4.2.1 The mean flow and Reynolds stress

The discussion now moves to the flow modulations caused by the presence of solid particles in each case. First, the bulk flow speeds in case SP, case PLL, case PLL–NR, case PLS, case PLS–NR, are $15.736\pm 0.027$ , $14.771\pm 0.009$ , $15.230\pm 0.036$ , $15.281\pm 0.019$ and $15.058\pm 0.023$ , respectively, when normalized by the friction velocity in the unladen case. Hereafter, the value after $\pm$ denotes the standard derivation of an averaged quantity. The mean velocity profiles are compared with these reported in the previous studies of particle-laden channel flows in figure 13. When normalized by the inner scale (friction velocity and wall unit), the mean velocity profiles of case PLS and case PLL are in very good agreement with the results reported by Picano et al. (Reference Picano, Breugem and Brandt2015) and Eshghinejadfard et al. (Reference Eshghinejadfard, Abdelsamie, Hosseini and Thévenin2017), respectively, when the particle sizes and volume fractions are similar. Unless specified otherwise, the inner scales used for normalization are taken from case SP. The mean velocity profiles in all the particle-laden cases are reduced, compared to the unladen flow, except in the near-wall region.

While the mean velocity profiles of the particle-laden flows were often reported, they had not been well understood. For the sake of discussion, we may express of local fluid velocity in the particle-laden case as

(4.3) $$\begin{eqnarray}\overline{\langle u_{x}\rangle }_{pl}=\int _{0}^{y}\frac{\text{d}\overline{\langle u_{x}\rangle }}{\text{d}y^{\prime }}_{pl}\,\text{d}y^{\prime }+\unicode[STIX]{x1D6FF}u.\end{eqnarray}$$

Particles may alter the local mean flow velocity in two ways. On the one hand, particles can directly modify the local fluid velocity through enforcing the no-slip condition on their surface. This mechanism is expressed as a modification $\unicode[STIX]{x1D6FF}u$ in (4.3). On the other hand, particles can also modify the local flow velocity by changing the fluid velocity gradient at $y^{\prime }$ . This mechanism might be pictured as a virtual thin plate that moves with the same speed of the local fluid speed. While the plate does not change the local fluid speed through the no-slip condition, it alters the viscous transport between its two sides, so the velocity gradient $\text{d}\overline{\langle u_{x}\rangle }/\text{d}y^{\prime }$ is modified. The overall modulation of the local fluid velocity is determined by the two mechanisms together.

Correspondingly in the single-phase flow, we have

(4.4) $$\begin{eqnarray}\overline{\langle u_{x}\rangle }_{sp}=\int _{0}^{y}\frac{\text{d}\overline{\langle u_{x}\rangle }}{\text{d}y^{\prime }}_{sp}\,\text{d}y^{\prime }.\end{eqnarray}$$

To further demonstrate this, a budget analysis of the streamwise momentum balance at the statistical stationary state in each case is analysed. When the flow reaches a statistical stationary state, the streamwise momentum balance equation of the fluid phase can be obtained as

(4.5) $$\begin{eqnarray}\underbrace{\overline{\unicode[STIX]{x1D6FC}\langle -u_{x}^{\prime }u_{y}^{\prime }\rangle }}_{\unicode[STIX]{x1D70F}_{R}}+\underbrace{\frac{1}{\unicode[STIX]{x1D70C}}\overline{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D707}\left\langle \frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}y}\right\rangle }}_{\unicode[STIX]{x1D70F}_{V}}+\underbrace{\frac{1}{\unicode[STIX]{x1D70C}}\int _{0}^{y}\overline{F_{x}}\,\text{d}y}_{\unicode[STIX]{x1D70F}_{P}}=\underbrace{\frac{\overline{\unicode[STIX]{x1D70F}_{w,pl}}}{\unicode[STIX]{x1D70C}}-g\int _{0}^{y}\overline{\unicode[STIX]{x1D6FC}}\,\text{d}y}_{\unicode[STIX]{x1D70F}_{T}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ is the volume fraction of fluid phase inside the control volume $V=L_{x}\times L_{z}\times \unicode[STIX]{x1D6FF}y$ ; $\overline{F_{x}}=1/V\int _{S_{I}\in V}n_{j}(-p\unicode[STIX]{x1D6FF}_{xj}+\unicode[STIX]{x1D70F}_{xj})\,\text{d}S$ is the hydrodynamic force in the streamwise direction exerted on the fluid phase by the solid particles, $S_{I}$ is the fluid–particle interface contained by $V$ ; $\overline{\unicode[STIX]{x1D70F}_{w,pl}}$ is the counterpart of mean wall stress $\overline{\unicode[STIX]{x1D70F}_{w}}$ in particle-laden cases. In the present simulation, $\overline{\unicode[STIX]{x1D70F}_{w,pl}}$ is obtained by replacing the integration limit $y$ with the half-channel width $H$ in (4.5), i.e. $\overline{\unicode[STIX]{x1D70F}_{w,pl}}=\unicode[STIX]{x1D70C}gH\unicode[STIX]{x1D6F7}_{f}+\int _{0}^{H}\overline{F_{x}}\,\text{d}y$ . The four terms in (4.5) from left to right are given the shorthand notations $\unicode[STIX]{x1D70F}_{R}$ , $\unicode[STIX]{x1D70F}_{V}$ , $\unicode[STIX]{x1D70F}_{P}$ and $\unicode[STIX]{x1D70F}_{T}$ , which represent the Reynolds stress, viscous stress, particle-induced stress and the total stress, respectively. The overbar in each term indicates the term is time averaged at the stationary state.

Figure 14. The streamwise momentum balance in the particle-laden channel flow simulations: (a) case PLL, (b) case PLL–NR, (c) case PLS, (d) case PLS–NR.

Figure 15. Profiles of hydrodynamic force exerted on the fluid phase by particles.

Equation (4.5) in the four particle-laden simulations is examined in figure 14. All terms in (4.5) are explicitly computed with the data obtained in the present simulations. Except case PLS, the other three cases capture well the streamwise momentum balance as the computed left-hand side of (4.5) matches quite well with the computed right-hand side. In case PLS, however, an approximately 8 % relative difference between the computed left-hand side and right-hand side is observed at the wall. This deviation is likely due to the insufficient local grid resolution near the channel wall. When a particle is very close to a channel wall, the flow in the gap between the particle surface and the channel wall could be under-resolved. Since particles can have strong slip near the channel wall, a large velocity gradient exists in the gap region between the particle surface and channel wall, which requires higher grid resolutions. The error in case PLS is particularly significant, because this case has the largest particle volume fraction and the largest total particle surface area near the wall, as shown in figure 10. Local grid refinement is needed to improve the results for case PLS. Fortunately, this inadequate grid resolution only affects the small region very close to the channel walls ( $y^{+}<5$ ), and the impact is not as significant in the other three cases. The well-balanced results of different stresses in the other three cases shown in figure 14 can be viewed as a good self-validation to the computed statistics in these simulations.

