Simulations of a porous particle settling in a density-stratified ambient fluid

We study numerically the settling of a porous sphere in a density-stratified ambient fluid. Simulations are validated against prior laboratory experiments and compared to two mathematical models. Two main effects cause the particle to slow down as it enters a density gradient: lighter fluid within the particle and entrainment of the density-stratified ambient fluid. The numerical simulations accurately capture the particle retention time. We quantify the delay in settling due to ambient fluid entrainment and lighter internal fluid becoming denser through diffusion as a function of the Reynolds, Péclet, and Darcy numbers, as well as the thickness of the transition layer and the ratio of the density difference between the lower and upper fluid layers to the density difference between the particle and the upper layer. A simple fitting formula is presented to describe the settling time delay as a function of each of those five nondimensional parameters.

Figure 1

Schematics of the domain under consideration. A porous particle settles in a cylindrical container filled with density-stratified fluid. (a) Fixed frame of reference where the salt concentration $c$ changes smoothly from zero in the upper layer to one in the lower layer with a transition layer thickness $\gamma$. (b) Moving frame of reference with the boundary conditions used in the simulations. Here $\vec{u}=\langle u_r,u_z\rangle =u_r{\widehat{e}}_r+u_z{\widehat{e}}_z$.

Figure 2

(a) Dependence of the vertical velocity on the resolution for different values of $\mathrm{Da}$. (b) Dependence of the vertical velocity on domain size for $\mathrm{\Delta }z=2^{-6}$ and $\mathrm{Da}=2.9\times {10}^{-4}$. The red circles indicate the values used in later simulations, unless otherwise specified.

Figure 3

(a) Time evolution of the average density for a sphere with porosity $\varphi =0.992$ and density ${\rho }_s=1$ filled with fluid of density ${\rho }_u=1$ that is instantly submerged in a fluid of density ${\rho }_l=1.1$ for $\mathrm{Pe}=6.67$. (b) Error of the density calculation, which is computed based on the results obtained with $\mathrm{\Delta }z=2^{-9}$.

Figure 4

Comparison of vertical velocity versus time simulated for different resolutions and domain sizes as a porous sphere settles through a two-layer density stratification. Here $\mathrm{Re}=2.22,\mathrm{Pe}=1746,\eta =0.023,\gamma =46.4$, and $\xi =5.5$.

Figure 5

Vertical velocity versus time for experimental data [17] in blue, reference model (17) in red, enhanced model (19) in yellow, and simulations in purple. (a) The porosity is $\varphi =0.992$, the value given in [17]. (b) The porosity is $\varphi =0.9896$, a value derived by accounting for inertial effects in a uniform ambient. The other parameter values are $\mathrm{Re}=2.22,\mathrm{Pe}=1746,\eta =0.023,\gamma =46.4$, and $\xi =5.5$.

Figure 6

Snapshots of a porous particle settling from a lighter to a heavier layer at several times, with blue showing low values of the concentration $c$ and yellow high values of $c$. The red circle indicates the porous sphere and the red dashed line the moving frame of reference. The domain shown is from $-4\le r\le 4$ and $38\le z\le 62$, and the parameters for this simulation are $\mathrm{Re}=4,\mathrm{Pe}=1746,\xi =1.8,\gamma =11.6,\eta =0.0058,\mathrm{Da}=5\times {10}^{-4}$, and $\varphi =0.992$.

Figure 7

(a) Nondimensional particle vertical velocity over time for several transition layer thicknesses $\gamma$. (b) Distance settled over time for the same values of $\gamma$. Other than $\gamma$, the parameters used are the same as in the case shown in Fig. 6.

Figure 8

Enlarged portion of Fig. 7 showing the particle's vertical velocity over time for $\gamma =4$ and the other parameters as in Fig. 6. We show the vertical velocity computed via the enhanced model (19) in blue and simulations in red.

Figure 9

(a) Nondimensional vertical velocity over time for several $\mathrm{Pe}$. (b) Distance settled over time for the same values of $\mathrm{Pe}$. Other than $\mathrm{Pe}$, the parameters used are the same as in the case shown in Fig. 6.

Figure 10

(a) Nondimensional vertical velocity over time for several $\mathrm{Da}$. (b) Distance settled over time for the same values of $\mathrm{Da}$. Other than $\mathrm{Da}$, the parameters used are the same as in the case shown in Fig. 6.

