Orientation of Finite Reynolds Number Anisotropic Particles Settling in Turbulence

In this subsection we present equations governing the orientation and settling velocity of small, slender fibers. It is assumed that the fiber Reynolds number $\textit{Re}_{\ell}=W_{min}l/\nu$ is small so that we include inertial effects only when they break the degeneracy of Stokes flow behavior. Here, $W_{min}$ is the settling velocity of the particle in a broadside orientation and $l=L/2$ is the half-length of the particle. The fiber length is much less than the Kolmogorov length scale $L\ll\eta$, so that the turbulent velocity field can be approximated as a local linear flow field. In the absence of particle inertia, fibers experience no net force or torque. In a quiescent fluid, their orientations, described by a unit vector $\boldsymbol{p}$, are independent of time and determined by the initial conditions. Fluid inertia will break this degeneracy and so we include the first effect of fluid inertia on the hydrodynamic torque experienced by a settling fiber even though $\textit{Re}_{\ell}$ is small.

Since a settling particle of length $\textit{L}=2\ell$ disturbs a fluid volume of $\textit{O}(L^{3})$, the particle mass is small compared with the fluid mass disturbed, if

From this relation we can see that for high aspect ratio particles, whenever the particle and fluid densities are comparable, the particle inertia is negligible compared to that of the fluid as is the case in the experimental study. We will use this observation to justify the neglect of particle inertia so that particles experience no net force or torque.

Any external force is balanced by drag and lift forces. Batchelor [3] derived analytical expressions for the drag and lift force valid for $\textit{Re}_{\ell}\ll 1$, $\textit{Re}_{D}\ll 1$ and $\kappa\gg 1$. The balance of forces expressed to leading order in aspect ratio at low $\textit{Re}_{\ell}$ is given by

where $\mathbb{1}$ is the identity matrix, $m=(\rho_{p}-\rho_{f})\pi LD^{2}/4$ is the mass difference between a cylindrical fiber and the displaced fluid, $\mu$ is the dynamic fluid viscosity and $\boldsymbol{g}$ is the gravitational acceleration. A fiber will therefore translate with a quasi-steady state velocity $\boldsymbol{W}$ relative to the local fluid velocity. Equation 2 yields a well-known result for the transverse and longitudinal settling velocities of a fiber

where $W^{f}_{\textit{max}}=|\boldsymbol{W}|_{\theta{=}0}$ and $W^{f}_{\textit{min}}=|\boldsymbol{W}|_{\theta{=}\pi/2}$, respectively. Here, $\theta$ is the angle between $\boldsymbol{p}$ and $\boldsymbol{g}$.

While one can neglect inertial effects on the settling velocity of small fibers, fluid inertia will break the degeneracy of particle orientation when a particle settles in a quiescent fluid. With the inclusion of fluid inertia, fibers experience inertial torques $\boldsymbol{G}_{sed}$ that rotate the particle to an equilibrium orientation where $\boldsymbol{p}$ is perpendicular to $\boldsymbol{W}$. Khayat and Cox [31] derived expressions for the torque experienced by a translating fiber $\boldsymbol{G}_{sed}$ (see their Eq. 6.12), which becomes in the low Reynolds number limit $\left(\textit{Re}_{\ell}\ll 1\right)$ and to leading order in small aspect ratio

The particle also experiences a rotational resistance $\boldsymbol{G}_{rel}$ to its relative rotation [3]:

Here, $\boldsymbol{\Omega}_{\textit{rel}}$ is the rotation of the particle relative to the local fluid rotation. In addition, fibers experience torques due to the fluid strain rate $\boldsymbol{S}=\frac{1}{2}(\boldsymbol{\Gamma}+\boldsymbol{\Gamma}^{T})$:

Here, $\Gamma_{ij}=\partial u_{i}/\partial x_{j}$ is the turbulent velocity gradient. For a symmetric fiber sedimenting in turbulence, in the absence of particle inertia, a torque balance reads:

The torque balance yields the following equation for the time rate of change $\boldsymbol{\dot{p}}$ of the fiber orientation

where the first two terms correspond to Jeffery rotation in the local linear flow field [29] and the last term is the rotation due to the inertial torque caused by the particles sedimentation. Without a background flow, the inertial torque acts to orient a sedimenting spheroidal particle broadside-on to gravity. However, in the presence of additional torque, such as a gravitational torque for an axisymmetric particle with mass asymmetry, the inertial torque can compete to create an oblique settling orientation [55]. In the current study, the torque due to turbulent shear competes with the inertial torque.

A time scale $\tau_{sed}$ of rotation due to sedimentation of a fiber at small $\textit{Re}_{\ell}$ may be defined as the inverse rotation rate of the particle in quiescent fluid at $\theta=45^{\circ}$, where $\theta$ is the angle between $\boldsymbol{p}$ and $\boldsymbol{g}$. When we generalize this definition to disk and triad particles, $\boldsymbol{p}$ will be the normal vector to the plane of the particle. Upon solving Eq. 2 and the component of the tumbling rate due to the inertial torque (from Eq. 8), at $\theta=45^{\circ}$ we get,

where we have used Eq. 2 to find $W_{min}$, the minimum settling velocity of a fiber which is achieved at $\theta=90^{\circ}$. The typical timescale of response of small fibers due to turbulence $\tau_{\textit{turb}}$ is the Kolmogorov timescale,

The orientation distribution of fibers may now be understood by comparing the two time scales $\tau_{\eta}$ and $\tau_{\textit{sed}}$. We define the settling factor as the ratio of these two time scales to be,

When $S_{F}\gg 1$, the rotation due to turbulence is weak compared to that due to the inertial torque and leads to a small deviation from the horizontal orientation, making $\boldsymbol{W}$ align parallel to gravity and perpendicular to $\boldsymbol{p}$. On the other hand at $S_{F}\ll 1$, turbulence is relatively stronger leading to an isotropic orientation distribution. For small fibers, ($L\ll\eta$), and $\textit{Re}_{\ell}\ll 1$ the expression for $S_{F}^{f}$ reduces to,

The superscript $f$ in Eq. 12 indicates the settling factor for small fibers. We will continue to use the general definition Eq. 11 to define settling factors for small triads and disks in the subsequent developments. In the cloud microphysics literature, the parameter Sv denotes the ratio of the Kolmogorov eddy turnover time to the time a particle takes to settle across an eddy, which can also be written as $S_{v}=W/u_{\eta}$ [12]. Thus, we can see that the settling factor at small Reynolds number is proportional to the square of this non-dimensional settling velocity, i.e., $S_{F}^{f}\propto S_{v}^{2}$.

To determine the fiber orientation, we must solve Eq. 2 and Eq. 8 in a reference frame translating with the particle. For slowly settling fibers $S_{F}\ll 1$, the fiber follows a Lagrangian path. For rapidly settling fibers $S_{F}\gg 1$, it will be shown in the next subsection that the fiber orientation arises from a quasi-steady balance of turbulent shear and inertial rotation so that the particle path does not change the orientation. One might then expect to obtain a reasonable estimate of the fiber orientation by simulating fibers experiencing the turbulence on a Lagrangian path for all $S_{F}$. For this purpose we employ a stochastic model. Meneveau [44] reviews stochastic models to describe the fluid velocity gradient along a Lagrangian path. We have employed the model developed by Girimaji and Pope [19] that captures the log-normal distribution of the pseudo-dissipation, the time scale for relaxation of the strain rate tensor on a Lagrangian path, and the tendency of the nonlinear inertial terms to align the vorticity with the strain axes. Shin and Koch [57] showed that this model predicts the rotational velocity variance of neutrally buoyant particles computed in direct-numerical simulations (DNS) with much greater accuracy than a simple Gaussian velocity gradient model (Brunk et al. [7]). Girimaji and Pope obtained favorable comparisons of many of the tensor invariants of turbulence with the DNS of Yeung and Pope [71]. The inputs to the model which include the correlation times for the pseudo-dissipation and the components of the strain rate and the variance of the logarithm of the pseudo-dissipation can be obtained from Yeung and Pope for several values of $Re_{\lambda}$, and we use $Re_{\lambda}=38$ and $93$ for our simulations. Recently, in another problem of particles in a turbulent flow, we have used the Lagrangian velocity gradient model of Girimaji and Pope [19] to explore the role of non-Gaussian statistics in collisions of hydrodynamically interacting particles settling in a turbulent flow [13].

