3.1 Particle distribution in two distinct layers

Mean particle volume concentrations in the inner layer and outer layer are shown in figure 3(a), exhibiting a non-monotonic behaviour with Stokes number but with an opposite trend. In the inner layer it is maximized for case 3, which corresponds to $St^{+}=24.2$ . At the same time, case 3 also exhibits the minimum in the outer layer, where the more relevant Stokes number is $St_{out}=0.8$ . Stokes numbers lower or higher than case 3 result in fewer particles in the inner layer but more particles in the outer layer. This often-observed behaviour is due to turbophoresis (Reeks Reference Reeks1983), which induces a net particle flux towards the wall resulting in higher particle volume concentration in the inner layer than in the outer layer.

It is also commonly accepted that inertial particles preferentially accumulate in low-speed streaks (Pan & Banerjee Reference Pan and Banerjee1996; Marchioli & Soldati Reference Marchioli and Soldati2002), which is also observed in this study. Particle numbers in ‘upwelling’ and ‘downwelling’ regions can be straightforwardly counted by testing whether $u_{f}^{\prime }<0$ or $u_{f}^{\prime }>0$ , where $u_{f}^{\prime }$ is the fluid fluctuating velocity seen by the particle. Then, the ratio of the number of particles in ‘upwelling’ and ‘downwelling’ regions can be used to compare across different wall-normal locations and Stokes numbers. This ratio, cast in terms of the effective volume concentration corresponding to these particle counts, is shown in figure 3(b). Here, there is a clear non-monotonic trend with the Stokes number and with $y$ . In the inner layer, there are more particles in the ‘upwelling’ fluid motions than in the ‘downwelling’ fluid motions, which is opposite compared to the outer layer. The lowest ratio appears for case 3 at $St^{+}=24.2$ in the inner layer, while the highest ratio appears for case 5 at $St_{out}=6.0$ in the outer layer. Thus in the inner layer, relatively low-inertia particles collect in the low-speed streaks, while in the outer layer, higher-inertia particles collect in the high-speed regions. The relevant Stokes numbers are different for each, since the respective fluid time scales are different.

Figure 3.

(a) The mean particle volume concentration in the inner layer (as a function of $St^{+}$ ) and outer layer (as a function of $St_{out}$ ), scaled by the bulk value. (b) The ratio between particle concentrations with $u_{f}^{\prime }>0$ and $u_{f}^{\prime }<0$ . For single-phase flow (case 1), Eulerian grid points with $u^{\prime }>0$ or $u^{\prime }<0$ are plotted.

Figure 4.

Mean fluid streamwise velocity fluctuation at particle locations defined in (3.1): (a) in LSMs ( $\unicode[STIX]{x1D706}_{x}<5h$ , $\unicode[STIX]{x1D706}_{z}<0.75h$ ) and (b) in VLSMs ( $\unicode[STIX]{x1D706}_{x}>5h$ , $\unicode[STIX]{x1D706}_{z}>0.75h$ ).

The relationship between particle distributions and the different turbulent structures (i.e. VLSMs and LSMs) is still poorly understood, however. To explore this, we introduce a mean fluid streamwise velocity fluctuation at the particle positions, but filtered according to the wavelengths associated with LSMs and VLSMs:

(3.1a,b ) $$\begin{eqnarray}\overline{u_{f\_p}^{\prime }}(\text{VLSMs})\equiv \frac{\overline{\unicode[STIX]{x1D6F7}_{v}u_{\unicode[STIX]{x1D706}_{x}>5h,\,\unicode[STIX]{x1D706}_{z}>0.75h}^{\prime }}}{\overline{\unicode[STIX]{x1D6F7}_{v}}},\quad \overline{u_{f\_p}^{\prime }}(\text{LSMs})\equiv \frac{\overline{\unicode[STIX]{x1D6F7}_{v}u_{\unicode[STIX]{x1D706}_{x}<5h,\,\unicode[STIX]{x1D706}_{z}<0.75h}^{\prime }}}{\overline{\unicode[STIX]{x1D6F7}_{v}}}.\end{eqnarray}$$

Here, the interpolated velocity $u_{f}$ is projected onto the Eulerian grid, allowing for the $u_{f}^{\prime }$ field to be transferred to Fourier space ( $\hat{u} _{f}^{\prime }$ ). The goal is to artificially filter out targeted turbulent structures (e.g. removing wavelengths with $\unicode[STIX]{x1D706}_{x}<5h$ ,  $\unicode[STIX]{x1D706}_{z}<0.75h$ to isolate VLSMs) in order to obtain $u_{f}^{\prime }$ contributed by specific turbulent structures. For single-phase flow, $\unicode[STIX]{x1D6F7}_{v}$ is set to one at all Eulerian grid points and the same procedure is followed. These quantities are shown in figure 4. In the inner layer, $\overline{u_{f\_p}^{\prime }}$ is negative in both VLSMs and LSMs, indicating that particles are more likely to reside in the large- and very-large-scale low-speed streaks compared to the high-speed streaks. The minimum values of $\overline{u_{f\_p}^{\prime }}$ for LSMs (case 3 with $St^{+}=24.2$ in figure 4 a) and VLSMs (case 4 with $St^{+}=60.5$ in figure 4 b) appear at different Stokes numbers, again since the flow time scales associated with LSMs and VLSMs are different.

