2.1. Inertial particles and PIV fluorescent tracer

The inertial particles are injected into the pipe through a magnetic mixer and precision peristaltic pump (Lead Fluid YZ15), which is calibrated for the current experimental conditions. To distinguish the fluid and solid phases, in-house fluorescent tracer is produced, which consists of Rhodamine 6G and epoxy (Pedocchi, Martin & García Reference Pedocchi, Martin and García2008). Mastersizer 3000 manufactured by Malvern Panalytical is employed to analyse the density and size distribution of fluorescent tracer. The probability density function (PDF) and cumulative fraction of particles are as shown in figures 16(a) and 16(b), respectively, in Appendix A, and they show that the mean diameter and size distribution (D50) are respectively 30.3 $\mathrm {\,\unicode{x03BC} m}$ and 26.1 $\mathrm {\,\unicode{x03BC} m}$ , corresponding to $St^+= 0.018$ (based on the mean diameter), which suggests that PIV particles follow the fluid reasonably well.

Polystyrene beads are chosen as the solid phase, and their density $\rho _p = 1050\,\mathrm {kg\,m^{-3}}$ is close to water but denser than water by approximately $5\,\%$ . This means that in pure water, the beads will tend to gravitate towards the lower side of the wall. Two sets of polystyrene spherical beads from Maxi-Blast Inc. are utilised, with mean diameters $d_{p} = 250\,\mathrm {\unicode{x03BC} m}$ and $437\,\mathrm {\unicode{x03BC} m}$ , corresponding to ratios of the pipe diameter to the mean particle diameter $D/d_{p}=82$ and 47, respectively. The size distributions of the small and large inertial particles are shown in figure 17 of Appendix B. Table 2 shows the relevant properties of fluorescent tracer and inertial particles at bulk velocity $0.3\,\mathrm {m\,s^-}{^1}$ . Also, table 2 presents the approximate settling velocity $u_{{ SV}}=(\rho _{p}-\rho _{f})g d_p^{2}/({18\mu })$ , which is obtained when the drag force equals the gravitational force in a stationary fluid. Notice that the large diameter particles result in larger $u_{{ SV}}$ .

Table 2. Parameters of inertial particles and fluorescent tracers. The values of $St=\tau _p/(\nu /u_b^2)$ are at $u_b=0.3\,{\rm m}\,{\rm s}^{-1}$ , which is the approximate value for most cases.

Separate experiments are conducted to measure the velocities of the fluid and the inertial particles. For the fluid phase, we use the standard PIV technique seeded with the in-house fluorescent tracer. The fluorescent tracer illuminated by the green laser light and emitting red colour is captured by the PIV camera with a filter that allows only red colour (thus isolating the bright green light from the inertial particles). Separate experiments with only inertial particles (i.e. without fluorescent tracer) are also conducted with PTV to obtain velocity statistics of the inertial particles. In the following, we discuss first the PIV for fluid and then the PTV technique for the inertial particles.

2.2. Particle image velocimetry

For the continuous fluid phase, in-house fluorescent tracer is illuminated by an approximately 1 mm thick laser sheet by the Innolas Spitlight Compact Nd:YAG continuous laser with energy 10 mW and wavelength 532 nm. To record the instantaneous fluid and particle position separately, the experiment utilises a PCO Dimax HS high-speed camera (16-bit frame) with resolution $2000\times2000$ pixels, and 180 mm Tamron optical lenses in measurement area $25.2\times25.2$ $\mathrm {mm^2}$ . The high-speed camera is operated at 2000 Hz. In order to improve PIV accuracy and alleviate the effects of weak contrast between background and tracer, uneven tracer intensity, and high background intensity from some reflective objects, three pre-processing steps are carried out. First, the ensemble average of more than 1500 PIV images is extracted and subtracted from each image. Second, a low-pass filter is utilised to artificially enlarge the tracer size to alleviate the peak-locking bias. Finally, an upper-intensity histogram threshold approach is employed to mitigate the error from much stronger intensity. In total, each experiment recorded 76 140 raw PIV images that were processed by an in-house PIV package. This package has been tested for a variety of turbulent flow cases (e.g. Chauhan et al. Reference Chauhan, Philip, De Silva, Hutchins and Marusic2014; Kevin, Monty & Hutchins Reference Kevin, Monty and Hutchins2019; Setiawan, Philip & Monty Reference Setiawan, Philip and Monty2022). Its post-processing approach utilises deformation of the window and multi-grid to mitigate bias. Although the majority of inertial particles are filtered out by the band-pass filter, a few inertial particles illuminated by fluorescent light are still captured. To eliminate these particles, a Matlab program was applied to detect any stray inertial particles, which were removed before PIV analysis. Also, considering that the streamwise velocity is larger than the vertical one, we selected the interrogation window size $64\times32$ pixels, with 50 % overlap.

Figure 2. Particle detection: (a) particle image template (top left) and the detected particles; (b,c) examples of detected particles in the cases $\rho _p/\rho _f =1$ and $1.05$ , respectively.

2.3. Particle tracking technique

To acquire the velocities of inertial particles, experiments are performed with only inertial particles (i.e. no PIV tracers). An in-house Matlab code is developed to capture the particle positions, and subsequently obtain the velocity field by utilising two particle positions. To identify particles, the particle mask correlation algorithm introduced by Takehara & Etoh (Reference Takehara and Etoh1998) is employed by selecting a threshold value for binarisation, which assumes a particle image template, and searches in the observation area obtaining correlation of pixel intensity. To ensure accuracy, only correlation results larger than 0.8 are processed further. For example, figure 2(a) shows a raw image where the laser enters from the top, and we select a particle template with $45\times45$ pixels as shown in the top left. Then the three particles (shown by red circles) are detected after using a correlation algorithm. Figures 2(b) and 2(c) show an example each for $\rho _p/\rho _f =1$ and $1.05$ , respectively, where detected particles are shown by white filled circles.

After identifying the spatial positions of particles, the cross-correlation method proposed by Hassan et al. (Reference Hassan, Blanchat, Seeley and Canaan1992) and Yamamoto et al. (Reference Yamamoto, Wada, Iguchi and Ishikawa1996) is employed on a processed image pair to calculate the velocity field. It is noted that the technique called binary cross-correlation is utilised since the particle images are recognised and binarised. In terms of the raw image pair, a circular area determined by centroid and radius in the first frame is applied to search in a potential moving area in the second frame. For example, assuming that the centre position is $(X,Y)$ in the first frame, and the radius of the particle was $r$ pixels, the rectangle searching area was $(X-r,X+r+50)$ and $(Y-r-15,Y+r+15)$ . Here, 50 and 30 pixels are the potential maximum moving distances of the particle for a PIV image pair in streamwise and vertical directions. A pre-processing thresholding procedure is employed to remove out-of-focus inertial particles, and background subtraction from the ensemble average is performed to eliminate the fixed and strong reflections from the pipe wall.

