5.1. The mean flow properties

At first, we briefly examine the mean flow properties to establish a basic impression of the particle inertial effects on turbulence modulation. Figure 3 $(a)$ depicts the mean flow velocity profiles for all cases. Compared with the single-phase result indicated by a solid grey curve, the linear scaling in the viscous sublayer remains, while the velocity profile around the logarithmic layer is significantly altered. In the logarithmic layer, the mean velocity is shifted downwards due to the introduction of particles at $St^+=3$ . As the Stokes number increases to $56{-} 100$ , the mean velocity increases successively, exceeding the single-phase one. With a further increase of $St^+$ to $300$ , the mean velocity decreases and tends to approach the single-phase profile. This variation of the mean flow velocity profile with regards to the Stokes number corresponds to a non-monotonic modification of wall shear stress, i.e. an intriguing flow transition from a drag-enhanced to a drag-reduced regime and subsequently approaching the single-phase state, as substantiated in the next section.

Figure 3.

Mean profiles of: $(a)$ streamwise flow velocity in wall units; $(b)$ streamwise particle velocity in outer units; $(c)$ streamwise particle velocity in wall units and $(d)$ normalised particle concentration.

Given its critical role in momentum and energy transport, the mean particle velocity, $\langle v_1\rangle$ , is also examined for different Stokes numbers in figure 3 $(b,c)$ . It is demonstrated that, as $St^+$ increases, the mean particle velocity profile flattens progressively, especially in the near-wall region. This trend has been explained by Johnson (Reference Johnson2020) and Liao et al. (Reference Liao, Chen, Wan, Liu and Lu2024) as follows: with the increase of $St^+$ , the particle inertia tends to dominate over the Stokes drag exerted by the background flow, leading to an increasingly stronger tendency for the particle phase to approach a so-called many-particle system that has an homogeneous and isotropic equilibrium state. Moreover, the particle concentration close to the wall exhibits a substantial decrease versus $St^+$ , see figure 3 $(d)$ . It deviates from the pattern observed in one-way coupled simulations, where turbophoresis is found leading to a dramatic increase of near-wall particle concentration over $2{-} 3$ orders of magnitude compared with the uniform distribution and the highest accumulation typically occurring at $St^+\approx 30$ (Marchioli & Soldati Reference Marchioli and Soldati2002; Jie et al. Reference Jie, Cui, Xu and Zhao2022). This deviation can be attributed to the synergistic effect of particle back-reaction and inter-particle collision. The former alters the intrinsic length and time scales of near-wall coherent structures, which are crucial for the preferential accumulation of inertial particles (Motoori, Wong & Goto Reference Motoori, Wong and Goto2022; Jie et al. Reference Jie, Cui, Xu and Zhao2022). The latter induces a strong diffusion of the particle phase, preventing excessive clustering and promoting a more uniform distribution.

5.2. The drag properties

We then focus on the drag properties, which are widely regarded as crucial in wall-bounded turbulence of practical relevance. The overall statistical results are summarised in table 2. It is intriguing to find that the drag modification depends non-monotonically on the Stokes number and the system experiences both drag-enhancement (DE, $St^+\leqslant 10$ ) and drag-reduction (DR, $St^+\geqslant 30$ ) regimes. Specifically, the maximum DE occurs at the lowest $St^+$ , with a smooth transition from DE to DR as $St^+$ increases, where $St^+ \approx 10$ marks the critical point. Further increasing the Stokes number, the flow statistics indicate a tendency to return to the single-phase state, resulting in a maximum DR at $St^+\approx 100$ .

Table 2.

Overall statistics of the present simulations. Here, the skin-friction coefficient is $C_f\equiv 2\tau _w/(\rho U_b^2)$ . The drag-modification coefficient is defined as $\text{DM}\equiv (C_f-C_f^{{SP}})/C_f^{{SP}}$ .

Such a DE–DR–SP transition can be substantiated by the shear stress balance (SSB, Costa et al. Reference Costa, Brandt and Picano2021; Dave & Kasbaoui Reference Dave and Kasbaoui2023)

(5.1) \begin{equation} \underbrace {\vphantom {\int _{0}^{y}}\mu \frac {\textrm{d} \bar {u}_{1}}{\textrm{d}y}}_{\bar {\tau }^{\textit{V}}} \underbrace {\vphantom {\int _{0}^{y}}-\rho \overline {u'_1u'_2}}_{\bar {\tau }^{\textit{T}}} +\underbrace {\vphantom {\int _{0}^{y}}\rho \int _{0}^{y}\overline {\mathcal{F}}_1\,\textrm{d}y}_{\bar {\tau }^{\textit{P}}} =\tau _w\left (1-\frac {y}{h}\right ). \end{equation}

Here, $\bar {\tau }^{\textit{V}}$ denotes the viscous shear stress of mean flow, $\bar {\tau }^{\textit{T}}$ the Reynolds shear stress of turbulent motions and $\bar {\tau }^{\textit{P}}$ the extra shear stress induced by particles. Essentially, these shear stresses reflect the transport of fluid-phase streamwise momentum in the wall-normal direction. Following Gualtieri et al. (Reference Gualtieri, Battista, Salvadore and Casciola2023), their contributions to the skin-friction coefficient can be quantified by the well-known FIK-type integrals (Fukagata, Iwamoto & Kasagi Reference Fukagata, Iwamoto and Kasagi2002), i.e. integrating (5.1) twice and reading as

