| 1. |
- Andersson, Björn, et al.
(author)
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Advective collisions
- 2007
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In: Europhysics Letters. - : IOP Publishing. - 0295-5075 .- 1286-4854. ; 80
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Journal article (peer-reviewed)
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| 2. |
- Biferale, L., et al.
(author)
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Zermelo's problem: Optimal point-to-point navigation in 2D turbulent flows using reinforcement learning
- 2019
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In: Chaos. - : AIP Publishing. - 1054-1500. ; 29:10
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Journal article (peer-reviewed)abstract
- To find the path that minimizes the time to navigate between two given points in a fluid flow is known as Zermelo's problem. Here, we investigate it by using a Reinforcement Learning (RL) approach for the case of a vessel that has a slip velocity with fixed intensity, Vs, but variable direction and navigating in a 2D turbulent sea. We show that an Actor-Critic RL algorithm is able to find quasioptimal solutions for both time-independent and chaotically evolving flow configurations. For the frozen case, we also compared the results with strategies obtained analytically from continuous Optimal Navigation (ON) protocols. We show that for our application, ON solutions are unstable for the typical duration of the navigation process and are, therefore, not useful in practice. On the other hand, RL solutions are much more robust with respect to small changes in the initial conditions and to external noise, even when V-s is much smaller than the maximum flow velocity. Furthermore, we show how the RL approach is able to take advantage of the flow properties in order to reach the target, especially when the steering speed is small. Published under license by AIP Publishing.
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| 3. |
- Gustafsson, Kristian, 1980
(author)
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Advective Collisions in Random Flows
- 2009
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Licentiate thesis (other academic/artistic)abstract
- In nature, suspensions of small particles in fluids are common. An important example are rain droplets suspended in turbulent clouds. Such clouds can start to rain very quickly and the reason for this is still not fully explained, but it is believed that the turbulent motion in the cloud plays an important role. This thesis gives an introduction to the model we use to describe this system and some results coming from this model. In particular, we will consider collisions between very small droplets that are so light that their motion can be approximated by that of the air flow in the cloud. Collisions between small particles suspended in a fluid can occur due to a macroscopic motion of the fluid [ Z. f. physik. Chemie, XCII, 129-168, (1917)]. This fact was used by Saffman & Turner to estimate the frequency of collisions between the small droplets in clouds [ J. Fluid I., 1, 16-30, (1956)]. We show that this estimate is not adequate, because it only describes a short lived initial transient. We will also discuss different asymptotic behaviors of the model in one spatial dimension. The type of motion of droplets or other particles governed by the model changes as the dimensionless parameters change. It is possible to divide the space spanned by the dimensionless parameters into different regimes, where the particles within each regime exhibit the same kind of motion. Some of these regimes have been discussed in earlier work, whereas others are not as well explored.
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| 4. |
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| 5. |
- Gustafsson, Kristian, 1980, et al.
(author)
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Distribution of velocity gradients and rate of caustic formation in turbulent aerosols at finite Kubo numbers
- 2013
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In: Physical Review E. - 1539-3755. ; 87:2
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Journal article (peer-reviewed)abstract
- In a one-dimensional model for a turbulent aerosol (inertial particles suspended in a random flow) we compute the distributions of particle-velocity gradients and the rate of caustic formation at finite but small Kubo numbers, Ku, for arbitrary Stokes numbers, St. Our results are consistent with those obtained earlier in the limit Ku -> 0 and St -> infinity such that Ku(2)St remains constant. We show how finite-time correlations and nonergodic effects influence the inertial-particle dynamics at finite but small Kubo numbers.
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| 6. |
- Gustafsson, Kristian, 1980
(author)
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Inertial collisions in random flows
- 2011
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Doctoral thesis (other academic/artistic)abstract
- In nature, suspensions of small particles in fluids are common. An important example are rain droplets suspended in turbulent clouds. Such clouds can start to rain very quickly and the reason for this is still not fully explained, but it is believed that the turbulent motion in the cloud plays an important role. This thesis gives an introduction to the model we use to describe inertial particles suspended in such systems and some results coming from this model. We identify a general behavior of the particle motion which is asymptotically correct independent of how the fluid velocity is generated and on the equation of motion of the suspended particles. This asymptotic behavior can be matched to other limiting cases where the details of the system are important. This allows us to calculate an asymptotically correct distribution of particle separations and relative velocities in a form which is universally valid. The form of the distribution depends on the phase-space fractal dimension, which describes the degree upon which particles cluster in phase-space, and on d scales at which the asymptotes are matched, where d is the spatial dimension. If the fluid velocity gradients consist of white-noise, the phase-space fractal dimension and the single matching scale can be calculated analytically in one spatial dimension. We introduce a new series expansion around deterministic particle trajectories. The expansion is done in terms of the magnitude of typical fluctuations of the fluid velocity at a fixed position. If typical fluctuations are small, we can calculate statistical quantities averaged along particle trajectories. In particular, we can calculate the degree of clustering for particles of general inertia in this limit.
