K. Gustavsson

Distribution of relative velocities in turbulent aerosols.

We compute the distribution of relative velocities for a one-dimensional model of heavy particles suspended in a turbulent flow, quantifying the caustic contribution to the moments of relative velocities. The same principles determine the corresponding caustic contribution in d spatial dimensions. The distribution of relative velocities Δv at small separations R acquires the universal form ρ(Δv,R)∼R(d-1)|Δv|(D(2)-2d) for large (but not too large) values of |Δv|. Here D(2) is the phase-space correlation dimension. Our conclusions are in excellent agreement with numerical simulations of particles suspended in a randomly mixing flow in two dimensions, and in quantitative agreement with published data on direct numerical simulations of particles in turbulent flows.

Shape-dependence of particle rotation in isotropic turbulence

We consider the rotation of neutrally buoyant axisymmetric particles suspended in isotropic turbulence. Using laboratory experiments as well as numerical and analytical calculations, we explore how particle rotation depends upon particle shape. We find that shape strongly affects orientational trajectories, but that it has negligible effect on the variance of the particle angular velocity. Previous work has shown that shape significantly affects the variance of the tumbling rate of axisymmetric particles. It follows that shape affects the spinning rate in a way that is, on average, complementary to the shape-dependence of the tumbling rate. We confirm this relationship using direct numerical simulations, showing how tumbling rate and spinning rate variances show complementary trends for rod-shaped and disk-shaped particles. We also consider a random but non-turbulent flow. This allows us to explore which of the features observed for rotation in turbulent flow are due to the effects of particle alignment i...

Tumbling of small axisymmetric particles in random and turbulent flows.

We analyze the tumbling of small nonspherical, axisymmetric particles in random and turbulent flows. We compute the orientational dynamics in terms of a perturbation expansion in the Kubo number, and obtain the tumbling rate in terms of Lagrangian correlation functions. These capture preferential sampling of the fluid gradients, which in turn can give rise to differences in the tumbling rates of disks and rods. We show that this is a weak effect in Gaussian random flows. But in turbulent flows persistent regions of high vorticity cause disks to tumble much faster than rods, as observed in direct numerical simulations [S. Parsa, E. Calzavarini, F. Toschi, and G. A. Voth, Phys. Rev. Lett. 109, 134501 (2012)]. For larger particles (at finite Stokes numbers), rotational and translational inertia affects the tumbling rate and the angle at which particles collide, due to the formation of rotational caustics.

Clustering of Particles Falling in a Turbulent Flow

Spatial clustering of identical particles falling through a turbulent flow enhances the collision rate between the falling particles, an important problem in aerosol science. We analyze this problem using perturbation theory in a dimensionless parameter, the so-called Kubo number. This allows us to derive an analytical theory quantifying the spatial clustering. We find that clustering of small particles in incompressible random velocity fields may be reduced or enhanced by the effect of gravity (depending on the Stokes number of the particles) and may be strongly anisotropic.

Correlation dimension of inertial particles in random flows

We obtain an implicit equation for the correlation dimension D 2 of dynamical systems in terms of an integral over a propagator. We illustrate the utility of this approach by evaluating D 2 for inertial particles suspended in a random flow. In the limit where the correlation time of the flow field approaches zero, taking the short-time limit of the propagator enables D 2 to be determined from the solution of a partial differential equation. We develop the solution as a power series in a dimensionless parameter which represents the strength of inertial effects.

Flow Navigation by Smart Microswimmers via Reinforcement Learning

Smart active particles can acquire some limited knowledge of the fluid environment from simple mechanical cues and exert a control on their preferred steering direction. Their goal is to learn the best way to navigate by exploiting the underlying flow whenever possible. As an example, we focus our attention on smart gravitactic swimmers. These are active particles whose task is to reach the highest altitude within some time horizon, given the constraints enforced by fluid mechanics. By means of numerical experiments, we show that swimmers indeed learn nearly optimal strategies just by experience. A reinforcement learning algorithm allows particles to learn effective strategies even in difficult situations when, in the absence of control, they would end up being trapped by flow structures. These strategies are highly nontrivial and cannot be easily guessed in advance. This Letter illustrates the potential of reinforcement learning algorithms to model adaptive behavior in complex flows and paves the way towards the engineering of smart microswimmers that solve difficult navigation problems.

Variable-Range Projection Model for Turbulence-Driven Collisions

We discuss the relative speeds DeltaV of inertial particles suspended in a highly turbulent gas when the Stokes number, a dimensionless measure of their inertia, is large. We identify a mechanism giving rise to the distribution P(DeltaV) approximately exp(-C|DeltaV|(4/3)) (for some constant C). Our conclusions are supported by numerical simulations, and by the analytical solution of a model equation of motion. The results determine the rate of collisions between suspended particles. They are relevant to the hypothesized mechanism for formation of planets by aggregation of dust particles in circumstellar nebula.

Collisions of particles advected in random flows

We consider collisions of particles advected in a fluid. As already pointed out by Smoluchowski (1917 Z. Phys. Chem. 92 129–68), macroscopic motion of the fluid can significantly enhance the frequency of collisions between the suspended particles. This effect was invoked by Saffman and Turner (1956 J. Fluid Mech. 1 16–30) to estimate collision rates of small water droplets in turbulent rain clouds, the macroscopic motion being caused by turbulence. Here, we show that the Saffman–Turner theory is unsatisfactory because it describes an initial transient only. The reason for this failure is that the local flow in the vicinity of a particle is treated as if it were a steady hyperbolic flow, whereas, in reality, it must fluctuate. We derive exact expressions for the steady-state collision rate for particles suspended in rapidly fluctuating random flows and compute how this steady state is approached. For incompressible flows, the Saffman–Turner expression is an upper bound.

Finding efficient swimming strategies in a three-dimensional chaotic flow by reinforcement learning

Abstract.We apply a reinforcement learning algorithm to show how smart particles can learn approximately optimal strategies to navigate in complex flows. In this paper we consider microswimmers in a paradigmatic three-dimensional case given by a stationary superposition of two Arnold-Beltrami-Childress flows with chaotic advection along streamlines. In such a flow, we study the evolution of point-like particles which can decide in which direction to swim, while keeping the velocity amplitude constant. We show that it is sufficient to endow the swimmers with a very restricted set of actions (six fixed swimming directions in our case) to have enough freedom to find efficient strategies to move upward and escape local fluid traps. The key ingredient is the learning-from-experience structure of the algorithm, which assigns positive or negative rewards depending on whether the taken action is, or is not, profitable for the predetermined goal in the long-term horizon. This is another example supporting the efficiency of the reinforcement learning approach to learn how to accomplish difficult tasks in complex fluid environments.Graphical abstract

Analysis of the Correlation Dimension for Inertial Particles

We obtain an implicit equation for the correlation dimension which describes clustering of inertial particles in a complex flow onto a fractal measure. Our general equation involves a propagator of a nonlinear stochastic process in which the velocity gradient of the fluid appears as additive noise. When the long-time limit of the propagator is considered our equation reduces to an existing large-deviation formalism from which it is difficult to extract concrete results. In the short-time limit, however, our equation reduces to a solvability condition on a partial differential equation. In the case where the inertial particles are much denser than the fluid, we show how this approach leads to a perturbative expansion of the correlation dimension, for which the coefficients can be obtained exactly and in principle to any order. We derive the perturbation series for the correlation dimension of inertial particles suspended in three-dimensional spatially smooth random flows with white-noise time correlations,...