Dynamics of the velocity fluctuations in sedimenting suspensions of rigid fibres

TL;DR

This study uses direct numerical simulations with immersed-boundary coupling to investigate how rigid fibre suspensions settling under gravity generate and organize fluid-phase velocity fluctuations. By varying the Galileo number $Ga$ and the fibre concentration $n\ell_f^3$, the authors show that high $Ga$ and low concentration promote gravity-aligned streamers that enhance small-scale fluid activity and shift energy toward finer scales, while also increasing anisotropy and nonlinear interactions. The work dissects energy budgets across scales, revealing distinct mechanisms sustaining vertical versus in-plane fluctuations and identifying intermittent energy transfer with forward cascades punctuated by backscatter, analyzed through the Kármán–Howarth–Monin–Hill framework. Local structure analyses of the velocity gradient tensor reveal a Ga-dependent transition from axisymmetric compression and 2D strain to axisymmetric extension and HIT-like dynamics, with fibres generally aligning with gravity and heavily clustering in descending streams. Overall, the findings illuminate how $Ga$ and fibre concentration shape the multiscale, anisotropic fluctuations in fibre-laden settling flows, offering a framework to extend to flexible fibres and non-Newtonian fluids.

Abstract

We use direct numerical simulations to investigate fluid-solid interactions in suspensions of rigid fibres settling under gravity in a quiescent fluid. The solid-to-fluid density ratio is $\mathcal{O}(100)$, while the Galileo number ($Ga$) and fibre concentration ($n\ell_f^3$) are varied over the ranges $Ga \in [180, 900]$ and $n\ell_f^3 \in [0.36, 23.15]$; $\ell_f$ denotes the fibre length and $n$ the number density. At high $Ga$ and/or low $n\ell_f^3$, fibres cluster into gravity-aligned streamers with elevated concentrations and enhanced settling velocities, disrupting the flow homogeneity. As $Ga$ increases and/or $n\ell_f^3$ decreases, the fluid-phase kinetic energy rises and the energy spectrum broadens, reflecting enhanced small-scale activity. The flow anisotropy is assessed by decomposing the energy spectrum into components aligned with and transverse to gravity. Vertical fluctuations are primarily driven by fluid-solid interactions, while transverse ones are maintained by pressure-strain effects that promote isotropy. With increasing $Ga$, nonlinear interactions become more prominent, producing a net forward energy cascade toward smaller scales, punctuated by localised backscatter events. Analysis of the local velocity gradient tensor reveals distinct flow topologies: at low $Ga$, the flow is dominated by axisymmetric compression and two-dimensional straining; at high $Ga$, regions of high fibre concentration are governed by two-dimensional strain, while voids are associated with axisymmetric extension. The fluid motion is predominantly extensional rather than rotational.

Figures (22)

• Figure 1: Settling velocity of the fibres. The filled symbols refer to the suspension of fibres with $N=10^4$. The empty symbols refer to the terminal velocity of single fibres settling in the same conditions. These velocities are obtained by running additional simulations with the same grid resolution and same domain size, but with $N=1$. Here and in the following, circles refer to cases where streamers are not formed or are rather weak, while squares refer to the cases where the streamers are evident and strong
• Figure 2: Effect of the Galileo number on streamer formation for $n\ell_f^3=2.89$. Fibres are coloured according to the vertical velocity of their midpoints: blue indicates downward motion (negative velocity), red indicates upward motion (positive velocity), and black corresponds to zero vertical velocity. From left to right: $Ga = 180$ ($w_{f,\mathrm{min}} = -0.6335\,u_g$, $w_{f,\mathrm{max}} = 0.3156\,u_g$), $Ga = 450$ ($w_{f,\mathrm{min}} = -1.4282\,u_g$, $w_{f,\mathrm{max}} = 0.4251\,u_g$), and $Ga = 900$ ($w_{f,\mathrm{min}} = -1.2117\,u_g$, $w_{f,\mathrm{max}} = 0.5019\,u_g$).
• Figure 3: Effect of concentration $n \ell_f^3$ on streamer formation for (top) $Ga = 180$ and (bottom) $Ga = 450$. Fibres are coloured according to the vertical velocity of their midpoint Lagrangian point: blue indicates negative velocity, red indicates positive velocity, and black corresponds to near-zero velocity. From left to right, the concentration is $n \ell_f^3 = 0.36$, $n \ell_f^3=2.89$ and $n \ell_f^3 = 23.15$. For $n \ell_f^3=0.36$, $w_{f,\mathrm{min}} = -1.3549\,u_g$, $w_{f,\mathrm{max}} = 0.2906\,u_g$ at $Ga=180$ and $w_{f,\mathrm{min}} = -2.0396\,u_g$, $w_{f,\mathrm{max}} = 0.1505\,u_g$ at $Ga=450$. For $n \ell_f^3 = 2.89$, $w_{f,\mathrm{min}} = -0.6335\,u_g$, $w_{f,\mathrm{max}} = 0.3156\,u_g$ at $Ga=180$ and $w_{f,\mathrm{min}} = -1.4282\,u_g$, $w_{f,\mathrm{max}} = 0.4251\,u_g$ at $Ga=450$. For $n \ell_f^3 = 23.15$, $w_{f,\mathrm{min}} = -0.4025\,u_g$, $w_{f,\mathrm{max}} = 0.1302\,u_g$ at $Ga=180$ and $w_{f,\mathrm{min}} = -0.4527\,u_g$, $w_{f,\mathrm{max}} = 0.1767\,u_g$ at $Ga=450$.
• Figure 4: Effect of $Ga$ and $n \ell_f^3$ on the (left) radial distribution function and (right) local concentration. Panel (a): variation of $Ga$ at fixed $n \ell_f^3 = 2.89$. Panel (b): variation of $n \ell_f^3$ at fixed $Ga = 180$ (top) and $Ga = 450$ (bottom). Symbols follow the convention in figure \ref{['fig:settling_velocity']}, where squares indicate cases with streamer formation, and circles indicate cases without.
• Figure 5: JPDF of fibre concentration $C$ and fibre velocity $w_f$ for $n \ell_f^3 = 2.89$, illustrating the effect of Galileo number. Panels (a–d) correspond to $Ga = 180$, 450, 675, and 900, respectively. Maximum JPDF values are $JPDF_{\mathrm{max}} = 30.59$, 16.94, 16.48, and 12.91 for $Ga = 180$, 450, 675, and 900.