Table 3. Parameters of fluid and solid phases in the present experiments. For the buoyant cases of $\rho _p/\rho _f=1.05$ , the approximate volume fractions in the upper and lower halves of the pipe are shown in parentheses. For example, in case 3A, $\phi _v=0.25\,\%$ is observed to be split up, respectively, as 0.18 % and 0.32 % in the upper and lower halves of the pipe. Here, $u_{{ SV}}^+=u_{{ SV}}/u_\tau = (1/18)(Ga^2/d_p^+) = (1/18)(d_p^+/Sh) = (1/18)(d_p^{+2} g^+)$ , $g^{\prime}\equiv (\rho _p/\rho _f-1)g$ , $Ga = \sqrt {g^{\prime}/(\nu ^2/d_p^3)}$ , $Sh = (u_\tau ^2/d_p)/g^{\prime}$ and $g^+ = g^{\prime}/(u_\tau ^3/\nu )$ .

2.4. Experimental conditions

In this study, two particle volume fractions $\phi _v=0.25\,\%$ and $1 \pm 0.05\,\%$ are tested. The particles are fed into the pipe (before the inlet expansion) using a peristaltic pump at a metered flow rate from a known concentrated particle and water mixture placed on a digital scale. This allows us to estimate $\phi _v$ during the experiments. Furthermore, a sieve with aperture 63 $\mathrm {\,\unicode{x03BC} m}$ (smaller than the particle diameter) is used to filter out the inertial particles, and a stopwatch is used to record the time duration. After measuring the weight of the dried particles, $\phi _v$ is gain estimated, which is in reasonable agreement with the results from the PTV post-processing. As mentioned, for the solid phase velocity measurements, the experiments are performed with only inertial particles without any fluorescent tracer, whereas the fluid phase measurements include both inertial and PIV fluorescent particles. As such, separate experiments are run for measuring fluid and particle velocities.

Table 3 lists the parameters of all the particle-laden experiments. Case numbers 1 and 3 indicate the small inertial particle of diameter 250 $\mathrm {\,\unicode{x03BC} m}$ , respectively in neutrally buoyant ( $\rho _p/\rho _f=1.0$ ) and buoyant ( $\rho _p/\rho _f=1.05$ ) scenarios. Similarly, case numbers 2 and 4 indicate the large inertial particle of diameter 437 $\mathrm {\,\unicode{x03BC} m}$ for the two different $\rho _p/\rho _f$ values. Also, a letter A or B indicates $\phi _v$ at 0.25 % or 1 %. All the friction Reynolds numbers $Re_{\tau }$ are close to 195, and bulk Reynolds numbers ( $Re_b = u_b D / \nu$ ) are approximately 6000 corresponding to $u_b\approx 0.30\,{\rm m}\,{\rm s}^{-1}$ . The current experiments span the range $St^+= 1.16{-}3.76$ . Table 3 also lists the ratio $u_{b}/u_{\mathrm { SV}}$ , which shows that particles (in a still fluid) can settle vertically about a distance of $D$ over pipe length $138D$ or $50D$ , suggesting that settling of particles would be expected in our set-up. Also, we define a relevant non-dimensional viscous settling velocity $u_{{SV}}^+=u_{{SV}}/u_\tau$ that ranges from 0.12 to 0.32, and is not too small, indicating a different mechanism by which $\rho _p/\rho _f=1.05$ particles can interact with wall turbulence. For comparison with other works, table 3 presents parameters such as $Ga$ , $Sh$ and $g^+$ . We note that in the channel experiments of Baker & Coletti (Reference Baker and Coletti2021), $Ga=11.3$ and $Sh=2$ for $\rho _p/\rho _f=1.016$ , whereas Lee & Lee (Reference Lee and Lee2019) in their numerical simulations take $g^+=0.077$ at $\rho _p/\rho _f=833$ , and some of these parameters are within the range of our experiments with $\rho _p/\rho _f=1.05$ . We will show that indeed some trends that we observe exhibit similarities to the results from the mentioned investigations.

Fresh water is the working fluid for $\rho _p/\rho _f=1.05$ , whereas water is made denser with salt to achieve the same fluid and particle density resulting in $\rho _p/\rho _f=1$ . The total acquisition time is $1114\,R/u_b$ , where $R/u_b$ is the eddy turnover time. To achieve this total acquisition time, for each case, the images are captured in six separate batches (with each batch capturing 12 690 time-resolved images) owing to the maximum camera storage capacity. Appendix C shows the particle number densities for both $\rho _p/\rho _f=1$ and $1.05$ that are evaluated from the images. As expected, $\rho _p/\rho _f=1.05$ shows a higher particle count at the bottom owing to settling, and although the $\rho _p/\rho _f=1$ case shows a more uniform particle distribution for the upper and lower parts of the pipe, there is a slightly higher particle count for the lower half, and the likely reasons are suggested in Appendix C. To test the efficacy of achieving $\rho _p/\rho _f=1$ in experiments, particles were suspended in a 10 cm high beaker of still salt water, and after approximately 10–20 min (for larger and smaller particles, respectively), some particles slowly started to float and some to settle. As such, we expect the particle and fluid density difference to be approximately $\pm 0.1\,\%$ . Here, we use $x$ and $r$ as the streamwise and radial directions, and subscripts $f$ and $p$ for fluid and particle velocities. For example, $U_{xf}$ and $U_{xp}$ are the mean streamwise velocities of fluid and particles, and $u^{\prime}_{xf}$ and $u^{\prime}_{xp}$ the corresponding fluctuating velocities.

Figure 3. Data on the standard Moody chart for pipe flow. Here, $f \equiv (2D \, \mathrm { d}P/\mathrm { d}x)/(\rho _{f} u^2_b)$ and $Re_b=u_b D/\nu$ . Cases S(nb) and S(b) indicate the unladen experiments in neutrally buoyant (salt water) and buoyant (fresh water) scenarios, respectively. Note that the error bar symbol indicates the maximum, mean and minimum values for a repeated set of experiments.

Apart from the PIV experiments listed in table 3, repeat experiments are conducted for nominally the same parameters where only pressure drop and flow rates are measured (i.e. no PIV images are acquired). The friction factors and Reynolds numbers from this collective set of experiments are plotted in figure 3. Here, the friction factor is defined as $f = (2D\, \mathrm { d}P/\mathrm { d}x)/(\rho _{f} u^2_b)$ , where $\mathrm { d}P/\mathrm { d}x$ is the pressure drop between the monitoring points as shown in figure 1. The ‘error bars’ are simply the ranges of $f$ and $Re_b$ observed in the experiments. A smaller symbol is used to specifically indicate the PIV experiments in table 3. Overall, there is a slight increase in frictional drag for $\rho _p/\rho _f=1$ , whereas a small drag reduction is noticeable for $\rho _p/\rho _f=1.05$ . This difference will be further made apparent when we discuss velocity profiles later.

