained from the laminar flow $J_{\textit{lam}}$ in figure 2. Despite the addition of particles, the contribution of particle feedback term $J_P$ to the angular velocity flux remains below 1 % across all cases. Therefore, this term has been neglected in the subsequent analysis. In contrast, the advection term predominantly governs the bulk region, while viscous diffusion dominates within the boundary layers of the TC system, consistent with the findings of Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013).

When particle settling is neglected (i.e. $\textit{Fr}^{-1}=0$ ), the contributions to angular velocity flux exhibit minimal variation. For light particles ( $\textit{St} = 0.088$ ), the introduction of effective gravity leads to a reduction in both advective and viscous diffusion contributions within their respective dominant regions, consistent with the observed drag reduction. In contrast, for heavy particles ( $\textit{St} = 0.876$ ), these contributions initially decrease but subsequently increase with increasing effective gravity, eventually surpassing the single-phase values when $\textit{Fr}^{-1}\gt 3.13$ .

Figure 3.

Dimensionless contributions of Taylor vortex and Reynolds stress as a function of radial position. (a) Taylor vortex contribution, $\textit{St}=0.088$ , (b) Reynolds stress contribution, $\textit{St}=0.088$ , (c) Taylor vortex contribution, $\textit{St}=0.876$ , (d) Reynolds stress contribution, $\textit{St}=0.876$ . The dashed lines present the single-phase case.

5. Modulation of flow structure by settling particles

Note that since TC turbulence consists of both turbulent Taylor vortices and background fluctuations, it is meaningful to decompose the advection contribution of flux $J_A = r^3\langle u_r \omega \rangle _{\theta ,z,t}$ into two components (Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007; Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025):

(5.1) \begin{gather} r^3\left \langle u_r \omega \right \rangle _{\theta ,z,t} = r^3\left \langle \left \langle u_r\right \rangle _{\theta ,t} \left \langle \omega \right \rangle _{\theta ,t} \right \rangle _{z}+ r^3\left \langle u_r^{\prime } \omega ^{\prime } \right \rangle _{\theta ,z,t}, \end{gather}

where $\langle \langle u_r\rangle _{\theta ,t} \langle \omega \rangle _{\theta ,t} \rangle _{z}$ and $\langle u_r^{\prime } \omega ^{\prime } \rangle _{\theta ,z,t}$ represent the contribution of the Taylor vortex and Reynolds stress to the angular velocity flux, respectively, where $u_r^{\prime } = u_r - \left \langle u_r \right \rangle _{\theta ,t}$ . Note that the Reynolds stress contribution is not identical to the Reynolds stress; see § 2 of the supplementary material for details. The dimensionless forms of Taylor vortex contribution $J_{\textit{T}V} = r^3\langle \langle u_r\rangle _{\theta ,t} \langle \omega \rangle _{\theta ,t} \rangle _z /J_{\textit{lam}}$ and Reynolds stress contribution $J_{RS} = r^3\langle u_r^{\prime }\omega ^{\prime } \rangle _{\theta ,z,t}/J_{\textit{lam}}$ are shown in figure 3. For the single-phase case, $J_{\textit{T}V}$ is larger than $J_{RS}$ , indicating that, under the present parameters, the angular velocity transport in the TC system is predominantly contributed by the Taylor vortices.

At $\textit{Fr}^{-1}=0$ , the addition of light particles exerts minimal influence on the overall angular velocity flux, due to a balance between two competing effects: the suppression of the Taylor vortex contribution $J_{\textit{T}V}$ and the simultaneous enhancement of flux from turbulent fluctuations $J_{RS}$ in the bulk region. This insensitivity differs notably from the case of heavy particles, which show only a weak influence on both $J_{\textit{T}V}$ and $J_{RS}$ under the same conditions. The distinction arises from the differing spatial distributions and transport dynamics of the particles: light particles are more uniformly dispersed throughout the bulk and are passively carried by the flow, facilitating the transfer of angular momentum from the fluid to the particles (see figure 4). This interaction suppresses the coherence of mean Taylor vortices while amplifying the intensity of background fluctuations. In contrast, heavy particles experience strong centrifugal forces, causing them to accumulate near the outer cylinder wall and interact less with the bulk flow, thereby exerting limited influence on advective transport.