In Figure 2, the orientational variance for a fiber settling in a turbulent flow with $Re_{\lambda}=38$ is shown as a function of the settling factor. It is seen that $\langle\cos^{2}(\theta)\rangle$, where $\theta$ is the angle between $\boldsymbol{p}$ and $\boldsymbol{g}$, shows a smooth transition from an isotropic orientation distribution $\langle\cos^{2}(\theta)\rangle=\frac{1}{3}$ to nearly aligned distribution in which orientational dispersion about the horizontal plane ($\cos\theta=0$) decays as a power law in the settling factor. As will be derived analytically in section 2.2, the decay in the orientational variance about the equilibrium orientation $\theta=90^{\circ}$ follows,

The scaling of the orientational variance with $S_{F}^{f}$ is in agreement with the findings of Gustavsson et al. [21] ($\langle\cos^{2}\theta\rangle\propto$ Sv^-4) and Anand et al. [1]. Anand et al. [1] evaluates the variance in terms of a Froude number, Fr_η, that is identical in definition to that of Sv.

The theory can be extended in an approximate way to larger particles by replacing the time scale of rotation due to sedimentation $\tau_{\textit{sed}}$ with $T_{\textit{sed}}$ extracted from experimental observations of a particle settling in a quiescent fluid

Moreover, the rotations of particles larger than the Kolmogorov length scale are dominated by turbulent eddies close to their size. Therefore, the appropriate turbulent time scale for rotations is the eddy turn over time at the scale of the particle (Parsa and Voth [53])

where $u_{L}^{T}=\sqrt{\langle(\Delta\mathbf{u}\cdot(\mathbb{1}-\boldsymbol{\hat{r}}\boldsymbol{\hat{r}}))^{2}\rangle}$ is the root-mean-square of the transverse components of the fluid velocity difference $\langle(\Delta\mathbf{u}=\mathbf{u}(\mathbf{x}+L\boldsymbol{\hat{r}})-\mathbf{u}(\mathbf{x})$) at the scale of the particle and the factor of $\sqrt{4/15}$ is chosen such that $\tau_{L}\to\tau_{\eta}$ in the limit $L\ll\eta$. The ratio of these two time scales allows us to define a settling factor

that will be used characterize relative rates of sedimentation-induced and turbulent-induced rotation rate in the experiments.

In this subsection, we will derive an analytical prediction for the variance of the orientation in the rapid settling limit, $S_{F}=\tau_{\eta}/\tau_{\textit{sed}}\gg 1$. This indicates that the relaxation of fiber orientation toward its equilibrium horizontal alignment occurs much more rapidly than the Kolmogorov time scale. From Eq. 12, it can be seen that this limit corresponds to one in which the time $\tau_{\textit{samp}}=\eta/W$ for a fiber to sample a Kolmogorov scale eddy is also much smaller than $\tau_{\eta}$. In particular, $\tau_{\eta}/\tau_{\textit{samp}}=W/u_{\eta}={S^{f}_{F}}^{1/2}[\ln(2\kappa)]^{1/2}\gg 1$. From these results it can be seen that the relaxation of fiber orientation is much faster than the sampling time

Thus, despite the rapid translational motion of the particle through the eddies, the fiber responds to changes in the local shear rate and achieves a new orientation sufficiently rapidly so that one may obtain the orientation from a quasi-steady balance of the rotation due to sedimentation and turbulent shear.

