3.1 Influence of particles on the flow stability

As will be further discussed in § 4.1, particles tend to accumulate in the large-scale vortex regions in Couette flow and in low-speed streak regions in channel flow. In order to understand how particles enhance or reduce the flow stability from their preferential spatial distribution, we carried out some simulations with specific flow configurations.

Figure 3. Particle effect on flow stability. (a,c,e,g) Couette flow starting from a fully turbulent regime. The turbulent state is stable under a single-phase condition; even when the streamwise velocity perturbations are suppressed, the flow recovers its fully turbulent nature. Adding particles damps the velocity fluctuations and makes the flow laminar. ——  $Re_{b}=430$ , single-phase flow removes $u^{\prime }$ ; – – –  $Re_{b}=430$ , $\unicode[STIX]{x1D6F7}=5\,\%$ and $L_{y}/d=10$ ; – - – -  $Re_{b}=455$ , single-phase flow removes $u^{\prime }$ ; $\cdots \cdots$   $Re_{b}=455$ , $\unicode[STIX]{x1D6F7}=10\,\%$ and $L_{y}/d=20$ . (b,d,f,g) Channel flow starting with a flow distribution according to (3.1) where an artificial streak is initially imposed. The single-phase flow tends towards the laminar state at $Re_{b}=2600$ which is above the laminar–turbulent transition. Adding a small number of particles in the flow triggers the transition to turbulence. —— single-phase flow; – – –  $\unicode[STIX]{x1D6F7}=0.5\,\%$ and $L_{y}/d=16$ ; – - – -  $\unicode[STIX]{x1D6F7}=0.75\,\%$ and $L_{y}/d=16$ ; $\cdots \cdots$   $\unicode[STIX]{x1D6F7}=0.5\,\%$ and $L_{y}/d=20$ .

The first test was done in Couette flow and it was inspired by the study of Hamilton et al. (Reference Hamilton, Kim and Waleffe1995). When the streamwise velocity perturbations were removed, while the linear streamwise velocity profile and streamwise vortices were maintained from a fully turbulent simulation, the authors observed that the flow evolved again to the fully turbulent regime. In a similar way, we considered for the initial configuration a snapshot from a steady single-phase turbulent Couette flow simulation at $Re_{b}=500$ where the large-scale $x$ -independent streak is maximum (from $M(0,\unicode[STIX]{x1D6FD})$ defined in (4.1)). We removed the streaks, and abruptly decreased the Reynolds number to $455$ and $430$ in two separate simulations.

The temporal evolution of root-mean-square (r.m.s.) velocity fluctuations shown in figure 3(a,c,e) as well as the mean velocity profile (not shown here) suggest that, despite the initial destabilization, the flow recovers its fully turbulent features after $200$ time units. Figure 3(g) shows the contours of velocity magnitude for single-phase flow; the left figure is taken at $t=0$ after removing $u^{\prime }$ and its evolution after $500$ time units can be seen in the right figure. The flow field plotted after nearly five regeneration cycles cannot be distinguished from the initial fully turbulent flow. Therefore the streamwise vortices, that were initially maintained, were strong enough to generate streaks through the lift-up effect and resume the regeneration cycle.

The main effect of the presence of particles in the Couette flow is stabilizing in nature. When we added small particles ( $L_{y}/d=20$ ) at $\unicode[STIX]{x1D6F7}=10\,\%$ and decreased the Reynolds number down to $455$ , without removing the streamwise velocity perturbations, the flow became laminar. Adding larger particles ( $L_{y}/d=10$ ) at $\unicode[STIX]{x1D6F7}=5\,\%$ and decreasing the Reynolds number to $430$ , the flow velocity fluctuations were significantly damped, and the flow became laminar after staying quasi-turbulent for around six regeneration cycles. These observations suggest that the particles were mainly enhancing dissipation in the flow.

The test of flow stability for the channel flow configuration has been done using an artificial finite-amplitude low-speed streak that was supplemented to a mean flow profile corresponding to wall-bounded turbulence. We used the same base flow as Schoppa & Hussain (Reference Schoppa and Hussain2002), who studied the streak transient growth mechanism in a two-dimensional streak configuration. The base flow is

(3.1) $$\begin{eqnarray}u(y,z)=U_{0}(y)+(\unicode[STIX]{x0394}u/2)\cos (\unicode[STIX]{x1D6FD}_{s}z)g(y)\end{eqnarray}$$

in the streamwise direction, and $v=w=0$ in the wall-normal and spanwise directions. $U_{0}(y)$ is the mean velocity and $g(y)$ is an amplitude function which satisfies the no-slip condition at $y=0$ and localizes the streak velocity defect at a single wall. A ‘single-sided’ turbulent mean velocity profile is imposed, analogous to that observed in minimal channel turbulence (Jiménez & Moin Reference Jiménez and Moin1991), with a parabolic profile $U_{lam}$ in the laminar top half of the channel, and a turbulent Reichardt profile $U_{turb}$ that respects the near-wall turbulence statistics, in the bottom half:

(3.2) $$\begin{eqnarray}U_{0}(y)=\left\{\begin{array}{@{}l@{}}\displaystyle U_{lam}=U_{c}[1-((y/h)-1)^{2}],\quad y_{m}\leqslant y\leqslant 2h,\\ \displaystyle U_{turb}=u_{\ast }\left[2.5\ln (1+0.4y/\unicode[STIX]{x1D6FF})+7.8\left(1-\text{e}^{(-y/11\unicode[STIX]{x1D6FF})}-\frac{y}{11\unicode[STIX]{x1D6FF}}\text{e}^{(-y/3\unicode[STIX]{x1D6FF})}\right)\right],\\ \hspace{30.00005pt}0\leqslant y<y_{m}.\end{array}\right.\end{eqnarray}$$

