Effects of the Saffman lift force on particle statistics and turbulence modulation in two-phase flow

Point-particle direct numerical simulations of horizontal open-channel turbulence two-way coupled with inertial particles (${\text{St}}^{+}=1,31,100$) are performed to investigate the effects of the Saffman force (the shear-induced lift force) on particle motion and turbulence modulation. The friction Reynolds numbers of particle-free wall turbulence are ${\text{Re}}_{\tau }=180$ and 580, respectively. The results show that for ${\text{St}}^{+}=31$ and ${\text{Re}}_{\tau }=180$, the near-wall negative lift force increases the rebound velocity of particles after particle-wall collisions and prevents them from following the fluid motions. Therefore, the accumulation of the particles near the wall and the preferential concentration in the low-speed streaks are suppressed by the lift force. As a result, the turbulent velocity fluctuations are decreased below the buffer layer. The destruction of the conditional hairpin vortex is observed because particles with the lift force are easy to cross the vortex core. In the outer layer, the particle-turbulence interaction is increased by the lift force because of higher particle concentration. The significance of the lift force on turbulence modulation decreases with the increasing Reynolds number, albeit the near-wall lift force itself becomes more important at high Reynolds number. Finally, even with the shear-induced lift force, the particle-turbulence interaction is still the strongest for the moderate-inertia particle in the ${\text{St}}^{+}$ range studied in this paper.

Figure 1

Schematics of the channel flow and the computational periodic cell.

Figure 2

Instantaneous snapshot of the three-dimensional vortex structures (isosurface for ${\lambda }_{ci}^{+}=0.01$) in the turbulent region with the particles and ${\lambda }_{ci}^{+}$ in horizontal plane at $y^{+}=15$ for (a) one-way coupled simulation with Saffman shear lift force, (b) two-way coupled simulation with lift force, and (c) without lift force. The vortex structures are colored by wall-normal height and the particles are indicated by small black spheres (${\text{Re}}_{\tau }=180$).

Figure 3

Particle statistics: (a) particle concentration, (b) mean wall-normal lift force, (c) mean streamwise particle velocity, and (d) RMS particle velocity fluctuations profiles for cases A–C (${\text{Re}}_{\tau }=180$).

Figure 4

(a) The ratio between particle concentrations in high-speed fluid ($u_1^{\prime +}>0$) and low-speed fluid ($u_1^{\prime +}<0$). (b) The mean streamwise vorticity magnitude $\langle \left|{\omega }_1^{+}\right|\rangle$ of the fluid (solid lines) and the absolute vorticity $\langle \left|{\omega }_1^{+}(u_{p2}^{+}>0)\right|\rangle$ conditionally averaged at the position of ascending particles ($u_{p2}^{+}>0$). (c) The probability density functions of positive $u_{p2}^{+}$ at $y^{+}=d_p^{+}/2$. (d) The angular distribution functions of particles in the $x$–$z$ slabs with thickness of $\mathrm{\Delta }{\mathrm{y}}^{+}=2.0$ in the streamwise and spanwise directions at $y^{+}=15$ (${\text{Re}}_{\tau }=180$).

Figure 5

Turbulence statistics: (a) mean streamwise fluid velocity and (b) RMS fluid velocity fluctuations profiles for cases B and $\mathrm{C}({\text{Re}}_{\tau }=180$). The inset in (a) represents the particle-induced stress.

Figure 6

Conditionally averaged hairpin vortex based on symmetric Q2 events ($u_1^{\prime }<0,u_2^{\prime }>0,u_3\approx 0$) at $y^{+}=50$ for ${\text{Re}}_{\tau }=180$. The hairpin vortices are identified by the isosurface of 0.3 (0.31 in Zhou et al. [60] and 0.39 in Richter and Sullivan [31]) times the maximum ${\lambda }_{ci}$ in the single-phase flow. Top row corresponds to single-phase flow, middle row corresponds to case C (without lift force), and bottom row corresponds to case B (with lift force). Horizontal slice in the left column is located at $y^{+}=15$ and the vertical slice in the right column is located at $z^{+}=0$. For the left column, the contours represent the conditionally averaged fluid streamwise velocity fluctuation. For the right column, the contours represent the conditionally averaged particle concentration.

Figure 7

Spanwise two-point correlations of the fluid streamwise velocity fluctuation at (a) $y^{+}=15$ and (b) $y^{+}=52$ for cases B and C. The dashed lines indicate $R_{u^{\prime }{u}^{\prime }}=0.2$.

Figure 8

Instantaneous flow field and particle distributions near a buffer-layer quasistreamwise vortex, which rotates clockwise, in the $y^{+}\text{−}{z}^{+}$ plane. The colored streamwise vorticity in all four diagrams are drawn as background, with a red bold arrow near the vortex core indicating its direction of rotation. Dot symbols indicate particle locations. The black arrows in panel (a) indicate the velocity vectors $(u_{p2}^{+},u_{p3}^{+})$ for inertial particles in case C, while those in panel (c) indicate resultant force vectors $(f_2^{+},f_3^{+})$ on them. The corresponding results for case B are shown in panels (b) and (d), respectively.

Figure 9

The particle distribution (a), the drag force (b), and the lift force (c) at the same instant as Figs. 8 and 8, but within an enlarged field of vision. The black arrows in panel (a) indicate the vectors of $(u_{p2}^{+},u_{p3}^{+})$. The same scale is used for the drag force and lift force vectors in panels (b) and (c).

Figure 10

(a) The mean ratio of the lift force to the drag force profiles with ${\text{St}}^{+}=31$ in the three directions for ${\text{Re}}_{\tau }=180$ and 580. (b) The particle concentration profiles with ${\text{St}}^{+}=31$ for ${\text{Re}}_{\tau }=580$ (inset represents the particle concentration ${\varphi }_{vp1}/{\varphi }_v$ at the first statistical height).

Figure 11

Same as described in the caption of Fig. 6 but for ${\text{Re}}_{\tau }=580$.

Figure 12

The relative difference for the streamwise component of (a) particle statistics and (b) fluid statistics with ${\text{St}}^{+}=31$ at ${\text{Re}}_{\tau }=180$ and 580.

Figure 13

(a) Particle concentration and (b) RMS streamwise fluid velocity fluctuation profiles with different inertial particle. The inset in panel (a) is the ratio profiles of the mean lift force to the mean drag force. The inset in panel (b) is the relative differences of RMS of streamwise fluid velocity fluctuation.

Figure 14

The profiles of particle and fluid statistics of (a) particle concentration, (b) mean particle velocity, (c) particle velocity fluctuations, and (d) fluid velocity fluctuations.

Figure 15

(a) The root-mean-square (RMS) of fluid velocity fluctuations and Reynolds shear stress for the single-phase flow, (b) the RMS of particle velocity fluctuations for additional one-way coupled simulation, and (c) the RMS of streamwise fluid velocity fluctuation for case C (${\text{Re}}_{\tau }=180$), compared with two previous simulations under similar parameter conditions.

Figure 16

The profiles of (a) mean fluid velocity and (b) RMS of fluid velocity fluctuations (${\text{Re}}_{\tau }=180, {\text{St}}^{+}=31$).

Figure 17

The profiles of (a) mean slip velocity and (b) ratio of the virtual lift force to the drag force (${\text{Re}}_{\tau }=180, {\text{St}}^{+}=31$).