Smoothed particle hydrodynamics and element bending group modeling of flexible fibers interacting with viscous fluids

This paper presents a smoothed particle hydrodynamics (SPH) and element bending group (EBG) coupling method for modeling the interaction of flexible fibers with moving viscous fluids. SPH is a well-developed mesh-free particle method for simulating viscous fluid flows. EBG is also a particle method for modeling flexible bodies. The interaction of flexible fibers with moving viscous fluids is rendered through the interaction of EBG particles for flexible fiber and SPH particles for fluid. In numerical simulation, flexible fibers of different lengths are immersed in a moving viscous fluid driven by a body force. The drag force on the fiber obtained from SPH-EBG simulation agrees well with experimental observations. It is shown that the flexible fiber demonstrates three typical bending modes, including the U-shaped mode, the flapping mode, and the closed mode, and that the flexible fiber experiences a drag reduction due to its reconfiguration by bending. It is also found that the $U$ drag scaling law for a flexible fiber is only valid for the U-shaped mode, but not valid for the flapping and closed modes. The results indicate that the reconfiguration of a flexible fiber is caused by the fluid force acting on it, while vortex shedding is of importance in the translations of bending modes.

Figure 1

A sketch of the computational settings. The midpoint of the fiber is fixed in the midline of the channel, while the two ends of the fiber are free to move. The system is initially at rest and the flow is driven by a body force $g$ in the $x$ direction. The periodic boundary condition is implemented in the $x$ direction, and there is a layer of porous media at the flow inlet area.

Figure 2

Mean flow velocity versus time for a rigid fiber of length $L=2.0\mathrm{cm}$ and three flexible fibers of length $L=3.3,5.0,\mathrm{and}8.0\mathrm{cm}$.

Figure 3

Sketch of kernel interpolation of SPH particles. As the distance between two particles increases, the mutual influence between the two particles reduces.

Figure 4

An EBG is made of two adjacent line segments connecting three neighboring particles. In the EBG model, the bending moment is converted to pairs of forces acting on particles.

Figure 5

Flow diagram of the SPH-EBG implementation. If the time step $d{t}_{\mathrm{EBG}}$ of the inner loop for EBG integration is smaller than the time step $d{t}_{\mathrm{SPH}}$ for SPH integration, the total time of one inner loop for EBG integration should be equal to one time step $d{t}_{\mathrm{SPH}}$.

Figure 6

Three typical bending modes of flexible fibers: (a) U-shaped mode $(L=3.3\mathrm{cm})$, (b) flapping mode $(L=5.0\mathrm{cm})$, and (c) closed mode $(L=8.0\mathrm{cm})$.

Figure 7

The $y$ coordinates of the two endpoints vs velocity for flexible fibers with different lengths: (a) $L=3.3\mathrm{cm}$, (b) $L=5.0\mathrm{cm}$, and (c) $L=8.0\mathrm{cm}$.

Figure 8

Comparisons of smoothed and unsmoothed drag for the flexible fiber of length $L=3.3\mathrm{cm}$.

Figure 9

Numerical results compared with the experimental results from Alben et al. [10]. The fiber of length $L=2.0\mathrm{cm}$ is a rigid fiber, and the corresponding converted thickness of the flow is $2.1\mu \mathrm{m}$. The fiber of length $L=3.3\mathrm{cm}$ is a flexible fiber, and the corresponding converted thickness of the flow is 1.6 μm. The thickness of soap film used in the experiment is in the range of 1–3 μm.

Figure 10

Fluid drag per unit profile length vs flow velocity for a rigid fiber of length $L=2.0\mathrm{cm}$. At a flow velocity of about 1.6 m/s, there is a drag transition because of vortex shedding.

Figure 11

Log-log plots of fluid drag per unit profile length versus flow velocity for the rigid fiber of length $L=2.0\mathrm{cm}$ (a) and the flexible fibers of length $L=3.3\mathrm{cm}$ (b), $L=5.0\mathrm{cm}$ (c), and $L=8.0\mathrm{cm}$ (d).

Figure 12

Drag coefficient versus flow velocity for the rigid fiber of length $L=2.0\mathrm{cm}$ and three flexible fibers of length $L=3.3,5.0,\mathrm{and}8.0\mathrm{cm}$.

Figure 13

The streamlines and vortices of the flexible fiber of length $L=3.3\mathrm{cm}$ at different times (and velocities): (a) $t=0.1\mathrm{s}$ $\left(u=0.091\mathrm{m}/\mathrm{s}\right)$, (b) $t=0.3\mathrm{s}$ $\left(u=0.335\mathrm{m}/\mathrm{s}\right)$, (c) $t=0.5\mathrm{s}$ $\left(u=0.616\mathrm{m}/\mathrm{s}\right)$, (d) $t=1.0\mathrm{s}$ $\left(u=1.403\mathrm{m}/\mathrm{s}\right)$, (e) $t=1.3\mathrm{s}$ $\left(u=1.988\mathrm{m}/\mathrm{s}\right)$, and (f) $t=1.5\mathrm{s}$ $\left(u=2.428\mathrm{m}/\mathrm{s}\right)$. The lines are streamlines and the color shows the angular velocity of SPH particles.

Figure 14

The streamlines and vortices of the flexible fiber of length $L=5.0\mathrm{cm}$ at different times (and velocities): (a) $t=1.0\mathrm{s}$ $\left(u=1.497\mathrm{m}/\mathrm{s}\right)$, (b) $t=1.5\mathrm{s}$ $\left(u=2.671\mathrm{m}/\mathrm{s}\right)$, (c) $t=2.0\mathrm{s}$ $\left(u=3.644\mathrm{m}/\mathrm{s}\right)$, (d) $t=3.0\mathrm{s}$ $\left(u=5.103\mathrm{m}/\mathrm{s}\right)$, (e) $t=3.8\mathrm{s}$ $\left(u=6.573\mathrm{m}/\mathrm{s}\right)$, and (f) $t=3.85\mathrm{s}$ $\left(u=6.586\mathrm{m}/\mathrm{s}\right)$. The lines are streamlines and the color shows the angular velocity of SPH particles.

Figure 15

The streamlines and vortices of the flexible fiber of length $L=8.0\mathrm{cm}$ at different times (and velocities): (a) $t=1.0\mathrm{s}$ $\left(u=1.609\mathrm{m}/\mathrm{s}\right)$, (b) $t=1.3\mathrm{s}$ $\left(u=2.198\mathrm{m}/\mathrm{s}\right)$, (c) $t=1.6\mathrm{s}$ $\left(u=2.780\mathrm{m}/\mathrm{s}\right)$, (d) $t=1.7\mathrm{s}$ $\left(u=2.992\mathrm{m}/\mathrm{s}\right)$, (e) $t=1.8\mathrm{s}$ $\left(u=3.271\mathrm{m}/\mathrm{s}\right)$, and (f) $t=1.85\mathrm{s}$ $\left(u=3.455\mathrm{m}/\mathrm{s}\right)$. The lines are streamlines and the color shows the angular velocity of SPH particles.