The friction velocity $u_{\ast }=\sqrt{\unicode[STIX]{x1D70F}_{w}/\unicode[STIX]{x1D70C}}$ and viscous length scale $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D708}/u_{\ast }$ are calculated using a wall shear stress estimated from Dean’s empirical correlation ( $C_{f}\equiv 2\unicode[STIX]{x1D70F}_{w}/\unicode[STIX]{x1D70C}\overline{u}^{2}=0.073Re_{b}^{-0.25}$ ) in a fully turbulent channel flow. For a given flow rate $Q$ , this leads the friction velocity to be $u_{\ast }=Q\sqrt{0.0365(Q/\unicode[STIX]{x1D708})^{-0.25}}/(2h)$ . The two profiles $U_{turb}$ and $U_{lam}$ are matched at a wall-normal distance $y_{m}$ in the turbulent half, with $y_{m}$ and $U_{c}$ determined so that the mean flow velocity and vorticity are continuous at the matching point, i.e.  $U_{lam}(y_{m})=U_{turb}(y_{m})$ and $\text{d}U_{lam}/\text{d}y\mid _{y=y_{m}}=\text{d}U_{turb}/\text{d}y\mid _{y=y_{m}}$ . Consequently, at $Re=Q/\unicode[STIX]{x1D708}=2600$ , $y_{m}=0.918h$ and $U_{c}=1.2Q/(2h)$ .

The function $g(y)\sim y\cdot \text{e}^{-\unicode[STIX]{x1D702}y^{2}}$ accounts for the streak velocity defect, and has been normalized to unity with $\unicode[STIX]{x1D702}$ specified such that the streak velocity defect $\unicode[STIX]{x0394}u$ and normal vorticity $\unicode[STIX]{x1D6FA}_{y}\mid _{max}=\unicode[STIX]{x1D6FD}_{s}\unicode[STIX]{x0394}u/2$ exhibit a plateau in the range $y^{+}=10-30$ , consistent with the observed lifted streaks and $\unicode[STIX]{x1D714}_{y,rms}$ statistics. Note that the amplitude function $g(y)$ in (3.1) determines the strength of the local streak upper bound $u(y)$ shear layer (e.g. local maximum of $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y$ ) residing on the crest of the lifted streak. Instability growth rates for the dominant sinuous modes are found to be relatively insensitive to the strength of this shear layer and hence to the amplitude function $g(y)$ . The value $\unicode[STIX]{x1D702}=20$ was used similarly to Schoppa & Hussain (Reference Schoppa and Hussain2002). The streak spanwise wavenumber $\unicode[STIX]{x1D6FD}_{s}$ in (3.1) is chosen as $2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FD}_{s}^{+}=100$ , corresponding to the well-accepted average spanwise spacing of adjacent low-speed streaks observed in many experimental and numerical studies.

Figure 3(h), left, shows the velocity magnitude contours of the flow according to (3.1). Schoppa & Hussain (Reference Schoppa and Hussain2002) found that this single-phase flow is stable and the energy of the artificial streaky perturbation will vanish in time due to viscous dissipation. A spanwise perturbation, following a sinuous profile in the flow direction, is necessary to trigger the growth in time of the perturbation. In the absence of such spanwise initial coherent motion, figure 3(b,d,f) confirms that the perturbation (3.1) is damped over time when the Reynolds number is equal to $2600$ (above the transition threshold).

Unlike the Couette flow test, the particles in this particular channel flow were not seeded throughout the entire domain. Of course this would lead the flow to undergo the transition to turbulence. Instead, we seeded a small number of particles only in the low-speed artificial streak region ( $u(y,z)/U_{bulk}\leqslant 1.5$ ) keeping the flow Reynolds number $Re_{b}=2600$ . Two different local concentrations (particles-to-streak volume) were considered ( $\unicode[STIX]{x1D6F7}=0.5$ and $0.75\,\%$ for the case of $L_{y}/d=16$ and $\unicode[STIX]{x1D6F7}=0.5\,\%$ for the case of $L_{y}/d=20$ ). In all cases, the particle presence triggered the transition to turbulence (this can be evidenced by the level of the r.m.s. velocity signals), and the particles were found after ${\sim}100$ time units spread all over the simulation domain. Figure 3(h), right, shows the contours of velocity magnitude for suspension flow in the case of $L_{y}/d=16$ with $\unicode[STIX]{x1D6F7}=0.75\,\%$ after $250$ time units. We can observe a quasi-fully turbulent state at $Re_{b}=2600$ (instead of the one-sided wall turbulence observed in single-phase flow noted by Jiménez & Moin Reference Jiménez and Moin1991).

Figure 4. Flow vorticity induced by a layer of particles seeded in the plane $y/d=0.8$ (near the wall) in the same flow configuration as 3(h). The total volume concentration is $\unicode[STIX]{x1D6F7}=0.5\,\%$ , and the size ratio $L_{y}/d=16$ is used. The snapshots are taken at three time instants $t=0.2$ , $2.7$ and $29.5$ (scaled by $h/U_{bulk}$ ) which correspond to $t^{+}=2$ , $27$ and $295$ (scaled by $\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}^{2}$ ).

