Numerical calculation of the particle–fluid–particle stress in random arrays of fixed particles

Based on the nearest particle statistics [Zhang, J. Fluid Mech. 910, A16 (2021)], the phase interaction force in a multiphase flow is decomposed into a particle–mean-field force and the divergence of the particle–fluid–particle (PFP) stress. The PFP stress is proportional to the correlation product of the distance from a particle to its nearest neighbor and the force on the particle conditionally averaged on the nearest-neighbor location. In this work, a functional form of the stress is obtained corrected to the first order of the ratio between the interparticle distance and the macroscopic length scale of the flow. Particle-resolved numerical simulations are used to calculate the PFP stress in random arrays of fixed particles and to explore the physics represented by the stress. The numerical results show that the PFP stress is attractive along the direction of the flow and is repulsive in the directions perpendicular to the flow. In the flow regime simulated, this PFP stress can be considered as a macroscopic representation of the drafting–kissing–tumbling mechanism. The Reynolds stress for the fluid phase is also calculated and compared with the PFP stress.

Figure 1

Sample packing with volume fraction $13.41\%$.

Figure 2

Comparison of radial distribution function $g\left(r\right)$ and probability ${\overline{h}}_2(\mathit{x},\mathit{y})$.

Figure 3

Comparison of drag coefficients for a single sphere with different Reynolds numbers.

Figure 4

Fluid velocity contour in a flow with ${\mathrm{Re}}_m=20$ and ${\theta }_p=13.41\%$.

Figure 5

Variation of normalized drag along flow direction

Figure 6

Comparison of numerically calculated normalized drag with the empirical relation [42] (23).

Figure 7

Decomposition of the antisymmetric force ${\mathit{f}}_a$.

Figure 8

Normalized ${\mathit{f}}_a$ when the nearest pairs are almost perpendicular (left) and parallel (right) to the flow.

Figure 9

Normalized ${\mathrm{\Sigma }}_{xx}$ and ${\mathrm{\Sigma }}_{yy}$ under different ${\theta }_p$ and ${\mathrm{Re}}_m$. In the $y$ labels, $v_r=|{\mathit{v}}_f-{\mathit{v}}_p|$ is the magnitude of the relative velocity

Figure 10

Coefficients $B_1$ and $B_2$ as functions of Re and ${\theta }_p$ (a) $B_1$ with different ${\theta }_p$ (b) $B_2$ as a function of Re.

Figure 11

Correlation coefficients between the nearest-neighbor distance and fluid-particle interaction

Figure 12

Particle pairs across a boundary.