From nearly homogeneous to core-peaking suspensions: Insight in suspension pipe flows using MRI and DNS

Magnetic resonance imaging (MRI) experiments have been performed in conjunction with direct numerical simulations (DNS) to study neutrally buoyant particle-laden pipe flows. The flows are characterized by the suspension liquid Reynolds number (${\mathrm{Re}}_s$), based on the bulk liquid velocity and suspension viscosity obtained from Eilers' correlation, the bulk solid volume fraction (${\varphi }_b$), and the particle-to-pipe diameter ratio ($d/D$). Six different cases have been studied, each with a unique combination of ${\mathrm{Re}}_s$ and $\varphi$, while $d/D$ is kept constant at 0.058. The selected cases ensure that the comparison is performed across different flow regimes, each exhibiting characteristic behavior. In general, an excellent agreement is found between experiment and simulation for the average liquid velocity and solid volume fraction profiles. Root-mean-square errors as low as 1.7% and 5.3% are found for the velocity and volume fraction profiles, respectively. This study presents accurate and quantitative velocity and volume fraction profiles of semidilute up to dense suspension flows using both experimental and numerical methods. Three different flow regimes are identified, based on the experimental and numerical solid volume fraction profiles. These profiles explain observations in the drag change. For low bulk solid volume fractions a drag increase (with respect to an equal ${\mathrm{Re}}_s$ single-phase case) is observed. For moderate volume fraction distributions the drag is found to decrease, due to particle accumulation at the pipe center. For high volume fractions the drag is found to decrease further. For solid volume fractions of 0.4 a drag reduction higher than 25% is found. This drag reduction is linked to the strong viscosity gradient in the radial direction, where the relatively low viscosity near the pipe wall acts as a lubrication layer between the pipe wall and the dense core.

Figure 1

Schematic overview of the experimental setup (a) and a photograph of the MRI system and part of the experimental facility (b). The Cartesian coordinate system used for data sampling is superimposed in the top left corner of the photograph, with the $x$ component in the streamwise direction.

Figure 2

Single-phase velocity profiles (markers) compared to reference data (solid lines) for $\mathrm{Re}\approx 900$, 5300, 10 000, and 24 600. For clarity, the cases corresponding to $\mathrm{Re}=10000$ and 24 600 are shown with a vertical offset of 0.25 and 0.5, respectively.

Figure 3

Instantaneous flow snapshots for cases 1–6 (${\mu }_c=0.39$) and 6* (${\mu }_c=0$) as obtained from initial DNS at a grid resolution corresponding to $d/\mathrm{\Delta }x=16$; ${\mathrm{Re}}_s$ values indicated in the snapshots therefore deviate slightly from the ${\mathrm{Re}}_s$ values in Table 3 for $d/\mathrm{\Delta }x=36$. The color denotes the local streamwise velocity in the fluid and solid phase, normalized with the intrinsic liquid bulk velocity. Contours indicate the particle positions.

Figure 4

Time-averaged intrinsic liquid velocity distributions for cases 1–6 obtained from (a) the MRI experiments and (b) the numerical simulations with ${\mu }_c=0.39$. The solid rings are added to identify different radial locations, $r/R=0.25$, 0.5, and 0.75, to assess the axisymmetry of the different cases.

Figure 5

Time-averaged volume fraction distributions for cases 1–6 obtained from (a) the MRI experiments and (b) the numerical simulations with ${\mu }_c=0.39$.

Figure 6

Normalized intrinsic liquid velocity and solid volume fraction profiles for (a) case 1, (b) case 2, (c) case 3, (d) case 4, (e) case 5, and (f) case 6. The experimental velocity (square markers) and solid volume fraction (round markers) results are corresponding to the left and right $y$ axis, respectively. The frictional and frictionless DNS results are represented by the dashed and solid curves, respectively. The flow conditions for the experiments are listed in the figures. Note that the DNS settings may slightly deviate from this (cf. Tables 2 and 3).

Figure 7

Normalized deviation between the experimental and numerical results for (a) the velocity and (b) the solid volume fraction for the frictional cases 1–6. The deviation for the velocity and the solid volume fraction for the frictionless cases 1*–6* is shown in (c) and (d), respectively. For each case every second marker is shown to improve readability.

Figure 8

Solid volume fraction distributions for three characteristic cases (a). The horizontal dashed lines represent the corresponding average solid volume fraction. The regime map corresponding to these cases is shown in the top panel in (b). The triangular markers are the six cases studied above. The round markers in the regime map are additional experimental results (exact conditions are listed in the Appendix). The square markers, connected with a dashed line, are numerical results for a similar pipe flow study ($d/D=1/15$) from Ardekani et al. [37]. The regimes are indicated with roman numerals, corresponding to the profiles in (a). Dashed dotted lines are added which serve as a visual indication of the different regimes. The bottom panel of (b) shows the drag change with respect to the single phase drag for increasing ${\varphi }_b$.

Figure 9

Similar regime map as shown in Fig. (a); here the additional MRI experiments are numbered. The corresponding flow conditions are listed in Table 5.