We will determine the rotation rate of fibers whose orientations exhibit small deviations from the horizontal plane, so that $\langle{p_{3}}^{2}\rangle\ll 1$ where the $3$ axis is parallel to gravity. We begin with an alternate mobility form of Eq. 2 written using Einstein notation as

where $F=mg$. For $\langle{p_{3}}^{2}\rangle\ll 1$, it can be seen that

Substituting this result into Eq. 8 yields,

Since the bodies will remain horizontal on average in this limit, as expected from theory and shown in simulation, we have $p_{3}\ll p_{1,2}$. Thus,

where $p^{t}_{i}$ denotes the transverse component ($i=1,2$) of the orientation vector, indicating that the fiber orientation samples the plane normal to gravity by turbulent shearing motions. This will lead to an isotropic distribution of orientation within the 1-2 plane. Since $\tau_{\textit{sed}}\ll\tau_{\eta}$, the motion within the 1-2 plane will be slow compared with the equilibration of the 3 component of the fiber orientation with the current turbulent shear flow. In the settling direction (3) there will exist a quasi-static equilibrium because $\tau_{sed}\ll\tau_{\textit{samp}}$. Thus,

Since $p_{3}\ll 1$ we can further simplify the above equation by balancing the second and third terms to obtain,

Thus, the variance characterizing the ”wiggle” out of the horizontal plane is

where the $\langle\rangle$ denotes ensemble averages and $R_{ij}=1/2(\Gamma_{ij}-\Gamma_{ji})$ is the antisymmetric part of the velocity gradient. Cross terms such as $\langle S_{ij}R_{mn}\rangle$ are zero due to isotropy of the turbulent field. In the above expression, we have assumed $\langle p^{t}_{j}p^{t}_{n}\Gamma_{3j}\Gamma_{3n}\rangle$ to be the corresponding product of mean of velocity gradient $\langle\Gamma_{3j}\Gamma_{3n}\rangle$ and orientation $\langle p^{t}_{j}p^{t}_{n}\rangle$ dyads. This is a valid assumption in the rapid settling limit because $\tau_{sed}\ll\tau_{\eta}$ and, as a result, $\boldsymbol{p}^{t}$ changes on a much larger time scale than the velocity gradient. $\langle S_{ij}S_{mn}\rangle$ and $\langle R_{ij}R_{mn}\rangle$ are fourth order isotropic tensors whose form can be deduced using the properties of symmetry and continuity, as shown in Brunk et al. [7] to obtain

where $S^{2}=\langle S_{ij}S_{ij}\rangle$ and $R^{2}=\langle R_{ij}R_{ij}\rangle$. Substituting Eq. 25b in Eq. 24 we have

where we have used the relations $S^{2}=R^{2}=\Gamma_{\eta}^{2}/2$ for homogeneous isotropic turbulence and $S^{f}_{F}$ from Eq. 12. Thus, we have for the rapid settling limit the following relation characterizing the departure of orientation from the horizontal plane due to turbulence,

In our simulations we use a Langrangian model of velocity gradient instead of a particle frame model of turbulence. In the large $S^{f}_{F}$ limit of the simulations, while the strong inertial torque tries to maintain a horizontal orientation, the fiber orientation continues to change on the Kolmogorov time scale in the 1-2 plane. This leads to a correlation of the transverse fiber orientation with the velocity gradient. As a result, our simulations are different from Eq. 27 by a factor corresponding to $\langle p^{t}_{i}p^{t}_{j}\Gamma_{3i}\Gamma_{3j}\rangle/\left(2\Gamma_{\eta}^{2}/15\right)$. In our simulations, this factor is observed to be around 0.66, making the asymptote

In this subsection, we will derive an analytical prediction for the variance of the orientation of rapidly settling disks before studying the rotational dynamics of triads in 2.4. Disks and triads are geometrically similar - triads have 3-fold rotational symmetry while disks have circular symmetry. This hints at possible similarities of the resistance tensors of the two objects. (See Brenner [6] for a general discussion on the equality of resistance tensors for objects based on symmetry considerations.)

