3.1. Modulation of coherent vortices

Figure 2. Visualisation of coherent vortices in turbulence at $Re_\tau = 512$ laden with particles with the same diameter ( $D/h=0.031$ , i.e. $D^+=16$ ) but different values of the Stokes number: (a) $St_h=0.056$ and (b) $3.6$ . Grey vortices are identified by positive isosurfaces of the second invariant $Q$ of the velocity gradient tensor. Yellow vortices are identified by the second invariant ${\widetilde {Q}}^{(\ell )}$ of the velocity gradient tensor coarse-grained at $\ell = 0.2h$ . We set the thresholds as $Q^+=7.0\times 10^{-3}$ and ${\,\widetilde {Q}}^{(\ell )+}=3.0\times 10^{-5}$ .

First, let us examine the modulation of coherent vortices in turbulence. Figure 2 shows vortices in turbulence at $Re_\tau =512$ laden with the smallest ( $D/h=0.031$ , i.e. $D^+=16$ ) particles. The Stokes number differs between the panels: $St_h=0.056$ (figure 2 a) and $3.6$ (figure 2 b). We show vortices at two different scales: the grey objects are the smallest vortices identified by positive isosurfaces of the second invariant $Q$ of the velocity gradient tensor. To extract channel-half-width-scale vortices (i.e. wall-attached vortices in the outer layer), we apply the three-dimensional Gaussian filter (Motoori & Goto Reference Motoori and Goto2019) with filter width $\ell = 0.2h$ to the fluctuation velocity. We then evaluate the second invariant ${\widetilde {Q}}^{(\ell )}$ of the coarse-grained velocity gradient tensor and show its positive isosurfaces in yellow. It is evident in figure 2(a) that the yellow outer-layer vortices are developed even in the presence of particles with smaller $St_h$ ( $=0.056$ ); whereas, in figure 2(b), these vortices are almost entirely attenuated due to particles with larger $St_h$ ( $=3.6$ ).

Figure 3. Magnification of subdomains in figure 2(b) for (a) $0 \leqslant y^+ \leqslant 50$ and (b) $0.3 \leqslant y/h \leqslant 0.75$ (i.e. $154 \leqslant y^+ \leqslant 384$ ). Particles are depicted by white spheres.

It is also important to observe in figure 2(b) that vortex rings are attached to particles. Figures 3(a) and 3(b) are the magnifications of subdomains in figure 2(b) in the buffer and outer layers, respectively. Here, particles are depicted by white spheres. There are many vortex rings around particles in the both layers. Since $St_h$ of these particles is larger than $1$ , they can follow neither the outer-layer vortices (with the longest time scale in the turbulence) nor the buffer-layer vortices. The particles also cannot follow the mean flow because its time scale is comparable to the turnover time of the wall-attached vortices at each height. This explains the reason why these vortex rings are perpendicular to the streamwise direction. If the particle Reynolds number were higher, the vortex rings would be shed from the particles. In contrast, for $St_h \ll 1$ (see figure 2 a), there are no vortex rings in the outer layer. These results therefore imply that the presence of particle-induced vortices results in the reduction of turbulent vortices. As is discussed in detail in the following (see § 3.4), these small vortices induced by particles are indeed important because they produce the additional energy dissipation, which is relevant to the turbulence attenuation.

Figure 4. Wall-normal profiles of the mean turbulent kinetic energy $K(y)$ (2.7) at $Re_\tau =512$ . (a) The Stokes-number dependence ( $St_h=0.056$ (black), $0.22$ (dark grey), $0.89$ (light grey) and $3.6$ (very light grey)) for the common particle diameter $D/h=0.031$ . (b) The particle-diameter dependence ( $D/h=0.13$ (black), $0.063$ (grey) and $0.031$ (light grey)) for the common Stokes number $St_h=3.6$ . The blue dashed line shows the result for the single-phase flow. The inset in (a) shows the results for $Re_\tau =180$ with $St_h=0.016$ (black), $0.63$ (dark grey), $2.5$ (light grey) and $10$ (very light grey).