In the outer layer, $\overline{u_{f\_p}^{\prime }}$ is positive for both VLSMs and LSMs, indicating that particles tend to reside in the ‘downwelling’ regions at the scales of both the LSMs and the VLSMs. The maximum value of $\overline{u_{f\_p}^{\prime }}$ for VLSMs (figure 4 b) appears for case 5 with $St_{out}=6.0$ . Comparing the inner layer with the outer layer, $\overline{u_{f\_p}^{\prime }}$ for LSMs is considerably stronger in the inner layer than in the outer layer, as shown in figure 4(a). Two possible reasons may explain this. One is that the intensity of LSMs is much weaker in the outer layer than in the inner layer (which will be shown later in figure 9 a); the other is that particle preferential concentration is not as strongly correlated with streaky motions in the outer layer as compared to the inner layer (which will be shown in figure 6 a). However, for $\overline{u_{f\_p}^{\prime }}$ in VLSMs (figure 4 b), the magnitude is comparable between the inner layer and the outer layer, albeit with opposite preferred signs.

In terms of particle clustering behaviour, it is well established that, for wall-bounded turbulent flow in the inner layer, low-inertia particles ( $St^{+}<O(0.1)$ ) tend to distribute homogeneously in wall-normal planes (Pan & Banerjee Reference Pan and Banerjee1996), while intermediate Stokes numbers ( $St^{+}=O(10)$ ) exhibit particle clustering in near-wall streaks (Marchioli & Soldati Reference Marchioli and Soldati2002; Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012; Richter Reference Richter2015; Wang & Richter Reference Wang and Richter2019) and high-inertia particles ( $St^{+}>O(100)$ ) behave with ballistic trajectories (thus eliminating much of the clustering). This qualitative transition with $St^{+}$ is observed within the inner region of the simulated open channel flow. Figure 5(a–e) present isosurfaces of particle concentration ( $2.5$ times the bulk $\overline{\unicode[STIX]{x1D6F7}_{v}}$ ) for cases 2–6 in the inner and outer regions for increasing Stokes numbers. The advantage of showing concentration isosurfaces as opposed to individual Lagrangian points is that this method better visualizes the high-concentration particle clusters. The three panels across the horizontal represent three slabs at progressively increasing wall-normal distances (at the same snapshot in time): layer 1, $30<y^{+}<130$ , layer 2, $150<y^{+}<250$ , and layer 3, $450<y^{+}<550$ .

Here we find particles accumulating in the inner-flow low-speed streaks at intermediate $St^{+}=24.2{-}60.5$ (cases 3 and 4 in layer 1 of figure 5 b,c); this is similar to many other studies, as noted above. At the same time, a new type of organized structure in the outer flow region is observed. These are especially clear at higher Stokes numbers, e.g.  $St^{+}=182$ ( $St_{out}=6$ ) for case 5 shown in layers 2 and 3 of figure 5(d). However, with a very high particle inertia, $St^{+}=908$ ( $St_{out}=30$ ), particles behave ballistically in the outer flow region as shown in layers 2 and 3 of figure 5(e), tending to distribute more homogeneously. These two distinct, non-monotonic particle accumulation behaviours in the inner and outer layers peak at different Stokes numbers ( $St^{+}=O(10)$ in the inner layer and $St_{out}=6$ in the outer layer) and have a strong influence on the non-monotonic modulation of the VLSMs in the outer region via the two routes indicated in figure 1. This will be discussed further in § 3.2.

Figure 5.

(a) Instantaneous streamwise velocity fluctuation $u^{\prime }$ (colour contours) in three $x{-}z$ planes (panels from left to right: $y^{+}=50$ , 150, 450) and isosurface of particle concentration ( $2.5$ times bulk $\overline{\unicode[STIX]{x1D6F7}_{v}}$ ) in three slabs (layer 1, $30<y^{+}<130$ , layer 2, $150<y^{+}<250$ , and layer 3, $450<y^{+}<550$ ). Panels (a–e) refer to cases 2–6, respectively. A supplementary movie is available online at https://doi.org/10.1017/jfm.2019.210.

Figure 6.

Standard deviation of the normalized Voronoï area $\unicode[STIX]{x1D70E}_{{\mathcal{V}}}$ , normalized by that of a random Poisson process, $\unicode[STIX]{x1D70E}_{RPP}$ : (a) as a function of height in wall-normal direction of five Stokes numbers and (b) as a function of $St_{K}$ at two wall-normal locations ( $y^{+}=50$ and $y^{+}=200$ are plotted based on $St_{K}$ of the inner layer and outer layer as in table 1). Experimental results of an isotropic turbulence from Monchaux, Bourgoin & Cartellier (Reference Monchaux, Bourgoin and Cartellier2010) are also shown.

