Transport of large particles through the transition to turbulence of a swirling flow

We investigate the behavior of large particles in a transitional swirling flow in a closed vessel, focusing on both their transport and flow sampling. We conduct a Lagrangian study of slightly buoyant particles, considering three particle diameters, keeping the fluid's density and velocity magnitude constant and varying the Reynolds number by changes of the fluid's viscosity. The literature lacks study of material particles in such flows, where chaos and bursts of turbulence can exist, despite applications in industrial and natural situations. One striking result is that particles beyond a certain size, between 10 and 18 mm, are subject to a strong trapping in the vicinity of the islands of the laminar flow, while large particles below this threshold sample the flow homogeneously, independently of the Reynolds number. The exploration of the flow by the large particles is widely different from the preferential sampling in fully developed turbulent conditions, in terms of characteristic times associated with motions escaping the regions and dimensions of the sampled regions. While the particle mean velocity is found to be independent of both the fluid's viscosity and particle size, the fluctuation magnitude strongly increases with decreasing flow viscosity and is marginally affected by size. We also characterize the intensity of the trapping through probability density functions of the particle positions, a measure of the dimensions of the sampled regions, and particle position autocorrelation functions as an attempt to quantify residence times. A qualitative origin for the trapping existence criterion based on a shear-induced lift force is proposed, as an argument for its sole dependence on particle size.

Figure 1

(a) Slices of the von Kármán square vessel showing a disk and the space between the disks with a schematic of the mean flow (only the poloidal component is shown in this plane). The median plane $(x,y,z=0$), separating each toroidal counterrotating cell, and the poloidal neutral lines are represented. The latter correspond to the center of the poloidal recirculations and are located around $\left|z\right|=0.7R$ and $r=\sqrt{x^2+y^2}=0.75R$. (b) Nondimensional torque of the motors $K_p=\varepsilon V/R^5{\left(2\pi \mathrm{\Omega }\right)}^3$ as a function of the large-scale Reynolds number $\text{Re}=2\pi R^2\mathrm{\Omega }/\nu$ for various fluid kinematic viscosities (symbols and colors) and increasing values of the disk frequency. The four vertical lines indicate the Reynolds number values for $\mathrm{\Omega }=4\mathrm{Hz}$ for the conditions of Table 1 (with two extra conditions to capture the trends, $◂$ and $★$, and water-glycerol mixtures of volume ratios 33:77 and 50:50 at temperatures $0{}^{\circ }\mathrm{C}$ and $20{}^{\circ }\mathrm{C}$). The dashed gray lines illustrate the asymptotic trends for low and high Reynolds numbers $K_p\sim 1/\text{Re}$ and $K_p=K_p^{\infty }$, respectively.

Figure 2

(a)–(c) Sample trajectory for a particle of 18 mm, with a fixed duration $T\mathrm{\Omega }=40$ and for the three Reynolds number values considered. The start and end points of the trajectories are indicated by $\blacksquare$ and $•$, respectively. The median plane is indicated by a semitransparent gray plane. (d) Standard deviation of the absolute value of the collection of $Z$ positions for all trajectories as a function of Re, for each particle diameter. (e) Sample trajectory for a particle of 10 mm, with a fixed duration $T\mathrm{\Omega }=40$ for $\text{Re}=200$. (f) Mean velocity field of a 10-mm particle at $\text{Re}=200$ in the half cross section of the vessel, obtained by an Eulerian conditioning of the Lagrangian tracks followed by an azimuthal integration (see [11] for more details). The white circles indicate the locations of the poloidal neutral lines.

Figure 3

Power spectral densities of the (a) transverse $X$ and (b) axial $Z$ particle positions, for a particle of 18 mm, and different Reynolds numbers, subsequently multiplied by a factor 10 for visibility. The peaks identified in (a) by dashed lines are at $f=[0.259,0.794,6.15]\mathrm{\Omega }$. The peaks identified in (b) by dot-dashed lines are at $f=[0.519,1.05,5.89]\mathrm{\Omega }$ and the dashed lines from (a) are reported as well.

Figure 4

(a) Local average and (b) standard deviation of the velocity extracted from an Eulerian conditioning of the Lagrangian tracks for a particle of 10 mm, for $r=0.8R$. The quantities are integrated azimuthally to increase statistical convergence, changing the frame of reference from Cartesian $(x,y,z)$ to cylindrical $(r,\theta ,z)$ for this figure. (a) Azimuthal component of the mean particle velocity and (b) particle velocity fluctuation magnitude $v^{\prime }=\sqrt{(v_r^{\prime 2}+v_{\theta }^{\prime 2}+v_z^{\prime 2})/3}$.

Figure 5

Standard deviation of the particle velocity fluctuations, after subtraction of a local Eulerian mean along the trajectories, for the (a) axial and (b) transverse components, averaged over both the $x$ and $y$ velocity components, as a function of Re, for each particle diameter. For comparison, the values for fully developed turbulent conditions, found independently of Re and $D$, are $0.13U$ and $0.15U$ for the axial and transverse components, respectively.

Figure 6

(a)–(c) Probability density functions of the particle position for a diameter of 18 mm and the three considered Reynolds number. The PDFs are integrated azimuthally to increase statistical convergence, changing the frame of reference from Cartesian $(x,y,z)$ to cylindrical $(r,\theta ,z)$ for this figure. (d) Axial position PDF for $\text{Re}=200$ and the three particle diameters considered (only one-fifth of the symbols are shown for better visibility). The PDF for a 10-mm particle in the fully developed turbulent flow is added for comparison (solid line). The probability to be on the sides of the vessel (corresponding to $\left|z\right|>0.25R$, half of the vessel's volume) goes from 75% to $61\%$ for $D=10\mathrm{mm}$ when Re increases, while it goes from 92% to $69\%$ for $D=24\mathrm{mm}$.

Figure 7

Characteristic residence time $T_0$ on one side of the vessel for a particle of 10 mm and as a function of the Reynolds number. The time is extracted by fitting the exponentially decreasing PDF of the residence time $\mathrm{\Delta }T$ (see the inset). We use the $95\%$ confidence bound to estimate the error bar for the extracted values of $T_0$. (b) Correlation time of the transverse position $T_x$ as a function of the Reynolds number. The time is extracted by fitting the exponential decrease of the envelope of the autocorrelation functions of the transverse position $R_x$. For $\text{Re}=701$, $T_x\simeq 0.5$ s for 10 mm and is roughly twice this value for both 18 and 24 mm. Also shown is the autocorrelation function of the transverse position $R_x$ as a function of the time lag $\tau$ for a (c) 10- and (d) 18-mm particle for the three values of the Reynolds number considered.