3.2. Turbulent kinetic energy

To quantify the degree of turbulence modulation at each height, we evaluate the mean turbulent kinetic energy $K(y)$ defined as (2.7). As mentioned below (2.7), when we take the spatial average, we only take into account the fluid region outside particles. Figure 4 shows the wall-normal profiles of $K(y)$ for turbulence at $Re_\tau =512$ . The blue dashed line indicates the value $K_\times$ in the single-phase flow, where the subscript $_\times$ denotes the value in the single-phase flow. First, let us look at figure 4(a), which shows the $St_h$ dependence for the smallest ( $D/h=0.031$ ) particles. Lighter and thicker lines indicate results for larger $St_h$ . We see that, irrespective of the height, turbulent kinetic energy is attenuated more significantly for larger $St_h$ . This is consistent with the observation in figure 2 that particles with larger $St_h$ more significantly attenuate outer-layer energetic vortices. More precisely, we can observe in figure 4(a) that as $St_h$ increases, $K$ is attenuated from a lower height. For example, particles with $St_h=0.056$ , that is, $St_+=28$ (the darkest and thinnest line), attenuate turbulence only for $y/h \lesssim 0.1$ (i.e. $y^+ \lesssim 50$ ). This is because the particle relaxation time is too long for them to follow the swirls of the buffer-layer vortices (whose time scale is of the order of $10\tau _+$ ) but short enough to follow the larger wall-attached vortices such as the outer-layer vortices (because $St_h \ll 1$ ). For the other $Re_\tau$ , we have also confirmed that particles with $St_+=28$ attenuate turbulent kinetic energy only in the buffer layer (figures omitted). In contrast, particles with $St_h = 3.6$ (the lightest line) attenuate the turbulent kinetic energy at any height. We observe similar trends with respect to $St_h$ for $Re_\tau =180$ (see the inset in figure 4 a) and for the results of smaller particles ( $D^+ \approx 10$ ) in Shen et al. (Reference Shen, Peng, Lu and Wang2024).

Figure 4(b) shows results for common $St_h=3.6$ but different values of $D/h$ (darker and thinner lines indicate larger particles). Since $St_h$ of these particles is larger than $1$ , they attenuate turbulence at all heights. It is, however, important to see in this panel that the attenuation rate depends on the particle size; more precisely, it is larger for smaller diameters for a fixed volume fraction of particles.

Thus, when particles cannot follow the wall-attached vortices (and therefore the mean flow) at a given height $y$ , the turbulent kinetic energy around $y$ is attenuated. The degree of the turbulence attenuation is larger for larger Stokes numbers (figure 4 a) and smaller diameters (figure 4 b).

Figure 5. Average attenuation rate $\mathcal{A}$ (3.1) of turbulent kinetic energy as a function of $St_h$ . Different symbols show the results for $Re_\tau = 512$ (circles), $360$ (triangles), $256$ (squares) and $180$ (diamonds). The colour indicates $D/h$ (lighter symbols indicate smaller particles).

Next, we consider spatially averaged quantities. Figure 5 shows the average attenuation rate,

(3.1) \begin{align} \mathcal{A} = 1-\frac {\langle K \rangle }{\langle K_\times \rangle } , \end{align}

of turbulent kinetic energy as a function of $St_h$ . Here, $\langle \:\cdot \: \rangle$ denotes the spatial average over the fluid region outside the particles. Lighter colours indicate smaller $D/h$ and different symbols represent different values of $Re_\tau$ . We can see that irrespective of $Re_\tau$ , the average attenuation rate gets larger for larger $St_h$ and smaller $D/h$ .

To explain these behaviours, we investigate how the kinetic energy transfers from the mean flow to turbulence. Turbulent kinetic energy $K(y)$ at a given height $y$ increases due to the production by the mean flow, transport from other heights and work done by particles; on the other hand, it decreases due to transport to other heights and dissipation into heat. Since the average of the spatial transport over the entire channel vanishes, the average budget of turbulent kinetic energy $\langle K \rangle$ reads

(3.2) \begin{align} \langle \mathcal{P}^{t \leftarrow M} \rangle + \langle \mathcal{W}^{t \leftarrow p} \rangle = \langle \epsilon \rangle , \end{align}

where $\mathcal{P}^{t \leftarrow M}$ is the turbulent energy production rate

(3.3) \begin{align} \mathcal{P}^{t \leftarrow M}(y) = -\overline {u^\prime v^\prime } \dfrac {{\textrm{d}} U}{{\textrm{d}} y} \end{align}

due to the mean flow and $\epsilon$ is the energy dissipation rate of turbulence. In (3.2), $\mathcal{W}^{t \leftarrow p}$ ( $=\mathcal{W}^{f \leftarrow p} - \mathcal{W}^{M \leftarrow p}$ ) is the rate of work done by particles on turbulence. Here, $\mathcal{W}^{f \leftarrow p}$ is the total rate of work by the force $\boldsymbol{f}^{f \leftarrow p}$ that particles exert on the fluid: namely $\mathcal{W}^{f \leftarrow p} = \overline {\boldsymbol{f}^{f \leftarrow p} \cdot \boldsymbol{u}}$ . By decomposing the velocity $\boldsymbol{u}$ on the particle surfaces into mean velocity $\boldsymbol{U}$ in the fluid phase and its fluctuation $\boldsymbol{u}^\prime$ , the total work can be separated into two contributions:

(3.4) \begin{align} \mathcal{W}^{f \leftarrow p} &= \overline {\boldsymbol{f}^{f \leftarrow p}}\cdot \boldsymbol{U} + \overline {{{\boldsymbol{f}}^{f \leftarrow p}}^\prime \cdot \boldsymbol{u}^\prime } \nonumber \\ &= \mathcal{W}^{M \leftarrow p} + \mathcal{W}^{t \leftarrow p} . \end{align}

In a statistically steady state, since the average kinetic energy of particles remains constant, the average rate of work between fluid and particles vanishes, i.e. $\langle \mathcal{W}^{f \leftarrow p} \rangle = 0$ . Therefore, when $\langle \mathcal{W}^{t \leftarrow p} \rangle$ is positive, $\langle \mathcal{W}^{M \leftarrow p} \rangle$ must be negative, meaning that particles transfer the kinetic energy from the mean flow to turbulence. Thus, the energy balance equation (3.2) describes the spatially averaged energy budget in turbulence with mean flow. In the following analyses, we investigate how the energy budget is modulated due to particles on the basis of (3.2). Incidentally, we need to take special care to evaluate the energy dissipation rate $\epsilon$ in (3.2) accurately in the DNS with IBM (see Appendix B).

3.3. Turbulent energy production by the mean flow

Figure 6. (a,b) Wall-normal profiles of the mean turbulent energy production rate $\mathcal{P}^{t \leftarrow M}(y)$ (3.3) due to the mean flow at $Re_\tau =512$ . The values are normalised by the mean turbulent energy dissipation rate $\epsilon _\times$ at each height in the single-phase flow. The lines in both panels indicate the same parameters as shown in figure 4.

We first evaluate the turbulent energy production rate (3.3) due to the mean flow. We show in figure 6 the wall-normal profiles of $\mathcal{P}^{t \leftarrow M}$ in turbulence at $Re_\tau =512$ . Here, we normalise $\mathcal{P}^{t \leftarrow M}(y)$ by the mean turbulent energy dissipation rate $\epsilon _\times (y)$ of the single-phase turbulence. Particle parameters in figure 6 are the same as figure 4; namely figure 6(a) shows results for the common $D/h$ ( $=0.031$ ) but different values of $St_h$ , while figure 6(b) shows those for the common $St_h$ ( $=3.6$ ) but different values of $D/h$ . Figure 6 shows that (i) when the Stokes number is larger or (ii) the particle size is smaller under the condition that the volume fraction is constant, the energy production rate tends to be more reduced. It is also worth noting that the Reynolds stress ( $-\overline {u^\prime v^\prime }$ ) is also reduced (see Appendix C). Since the Reynolds stress is produced by wall-attached vortices (Lozano-Durán et al. 2012; Motoori & Goto Reference Motoori and Goto2021), turbulent kinetic energy, which is related to these energetic vortices, is also attenuated.

Next, we show in figure 7(a) the spatial average $\langle \mathcal{P}^{t \leftarrow M} \rangle$ of the energy production rate normalised by the value $\langle \mathcal{P}^{t \leftarrow M}_\times \rangle$ in the single-phase flow as a function of $St_h$ . The symbols are the same as in figure 5; namely those in lighter colours indicate smaller $D/h$ , and different shapes represent different values of $Re_\tau$ . We see that the average production rate gets smaller for (i) larger $St_h$ and (ii) smaller $D/h$ . These behaviours are similar to those for turbulent kinetic energy shown in figure 5.

Figure 7. Spatial average of (a) turbulent energy production rate $\langle \mathcal{P}^{t \leftarrow M} \rangle$ by the mean flow and (b) turbulent energy dissipation rate $\langle \epsilon \rangle$ as functions of $St_h$ . The values are normalised by those in the single-phase flow. The symbols are the same as in figure 5.