The influence of the particles on the transition is also illustrated in figure 4, showing the temporal evolution of the streamwise vorticity generated by the particles. The initial condition is equivalent to figure 3(h), except that $60$ particles were initially seeded in a plane parallel to the wall (instead of being located in the artificial streak). At the first instants (figure 4 a), streamwise vorticity is generated around finite-size particles due to the secondary flows occurring at finite $Re_{p}$ . As time goes on and particles move, this streamwise vorticity is stretched in the streamwise direction, as in figure 4(b). Furthermore, these structures are tilted due to the mean shear through the streamwise vorticity generation term $-(\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}x)(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y)$ (explained in § 5) which is large near the wall. They further interact with each other to form larger-scale streamwise vortical structures, as shown in figure 4(c). Clearly the generated vortical structures are comparable to the near-wall vorticity layers induced by large-scale vortices, which are the essential ingredients of the regeneration cycle for channel flow.

4.1 Particle dispersion

Figure 5(a,b) shows the average particle distribution over the cross-sectional plane. The contours of concentration are averaged over $80$ time units ( $h/U_{bulk}$ ) whereas the wall-normal concentration profiles were averaged over $500$ time units. The maximum concentration is located in the core region of the Couette flow, whereas two peaks can be observed near the walls of the channel flow. The average concentration profiles are the result of a balance between the lift force on the finite-size particles, the hydrodynamic repulsion from the wall and the shear-induced turbulent diffusion. To be more specific, in laminar plane Couette flow (linear flow), a finite-size neutrally buoyant particle exhibits a (rotational and translational) slip with respect to the local flow field. The pressure is larger on the side of the particle closer to the wall, and thereby the particle is pushed away from the wall (Ho & Leal Reference Ho and Leal1974). When the velocity profile is not linear (quadratic for Poiseuille flow), the particle experiences an inertia-induced lift force (Asmolov Reference Asmolov1999), resulting from its perturbation (as a rigid body) to the quadratic velocity profile (see a review paper by Matas, Morris & Guazzelli Reference Matas, Morris and Guazzelli2004). This force is oriented towards the wall. This is known as the Segré–Silberberg or pinch-off effect. The particle also experiences a hydrodynamic viscous repulsion when it is very close to the wall. The balance between these two opposite forces leads to equilibrium positions (in laminar channel flow) that depend on the particle Reynolds number, and yield segregation of a suspension in channel flow (see Matas et al. Reference Matas, Morris and Guazzelli2004; Loisel et al. Reference Loisel, Abbas, Masbernat and Climent2015)

Figure 5. (a,b) Particle distribution in the cross-section ( $y{-}z$ plane) of Couette $C500-5$ and pressure-driven $P2600-5$ flows respectively. The concentration contours are averaged in the streamwise direction, over 80 time units. The corresponding concentration profiles, averaged over 500 time units, are also shown for ——  $C500-5$ , – – –  $C500-10$ in (a), and ——  $P2600-1$ , – – –  $P2600-5$ in (b).

Figure 6. (a,b) Particle distribution in a plane parallel to the wall ( $x{-}z$ plane) in Couette $C500-5$ and pressure-driven $P2600-5$ flows respectively. The concentration contours taken at $y/L_{y}=0.2$ are averaged over $80$ time units. The isolines show $u^{\prime }/U_{bulk}$ in the $x$ – $z$ plane, where dashed lines stand for negative and solid lines for positive values, the interval being $0.04$ in (a) and $0.03$ in (b). The spanwise variation of the fluid (black) and particle (dashed blue) fluctuation velocity $u^{\prime }/U_{bulk}$ averaged in the streamwise direction is also plotted on the right sides of (a,b), separately. (c,d) The streamwise velocity profiles of —— fluid phase (black line), particle phase (blue dashed line) and slip (red dot-dashed line) velocities (defined as the fluid velocity with the particulate phase velocity subtracted). The cross symbols indicate the velocity profile for the single-phase flow.

The instantaneous spatial distribution of particles is shown together with the streamwise velocity fluctuation contours in the $x$ – $z$ plane (figure 6 a,b). These figures show a strong correlation between the particle spatial distribution and the flow coherent structures. In Couette flow, particles are pulled away from the walls by turbulent ejection and towards the wall by the sweep events. On average, they are more present in the sweep and core regions in Couette flow. In channel flow, the particles are accumulated in the ejection region, near the wall. This can be understood as follows: the inertial lift force drives the particles to be preferentially located near the walls, where high- and low-speed flow regions are encountered. Sweeps are regions of spanwise divergence near the wall which drive the particles to leave the high-speed (sweep) region towards the low-speed (ejection) region. When a particle is dragged away from the wall by an ejection event, there is an opposite hydrodynamic contribution that reduces this effect. Lubrication force acting on a particle moving away from a flat wall is attractive to the wall (and decreases when the particle is far from the wall). The resulting particle migration should depend on the particle size and the distance from the wall. Thus the outward wall-normal flow in the ejection region is most probably insufficient to pull the neutrally buoyant particles, leading to particle accumulation in that region.

The profiles of the fluid, particle and slip velocity (fluid velocity minus particulate-phase velocity) are shown in figure 6(c,d). Also, the spanwise variation of the slip velocity is shown in this figure. The velocity of the particulate phase is always smaller than the local fluid velocity in both Couette and pressure-driven flow configurations (on average particles are lagging the flow), especially near the walls, in both flow configurations.

4.2 Quadrant analysis of velocity r.m.s.

The coherent structures are the major contributors to the Reynolds shear stress. They play an essential role in the active motion of wall turbulence. According to the quadrant analysis, the flow fluctuations can be divided into $Q1(u^{\prime }>0,v^{\prime }>0)$ , $Q2(u^{\prime }<0,v^{\prime }>0)$ , $Q3(u^{\prime }<0,v^{\prime }<0)$ and $Q4(u^{\prime }>0,v^{\prime }<0)$ . $Q2$ and $Q4$ correspond to the ejection and sweep events respectively, $Q1$ and $Q3$ contain the outward and inward interactions respectively (see a recent review on quadrant analysis by Wallace Reference Wallace2016). We give details of the impact of particles on the different Reynolds stress components, by considering separately the different contributions according to the quadrant analysis.