(5.2) \begin{equation} \underbrace {\vphantom {\frac {6}{h^2U_b^2}}\frac {6}{Re_b}}_{C_f^{\textit{V}}} +\underbrace {\vphantom {\frac {6}{h^2U_b^2}}\frac {6}{h^2U_b^2}\int _{0}^{h}(h-y)\big(-\overline {u'_1u'_2}\big)\,\textrm{d}y}_{C_f^{\textit{T}}} +\underbrace {\vphantom {\frac {6}{h^2U_b^2}}\frac {6}{h^2U_b^2}\int _{0}^{h}\int _{0}^{y}\int _{0}^{y_1}\overline {\mathcal{F}}_1(y_2)\,\textrm{d}y_2\textrm{d}y_1\textrm{d}y}_{C_f^{\textit{P}}} =C_f. \end{equation}

Since the bulk Reynolds number is fixed in the present simulations, $C_f^{\textit{V}}$ remains constant. Hence, the frictional drag is determined by an effective shear stress $\bar {\tau }^{\textit{E}}$ defined as the summation of $\bar {\tau }^{\textit{T}}$ and $\bar {\tau }^{\textit{P}}$ .

Figure 4.

Shear stress balance profiles for four typical cases: $(a)$ St003; $(b)$ St010; $(c)$ St100 and $(d)$ St300. In panel $(d)$ , the SP results are also shown as dashed lines of corresponding colours, representing the $St^+\to \infty$ limit of SSB.

We depict four typical SSB profiles (St003, St010, St100 and St300) in figure 4. Specifically, in the St003 case, the introduction of $\bar {\tau }^{\textit{P}}$ exceeds the suppression of $\bar {\tau }^{\textit{T}}$ , resulting in a DE of $20.6\,\%$ . In the St010 case, $\bar {\tau }^{\textit{P}}$ compensates for the suppression of $\bar {\tau }^{\textit{T}}$ , leading to a flow resistance close to the single-phase case. In the St100 case, $\bar {\tau }^{\textit{P}}$ drops substantially along with a further suppression of $\bar {\tau }^{\textit{T}}$ , resulting in a DR of $-16.4\,\%$ . In the St300 case, the degree of suppression of $\bar {\tau }^{\textit{T}}$ is reduced, accompanied by a relatively small $\bar {\tau }^{\textit{P}}$ , causing the flow system to tend closer to the SP situation. To better illustrate the Stokes number dependence of each term, the $\bar {\tau }^{\textit{T}}$ , $\bar {\tau }^{\textit{P}}$ and $\bar {\tau }^{\textit{E}}$ profiles of all cases are shown in figure 5 $(a$ – $c)$ . As $St^+$ increases, it is demonstrated that basically the Reynolds shear stress decreases as $St^+\leqslant 56$ and then increases when $St^+\geqslant 100$ (figure 5 a), while the particle-induced shear stress has an overall monotonically decreasing trend despite a slight increase of $\bar {\tau }^{\textit{P}}$ for $y\gtrsim 0.4 h$ at high $St^+\geqslant 100$ (figure 5 b). Their superposition leads to a non-monotonic variation of the effective shear stress versus $St^+$ (figure 5 c). Such a competition between the particle-induced extra transport and the suppressed turbulent transport is clearly demonstrated by the $C_f$ – $St^+$ curves depicted in figure 5 $(d)$ . In short, by exploring the drag modification properties over a wide range of friction Stokes numbers (covering two orders of magnitude), we confirm and significantly extend the competition arguments proposed by Dave & Kasbaoui (Reference Dave and Kasbaoui2023). A return-to-SP regime is found at high $St^+$ , with the critical Stokes numbers for the DE–DR transition and the maximum DR being identified (at $\Phi _m=0.6$ and $Re_\tau \approx 300$ ).

Figure 5.

Profiles of $(a)$ $\bar {\tau }^{\textit{T}}$ , $(b)$ $\bar {\tau }^{\textit{P}}$ and $(c)$ $\bar {\tau }^{\textit{E}} = \bar {\tau }^{\textit{T}} + \bar {\tau }^{\textit{P}}$ , and the $(d)$ $C_f$ – $St^+$ curves.

Figure 6.

$(a)$ Streamwise momentum coupling term and $(b)$ mean streamwise slip velocity.

To gain some new insights into the particle-induced shear stress, we next direct our focus to the particle-phase transport. The interphase momentum coupling term $C\langle \widehat {F}_1\rangle$ in (3.5) is depicted in figure 6 $(a)$ , which is the opposite of the particle feedback force $\overline {\mathcal{F}}_1$ according to (3.6). As shown, in the statistical sense, particles always gain high streamwise momentum from the background flow in the bulk ( $C\langle \widehat {F}_1\rangle \gt 0$ ), transport it towards the wall and then release high momentum to the near-wall low-speed fluids ( $C\langle \widehat {F}_1\rangle \lt 0$ ). This bulk-to-wall transport of high momentum undertaken by particles results from the intrinsic wall-normal inhomogeneity of carrying fluid flow and the flattening of particle velocity profile that we have discussed in § 5.1. Given the fact that interphase momentum coupling is proportional to the slip velocity, $s_i=u_i-v_i$ , shown in figure 6 $(b)$ , $C\langle \widehat {F}_1\rangle$ should always be negative near the wall and positive in the bulk, while $\overline {\mathcal{F}}_1$ has just the opposite feature. Further, since there is no mean streamwise momentum flux at the wall and the channel centreline, i.e. $\langle v_1v_2\rangle |_{y=0,\,h}\equiv 0$ , integrating (3.5) over the channel yields the vanishing of $\int _{0}^{h}C\langle \widehat {F}_1\rangle \,\textrm{d}y$ , as well as $\int _{0}^{h}\overline {\mathcal{F}}_1\,\textrm{d}y=\bar {\tau }^{\textit{P}}(h)=0$ . Combining the above mentioned properties, we have $\bar {\tau }^{\textit{P}}(y)\gt 0$ with $0\lt y\lt h$ and $C_f^{{P}}\gt 0$ , consistent with the numerical results plotted in figure 5 $(b,d)$ . In other words, inertial particles provide an additional way to transport high streamwise momentum of fluid flow from the bulk to the wall, in addition to that by turbulent motions, contributing a positive part like the Reynolds shear stress to the frictional drag.