It is also important to observe that the turbulent energy dissipation rate $\langle \epsilon \rangle$ is not modulated by particles as much as $\langle \mathcal{P}^{t \leftarrow M} \rangle$ . Figure 7(b) shows $\langle \epsilon \rangle$ normalised by $\langle \epsilon _\times \rangle$ as a function of $St_h$ . Note that $\epsilon$ includes the two contributions from the energy dissipation through the energy cascade and from the wake behind added particles. We see that $\langle \epsilon \rangle / \langle \epsilon _\times \rangle$ is close to unity. This result means that the average energy input rate $\langle I^M \rangle$ minus the average energy dissipation rate $\langle \epsilon ^M \rangle$ of the mean flow is not significantly altered from that of the single-phase flow. More concretely, $\langle \mathcal{I}^M \rangle$ and $\langle \epsilon ^M \rangle$ show similar monotonic trends with respect to $St_h$ (see Appendix D).

Figure 8. Spatial average of (a) rate of work $\mathcal{W}^{t \leftarrow p}$ done by particles on turbulence evaluated by (3.2) and (b) energy dissipation rate $\epsilon _p$ around particles defined with (3.7) as functions of $St_h$ . The values are normalised by the spatial average $\langle \epsilon _\times \rangle$ of turbulent energy dissipation rate in the single-phase flow. The symbols are the same as in figure 5.

3.4. Additional energy dissipation due to particles

We have demonstrated in figures 6 and 7(a) that particles can reduce the energy production rate $\mathcal{P}^{t \leftarrow M}$ , which relates to the attenuation of turbulent kinetic energy. We have also shown in figures 2 and 3 that particle-induced vortices seem relevant to turbulence attenuation. In this subsection, we discuss how these vortices contribute to the reduction of $\mathcal{P}^{t \leftarrow M}$ .

For the spatial average in the single-phase flow, the energy production rate due to the mean flow is balanced by the energy dissipation rate, i.e. $\langle \mathcal{P}^{t \leftarrow M} \rangle = \langle \epsilon \rangle$ . However, particles can break this balance due to the rate of work $\mathcal{W}^{t \leftarrow p}$ done by particles on turbulence, as described in (3.2). Figure 8(a) shows the average of $\mathcal{W}^{t \leftarrow p}$ evaluated by (3.2). We see that $\langle \mathcal{W}^{t \leftarrow p} \rangle$ gets larger for particles with larger $St_h$ and smaller $D/h$ . This implies that particles play a role in transferring turbulent kinetic energy from the mean flow to turbulence. As a result, vortex rings shown in figures 2 and 3 are generated. Then, the turbulent kinetic energy associated with these vortices is rapidly dissipated, leading to a substantial energy dissipation rate $\epsilon _p$ around the particles. Therefore, combining the local balance $\mathcal{W}^{t \leftarrow p} = \epsilon _p$ with (3.2), we can describe the energy production rate as

(3.5) \begin{align} \langle \mathcal{P}^{t \leftarrow M} \rangle = \langle \epsilon \rangle - \langle \epsilon _p \rangle . \end{align}

Since $\langle \epsilon \rangle = \langle \epsilon _\times \rangle$ approximately holds in the present system as shown in figure 7(b), we can rewrite (3.5) as

(3.6) \begin{align} 1 - \frac { \langle \mathcal{P}^{t \leftarrow M} \rangle }{\langle \mathcal{P}^{t \leftarrow M}_\times \rangle } = \frac { \langle \epsilon _p \rangle }{ \langle \epsilon _\times \rangle }. \end{align}

This equation means that the degree of the reduction of $\langle \mathcal{P}^{t \leftarrow M} \rangle$ is proportional to the energy dissipation rate $\langle \epsilon _p \rangle$ due to vortices induced by particles.