In order to further quantify the particle clustering behaviour, we employ a Voronoï diagram analysis, which compares the distribution of the tessellation areas in the particle-laden cases with the expected Poisson distribution if the particles were randomly distributed (see e.g. Monchaux, Bourgoin & Cartellier Reference Monchaux, Bourgoin and Cartellier2012). A maximum clustering effect is typically observed for $St_{K}$ around unity in isotropic turbulence (Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2010; Baker et al. Reference Baker, Frankel, Mani and Coletti2017) and buoyancy-driven wall-bounded turbulence (Park, O’Keefe & Richter Reference Park, O’Keefe and Richter2018). Figure 6(a,b) show the standard deviation ( $\unicode[STIX]{x1D70E}_{{\mathcal{V}}}$ ) of the distribution of the normalized Voronoï area ${\mathcal{V}}=A/\overline{A}$ , where the inverse of the average Voronoï area $\overline{A}$ indicates the mean particle concentration. The $\unicode[STIX]{x1D70E}_{{\mathcal{V}}}$ value is scaled by the standard deviation of a random Poisson process (RPP; $\unicode[STIX]{x1D70E}_{RPP}=0.52$ ). The ratio $\unicode[STIX]{x1D70E}_{{\mathcal{V}}}/\unicode[STIX]{x1D70E}_{RPP}$ exceeding unity indicates that particles are accumulating in clusters as compared to truly randomly distributed particles.

Figure 6(a) shows the ratio $\unicode[STIX]{x1D70E}_{{\mathcal{V}}}/\unicode[STIX]{x1D70E}_{RPP}$ for multiple heights across all Stokes numbers, while figure 6(b) shows the ratio as a function of $St_{K}$ for two different representative heights ( $y^{+}=50$ and $y^{+}=200$ are plotted based on $St_{K}$ of the inner layer and outer layer, respectively, as provided in table 1). In the inner layer ( $y^{+}=50$ ), from case 2 to case 6 ( $St^{+}=2.42-908$ or $St_{K}=0.58-220$ ), the clustering effect experiences a non-monotonic evolution as a function of Stokes number. The largest value appears at $St^{+}=24.2$ and it gradually decreases to unity with a higher Stokes number. This is similar to the result of the investigation of Wang & Richter (Reference Wang and Richter2019) that particles with $St^{+}=29.5$ preferentially accumulate in the streaks whereas higher-inertia particles tend to spread throughout the inner layer. With increasing wall-normal distance, the ratio $\unicode[STIX]{x1D70E}_{{\mathcal{V}}}/\unicode[STIX]{x1D70E}_{RPP}$ decreases at very low Stokes number (case 2) whereas it increases at higher Stokes numbers (cases 3–6). At still higher Stokes numbers (case 7), the ratio $\unicode[STIX]{x1D70E}_{{\mathcal{V}}}/\unicode[STIX]{x1D70E}_{RPP}$ again approaches unity. These very heavy particles cannot follow the streamlines, resulting in a nearly random particle distribution. In the outer layer ( $y^{+}=200$ ), the largest value appears at $St_{K}=2.11$ ( $St_{out}=0.8$ ), similar in magnitude to previous investigations of clustering in isotropic turbulence; peaks were found for $St_{K}=3.0$ and $St_{K}=1.0$ by Monchaux et al. (Reference Monchaux, Bourgoin and Cartellier2010) and Baker et al. (Reference Baker, Frankel, Mani and Coletti2017), respectively.

3.2 Particle modulation of TKE in LSMs and VLSMs

The premultiplied, two-dimensional energy spectrum of streamwise velocity, $k_{x}k_{z}\unicode[STIX]{x1D6F7}_{u^{\prime }u^{\prime }}$ , where $\unicode[STIX]{x1D6F7}_{u^{\prime }u^{\prime }}=\langle \hat{u} ^{\prime }(k_{x},k_{z},y){\hat{u} ^{\prime }}^{\ast }(k_{x},k_{z},y)\rangle$ , is shown in figure 7 for $Re_{\unicode[STIX]{x1D70F}}=550$ as a function of wall-normal distance ( $\hat{u} ^{\prime }$ is the Fourier coefficient of $u^{\prime }$ ). Figure 7 exhibits a ‘boot-shaped’ structure, particularly well defined for case 2 (figure 7 b) and case 5 (figure 7 e). The ‘forefoot’ corresponds to the LSMs in the near-wall region, whereas the ‘bootleg’ corresponds to the VLSMs. The signature of VLSMs indeed appears at the upper-left corner (long and wide wavelengths in the streamwise and spanwise directions) for single-phase flow in figure 7(a). This VLSM signature is nearly unchanged for case 3 and case 6 whereas it is slightly weakened in case 4. It is clear, however, that for low Stokes number ( $St^{+}=2.42$ of case 2 in figure 7 b) and high Stokes number ( $St_{out}=6$ of case 5 in figure 7 e), energy contained by the VLSMs is enhanced by the presence of particles. In addition to the enhanced VLSMs observed in the outer layer, the large-scale energetic structures (e.g. the ‘bootleg’ within $y^{+}<100$ in figure 7 b,e) extend into the inner layer, which are referred to as deep $u$ -modes (del Álamo & Jiménez Reference del Álamo and Jiménez2003) or VLSM ‘footprints’ (Hutchins & Marusic Reference Hutchins and Marusic2007), and are a possible path for the inverse scale transfer from LSMs to VLSMs found by Lee & Moser (Reference Lee and Moser2019).