Based on the above mentioned analysis, we further attempt to explain why a DE regime emerges at low Stokes numbers and a return-to-SP regime at high Stokes numbers. As is well known, low-inertia ( $St^+\to 0$ ) particles act as flow tracers, while high-inertia ( $St^+\to \infty$ ) particles have a ballistical behaviour (Brandt & Coletti Reference Brandt and Coletti2022). In both situations, due to the mismatch between the time scales of particles and flow structures, particles will not preferentially accumulate in carrying fluids (Motoori et al. Reference Motoori, Wong and Goto2022; Jie et al. Reference Jie, Cui, Xu and Zhao2022). Theoretically, this results in a uniform distribution of particles, namely, $C\to {\Phi }_m$ . Consequently, the uniformly distributed particles sample the background flow field unbiasedly, degrading the particle-averaged flow variables to their Reynolds-averaged counterparts, i.e. $\langle A_f\rangle \to \overline {A_f}$ . When $St^+\to 0$ or $\tau \to 0$ , particles respond to the fluid flow immediately; thus, all the particle statistics reduce to their fluid-phase counterparts. Equation (3.5) leads to $\langle \widehat {F}_i\rangle _0=\textrm{d}(\overline {u'_i u'_2})/\textrm{d}y$ , where $\langle \,\cdot \,\rangle _0$ denotes the particle average at the $St^+\to 0$ limit. This means that the particle-induced shear stress becomes

(5.3) \begin{equation} \bar {\tau }^{\textit{P}} = \rho \int _{0}^{y} \overline {\mathcal{F}}_1 \,\textrm{d} y = -\rho C \langle v_1 v_2 \rangle \to -\rho {\Phi }_m \overline {u'_1 u'_2}, \quad St^+\to 0, \end{equation}

and the modified SSB becomes

(5.4) \begin{equation} {\mu \frac {\textrm{d}\bar {u}_{1}}{\textrm{d}y}} {-\rho (1+{\Phi }_m)\overline {u'_1u'_2}} =\tau _w\left (1-\frac {y}{h}\right ), \quad St^+\to 0. \end{equation}

It indicates that the effect of particle tracers is to amplify the turbulent transport, namely, the Reynolds shear stress, by a factor of $1+{\Phi }_m$ , corresponding to a DE asymptotic state expected for $St^+\to 0$ . In contrast, for the other limit of $St^+\to \infty$ or $\tau \to \infty$ , all flow properties discussed exhibit a tendency to approach the SP case. It is straightforward to rationalise this from the Stokes drag force formulated as $F_i\approx s_i/\tau$ . Since the slip velocity is certain to be finite, as $\tau$ approaches infinity and ${\Phi }_m$ remains fixed, the feedback force acting on the carrying fluids vanishes (Saffman Reference Saffman1962). Hence, the flow modulation by the particle phase should be trivial. Clearly, these two limits are deemed to set bounds for the particle-induced shear stress. Thus, it makes sense that for a finite Stokes number, the positive-valued particle-induced shear stress follows a continuous change from $-\rho {\Phi }_m\overline {u'_1u'_2}$ to zero, leading to the monotonically decreasing $C_f^{\textit{P}}$ , as observed in figure 5 $(d)$ .

5.3. The production and dissipation rates

This subsection is directed to examine the turbulent kinetic energy, clarifying how inertial particles affect its production and dissipation. In fully developed particle-laden channel flows, the budget equation for TKE ( $k=\overline {u'_iu'_i}/2$ ) reads (Pan et al. Reference Pan, Xiang, Wang, Andersson and Zhao2020)

(5.5) \begin{align} 0= & \, \underbrace {\vphantom {\overline {\frac {\partial u'_i}{\partial x_j}}}-\overline {u'_1u'_2}\frac {\textrm{d}\bar {u}_1}{\textrm{d}y}}_{P} \underbrace {\vphantom {\overline {\frac {\partial u'_i}{\partial x_j}}}-\nu \overline {\frac {\partial u'_i}{\partial x_j}\frac {\partial u'_i}{\partial x_j}}}_{\epsilon } \underbrace {\vphantom {\overline {\frac {\partial u'_i}{\partial x_j}}}-\frac {\textrm{d}\left(\overline {u'_iu'_iu'_2}/2\right)}{\textrm{d}y}}_{D^t} +\underbrace {\vphantom {\overline {\frac {\partial u'_i}{\partial x_j}}}\nu \frac {\textrm{d}^2 k}{\textrm{d}y^2}}_{D^v} \underbrace {\vphantom {\overline {\frac {\partial u'_i}{\partial x_j}}}-\frac {1}{\rho }\overline {u'_i\frac {\partial p'}{\partial x_i}}}_{\mathit{\Pi }} +\underbrace {\vphantom {\overline {\frac {\partial u'_i}{\partial x_j}}}\overline {\mathcal{F}'_iu'_i}}_{{\mathcal{W}}^F}, \end{align}