As discussed earlier, the presence of particles can modify the local mean flow velocity by introducing a non-zero $\unicode[STIX]{x1D6FF}u$ in (4.3). This correction term $\unicode[STIX]{x1D6FF}u$ is associated with the streamwise particle force, i.e. the force exerted on the fluid phase by particles, $\overline{F_{x}}^{+}=\overline{F_{x}}\unicode[STIX]{x1D6FF}y/(\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}^{2})$ , which is shown for each particle-laden case in figure 15. Note that the normalization shown here is different from that used in Yu et al. (Reference Yu, Lin, Shao and Wang2017), the latter gave $\overline{F_{x}}H/(\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}^{2})$ so the value differs by a factor $H/\unicode[STIX]{x1D6FF}y$ . In the near-wall region $y^{+}\leqslant 10$ , $\overline{F_{x}}$ in each case is positive, meaning particles push fluid forward. This is because particles can slip on the wall while fluid has to follow the no-slip condition on the wall, thus the particle velocity is much larger than the corresponding fluid velocity. As a result, particles accelerate the fluid velocity locally. Outside the near-wall region, $\overline{F_{x}}$ in each case changes the sign at some point, which may contribute to the reduced fluid velocity in the particle-laden cases compared to the single-phase case in the same region. It is worth emphasizing that when the velocities of the particle phase and fluid phase are close, the negative particle force $\overline{F_{x}}$ does not necessarily imply that a faster fluid-phase velocity than the particle-phase velocity, since the results shown in figure 15 are the statistical averages and the local particle force does not scale linearly with the velocity difference in general. However, we do observe that $\overline{F_{x}}$ is more negative in case PLL and case PLS compared to the corresponding no-rotation cases as the fluid-phase velocities in the former two cases surpass the particle-phase velocities in a small region. Further approaching the channel centre, the hydrodynamic force is almost identically zero, which is probably because the velocities of the particle phase and the fluid-phase converge in the same region, as shown in figure 12.

Figure 16. The Reynolds stress profiles in single-phase and particle-laden turbulent channel flows. All results are normalized by the inner scale of the corresponding single-phase simulations. The result of Eshghinejadfard is extracted from figure 7 in Eshghinejadfard et al. (Reference Eshghinejadfard, Abdelsamie, Hosseini and Thévenin2017), the result of Picano is extracted from figure 5 in Picano et al. (Reference Picano, Breugem and Brandt2015) and rescaled.

Besides providing a non-zero particle force to modify the local fluid velocity, the presence of particles also alters the fluid velocity gradient significantly. The profiles of the fluid velocity gradient in the single-phase and particle-laden cases are presented in figure 11. Except in a small region very close to the wall in case PLS, where the accuracy can be affected by the insufficient grid resolution, particles generally reduce the fluid velocity gradient $\text{d}\overline{\langle u_{x}\rangle }/\text{d}y$ , as shown in figure 11. Part of the reason is that particles have a much smaller velocity gradient (i.e. twice its spanwise angular velocity) than the velocity gradient of the single-phase channel flow. Very close to the wall, the flow acceleration brought about by the positive particle force could be partially cancelled out by the negative impact due to the reduction of $\text{d}\overline{\langle u_{x}\rangle }/\text{d}y$ , but may be still dominating. Thus slightly larger mean flow velocities are observed in the particle-laden cases in this region. A special case happens in case PLS where both mechanisms are positive, thus the flow acceleration is more significant. Outside the viscous sublayer $y^{+}\geqslant 8$ , the reduction of $\text{d}\overline{\langle u_{x}\rangle }/\text{d}y$ starts to dominate, thus reduced local fluid velocities compared to the single-phase case are observed. Near the channel centre, the fluid velocity gradient $\text{d}\overline{\langle u_{x}\rangle }/\text{d}y$ in the particle-laden cases becomes slightly larger than the single-phase case. Since this phenomenon occurs even in case PLL–NR and case PLS–NR where the velocity gradient of the particle phase is forced to zero, the enhancement of $\text{d}\overline{\langle u_{x}\rangle }/\text{d}y$ is probably because particles enhanced the viscous effect there. Costa et al. (Reference Costa, Picano, Brandt and Breugem2016) argued that, near the channel centre, the particle distribution was rather homogeneous such that the effective viscosity would be enhanced due to the particle suspensions. They proposed a unified log law applicable to both single-phase and particle-laden channel flows, using a rescaling for the latter based on the effective viscosity and the effective distance from the virtual wall formed by local particle concentration, and validated the rescaling against the DNS datasets. We have examined their scaling against our simulation results in both particle-laden turbulent channel and pipe flows (Peng & Wang Reference Peng and Wang2018). Under Costa et al.’s (Reference Costa, Picano, Brandt and Breugem2016) scaling, the slope of the velocity profiles away from the channel wall can be made unified for both single-phase and particle-laden turbulent channel flows. Here we focus on understanding how certain local modulation of the mean flow velocity happens, so the similar investigation is not repeated.

The Reynolds stress profiles of the single-phase and particle-laden simulations are presented in figure 16. In comparison to the Reynolds stress in case SP, the small particles in case PLS result in larger Reynolds stress while the large particles in case PLL lead to smaller Reynolds stress. This particle size dependence was also observed in previous simulations by Shao et al. (Reference Shao, Wu and Yu2012) and Eshghinejadfard et al. (Reference Eshghinejadfard, Abdelsamie, Hosseini and Thévenin2017). Note that in figure 16, the Reynolds stress of Picano et al. (Reference Picano, Breugem and Brandt2015) is quantitatively larger than that in case PLS because the total driving force was increased in their simulations, but it is unchanged in the present simulations. When particle rotation is restricted, the magnitude of Reynolds stress decreases relative to the corresponding unrestricted case, as shown in figure 16. This comparison reveals two opposite effects of the solid particles on the modification of Reynolds stress. It is well known from the quadrant analysis in a single-phase turbulent channel flow that the Reynolds stress $(-u_{x}^{\prime }u_{y}^{\prime })$ has the statistical preference in the second (ejection) and fourth (sweeping) quadrants (Kim, Moin & Moser Reference Kim, Moin and Moser1987). When particles are present, on the one hand, due to their finite size, particles can filter out the small-scale fluctuations in the fluid phase. This results in a reduction of Reynolds stress and the reduction scales with the particle size. This effect can be seen from the comparisons between case PLL and case PLS, and between case PLL–NR and case PLS–NR. On the other hand, particle rotation in the spanwise direction induces additional ejection and sweeping events through entraining high speed fluid from the channel centre to the channel wall, and promoting ejection of low speed fluid from the wall. This effect is more significant with small particles, since small particles not only have larger spanwise rotation (see figure 11), but also have a larger total surface area than large particles when the particle volume fractions are the same. This explains the larger drop of Reynolds stress from case PLS to case PLS–NR compared to the drop from case PLL to case PLL–NR.

Table 4. The simulated TKEs and dissipation rates. The values in parentheses represent the relative percentages.

4.2.2 The TKE profile and budget analysis

The average fluid-phase TKE over the whole channel in each case is listed in table 4, and the profiles of TKE are compared in figure 6. Quantities shown in table 4 have been averaged over both the phase and time. In general, the presence of particles results in more homogeneous TKE distributions in the wall-normal direction, with enhanced TKE in the viscous sublayer and reduced TKE in the buffer region, compared to the single-phase case. The same observation was also reported in the other studies of particle-laden turbulent channel flow simulations by Picano et al. (Reference Picano, Breugem and Brandt2015), Wang et al. (Reference Wang, Peng, Guo and Yu2016b ) and Eshghinejadfard et al. (Reference Eshghinejadfard, Abdelsamie, Hosseini and Thévenin2017). A part of this change is directly related to the particle TKE, which is smaller than the TKE of the fluid phase in most regions except near the wall. This is because the finite size of particles makes them respond less to the rapidly changing small-scale fluctuating motions. The restriction of particle rotation further reduces the TKE in the particle phase, as the whole volume occupied by a particle moves with the same velocity. In the near-wall region, particles have larger TKE than the corresponding fluid phase in all four cases, due to the no-slip constraint on the channel walls. The lift forces could also lead to stronger fluctuating motions in the wall-normal direction. The modulation of TKE near the channel centre is small. A slightly increased TKE in case PLS and an almost unchanged TKE in the other three cases can be observed. Overall, TKE in case PLS is always larger than that in case PLL for all wall-normal locations. The same observation applies to the comparison between case PLS–NR and case PLL–NR.