Figure 7.

Premultiplied two-dimensional energy spectrum $k_{x}k_{z}\unicode[STIX]{x1D6F7}_{u^{\prime }u^{\prime }}/u_{\unicode[STIX]{x1D70F}}^{2}$ as a function of $\unicode[STIX]{x1D706}_{x}$ and $\unicode[STIX]{x1D706}_{z}$ and the wall-normal direction $y$ . An isosurface of $0.1$ times the maximum value of the single-phase flow is illustrated. Panels (a–f) refer to cases 1–6, respectively.

Figure 8.

Premultiplied one-dimensional $u$ -spectra as functions of spanwise wavelength and wall-normal direction. The contours are $0.2,0.4,0.8$ times the maximum value based on the single-phase flow. Panels (a–f) are for cases 1–6 at $Re_{\unicode[STIX]{x1D70F}}=550$ ; and panels (g–i) are for cases 8–10 at $Re_{\unicode[STIX]{x1D70F}}=950$ . In (a), the shaded contours are for case 1 and line types represent case 7, which doubles the domain size of case 1 in the streamwise direction. The ‘ $+$ ’ symbols denote isolines of ( $0.2,0.4,0.8$ ) from del Álamo & Jiménez (Reference del Álamo and Jiménez2003), containing modes only with $\unicode[STIX]{x1D706}_{x}<5h$ (VLSMs with $\unicode[STIX]{x1D706}_{x}>5h$ are artificially removed).

Figure 8 displays as a function of $y^{+}$ the premultiplied, one-dimensional $u$ -spectra $k_{z}\unicode[STIX]{x1D719}_{u^{\prime }u^{\prime }}(k_{z})$ , where $\unicode[STIX]{x1D719}_{u^{\prime }u^{\prime }}(k_{z})=\langle \hat{u} ^{\prime }(k_{z}){\hat{u} ^{\prime }}^{\ast }(k_{z})\rangle$ , as a function of the normalized spanwise wavelength $\unicode[STIX]{x1D706}_{z}^{+}$ . As a reference, we compare with the results of del Álamo & Jiménez (Reference del Álamo and Jiménez2003) for wall-bounded channel flow at the same $Re_{\unicode[STIX]{x1D70F}}=550$ , who find that the turbulence in the outer flow behaves roughly isotropically if VLSMs are artificially removed (i.e. VLSMs introduce anisotropy). Comparing del Álamo & Jiménez (Reference del Álamo and Jiménez2003) (who filter out high wavelengths, so $k_{z}\unicode[STIX]{x1D719}_{u^{\prime }u^{\prime }}(k_{z})$ with only $\unicode[STIX]{x1D706}_{x}<5h$ are plotted) and the present (unfiltered) simulation in figure 8(a), we observe that, for unladen flow, with or without the contribution from turbulent structures of $\unicode[STIX]{x1D706}_{x}>5h$ , the spectral signature of LSMs is hardly affected (see the spectrum below $y^{+}=100$ with $\unicode[STIX]{x1D706}_{z}^{+}<200$ ) (Jiménez & Pinelli Reference Jiménez and Pinelli1999). As noted previously, any effect of a limited streamwise domain extent $L_{x}$ is minimal, since figure 8(a) shows that the energy contained in VLSMs in a short domain (contour represents for case 1) is nearly identical to the long domain (case 7 is shown by lines). Thus overall our unladen simulations are consistent with the current understanding of VLSMs. We point out that in single-phase flow, only $(0.2,0.4,0.8)$ times the maximum value based on the single-phase flow are displayed in figure 8(a); therefore the second peak in the outer layer is not as readily observed as in figure 7(a). For case 2 (figure 8 b) and case 5 (figure 8 e), the contribution from VLSMs forms a bimodal spanwise spectrum at $600<\unicode[STIX]{x1D706}_{z}^{+}<1000$ ( $h<\unicode[STIX]{x1D706}_{z}<2h$ ) at heights above the inner layer. The enhancement of the VLSM signature is also found at $Re_{\unicode[STIX]{x1D70F}}=950$ , $St_{out}=8.0$ as shown in figure 8(g–i) for cases 8–10. The Stokes number based on the outer time scale for case 9 is similar to that of case 5 at $Re_{\unicode[STIX]{x1D70F}}=550$ . By investigating inertial particle modulation of the regeneration cycle of LSMs, Wang & Richter (Reference Wang and Richter2019) found that particle inertia has a non-monotonic effect on LSMs in the inner region: low inertia ( $St^{+}=O(1)$ , e.g.  $St^{+}=2.42$ of case 2) promotes the regeneration cycle whereas high inertia ( $St^{+}=O(10)$ , i.e.  $St^{+}=24.2-182$ of cases 3–5) attenuates the regeneration cycle. However, we see from figure 8 that VLSM enhancement occurs at both low and high Stokes numbers (i.e. case 2 and case 5).