where the right-hand side terms are in sequence: $P$ the turbulence production rate; $\epsilon$ the dissipation rate; $D^t$ the turbulent diffusion rate; $D^v$ the viscous diffusion rate; $\mathit{\Pi }$ the velocity pressure-gradient term; and ${\mathcal{W}}^F$ the fluctuating particle–fluid work. The profiles of these terms are depicted for typical cases in figure 7. Following previous studies (Lee & Lee Reference Lee and Lee2015; Gualtieri et al. Reference Gualtieri, Battista, Salvadore and Casciola2023; Cui, Zhang & Zheng Reference Cui, Zhang and Zheng2024), special interest is focused on interpreting the key terms, i.e. the turbulence production rate $P$ and the fluctuating particle–fluid work ${\mathcal{W}}^F$ . In comparison with the SP case, it is clear that ${\mathcal{W}}^F$ plays a critical role in the production–dissipation balance. This term is positive for low-inertia particles and becomes negative in almost the entire channel for high-inertia particles. Moreover, the ‘native’ turbulence production rate $P$ is reduced for all $St^+$ as compared with the SP case.

Figure 7.

TKE budget profiles of three typical cases: $(a)$ St003, $(b)$ St100 and $(c)$ St300 and $(d)$ the SP results. Hereafter, all the budget terms are scaled by $(u_\tau ^{{SP}})^4/\nu$ .

Figure 8.

PFKE budget profiles of three typical particle-laden cases: $(a)$ St003, $(b)$ St100 and $(c)$ St300.

Figure 9.

Terms that are critical to the TKE, PFKE budgets and TKE–PFKE transfer, including $(a)$ $P$ , $(b)$ ${\mathcal{W}}^F$ , $(c)$ $P_p$ , $(d)$ $W^F$ , $(e)$ $\epsilon ^{p,F}$ in solid and $\alpha$ in dashed lines. For clarity, only the near-wall region is displayed, as their values in the outer layer gradually decay to zero. Panel $(f)$ shows the integrals of dominant terms, where $P^E\equiv P+{\mathcal{W}}^F$ denotes the effective production rate, and all the quantities are normalised using the integral of single-phase production rate $\mathcal{I}(P^{{SP}})$ .

To further understand the role of particle–fluid work on the TKE production/dissipation, we examine the corresponding PFKE ( $k_p \equiv \langle v_i^\circ v_i^\circ \rangle /2$ ) budget (Appendix A.3),

(5.6) \begin{align} 0= & \, \underbrace {-C\big\langle v_1^\circ v_2^\circ \big\rangle \frac {\textrm{d}\langle v_1\rangle }{\textrm{d} y}}_{P_p} \underbrace {-\frac {\textrm{d}\big(C\big\langle v_i^\circ v_i^\circ v_2^\circ \big\rangle /2\big)}{\textrm{d} y}}_{D_p} +\underbrace {\vphantom {\frac {\textrm{d}\langle v_1\rangle }{\textrm{d} y}}C\big\langle \widehat {F}_i^\circ v_i^\circ \big\rangle }_{W^F}, \end{align}

where the right-hand side terms are identified in sequence as: $P_p$ the particle production rate; $D_p$ the particle diffusion rate; and $W^F$ the fluctuating fluid–particle work. As demonstrated in figure 8, the PFKE balance is achieved by the positive-valued $P_p$ , which transfers energy from the mean particle motion to the fluctuating counterpart, and the negative-valued work $W^F$ done by carrying fluids. The particle diffusion $D_p$ serves just to redistribute the PFKE along the wall-normal direction. Of fundamental importance, the PFKE budget is inherently connected to the TKE budget through the interphase coupling relation (Appendix A.4),

(5.7) \begin{equation} {\mathcal{W}}^F=-W^F+N^F=-W^F+\epsilon ^{p,F}+\alpha . \end{equation}

Here, $N^F$ stands for the net energy loss during fluctuating kinetic energy exchange, $\epsilon ^{p,F}=-C\langle \widehat {F}^\circ _i s^\circ _i\rangle$ denotes the fluctuating particle dissipation (Zhao, Andersson & Gillissen Reference Zhao, Andersson and Gillissen2013) and $\alpha =C\langle \widehat {F}_i\rangle (\bar {u}_i-\langle u_i\rangle )$ denotes an extra energy transfer term. Provided with the velocity-dependent Stokes drag, the dissipative nature of the $\epsilon ^{p,F}$ is straightforward in the physical sense as ${\epsilon }^{p,F}\approx -C\langle s^\circ _is^\circ _i\rangle /\tau \leqslant 0$ . In contrast, $\alpha$ is non-dissipative, which transfers kinetic energy between the mean and fluctuating motions involving both fluid and particle phases. This energy transfer term is non-trivial when there exists a non-zero mean slip velocity accompanied by the preferential accumulation (clustering) of inertial particles (Capecelatro et al. Reference Capecelatro, Desjardins and Fox2015, Reference Capecelatro, Desjardins and Fox2016). According to Capecelatro et al. (Reference Capecelatro, Desjardins and Fox2018), $\alpha$ – referred to as drag production in their study – becomes dominant only when the mass loading is sufficiently high ( ${\Phi }_m \geqslant 10$ ) and gravity is imposed on the particles, which together give rise to a large interphase slip velocity.

Figure 9 highlights the dominant terms in (5.5)–(5.7) that govern the net energy exchange. These terms include $P$ , ${\mathcal{W}}^F$ , $P_p$ , $W^F$ , $\epsilon ^{p,F}$ and $\alpha$ . The Stokes number dependence of each term is clearly illustrated. The production rate $P$ decreases as $St^+$ increases from $3$ to $56$ , before increasing again. Both $\mathcal{W}^F$ and $P_p$ decrease monotonically, while $W^F$ increases monotonically. Furthermore, the particle dissipation initially decreases with increasing $St^+$ and then increases, with its extrema obtained at a moderate Stokes number around $30$ . It is noted that compared with ${\epsilon }^{p,F}$ , $\alpha$ is indeed negligibly small. Furthermore, combining (5.6) and (5.7) and integrating the terms over the half-channel, one obtains the following integral relation:

(5.8) \begin{equation} \mathcal{I}({\mathcal{W}}^F)=\mathcal{I}(P_p)+\mathcal{I}(N^{F})\approx \mathcal{I}(P_p)+\mathcal{I}({\epsilon }^{p,F}). \end{equation}

Here, $\mathcal{I}(A)\equiv \int _{0}^{h} A\,\textrm{d}y$ denotes the wall-normal integral of arbitrary function $A(y)$ . That is to say, the net contribution of the fluctuating particle–fluid work to the TKE budget can be formally decomposed into two dominant components. As shown in figure 9 $(f)$ , they include a positive-valued particle production term $\mathcal{I}(P_p)$ , which decreases almost monotonically with $St^+$ , and a negative-valued particle dissipation term $\mathcal{I}({\epsilon }^{p,F})\approx \mathcal{I}(N^{F})$ , which varies non-monotonically with $St^+$ .

The underlying mechanisms for the $St^+$ dependence of the above mentioned terms are explained as follows. First, the decrease of $\mathcal{I}(P_p)$ is attributed to the flattening of the mean particle velocity profile. As shown in figure 3 $(b)$ , the mean particle velocity profile flattens as particle inertia increases, leading to a decrease in particle velocity gradient, especially in the near-wall region. Consequently, $\mathcal{I}(P_p)$ decreases monotonically with the continuous increase of $St^+$ , considering the fact that $P_p$ is proportional to $\textrm{d}\langle v_1\rangle /\textrm{d} y$ . This positive term signifies that inertial particles provide a new way for transferring kinetic energy from the mean flow to turbulent motions through the particle phase, namely, a bypass way termed MKE–PMKE–PFKE–TKE. Here, MKE stands for the mean kinetic energy ( $K\equiv \bar {u}_i\bar {u}_i/2$ ) and PMKE stands for the particle-phase mean kinetic energy ( $K_p \equiv \langle v_i\rangle \langle v_i\rangle /2$ ). Second, the non-monotonic $St^+$ dependence of particle dissipation $\mathcal{I}({\epsilon }^{p,F})$ can be interpreted by the slip velocity–relaxation time scaling, as indicated by the formulation ${\epsilon }^{p,F}\approx -C\langle s_i^\circ s_i^\circ \rangle /\tau$ . Specifically, when $St^+\to \infty \,(\tau \to \infty )$ , the fluctuating slip velocity variance is bound to converge to a finite value, leading to a limit of ${\epsilon }^{p,F}_\infty \to 0$ . While $St^+\to 0 \,(\tau \to 0)$ , the leading order of slip velocity is proportional to the relaxation time, $|\boldsymbol{s}|\sim O(\tau )$ (Maxey Reference Maxey1987), also leading to ${\epsilon }^{p,F}_0\sim O(\tau ^2)/\tau \to 0$ . Therefore, it is expected that the fluctuating particle dissipation term exhibits a non-monotonic variation as $St^+$ gradually increases, i.e. initially decreasing from zero, reaching its extrema at a moderate $St^+$ and then recovering back to zero. Intriguingly, a very recent study offers convincing support to the above mentioned analysis (Berk & Coletti Reference Berk and Coletti2024). An analytical model is proposed by Berk & Coletti (Reference Berk and Coletti2024) to demonstrate that the slip velocity variance $\langle s^\circ _is^\circ _i\rangle$ follows a piecewise scaling of $\tau ^2$ – $\tau ^1$ – $\tau ^0$ as $\tau$ increases continuously. As a consequence of the interplay of these two competing mechanisms, $\mathcal{I}(P_p)+\mathcal{I}({\epsilon }^{p,F})$ , the fluctuating particle–fluid work acts as a TKE source for low Stokes numbers, while behaves like an energy sink for moderate-to-high Stokes numbers.

5.4. Self-sustaining processes

As discussed previously, two mechanisms through which particles directly affect the flow properties have been identified by examining the shear stress balance and TKE budget. The first is the particle-induced extra transport of momentum and kinetic energy, which is positive and manifests as the particle shear stress and the particle production rate. The second is the particle-induced extra dissipation, which is negative and plays a role in the interphase energy exchange. Beyond these two, there appears to be an indirect effect that accounts for the suppression of Reynolds shear stress and turbulence production rate. To clarify this effect and fulfil the complete physical picture underlying turbulence modulation, we revisit the self-sustaining processes (SSPs) of particle-laden channel flows, examining the Reynolds stress ( $r_{ij}=\overline {u'_iu'_j}$ ) budget (Chou Reference Chou1945; Mansour, Kim & Moin Reference Mansour, Kim and Moin1988; Pope Reference Pope2000),

(5.9) \begin{align} \frac {\partial r_{ij}}{\partial t}= & \underbrace {\vphantom {\left (\frac {\partial \bar {u}_i}{\partial x_k}\right )}- \frac {\partial (r_{ij}\bar {u}_k)}{\partial x_k}}_{C_{ij}} \underbrace {\vphantom {\left (\frac {\partial \bar {u}_i}{\partial x_k}\right )}-\left (\overline { u'_j u'_k}\frac {\partial \bar {u}_i}{\partial x_k}+\overline { u'_i u'_k}\frac {\partial \bar {u}_j}{\partial x_k}\right )}_{P_{ij}} \underbrace {\vphantom {\left (\frac {\partial \bar {u}_i}{\partial x_k}\right )}-\frac {\partial \overline {u'_i u'_j u'_k}}{\partial x_k}}_{D^t_{ij}} +\underbrace {\vphantom {\left (\frac {\partial \bar {u}_i}{\partial x_k}\right )}\nu \frac {\partial ^2r_{ij}}{\partial x_k\partial x_k}}_{D^v_{ij}} \notag \\[8pt] & \underbrace {-\frac {1}{\rho }\left (\overline {{u'_i}\frac {\partial p'}{\partial x_j}}+\overline {{u'_j}\frac {\partial p'}{\partial x_i}}\right )}_{\mathit{\Pi }_{ij}} \underbrace {\vphantom {\left (\overline {{u'_j}\frac {\partial p'}{\partial x_i}}\right )}-2\nu \overline {\frac {\partial u'_i}{\partial x_k}\frac {\partial u'_j}{\partial x_k}}}_{\epsilon _{ij}}, \end{align}

where the right-hand side terms are identified in sequence as: $C_{ij}$ the convection term; $P_{ij}$ the production rate; $\epsilon _{ij}$ the dissipation rate; $D^v_{ij}$ the viscous diffusion rate; $D^t_{ij}$ the turbulent diffusion rate and $\mathit{\Pi }_{ij}$ the velocity pressure-gradient term. It should be noted that the normal Reynolds stresses are physically equivalent to the components of TKE, given by $k=\overline {u'_iu'_i}/2=r_{ii}/2$ . The SSP in canonical particle-free wall-bounded turbulence has been established as the following three phases (Mansour et al. Reference Mansour, Kim and Moin1988; Pope Reference Pope2000; Zhang et al. Reference Zhang, Zhang, Wang, Li, Yu and Li2022): (1) the Reynolds shear stress $-r_{12}$ acts to pump the kinetic energy of mean motions to $r_{11}$ through $P_{11}$ ; (2) part of this kinetic energy is redistributed to $r_{22}$ through $\mathit{\Pi }_{22}$ ; (3) in turn, $r_{22}$ works to reproduce $-r_{12}$ through $P_{12}$ , closing the TKE regeneration cycle. During this cyclic SSP, $r_{33}$ is also generated in the phase of kinetic energy redistribution. Such an intriguing regeneration cycle of Reynolds stress components in the single-phase channel turbulence is clearly illustrated in figure 10.

Figure 10.

Reynolds stress budgets of the present single-phase channel flow: $(a)$ $r_{11}$ ; $(b)$ $r_{22}$ ; $(c)$ $r_{12}$ . Similar to the previous section, the budget terms are scaled by $(u_\tau ^{{SP}})^4/\nu$ .

For particle-laden channel flow, the Reynolds stress budget can be derived accordingly from the momentum equation (2.2) and reads

(5.10) \begin{align} \frac {\partial r_{ij}}{\partial t}\equiv 0= & \underbrace {\vphantom {\left (\frac {\partial \bar {u}_i}{\partial x_k}\right )}-\left (\overline { u'_j u'_2} \delta _{i1}+\overline { u'_i u'_2} \delta _{j1}\right ) \frac {\textrm{d}\bar {u}_1}{\textrm{d}y} }_{P_{ij}} \underbrace {\vphantom {\left (\frac {\textrm{d}\bar {u}_i}{\textrm{d}y}\right )}-\frac {\textrm{d}\overline {u'_i u'_j u'_2}}{\textrm{d}y}}_{D^t_{ij}} +\underbrace {\vphantom {\left (\frac {\textrm{d}\bar {u}_i}{\textrm{d}y}\right )}\nu \frac {\textrm{d}^2r_{ij}}{\textrm{d}y^2}}_{D^v_{ij}} \notag \\[8pt] & \underbrace {-\frac {1}{\rho }\left (\overline {{u'_i}\frac {\partial p'}{\partial x_j}}+\overline {{u'_j}\frac {\partial p'}{\partial x_i}}\right )}_{\mathit{\Pi }_{ij}} \underbrace {\vphantom {\left (\overline {{u'_j}\frac {\partial p'}{\partial x_i}}\right )}-2\nu \overline {\frac {\partial u'_i}{\partial x_k}\frac {\partial u'_j}{\partial x_k}}}_{\epsilon _{ij}} +\underbrace {\vphantom {\left (\overline {{u'_j}\frac {\partial p'}{\partial x_i}}\right )}\left (\overline {\mathcal{F}'_iu_j'}+\overline {\mathcal{F}'_ju_i'}\right )}_{{\mathcal{W}}^F_{ij}}, \end{align}