2.5. Particle-free flow

The experiment free of any inertial particles is first performed as the baseline. Velocity statistics are presented in viscous or inner scaling, i.e. non-dimensionalised by the friction velocity $u_\tau$ (obtained from measured pressure drop) and kinematic viscosity $\nu$ , and denoted by a superscript $+$ . Figure 4 shows the time-averaged statistics of unladen flow, including mean streamwise fluid velocity ( $U^{+}_{xf}$ ), streamwise and radial velocity fluctuation root mean square (r.m.s.) ( $u^{\prime +}_{xf,\,rms}$ and $u^{\prime +}_{rf,\,rms}$ , respectively) and Reynolds shear stress $\langle {u^{\prime}_{xf}} {u^{\prime}_{rf}} \rangle ^{+}$ . There is reasonable agreement with DNS results at $Re_{\tau }=180$ (El Khoury et al. Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013) presented in figure 4 as solid red lines and experimental data as symbols, which demonstrates that the turbulent pipe flow is fully developed. The attenuation in the turbulence statistics is due to the PIV window averaging, which is approximately 7.8 viscous length scales in the wall-normal direction.

Figure 4. Velocity profiles of particle-free flow normalised by inner units at the considered Reynolds number: (a) streamwise mean velocity, (b) streamwise and vertical fluctuating r.m.s., (c) Reynolds shear stress. Lines and symbols denote DNS and the current experiment, respectively. The subscripts $x$ and $r$ mean the streamwise and vertical directions. The subscript $f$ represents the fluid phase.

Figure 5. Fluid velocity statistics for $\rho _p/\rho _f=1$ : (a,c,e,g) normalised in inner units, and (b,d,f,h) normalised in outer units ( $U_c$ and $R$ ). Mean streamwise velocity ( $U_{xf}$ ), turbulence intensities ( $u^{\prime}_{xf,rms}$ and $u^{\prime}_{rf,rms}$ ) and Reynolds shear stress ( $\langle {u^{\prime}_{xf}} {u^{\prime}_{rf}} \rangle$ ) profiles at $St^+ \approx 1.2$ ( $\phi _v= 0.25\,\%$ , $\phi_v= 1\,\%$ ) of the small particles, and $St^+ \approx 3.8$ ( $\phi _v= 0.25\,\%$ , $\phi_v= 1\,\%$ ) of the large particles. Unladen cases: DNS at $Re_\tau =180$ , and experiments. For clarity, only one-fifth of the data point symbols are shown.

3.1. Fluid statistics

As mentioned briefly before, we choose the same particles but increase the fluid density by using salt to ensure $\rho _p/\rho _f=1$ . Figures 5(a,c,e,g) show streamwise mean velocity, turbulent fluctuating intensities and Reynolds shear stress of fluid phase in inner units (coloured symbols) compared with the experiments and DNS of unladen flow (in black symbols and solid lines, respectively). With increasing $\phi _v$ and $St^+$ , the mean profiles in figure 5(a) decrease monotonically within the (small) log region and outer layer. Compared with $St^{+} \approx 1.2$ (cases 1A and 1B), the particles at $St^{+} \approx 3.8$ (cases 2A and 2B) have a clearer effect on the reduction of mean velocity. Furthermore, figures 5(c) and 5(e) show that with an increase of volume fraction and viscous Stokes number, streamwise intensities monotonically reduce mostly close to the wall, especially around the peak, while vertical intensities also decrease substantially near the centreline. Reynolds stress in figure 5(g) shows attenuation closer to the wall that is more severe than the r.m.s. components; however, there is reduced attenuation close to the centreline. In particular, the large particles produce more turbulence attenuation than the small ones at $\phi _v=0.25\,\%$ . However, they show similar contributions to turbulence damping at $\phi _v=1\,\%$ , suggesting a larger effect on turbulence as $\phi _v$ changes from $0.25\,\%$ to $1\,\%$ . These results are consistent with Picano et al. (Reference Picano, Breugem and Brandt2015), although their lowest $\phi _v=5\,\%$ is larger than ours.

One of the physical reasons for turbulence attenuation in vertical particle-laden pipes and channels suggested by Vreman (Reference Vreman2007, Reference Vreman2015) is the non-uniform drag force owing to the different mean fluid and particle velocities in the core and wall region. In the present scenario also (cf. Figure 6(a) – to be discussed later), there is indeed evidence of non-uniform drag, which could partially explain the turbulence attenuation that we observe in figure 5. Note that the numerical value of reduction in turbulence, when presented in viscous units, is the result of a slight increase in $u_\tau$ (owing to overall drag increase) as well as any actual attenuation of turbulence intensity by the particles. Distributions in outer units can clarify this.

Figures 5(b,d,f,h) display the mean velocity, fluctuating intensities and Reynolds stress in outer units (normalised by centreline velocity $U_c$ and $R$ ). The analogous mean profiles indicate that the presence of inertial particles is not enough to modify streamwise velocity substantially. This is consistent with the numerical simulation of Picano et al. (Reference Picano, Breugem and Brandt2015), where they show that the effect of the particles at $\phi _{v}=5\,\%$ on streamwise mean velocity in outer units is negligible. The mean profiles slightly reduce for $y/R \lessapprox 0.5$ , and this changes the velocity profile closer to a laminar flow, which suggests a trend towards laminarisation of the turbulent flow. (Note that the overall drag can still increase owing to the particle-induced stress that we discuss later.) The streamwise intensities decrease monotonically at the peak with increasing $\phi _v$ . Interestingly, cases 1A and 2A display a similar magnitude of streamwise and vertical intensities at lower volume fraction ( $\phi _{v}=0.25\,\%$ ), whereas the large particles (2B) contribute to increased suppression compared to small particles (2A) at high volume fraction ( $\phi _{v}=1\,\%$ ). Figure 5(h) shows that case 2B with high $\phi _{v}$ and $St^+$ experiences the largest turbulence attenuation only below $y/R \lessapprox 0.5$ .

