Hydrodynamic forces on randomly formed marine aggregates

We study numerically the fluid forces acting on aggregates formed by a collation of cubic particles as a model of marine aggregates in the ocean. The flow around the aggregates and the resulting stresses on the surface of the aggregates are computed in the limit of zero Reynolds number using a boundary integral method, resulting in an accurate evaluation of the flow around fractal objects. We compare a single- and double-layer integral method to compute the velocity, and we determine that the single-layer approach is more suitable to capturing the flow around aggregates. We then characterize the drag of translation flows, the torque of rotational flows, and the straining force of extensional flows acting on aggregates as a function of their size and mode of formation. We determine that the force and torque are best characterized using the gyration radius of the aggregates, and the straining force is better characterized by the maximal radius.

Figure 1

We show typical aggregates as formed by two different methods. Top row: individually added aggregates (IAAs) containing (a) 50, (b) 100, and (c) 200 cubes. Bottom row: cluster-to-cluster aggregates (CCAs) containing (d) 50, (e) 100, and (f) 200 cubes.

Figure 2

Distribution of the nondimensional distance between each constituting cube and the aggregate's center of mass. Here each aggregate is made of $N=100$ cubes, and the periodic domain size, $P$, is varied. The density was calculated using 400 aggregates of each type. We show in (a) individually added aggregates (IAAs) and in (b) cluster-to-cluster aggregates (CCAs). The circles indicate the maximum distance to the center of mass within a given aggregate, averaged over 400 aggregates.

Figure 3

Nondimensional gyration radius of aggregates, $R_g^{\prime }$, as a function of the number of cubes within the aggregates. We present in (a) IAAs and in (b) CCAs. Each box has height given by one standard deviation and is centered around the mean as observed in 400 sample aggregates. The dashed lines are the log-log fits of the mean gyration radius as a function of $N$.

Figure 4

Schematic of the domain under consideration. We solve for the fluid velocity at points ${\vec{x}}_s$ that lie on the surface, $S$, and at points, ${\vec{x}}_0$, that are exterior to $S$. The outward normal to the surface is denoted by $\widehat{n}$.

Figure 5

Sample streamlines of flow past a 20-cube aggregate. The aggregate is assumed to move horizontally, into the page and to the right.

Figure 6

Cubes of various resolutions used for validation. All three cubes have the same volume and contain (a) one interior cube with length $\mathrm{\Delta }x=2$, (b) $5^3=125$ interior cubes with length $\mathrm{\Delta }x=2/5$, and (c) $9^3=729$ interior cubes with length $\mathrm{\Delta }x=2/9$.

Figure 7

(a) Drag on a cube as a function of $\mathrm{\Delta }x$ for both the single- and double-layer methods. (b) Error of the drag as $\mathrm{\Delta }x$ is varied, shown on a log-log scale. Dashed lines are linear fits to the data. The error is computed by taking the difference between the limit shown in (a), $D_{\text{ex}}^{\prime }=25.401$, and drag values, $D^{\prime }$, when $\mathrm{\Delta }x=2,1,0.5,0.25,0.125,$ and 0.0625.

Figure 8

The shaded face is the domain where the stress vector and density shown in Figs. 9 and10 are computed. The red arrow shows the $x$-axis used in Fig. 11. Sample streamlines, discussed in Sec. 4b, are shown as dashed lines. We also show the translation vector, ${\vec{U}}_a^{\prime }$.

Figure 9

Vertical ($z^{\prime }$) component of the stress vector, ${\vec{f}}^{\prime }$, computed using the single-layer potential shown at $z^{\prime }=1$, as illustrated in Fig. 8. The resolution is (a) $\mathrm{\Delta }x=2$, (b) $\mathrm{\Delta }x=0.5$, and (c) $\mathrm{\Delta }x=0.0625$. The color bar is the same for all three figures.

Figure 10

Vertical ($z^{\prime }$) component of the density, ${\vec{\psi }}^{\prime }$, computed using the double-layer potential, shown at $z^{\prime }=1$, as illustrated in Fig. 8. The resolution is (a) $\mathrm{\Delta }x=2$, (b) $\mathrm{\Delta }x=0.5$, and (c) $\mathrm{\Delta }x=0.0625$. The color bar is the same for all three figures and the same as in Fig. 9.

Figure 11

Vertical ($z^{\prime }$) component of the velocity, denoted by $w^{\prime }$, along the line $(x^{\prime },0,1)$ as illustrated in Fig. 8. We show three different resolutions for the (a) single-layer method and (b) double-layer method.

Figure 12

Comparison of the condition number for the linear system of both the single-layer and double-layer methods as the resolution of the cube is varied.

Figure 13

Streamlines around a cube with varying resolution moving in the $z^{\prime }$-direction. The resolution is (a) $\mathrm{\Delta }x=2$, (b) $\mathrm{\Delta }x=1$, (c) $\mathrm{\Delta }x=0.5$, and (d) $\mathrm{\Delta }x=0.25$.

Figure 14

Nondimensional drag of aggregates in a constant flow, ${\vec{U}}_{\text{bg}}^{\prime }=-{\vec{U}}_a^{\prime }=(0,0,1)$, as a function of gyration radius $R_g^{\prime }$ for aggregates of different sizes, $N$. We present in (a) the drag of IAAs and in (b) the drag of CCAs. The dashed lines are least-squares linear fits: in (a) $D^{\prime }=21.47R_g^{\prime }$ and in (b) $D^{\prime }=18.04R_g^{\prime }$.

Figure 15

Rescaled nondimensional drag, $D_r^{\prime }$, as a function of gyration radius $R_g^{\prime }$ for (a) IAAs and (b) CCAs. The rescaled drag, $D_r^{\prime }$, is defined in Eq. (29), and the fits to the data, plotted as dashed lines, are given in Eq. (30) for (a) and in Eq. (31) for (b).

Figure 16

Nondimensional torque, $T^{\prime }$, defined in Eq. (32), as a function of gyration radius, $R_g^{\prime }$, for aggregates composed of a varying number of cubes, $N$. In (a) IAAs are presented, and in (b) CCAs are presented. The dashed lines are least-squares cubic fits: in (a) $T^{\prime }=44.12{\left(R_g^{\prime }\right)}^3$ and in (b) $T^{\prime }=24.98{\left(R_g^{\prime }\right)}^3$.

Figure 17

Nondimensional straining force, $S_f^{\prime }$, defined using Eqs. (33) and (34), as a function of the maximum radius, $R_m^{\prime }$, for aggregates of different sizes $N$. In (a) IAAs and in (b) CCAs are presented. The dashed lines are least-squares quadratic fits: in (a) for IAAs, $S_f^{\prime }=7.48{\left(R_m^{\prime }\right)}^2$ and in (b) for CCAs, $S_f^{\prime }=6.64{\left(R_m^{\prime }\right)}^2$.

Figure 18

Rescaled nondimensional straining force, $S_r^{\prime }$, as a function of maximum radius $R_m^{\prime }$ for aggregates of different sizes $N$. In (a) IAAs and in (b) CCAs are presented. The dashed lines are least-square quadratic fits: in (a) for IAAs, $S_r^{\prime }=8.24{\left(R_m^{\prime }\right)}^2$, in (b) for CCAs, $S_r^{\prime }=7.47{\left(R_m^{\prime }\right)}^2$.

Figure 19

Schematic of the mapped domain where numerical integration is performed.