Reorientation of elongated particles at density interfaces

Density interfaces in the water column are ubiquitously found in oceans and lakes. Interaction of settling particles with pycnoclines plays a pivotal function in nutrient transport between ocean layers and settling rates of marine particles. We perform direct numerical simulations of an elongated particle settling through a density interface and scrutinize the role of stratification on the settling dynamics. It is found that the presence of the density interface tends to turn the long axis of an elongated particle parallel to the settling direction, which is dramatically different from its counterpart in a homogeneous fluid. Although broadside-on settling of the elongated particle is enhanced upon approaching the interface, the long axis rotates toward the settling direction as the particle passes through the interface. We quantify turning couples due to stratification effects, which counteract the pressure-induced torques due to the fluid inertia. A similar behavior is observed for different initial orientations of the particle. It is shown that the reorientation of an elongated particle occurs in both sharp and linear density stratifications.

Figure 1

The schematic of the problem setup.

Figure 2

The temporal variation of the angular velocity around the $y$ axis for an ellipsoid settling in a homogeneous fluid.

Figure 3

Convergence test for the angular velocity of the ellipsoid settling in a sharply stratified fluid for the initial orientation of ${\theta }_0=20$° and buoyancy jump of $b=0.04$. The $L_2$ norm of the error is plotted versus the grid size.

Figure 4

The effect of the buoyancy jump at the density interface on the (a) orientation of the ellipsoid and (b) settling velocity, where $z$ is the vertical position of the particle, $z_l$ is the position of the transition layer, and $(z-z_l)/r=0$ is the density interface. The Reynolds number based on the maximum velocity of the ellipsoid in the upper layer $W_u$ is $\text{Re}={\rho }_u{W}_{u}r/\mu =2.9$.

Figure 5

The snapshots of the settling of an ellipsoid with the initial orientation of ${\theta }_0=20$°, at a density interface with buoyancy jump of $b=0.04$. The ellipsoid is initially at rest and the time increment between frames is $\Delta t/\tau =19.8$, where $\tau =\sqrt{r/g}$.

Figure 6

The magnified view of the ellipsoidal particle at the density interface for $b=0.06$ and ${\theta }_0=20$° at $t/\tau =446$. Color maps in (a) and (b) show contours of dimensionless vorticity and dimensionless shear rate, respectively. Black solid lines illustrate isopycnals at $\rho /{\rho }_u=1.0,1.01,1.02,1.03,1.04,1.05,1.06$.

Figure 7

Color maps show the dimensionless shear rate and solid lines show dimensionless vorticity contours $\omega \tau =0.2,0.3$ for $b=0.06$ and ${\theta }_0=20$°. (a) $t/\tau =495$, (b) $t/\tau =545$, and (c) $t/\tau =594$.

Figure 8

A sequence of images showing the settling of an ellipsoid at a density interface for ${\theta }_0=20$° and $b=0.06$. The ellipsoid is initially at rest and the time increment between frames is $\Delta t/\tau =29.7$.

Figure 9

The orientation (a, c) and settling velocity (b, d) of the ellipsoid as a function of the fallen distance for different initial angles. The results are shown for $b=0.04$ in (a, b) and $b=0.06$ in (c, d).

Figure 10

The effect of the aspect ratio of the ellipsoid on (a) the orientation and (b) settling velocity for $b=0.04$.

Figure 11

The orientation of the particle at the interface is insensitive to the thickness of the transition layer for a thickness comparable to the size of the particle. The results are shown for $b=0.04$.

Figure 12

Settling of an ellipsoid through a density interface in a viscous regime for ${\text{Re}}_t={\rho }_u{W}_{t}d/\mu =0.1$ and $b=0.04$. (a) Orientation of the particle and (b) settling velocity.

Figure 13

Pressure distribution on the surface of the ellipsoid for ${\text{Re}}_t={\rho }_u{W}_{t}d/\mu =0.1$ and $b=0.04$ at three stages of the settling: before reaching the interface where the ellipsoid is $8.5r$ above the interface; close to the interface where the particle is $2r$ above the interface; and after passing through the interface where the ellipsoid is $5r$ below the interface.

Figure 14

Settling of an ellipsoid through a density interface in a viscous regime for ${\text{Re}}_t={\rho }_u{W}_{t}d/\mu =0.1$ and $b=0.04$. (a) The torque acting on the particle as a function of the fallen distance for homogeneous and sharply stratified fluids. (b) The contribution of pressure and viscous stress to the torque acting on the ellipsoid in a sharply stratified fluid. The superscript $*$ refers to the dimensionless torque, where $T^{*}=T/\left(\mu r^2{W}_t\right)$.

Figure 15

The effect of stratification on the settling of an ellipsoid through a linearly stratified fluid in a viscous regime for ${\text{Re}}_t=0.1$ and ${\theta }_0=20$°. (a) The settling velocity versus the fallen distance. (b) The orientation angle versus the fallen distance.

Figure 16

Snapshots of the particle descent in a linearly stratified fluid for ${\text{Re}}_t=0.1$, ${\theta }_0=20$°, and $\text{Fr}=1.03$. The time step between frames is $\Delta t/\tau =297$. The density difference between two adjacent black solid lines is $\Delta \rho /\Lambda r=2$.