Inertial effects on the dynamics of rigid heavy fibers in isotropic turbulence

While fiber-turbulence interactions are common in industrial and environmental applications, little is known about inertial fiber dynamics in isotropic turbulence. Here, rotational and translational dynamics of rigid, heavy fibers in air isotropic turbulence were measured using two-orthogonal-view, holographic cinematography. Measurements were conducted in a turbulence chamber (${\mathrm{Re}}_{\lambda }$ = 115). Several batches of nylon fibers with different diameters and lengths were investigated, resulting in Stokes numbers ranging between 1.0 $\le$ St $\le$ 32.5, and fiber length to Kolmogorov length scale ratios ranging between 3.6 $\le \overline{L}/{\eta }_k\le$ 17.3. Ratios between fiber settling velocities in turbulence and quiescent conditions, ${\overline{V}}_2/V_s$, scaled with the ratio of the rms of air fluctuating velocities, $u^{\prime }$, and $V_s$, similar as for spherical particles. Fiber inertia (as indicated by St) decreased the response of the fibers to the fluctuating air velocities, and ratios of the rms values of fluctuating fiber centroid velocities and air velocities dropped from 0.96 to $\sim 0.77$ for the highest St and were well predicted by the model of Wang and Stock [J. Atmos. Sci. 50, 1897 (1993)]. Furthermore, with increasing St, probability density functions (PDFs) of fiber centroid velocities that were well described by a normal distribution narrowed in comparison to the those of the air velocity. PDFs of in-plane fiber rotation rates could not be described by a normal distribution. In the absence of significant length effects, fiber rotation rates were governed by St. Our results indicate that the fiber “tumbling” rate peaks at around St $\approx$ 4, most likely as a result of reduced fiber alignment with the vorticity vector compared to fibers having lower St. At the highest investigated St, decreasing “tumbling” rates are the result of increasingly limited response to the fluctuating flow field, well predicted by the slightly modified model proposed by Bounoua et al. [Phys. Rev. Lett. 121, 124502 (2018)].

Figure 1

Schematic layout of (a) the experimental facility and (b) the volume of interest (VOI) and employed coordinate system. The gravitational acceleration and its direction are indicated by $g$ and an arrow, respectively, in (b).

Figure 2

(a) CDF of $(L^{\left(i\right)}-L^{\left(ii\right)})/L^{\left(ii\right)}$ for all fiber types and examples of fiber curvature for $(L^{\left(i\right)}-L^{\left(ii\right)})/L^{\left(ii\right)}$ = (b) 0.001, (c) 0.01, and (d) 0.03.

Figure 3

Schematic of fiber and associated coordinate systems (a) projected onto the fields of view of cameras I and II (not to scale); (b) definition of the fiber polar angle and unit orientation vector in a comoving coordinate system.

Figure 4

Ratio of the mean settling velocity in isotropic turbulence and the settling velocity in a quiescent fluid ${\overline{V}}_2/V_s$ plotted vs the ratio between the rms of the fluid fluctuating velocity and $V_s$. For a legend of the present results, see Table 2. For clarity, the present results are plotted as filled symbols. Literature results: $+$ [45], $\circ$ [46], $\square$ [47], Friedman and Katz [48]: St = $▵$ 8, $◊$ 4, $\times$ 2, $*$ 1, $○$ 0.5, $▹$ 0.25. Linear relation ${\overline{V}}_2=a{u}^{\prime }$ with $a$ = 0.4 (solid line), 0.25 (dash-dot line), and 0.29 (least-squares fit to present data, dash-dash line).

Figure 5

Root-mean-square ratios of fluctuating fiber and air velocities $v_2^{\prime }/u^{\prime }$ vs ${\mathrm{St}}_T{V}_s/u^{\prime }$. For legend of present results, see Table 2; literature results, experimental data: $○$ Good $\mathit{et}$ al. [47] (E2), and Sabban et al. [33] $▿$ ragweed pollen, $\diamond$ pine pollen, $\square$ corn pollen, $▹$ polystyrene spheres (open and closed symbols denote ${\mathrm{Re}}_{\lambda }$ = 144 and 162, respectively). Dash curve: model by Wang and Stock [54].

Figure 6

Root-mean-square ratios of fluctuating fiber and air velocities $v^{\prime }/u^{\prime }$ vs (a) Stokes number and (b) $\overline{L}/{\eta }_k$. Legend: present results see Table 2; literature results [31]: $\circ$ (St = 1.35, $\overline{L}/{\eta }_k$ = 2.8), $•$ (St = 2.44, $\overline{L}/{\eta }_k$ = 2.8).

Figure 7

PDFs of fluctuating fiber centroid velocities in the transverse directions ($i$ = 1, 3, taken together) and those of the fluid fluctuations normalized by $u^{\prime }$. (a) Group I (fiber types 2 and 5) and (b) group II (fiber types 3, 4, and 6) and type 7. Legend see Table 2; black filled squares: $u_1/u^{\prime }$. Solid, dash, dotted and dash-dot curves are normal distributions fitted to the measurements.

Figure 8

PDFs of fiber in-plane orientation angles measured by camera I for all fiber types. Legend see Table 2.

Figure 9

PDFs of the fibers' in-plane orientation rate ${\dot{\varphi }}_{1-2}$ plotted for several fiber types and normalized by ${\dot{\varphi }}_{1-2}^{\prime }$ of fiber type 1. Fiber type: $▵$ 1 (St = 1.0), $▹$ 2 (St = 2.1), $\star$ 5 (St = 6.1), $◊$ 6 (St = 20.7), and $◃$ 7 (St = 32.5).

Figure 10

Root-mean-square values of the in-plane fiber rotation rates ${\dot{\varphi }}_{1-2}$ and ${\dot{\varphi }}_{2-3}$ measured by cameras I and II, respectively, normalized by ${\overline{\omega }}_3/2$ and plotted vs (a) St and (b) $\overline{L}/{\eta }_k$. Error bars denote uncertainty based on clustered bootstrap analysis [43]. Present data: legend see Table 2, $i$ = 1, $j$ = 2 (open symbols) and $i$ = 2, $j$ = 3 (filled symbols); literature data [31] ($i$ = 1, $j$ = 2): $\circ$ (St = 1.35, $\overline{L}/{\eta }_k$ = 2.8), $•$ (St = 2.44, $\overline{L}/{\eta }_k$ = 2.8), ($i$ = 2, $j$ = 3), gray filled symbols.

Figure 11

PDFs of the normalized, instantaneous squared fiber rotation rate and the normalized, squared out-of-plane component of the vorticity. Present results: fiber type $▵$ 1 (1.0, 3.6), $▹$ 2 (2.1, 5.6), $\square$ 3 (2.3, 8.1), $▿$ 4 (4.1, 8.6),$\star$ 5 (6.1, 5.4), $◊$ 6 (20.7, 9.4), $◃$ 7 (32.5, 17.3); literature results [31]: $\circ$ (1.35, 2.8), $•$ (2.44, 2.8). Values in parentheses denote (St, $\overline{L}/{\eta }_k$). Blue squares: ${\omega }_3^2/\overline{{\omega }_3^2}$. Data for fiber types 2 and 7 as well as ${\omega }_3^2/\overline{{\omega }_3^2}$ are replotted in the inset.

Figure 12

The normalized variance of the tumbling rate $\overline{{\dot{p}}_i{\dot{p}}_i}{\tau }_k^2$ vs the fiber's aspect ratio. Present data: legend see Table 2, mean values and error bars based on clustered bootstrap data analysis [43]. Literature data: DNS: dash-dash lines [25], $+$ [21], solid line (${\mathrm{Re}}_{\lambda }$ = 180) [55]; experiments: $\times ,\square$ [55] (neutrally buoyant), $\circ$ (St = 1.35, $\overline{L}/{\eta }_k$ = 2.8, ${\tau }_k$ = 1.98 ms), $•$ (St = 2.44, $\overline{L}/{\eta }_k$ = 2.8, ${\tau }_k$ = 1.78 ms) [31].

Figure 13

Normalized variance of the tumbling rate $\overline{{\dot{p}}_i{\dot{p}}_i}{\tau }_k^2$ vs Stokes number. Present data: legend see Table 2, mean values and error bars based on clustered bootstrap data analysis [43]. Solid line denotes least-squares fit of Eq. (7) ($A$ = 0.083, $B$ = 0.015) to the data points of fiber types 4–7.