To verify the above argument, we show in figure 8(b) the spatial average $\langle \epsilon _p \rangle$ of the energy dissipation rate $\epsilon _p$ around particles normalised by $\langle \epsilon _\times \rangle$ . Here, to estimate $\langle \epsilon _p \rangle$ , we numerically compute the local average of the turbulent energy dissipation rate around all particles:

(3.7) \begin{align} \langle \epsilon _p \rangle = \frac {1}{V_{box}}\left ( \left \langle \int _{\Omega _\circledcirc } \epsilon (\boldsymbol{x}^\prime ,t) \: \textrm{d} V^\prime \right \rangle - V_\circledcirc \: \langle \epsilon \rangle \right ). \end{align}

Here, $\Omega _\circledcirc$ denotes the region between a spherical particle and another sphere with diameter $3D$ concentric with the particle (Sun et al. Reference Sun, Peng, Wang, Chen and Zhu2024), $V_{\circledcirc }$ is the volume of $\Omega _\circledcirc$ and $V_{box}$ is the volume of the computational box. The first term on the right-hand side of (3.7) captures the local average of the energy dissipation rate $\epsilon$ around particles, where we evaluate $\epsilon$ by (B2) in Appendix B. Since this quantity includes the dissipation rates due to not only the vortices created around particles but also vortices generated by energy cascade or the mean shear, we subtract the latter contribution expressed by the second term. Figure 8(b) shows that $\langle \epsilon _p \rangle$ gets larger for larger $St_h$ or smaller $D/h$ , which is similar to the behaviour of $\langle \mathcal{W}^{t \leftarrow p} \rangle$ in figure 8(a). We can explain these dependences by noting that the local energy dissipation rate per unit mass around a single particle is approximated by $|\Delta u|^3/D$ , where $\Delta u$ is the magnitude of the relative velocity. Hence, when (i) the Stokes number is larger, and therefore the relative velocity is larger or (ii) the diameter is smaller, $\epsilon _p$ gets larger under the condition that the particle volume fraction is fixed. Recall that figure 7(a) shows a similar tendency for the average reduction rate $\langle \mathcal{P}^{t \leftarrow M} \rangle$ . To verify this similarity, we show $1-\langle \mathcal{P}^{t \leftarrow M} \rangle /\langle \mathcal{P}^{t \leftarrow M}_\times \rangle$ as a function of $\langle \epsilon _p \rangle / \langle \epsilon _\times \rangle$ in figure 9(a). The data approximately collapse on the red dotted line. This implies that we can describe the average reduction rate of the energy production $\langle \mathcal{P}^{t \leftarrow M} \rangle$ due to the mean flow in terms of the additional energy dissipation rate $\langle \epsilon _p \rangle$ by particles, as indicated by (3.6). Incidentally, although some plots lie beneath the red line, this deviation arises from the slight modulation of $\langle \epsilon \rangle$ from the single-phase flow (see figure 7 b). Taking this effect into account, when we replot the attenuation rate of $\langle \mathcal{P}^{t \leftarrow M} \rangle$ as a function of the energy flux $\langle \epsilon \rangle - \langle \epsilon _p \rangle$ in figure 9(b), the plots collapse better.

Figure 9. Average attenuation rate of turbulent energy production rate $\langle \mathcal{P}^{t \leftarrow M} \rangle$ by the mean flow as functions of (a) $\langle \epsilon _p \rangle /\langle \epsilon _\times \rangle$ and (b) $1-(\langle \epsilon \rangle -\langle \epsilon _p \rangle )/\langle \epsilon _\times \rangle$ . (c) Average attenuation rate $\mathcal{A}$ of turbulent kinetic energy as a function of $\langle \epsilon _p \rangle /\langle \epsilon _\times \rangle$ . The symbols are the same as in figure 5. The proportional coefficients of the red dotted lines are (a,b) $1$ and (c) $1.8$ .

Moreover, the additional energy dissipation rate is also important in describing the average attenuation rate of turbulent kinetic energy. We show in figure 9(c) the average attenuation rate $\mathcal{A}$ , defined as (3.1), of turbulent kinetic energy. Although the turbulence attenuation rate depends on $St_h$ , $D/h$ and $Re_\tau$ (see figure 5), when plotting $\mathcal{A}$ as a function of $\langle \epsilon _p \rangle$ / $\langle \epsilon _\times \rangle$ , the data approximately collapse on the red dotted line. We conclude therefore that the turbulence attenuation rate is related to the additional energy dissipation rate $\langle \epsilon _p \rangle$ due to particles. This is consistent with previous studies on modulation of wall-bounded turbulence by pointwise particles (Zhao et al. Reference Zhao, Andersson and Gillissen2013) and fixed particles with restricted motion (Peng et al. Reference Peng, Ayala and Wang2019b ), which emphasised the importance of the additional energy dissipation around particles. Incidentally, in figure 9(c), we observe the proportionality between $\mathcal{A}$ and $\langle \epsilon _p \rangle / \langle \epsilon _\times \rangle$ , indicating that $\langle \mathcal{P}^{t \leftarrow M} \rangle$ is proportional to $\langle K \rangle$ . However, there may be a better expression for the relationship between $\mathcal{P}^{t \leftarrow M}$ and $K$ for higher-Reynolds-number turbulence. We further discuss this in § 4.3.

