Rotational dynamics of a particle in a turbulent stream

The paper presents results for the resolved numerical simulation of a turbulent flow past a homogeneous sphere and a spherical shell of equal mass and radius (and, therefore, with a larger moment of inertia) free to rotate around a fixed center. This situation approximates the behavior of a particle whose relative motion with respect to the fluid is driven by external forces, such as a density difference in a gravitational field. Holding the center fixed makes it possible to have precise information on the turbulent flow incident on the particle by repeating the same simulations without the particle. Two particle Reynolds numbers based on the mean velocity, ${\text{Re}}_p=80$ and 150, are investigated; the incident turbulence has ${\text{Re}}_{\lambda }=36$ and 31, respectively. The particle diameter is an order of magnitude larger than the Kolmogorov length scale and close to the integral length scale. The turbulent eddies that interact most strongly with the particle are characterized. Their size is found to increase with ${\text{Re}}_p$ due to the interplay of the convection timescale, the particle timescale, and the eddy timescale, but it remains of the order of the particle diameter. The sign of the hydrodynamic torque is likely to persist much less than the convection time, although longer durations are also found, revealing the effect of occasional interactions with larger eddies. The autocorrelation of the torque changes sign at shorter and shorter fractions of the convection time as the Reynolds number increases. Significant cross-stream forces are found. An analysis of their magnitude shows that they are mostly due to induced vortex shedding combined with a weaker Magnus-like mechanism.

Figure 1

Schematic illustration of how the turbulence affecting the particle is generated. Zero-mean-velocity homogeneous isotropic turbulence is generated in the cubic domain on the left. The velocity field seen by a plane moving to the left in the cubic domain with velocity $U$ is augmented by a constant velocity $U$ along the long dimension and used as inlet flow velocity for the longer domain containing the particle; see Ref. [25] for details.

Figure 2

PDF of the angular velocity in the cross-stream plane $(x,y)$ normalized as in Eq. (11) for ${\text{Re}}_p=80$ (above) and 150. The highest-peaked (red) and second-highest peaked (black) curves are for the shell and the sphere, respectively. The broadest curve (blue) is for the fluid angular velocity (i.e., $\frac{1}{2}\times \text{vorticity}$) averaged over a sphere of radius $d/2$ (RMS value 3.7179); the other curve (yellow) is for the fluid angular velocity averaged over a sphere of radius $d$ (RMS value 1.8361), both centered at the position of the particle center in the absence of the particle. The dashed lines are Gaussian fits.

Figure 3

PDF of the dimensionless torque in the cross-stream plane $(x,y)$ (black lines, squares) and in the direction of the incident flow (red lines, circles) normalized as in Eq. (13) for ${\text{Re}}_p=80$ (above) and 150. The solid lines and solid symbols are for the solid sphere and the dashed lines and open symbols for the spherical shell.

Figure 4

Time history of the dimensionless torque $L_x^{*}$ for the sphere with ${\text{Re}}_p=150$. Time is normalized by the convection time ${\tau }_c=d/U$. The persistence time of the sign of $L_x^{*}$ is denoted by $\mathrm{\Delta }$.

Figure 5

The solid lines show the PDF's of the sign persistence of the components of the torque in the cross-stream plane acting on the sphere for ${\text{Re}}_p=80$ (upper panel) and ${\text{Re}}_p=150$. The dashed lines are the sign persistence of the cross-stream vorticity in the absence of the particle averaged over spherical volumes with radius $d/4$ (solid square), $d/2$ (solid circle), $d$ (open square), and $1.5d$ (open circle).

Figure 6

The solid lines show the autocorrelation of the cross-stream components of the torque acting on the sphere for ${\text{Re}}_p=80$ (upper panel) and ${\text{Re}}_p=150$. The dashed lines are the autocorrelations of the vorticity averaged over spherical volumes with radius $d/4$ (solid square), $d/2$ (solid circle), $d$ (open square), and $1.5d$ (open circle) in the absence of the particle.

Figure 7

Autocorrelation of the cross-stream components of the particle angular velocity. The solid lines are for the sphere and the dashed lines for the spherical shell. The squares are for ${\text{Re}}_p=80$ and the circles for ${\text{Re}}_p=150$.

Figure 8

The shedding of counterclockwise (negative) vorticity on the right imparts to the particle a (negative) reaction force $F_x$ to the left and, by conservation of angular momentum, a clockwise (positive) rotation ${\mathrm{\Omega }}_y$. As a consequence, this process is associated with a negative value of the products $F_x{\mathrm{\Omega }}_y$ and $F_x{L}_y$.

Figure 9

PDF of the normalized product $F_x{\mathrm{\Omega }}_y$ for the sphere with ${\text{Re}}_p=150$. The slight bias toward negative values is compatible with a cross-stream force component due to vortex shedding.

Figure 10

Scatter plot of $L_y^{*}$ vs. $F_x^{*}$ (left) and $L_x^{*}$ vs. $F_y^{*}$ for the sphere with ${\text{Re}}_p=150$ normalized as in Eqs. (13) and (18), respectively. In spite of the large scatter, a trend compatible with vortex shedding from the particle is clearly visible.

Figure 11

PDF of the product $F_x{L}_y$ normalized by the respective RMS values of the two factors for the sphere with ${\text{Re}}_p=150$. The prevalence of negative values is compatible with the effect of vortex shedding in generating the cross-stream force component $F_x$.

Figure 12

Six successive images separated by 0.1875 ${\tau }_c$ showing an example of the induced shedding of positive (red) vorticity ${\mathrm{\Omega }}_y^{*}$ from the sphere for ${\text{Re}}_p=150$. The range of the blue to red color scale is $-2.5$ $\le {\mathrm{\Omega }}_y^{*}\le 2.5$. The mean incident flow is vertically upward in the $z$ direction; the $x$ axis is horizontal to the right and the $y$ axis into the page. This sequence is part of the movie clip available as Supplemental Material [38]; the first image (a) corresponds to approximately $t=31.7{\tau }_c$ after the start of the movie and the last one (f) to about $t=32.6{\tau }_c$.