Figure 6. Particle velocity statistics (in symbols) for $\rho _p/\rho _f=1$ in outer units for different $\phi _v$ and $St^+$ . For comparison, lines show the corresponding fluid statistics from figure 5. (a) Streamwise mean velocity; (b) streamwise and (c) vertical fluctuating r.m.s.; and (d) Reynolds shear stress. The subscript $p$ represents the particle phase. Here, $St^+ \approx 1.2$ ( $\phi _v= 0.25\,\%$ , $\phi_v= 1.2\,\%$ ) for the smaller particles, and $St^+ \approx 3.8$ ( $\phi _v= 0.25\,\%$ , $\phi_v= 1\,\%$ ) for the larger particles.

3.2. Particle statistics

Figure 6 shows the mean streamwise velocity profiles, streamwise and vertical intensities, and Reynolds stress of solid phase in outer units, where fluid statistics with dotted lines are displayed for comparison. The mean streamwise velocity profiles have lower values than the fluid phase in the bulk of the flow, except in the near-wall region, where particles have a higher velocity. Costa, Brandt & Picano (Reference Costa, Brandt and Picano2021), Noguchi & Nezu (Reference Noguchi and Nezu2009) and Nezu & Azuma (Reference Nezu and Azuma2004) have also observed the same phenomenon that particle velocity is higher than the fluid velocity in the near-wall region. Particles maintain a relatively high tangential velocity owing to their inertia as they approach the wall, while the motion of the fluid is restricted by the no-slip boundary condition, and this causes higher streamwise velocities for inertial particles compared to their surrounding fluid. The opposite is true in the outer regions, where particles drag behind the fluid with a lower velocity. Figures 6(b) and 6(c) show that the streamwise and vertical intensities of the non-sedimenting ( $\rho _p/\rho _f=1$ ) solid phase have a lower magnitude than the fluid phase, with the exception of the near-wall region. In the near-wall region, the particles have higher streamwise intensities than the fluid, which is likely a result of the no-slip fluid boundary condition and particles that are free to move, as well as owing to a larger mean particle velocity compared to the fluid in the near-wall region (cf. figure 6 a).

Furthermore, figure 6(d) demonstrates that owing to the small inertia, the solid phase has a level of turbulent activity similar to that of the fluid phase around the peak, but slightly higher solid phase intensities closer to the walls compared to the fluid phase (see also Picano et al. Reference Picano, Breugem and Brandt2015). In § 4.2, we will discuss these results against the case $\rho _p/\rho _f=1.05$ .

3.3. Relative drag contributions

Over the past few years, it has become clearer that even though turbulent stresses are reduced in the particle-laden flow, the overall stress (or drag) can still increase owing to additional contributions from the particles (for channel flows, see e.g. Picano et al. (Reference Picano, Breugem and Brandt2015); Costa et al. (Reference Costa, Brandt and Picano2021); Gualtieri et al. (Reference Gualtieri, Battista, Salvadore and Casciola2023)). The starting point is usually the mean particle–fluid mixture equation, which is written here for the case of an axisymmetric pipe where the velocity is decomposed into into mean and fluctuations (e.g. Chin & Philip Reference Chin and Philip2021) in the axial direction:

(3.1) \begin{align} & \frac {1}{r}\,\frac {\mathrm {d}}{\mathrm {d r}}\left [ r \left ( (1-\phi _v)\left ( \mu\, \frac {\mathrm {d} U_x}{\mathrm {d} r} - \rho _f \big\langle u^{\prime}_{fx} u^{\prime}_{fr} \big\rangle \right ) - \phi _v \rho _p \big\langle u^{\prime}_{px} u^{\prime}_{pr} \big\rangle \right ) \right ]\nonumber\\& \quad - \frac {\mathrm {d} ((1-\phi _v) P)}{\mathrm {d} x} - F_m= 0, \end{align}

where $F_m$ is the resultant mixture drag force in the axial direction. Note that a rigorous derivation for the above equation is not completely clear, although suggestions have been made; for example, see the two perspectives in Jackson (Reference Jackson1997) and Marchioro, Tanksley & Prosperetti (Reference Marchioro, Tanksley and Prosperetti1999). Nevertheless, for ‘point particles’ (such as in many DNS simulations), $\phi _v=0$ , and even in the present experiments, $\phi _v\approx 0.01$ implies that the particle Reynolds stresses contribute negligibly to the overall momentum balance. Also, it is common (e.g. Picano et al. Reference Picano, Breugem and Brandt2015; Lee & Lee Reference Lee and Lee2019) to write $F_m$ as a gradient of some ‘particle stress’, which can simplify the analysis slightly. We, however, retain a more generic form. Integrating (3.1) in the radial direction, and substituting $u_\tau ^2 \equiv -(R/2\rho _f) (\mathrm {d} ((1-\phi _v) P)/\mathrm {d} x)$ (obtained by integrating (3.1) twice) leads to

(3.2) \begin{equation} \underbrace { (1-\phi _v) \nu \frac {\mathrm {d} U_x}{\mathrm {d} r}}_{-\tau _{\mathrm{V}} (r)} - \underbrace { \left ( (1-\phi _v)\big\langle u^{\prime}_{fx} u^{\prime}_{fr} \big\rangle + \phi _v \frac {\rho _p}{\rho _f } \big\langle u^{\prime}_{px} u^{\prime}_{pr} \big\rangle \right ) }_{\tau _T (r)} + \underbrace { \frac {r}{R} u_\tau ^2 }_{\tau } - \underbrace {\frac {1}{r \rho _f }\!\int _0^r\! F_m r_1\,\textrm {d} r_1}_{\tau _p (r)}= 0, \end{equation}

where $r_1$ is a dummy variable, and the total stress $\tau = \tau _{\mathrm{V}} + \tau _T + \tau _p$ is the sum of viscous, turbulent and particle-induced stress at a radial location $r$ . The various stress contributions in cases 2A and 2B ( $St^+\approx 3.8$ and $\phi _v = 0.25\,\%, 1\,\%$ ) are shown in figures 7(a) and 7(b), where $y=R-r$ . The lower $St^+$ cases (1A and 1B) are similar and not shown here. Particle-induced stress ( $\tau _p$ ) is indirectly calculated based on the total stress balance. Compared with the unladen case, the presence of particles slightly decreases the viscous stress near the walls, but the main reduction is in the turbulent Reynolds stresses ( $\tau _T$ ). The values of $\tau _p$ in figure 7(a) are similar to those obtained by Picano et al. (Reference Picano, Breugem and Brandt2015) and Yu et al. (Reference Yu, Lin, Shao and Wang2017) for low volume fractions in channel flow DNS. Figure 7(b) for $\phi _v=1\,\%$ shows elevated reduction $\tau _T$ compared to $\phi _v=0.25\,\%$ . In fact, for $\phi _v=1\,\%$ , the particle-induced stress and turbulent Reynolds stress are approximately the same magnitude, each amounting to nearly 40 % of the total stress in the near-wall region, and this percentage of particle-induced stress reduces towards the outer wall. It is worth noting that (although not shown here) the particle-induced stress of the large particles ( $St^{+} \approx 3.8$ ) is slightly larger than that of small particles ( $St^{+} \approx 1.2$ ).