We summarise the mechanism of attenuation of wall-bounded turbulence. When particles cannot follow the fluid motions and create small vortices around them (figures 2 and 3), turbulent kinetic energy $K$ is attenuated (figure 4). This occurs because these particle-induced vortices acquire a part of the kinetic energy from the mean flow (figure 8 a), and produce the additional energy dissipation rate (figure 8 b), thereby preventing energy transfer rate $\mathcal{P}^{t \leftarrow M}$ from the mean flow to turbulent vortices (figures 6 and 7 a). This is the reason why we can describe the average attenuation rates of $\mathcal{P}^{t \leftarrow M}$ and $K$ in terms of the additional energy dissipation rate $\langle \epsilon _p \rangle$ due to particles (figure 9).

Incidentally, it is also worth mentioning that the Reynolds stress is also reduced (figure 15 b in Appendix C), leading to a reduction of the frictional drag coefficient. This is because the mean bulk velocity increases (figure 15 a) as a consequence of the reduction of fluid momentum transfer in the wall-normal direction. We show this result in Appendix E.

4.1. Estimation of particle energy dissipation

We first consider the magnitude of the force exerted on a particle by the fluid:

(4.1) \begin{align} F^{p \leftarrow f} = \frac {1}{2} C_D \rho _f |\Delta u|^2 A, \end{align}

where $C_D$ is the drag coefficient, $A$ ( $=\pi D^2/4$ ) is the cross-sectional area of the sphere and $\Delta u$ is the magnitude of the relative velocity. The particle subjected to this force leads to energy dissipation at the rate of $F^{p \leftarrow {f}} \Delta u$ . Therefore, the energy dissipation rate due to all the particles in the system can be expressed as

(4.2) \begin{align} \langle \epsilon _p^\# \rangle = \frac {3}{4} \varLambda _0 C_D \frac {|\Delta u|^3}{D}. \end{align}

Thus, to obtain $\langle \epsilon _p^\# \rangle$ , we need to estimate the relative velocity $\Delta u$ . We discuss the estimation of $\Delta u$ in the next subsection.

Moreover, by estimating the average energy dissipation rate in the single-phase flow as $\langle \epsilon _\times ^\# \rangle = u_L^3/h$ with $u_L$ ( $\sim \hspace {-1mm}u_\tau$ ) being the characteristic velocity at the largest scale in the outer layer, we obtain

(4.3) \begin{align} \frac {\langle \epsilon _p^\# \rangle }{\langle \epsilon _\times ^\# \rangle } = \frac {3}{4} \varLambda _0 C_D \frac {h}{D} \frac {{|\Delta u|^3}}{u_L^3} . \end{align}

For the drag coefficient $C_D$ , we use the experimental law (Schiller Reference Schiller1933):

(4.4) \begin{align} C_D = \frac {24}{Re_D}\left(1+0.15Re_D^{0.687}\right) , \end{align}

where $Re_D$ ( $=|\Delta u| D/\nu$ ) is the particle Reynolds number. Thus, we can describe the additional energy dissipation rate (4.3) by using the particle parameters ( $\varLambda _0$ and $D/h$ ) and the relative velocity $\Delta u$ .

4.2. Relative velocity

Figure 10. Relative velocity $\langle | \Delta u | \rangle _o$ between particles and their surrounding fluid averaged in the outer layer. The symbols are the same as in figure 5. The grey and black dashed lines are evaluation by (4.5a ) and (4.5b ), respectively, with coefficients of $u_L=1.5u_\tau$ and $\tau _L=2.5\tau _h$ .

We show in figure 10 the relative velocity $\langle |\Delta {u}| \rangle _{o}$ averaged in the outer layer ( $y/h \geqslant 0.3$ ) as a function of $St_h$ . Note that since $\Delta u$ strongly depends on height $y$ , we focus on the outer layer where $\Delta u$ weakly depends on $y$ (see Appendix F). Here, we evaluate the relative velocity as , where we define the surrounding fluid velocity for each particle by the average fluid velocity on the surface of a sphere with diameter $2D$ concentric with the particle (Kidanemariam et al. Reference Kidanemariam, Chan-Braun, Doychev and Uhlmann2013; Uhlmann & Chouippe Reference Uhlmann and Chouippe2017). Symbols in lighter colours indicate smaller $D/h$ , and different shapes represent different values of $Re_\tau$ . We see in the figure that $St_h$ determines the relative velocity irrespective of the other parameters. We also see that the functional forms of the relative velocity are well approximated by the dashed lines, which are defined as

\begin{align*} \qquad\qquad {|\Delta {u}|} &= \left \{\begin{array}{l}\displaystyle \frac {(\tau _p/\tau _L)^{1/2}}{\sqrt {2}} \: u_L \quad \quad \qquad \qquad (\tau _\eta \leqslant \tau _p \leqslant \tau _L),\: \qquad \qquad \qquad \ \, {(4.5a)}\\ \displaystyle \frac {\tau _p/\tau _L}{\sqrt {1+(\tau _p/\tau _L)^2}} \: u_L \quad \quad \qquad \qquad (\tau _p \gt \tau _L).\: \qquad \qquad \qquad \quad {(4.5b)}\end{array}\right . \end{align*}

Here, (4.5a ) holds when the particle relaxation time $\tau _p$ is longer than the Kolmogorov time $\tau _\eta$ and shorter than the integral time $\tau _L$ , whereas (4.5b ) holds for $\tau _p \gt \tau _L$ . We can derive (4.5) by assuming pointwise heavy particles based on the argument by Balachandar (Reference Balachandar2009) as follows. As considered in our previous studies (Oka & Goto Reference Oka and Goto2021; Motoori, Wong & Goto Reference Motoori, Wong and Goto2022), we first assume that the motion of particles with $\tau _p$ is independent of fluid motions smaller than $\ell$ , where $\ell$ is the length scale such that the turnover time $\tau ^{(\ell )}$ ( $=\langle \epsilon _\times \rangle ^{-1/3} \ell ^{2/3}$ ) of vortices with size $\ell$ is approximately $\tau _p$ , i.e. $\ell = \langle \epsilon _{\times }\rangle ^{1/2} \tau _p^{3/2}$ . Then, considering a particle in the oscillating flow with frequency $\tau ^{(\ell )}$ (Balachandar Reference Balachandar2009), we obtain

(4.6) \begin{align} |\Delta u| &= \frac {\tau _p/\tau ^{(\ell )}} {\sqrt {1+(\tau _p/\tau ^{(\ell )})^2}} \: \widetilde {u}^{({\ell })} . \end{align}

Here, $\widetilde {u}^{(\ell )}$ is the fluid velocity coarse-grained at scale $\ell$ . For $\tau _\eta \leqslant \tau _p \leqslant \tau _L$ , when assuming that the relative velocity is determined by vortices whose turnover time is comparable to the particle relaxation time (i.e. $\tau ^{(\ell )} = \tau _p$ ), we can obtain (4.5a ). In this derivation, we use $\widetilde {u}^{(\ell )} = (\tau ^{(\ell )}/\tau _L)^{1/2} u_L$ . On the other hand, we can obtain (4.5b ) for $\tau _p \gt \tau _L$ by assuming that the relative velocity is determined by the largest-scale vortices (i.e. $\tau ^{(\ell )} = \tau _L$ ).

Figure 10 shows that (4.5) is in good agreement with our DNS data. Here, we choose the parameters $u_L$ and $\tau _L$ in (4.5) as in the order of $u_\tau$ and $\tau _h$ , respectively, so that the curves expressed by (4.5) fit our data. Incidentally, the relative velocity for $\tau _p \lt \tau _\eta$ is determined by vortices at the Kolmogorov time scale (i.e. $\tau ^{(\ell )}= \tau _\eta$ ), although in the present study, we do not simulate particles with $\tau _p \lt \tau _\eta$ .

Incidentally, it is not always possible to estimate the relative velocity using (4.5). This evaluation is valid when the particle diameter is smaller than the length scale $\ell$ of vortices with the turnover time $\tau ^{(\ell )} \approx \tau _p$ , i.e.

\begin{align*}\qquad\qquad D \lesssim \ell \quad \Leftrightarrow \quad \frac {D}{h} \lesssim \left \{ \begin{array}{l} \displaystyle (\tau _p/\tau _L)^{3/2} \approx St_h^{3/2} \quad \quad (\tau _\eta \leqslant \tau _p \leqslant \tau _L),\: \qquad\quad \ \ \ {(4.7a)}\\ 1 \quad \quad \quad \quad \qquad \quad \quad \quad \quad (\tau _p \gt \tau _L).\: \qquad \quad \qquad\, {(4.7b)}\end{array}\right . \end{align*}

The examined particles satisfy this condition.