where an additional term, named fluctuating particle–fluid interaction, ${\mathcal{W}}^F_{ij}=\overline {\mathcal{F}'_i u'_j}+\overline {\mathcal{F}'_j u'_i}$ , appears on the right-hand side. In this situation, ${\mathcal{W}}^F_{ij}$ could be viewed as the ‘input’ imposed by particles, while other terms could be understood as the ‘response’ of the channel flow system. From this point, we will mainly focus on the behaviour of ${\mathcal{W}}^F_{ij}$ and the resulting changes in the response. Other terms – the production, diffusion, dissipation, etc. – will not be addressed individually, since they all exhibit a similar response to the particle–fluid interaction, as will be shown later. The budget profiles of the St003 and the St100 cases are depicted in figure 11 to demonstrate two typical SSP modulation patterns. Specifically, for the St003 case, the particle–fluid interaction augments $r_{11}$ and $-r_{12}$ as a source while suppresses $r_{22}$ as a sink. The extra gains in the streamwise and shear stresses compensate for the loss in the wall-normal stress, allowing the SSP to be maintained with considerable strength. In contrast, for the St100 case, ${\mathcal{W}}^F_{ij}$ plays a dissipative role in all the Reynolds stress components, resulting in a drastic attenuation of SSP.

Figure 11.

Fluid-phase Reynolds stress budgets of the $(a, b, c)$ St003 and $(d, e, f)$ St100 cases: $(a, d)$ $r_{11}$ ; $(b, e)$ $r_{22}$ ; $(c, f)$ $r_{12}$ .

Deeper insights into the particle–fluid interactions can be gained by examining the budget equations of particle-phase Reynolds stresses, $r_{p,ij}=\langle v_i^\circ v_j^\circ \rangle$ , given as (Appendix A.3)

(5.11) \begin{align} \frac {\partial \big(C r_{p,ij}\big)}{\partial t}\equiv 0= & \, \underbrace {-C \left ( \big\langle v_i^\circ v_2^\circ \big\rangle \delta _{j1}+ \big\langle v_j^\circ v_2^\circ \big\rangle \delta _{i1}\right ) \frac {\textrm{d} \langle v_1\rangle }{\textrm{d} y} }_{P_{p,ij}} \underbrace {\vphantom {\left (\frac {\textrm{d} \langle v_j\rangle }{\textrm{d} y}\delta _{j1}\right )}-\frac {\textrm{d} \big(C\big\langle v_i^\circ v_j^\circ v_2^\circ \big\rangle \big)}{\textrm{d} y}} _{D_{p,ij}}\notag \\[2pt] & \,+\underbrace {C\left (\big\langle { \widehat F_i^\circ } v_j^\circ \big\rangle + \big\langle { \widehat F_j^\circ } v_i^\circ \big\rangle \right )}_{W^F_{ij}} +\underbrace {\vphantom {C\left (\big\langle { \widehat F_i^\circ } v_j^\circ \big\rangle \right )} \big\{v_i^\circ v_j^\circ \big\}_{\textit{coll}}}_{\mathit{\Pi }_{p,ij}}, \end{align}

where the right-hand side terms are ordered as: $P_{p,ij}$ the particle production rate; $D_{p,ij}$ the particle diffusion rate; $W^F_{ij}$ the fluctuating fluid–particle interaction term and $\mathit{\Pi }_{p,ij}$ the inter-particle collision term. Similar to the Reynolds stress, the normal particle-phase Reynolds stresses are physically equivalent to the FPKE components as $k_p \equiv \langle v_i^\circ v_i^\circ \rangle /2=r_{p,ii}/2$ . Corresponding to figure 11, the budget profiles of the St003 and St100 cases are provided as typical results in figure 12. For $r_{p,11}$ , the budget balance is mainly achieved by a positive particle production rate and a negative fluid–particle interaction term. A small portion of $r_{p,11}$ is redistributed to the wall-normal and spanwise components $r_{p,22},\,r_{p,33}$ by inter-particle collisions. For $r_{p,22}$ , the particle production disappears due to the absence of mean wall-normal velocity $\langle v_2\rangle$ and its gradient. Hence, the net contribution of $\mathit{\Pi }_{p,22}$ is directly balanced by $W^F_{22}$ (Caraman et al. Reference Caraman, Borée and Simonin2003). For $-r_{p,12}$ , the budget is mainly balanced by three elements: particle production, fluid–particle interaction and particle diffusion.

Figure 12.

Particle-phase Reynolds stress budgets of the $(a, b, c)$ St003 and $(d, e, f)$ St100 cases: $(a, d)$ $r_{p,11}$ ; $(b, e)$ $r_{p,22}$ ; $(c, f)$ $r_{p,12}$ .

Figure 13.

Decomposition of particle–fluid interaction term (5.13) versus Stokes number: $(a)$ $ij=11$ ; $(b)$ $ij=22$ ; $(c)$ $ij=12$ .

Similar to (5.7), the interphase coupling relation with regards to each Reynolds stress component can be further characterised as follows:

(5.12) \begin{align} N_{ij}^F=W_{ij}^F+{\mathcal{W}}_{ij}^F = {-C\big\langle \widehat {F}^\circ _i s^\circ _j\big\rangle -C\big\langle \widehat {F}^\circ _j s^\circ _i\big\rangle } +{C\big[\big\langle \widehat {F}_i\big\rangle (\bar {u}_j-\langle u_j\rangle )+\big\langle \widehat {F}_j\big\rangle (\bar {u}_i-\langle u_i\rangle )\big]}.\nonumber\\[1pt] \end{align}

By taking the integral of (5.11) and (5.12), we obtain a decomposition that facilitates the identification of particle–fluid interaction mechanisms, expressed as

(5.13) \begin{equation} \mathcal{I}\big({\mathcal{W}}_{ij}^F\big)=\mathcal{I}\big(P_{p,ij}\big)+\mathcal{I}\big(\mathit{\Pi }_{p,ij}\big)+\mathcal{I}\big(D_{p,ij}\big)+\mathcal{I}\big(N^{F}_{ij}\big). \end{equation}

Figure 13 depicts these integrals as a function of Stokes numbers. For the $11$ and $12$ components; a pleasing outcome is that the Stokes number dependencies of the two dominant terms, the particle production rate $\mathcal{I}(P_{p,ij})$ and the net energy loss $\mathcal{I}(N^{F}_{ij})$ (with the particle dissipation rate being the dominant contributor) remain consistent with the understanding established in the previous section. In contrast, the particle diffusion rate plays a secondary role. Their superposition leads to a shift of the contribution of  ${\mathcal{W}}_{11,12}^F$ from positive to negative as $St^+$ increases. For the $22$ component, it is shown that the particle dissipation dominates over the collision-induced redistribution throughout the considered parameter space, making ${\mathcal{W}}_{22}^F$ always act as an energy sink. To sum up, since the particle-induced extra transport requires a mean gradient, ${\mathcal{W}}_{ij}^F$ acts on the carrying flow anisotropically. The resulting intrinsic anisotropy of particle-laden wall-bounded turbulence has been confirmed in both statistical and structural senses by numerous studies (Zhao et al. Reference Zhao, Andersson and Gillissen2010, Reference Zhao, Andersson and Gillissen2013; Dritselis Reference Dritselis2016; Zhou et al. Reference Zhou, Zhao, Huang and Xu2020; Costa et al. Reference Costa, Brandt and Picano2021; Dave & Kasbaoui Reference Dave and Kasbaoui2023).

Integrating all the previously discussed insights, we turn our attention back to the Reynolds stress budgets and the modification of fluid-phase SSP. Figure 14 $(a$ – $c)$ depicts the budget integrals versus Stokes numbers. We observed that almost all the native terms in Reynolds stress budgets respond to the non-monotonic and anisotropic input in a very similar manner. However, these results contradict the previously established understanding of particle dissipation in the following two aspects. As shown in figure 14 $(b)$ , the dissipation-dominated particle–fluid interaction in the wall-normal direction, ${\mathcal{W}}^F_{22}$ , increases monotonically, rather than initially decreasing before increasing. Furthermore, at high Stokes numbers, all the particle–fluid interaction components ${\mathcal{W}}^F_{ij}$ remain nearly unchanged, while the amplitudes of the responses, $P_{ij}$ , $\epsilon _{ij}$ and $\mathit{\Pi }_{ij}$ , increase significantly.

Figure 14.

Reynolds stress budget integrals: $(a, d)$ $r_{11}$ ; $(b, e)$ $r_{22}$ ; $(c, f)$ $r_{12}$ . In panels $(d, e, f)$ , the budget integrals are normalised such that, in each case, the summation of their squares equals one. Here, we denote $D_{ij}=D_{ij}^v+D_{ij}^t$ .

These counterintuitive results are actually attributed to the fluid–particle coupling nature of PLT and will be explained as follows. As Stokes number (relaxation time) increases from a moderate value, the strength of particle dissipation $|\langle s^\circ _i s^\circ _j\rangle |/\tau$ begins to decrease. This leads to a weaker suppression of fluid-phase SSP and, in turn, stronger flow velocity fluctuations. Notice that $|\langle s^\circ _i s^\circ _j\rangle |/\tau =|\langle (u^\circ _i-v^\circ _i)(u^\circ _j-v^\circ _j)\rangle |/\tau$ depends on the fluctuation of background fluid flow through ${u}_i^\circ$ , the absolute strength of particle dissipation should thus be augmented. In other words, the interphase coupling brings about a ‘feedback loop’ for the flow-velocity-dependent particle dissipation. Hence, to correctly understand the genuine Stokes number dependence of particle dissipation, one should focus on its relative importance in SSP. To that end, we redraw the budgets in figure 14 $(d$ – $f)$ , where the integrals are normalised such that the summation of their squares equals one (a similar treatment on Reynolds stress budgets can be found from Pope Reference Pope2000). Taking the feedback nature into account, we see immediately that the system behaves exactly as expected. For $r_{11}$ and $-r_{12}$ , both extra transport and extra dissipation take effect, leading to the net effect of the particle–fluid interaction term varying from positive to negative and finally returning to zero. For $r_{22}$ , the extra dissipation dominates, the negative-valued particle–fluid interaction decreases at first and then increases back to zero. Moreover, by plotting the normalised integrals in the single-phase case as dashed lines, the system’s departure from and return to this reference state in response to the input ${\mathcal{W}}^F_{ij}$ are clearly illustrated.

Overall, the SSP undergoes three distinct stages: (1) at low $St^+$ , both $r_{11}$ and $-r_{12}$ are augmented while $r_{22}$ is suppressed (one could expect that, at a lower Stokes number, $St^+\lt 3$ , the SSP is likely to be augmented); (2) at moderate $St^+$ , the particle dissipation dominates, leading to significant suppression of all Reynolds stress components; (3) at high $St^+$ , the suppression effects decline and the cycling strength recovers, returning to the particle-free state. This understanding rationalises the aforementioned indirect effects, as reflected in the non-monotonic modification of the Reynolds shear stress and turbulence production rate. Up to this point, the particle inertial effects of turbulence modulation at various aspects – momentum transport, kinetic energy transfer and self-sustaining processes – have been comprehensively addressed, leading to the closure of our physical picture.