Figure 7. Momentum budget for different Stokes numbers and volume fractions of neutrally buoyant scenarios $St^+\approx 3.5$ : (a) case 2A ( $\phi _v=0.25\,\%$ ), and (b) case 2B $\phi _v=1\,\%$ . Note that $y/R=0$ represents the wall, and $y/R=1$ is the pipe centreline; $\tau ^{S}_V$ and $\tau ^{S}_T$ are the unladen viscous and turbulent Reynolds shear stresses, respectively. The solid and empty triangle symbols represent raw (, ) and modified ( , ) data. To ensure that the sum of the unladen stresses equals 1, using DNS data, the turbulent shear stress is multiplied by 1.1 to account for the PIV attenuation, and we obtain the modified data. Subsequently, this ratio is used to modify the turbulent shear stresses of particle-laden cases. (c) Contributions to the normalised friction factor with terms from (3.5).

Following Fukagata, Iwamoto & Kasagi (Reference Fukagata, Iwamoto and Kasagi2002), (3.2) is integrated twice, i.e. $(2/R^2)\int _0^{R} ( \int _R^{r}$ (3.2) $\textrm {d} r_1 ) r\,\textrm {d} r$ , and noting that for an arbitrary function $f(r)$ , $\int _0^{R} ( \int _R^{r} f(r_1) \,\textrm {d} r_1 ) r\,\textrm {d} r = - \int _0^R f(r)\, (r^2/2)\, \textrm {d} r$ , leads to

(3.3) \begin{equation} \frac {16(1-\phi _v)}{Re_b} + 8\int _0^1\left ( (1-\phi _v)\frac {\big\langle u^{\prime}_{fx} u^{\prime}_{fr} \big\rangle }{u_b^2} + \phi _v \frac {\rho _p}{\rho _f}\frac {\big\langle u^{\prime}_{px} u^{\prime}_{pr} \big\rangle }{u_b^2} + \frac {\tau _p}{u_b^2} \right ) \left (\frac {r}{R}\right )^2 \textrm {d}\! \left (\frac {r}{R}\right ) = C_f, \end{equation}

where the bulk velocity is $u_b \equiv (2/R^2)\int _0^R U_x\,r\,\textrm {d} r$ , and $C_f \equiv 2 u_\tau ^2/u_b^2$ . We note that for the unladen case (i.e. $\phi _v=0$ and $\tau _p=0$ ), (3.3) reduces to

(3.4) \begin{equation} \frac {16}{Re_b} + 8\int _0^1\left ( \frac {\big\langle u^{\prime}_{fx} u^{\prime}_{fr} \big\rangle }{u_b^2} \right ) \left (\frac {r}{R}\right )^2 \textrm {d}\! \left (\frac {r}{R}\right ) = C_f. \end{equation}

Under viscous normalisation, (3.3) takes the form

(3.5) \begin{align} \underbrace {4(1-\phi _v)\!\left (\frac {u_b^+}{Re_\tau }\right )}_{C_{fl}^+} + \underbrace { 4\int _0^1\!\left ( (1-\phi _v)\big\langle u^{\prime}_{fx} u^{\prime}_{fr} \big\rangle ^+\! + \phi _v \frac {\rho _p}{\rho _f}\big\langle u^{\prime}_{px} u^{\prime}_{pr} \big\rangle ^+ + \tau _p^+ \right )\! \left (\frac {r}{R}\right )^2 \!\textrm {d}\! \left (\frac {r}{R}\right )} _{C_{fR}^+ \,\,\,\,\,\,\,\, + \,\,\,\,\,\,\,\, C_{pR}^+ \,\,\,\,\,\,\,\, + \,\,\,\,\,\,\,\, C_{pi}^+} = 1. \end{align}

The different terms in (3.5) are plotted in figure 7(c) for different cases with $\rho _p/\rho _f=1$ , where fluid laminar contribution $C_{fl}^+$ denotes the first term in (3.5), and $C_{fR}^+$ , $C_{pR}^+$ and $C_{pi}^+$ are respectively the contributions from fluid and particle Reynolds stress and particle-induced stress represented by the terms within the integral representation in (3.5). Figure 7(c) shows that $C_{pR}^+$ is negligible owing to small $\phi _v$ values, hence (3.5) could be conveniently studied with $\phi _v=0$ , leaving only three major contributors, and (3.5) reduces to

(3.6) \begin{equation} 4\left (\frac {u_b^+}{Re_\tau }\right ) + 4\int _0^1\left ( \big\langle u^{\prime}_{fx} u^{\prime}_{fr} \big\rangle ^+ + \tau _p^+ \right ) \left (\frac {r}{R}\right )^2 \textrm {d}\! \left (\frac {r}{R}\right ) = 1. \end{equation}

It is evident that the particle-induced stress ( $C_{pi}^+$ ) increases with increasing $St^+$ and $\phi _v$ , contributing increasingly towards the total drag. Therefore, as mentioned before, even though the turbulent stresses are reduced, it is more than compensated by the particle-induced stresses, and can result in an overall drag increase. In their channel flow DNS, Costa et al. (Reference Costa, Brandt and Picano2021) and Gualtieri et al. (Reference Gualtieri, Battista, Salvadore and Casciola2023) have also found similar significant contributions from particle-induced stress, although their focus is on $\rho _p/\rho _f$ of $O(100){-}O(1000)$ , neglecting the effect of gravity.

A similar stress analysis is not readily possible for $\rho _p/\rho _f=1.05$ because of the non-axisymmetric nature of the flow. Nevertheless, as we discuss below, fluid and particle velocity statistics provide evidence that even relatively small density differences can lead to substantial differences in flow characteristics.

4.1. Fluid statistics

Figure 8 shows streamwise mean velocity, turbulent fluctuating intensities and Reynolds shear stress (in coloured symbols, cf. table 3) as well as the unladen case (in black symbols) in the upper and lower parts of the pipe. Data for the upper and lower parts are shifted along the ordinate for clarity. The left-hand and right-hand plots show the same data, but are respectively normalised by inner and outer units. Owing to particle deposition, the lower part has a higher particle volume fraction than the upper part as shown in table 3 and Appendix C. In particular, large-diameter particles show a higher concentration than the smaller ones in the lower part, although $\rho _p/\rho _f$ is the same, and this is due to the increased settling velocity of the larger particles (see table 2).