In order to find out what mechanisms are responsible for the TKE modulation described above, a budget analysis of TKE in a particle-laden turbulent flow is conducted. Several previous attempts to investigate the full budget equation of TKE suffered from either difficulty in resolving the flow around the particles or accessing the data or both. In the cases where the particles were small compared to the flow structures, the flow was assumed continuous, with an additional term representing the force acting on the particles. The effect of the particles on the fluid TKE is then simplified as the work done by this particle force (e.g. Kulick et al. Reference Kulick, Fessler and Eaton1994; Li et al. Reference Li, McLaughlin, Kontomaris and Portela2001). This simple analysis was sometimes extended to the flows laden with large-size particles. For example, Lucci et al. (Reference Lucci, Ferrante and Elghobashi2010) analysed the sign of $\langle f_{i}u_{i}\rangle$ in their simulations of a decaying homogeneous isotropic turbulence laden with finite-size heavy particles, where $f_{i}$ was defined as the local immersed boundary force due to the presence of particles, $u_{i}$ was the local fluid velocity. They reported that $\langle f_{i}u_{i}\rangle$ was positive, i.e. the presence of particles acted as an extra source rather than a sink of the TKE. The major problem of their study is the lack of clear physical interpretation of the coupling term $\langle f_{i}u_{i}\rangle$ , since $f_{i}$ is an algorithm generated term according to the immersed boundary method, instead of a term that is physically determined. Full TKE budget analyses in FRSs became possible only recently. Vreman (Reference Vreman2016) performed a budget analysis in homogeneous isotropic turbulence with an array of fixed particles. In this study, since the particles were well defined and not changing, TKE budget equations of the single-phase flow could be applied to the coordinate attached to the particles. Vreman (Reference Vreman2016) found that the presence of particles triggered the conveyance of TKE from the far field to the particle surfaces to sustain the enhanced dissipation there. Two other relevant studies were reported by Kajishima et al. (Reference Kajishima, Takiguchi, Hamasaki and Miyake2001) and Wang et al. (Reference Wang, Ardila, Ayala, Gao and Peng2016a ). The former was performed in a simulation of particle-laden turbulent flow in a vertical channel with $Re_{p}\approx 500$ , but no details were provided on what the budget equation was and how this equation was examined. The latter effort calculated terms in a budget equation of TKE in coordinates attached to moving particles in particle-laden homogeneous isotropic turbulence. A direct investigation of the TKE budget equations of a fully resolved particle–fluid two-phase system was performed by Santarelli, Roussel & Fröhlich (Reference Santarelli, Roussel and Fröhlich2016) for a turbulent channel flow laden with small bubbles acting as rigid particles. This effort provided useful data to understand the turbulence modulation by light particles, but only the budget equation of the total TKE was investigated. In an anisotropic turbulent channel flow, the inter-component TKE transfer could also be affected by the solid particles, which should be studied. Very recently, Vreman & Kuerten (Reference Vreman and Kuerten2018) conducted a TKE budget analysis in their simulation of an array of particles moving at a constant velocity in a turbulent channel flow. Their study only involved the budget analysis of the total TKE, but the analysis was performed in both a moving frame attached the particles and a fixed frame attached to the channel wall. Since Vreman & Kuerten (Reference Vreman and Kuerten2018) conducted the simulation with an over-set grid and the fluid–particle interfaces were well defined, their results are accurate and provided a lot of insights into the local turbulence modulation near particle surfaces. However, the wall-normal locations of the particles in this study were quite far away from the channel wall ( $y^{+}=90$ and $150$ ), the interactions between the particles and the most inhomogeneous and anisotropic near-wall turbulence were not fully explored. du Cluzeau, Bois & Toutant (Reference du Cluzeau, Bois and Toutant2019) reported a full TKE budget analysis in a bubbly turbulent channel flow for both the total TKE and component-wise TKE. However, not every term was calculated explicitly in their work, so it is not possible to assess the overall reliability of their budget analysis.

The budget equations in a particle-laden turbulent channel flow can be derived using the theorem of volume averaging (Prosperetti & Tryggvason Reference Prosperetti and Tryggvason2009; Crowe et al. Reference Crowe, Schwarzkopf, Sommerfeld and Tsuji2011) and from the phase-field theory (Kataoka & Serizawa Reference Kataoka and Serizawa1989; Vreman & Kuerten Reference Vreman and Kuerten2018). These two methods essentially reach the same budget equations that are expressed as

total:

(4.6a ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D6FC}\frac{1}{2}\langle u_{i}^{\prime }u_{i}^{\prime }\rangle =\underbrace{-\unicode[STIX]{x1D6FC}\langle u_{x}^{\prime }u_{y}^{\prime }\rangle \frac{\unicode[STIX]{x2202}\langle u_{x}\rangle }{\unicode[STIX]{x2202}y}}_{E_{P}}\underbrace{-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\unicode[STIX]{x1D6FC}\frac{1}{2}\langle u_{i}^{\prime }u_{i}^{\prime }u_{y}^{\prime }\rangle }_{E_{TT}}\underbrace{-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\unicode[STIX]{x1D6FC}\langle p^{\prime }u_{y}^{\prime }\rangle }_{E_{PT}}\underbrace{+\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\unicode[STIX]{x1D6FC}\langle \unicode[STIX]{x1D70F}_{iy}^{\prime }u_{i}^{\prime }\rangle }_{E_{VT}}\nonumber\\ \displaystyle & & \displaystyle \qquad \underbrace{-\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\left\langle \unicode[STIX]{x1D70F}_{ij}^{\prime }\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{j}}\right\rangle }_{E_{VD}}\underbrace{+\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}(\langle -p\unicode[STIX]{x1D6FF}_{ij}+\unicode[STIX]{x1D70F}_{ij}\rangle )\left(\frac{\unicode[STIX]{x2202}\langle u_{i}\rangle }{\unicode[STIX]{x2202}x_{j}}-\left\langle \frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{j}}\right\rangle \right)}_{E_{IM}}\nonumber\\ \displaystyle & & \displaystyle \qquad \underbrace{+\frac{1}{\unicode[STIX]{x1D70C}V}\int _{S_{I}}n_{j}(-p\unicode[STIX]{x1D6FF}_{ij}u_{i}+u_{i}\unicode[STIX]{x1D70F}_{ij})\,\text{d}S}_{E_{IW}}-\underbrace{\frac{1}{\unicode[STIX]{x1D70C}V}\langle u_{x}\rangle \int _{S_{I}}n_{j}(-p\unicode[STIX]{x1D6FF}_{xj}+\unicode[STIX]{x1D70F}_{xj})\,\text{d}S}_{E_{IP}},\end{eqnarray}$$

streamwise:

(4.6b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D6FC}\frac{1}{2}\langle u_{x}^{\prime }u_{x}^{\prime }\rangle =\underbrace{-\unicode[STIX]{x1D6FC}\langle u_{x}^{\prime }u_{y}^{\prime }\rangle \frac{\unicode[STIX]{x2202}\langle u_{x}\rangle }{\unicode[STIX]{x2202}y}}_{E_{Px}}\underbrace{-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\unicode[STIX]{x1D6FC}\frac{1}{2}\langle u_{x}^{\prime }u_{x}^{\prime }u_{y}^{\prime }\rangle }_{E_{TTx}}\underbrace{+\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\unicode[STIX]{x1D6FC}\langle \unicode[STIX]{x1D70F}_{xy}^{\prime }u_{x}^{\prime }\rangle }_{E_{VTx}}\nonumber\\ \displaystyle & & \displaystyle \quad \underbrace{+\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\left\langle p^{\prime }\frac{\unicode[STIX]{x2202}u_{x}^{\prime }}{\unicode[STIX]{x2202}x}\right\rangle }_{E_{PSx}}\underbrace{-\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\left(\left\langle \unicode[STIX]{x1D70F}_{xx}^{\prime }\frac{\unicode[STIX]{x2202}u_{x}^{\prime }}{\unicode[STIX]{x2202}x}\right\rangle +\left\langle \unicode[STIX]{x1D70F}_{xy}^{\prime }\frac{\unicode[STIX]{x2202}u_{x}^{\prime }}{\unicode[STIX]{x2202}y}\right\rangle +\left\langle \unicode[STIX]{x1D70F}_{xz}^{\prime }\frac{\unicode[STIX]{x2202}u_{x}^{\prime }}{\unicode[STIX]{x2202}z}\right\rangle \right)}_{E_{VDx}}\nonumber\\ \displaystyle & & \displaystyle \quad \underbrace{+\!\left[\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\langle p\rangle \left\langle \frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}x}\right\rangle +\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\langle \unicode[STIX]{x1D70F}_{xy}\rangle \frac{\unicode[STIX]{x2202}\langle u_{x}\rangle }{\unicode[STIX]{x2202}y}-\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\!\left(\!\langle \unicode[STIX]{x1D70F}_{xx}\rangle \!\left\langle \frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}x}\right\rangle +\langle \unicode[STIX]{x1D70F}_{xy}\rangle \left\langle \frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}y}\right\rangle +\langle \unicode[STIX]{x1D70F}_{xz}\rangle \!\left\langle \frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}z}\!\right\rangle \!\right)\!\right]}_{E_{IMx}}\nonumber\\ \displaystyle & & \displaystyle \quad \underbrace{+\frac{1}{\unicode[STIX]{x1D70C}V}\int _{S_{I}}n_{j}\left(-p\unicode[STIX]{x1D6FF}_{xj}u_{x}+u_{x}\unicode[STIX]{x1D70F}_{xj}\right)\text{d}S}_{E_{IWx}}-\underbrace{\frac{1}{\unicode[STIX]{x1D70C}V}\langle u_{x}\rangle \frac{1}{\unicode[STIX]{x1D70C}V}\int _{S_{I}}n_{j}\left(-p\unicode[STIX]{x1D6FF}_{xj}+\unicode[STIX]{x1D70F}_{xj}\right)\text{d}S}_{E_{IPx}},\end{eqnarray}$$

transverse:

(4.6c ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D6FC}\frac{1}{2}\langle u_{y}^{\prime }u_{y}^{\prime }\rangle =\underbrace{-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\unicode[STIX]{x1D6FC}\frac{1}{2}\langle u_{y}^{\prime }u_{y}^{\prime }u_{y}^{\prime }\rangle }_{E_{TTy}}\underbrace{-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\unicode[STIX]{x1D6FC}\langle p^{\prime }u_{y}^{\prime }\rangle }_{E_{PTy}}\underbrace{+\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\unicode[STIX]{x1D6FC}\langle \unicode[STIX]{x1D70F}_{yy}^{\prime }u_{y}^{\prime }\rangle }_{E_{VTy}}\nonumber\\ \displaystyle & & \displaystyle \qquad \underbrace{+\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\left\langle p^{\prime }\frac{\unicode[STIX]{x2202}u_{y}^{\prime }}{\unicode[STIX]{x2202}y}\right\rangle }_{E_{PSy}}\underbrace{-\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\left(\left\langle \unicode[STIX]{x1D70F}_{xy}^{\prime }\frac{\unicode[STIX]{x2202}u_{y}^{\prime }}{\unicode[STIX]{x2202}x}\right\rangle +\left\langle \unicode[STIX]{x1D70F}_{yy}^{\prime }\frac{\unicode[STIX]{x2202}u_{y}^{\prime }}{\unicode[STIX]{x2202}y}\right\rangle +\left\langle \unicode[STIX]{x1D70F}_{yz}^{\prime }\frac{\unicode[STIX]{x2202}u_{y}^{\prime }}{\unicode[STIX]{x2202}z}\right\rangle \right)}_{E_{VDy}}\nonumber\\ \displaystyle & & \displaystyle \qquad \underbrace{+\left[\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\langle p\rangle \left\langle \frac{\unicode[STIX]{x2202}u_{y}}{\unicode[STIX]{x2202}y}\right\rangle -\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\left(\langle \unicode[STIX]{x1D70F}_{xy}\rangle \left\langle \frac{\unicode[STIX]{x2202}u_{y}}{\unicode[STIX]{x2202}x}\right\rangle +\langle \unicode[STIX]{x1D70F}_{yy}\rangle \left\langle \frac{\unicode[STIX]{x2202}u_{y}}{\unicode[STIX]{x2202}y}\right\rangle +\langle \unicode[STIX]{x1D70F}_{yz}\rangle \left\langle \frac{\unicode[STIX]{x2202}u_{y}}{\unicode[STIX]{x2202}z}\right\rangle \right)\right]}_{E_{IMy}}\nonumber\\ \displaystyle & & \displaystyle \qquad \underbrace{+\frac{1}{\unicode[STIX]{x1D70C}V}\int _{S_{I}}n_{j}(-p\unicode[STIX]{x1D6FF}_{yj}u_{y}+u_{y}\unicode[STIX]{x1D70F}_{yj})\,\text{d}S}_{E_{IWy}},\end{eqnarray}$$

and spanwise:

(4.6d ) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D6FC}\frac{1}{2}\langle u_{z}^{\prime }u_{z}^{\prime }\rangle =\underbrace{-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\unicode[STIX]{x1D6FC}\frac{1}{2}\langle u_{z}^{\prime }u_{z}^{\prime }u_{y}^{\prime }\rangle }_{E_{TTz}}\underbrace{+\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\unicode[STIX]{x1D6FC}\langle \unicode[STIX]{x1D70F}_{yz}^{\prime }u_{z}^{\prime }\rangle }_{E_{VTz}}\nonumber\\ \displaystyle & & \displaystyle \qquad \underbrace{+\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\left\langle p^{\prime }\frac{\unicode[STIX]{x2202}u_{z}^{\prime }}{\unicode[STIX]{x2202}z}\right\rangle }_{E_{PSz}}\underbrace{-\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\left(\left\langle \unicode[STIX]{x1D70F}_{xz}^{\prime }\frac{\unicode[STIX]{x2202}u_{z}^{\prime }}{\unicode[STIX]{x2202}x}\right\rangle +\left\langle \unicode[STIX]{x1D70F}_{yz}^{\prime }\frac{\unicode[STIX]{x2202}u_{z}^{\prime }}{\unicode[STIX]{x2202}y}\right\rangle +\left\langle \unicode[STIX]{x1D70F}_{zz}^{\prime }\frac{\unicode[STIX]{x2202}u_{z}^{\prime }}{\unicode[STIX]{x2202}z}\right\rangle \right)}_{E_{VDz}}\nonumber\\ \displaystyle & & \displaystyle \qquad \underbrace{+\left[\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\langle p\rangle \left\langle \frac{\unicode[STIX]{x2202}u_{z}}{\unicode[STIX]{x2202}z}\right\rangle -\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70C}}\left(\langle \unicode[STIX]{x1D70F}_{xz}\rangle \left\langle \frac{\unicode[STIX]{x2202}u_{z}}{\unicode[STIX]{x2202}x}\right\rangle +\langle \unicode[STIX]{x1D70F}_{yz}\rangle \left\langle \frac{\unicode[STIX]{x2202}u_{z}}{\unicode[STIX]{x2202}y}\right\rangle +\langle \unicode[STIX]{x1D70F}_{zz}\rangle \left\langle \frac{\unicode[STIX]{x2202}u_{z}}{\unicode[STIX]{x2202}z}\right\rangle \right)\right]}_{E_{IMz}}\nonumber\\ \displaystyle & & \displaystyle \qquad \underbrace{+\frac{1}{\unicode[STIX]{x1D70C}V}\int _{S_{I}}n_{j}\left(-p\unicode[STIX]{x1D6FF}_{zj}u_{z}+u_{z}\unicode[STIX]{x1D70F}_{zj}\right)\text{d}S}_{E_{IWz}}.\end{eqnarray}$$

Here, $E_{P}$ is the production due to mean flow shear, $E_{TT}$ , $E_{PT}$ , $E_{VT}$ are the transport of TKE in the inhomogeneous direction due to turbulence fluctuation, pressure fluctuation and viscous diffusion, respectively; $E_{VD}$ is the viscous dissipation rate; $E_{IM}$ is the change of TKE due to the unsteadiness of the fluid volume fraction in the fixed control volume $V$ , which arises due to the fact that the spatial gradient of a phase-averaged quantity is not equal to the phase average of the spatial gradient of the same quantity. At the statistically stationary stage, the long-time average of $E_{IM}$ should be zero as the long term average of fluid volume fraction in $V$ is constant. Here, $E_{IW}$ is the rate of work done by the particles on the fluid phase; $E_{IP}$ represents the part of $E_{IW}$ contributing to the kinetic energy of the mean flow; $E_{IW}-E_{IP}$ is therefore the part of particle-induced work that modifies TKE. The last three equations are the component-wise TKE budget equation of the streamwise ( $x$ ), wall-normal ( $y$ ) and spanwise velocity ( $z$ ) components, respectively; $E_{PSx}$ , $E_{PSy}$ and $E_{PSz}$ are the inter-component transfer of TKE due to the pressure strain rate coupling in the streamwise, wall-normal and spanwise directions, respectively; $\unicode[STIX]{x1D6FC}$ is the volume fraction of the fluid phase inside the control volume $V$ ; $S_{I}$ is the fluid–particle interface inside $V$ . In the present particle-laden turbulent channel flow simulations, the control volume $V$ is naturally chosen as thin layers stretching over the two homogeneous directions, i.e. the streamwise and spanwise directions. Therefore, the spatial derivatives of any quantity in the streamwise and spanwise directions are zero, i.e. $\unicode[STIX]{x2202}\langle \cdots \!\,\rangle /\unicode[STIX]{x2202}x=\unicode[STIX]{x2202}\langle \cdots \!\,\rangle /\unicode[STIX]{x2202}z=0$ . Furthermore, the fluid shall have zero mean velocity in the wall-normal and spanwise directions, with and without particles. All these trivial terms are already crossed out in the four TKE budget equations in (4.6). After careful cross-examination, equation (4.6) is found to be identical with the budget equation derived by Vreman & Kuerten (Reference Vreman and Kuerten2018) with the assumption of constant driven force and the above simplifications. At the statistical stationary state, the time average of the TKE budget equations in (4.6) are investigated.

Figure 17. Budgets of turbulent kinetic energy in the particle-laden turbulent channel flow simulations: (a) case PLL, (b) case PLL–NR, (c) case PLS, (d) case PLS–NR.

Figure 18. Time-averaged rates of TKE input in the single-phase and particle-laden turbulent channel flows. (a) $E_{P}$ , (b) $E_{W}$ , (c) $E_{P}+E_{W}$ .

The time-averaged values of various terms in the total TKE budget equation are presented in figure 17 for all simulated cases, where the solid black line in each plot represents summation of all terms on the right-hand side of (4.6a ); $E_{IW}$ and $-E_{IP}$ are combined as $E_{W}$ for simplicity. The overall balance in each case is well captured except for a significant deviation in a small region very close to the channel wall in case PLL and case PLS. The imbalances are contained in the region $y^{+}<5$ , which roughly correspond to the first three grid points from the wall. They are likely numerical errors due to the insufficient local grid resolution very close to the wall. Without particle rotation, the chance that particles enter the near-wall region is much smaller, thus the numerical errors described above might be small. The fact that the whole particle moves with the same translational velocity may also help reduce the numerical error. The imbalances in case PLL–NR and case PLS–NR are therefore much smaller. Keeping these numerical errors in mind, the near-wall results in case PLL and case PLS must be taken as preliminary results.

A significant volume (5 %) is occupied by particles in each particle-laden case. To fairly compare the volumetric intensity of each mechanism, results in figure 17 should be scaled by normalizing the time-averaged local fluid volume fraction in each case, which essentially means the phase average, and is expressed by $\langle \cdots \!\,\rangle$ in the rest of the paper. For the sake of conciseness in the ongoing discussion, terms in (4.6a ) are grouped into three categories. The TKE production $E_{P}$ and the particle work $E_{W}$ are discussed together as the TKE source terms; $E_{VD}$ is the viscous dissipation rate that converts TKE into heat, which alone represents the intensity of TKE sink. Finally, $E_{TT}$ , $E_{PT}$ , $E_{VT}$ and $E_{IM}$ are combined as TKE transport along the wall-normal direction due to flow inhomogeneity.

Time-averaged TKE source terms in the single-phase and particle-laden cases are shown in figure 18. In the viscous sublayer $y^{+}\leqslant 8$ , $\overline{\langle E_{P}\rangle }$ is slightly enhanced in case PLS and almost unaltered in the other three cases. By definition, $\overline{\langle E_{P}\rangle }$ is the product of the Reynolds stress $\overline{\langle -u_{x}^{\prime }u_{y}^{\prime }\rangle }$ and the mean velocity gradient $\text{d}\overline{\langle u_{x}\rangle }/\text{d}y$ . The slight enhancement inside the viscous sublayer in case PLS is a result of the increase of mean velocity gradient in the same region, as shown in figure 11. Again, caution should be taken as this enhancement may not be quantitatively accurate due to inadequate local grid resolution, as discussed earlier. In the other three cases, the Reynolds stress and mean velocity gradient in the viscous sublayer are almost unaltered, so is $\overline{\langle E_{P}\rangle }$ . In the buffer region, $\overline{\langle E_{P}\rangle }$ is reduced by the presence of particles in all four particle-laden cases. In case PLS, while the mean velocity gradient in the buffer region is reduced by particles, the Reynolds stress is enhanced. Therefore, the reduction of $\overline{\langle E_{P}\rangle }$ in case PLS is not as significant as in the other three case, where both the Reynolds stress and mean velocity gradient are reduced. The reduction of $\overline{\langle E_{P}\rangle }$ in case PLS also occurs over a smaller range of wall-normal locations than these in the other three cases.

Particle rotation can either increase or decrease $\overline{\langle E_{P}\rangle }$ in the buffer region, depending on the particle size. For large particles, although the restriction of particle rotation results in a further reduction of Reynolds stress, it yields a smaller reduction of the mean velocity gradient. The combination leads to a higher $\overline{\langle E_{P}\rangle }$ in case PLL–NR than that in case PLL in the buffer region. On the contrary, for small particles, the restriction of particle rotation causes a significant drop of Reynolds stress and almost unaltered mean velocity gradient. Thus a smaller $\overline{\langle E_{P}\rangle }$ is observed in case PLS–NR in the buffer region. Moving closer to the channel centre, $\overline{\langle E_{P}\rangle }$ is slightly increased in all four particle-laden cases. This change is attributed to the enhanced mean velocity gradient brought by particles, as shown in figure 11. The enhancement in case PLS is greater than the other three cases probably because the Reynolds stress is also increased in this case.

In the present particle-laden flows, particle work $E_{W}$ always acts as an additional production mechanism of the TKE. The same observation was also made by Lucci et al. (Reference Lucci, Ferrante and Elghobashi2010) in their simulation of decaying homogeneous isotropic turbulence with finite-size particles. In that work, the extra TKE input due to the presence of particles was interpreted as the two-way coupling term $\langle u_{i}^{\prime }f_{i}\rangle$ . As commented earlier, the physical meaning of this two-way coupling term is not as clear as $E_{W}$ derived in the TKE budget. The magnitude of this two-way coupling term also largely depends on the numerical algorithm that defines the coupling force $f_{i}$ . In the present simulations, $E_{W}$ contributes to TKE of the fluid phase by either the particle translation or the particle rotation. The comparison of $\overline{\langle E_{W}^{+}\rangle }$ between case PLL(PLS) and case PLL–NR(PLS–NR) in figure 18(b) shows that both mechanisms are important. With the increase of particle size, $\overline{\langle E_{W}^{+}\rangle }$ due to particle rotation takes a larger percentage, which can be seen from the larger drop of $\overline{\langle E_{W}^{+}\rangle }$ in the two large particle cases. This increased importance of particle rotation has two reasons. On the one hand, particle rotation becomes increasingly important when the particle size increase. The effects of particle rotation are usually ignored for small particles. On the other hand, large particles tend to have smaller translational velocity fluctuations than small particles due to their larger size. Comparing the two PLL profiles, there is a small region around $y^{+}=10$ where $\overline{\langle E_{W}^{+}\rangle }$ in the free-rotation case is slightly smaller than that in the corresponding no-rotation case. This is not because particle rotation does negative work on the fluid phase, but rather due to the change of particle distribution when particle rotation is turned off. Unlike the other terms in the budget equations that are essentially volume integrals, $E_{W}$ is a surface integral that may not be scaled properly by normalizing the fluid volume fraction. The combination of $E_{P}$ and $E_{W}$ is the total TKE source. With the addition of $E_{W}$ , the net TKE input is increased at all wall-normal locations in case PLS, compared to the unladen case. In the other three particle-laden cases, the total rates of TKE input are larger than the unladen case except in the region of $10\leqslant y^{+}\lessapprox 25$ .

Figure 19. Time-averaged rates of viscous dissipation of TKE in the single-phase and particle-laden turbulent channel flows.

Figure 20. Time-averaged rates of turbulent kinetic energy transportation in the single-phase and particle-laden turbulent channel flows.

The time-averaged dissipation rates $\overline{\langle E_{VD}\rangle }$ in different cases are compared in figure 19. The averaged dissipation rate over the whole channel in each case is also listed in table 4. The presence of particles enhances the dissipation rate in case PLS, but has a negligible effect in the other three cases. In the latter three cases, the magnitudes of $\overline{\langle E_{VD}\rangle }$ are increased near the wall at $y^{+}\leqslant 10$ , and only marginally altered elsewhere. It should be clear that at the stationary stage, the local dissipation rate of TKE must balance the local TKE source and the TKE transported from other places. Therefore, the presence of particles does not necessarily lead to an enhanced total dissipation rate, as reported in earlier studies of homogeneous isotropic particle-laden turbulent flows (Burton & Eaton Reference Burton and Eaton2005; Lucci et al. Reference Lucci, Ferrante and Elghobashi2010).

Finally, the TKE transport due to the flow inhomogeneity in the wall-normal direction is examined in figure 20. Compared to the single-phase case, less TKE is transported out from $10\leqslant y^{+}\leqslant 30$ in the particle-laden cases. As shown in figure 17, the TKE transport in this region is mainly contributed by the turbulent transport $E_{TT}$ , which is suppressed in the particle-laden cases due to the local reduction of turbulence intensity. Very close to the wall $y^{+}\leqslant 2$ , more TKE is transported to this region to balance the enhanced viscous dissipation. In this near-wall region, most TKE is transported through viscous diffusion, as the viscous effect dominates locally. The viscous effect in this region is further enhanced by the presence of particles. Since the restriction of particle rotation makes it harder for particles to approach the channel walls, the enhancements of TKE transport in $y^{+}\leqslant 2$ are smaller in cases PLL–NR and PLS–NR, compared to the two freely rotating particle cases. In the region of $2\leqslant y^{+}\leqslant 10$ , the TKE transport is still mainly carried by the viscous diffusion, as shown in figure 17. The intensity of TKE transport in the particle-laden cases is either reduced ( $2\leqslant y^{+}\leqslant 5$ ) or enhanced ( $5\leqslant y^{+}\leqslant 10$ ) compared to unladen case, but the overall effect of particles is to reduce the net TKE received by this region. This implies that particles may have two opposite impacts on the overall viscous transport in this region. On the one hand, the presence of particles always enhances the effective viscosity. On the other hand, particles also increase the homogeneity of TKE distribution in the wall-normal direction, which lowers the transport along the wall-normal direction. The contribution of $\overline{\langle E_{IM}\rangle }$ to the TKE transport is almost negligible. This is expected since although the instantaneous TKE transport due to the dynamic redistribution of particles is non-zero, the long-term average of $\langle E_{IM}\rangle$ should be zero as there is no net particle volume flux in the wall-normal direction.

So far, each mechanism in the budget equation of the total TKE has been quantified using the DNS data. This analysis provides a better understanding of the TKE modulations by the particles. However, it must be emphasized that terms the budget equation only represent the changing rates of TKE instead of TKE itself. An increased rate of TKE production does not necessarily lead to the same increase on TKE itself. The budget analysis based on the statistically stationary condition may also be insufficient to reveal how a new equilibrium state of TKE is established after particles are released. Further insights could be drawn if the dynamic transition from the stationary state of single-phase flow to that of particle-laden flow is examined, which could be a subject for future study. Based on the above analysis, the reduction of turbulent intensity in the region $5\leqslant y^{+}\leqslant 30$ in the particle-laden cases is probably due to the reduced TKE production by the flow shear, i.e. $E_{P}$ , resulting from the reduced mean velocity gradient in the same region, with an exception in case PLS. A same observation was made in the earlier experimental investigation of a turbulent channel flow laden with small particles by Paris (Reference Paris2001), where TKE was found to be significantly reduced while the dissipation rate was only slightly enhanced. In case PLS, on the other hand, adding the particle work $E_{W}$ , the net TKE production becomes larger compared to that in the single-phase case. In this case, the attenuated TKE in the region of $5\leqslant y^{+}\leqslant 30$ might be associated with the enhanced dissipation induced by particles.

Figure 21. The profiles of phase-partitioned streamwise r.m.s. velocity in single-phase and particle-laden turbulent channel flows: (a) streamwise r.m.s. velocity, (b) the modulation of the fluid streamwise r.m.s. velocity compared to single phase. The result of Eshghinejadfard is extracted from figure 6 in Eshghinejadfard et al. (Reference Eshghinejadfard, Abdelsamie, Hosseini and Thévenin2017), the result of Picano is extracted from figure 5 in Picano et al. (Reference Picano, Breugem and Brandt2015).

4.2.3 The r.m.s. velocities and the budget analyses of component TKEs

Besides the modulation of the total TKE, it is also interesting to investigate how the presence of particles redistributes TKE among different spatial directions. The component-wise TKE averaged over the whole channel in each case is shown in table 4. When particle rotation is not restricted, small particles tend to yield a larger r.m.s. velocity than the large particles in each spatial direction. Such a particle-size effect is much weaker for non-rotating particles. The presence of freely rotating particles also generates a net flux of TKE from the streamwise velocity component to the spanwise component. The same trend is less obvious for non-rotating particles. The profiles of the streamwise, wall-normal and spanwise r.m.s. velocities, computed as $\sqrt{\overline{\langle u_{x}^{\prime 2}\rangle }}$ , $\sqrt{\overline{\langle u_{y}^{\prime 2}\rangle }}$ , $\sqrt{\overline{\langle u_{z}^{\prime 2}\rangle }}$ , respectively, in the single-phase and particle-laden cases are presented in figures 21–23, respectively. These results obtained in the present simulations are in good qualitative agreement with those reported in earlier studies of Picano et al. (Reference Picano, Breugem and Brandt2015) and Eshghinejadfard et al. (Reference Eshghinejadfard, Abdelsamie, Hosseini and Thévenin2017). Similar to the effect on total TKE, particles generally result in a more uniform streamwise TKE distribution along the wall-normal direction. For the other two velocity components, the presence of particles enhances the TKE near the channel wall but attenuates the TKE near the channel centre, with an exception of case PLS where the wall-normal and spanwise TKEs are enhanced everywhere. As we shall see shortly, this exception is probably related to the strongest enhancement of the inter-component TKE transfer from the streamwise velocity component to the other two velocity components near the channel centre in case PLS compared to other cases.

Figure 22. The profiles of phase-partitioned wall-normal r.m.s. velocity in single-phase and particle-laden turbulent channel flows: (a) wall-normal r.m.s. velocity, (b) the modulation of the fluid wall-normal r.m.s. velocity compared to single phase. The result of Eshghinejadfard is extracted from figure 6 in Eshghinejadfard et al. (Reference Eshghinejadfard, Abdelsamie, Hosseini and Thévenin2017), the result of Picano is extracted from figure 5 in Picano et al. (Reference Picano, Breugem and Brandt2015).

Figure 23. The profiles of phase-partitioned spanwise r.m.s. velocity in single-phase and particle-laden turbulent channel flows: (a) spanwise r.m.s. velocity, (b) the modulation of the fluid spanwise r.m.s. velocity compared to single phase. The result of Eshghinejadfard is extracted from figure 6 in Eshghinejadfard et al. (Reference Eshghinejadfard, Abdelsamie, Hosseini and Thévenin2017), the result of Picano is extracted from figure 5 in Picano et al. (Reference Picano, Breugem and Brandt2015).

Figure 24. Budgets of streamwise turbulent kinetic energy in the particle-laden turbulent channel flow simulations: (a) case PLL, (b) case PLL–NR, (c) case PLS, (d) case PLS–NR.

Figure 25. Budgets of wall-normal turbulent kinetic energy in the particle-laden turbulent channel flow simulations: (a) case PLL, (b) case PLL–NR, (c) case PLS, (d) case PLS–NR.

Figure 26. Budgets of spanwise turbulent kinetic energy in the particle-laden turbulent channel flow simulations: (a) case PLL, (b) case PLL–NR, (c) case PLS, (d) case PLS–NR.

Figure 27. Terms in the streamwise turbulent kinetic energy budget equation (4.6b ): (a) source, (b) transport, (c) inter-component transfer, (d) viscous dissipation.

To better understand the above modulations, the component-wise TKE budget equations, equation (4.6b )–(4.6d ), are examined in figures 24–26, respectively. The overall balance shown in (4.6b )–(4.6d ) is well captured except that, in case PLL and case PLS, certain errors appear on the two node points next to the channel walls, especially in the streamwise and spanwise directions. Again, these errors are likely related to the inadequate grid resolution locally because of the small gap between the particles and channel walls. The same errors almost disappear when particle rotation is restricted. For the sake of the simplicity in the following discussion, terms in the TKE budget equations are grouped according to their physical significance. Besides the terms in the budget equation of total TKE, $E_{PSi}$ represents the inter-component transfer of TKE. A positive $E_{Psi}$ indicates the $i$ velocity component is receiving net TKE from other velocity components. A negative $E_{PSi}$ means the TKE in the $i$ th velocity component is being transferred out to other components.

The time-averaged profiles of the source, the transport, the inter-component transfer and the sink of the streamwise TKE in the single-phase and particle-laden flows are presented in figure 27. In all four particle-laden cases, particle work $E_{Wx}$ serves as an additional source of TKE throughout the channel. This additional TKE production is contributed by both the translational velocity fluctuation and rotation of the particles. Although $E_{Wx}$ provides additional TKE input, it is not enough to compensate the reduction of $E_{P}$ in the buffer region, which is due to the reduced flow shear rate. This leads to the attenuation for the streamwise r.m.s. velocity in the same region. The TKE source at other wall-normal locations is, however, increased by the presence of particles. This causes augmentation of the streamwise r.m.s. velocity in the viscous sublayer and outside the buffer region relative to the value in case SP, as shown in figure 21(b). The modification of the streamwise TKE transport is similar to that observed for the total TKE. In the region $10\leqslant y^{+}\leqslant 30$ where the turbulent transport $E_{TTx}$ serves as the major mechanism, less streamwise TKE is transported out because of the reduced turbulence intensity there. Further close to the channel walls, the viscous transport $E_{VT}$ dominates, and the intensity of the transport of streamwise TKE can be either enhanced ( $y^{+}\leqslant 2$ and $5\leqslant y^{+}\leqslant 10$ ) or attenuated ( $2\leqslant y^{+}\leqslant 5$ ), probably due to the dual effects of particles on the viscous diffusion, i.e. particles result in larger effective viscosity but at the same time create a more homogeneous distribution of TKE.