As the effect of gravitational settling increases ( $\textit{Fr}^{-1}$ rises), the contribution from coherent Taylor vortices progressively diminishes, becoming minimal in the case of heavy particles. In contrast, the contribution from background turbulent fluctuations increases significantly with rising $\textit{Fr}^{-1}$ , indicating a transition towards a flow regime dominated by particle-induced turbulence. In this regime, particles are more likely to be transported from the outer-cylinder region towards the bulk under the action of strong turbulent fluctuations.

Figure 4.

Top-view instantaneous snapshots of particle distribution for varying Stokes numbers $\textit{St} = 0.088$ ( $a$ – $c$ ), $\textit{St} = 0.176$ ( $d$ – $f$ ) and $\textit{St} = 0.876$ ( $g$ – $i$ ), and varying inverse Froude numbers $\textit{Fr}^{-1} = 0$ ( $a$ , $d$ , $g$ ), $\textit{Fr}^{-1} = 2.34$ ( $b$ , $e$ , $h$ ) and $\textit{Fr}^{-1} = 5.26$ ( $c$ , $f$ , $i$ ). Only the angular range $\theta \in (0,\pi /6)$ is shown.

Figure 5.

The ( $r$ – $z$ ) plane angular velocity contours (left) and radial velocity contours (right) for varying $\textit{St}$ : $(a{-}c)$ $\textit{St}=0.088$ , $(d{-}f)$ $\textit{St}=0.176$ , $(g-i)$ $\textit{St}=0.375$ and $(j{-}l)$ $\textit{St}=0.876$ ; and varying $\textit{Fr}^{-1}$ : (a,d,g, j) $\textit{Fr}^{-1}=0$ , (b,e,h,k) $\textit{Fr}^{-1}=0.78$ and (c,f,i,l) $\textit{Fr}^{-1}=5.26$ . Only the axial domain $z \in (0,2)$ is shown. In $(a)$ , the red box indicates the Taylor vortex region, while in $(d)$ , the green boxes mark the Görtler vortex regions generated around the plume due to centrifugal instability.

Figure 6.

The ( $r$ – $z$ ) plane mean angular velocity contours. $(a)$ Single-phase case, $(b)$ $\textit{St}=0.876,\ \textit{Fr}^{-1}= 0$ , $(c)$ $\textit{St}=0.876,\ \textit{Fr}^{-1}= 0.78$ and $(d)$ $\textit{St}=0.876,\ \textit{Fr}^{-1}= 5.26$ .

Figure 7.

Viscous dissipation rate near the cylinder wall, normalised by the single-phase (SP) case $\epsilon /\epsilon _{s,i}$ , where $\epsilon _{s,i}$ represents the viscous dissipation rate of the single phase at the inner cylinder surface, for ( $a{,}b$ ) $\textit{St} = 0.088$ and ( $c{,}d$ ) $\textit{St} = 0.876$ . (a,c) The region near the inner cylinder and (b,d) the region near the outer cylinder.

The vanishing of the Taylor vortex contribution ( $J_{\textit{T}V}$ ) does not imply the complete absence of Taylor vortices in the TC system. When the vortices exhibit axial migration or strong temporal fluctuations, the averaging operation can cause $J_{\textit{T}V}$ to vanish despite the vortices still being present. To clarify the underlying flow states in the two-phase TC system, we present instantaneous snapshots of angular velocity and radial velocity in the radial–axial plane, as shown in figure 5. In the absence of settling effect, Taylor vortices are clearly visible in the bulk region, where they dominate the flow. At small values of $\textit{Fr}^{-1}$ , increasing particle inertia causes the edges of the Taylor vortices to become increasingly blurred, accompanied by axial oscillations. In this case, velocity plumes near the wall are primarily governed by small-scale Görtler vortices (Dong Reference Dong2007; Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025). At $\textit{St} = 0.176$ and $\textit{Fr}^{-1} = 0.78$ , Taylor rolls are still discernible in the instantaneous snapshots, although the contribution from $J_{\textit{T}V}$ has already diminished.

For heavy particles ( $\textit{St} = 0.375, 0.876$ ), even a small effective gravity disrupts the Taylor vortices, as shown in figure 5( $h\textrm{,}k$ ), leading to rapid suppression of the mean Taylor vortex structures and a concurrent increase in background turbulent fluctuations. However, this enhancement of turbulence is weaker than the suppression of the vortex contribution, resulting in significant drag reduction in the system. The same phenomenon can also be observed in the time-averaged angular velocity field, as shown in figure 6. When gravity is taken into account, the time-averaged Taylor vortex structures also disappear. To further analyse this, we computed $\langle u_r\rangle _{\theta ,t}$ in accordance with (5.1), and found that when the mean Taylor vortex effect is present, the averaged $\langle u_r\rangle _{\theta ,t}$ field clearly exhibits the characteristic structure of Taylor vortices. In contrast, when the mean Taylor vortex contribution is negligible, the averaged $\langle u_r\rangle _{\theta ,t}$ field is nearly zero, indicating that the flow in the bulk region is predominantly governed by turbulent fluctuations, as shown in figure 5 for large $\textit{Fr}^{-1}$ .

Figure 8.

Azimuthal velocity profile near the cylinder wall for ( $a{,}b$ ) $\textit{St} = 0.088$ and ( $c{,}d$ ) $\textit{St} = 0.876$ . (a,c) The region near the inner cylinder and (b,d) the region near the outer cylinder.

6. Near-wall properties modulated by particles

Next, we examine how changes in the flow structure lead to distinct near-wall properties, such as viscous dissipation rates. This analysis is essential, as drag reduction is typically associated with a decrease in viscous dissipation (Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007; Sugiyama et al. Reference Sugiyama, Calzavarini and Lohse2008; Spandan et al. Reference Spandan, Lohse and Verzicco2016a ). To understand how settling particles influence drag in the near-wall region, figure 7 presents the dissipation rate normalised by the single-phase dissipation at the inner cylinder $\epsilon /\epsilon _{s,i}$ , where

(6.1) \begin{gather} \epsilon =\frac {1}{2}\nu \sum _{\textit{ij}}\left(\frac {\partial u_j}{\partial x_i} + \frac {\partial u_i}{\partial x_j}\right)^2. \end{gather}

We compare the two-phase dissipation profiles for light particles ( $\textit{St} = 0.088$ ) and heavy particles ( $\textit{St} = 0.876$ ).

For light particles, the near-wall viscous dissipation is significantly reduced, except in the case of $\textit{Fr}^{-1}=6.25$ near the outer cylinder. However, in the outer part of the near-wall region, particle settling leads to enhanced viscous dissipation. In contrast, for heavy particles under the drag-enhanced cases, the near-wall dissipation is significantly elevated, with peak values reaching up to 30 times those of the single-phase system. To further assess how particles modulate the near-wall velocity field, we plot the azimuthal velocity profiles near the inner cylinder in figure 8. Light particles enhance the velocity near the inner cylinder wall while reducing it near the outer cylinder, leading to a smaller velocity gradient in the near-wall region. In contrast, under strong effective gravity, heavy particles cause a sharp drop in the near-wall azimuthal velocity. This sharp change in velocity implies that the near-wall velocity gradient can be significantly altered by the presence of settling particles, suggesting that particles can effectively modify the boundary-layer thickness.

Finally, we estimated the inner and outer velocity boundary-layer thickness from the azimuthal velocity ( $u_{\theta }$ ) profile, as shown in figure 9, following the method of Ostilla et al. (Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013).To determine the boundary-layer thickness, we first obtain the boundary-layer slope by a fitting through the first three grid points near the wall. For the bulk, the slope is taken from a fitting around the inflection point, using two points on each side. The intersection of the respective boundary-layer line and bulk line defines the boundary-layer thickness. Our results reveal that the variation in boundary-layer thickness closely correlates with the observed drag reduction: an increase in boundary-layer thickness corresponds to reduced drag, while a thinner boundary layer is associated with increased drag. This trend suggests that particle-induced modifications to the near-wall shear and momentum transport play a key role in altering the boundary-layer dynamics. Specifically, a thinner boundary layer intensifies velocity gradients, which can increase turbulent momentum transport and lead to drag enhancement. Conversely, a thicker boundary layer may reflect a damping of near-wall turbulence and a weakening of Taylor vortices, thereby reducing the exchange of angular momentum and contributing to drag reduction.

These results indicate that particle settling modulates drag not only directly by altering the large-scale coherent structures in the bulk region, but also indirectly by influencing energy dissipation and the boundary-layer thickness near the inner cylinder wall – even in regions with relatively low particle number density.

Figure 9.

Boundary-layer thickness normalised by the single-phase (SP) case. Here $\delta _{i}$ and $\delta _{o}$ represent inner and outer boundary-layer thickness, respectively. The subscript ‘ $s$ ’ denotes the single-phase case. $(a)$ Dimensionless inner boundary-layer thickness. $(b)$ Dimensionless outer boundary-layer thickness.