The mean fluid velocity distributions $U_{xf}$ are presented in figures 8(a,b) for $\rho _p/\rho _f=1.05$ . The inner normalised distributions in figure 8(a) are quite different to the case $\rho _p/\rho _f=1$ in figure 5(a). In the inner units (figure 8 a), it is clear that there is hardly any drag increase, i.e. no downward shift of profiles is observed in $\rho _p/\rho _f=1.05$ , unlike the case $\rho _p/\rho _f=1$ , where profiles are shifted down (observed, for example, by taking the DNS red line as a reference). In fact, the profiles are slightly shifted upwards for $\rho _p/\rho _f=1.05$ , implying a slight drag reduction. Also, there are subtle differences between the upper and lower halves of the pipe. Owing to the non-zero settling velocity, the numbers of inertial particles are considerably reduced in the upper part of the pipe, and consequently the mean profiles are similar to the unladen case, although with a slight drag reduction. The drag reduction (upward shift of profiles) are more noticeable in the lower part. The effect becomes clearer in the outer normalised form (figure 8 b) where the fluid velocity drops below the unladen case close to the wall, and here the particle number density is the highest. This trend towards laminarisation seems slightly enhanced for $\rho _p/\rho _f=1.05$ compared to $\rho _p/\rho _f=1$ . Furthermore, the noticed velocity reduction is consistent with previous studies (e.g. Nezu & Azuma Reference Nezu and Azuma2004; Lee & Lee Reference Lee and Lee2019) with the settling particles where $\rho _p/\rho _f \gt 1$ .

Streamwise turbulence intensities in figures 8(c,d) exhibit a general reduction compared to the unladen case, but the reduction in the lower half is similar to the case $\rho _p/\rho _f=1$ (cf. figures 5 c,d); however, the upper part shows a higher $u^{\prime +}_{xf,\,rms}$ – a reduced particle density that is not able to suppress turbulence in the upper half is a possible cause. The radial fluctuations $u^{\prime +}_{rf,\,rms}$ (cf. figures 8 e,f), on the other hand, show quite a different scenario – the intensities are large in both the upper and lower halves, and they are substantially higher than the $\rho _p/\rho _f=1$ cases. The upper half again shows increased radial fluctuations relative to $u_\tau$ (and note that $u_\tau$ is reduced in $\rho _p/\rho _f=1.05$ ). In the upper half, the smaller particles ( $St^+\approx 1.2$ ) seem to be less effective in turbulence suppression compared to the larger particles ( $St^+\approx 3.8$ ), which is perhaps anticipated. The Reynolds shear stress in figures 8(g,h) follows a similar trend, with elevated values in the upper half, although in the lower half the shear stress is suppressed, which has similarities to $\rho _p/\rho _f=1$ cases. In all cases with $\rho _p/\rho _f=1.05$ , the effects are more dramatic for higher $St^+$ and $\phi _v$ . The increased radial and shear stress (compared to $\rho _p/\rho _f=1$ ) is perhaps not surprising given that the settling velocity is not insignificant (cf. table 2), and is approximately $(1/3)u_\tau$ for the highest volume fraction.

Figure 9. Particle velocity statistics (in symbols) for $\rho _p/\rho _f=1.05$ in outer units for different $\phi _v$ and $St^+$ , and for upper and lower halves of the pipe. For comparison, lines show the corresponding fluid statistics from figure 8: (a) streamwise mean velocity, (b) streamwise intensities, (c) vertical fluctuating intensities, (d) Reynolds shear stress. Mean and turbulence profiles at $St^+ \approx 1.2$ ( $\phi _v= 0.25\,\%$ , $\phi_v=1\,\%$ ) for smaller particles, and $St^+ \approx 3.8$ ( $\phi _v= 0.25\,\%$ , $\phi_v=1\,\%$ ) for larger particles.

Interestingly, Lee & Lee (Reference Lee and Lee2019) observed augmented and suppressed stresses, respectively, on the upper and lower walls in their numerically simulated channel flow, which has similarities to our results. As mentioned, in their cases $g^+=0.077$ , which is similar to ours, but $\rho _p/\rho _f=833$ . It is noteworthy that the Kussin & Sommerfeld (Reference Kussin and Sommerfeld2002) horizontal air channel flow experiments (with estimated $g^+\approx 0.58$ , $u_b/u_{{ SV}}\approx 23$ and $u_{{ SV}}^+\approx 0.96$ for a nominal case) do not show such explicit differences in the upper and lower wall statistics. They observe consistent suppression of turbulence on both walls, possibly because of their large $\rho _p/\rho _f=2500$ , and also because they found wall collisions to be important, reducing the particle settling effect. A density ratio closer to ours was utilised by Nezu & Azuma (Reference Nezu and Azuma2004) in their open channel water experiments with $\rho _p/\rho _f$ ranging from 1.02 to 1.5 (with $u_{{ SV}}^+ \approx0.0005{-}18$ and $g^+ \approx0.06{-}1.45$ ), and they observe streamwise turbulent suppression similar to our bottom half of the pipe; however, they have no analogue of our top half owing to geometrical differences. One of their suggestions is that suppression happens when $d_p/\eta \lt 1$ (where $\eta$ is the Kolmogorov length scale), i.e. small particles, whereas turbulence is enhanced for larger ( $d_p/\eta \gt 1$ ) particles. To test this, an approximation of $\eta =(\nu ^3/\epsilon )^{1/4}$ requires dissipation rate ( $\epsilon$ ) estimates, which with a log-law mean velocity profile can be approximated as $\epsilon \approx u_\tau ^3/(\kappa y)$ , where $\kappa =0.39$ is the von Kármán constant (e.g. Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Morrill-Winter, Philip & Klewicki Reference Morrill-Winter, Philip and Klewicki2017), and $y$ is the distance from the wall. An estimate at, say, $y=0.1 R$ provides $\eta ^+ = \eta /(\nu /u_\tau ) \approx ( 0.1 \kappa (u_\tau R/\nu ) )^{1/4} = 0.4444\, Re_\tau ^{1/4}$ , which for $Re_\tau = 190$ leads to $d_p/\eta \approx 2.7$ and 4.8, which are greater than 1. Note that, strictly, $\eta$ increases with increasing $y$ . Nevertheless, this approximate calculation shows that in our case $d_p/\eta \gt 1$ , and it does result in a slight turbulence enhancement mostly in the upper half, but in the lower half we largely observe turbulence suppression. This difference in statistics between upper and lower halves suggests that $d_p/\eta$ by itself cannot fully capture the turbulence attenuation/suppression; rather, a consideration of settling velocity is necessary.

4.2. Particle statistics

Particle velocity statistics (in outer units) for $\rho _p/\rho _f=1.05$ are presented in figure 9 using symbols, and the corresponding fluid statistics (taken from figure 8) are shown by lines of the same colour. Similar to $\rho _p/\rho _f=1$ , the mean particle velocities (cf. figure 9 a) are smaller than the fluid velocities in the bulk, but become larger closer to the wall, with the exception of the near-wall region, which is consistent with other studies (e.g. Noguchi & Nezu Reference Noguchi and Nezu2009; Costa et al. Reference Costa, Brandt and Picano2021). This is again owing to the particles carrying their inertia into the wall regions, and thus retaining their increased velocity from the bulk region.

The turbulence intensities for $\rho _p/\rho _f=1.05$ (cf. figures 9 b–d), especially for the lower half, depict a substantially different scenario compared to $\rho _p/\rho _f=1$ . Particle turbulent stresses are higher than their corresponding fluid counterparts in the lower half, and even in the upper half, the stresses are higher compared to the neutrally buoyant cases. Settling velocity of the particles owing to their larger-than-fluid density is the main reason for this heightened particle stress. Thus we can surmise that as the particles settle, they constantly interact with the incoming larger mean axial flow, and eventually lead to higher stresses.

In the following, we turn our attention to the differences in local structure and statistics of the velocity fluctuations for $\rho _p/\rho _f=1$ and $1.05$ .

6.1. Streamwise velocity correlation

Two-point spatial autocorrelation of streamwise velocity separated by a streamwise distance $\Delta x$ at a particular wall-normal location ( $y$ ) is defined as

(6.1) \begin{equation} R_{uu}(\Delta {x},y)=\frac {\left \langle u^{\prime}_{xf}(x,y)\, u^{\prime}_{xf}(x+\Delta {x},y) \right \rangle }{ {u^{\prime 2}_{xf,rms}} }, \end{equation}

where the averaging $\langle \, \rangle$ is over $x$ and different ensembles. It is well known that the autocorrelation reflects the typical turbulent structure, including near-wall streaks and quasi-streamwise vortices (e.g. Kim, Moin & Moser Reference Kim, Moin and Moser1987; Pope Reference Pope2000).

Figures 12(a,b,c) show the data for $\rho _p/\rho _f = 1$ , of which figures 12(a,b) respectively show $R_{uu}$ at two different vertical positions of $y^{+}=20$ and $100$ . The coloured lines show different $St^+$ and $\phi _v$ , and the black solid line shows data for the unladen case that has a negative minimum value at $\Delta {x}^+ \approx 160$ . For small $ \Delta x^+$ , the presence of the particles reduces $R_{uu}$ , which is clearer in figure 12(b); however, at larger $\Delta x^+$ (around the $R_{uu}$ minima), the correlations shift towards larger $\Delta x^+$ , which implies that the particles lead to a less correlated but longer turbulent structure. These trends have similarities to the DNS results of Vreman (Reference Vreman2007) for $\rho _p/\rho _f = 2058$ in vertical particle-laden pipe flows, and Picano et al. (Reference Picano, Breugem and Brandt2015) for $\rho _p/\rho _f = 1$ in channel flows.

A surrogate for the length of turbulent structures is the correlation length, which is usually the integral of $R_{uu}$ from $0$ to $\infty$ . In our case, where $R_{uu}$ can be negative, one has to use other options, such as the locations of the minima or the zero-crossing of $R_{uu}$ . Figure 12(c) shows that $\Delta x ^+$ at $R_{uu}=0$ for changing $y^+$ as well as for different $St^+$ and $\phi _{v}$ , which reinforces the point that the particles result in slightly longer dominant structures. Furthermore, larger $\phi _v$ (and $St^+$ ) seem to favour larger structures.

Cases with $\rho _p/\rho _f = 1.05$ are presented in figures 12(d,e,f) (upper pipe half) and figures 12(g,h,i) (lower pipe half). Both the top and bottom half $R_{uu}$ differs from $\rho _p/\rho _f = 1$ cases in figures 12(a,b,c). With increasing $\Delta x^+$ , the correlation continues to drop and crosses zero before the unladen case, which is in contrast to $\rho _p/\rho _f = 1$ . The extent of zero-crossing is evident in figures 12(f) and 12(i) for upper and lower halves, respectively. This suggests that although both $\rho _p/\rho _f = 1$ and $\rho _p/\rho _f = 1.05$ result in a lower absolute correlation, the settling particles tend to promote smaller dominant turbulent structures, likely because the falling particles will generate length scales that are restricted in the vertical direction (unlike the streamwise elongated structures).

6.2. Energy spectra

Streamwise energy spectra of fluid velocity ( $\Phi _{uu}$ ) provide a relatively well-established descriptor of energy-containing scales. The spectra at each $y$ location are constructed by taking the time-resolved PIV velocity field of a particular position and calculating $\Phi _{uu}$ from the time series of $u^{\prime}_{xf}$ . The wavenumber is constructed by the usual Taylor’s frozen hypothesis of turbulence using $U_{xf}(y)$ as the convection velocity. The spectra at various $x$ -locations are then averaged, and normalised such that $\int _{0}^{\infty } \Phi _{uu} (k_x)\,\textrm {d} k_x = u^{2}_{xf,rms}$ . The pre-multiplied energy spectra $k_x\Phi _{uu}$ for the unladen case are compared with the DNS data of Ahn et al. (Reference Ahn, Lee, Jang and Sung2013) and Chan et al. (Reference Chan, Zahtila, Ooi and Philip2021) at similar $Re_\tau$ in figure 19 of Appendix D, and we find that experiments and DNS are in reasonable agreement.

The premultiplied one-dimensional spectra of streamwise turbulence fluctuation $k_{x} \Phi _{uu}$ normalised by $u_{\tau }^2$ against wavelength ( $\lambda _x = 2\pi /k_x$ ) are presented on a linear-log scale in figure 13 for cases $\rho _p/\rho _f=1$ in the top row and for 1.05 in the middle (upper pipe half) and bottom (lower pipe half) rows. The left and right columns have spectra at $y^+=20$ and 100, respectively. Note that since $\int k_x \Phi _{uu} \,\textrm {d} (\mathrm { ln} \lambda _x) = u^{2}_{xf,rms}$ , premultiplied spectra on a log-linear scale allow a visual assessment of the energy contribution at different $\lambda _x$ . The unladen pipe flow spectra are shown in black solid lines, and at $y^+=20$ , they peak at $\lambda _x \approx 1000$ , representing the usual near-wall streaks. For $\rho _p/\rho _f=1$ , the turbulence energy is suppressed in the presence of the particles at almost all wavelengths. At $y^{+}=20$ (cf. figure 13 a), the inertial particles cut off energy at peak $\lambda _x$ , producing a flatter plateau with increasing volume fraction, and particles damp energy up to large wavelengths. At $y^+=100$ (cf. figure 13 b), there is substantially less damping, and the energy reduction is confined around peak $\lambda _x$ , with almost no attenuation at larger $\lambda _x$ . For $\rho _p/\rho _f=1.05$ , the upper half of the pipe has less energy attenuation than the lower half (owing mostly to the lower particles in the upper half) with similar characteristics as for the neutrally buoyant particles. The main difference between $\rho _p/\rho _f=1$ and $\rho _p/\rho _f=1.05$ (apart from higher attenuation in $\rho _p/\rho _f=1$ , consistent with figures 5 and 8) is that with the inertial particles, peak energy is at a lower wavelength for $\rho _p/\rho _f=1.05$ compared to the unladen case as well as $\rho _p/\rho _f=1$ . This is consistent with the settling particles leading to energy concentration at lower length scales (as discussed in relation to figure 12). Furthermore, as perhaps should be expected, higher $St^+$ and $\phi _v$ generally result in an increased reduction in energy across most wavelengths.

Figure 13. Pre-multiplied energy spectra of streamwise velocity fluctuation ( $k_x \Phi _{uu}$ ) versus streamwise wavelength ( $\lambda _x$ ): (a,b) $\rho _{p}/\rho _{f} = 1$ ; (c,d) upper half for $\rho _{p}/\rho _{f} = 1.05$ ; and (e,f) lower half for $\rho _{p}/\rho _{f} = 1.05$ ; at (a,c,e) $y^{+}=20$ , and (b,d,f) $y^{+}=100$ . Line colours are the same as in figure 12 and table 3.

The variation in $k_x\Phi _{uu}$ across all wall-normal locations ( $y^+$ ) is better visualised as colour contours shown in figure 14 for $\rho _{p}/\rho _{f} = 1$ , and in figure 15 for $\rho _{p}/\rho _{f} = 1.05$ . In both these figures, the black dashed-line contours are for unladen pipe flow, where the energetic peak associate with near-wall streaks is located at $y^+ \approx 18$ and $\lambda ^+_x \approx 1000$ . The five contour lines of the unladen case divide the map into different regions, with inner contour lines representing a region with values greater than 1.5.

In figure 14(a) for $\rho _{p}/\rho _{f} = 1$ ( $St^+\approx 1.2$ , $\phi _v = 0.5\,\%$ ), the presence of the particles causes the disappearance of the innermost region with energy higher than 1.5. This is also true for the higher $St^+$ case (figure 14 c) at the same $\phi _v$ . Increasing $\phi _v$ (figures 14 b,c), however, seems to have a more dramatic effect on the energy distribution than $St^+$ . In all cases, the modification to energy is more pronounced in the inner region ( $y^+ \lessapprox 30$ ) than in the outer region. Also, the presence of the particles causes the fluctuating energy to become more evenly distributed throughout the pipe. Nonetheless, the wavelength of peak energy location is more or less unaltered compared to the unladen pipe.

Figure 14. For $\rho _p/\rho _f=1$ , isocontours of pre-multiplied one-dimensional spectra as functions of wall-normal location ( $y^+$ ) and streamwise wavelength ( $\lambda ^{+}_x$ ). The range of contour lines covers 0.3–1.5 in increments of 0.3: (a) case 1A ( $St^+ \approx 1$ , $\phi _v= 0.25\,\%$ ), (b) case 1B ( $St^+ \approx 1$ , $\phi _v= 1\,\%$ ), (c) case 2A ( $St^+ \approx 3.5$ , $\phi _v= 0.25\,\%$ ), and (d) case 2B ( $St^+ \approx 3.5$ , $\phi _v= 1\,\%$ ). The black dashed lines indicate the unladen case.

Figure 15. For $\rho _p/\rho _f=1.05$ , isocontours of premultiplied one-dimensional spectra as functions of wall-normal location ( $y^+$ ) and streamwise wavelength ( $\lambda ^{+}_x$ ). The range of contour lines covers 0.3–1.5 in increments of 0.3: (a–d) upper pipe half, and (e–h) lower pipe half, for (a,e) case 3A ( $St^+ \approx 1.2$ , $\phi _v= 0.25\,\%$ ), (b,f) case 3B ( $St^+ \approx 1.2$ , $\phi _v= 1\,\%$ ), (c,g) case 4A ( $St^+ \approx 3.8$ , $\phi _v= 0.25\,\%$ ), and (d, h) case 4B ( $St^+ \approx 3.8$ , $\phi _v= 1\,\%$ ). The black dashed lines indicate the unladen case.

Premultiplied spectra for $\rho _p/\rho _f=1.05$ in the top half of the pipe are presented in figures 15(a–d), whereas the lower half spectra are in figures 15(e–h). The difference between the premultiplied spectra for $\rho _p/\rho _f=1.05$ and the unladen case in figure 15 is much greater when compared to the difference between the density ratio 1 and the unladen case in figure 14. Furthermore, compared to density ratio 1 in figure 14, the peak energy wavelength location is shifted to smaller wavelengths at $\rho _p/\rho _f=1.05$ in figure 15. This is more evident with increasing $\phi _v$ (right-hand column of figure 15). As suggested before while considering the two-point correlations, the settling particles is the main cause for this, which results in shifting the flow features to smaller length scales, and for the highest $St^+$ and $\phi _v$ , the peak wavelength could be reduced to approximately half.

The anisotropy particle distribution within the pipe for $\rho _p/\rho _f=1.05$ invariably leads to differences in the upper and lower halves of the pipe. The increased energy attenuation in the lower half of the pipe (figures 15 e–h) compared to the upper half (figures 15 a–d), owing to increased particle number density in the lower half, is evident from the reduction in colour levels. Furthermore, upper half energy seems to shift towards the wall more prominently compared to the lower half, which seems to shift energy to large $y^+$ , especially at higher $St^+$ and $\phi _v$ (cf. figure 15 h). In fact, compared to the upper half, the lower half energy contours are also slightly elongated in the wall-normal direction, which is best seen in figure 15(h), where the peak location shifts to slightly larger $y^+$ – in contrast to the upper half, where energy moves to smaller $y^+$ . Note that the upper half at lower $St^+$ and $\phi _v$ shows more similarities to the neutrally buoyant particle. Nevertheless, overall there is a dominance of smaller wavelengths at $\rho _p/\rho _f=1.05$ , which is felt across a larger number of wall-normal locations – and this is not incompatible with particles settling across the pipe towards the lower wall. Finally, similar to the density ratio 1, at $\rho _p/\rho _f=1.05$ , the particle volume fraction seems to have a dominant effect over $St^+$ .