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| 7. |
- Gustafsson, Kristian, 1980, et al.
(author)
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Lyapunov Exponents for Particles Advected in Compressible Random Velocity Fields at Small and Large Kubo Numbers
- 2013
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In: Journal of Statistical Physics. - : Springer Science and Business Media LLC. - 0022-4715 .- 1572-9613. ; 153:5, s. 813-827
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Journal article (peer-reviewed)abstract
- We calculate the Lyapunov exponents describing spatial clustering of particles advected in one- and two-dimensional random velocity fields at finite Kubo numbers (a dimensionless parameter characterising the correlation time of the velocity field). In one dimension we obtain accurate results up to by resummation of a perturbation expansion in . At large Kubo numbers we compute the Lyapunov exponent by taking into account the fact that the particles follow the minima of the potential function corresponding to the velocity field. The Lyapunov exponent is always negative. In two spatial dimensions the sign of the maximal Lyapunov exponent lambda (1) may change, depending upon the degree of compressibility of the flow and the Kubo number. For small Kubo numbers we compute the first four non-vanishing terms in the small- expansion of the Lyapunov exponents. By resumming these expansions we obtain a precise estimate of the location of the path-coalescence transition (where lambda (1) changes sign) for Kubo numbers up to approximately . For large Kubo numbers we estimate the Lyapunov exponents for a partially compressible velocity field by assuming that the particles sample those stagnation points of the velocity field that have a negative real part of the maximal eigenvalue of the matrix of flow-velocity gradients.
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| 8. |
- Gustafsson, Kristian, 1980, et al.
(author)
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Relative velocities of inertial particles in turbulent aerosols
- 2014
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In: Journal of turbulence. - : Informa UK Limited. - 1468-5248. ; 15:1, s. 34-69
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Journal article (peer-reviewed)abstract
- We compute the joint distribution of relative velocities and separations of identical inertial particles suspended in randomly mixing and turbulent flows. Our results are obtained by matching asymptotic forms of the distribution. The method takes into account spatial clustering of the suspended particles as well as singularities in their motion (so-called 'caustics'). It thus takes proper account of the fractal properties of phase space and the distribution is characterised in terms of the corresponding phase-space fractal dimension D_2. The method clearly exhibits universal aspects of the distribution (independent of the statistical properties of the flow): at small particle separations R and not too large radial relative speeds |V_R|, the distribution of radial relative velocities exhibits a universal power-law form \rho(V_R,R) \sim |V_R|^{D_2-d-1} provided that D_2 < d+1 (d is the spatial dimension) and that the Stokes number St is large enough for caustics to form. The range in V_R over which this power law is valid depends on R, on the Stokes number, and upon the nature of the flow. Our results are in good agreement with results of computer simulations of the dynamics of particles suspended in random velocity fields with finite correlation times. In the white-noise limit the results are consistent with those of [Gustavsson and Mehlig, Phys. Rev. E84 (2011) 045304].
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| 9. |
- Gustafsson, Kristian, 1980, et al.
(author)
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Statistical models for spatial patterns of heavy particles in turbulence
- 2016
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In: Advances in Physics. - : Informa UK Limited. - 0001-8732 .- 1460-6976. ; 65:1, s. 1-57
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Research review (peer-reviewed)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.
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| 10. |
- Gustafsson, Kristian, 1980, et al.
(author)
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Tumbling of small axisymmetric particles in random and turbulent flows
- 2014
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In: Physical Review Letters. - 0031-9007 .- 1079-7114. ; 112:1
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Journal article (peer-reviewed)abstract
- We analyse the tumbling of small non-spherical, 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 [Parsa et al., Phys. Rev. Lett. 109 (2012) 134501]. 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.
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