An oblate spheroid of semi-major axis length $l$ spinning with relative angular velocity $\bm{\Omega}_{rel}$ with respect to the local fluid rotation in a linear flow experiences a net torque

Here $X^{C},Y^{C}$ and $Y^{H}$ are the scalar resistance functions associated with the dynamics of a spheroidal particle in Stokes flow [32]. For thin disks they take the values

With the inclusion of fluid inertia, disks, like fibers, experience inertial torques that rotate toward an equilibrium orientation with the large dimensions of the particle perpendicular to the velocity. In the case of disks, this corresponds to $\mathbf{p}$ aligned with $\mathbf{W}$. Dabade et al. [11] derived expressions for the torque experienced by a sedimenting spheroidal particle, assuming fluid inertia to be weak. For oblate spheroids, they derive the torque on a thin disk of semi-major axis length $l$

For a thin disk sedimenting in turbulence, in the absence of particle inertia, a torque balance reads:

The zero torque balance yields the following equation for the time rate of change $\dot{\mathbf{p}}$ of the disk orientation

where $c=(19/32-269/105\pi^{2})/12\approx 0.028$. The first three terms on the right-hand side of Equation 33 are the Jeffery rotation rate [29] for a particle with $\kappa\ll 1$ and the third term is the rotation due to the settling-induced inertial torque.

We will now determine the rotation rate of rapidly settling disks that remain nearly aligned with the 3 axis with small fluctuations in the horizontal plane, so that $<p_{3}^{2}>\sim 1$. We begin with the mobility expression giving the sedimentation velocity

where $F=mg$. Equation 34 yields the following well-known result for the transverse and longitudinal settling velocities of a thin disk

where $W^{d}_{max}=|\mathbf{W}|_{\theta=\pi/2}$ and $W^{d}_{min}=|\mathbf{W}|_{\theta=0}$, respectively. For $<p_{3}^{2}>\sim 1$

Substituting this result into Eq. 33 yields

Since the disk remains nearly horizontal, we can write $p_{i}=\delta_{i3}+p_{i}^{t}$ where $p_{i}^{t}\ll 1$. The rotation rate of the transverse component of the orientation vector, $p^{t}_{i}$ ($i=1,2$), then assumes the following simplified form

Thus the variance characterizing the “wiggle” out of the vertical axis is

Similar to fibers we can define a settling factor for disks as

where $\tau_{sed}=4/(3cW^{d\,2}_{min})$ is the time scale of rotation due to sedimentation for a disk whose axis is at a 45^∘ angle to gravity. The small orientation variance in the high $S^{d}_{F}$ limit is

To compare orientational behavior of disks with fibers, we define the average deviation of a disk away from horizontal, similar to the $\langle\cos^{2}\theta\rangle$ for fibers,

In this subsection, the theory and simulation are extended to triads - three armed ramified particles where all three fiber arms lie in the same plane at equal separation. We model ramified particles as hydrodynamically independent fibers connected together to translate and rotate as a single rigid body. Thus, the force and torque balances on the triad are:

where $L=2\ell$ is the arm length,$\boldsymbol{W}^{c}$ is the relative velocity of the triad center of mass with the fluid and $\boldsymbol{W}_{n}=\boldsymbol{W}^{c}+\ell\boldsymbol{\Omega}^{c}\times\boldsymbol{p}_{n}^{\prime}-\ell\boldsymbol{p}_{n}^{\prime}.\boldsymbol{\Gamma}$ is the relative velocity of the nth arm with the fluid. The orientation of each arm is defined by $\boldsymbol{p}_{n}^{\prime}$ and the orientation of a ramified particle is defined by $\boldsymbol{p}$, which is perpendicular to the plane spanned by the arms. From the symmetry of the particle, the gravitational torque sums to zero. As in the case of disks, the minimum velocity occurs when $\boldsymbol{p}$ is parallel to gravity, so that $W^{t}_{\textit{max}}=|\boldsymbol{W}|_{\theta{=}\pi/2}$ and $W^{t}_{\textit{min}}=|\boldsymbol{W}|_{\theta{=}0}$.

This model neglects hydrodynamic interactions among the rods. The influence of hydrodynamic interactions on the drag and the torques due to relative rotation and straining motions are of higher order in the small parameter $1/\ln(2\kappa)$. However, hydrodynamic interactions would influence the inertial torque at the same order of magnitude as the terms retained and the present model that includes only the torque on each arm acting independently is likely an underestimate of the triads’ inertial torque. When comparing with experimental measurements of orientation in turbulent flows, the experimentally observed inertial rotation rate of a large triad is used to correct for this discrepancy.

It is important to note that our theory applies to a case where the Reynolds number is small $Re_{\ell}\ll 1$. In this limit, the rotation of the triad toward horizontal orientations is slow. The competition between turbulent shear and inertial rotation leads to intermediate orientation distributions between isotropic and full alignment when the turbulence is weak $G=\ell\Gamma_{\eta}/W^{f}_{min}\ll 1$ and the Reynolds number is small $Re_{\ell}\ll 1$, but the settling parameter $S^{f}_{F}=\left(\frac{5}{16}\right)\frac{Re_{\ell}}{G}=O(1)$. In this limit the velocity of the triad center of mass $\boldsymbol{W}^{c}$ is much larger than the relative velocity of the arms with respect to the triad center of mass, so that the inertial torque due to the translational motion of the particle dominates that due to the triad’s rotation. In order to ensure these conditions in our simulations, especially at higher settling rates, we scale our force and torque balance equations. We scale Eq. 44a using ${\mu}W^{f}_{min}\ell^{2}$ and Eq. LABEL:eq:torque-balance-ram using ${\mu}{\ell^{3}}\Gamma_{\eta}$, and express Eq. 44 with $\boldsymbol{W}_{n}=\boldsymbol{W}^{c}+\boldsymbol{w}_{n}$, where $\boldsymbol{W}^{c}$ is the velocity of triad center of mass and $\boldsymbol{w}_{n}\propto\ell\Gamma_{\eta}$ is the disturbance velocity experienced by arms $n=1,2,3$. In the limit of $Re_{\ell}\ll 1$ and $G=\ell\Gamma_{\eta}/W^{f}_{min}\ll 1$, the triad equations reduce to

In Eq. 45a, $W^{f}_{min}=\frac{mg\ln 2\kappa}{4\pi\mu L}$ is transverse velocity of a settling fiber, $\bar{\boldsymbol{W}}^{c}=\frac{\boldsymbol{W}^{c}}{W^{f}_{min}}$, and $\hat{\boldsymbol{e}}_{g}$ is the gravitational unit vector. In Eq. 45b, the settling factor is defined using the definition for fibers in Eq. 12, $\bar{\boldsymbol{\Omega}}^{c}=\frac{\boldsymbol{\Omega}^{c}}{\Gamma_{\eta}}$, and $\bar{\boldsymbol{S}}=\frac{\boldsymbol{S}}{\Gamma_{\eta}}$. The symmetry of the triad leads to $\sum_{n=1}^{3}\boldsymbol{p}^{\prime}_{n}\boldsymbol{p}^{\prime}_{n}=3/2(\mathbb{1}-\boldsymbol{pp})$, the signature of a body whose hydrodynamic response is transversely isotropic. The triad equations (Eq. 45a-45b) then assume the following simplified forms -