Figure 9.

Streamwise TKE ( $k_{u^{\prime }u^{\prime }}$ ) contributed by (a) LSMs, $\unicode[STIX]{x1D706}_{x}<5h$ , and (b) VLSMs, $\unicode[STIX]{x1D706}_{x}>5h$ , of cases 1–6. Here $y^{+}=15$ and $y^{+}=157$ are used to represent the inner layer and outer layer, respectively. The corresponding value of $k_{u^{\prime }u^{\prime }}$ at the two heights is plotted against (c)  $St^{+}$ and (d)  $St_{out}$ .

Figure 10.

Instantaneous contours of streamwise velocity fluctuation scaled by $u_{\unicode[STIX]{x1D70F}}$ on a wall-parallel plane at $y^{+}=157$ at the same time step. Panels (a,c,e) contain all modes; panels (b,d,f) are for the same flow field but only associating with VLSMs, containing modes with $\unicode[STIX]{x1D706}_{x}>10h$ , $\unicode[STIX]{x1D706}_{z}>0.75h$ ( $\unicode[STIX]{x1D706}_{x}^{+}>5500$ , $\unicode[STIX]{x1D706}_{z}^{+}>412$ ). (a,b) Case 1; (c,d) case 2; and (e,f) case 5.

In order to quantify the TKE of VLSMs and LSMs modulated by particles and to demonstrate their effect as a function of Stokes number, we show $k_{u^{\prime }u^{\prime }}\equiv \overline{u^{\prime }u^{\prime }}$ for LSMs (represented by $\unicode[STIX]{x1D706}_{x}<5h$ ) and VLSMs (represented by $\unicode[STIX]{x1D706}_{x}>5h$ ) in figure 9(a) and (b), respectively. The TKE of LSMs is 4–5 times larger than the TKE of VLSMs. The $k_{u^{\prime }u^{\prime }}$ contributed by LSMs and VLSMs is shown as a function of Stokes number for the representative inner-layer height $y^{+}=15$ in figure 9(c) and the representative outer-layer height $y^{+}=157$ in figure 9(d). Figure 9(c) shows that, in the inner layer, the minimum TKE of LSMs appears for case 3 ( $St^{+}=24.2$ ), corresponding to the strongest turbulence attenuation observed by Wang & Richter (Reference Wang and Richter2019). In the outer layer, the minimum TKE of LSMs appears at a higher Stokes number, i.e. case 5 ( $St_{out}=6.0$ ). However, the TKE modulation of VLSMs is distinct from, and even opposite in behaviour to, that of LSMs. As shown in figure 9(c), in the inner layer, the intensity of VLSMs reaches its maximum for case 3 ( $St^{+}=24.2$ ) whereas the TKE of VLSMs has two peaks in the outer layer: case 2 ( $St_{out}=0.08$ ) and case 5 ( $St_{out}=6.0$ ). As argued below, these two peaks correspond to indirect and direct modulation mechanisms, respectively, and will be discussed in § 3.3.

Additional qualitative evidence of VLSM enhancement can be seen in figure 10, which provides a representative snapshot of the streamwise velocity fluctuation in the $x{-}z$ plane at $y^{+}=157$ . Comparing the snapshots in figure 10(c,e) and figure 10(a), it may not be easy to detect VLSM modulation. However, when the flow field is filtered by a threshold $\unicode[STIX]{x1D706}_{x}>10h$ and $\unicode[STIX]{x1D706}_{z}>0.75h$ , as displayed in figure 10(b,d,f), it is evident that VLSMs are stronger and more coherent in case 2 (figure 10 d) and case 5 (figure 10 f) as compared to single-phase flow. This is a more qualitative confirmation of the results shown in figures 7, 8 and 9. We now turn our investigation to understanding why the VLSM modulation appears to have two distinct peaks in Stokes number – in particular, demonstrated by case 2 and case 5.