The inter-component transfer of TKE is significantly enhanced by the presence of particles. More TKE is given out by the streamwise velocity component to the other two components in both the viscous sublayer and the buffer region, except a small region attached to the wall, as shown in figure 27(c). The strengthened inter-component transfer of TKE results from not only the spherical shape of the particles but also the particle rotation. It eventually leads to a more isotropic TKE distribution in the particle-laden cases in the buffer region. In a small region attached to the channel walls, the streamwise velocity component actually receives net TKE from the other two components when particle rotation is allowed. This phenomenon is not observed in the single-phase case and is very weak in case PLL–NR and case PLS–NR. As will become clear later, this phenomenon occurs essentially because more TKE is transferred from the streamwise velocity component to the wall-normal component in the buffer region. Then, through a significantly enhanced pressure transport mechanism, the received TKE in the wall-normal component is carried from the buffer region to the viscous sublayer. When this wall-normal TKE arrives the near-wall region, it is returned back to the streamwise velocity component, so a positive $E_{Px}$ is observed. Finally, the dissipation rates $E_{VDx}$ of streamwise TKE in the particle-laden cases are enhanced in the viscous sublayer but reduced in the buffer region. The modulation of $E_{VDx}$ balances with the modulations of TKE source, transport and inter-component transfer in each particle-laden turbulent flow case.

Figure 28. Terms in the transverse turbulent kinetic energy budget equation (4.6c ): (a) source, (b) transport, (c) inter-component transfer, (d) viscous dissipation.

Figure 29. Terms in the spanwise turbulent kinetic energy budget equation (4.6d ): (a) source, (b) transport, (c) inter-component transfer, (d) viscous dissipation.

The detailed investigation of the terms in the budget equation of wall-normal TKE is presented in figure 28. Different from the single-phase case, where the only source of wall-normal TKE is inter-component transferred from the streamwise TKE, in the particle-laden cases, wall-normal velocity fluctuations also receive energy input directly from the particles work $E_{Wy}$ , as shown in figure 28(a). However, unlike $E_{Wx}$ in the streamwise TKE budget equation that is contributed by both particle translational velocity fluctuation and particle rotation, $E_{Wy}$ in the wall-normal TKE balance almost purely comes from the particle rotation. This observation is made since in case PLL–NR and case PLS–NR where particle rotation is restricted, $E_{Wy}$ is almost zero; $E_{Wy}$ in case PLL and case PLS is significant only in the region of $y^{+}\leqslant 40$ . As shown in figure 28(c), in the region of $10\leqslant y^{+}\leqslant 30$ , particles also enhance the inter-component transfer mechanism and convert more TKE to the wall-normal velocity component. The above two modulations together result in an enhanced wall-normal r.m.s. velocity in the buffer region. Since the transport mechanisms, especially the pressure transport $E_{PTy}$ are significantly augmented, the received wall-normal TKE in the buffer region is transported to the viscous sublayer, as shown in figure 28(b). The enhanced wall-normal TKE transport mechanisms are mainly related to the particle rotation, probably associated with the sweeping events created by the particle rotation. When the particle rotation is restricted, the profiles of TKE transport rates for case PLL–NR and case PLS–NR in figure 28(b) appear similar to that of the single-phase case. When the transported TKE reaches the channel walls, it is transferred to the streamwise and spanwise components rather than dissipated directly into heat, as the dissipation rate of the wall-normal TKE is zero at the wall. The wall-normal TKE transferred back to the streamwise component results in the positive $E_{PSx}$ observed in figure 27(c), while the part of wall-normal TKE transferred to the spanwise component further enhanced the spanwise TKE in the near-wall region. At last, the dissipation rates of the wall-normal TKE in the particle-laden cases are significantly enhanced throughout the channel. This can also be seen in table 4. The relative enhancements of $\overline{\langle \unicode[STIX]{x1D700}_{y}^{+}\rangle }$ are 35.6 %, 7.5 %, 71.7 % and 15.1 % in case PLL, case PLL–NR, case PLS and case PLS–NR, respectively.

Finally, each term in the spanwise TKE budget equation, equation (4.6d ), is examined in figure 29. For the spanwise velocity component, the TKE input brought by the particle work $E_{Wz}$ is almost negligible. A nearly zero $E_{Wz}$ together with non-zero $E_{Wx}$ and $E_{Wy}$ implies that only the particle rotation in the spanwise direction contributes to the net turbulence modulation. The enhancement of spanwise r.m.s. velocity in the particle-laden cases is due to the enhanced inter-component transfer of TKE from the other two components. This mechanism mainly happens in the region $y^{+}\leqslant 40$ , which is in good correspondence with the region of major spanwise r.m.s. velocity enhancement, as shown in figure 23. The zigzag shape of the pressure–strain rate profile in the single-phase case is probably due to the numerical error resulting from the weak compressibility of LBM, which does not affect the conclusions of the budget analyses. The maximum values of $\overline{E_{PSz}}$ in the particle-laden cases occur around $5\leqslant y^{+}\leqslant 20$ . This results in an increased level of inhomogeneity that transports more TKE out from $5\leqslant y^{+}\leqslant 20$ to $y^{+}\leqslant 5$ , in order to balance the enhanced local dissipation in the latter region. Similar to the wall-normal component, the dissipation rate of the spanwise TKE is enhanced throughout the channel in the particle-laden cases.

Figure 30. Diagram to summarize the results of turbulence modulation by particles from this study. Blue arrows: mechanisms that exist in the single-phase turbulent channel flow, where double arrows represent mechanisms are strengthened by the presence of particles, dash arrows represent mechanisms are weakened by the presence of particles. Red arrows: new mechanisms that are not present in the single-phase turbulent channel flow. Mechanisms: (1) production, (2) particle work, (3a) turbulent transport, (3b) pressure transport, (3c) viscous transport, (4) inter-component transfer.

4.2.4 Summary of TKE modulation by solid particles

The results of various aspects of TKE modulation by solid particles can now be summarized in figure 30. First, the double solid blue arrows represent the existing mechanisms that are strengthened by the presence of particles. These mechanisms include: (1) the inter-component transfer of TKE from the streamwise velocity component to the wall-normal and spanwise components in the buffer region, (2) the production of streamwise TKE from the mean flow in the viscous sublayer, (3) the pressure transport of the wall-normal TKE from the buffer region to the viscous sublayer, (4) the viscous transport of the spanwise TKE from buffer region to the viscous sublayer, and (5) the inter-component transfer of TKE from the wall-normal component to the spanwise component in the viscous sublayer. Second, the blue dash arrows mark these existing mechanisms that are suppressed by the presence of particles. These mechanisms include (1) the production of the streamwise TKE from the mean flow in the buffer region, and (2) the viscous and turbulent transport of streamwise TKE from the buffer region to the viscous sublayer. Finally, the presence of particles also creates new mechanisms that are not present in the single-phase turbulent channel flow; these are labelled with red solid arrows in figure 30: (1) the production of the wall-normal TKE directly from the mean flow through particle work in both the viscous sublayer and buffer region, and (2) the inter-component TKE transfer from the wall-normal component back to the streamwise component very close to the channel wall. The overall effect of all the above modulations is to yield a more homogeneous distribution of TKE in the wall-normal direction, and a slightly more isotropic distribution of TKE among different velocity components. Since more TKE is transferred from the streamwise velocity component to the other two components when particles are present, the overall dissipation rates of the wall-normal and spanwise TKE take a larger percentage of the total dissipation rate, which is evident in table 4.