Statistical models for spatial patterns of heavy particles in turbulence

Abstract

The dynamics of heavy particles suspended in turbulent flows is of fundamental importance for a wide range of questions in astrophysics, atmospheric physics, oceanography, and technology. Laboratory experiments and numerical simulations have demonstrated that heavy particles respond in intricate ways to turbulent fluctuations of the carrying fluid: non-interacting particles may cluster together and form spatial patterns even though the fluid is incompressible, and the relative speeds of nearby particles can fluctuate strongly. Both phenomena depend sensitively on the parameters of the system. This parameter dependence is difficult to model from first principles since turbulence plays an essential role. Laboratory experiments are also very difficult, precisely since they must refer to a turbulent environment. But in recent years it has become clear that important aspects of the dynamics of heavy particles in turbulence can be understood in terms of statistical models where the turbulent fluctuations are approximated by Gaussian random functions with appropriate correlation functions. In this review we summarise how such statistical-model calculations have led to a detailed understanding of the factors that determine heavy-particle dynamics in turbulence. We concentrate on spatial clustering of heavy particles in turbulence. This is an important question because spatial clustering affects the collision rate between the particles and thus the long-term fate of the system.

Scale-dependent colocalization in a population of gyrotactic swimmers

“…Recent works have shown that gyrotaxis also produces clustering at very small scales (comparable with the Kolmogorov scale) in nonstationary turbulent flows [17][18][19][20]. In this case cells are found to accumulate on fractal dynamical clusters characterized by a fractal dimension which depends on the cell and flow parameters [17,18,21].…”

Acceleration of raindrop formation due to the tangling-clustering instability in a turbulent stratified atmosphere

“…For example, when the dissipation rate of turbulence is increased from 100 to 400 cm 2 s −3 , the droplet coalescence rate (between droplets with the sizes 18 µm and 20 µm) increases by a factor of 3.5 [9]. The increase of droplet relative velocity and local accumulation of inertial droplets near the periphery of turbulent eddies due to centrifugal effect, can increase droplet collision rate (see, e.g., [1,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]). Numerical simulations showed that due to the effect of preferential concentration of inertial particles in turbulent flows their settling rate is about 20% larger than the terminal fall velocity in the quiescent atmosphere (see, e.g., [1,10,14,15,18,20,21,26]).…”

Fractal catastrophes

“…Examples are particles in turbulence, such as water droplets in turbulent clouds [1], dust in the turbulent gas of protoplanetary disks [2,3], or small particles floating on the free surface of a fluid in motion [4]. When the particle momenta are damped by friction, the phase-space dynamics is dissipative, leading to spatial clustering in the form of fractal patterns in the particle distribution in configuration space [5][6][7]. Spatial clustering has been observed in experiments [8][9][10][11][12][13][14][15] and in numerical simulations of particles in turbulence [15][16][17][18][19][20][21][22][23][24][25][26].…”

“…The fractal nature of spatial clustering is quantified by fractal dimensions [7,[31][32][33][34][35][36] that describe how the fractal patterns fill out configuration space. These dimensions, in turn, are determined by the large-deviation statistics [37][38][39] of finite-time Lyapunov exponents (FTLEs) [35,36,40], measuring the evolution of infinitesimal volumes spanned by nearby particle trajectories.…”