Figure 7. Velocity r.m.s. of single- and two-phase flows in different quadrants: (a–c) Couette flow; (d–f) channel flow. The line style denotes the quadrant: – – –  $Q1$ , ——  $Q2$ , – - – -  $Q3$ , $\cdots \cdots$   $Q4$ . The symbols refer to the flow concentration: open circles for single-phase flow and filled circles for $\unicode[STIX]{x1D719}=5\,\%$ . (a,d), (b,e), (c,f) $-\overline{u^{\prime }v^{\prime }}$ , $\overline{u^{\prime }u^{\prime }}$ and $\overline{v^{\prime }v^{\prime }}$ , respectively.

The r.m.s. velocity profiles are displayed in figure 7, according to the quadrant analysis, for both single-phase and suspension flow at $\unicode[STIX]{x1D6F7}=5\,\%$ . The Reynolds stress components are not significantly influenced when particles are present in Couette flow at $5\,\%$ (figure 7 a–c). However, in channel flow, profiles of the Reynolds shear stress (figure 7 d) reveal that the particles enhance significantly the shear stress in the sweep ( $Q4$ ) part of the logarithmic region, where $Q2$ and $Q4$ events are dominant. The streamwise Reynolds stress is decreased by the particles, especially in the ejection regions, whereas the wall-normal Reynolds stress is increased in both sweep and ejection regions. The peaks of the profiles are also closer to the wall.

Figure 8. As sketched in (a), the velocity r.m.s. of single- and two-phase flows in $Q2$ and $Q4$ are split into contributions near the particle surface ( $\overline{u_{p1}^{\prime }u_{p1}^{\prime }}$ within $a<r<1.5a$ using dot-dashed lines and $\overline{u_{p2}^{\prime }u_{p2}^{\prime }}$ within $1.5a<r<2a$ using dashed lines), and far from the particle surface ( $\overline{u_{f}^{\prime }u_{f}^{\prime }}$ within $r>2a$ using solid lines). The profiles in (b,c) are from Couette and channel flow respectively. Here the circles correspond to the ejection and the plus signs to the sweep events.

These turbulent stress modifications suggest a more isotropic turbulence in a channel flow compared to a single-phase flow, in agreement with previous numerical results obtained using the total profiles of the Reynolds stress components (see Shao, Wu & Yu Reference Shao, Wu and Yu2012; Loisel et al. Reference Loisel, Abbas, Masbernat and Climent2013; Picano, Breugem & Brandt Reference Picano, Breugem and Brandt2015; Fornari et al. Reference Fornari, Formenti, Picano and Brandt2016; Yu, Vinkovic & Buffat Reference Yu, Vinkovic and Buffat2016), the transfer between different directions being promoted by the Reynolds shear stress. Particular attention is drawn here to the reduction of the streamwise Reynolds stress component. Fornari et al. (Reference Fornari, Formenti, Picano and Brandt2016) related this observation to the fluid squeezed between the layer of particles near the wall and the wall itself. Shao et al. (Reference Shao, Wu and Yu2012) associated this with the weakening intensity of the large-scale streamwise vortices, a phenomenon that is not observed in Fourier space, as shown later in figure 10(d) and figure 9(e,f). In an attempt to understand why the streamwise Reynolds stress is decreased, we calculated the velocity fluctuations in the fluid far from and close to the particles separately, as sketched in figure 8(a). $\overline{u_{f}^{\prime }u_{f}^{\prime }}$ is averaged within the fluid region located at $r>2a$ relative to each particle centre. $\overline{u_{p1}^{\prime }u_{p1}^{\prime }}$ and $\overline{u_{p2}^{\prime }u_{p2}^{\prime }}$ are averaged in the neighbourhood of the particles $a<r<1.5a$ and $1.5a<r<2a$ , respectively. The profiles of $\overline{u_{f}^{\prime }u_{f}^{\prime }}$ , $\overline{u_{p1}^{\prime }u_{p1}^{\prime }}$ and $\overline{u_{p2}^{\prime }u_{p2}^{\prime }}$ are plotted in different quadrants in figure 8(b,c). The profiles in figure 6 indicate only the intensity of the events, and not their contribution to the total suspension flow fluctuations. Thereby they do not give an indication of the flow modulation by the particles, but they allow description of the particle behaviour compared to that of the fluid. In the sweep region, the velocity fluctuations near the particles and far from the particles $(r>2a)$ are almost the same. In the ejection region, the difference in the velocity fluctuations depends on the flow configuration. On the one hand in Couette flow, the particle velocity fluctuations are well below the fluid fluctuations. This should not have an a significant impact on the overall suspension since particles are rarely located in that region. On the other hand in pressure-driven flow, the particles are preferentially accumulated in the ejection region, where their velocity fluctuations are stronger than those of the fluid.

Figure 9. (a–c) Couette flow; (d–f) channel flow. (a,d) The one-dimensional streamwise and spanwise wavenumber energy spectra of the streamwise velocity $E_{uu}$ averaged in the wall-normal direction: (a) ——  $C500-0$ , – – –  $C500-5$ , – - – -  $C500-10$ ; (d) ——  $P2600-0$ , – – –  $P2600-5$ , – - – -  $P2600-10$ . Contour figures show the two-dimensional contours of the energy spectra. (b,e) Single-phase flow with $C500-0$ and $P2600-0$ . (c,f) Two-phase flow with $C500-5$ and $P2600-5$ .

4.3 Energy spectra

The average streamwise energy spectrum $E_{uu}^{\unicode[STIX]{x1D6F7}}$ is plotted in figure 9(a,d) as a function of both streamwise and spanwise wavelengths, for both suspension and single-phase flows. One advantage of FCM is that particles are represented in the fluid equations by smooth Gaussian envelope forcing, making this method well suited for spatial Fourier analysis of mixture flow.

Figure 10. Simultaneous temporal evolution of the friction coefficient $C_{f}$ and the near-wall streamwise vorticity, in Couette flow (a,b), and channel flow (c,d). (a,c) Plots of the summation of the amplitude squared of the $x$ -independent vortices ( $m=0$ ) for different spanwise wavenumbers ( $1\leqslant n\leqslant N_{z}/2$ ) and integrated in the near-wall region ( $y^{+}<15$ ). (b,d) The $x$ -independent vortices ( $m=0$ ) in the near-wall region ( $y^{+}<15$ ) as a function of spanwise wavelength ( $2L_{z}/N_{z}\leqslant \unicode[STIX]{x1D706}_{z}\leqslant L_{z}$ ). The line style indicates single-phase (solid) and two-phase (dashed line) flows: (a,b) ——  $C500-0$ , – – –  $C500-5$ ; (c,d) ——  $P2600-0$ , – – –  $P2600-5$ .

Figure 9(a,d) displays the energy spectra in both flow configurations and for both streamwise and spanwise wavenumbers averaged over the whole domain. Particles strengthen the energy of the flow structures at intermediate scales, especially in the streamwise direction of the channel flow. Moreover, particles hardly affect the energy of the large-scale structures of wavelength $3h<\unicode[STIX]{x1D706}_{x}<5.7h$ in Couette flow whereas particles increase the energy contained in $h<\unicode[STIX]{x1D706}_{x}<3.1h$ for channel flow. Note that the energy of the fluctuations at small scales in two dimensions ( $\unicode[STIX]{x1D706}_{x}<d_{p}$ and $\unicode[STIX]{x1D706}_{z}<d_{p}$ ) might be over-estimated by the numerical method. However, the energy contained in these scales obtained from two-dimensional modal analysis is $10^{-7}\,\%$ (respectively $10^{-4}\,\%$ ) of the energy of the largest scales in Couette (respectively channel) flow, and therefore the errors introduced by small scales on the analysis of energy spectra can be neglected.

The energy spectra in the streamwise direction are plotted as a function of the wall-normal position in figure 9(b,c,e,f). It is interesting to note in figure 9(b,c) that the most energetic large-scale motion in Couette flow is found in the range $20<y^{+}<40$ , which corresponds to the buffer layer region. The extent of the most energetic eddies is slightly shrunk towards the Couette flow centre by the presence of the particle.

In channel flow the most energetic flow structures ( $h<\unicode[STIX]{x1D706}_{x}<3h$ ) are located at $10<y^{+}<50$ as shown in figure 9(e,f). Contrary to Couette flow, finite-size particles in channel flow enhance the strength of moderate streamwise vortices in comparison to single-phase flow. The energy of these streamwise vortical structures subsequently increases the flow velocity gradient near the wall, as will be shown in figure 10. The energy modulation near the walls ( $y^{+}<20$ ) is due to the interaction of the particles with the streaks rather than their interaction with the large-scale vortices. Two indirect pieces of evidence may support this conclusion. First, particles in Couette flow do not generate significant modulation near the walls ( $y^{+}<20$ ) in figure 9(c) when particles are in large-scale vortices. Second, in channel flow, we also observe the generation of vortical energetic structures when particles are seeded in the bottom wall only with artificial streaks (as shown in figure 4 c).

5.1 Wall friction coefficient and streamwise vorticity

The friction coefficient is a dimensionless measure of the wall shear stress. The temporal evolution of the friction coefficient is displayed in figure 10. For Couette flow, the average friction coefficient and temporal fluctuations are slightly increased by the presence of particles, whereas for channel flow the increase of the average friction coefficient is more significant, and the fluctuation amplitude is slightly reduced. The increase of friction coefficient cannot be exclusively related to the increase of the suspension effective viscosity. Indeed the ratio of the time-averaged friction coefficient of the suspension to single-phase flow is around $1.4$ , whereas the viscosity increase due to the particle presence is around $1.14$ . Recent work from Costa et al. (Reference Costa, Picano, Brandt and Breugem2016) provided a theoretical prediction of the total suspension drag. Predicted $Re_{\unicode[STIX]{x1D70F}}$ is $103$ based on suspension viscosity, where $Re_{\unicode[STIX]{x1D70F}}$ is $109$ based on DNS in table 1.

Jiménez & Moin (Reference Jiménez and Moin1991) have explicitly shown that the maximum (in time) of the wall shear stress is synchronous with the maximum near-wall vorticity ( $0<y^{+}<10$ ). Using two-dimensional numerical simulations (neglecting the variation in the streamwise direction), Orlandi & Jiménez (Reference Orlandi and Jiménez1994) showed that the transport of fluid from longitudinal vortices to the high- and low-speed streaks is the origin of the higher wall friction in turbulent layers, especially in the sweep region where high-speed fluid is transported towards the wall. Therefore large-scale streamwise vortical structures control the near-wall velocity gradient. Figure 10(a,c) shows simultaneously the evolution of the wall friction coefficient and the summation over all of the spanwise wavenumbers for $x$ -independent vorticity $\hat{|\unicode[STIX]{x1D714}|}_{x,wall}^{2}\equiv \sum _{n=1}^{N_{z}/2}\int _{0}^{y^{+}=15}\hat{|\unicode[STIX]{x1D714}|}_{x}^{2}(0,n\unicode[STIX]{x1D6FD})\,\text{d}y$ , where the mode $(0,n\unicode[STIX]{x1D6FD})$ with $n\neq 0$ corresponds to an $x$ -independent structure. The cross-correlation between $C_{f}$ and $\hat{|\unicode[STIX]{x1D714}|}_{x,wall}^{2}$ is calculated following (5.1), where the prime denotes the temporal fluctuation of a quantity and the overline indicates the average over time:

(5.1) $$\begin{eqnarray}R=\frac{\overline{(C_{f})^{\prime }\cdot (\hat{|\unicode[STIX]{x1D714}|}_{x,wall}^{2})^{\prime }}}{(C_{f})_{rms}\cdot (\hat{|\unicode[STIX]{x1D714}|}_{x,wall}^{2})_{rms}}.\end{eqnarray}$$

The cross-correlation gives $R=0.51,0.49,0.63,0.68$ for the respective flow configurations $C500-0,C500-5,P2600-0,P2600-5$ . Clearly, there is a correlation in time between the instantaneous wall friction coefficient and the near-wall streamwise vorticity in all cases. The spectra of near-wall streamwise vorticity $\hat{|\unicode[STIX]{x1D714}|}_{x,wall}^{2}$ are shown in figure 10(b,d) for both single-phase and suspension flows. The ratio of the vorticity at small scales ( $\unicode[STIX]{x1D706}_{z}/L_{y}<0.2$ ) to the largest-scale streamwise vortices is much smaller in Couette flow than in channel flow (the ratios are $O(0.01)$ and $O(0.1)$ respectively). In two-phase flows, turbulence becomes more isotropic, because particles inject energy in small scales, which is transferred back to intermediate scales. This is for example the case in the work of Elghobashi & Truesdell (Reference Elghobashi and Truesdell1993), even though the origin of momentum transfer is not the same; there is slip between the phases in their work, whereas in our work the interactions are mainly due to the finite size of the particles. The streamwise vorticity is enhanced at low spanwise wavelengths ( $\unicode[STIX]{x1D706}_{z}/L_{y}<0.2$ , which corresponds to $\unicode[STIX]{x1D706}_{z}/d_{p}<4$ ) in both configurations. The enhancement is of one order of magnitude in Couette flow, and of two orders of magnitude in channel flow.

5.2 Reynolds shear stress

From the investigation of total energy input and dissipation rate, Kawahara & Kida (Reference Kawahara and Kida2001) evidenced the temporal evolution of spatial structures in a cyclic sequence consistent with the regeneration cycle proposed by Hamilton et al. (Reference Hamilton, Kim and Waleffe1995). A strong ejection event is followed by a gradual decrease of intensity over a certain period of time. The maximum (in time) of the Reynolds stress occurs when the dissipation rate is large along the periodic orbit. The quasi-periodicity of the turbulent events can be represented by the spatio-temporal evolution of the Reynolds stress $-\overline{u^{\prime }v^{\prime }}(y,t)$ across the Couette gap or channel height, as shown in figure 11. Note that the period of turbulent events is of $O(100)$ time units in both flows. This characterizes the time needed for the velocity fluctuations to become uncorrelated in time. The particle Stokes number which can be based on this time scale is very small compared to the one related to the shear.

Figure 11. Spatio-temporal evolution of the Reynolds shear stress ( $-\overline{u^{\prime }v^{\prime }}/U_{bulk}^{2}$ ) averaged in the homogeneous directions (streamwise and spanwise) in (a–d); its average within $5<y^{+}<30$ is shown in (e,f). (a,c,e) Couette flow and (b,d,f) channel flow. Panels (a,b) correspond to single-phase flows whereas (c,d) correspond to two-phase flows; (e) ——  $C500-0$ and – – –  $C500-5$ ; (f) ——  $P2600-0$ and – – –  $P2600-5$ .

For Couette flow, the maximum of the Reynolds stress is located in the centre of the gap. The two walls share one buffer layer and a couple of central large-scale vortices, with a strong coupling between the streaks near the walls. The low-speed streak near one wall ejects fluid to the other wall, acting there as a high-speed streak. It is revealed by figure 11(a,c,e) that neutrally buoyant particles have a negligible effect on both the intensity and intermittency of the Reynolds stress in the Couette flow configuration (Wang et al. Reference Wang, Abbas and Climent2017). The channel flow contains a log-law region and the central region is ruled by the velocity defect law. Figure 11(b) shows that the strongest shear stress bursts are located close to the channel walls, and that the frequency of these bursts is of the same order of magnitude as in Couette flow. In the presence of neutrally buoyant particles, the intensity of the Reynolds shear stress is enhanced, as shown in figure 11(d), and the frequency of these events is decreased, as shown in figure 11(f). The increase of Reynolds shear stress is closely correlated with the sweep events, as indicated in figure 7(d), making the friction coefficient and Reynolds shear stress fluctuations synchronous.

5.3 Streak formation: the lift-up mechanism

The streaks form on both sides of a vortex. Low-speed fluid is lifted up away from the wall by the vortex into a region of higher-speed fluid, producing a low-speed streak, while on the other side of the vortex, high-speed fluid is pushed towards the wall, creating a high-speed streak. Ellingsen & Palm (Reference Ellingsen and Palm1975) have shown, using a linear stability analysis, that the $x$ -independent streamwise perturbations grow linearly in time as $-v(\text{d}\overline{u}/\text{d}y)t$ (the so-called lift-up effect), making any shear flow $\overline{u}(y)$ unstable to $x$ -independent (transverse) perturbations. Consequently, in shear flows, the main linear mechanism for transient disturbance growth is the lift-up effect which produces high- and low-speed streaks in the streamwise velocity. Bech et al. (Reference Bech, Tillmark, Alfredsson and Andersson1995) stated that the inner shear layer is formed through the lift-up of low-speed streaks from the viscous sublayer; then the shear layers are coupled to an instantaneous velocity profile with inflectional character and they have been observed to become unstable and break up into chaotic motion, so-called ‘bursting’. The lift-up effect or advection was identified as a robust mechanism for the generation of the streaky motions in both transitional and turbulent flows (Ellingsen & Palm Reference Ellingsen and Palm1975; Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Del Álamo & Jiménez Reference Del Álamo and Jiménez2006).

Figure 12. Spatio-temporal evolution of the lift-up term ( $-v\text{d}\overline{u}/\text{d}y$ ) scaled by $U_{bulk}^{2}/h$ . The green isoline of the Reynolds shear stress ( $-\overline{u^{\prime }v^{\prime }}/U_{bulk}^{2}$ ) is added on the top of these figures; the interval in (a,c) is $0.003$ , and the interval in (b,d) is $0.005$ . (a,c,e) Couette flow and (b,d,f) channel flow. (a,b) Single-phase flows; (c,d) two-phase flows. The average of the lift-up term within $5<y^{+}<30$ is shown in (e,f): (e) ——  $C500-0$ and – – –  $C500-5$ ; (f) ——  $P2600-0$ and – – –  $P2600-5$ .

Klinkenberg et al. (Reference Klinkenberg, Sardina, De Lange and Brandt2013) have shown that small pointwise inertial particles do affect the transition to turbulence, not by altering the lift-up effect but rather by modifying the dynamics of the oblique waves necessary for the streak regeneration and breakdown. In order to show whether finite-size particles modify the lift-up term, the contours of $-v\,\text{d}\overline{u}/\text{d}y$ are displayed in figure 12 together with the isolines of the Reynolds shear stress (from figure 11). The lift-up term is important near the walls in both flow configurations. For Couette flow, the contours shown in figure 11(a,c) are not significantly modified by the presence of the particles. Based on figure 11(a,c), which plots the averaged value within $5<y^{+}<30$ , we can observe that the strongest effects are nearly the same with or without particles, whereas the weakest effects are enhanced by the particles. However, in channel flow within the buffer layer ( $5<y^{+}<30$ ), the particles not only enhance the lift-up significantly, but also let it act continuously (the lift-up effect is less frequent in suspension flow, as shown in figure 11 f).

5.4 Streak breakdown: modal decomposition of the fluctuating energy

The subsequent process is the instability of $x$ -independent streaks, the so-called streak breakdown. Hamilton et al. (Reference Hamilton, Kim and Waleffe1995) have shown that it is the instability of the streaks (through a nonlinear process) which causes breakdown. We investigated the temporal evolution of the energy contained in the dominant flow fluctuation modes, since it can give evidence on the dynamics of the streak breakdown process, and on the particle modulation of this process.

Figure 13. Modal decomposition as in (4.1): —— mode $M(0,\unicode[STIX]{x1D6FD})$ ; $\cdots \cdots$  mode $M(\unicode[STIX]{x1D6FC},0)$ . (a) Couette flow in the whole domain, (b) channel flow in the whole domain, (c,d) channel flow in the upper half-domain and bottom half-domain. (a) Black is $C500-0$ and red is $C500-5$ . (b–d) Black circles are $P2600-0$ and red discs are $P2600-5$ .

The temporal evolution of the most energetic modes is shown in figure 13. In figure 13(a,b), (4.1) is integrated between the two walls ( $Y_{1}=0$ and $Y_{2}=L_{y}$ ), whereas in figure 13(c,d), the integration is performed in the vicinity of one single wall, which is regarded as an individual shear layer ( $Y_{1}=0\rightarrow Y_{2}=L_{y}/2$ near the bottom wall, and $Y_{1}=L_{y}/2\rightarrow Y_{2}=L_{y}$ near the upper wall). The quasi-periodic fluctuations of these modes, with period ${\sim}100h/U_{bulk}$ for Couette flow, are related to the regeneration cycle. The strongest mode is $M(0,\unicode[STIX]{x1D6FD})$ which corresponds to $x$ -independent streaks. As a general trend, neutrally buoyant particles decrease the amplitude of the fluctuations of this mode, whereas they do not have significant impact on its period, which is related to the regeneration cycle. However, in channel flow, it can be noted that both the $(0,\unicode[STIX]{x1D6FD})$ mode and $(\unicode[STIX]{x1D6FC},0)$ mode are of the same strength and period compared between single- and two-phase flow.

In Couette flow, one can note the relation of the intermittency of modes $(0,\unicode[STIX]{x1D6FD})$ ( $x$ -independent) and $(\unicode[STIX]{x1D6FC},0)$ ( $x$ -dependent), when integrated over the entire gap. The correlation between the different modes calculated similarly to (5.1) gives $R=-0.64$ for single-phase Couette flow ( $C500-0$ ) and $-0.59$ for suspension flow ( $C500-5$ ). The values of $R$ are negative, showing the clear anti-correlation between the two modes. The peaks of $M(0,\unicode[STIX]{x1D6FD})$ correspond to instants at which the streaks have the least $x$ -dependence. As the streaks become wavy, $M(0,\unicode[STIX]{x1D6FD})$ decreases, while the energy of $M(\unicode[STIX]{x1D6FC},0)$ (the fundamental mode in the $x$ -direction with no spanwise variation) sharply increases. The other $(\unicode[STIX]{x1D6FC},n\unicode[STIX]{x1D6FD}),n\neq 0$ modes can hardly be distinguished. Breakdown occurs when $M(0,\unicode[STIX]{x1D6FD})$ reaches a minimum. The amplitude of both mode fluctuations is slightly damped by the particle presence, as shown in figure 13(a).

For channel flow, figure 13(b) shows higher frequency fluctuations than in Couette flow, and less correlation between the $(\unicode[STIX]{x1D6FC},0)$ and $(0,\unicode[STIX]{x1D6FD})$ modes, when integrated over the whole domain. This is due to two coexisting shear layers (one at each wall) which are relatively independent of each other (turbulent mixing is weak in the core region between the two shear layers at low Reynolds number). When the modal energy is integrated over half of the channel section, as shown in figure 13(c,d), one can notice a stronger correlation between the $(0,\unicode[STIX]{x1D6FD})$ mode and $(\unicode[STIX]{x1D6FC},0)$ mode, as in Couette flow, although this is less pronounced in channel flow. The particles do not have a strong effect on the temporal evolution of these modes, suggesting that particles do not significantly alter the breakdown process.

5.5 Vortex regeneration: nonlinear interaction and vortex stretching

During streak breakdown, a complex set of interactions reinforces the streamwise vortices, leading to the formation of a new set of streaks, and completing the regeneration cycle. Hamilton et al. (Reference Hamilton, Kim and Waleffe1995) proposed that the vortex strengthening is due to interactions among the $\unicode[STIX]{x1D6FC}$ -modes, which grow during the streak breakdown. Schoppa & Hussain (Reference Schoppa and Hussain2002) suggested that the vortex formation is inherently three-dimensional, with a direct stretching (inherent to streak $(x,z)$ -waviness) of the near-wall $\unicode[STIX]{x1D714}_{x}$ sheets, leading to the collapse of streamwise vortices. They provided insights into the dynamics of near-wall vortex formation through the inviscid balance of the streamwise vorticity:

(5.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{x}}{\unicode[STIX]{x2202}t}=\underbrace{-u\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{x}}{\unicode[STIX]{x2202}x}-v\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{x}}{\unicode[STIX]{x2202}y}-w\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{x}}{\unicode[STIX]{x2202}z}}_{advection}+\underbrace{\unicode[STIX]{x1D714}_{x}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}}_{stretching}+\underbrace{\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}-\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}}_{tilting}.\end{eqnarray}$$

In fully developed turbulence, the greatest contribution, in magnitude, to the temporal evolution of the vorticity $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{x}/\unicode[STIX]{x2202}t$ comes from the tilting term. This is confirmed by our simulations (not shown here). However Schoppa & Hussain (Reference Schoppa and Hussain2002) have stated that this term contributes to the thin tail of the near-wall $\unicode[STIX]{x1D714}_{x}$ layer, and is not responsible for the formation of $x$ -independent streamwise vortices ( $(0,\unicode[STIX]{x1D6FD})$ mode in miniunit). Instead, the vortex formation is dominated by the stretching of the streamwise vorticity. The local $\unicode[STIX]{x1D714}_{x}$ stretching is sustained in time and is mainly responsible for the vortex collapse, whose location coincides with the $+\unicode[STIX]{x1D714}_{x}\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ peak. The streak meandering provides the generation of $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ . Then direct stretching of positive and negative $\unicode[STIX]{x1D714}_{x}$ occurs in regions where $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ is generated across the wavy streak flanks during the streak breakdown process. The stretching term is active only during the peaks of the cycle when local three-dimensionality is induced after streak breakdown (see Jiménez & Moin Reference Jiménez and Moin1991).

Figure 14. Evidence of the particle influence on the vorticity stretching in channel flow. (a,c,e,) Single-phase flow $P2600-0$ ; (b,d,f) suspension flow $P2600-1$ . (a,b) The evolution in time of both —— the circulation and $\cdots \cdots$  the vorticity stretching $|\unicode[STIX]{x1D714}_{x}\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x|$ . Both are averaged within the region $0.1<y/L_{y}<0.4$ where large-scale vortices take place, as seen in figure 5(b). (c–f) The streamwise velocity fluctuations in the plane $y/L_{y}=0.2$ , using $\cdots \cdots$  for negative isolines and – – – for positive isolines, with a step equal to 0.04. The contours indicate the stretching term $\unicode[STIX]{x1D714}_{x}\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ . The instants corresponding to the different snapshots are marked in (a,b). Panels (c,d) correspond to a trough of the vorticity stretching whereas (e,f) correspond to a peak of the vorticity stretching.

In our previous paper (Wang et al. Reference Wang, Abbas and Climent2017), we have shown that the streak waviness and vorticity stretching are almost unchanged for Couette flow in the presence of neutrally buoyant particles. However, in channel flow, the particles have some impact. Figure 14(a,b) shows the temporal evolution of the circulation as well as the vorticity stretching term, for both single- and two-phase channel flows. The vorticity stretching (averaged value is $0.17$ (respectively $0.11$ ) for $P2600-1$ (respectively $P2600-0$ )) and the circulation (averaged value is $0.025$ (respectively $0.02$ ) for $P2600-1$ (respectively $P2600-0$ )) are enhanced near the channel walls due to the presence of the particles. The cross-correlation between them was calculated, as in (5.1), giving $0.576$ for $P2600-0$ and $0.624$ for $P2600-5$ . This shows that high flow circulation is synchronized with the appearance of $x$ -dependent flow structures. Figure 14(c–f) shows instantaneous snapshots containing the contours of the streamwise velocity fluctuations (that illustrate the streak shape), as well as the vorticity stretching. These snapshots are taken at high vorticity stretching (high flow circulation). They show clear evidence that nonlinear processes such as streak breakdown, and thereby vortex regeneration, take place in the suspension flow as in single-phase flow. They also show that the wavy streaks are smaller and more numerous in suspension flow, when compared to the single-phase flow.