Mixing and collisions of inertial particles at the small scales of turbulence can be investigated by considering how pairs of particles move relative to each other. In real problems the two particles will have different sizes, i.e. they are bidisperse, and the effect of gravity on their motion is often important. However, how turbulence and gravity compete to control the motion of bidisperse inertial particles is poorly understood. Motivated by this, we use direct numerical simulations (DNS) to investigate the dynamics of settling, bidisperse particles in isotropic turbulence. In agreement with previous studies, we find that without gravity (i.e. $Fr=\infty$, where $Fr$ is the Froude number), bidispersity leads to an enhancement of the relative velocities, and a suppression of their spatial clustering. For $Fr<1$, the relative velocities in the direction of gravity are enhanced by the differential settling velocities of the bidisperse particles, as expected. However, we also find that gravity can strongly enhance the relative velocities in the ‘horizontal’ directions (i.e. in the plane normal to gravity). This non-trivial behaviour occurs because fast settling particles experience rapid fluctuations in the fluid velocity field along their trajectory, leading to enhanced particle accelerations and relative velocities. Indeed, the results show that even when $Fr\ll 1$, turbulence can still play an important role, not only on the horizontal motion, but also on the vertical motion of the particles. This is related to the fact that $Fr$ only characterizes the importance of gravity compared with some typical acceleration of the fluid, yet accelerations in turbulence are highly intermittent. As a consequence, there is a significant probability for particles to be in regions of the flow where the Froude number based on the local, instantaneous fluid acceleration is ${>}1$, even though the typically defined Froude number is $\ll 1$. This could imply, for example, that extreme events in the mixing of settling, bidisperse particles are only weakly affected by gravity even when $Fr\ll 1$. We also find that gravity drastically reduces the clustering of bidisperse particles. These results are strikingly different to the monodisperse case, for which recent results have shown that when $Fr<1$, gravity strongly suppresses the relative velocities in all directions, and can enhance clustering. Finally, we consider the implications of these results for the collision rates of settling, bidisperse particles in turbulence. We find that for $Fr=0.052$, the collision kernel is almost perfectly predicted by the collision kernel for bidisperse particles settling in quiescent flow, such that the effect of turbulence may be ignored. However, for $Fr=0.3$, turbulence plays an important role, and the collisions are only dominated by gravitational settling when the difference in the particle Stokes numbers is ${\geqslant}O(1)$.
Expanding recent observations by Hammond & Meng (J. Fluid Mech., vol. 921, 2021, A16), we present a range of detailed experimental data of the radial distribution function (r.d.f.) of inertial particles in isotropic turbulence for different Stokes number, $St$ , showing that the r.d.f. grows explosively with decreasing separation r, exhibiting $r^{-6}$ scaling as the collision radius is approached, regardless of $St$ or particle radius $a$ . To understand such explosive clustering, we correct a number of errors in the theory by Yavuz et al. (Phys. Rev. Lett., vol. 120, 2018, 244504) based on hydrodynamic interactions between pairs of small, weakly inertial particles. A comparison between the corrected theory and the experiment shows that the theory by Yavuz et al. underpredicts the r.d.f. by orders of magnitude. To explain this discrepancy, we explore several alternative mechanisms for this discrepancy that were not included in the theory and show that none of them are likely the explanation. This suggests new, yet-to-be-identified physical mechanisms are at play, requiring further investigation and new theories.
Using Direct Numerical Simulations (DNS), we examine the effects of Taylor Reynolds number, R λ , and Froude number, F r, on the motion of settling, bidisperse inertial particles in isotropic turbulence. Particle accelerations play a key role in the relative motion of bidisperse particles, and we find that reducing F r leads to an enhancement of the accelerations, but a suppression of their intermittency. For Stokes numbers St > 1, the effect of R λ on the accelerations is enhanced by gravity, since settling causes the particle accelerations to be affected by a larger range of flow scales. The results for the Probability Density Function (PDF) of the particle relative velocities show that for bidisperse particles, decreasing F r leads to an enhancement of their relative velocities in both the vertical (parallel to gravity) and horizontal directions. Importantly, our results show that even when the particles are settling very fast, turbulence continues to play a key role in their vertical relative velocities, and increasingly so as R λ is increased. This occurs because although the settling velocity may be much larger than typical velocities of the turbulence, due to intermittency, there are significant regions of the flow where the turbulence contribution to the particle motion is of the same order as that from gravitational settling. Increasing R λ enhances the non-Gaussianity of the relative velocity PDFs, while reducing F r has the opposite effect, and for fast settling particles, the PDFs become approximately Gaussian. Finally, we observe that low-order statistics such as the Radial Distribution Function (RDF) and the particle collision kernel, are strongly affected by F r and St, and especially by the degree of bidispersity of the particles. Indeed, even when the difference in the value of St of the two particles is ≪ 1, the results can differ strongly from the monodisperse case, especially when F r ≪ 1. However, we also find that these low-order statistics are very weakly affected by R λ when St O(1), irrespective of the degree of bidispersity. Therefore, although the mechanisms controlling the collision rates of monodisperse and bidisperse particles are different, they share the property of a weak sensitivity to R λ when St O(1).