4.3. Prediction of turbulence attenuation rate

Figure 11. Average attenuation rate $\mathcal{A}$ of turbulent kinetic energy as a function of $\langle \epsilon _p^\# \rangle /\langle \epsilon _\times ^\# \rangle$ , which is estimated by (4.8) using only particle parameters. Particle parameters for the filled symbols are the same as in figure 5. The open circles show the turbulence attenuation rate at the channel centre measured in experiments by Kulick et al. (Reference Kulick, Fessler and Eaton1994). The brown and light blue symbols indicate the results for copper and glass particles (with different volume fractions $\varLambda _0$ ), respectively. The proportional coefficient of the red dotted line is $1.5$ .

By substituting the relative velocity prediction (4.5) into (4.3), we can estimate the additional energy dissipation rate by

\begin{align*}\qquad\qquad \frac {\langle \epsilon _p^\# \rangle }{\langle \epsilon _\times ^\# \rangle } = \left \{ \begin{array}{l} \displaystyle \frac {3}{4} \varLambda _0 C_D \frac {h}{D} \frac {(\tau _p/\tau _L)^{3/2}}{2\sqrt {2}}\quad \quad (\tau _\eta \leqslant \tau _p \leqslant \tau _L),\: \quad \qquad \qquad \qquad\, {(4.8a)}\\[15pt] \displaystyle \frac {3}{4} \varLambda _0 C_D \frac {h}{D} \frac {(\tau _p/\tau _L)^{3}}{(1+(\tau _p/\tau _L)^2)^{3/2}} \quad \quad (\tau _p \gt \tau _L).\: \ \ \, \qquad \qquad \qquad {(4.8b)}\end{array}\right . \end{align*}

We plot in figure 11 the average attenuation rate $\mathcal{A}$ as a function of $\langle \epsilon _p^\# \rangle /\langle \epsilon _\times ^\# \rangle$ . We see that our DNS data collapse onto a single line, which is approximated by the red dotted line. In this figure, we also plot the experimental results (Kulick et al. Reference Kulick, Fessler and Eaton1994) for turbulent channel flow. The brown and light blue open symbols indicate the attenuation rates at the channel centre by copper and glass particles, respectively. Despite the different particle types and volume fractions, the experimental data align closely with our DNS results. Thus, we can use (4.8) to estimate the additional energy dissipation rate $\langle \epsilon _p^\# \rangle /\langle \epsilon _\times ^\# \rangle$ , and by using this estimation, we can describe the turbulence attenuation rate.

Before concluding this article, we discuss the relevance to the study on periodic turbulence by Oka & Goto (Reference Oka and Goto2022). They derived the formula

(4.9) \begin{align} 1-\left ( 1-\frac {\mathcal{A}}{1+\alpha } \right )^{3/2} = \frac {\langle \epsilon _p \rangle }{\langle \epsilon _\times \rangle } \end{align}

for describing the turbulence attenuation rate, and then verified it using their DNS results of periodic turbulence. Here, $\alpha$ is the ratio of the kinetic energy of the mean flow to turbulent energy for the single-phase flow. When deriving this formula, they first assumed that the additional energy dissipation rate bypasses the energy cascade. Then, they used the dissipation law of Taylor (Reference Taylor1935) to relate average turbulent kinetic energy $\langle K \rangle$ to its dissipation rate $\langle \epsilon \rangle$ . We may use a similar relation between $K$ and $\epsilon$ ( $= \mathcal{P}^{t \leftarrow M}$ ) in the log layer for wall turbulence at sufficiently high Reynolds numbers. However, since the present turbulence does not have a large scale separation to discuss the buffer, log and outer layers individually, we have argued the spatially averaged quantities. Nevertheless, our DNS results show that the turbulence attenuation rate increases monotonically with respect to $\langle \epsilon _p \rangle / \langle \epsilon _\times \rangle$ (see figure 9 c). Moreover, we have demonstrated in figure 11 that $\langle \epsilon _p \rangle / \langle \epsilon _\times \rangle$ can be estimated based solely on particle parameters using (4.8). This estimation can be applicable to developed turbulence laden with small particles satisfying (4.7) in a dilute regime.