Figure 11

(a) Nondimensional vertical velocity over time for several $\xi$. (b) Distance settled over time for the same values of $\xi$. Other than $\xi$, the parameters used are the same as in the case shown in Fig. 6.

Figure 12

(a) Nondimensional vertical velocity over time for several $\mathrm{Re}$. The particle experiences bouncing for $\mathrm{Re}\ge 8$. (b) Distance settled over time for the same values of $\mathrm{Re}$. Other than $\mathrm{Re}$, the parameters used are the same as in the case shown in Fig. 6.

Figure 13

(a) Time required to settle a distance of 100 radii across a density gradient for several transition layer thicknesses $\gamma$. The reference time $t_{\text{ref}}$ (blue) was computed using Eq. (17), the enhanced model time $t_{\text{enh}}$ (red) using Eq. (19), and the simulation time $t_{\text{sim}}$ (yellow) using our full numerical simulations. (b) Delay in settling time due to diffusion within the particle $(t_{\text{enh}}-t_{\text{ref}}$, purple), diffusion within the entrained fluid $(t_{\text{sim}}-t_{\text{enh}}$, orange), and both $(t_{\text{sim}}-t_{\text{ref}}$, green). The fit of Eq. (20) is shown as a black dashed line.

Figure 14

(a) Time required to settle a distance of 100 radii across a density gradient for several Péclet numbers $\mathrm{Pe}$. The reference time $t_{\text{ref}}$ (blue) was computed using Eq. (17), the enhanced model time $t_{\text{enh}}$ (red) using Eq. (19), and the simulation time $t_{\text{sim}}$ (yellow) using our full numerical simulations. (b) Delay in settling time due to diffusion within the particle $(t_{\text{enh}}-t_{\text{ref}}$, purple), diffusion within the entrained fluid $(t_{\text{sim}}-t_{\text{enh}}$, orange), and both $(t_{\text{sim}}-t_{\text{ref}}$, green). The fit of Eq. (21) is shown as a black dashed line.

Figure 15

(a) Time required to settle a distance of 100 radii across a density gradient for several Darcy numbers $\mathrm{Da}$. The reference time $t_{\text{ref}}$ (blue) was computed using Eq. (17), the enhanced model time $t_{\text{enh}}$ (red) using Eq. (19), and the simulation time $t_{\text{sim}}$ (yellow) using our full numerical simulations. (b) Delay in settling time due to diffusion within the particle $(t_{\text{enh}}-t_{\text{ref}}$, purple), diffusion within the entrained fluid $(t_{\text{sim}}-t_{\text{enh}}$, orange), and both $(t_{\text{sim}}-t_{\text{ref}}$, green). The fit of Eq. (22) is shown as a black dashed line.

Figure 16

(a) Time required to settle a distance of 100 radii across a density gradient for several values of $\xi =\frac{{\rho }_l-{\rho }_u}{({\rho }_s-{\rho }_u)(1-\varphi )}$. The reference time $t_{\text{ref}}$ (blue) was computed using Eq. (17), the enhanced model time $t_{\text{enh}}$ (red) using Eq. (19), and the simulation time $t_{\text{sim}}$ (yellow) using our full numerical simulations. (b) Delay in settling time due to diffusion within the particle $(t_{\text{enh}}-t_{\text{ref}}$, purple), diffusion within the entrained fluid $(t_{\text{sim}}-t_{\text{enh}}$, orange), and both $(t_{\text{sim}}-t_{\text{ref}}$, green). The fit of Eq. (23) is shown as a black dashed line.

Figure 17

(a) Time required to settle a distance of 100 radii across a density gradient for several Reynolds numbers $\mathrm{Re}$. The reference time $t_{\text{ref}}$ (blue) was computed using Eq. (17), the enhanced model time $t_{\text{enh}}$ (red) using Eq. (19), and the simulation time $t_{\text{sim}}$ (yellow) using our full numerical simulations. (b) Delay in settling time due to diffusion within the particle $(t_{\text{enh}}-t_{\text{ref}}$, purple), diffusion within the entrained fluid $(t_{\text{sim}}-t_{\text{enh}}$, orange), and both $(t_{\text{sim}}-t_{\text{ref}}$, green). The fit of Eq. (24) is shown as a black dashed line.