The probability density function (PDF) kinetic equation describing the relative motion of inertial particle pairs in a turbulent flow requires closure of the phase-space diffusion current. A novel analytical closure for the diffusion current is presented that is applicable to high-inertia particle pairs with Stokes numbers St r 1. Here St r is a Stokes number based on the time scale τ r of eddies whose size scales with pair separation r. In the asymptotic limit of St r 1, the pair PDF kinetic equation reduces to an equation of Fokker-Planck form. The diffusion tensor characterizing the diffusion current in the Fokker-Planck equation is equal to 1/τ 2 v multiplied by the time integral of the Lagrangian correlation of fluid relative velocities along particle-pair trajectories. Here, τ v is the particle viscous relaxation time. Closure of the diffusion tensor is achieved by converting the Lagrangian correlations of fluid relative velocities 'seen' by pairs into Eulerian fluid-velocity correlations at pair separations that remain essentially constant during time scales of O(τ r ); the pair centre of mass, however, is not stationary and responds to eddies with time scales comparable to or smaller than τ v . For isotropic turbulence, Eulerian fluid-velocity correlations may be expressed as Fourier transforms of the velocity spectrum tensor, enabling us to derive a closed-form expression for the diffusion tensor. A salient feature of this closure is that it has a single, unique form for pair separations spanning the entire spectrum of turbulence scales, unlike previous closures that involve velocity structure functions with different forms for the integral, inertial subrange, and Kolmogorov-scale separations. Using this closure, Langevin equations, which are statistically equivalent to the Fokker-Planck equation, were solved to evolve particle-pair relative velocities and separations in stationary isotropic turbulence. The Langevin equation approach enables the simulation of the full PDF of pair relative motion, instead of only the first few moments of the PDF as is the case in a moments-based approach. Accordingly, PDFs Ω(U|r) and Ω(U r |r) are computed and presented for various separations r, where the former is the PDF of relative velocity U and the latter is the PDF of the radial component of relative velocity U r , both conditioned upon the separation r. Consistent with the direct numerical simulation (DNS) study of Sundaram & Collins (J.
Using Direct Numerical Simulations (DNS), we investigate how gravity modifies the multiscale dispersion of bidisperse inertial particles in isotropic turbulence. The DNS has a Taylor Reynolds number R λ = 398, and we simulate Stokes numbers (based on the Kolmogorov timescale) in the range St ≤ 3 , and consider Froude numbers F r = 0.052 and ∞, corresponding to strong gravity and no gravity, respectively. The degree of bidispersity is quantified by the difference in the Stokes number of the particles |∆St|. We first consider the mean-square separation of bidisperse particle-pairs and find that without gravity (i.e. F r = ∞), bidispersity leads to an enhancement of the the mean-square separation over a significant range of scales. When |∆St| ≥ O(1), the relative dispersion is further enhanced by gravity due to the large difference in the settling velocities of the two particles. However, when |∆St| 1, gravity suppresses the relative dispersion as the settling velocity contribution is small, and gravity suppresses the non-local contribution to the particle dynamics. In order to gain further insights, we consider separately the relative dispersion in the vertical (parallel to gravity) and horizontal directions. As expected, the vertical relative dispersion can be strongly enhanced by gravity due to differences in the settling velocities of the two particles. However, a key finding of our study is that gravity can also significantly enhance the horizontal relative dispersion. This non-trivial effect occurs because fast settling particles experience rapid fluctuations in the fluid velocity field along their trajectory, leading to enhanced particle accelerations and relative velocities. For sufficiently large initial particle separations, however, gravity can lead to a suppression of the horizontal relative dispersion. We also compute the Probability Density Function (PDF) of the particle-pair dispersion. Our results for these PDFs show that even when F r 1 and |∆St| ≥ O(1), the vertical relative dispersion of the particles can be strongly affected by turbulence. This occurs because although the settling velocity contribution to the relative motion is much larger than the "typical" velocities of the turbulence when F r 1 and |∆St| ≥ O(1), due to intermittency, there are significant regions of the flow where the turbulent velocities are of the same order as the settling velocity. These findings imply that in many applications where R λ ≫ 1, the effect of turbulence on the vertical relative dispersion of settling bidisperse particles may never be ignored, even if the particles are settling rapidly.
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