3.3 Two mechanisms of VLSM enhancement by particles

In order to verify the above hypothesis that the VLSM enhancement in case 2 is due to particle modulation of LSMs in the inner flow (we refer to this as indirect modulation of VLSMs) whereas in case 5 it is due to the particle direct modulation on the VLSMs in the outer flow, we perform a conditional numerical test to identify the particles’ effective region of influence regarding VLSM enhancement by artificially applying the particle feedback force separately in only one of three locations: (1) the viscous sublayer ( $y^{+}<15$ ), (2) the regeneration cycle region ( $15<y^{+}<100$ ), or (3) the outer flow ( $y^{+}>100$ ). The premultiplied two-dimensional energy spectrum of streamwise velocity $k_{x}k_{z}\unicode[STIX]{x1D6F7}_{u^{\prime }u^{\prime }}$ is shown in figure 11 as a function of $y^{+}$ . Compared with figure 7(a), it can be seen that the VLSMs only experience enhancement in case 2 when particle coupling is included in the regeneration cycle region. For case 5, on the other hand, the opposite is true: VLSM enhancement is found only when particle coupling effects are included in the outer region. Both of these effects are observed throughout the entire range of $y^{+}$ . The tests of two-way coupling applied for $y^{+}<15$ are not shown, but we find the spectrum in the range $3.77h<\unicode[STIX]{x1D706}_{x}<6.28h$ ( $2073<\unicode[STIX]{x1D706}_{x}^{+}<3454$ ) and $0.78h<\unicode[STIX]{x1D706}_{z}<1.0h$ ( $429<\unicode[STIX]{x1D706}_{z}^{+}<550$ ) is stronger than in the single-phase flow, but shorter and narrower than the streamwise and spanwise scale of the second peak of the TKE spectrum.

Figure 11.

Premultiplied two-dimensional spectrum $k_{x}k_{z}\unicode[STIX]{x1D6F7}_{u^{\prime }u^{\prime }}/u_{\unicode[STIX]{x1D70F}}^{2}$ as a function of $\unicode[STIX]{x1D706}_{x}$ and $\unicode[STIX]{x1D706}_{z}$ in the wall-normal direction $y$ . An isosurface of $0.1$ times the common maximum value based on the single-phase flow is illustrated. (a,c) Only with particle coupling in the outer flow for case 2 and case 5. (b,d) Only with particle coupling in the regeneration cycle region for case 2 and case 5.

Figure 12.

Production contribution to the streamwise TKE budget and Reynolds shear stress budget in spectral space normalized by $u_{\unicode[STIX]{x1D70F}}^{3}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ : (a–c)  $\hat{P}_{11}$ and (d–f)  $\hat{P}_{12}$ ; (a,d) case 1, (b,e) case 2 and (c,f) case 5.

In spectral space, the modulation of LSMs and VLSMs by a dispersed phase is at least partially related to the direct influence on velocity fluctuations, which in turn can modify the production of TKE and/or Reynolds shear stress. This is demonstrated in figure 12, where we present the $\overline{u^{\prime }u^{\prime }}$ spectral production term $\hat{P}_{11}=-\langle \hat{u} ^{\prime }(k_{z},y){\hat{v}^{\prime }}^{\ast }(k_{z},y)\rangle \,\text{d}U/\text{d}y$ as well as the $\overline{u^{\prime }v^{\prime }}$ spectral production term $\hat{P}_{12}=-\langle \hat{v}^{\prime }(k_{z},y){\hat{v}^{\prime }}^{\ast }(k_{z},y)\rangle \,\text{d}U/\text{d}y$ as functions of $\unicode[STIX]{x1D706}_{z}^{+}$ and the wall-normal direction $y^{+}$ for two different Stokes numbers both previously seen to enhance VLSMs: the low-Stokes-number case 2 and high-Stokes-number case 5. These are shown in comparison with single-phase flow case 1. Throughout the wall-normal direction, $\hat{P}_{11}$ is positive whereas $\hat{P}_{12}$ is negative. Comparing figures 12(a–c) with figures 8(a,b,e), respectively, we find that they have a similar overall shape, and the bimodal spectrum appears both in the premultiplied $u$ -spectra as well as the production term $\hat{P}_{11}$ , which is enhanced in case 2 and case 5 in comparison with the single-phase flow. In regards to $\hat{P}_{12}$ , a similar overall shape as compared to premultiplied $uv$ -cospectra (figure not shown) is observed even though the bimodal spectrum is not as obviously established as that for $\hat{P}_{11}$ . The intensity of $\hat{P}_{12}$ ( $\hat{P}_{12}$ is negative in the domain) is weakened in case 2 and case 5 in comparison with the single-phase flow by the presence of particles. Thus figure 12 indicates that production of $\overline{u^{\prime }u^{\prime }}$ is enhanced at the heights and wavenumbers associated with VLSMs, while at the same time, particularly for case 5, the production of Reynolds shear stress is diminished at the same wavelengths and heights.

Figure 13.

Particle feedback term $\unicode[STIX]{x1D6F9}_{11}$ contribution to the streamwise TKE budget. (a) Mean value of all modes in the inner and outer layers as a function of Stokes number, normalized by $u_{\unicode[STIX]{x1D70F}}^{3}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ and bulk mass fraction $\overline{\unicode[STIX]{x1D6F7}_{m}}$ . (b–f) Contribution of $\hat{P}_{11}$ in spectral space of cases 2–6 as functions of spanwise wavelength and wall-normal direction, normalized by $u_{\unicode[STIX]{x1D70F}}^{3}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ and local mass fraction $\overline{\unicode[STIX]{x1D6F7}_{m}}(y)$ .

Figure 14.

Particle feedback term $\unicode[STIX]{x1D6F9}_{12}$ contribution to the Reynolds shear stress budget. Figure legend is the same as figure 13.

In addition to the modifications to streamwise TKE and Reynolds shear stress production, particles can also act as a direct source/sink in the spectral TKE and Reynolds stress budgets. In the spectral energy budget, particle sources to the $\overline{u^{\prime }u^{\prime }}$ budget are denoted as $\hat{\unicode[STIX]{x1D6F9}}_{11}=\text{Re}\langle \hat{F_{x}^{\prime }}(k_{z},y){\hat{u} ^{\prime }}^{\ast }(k_{z},y)\rangle$ and to the $\overline{u^{\prime }v^{\prime }}$ budget as $\hat{\unicode[STIX]{x1D6F9}}_{12}=\text{Re}\langle \hat{F_{x}^{\prime }}(k_{z},y){\hat{v}^{\prime }}^{\ast }(k_{z},y)+\hat{F_{y}^{\prime }}(k_{z},y){\hat{u} ^{\prime }}^{\ast }(k_{z},y)\rangle$ , where $\text{Re}$ stands for the real part and $\hat{F}$ is the Fourier transform of the particle coupling force. The mean value of $\unicode[STIX]{x1D6F9}_{11}=F_{x}^{\prime }u^{\prime }$ and $\unicode[STIX]{x1D6F9}_{12}=F_{x}^{\prime }v^{\prime }+F_{y}^{\prime }u^{\prime }$ (where $F_{i}^{\prime }$ , $i=x,y,z$ , is the fluctuation of the particle feedback force on the carrier phase) in the inner layer and outer layer of all modes normalized by the bulk mass fraction $\unicode[STIX]{x1D6F7}_{m}$ , is shown in figure 13(a) and figure 14(a), respectively. In both the inner layer and outer layer, the sign of $\unicode[STIX]{x1D6F9}_{11}$ is positive for case 2 whereas it becomes negative for cases 3–6. This is opposite when compared to $\unicode[STIX]{x1D6F9}_{12}$ , indicating that $\unicode[STIX]{x1D6F9}_{11}$ and $\unicode[STIX]{x1D6F9}_{12}$ play opposite roles in the streamwise TKE budget and Reynolds stress budgets for the same Stokes number. In the spectral energy budget, the particle source to the $\overline{u^{\prime }u^{\prime }}$ budget is denoted by $\hat{\unicode[STIX]{x1D6F9}}_{11}$ and to the $\overline{u^{\prime }v^{\prime }}$ budget by $\hat{\unicode[STIX]{x1D6F9}}_{12}$ (both normalized by local particle mass fraction $\unicode[STIX]{x1D6F7}_{m}$ in figures 13 and 14). With increasing Stokes number as shown in figure 13(b–f) and figure 14(b–f), the regions of the highest magnitudes of $\hat{\unicode[STIX]{x1D6F9}}_{11}$ and $\hat{\unicode[STIX]{x1D6F9}}_{12}$ shift from the inner layer to the outer layer, and at the same time from low to high wavelength. In particular, it is found that, for case 2, a region of large positive $\hat{\unicode[STIX]{x1D6F9}}_{11}$ (figure 13 b) appears in the inner layer with wavelengths associated with LSMs, whereas in case 5, a region of large positive $\hat{\unicode[STIX]{x1D6F9}}_{12}$ region instead develops at wavelengths associated with VLSMs (figure 14 e). In contrast, case 2 exhibits a negative source of $\hat{\unicode[STIX]{x1D6F9}}_{12}$ at these wavelengths (figure 14 b), but for case 5 there is at the same time a noticeable change in the sign of the contribution and location of $\hat{\unicode[STIX]{x1D6F9}}_{11}$ (figure 13 e). This picture is consistent with the conditional tests, i.e. case 2 works with LSMs in the inner layer whereas case 5 works with VLSMs in the outer layer. This nearly opposite behaviour indicates that there might be two underpinning mechanisms of VLSM enhancement induced by low- and high-Stokes-numbers particles.

In particle-laden flow, Richter (Reference Richter2015) finds that the particle-induced, Stokes-number-dependent source/sink of TKE varies with wavelength, and is possibly associated with the particle clusters themselves (Capecelatro et al. Reference Capecelatro, Desjardins and Fox2018). Focusing exclusively on the inner region, Wang & Richter (Reference Wang and Richter2019) find that low-inertia particles enhance LSMs whereas high-inertia particles attenuate LSMs, which can help explain the positive particle feedback $\hat{\unicode[STIX]{x1D6F9}}_{11}$ for case 2 (figure 13 b) but negative for case 5 (figure 13 e). In addition, as shown in figure 14(b), the particle source $\hat{\unicode[STIX]{x1D6F9}}_{12}$ in case 2 always attenuates the generation of $\overline{u^{\prime }v^{\prime }}$ . Therefore, in case 2, the positive feedback $\hat{\unicode[STIX]{x1D6F9}}_{11}$ in the inner region is the most likely mechanism responsible for the enhancement of VLSMs in the outer region. This process of particles inducing upscale energy transfer (or a reverse cascade), tending to build up the energy level at high wavelengths due to the modulation of small-scale turbulent motions, is also observed in homogeneous turbulence (see Elghobashi & Truesdell Reference Elghobashi and Truesdell1993; Carter & Coletti Reference Carter and Coletti2018) and in turbulent Couette flow (see Richter Reference Richter2015).

At low Stokes number ( $St^{+}=O(1)$ ), Klinkenberg et al. (Reference Klinkenberg, Sardina, De Lange and Brandt2013) found that the disturbance energy needed to induce turbulence is low at small mass fractions (i.e.  $\overline{\unicode[STIX]{x1D6F7}_{m}}=0.02{-}0.06$ ), whereas it is high at large mass fraction (i.e.  $\overline{\unicode[STIX]{x1D6F7}_{m}}=0.138$ ), indicating a subtle dependence of two-way coupling effects on mass fraction at low Stokes number. In the current context, this understanding can be used to better interpret and understand the indirect modulation of VLSMs by low-Stokes-number particles. Figure 15 shows one-dimensional $u$ -spectra $k_{z}\unicode[STIX]{x1D719}_{u^{\prime }u^{\prime }}(k_{z})$ with increasing mass fraction ranging from $\overline{\unicode[STIX]{x1D6F7}_{m}}=1.2\times 10^{-2}$ to $12\times 10^{-2}$ , all compared with the unladen flow. We observe that the modulation of VLSMs as a function of mass fraction is consistent with the behaviour shown by Klinkenberg et al. (Reference Klinkenberg, Sardina, De Lange and Brandt2013): VLSMs are slightly enhanced by a small mass fraction of low-Stokes-number particles ( $\overline{\unicode[STIX]{x1D6F7}_{m}}=1.2\times 10^{-2}$ in figure 15 b) and significantly promoted by increasing the mass fraction to $\overline{\unicode[STIX]{x1D6F7}_{m}}=2.4\times 10^{-2}$ (figure 15 c). With further increases in mass fraction, however ( $\overline{\unicode[STIX]{x1D6F7}_{m}}=12\times 10^{-2}$ in figure 15 d), this enhancement begins to diminish. Again, the non-monotonic response of LSM regeneration found in Klinkenberg et al. (Reference Klinkenberg, Sardina, De Lange and Brandt2013) and Wang & Richter (Reference Wang and Richter2019) appears to be linked to the underlying non-monotonic response of VLSM enhancement to low-Stokes-number particles.

Figure 15.

Premultiplied one-dimensional $u$ -spectra as functions of spanwise wavelength and wall-normal direction, normalized by $u_{\unicode[STIX]{x1D70F}}^{2}$ . (a) Single-phase flow case 1. (b,d) Stokes number $St^{+}=2.42$ of case 2 with three mass fractions: (b)  $\overline{\unicode[STIX]{x1D6F7}_{m}}=1.2\times 10^{-2}$ , (c)  $\overline{\unicode[STIX]{x1D6F7}_{m}}=2.4\times 10^{-2}$ , and (d)  $\overline{\unicode[STIX]{x1D6F7}_{m}}=14\times 10^{-2}$ .

Finally, in contrast to case 2, it is more straightforward to understand the VLSM modulation in case 5. As shown in figure 14(e), throughout the whole domain we observe that $\hat{\unicode[STIX]{x1D6F9}}_{12}$ for case 5 always has a positive contribution to the $\overline{u^{\prime }v^{\prime }}$ budget and is at the same spatial locations as the VLSMs of the $u$ -spectra (seen also at $Re_{\unicode[STIX]{x1D70F}}=950$ in case 9; figure not shown). On the other hand, $\hat{\unicode[STIX]{x1D6F9}}_{11}$ for case 5 in figure 13(e) tends to suppress the generation of $\overline{u^{\prime }u^{\prime }}$ . This ultimately results in $\hat{\unicode[STIX]{x1D6F9}}_{12}$ exerted in the outer region as being the most likely explanation for the enhancement of VLSMs. As the source of Reynolds shear stress $-\overline{u^{\prime }v^{\prime }}$ , the ‘upwelling’ and ‘downwelling’ cycles at very large scales (Adrian & Marusic Reference Adrian and Marusic2012) are directly enhanced by the presence of high-inertia particles ( $St_{out}=6$ of case 5 at $Re_{\unicode[STIX]{x1D70F}}=550$ and $St_{out}=8.2$ of case 10 at $Re_{\unicode[STIX]{x1D70F}}=950$ ) in the outer flow. These very large upwelling/downwelling structures further extract energy from the mean flow by working with local mean shear, as the production of streamwise turbulent kinetic energy budget ( $P_{11}=-\overline{u^{\prime }v^{\prime }}\,\text{d}U/\text{d}y$ ); see for example the physical explanation in Nezu (Reference Nezu2005).