In the low Reynolds number limit, this ramified particle model (Eq. 46) predicts a different ratio of maximum and minimum sedimentation velocities for triads than for disks. In place of Eqn. 35, we have

In the rapid settling limit the triad is approximated to be in a quasi-steady nearly horizontal orientation, allowing the angular velocity that would rotate the triad out of 12-plane to be neglected. Eq. (47) then simplifies to

Similar to disks, the variance characterizing the “wiggle” of a triad out of the vertical axis is

where $p^{t}_{i}$ ($i=1,2$) is the transverse component of the orientation vector. The asymptotic expression for a triad is qualitatively similar to that for a disk, with a power law dependence.

However, we see a difference of coefficient compared to Eq.  (42) for the disk orientational moment. This reflects differences in the inertial rotation rate of triads and disks as well as the fact that we have used $S^{f}_{F}$ for the settling factor in the preceding development. To define a settling factor for triads, we use Eq. (47) to obtain an expression for ${\boldsymbol{\dot{p}}}$ in a quiescent fluid and find the rotational time scale for a triad oriented at a 45^∘ angle to gravity to be $\tau_{sed}=6/(S_{F}^{f}\Gamma_{\eta})$. Thus, the triad settling factor is

Using Eq. 52, we rewrite Eq. 51 to obtain the average deviation of arms away from the horizontal, a definition similar to $\langle\cos^{2}\theta\rangle$ for fibers, as

Comparing Eqs. 27, 43 , and 53, it is seen that the mean-square deviation of the orientation from horizontal is the same for fibers, disks and triads when defined in terms of settling factors based on the inertial rotation of the respective particles in a quiescent fluid at a 45^∘ angle to gravity.

Fig. 3 presents simulation results for the orientational variance of settling triads obtained by solving the triad velocity, rotation rate and orientational dynamics using Eqs. 45a and 45b in the Lagrangian stochastic fluid velocity gradient model. The simulation results are in good agreement with the rapid settling theory (Eq. 53) for $S^{t}_{F}\gg 1$. Unlike in the case of fibers, correlations between the transverse orientation of triads and disk-like particles do not affect the high $S_{F}$ limit, because the transverse orientation for disk-like particles is very small. Thus, there is no difference between Lagrangian and rapidly settling particle frame models for the high $S_{F}$ behavior of disk-like particles.

Fig. 4 compares stochastic simulation results for triads and fibers at different $Re_{\lambda}$. It may be noted that the general definition of the settling factor $S_{F}$ based on each particle’s inertial rotation rate nearly collapses the results for different particle shapes. The results are also nearly independent of $Re_{\lambda}$.

The theory outlined above can be extended to finite $\textit{Re}_{\ell}$ as long as $\textit{Re}_{D}\ll 1$ by using the full expressions derived by Khayat and Cox [31]. However, the resulting non-linear Reynolds number dependency couples with particle orientation in a non-trivial way when including terms of order $\mathcal{O}(\ln(\kappa)^{2})$. Solving the force and torque balance equations is therefore computationally very expensive. Lopez and Guazzelli [40] suggested a convenient way to handle this case while at the same time keeping computational cost at a minimum. Since the first-order term in aspect ratio nicely decouples drag and lift, we can define two constants, $C_{\perp}$ and $C_{R}$, that account for finite Reynolds number and aspect ratio effects:

Here, $C_{\perp}$ accounts for the overall change in drag on a particle sedimenting at non-zero Reynolds number and $C_{R}$ accounts for the change of the drag ratio between a particle sedimenting with $\theta=0$ and $\theta=\pi/2$. In the low Reynolds number limit, $C_{\perp}=1$ and $C_{R}=1/2$ (see Eq. 3). The expressions for $C_{\perp}$ and $C_{R}$ include the full analytical expressions given by Khayat and Cox [31] and the only approximation comes from interpolating the angular dependence at intermediate orientations. They are defined as:

where $\mathcal{F}_{\parallel}=\mathcal{F}_{D}(\textit{Re}_{\ell},\theta=0)$ and $\mathcal{F}_{\perp}=\mathcal{F}_{D}(\textit{Re}_{\ell},\theta=\pi/2)$. The above expression has two small parameters present - (ln$(2\kappa))^{-1}$ and (ln$(\kappa))^{-1}$. This is due to the difference in the choice of perturbation parameters in the slender body theories of [3] and Khayat and Cox [31]. The expressions for $\mathcal{F}_{D}(\textit{Re}_{\ell},\theta)$ are given by Khayat and Cox [31]. The two constants $C_{\perp}$ and $C_{R}$ are plotted as functions of Reynolds number in Fig. 5 for different aspect ratios. The unexpected behavior of these functions at low Reynolds numbers shows that the theory is very sensitive to the high aspect ratio requirement and that the Stokes flow limit can only be recovered when $\kappa\to\infty$ as $\textit{Re}_{\ell}\to 0$.

Solving Eq. 54 for the velocity of the fiber $\boldsymbol{W}$ yields the same expression as derived by Lopez and Guazzelli [40], who have approached this problem in terms of the mobility matrix.

For finite Reynolds numbers, the experimental measurements of the sedimentation rate of horizontal fibers in quiescent fluid $W_{\textit{min}}=|\boldsymbol{W}|_{\theta{=}\pi/2}$ described in section 3 enable us to determine $C_{\perp}$. This also includes the uncertainties in particle dimensions and density. With

we measure $C_{\perp}\approx 3.5$ and 3.0 for our small fibers and triads, and $C_{\perp}\approx 7.8$ and 7.6 for our large fibers and triads, respectively. In order to determine $C_{R}$, one has to measure the velocity of vertical fibers $W_{\textit{max}}=|\boldsymbol{W}|_{\theta{=}0}$. It is well known that $W_{\textit{max}}=2~{}W_{\textit{min}}$ for slender fibers in the Stokes flow limit. Finite aspect ratio effects lower this ratio, but it increases again slowly when considering finite Reynolds number effects. Our particles fall into the range where this ratio is approximately 2. As seen in Fig. 5, the theoretical predictions for $C_{\perp}$ and $C_{R}$ (from Eqs. 55 and 56) are less than our experimental measurements. This may be due to the finite value of $\textit{Re}_{D}$ in the experiments. Thus measurements of $C_{\perp}$ and $C_{R}$ from quiescent fluid experiments are used for the analysis.

Similar to the finite Reynolds number force corrections ($C_{\perp}$ and $C_{R}$), we introduce a constant $C_{G}$ to correct the inertial torque for large particles. Here, $C_{G}$ is defined as the ratio of inertial torque from the full expression $\mathcal{F}_{G}(\textit{Re}_{\ell},\theta)$ (Eq. 6.13) given by [31] to the low Reynolds number expression $\mathcal{F}_{G}(\textit{Re}_{\ell}\ll 1,\theta)$ (Eq. 6.22). The behavior of $C_{G}$ is plotted in Fig. 6 and shows that the inertial torques decrease quickly with increasing Reynolds number. The value of $C_{G}$ for the experiments can be determined by comparing the time scales from the low Reynolds number model Eq. 9 and the empirically determined time scale Eq. 14:

and we find $C_{G}=0.007$ for small fibers and triads and $C_{G}=0.002$ for large fibers and triads. These values are larger than the theoretical value of $C_{G}$ at the corresponding particle Reynolds numbers and $\theta=45^{\circ}$, where $C_{G}=0.002$ for small fibers and $C_{G}=0.0002$ for large fibers.

With the force and torque correction factors $C_{\perp},C_{R}$ and $C_{G}$, the force and torque balance equations for large triads can be written as: