Statistics of rigid fibers in strongly sheared turbulence

Practically all flows are turbulent in nature and contain some kind of irregularly shaped particles, e.g., dirt, pollen, or life forms such as bacteria or insects. The effects of the particles on such flows and vice versa are highly nontrivial and are not completely understood, particularly when the particles are finite sized. Here, we report an experimental study of millimetric fibers in a strongly sheared turbulent flow. We find that the fibers show a preferred orientation of $-0.38\pi \pm 0.05\pi$ ($-68\pm 9^{\circ }$) with respect to the mean flow direction in high-Reynolds-number Taylor-Couette turbulence, for all studied Reynolds numbers, fiber concentrations, and locations. Despite the finite size of the anisotropic particles, we can explain the preferential alignment by using Jefferey's equation, which provides evidence of the benefit of a simplified point-particle approach. Furthermore, the fiber angular velocity is strongly intermittent, again indicative of point-particle-like behavior in turbulence. Thus large anisotropic particles still can retain signatures of the local flow despite classical spatial and temporal filtering effects.

Figure 1

(a) Schematic of the experimental apparatus (not to scale). The flow is confined between two concentric independently rotating cylinders with radii $r_i$ and $r_o$. Only the inner cylinder (IC) rotates with an angular velocity ${\omega }_i$. A mirror and a window in the bottom plate provide optical access to the $r\text{−}\theta$ plane for a high-speed camera. (b) A typical still image with the inner and outer cylinder highlighted in orange. The fibers (aspect ratio $\mathrm{\Lambda }=5.3$) are clearly visible as white rods. ${\text{Re}}_i=1.7\times {10}^5$ and $\alpha =0.05\%$. (c) Schematic of the $r\text{−}\theta$ plane. The orientation of the particle ${\theta }_p$ is zero when it is aligned with the IC. Fibers with their center in the red areas are removed from all statistics. (d) Definition of the orientation ${\theta }_p$ and the orientation vector $p_i$ of a fiber. ${\theta }_p$ is measured with respect to the azimuthal direction and is defined positive in the counterclockwise direction. See Movies S1 and S2 in the Supplemental Material for typical recordings showing the movement and tracking of the fibers [43].

Figure 2

Fiber velocity as a function of the dimensionless radius $\tilde{r}=(r-r_i)/d$ for ${\text{Re}}_i=1.7\times {10}^5,\alpha =0.05\%$, and $\tilde{z}=z/L=0.24$. All velocities are normalized using the velocity of the IC $u_i$. For comparison, the azimuthal flow profile is included as a red dashed line. We find that the azimuthal velocity of the fibers is very close to the velocity of the flow.

Figure 3

PDF of the fiber orientation ${\theta }_p$ measured at $\tilde{z}=0.24$. Different $\alpha$ are indicated by different hues and different ${\text{Re}}_i$ are shown with different shades. A representation of the fiber alignment is shown at the top of the figure. Independent of $\alpha$ and ${\text{Re}}_i$ there is a clear preference for an alignment around $-0.38\pi \pm 0.05\pi \left(-68\pm 9^{\circ }\right)$. A large $40\%$ difference between the most and least probable orientation is observed.

Figure 4

(a) PDF of the fiber orientation ${\theta }_p$ at various radial bins, indicated by different colors. $\alpha$ is fixed to 0.05%, ${\text{Re}}_i=2.5\times {10}^5$, and the measurement is performed at $\tilde{z}=0.24$. (b) Axial dependence of the PDF of ${\theta }_p$, indicated by different colors. For these measurements, $\alpha =0.05\%$ and ${\text{Re}}_i=8.3\times {10}^4$. The diagram on the right indicates the position of the weak vortical structures [50, 51]. The distribution is found to be nearly independent of radial and axial positions, and all show similar alignment.

Figure 5

Averaged PDF of the experimentally found fiber orientation (dashed) compared to the alignment found from integrating Jeffery's equations (solid lines). The legend indicates the integration timescale as multiples of ${\tau }_{\ell }$.

Figure 6

PDF of the rotation rate of the fibers for $\alpha =0.05\%$ and $z/L=0.24$. Rotational velocities are normalized using the angular velocity of the IC. The PDF is independent of ${\text{Re}}_i$ and shows a slight preference for retrograde rotation (blue). Note that the icons hold for clockwise (CW) rotation of the inner cylinder. The mean rotation is $\langle {\dot{\theta }}_p/{\omega }_i\rangle \approx -0.42$ with a standard deviation of $\sigma ({\dot{\theta }}_p/{\omega }_i)=3.13$, which reveals that a large number of fibers rotate much faster than the inner cylinder. For comparison, a Gaussian distribution with the same mean and variance is added. The skewness and kurtosis are found to lie in the range $[-0.14,0.24]$ and [34, 40], respectively. The inset shows the same data on a linear scale.