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Register a new userStochastic dynamics of particles trapped in turbulent flows
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The long time dynamics of large particles trapped in two inhomogeneous turbulent shear flows is studied experimentally. Both flows present a common feature, a shear region that separates two colliding circulations, but with different spatial symmetries and temporal behaviors. Because large particles are less and less sensitive to flow fluctuations as their size increases, we observe the emergence of a slow dynamics corresponding to back-and-forth motions between two attractors, and a super-slow regime synchronized with flow reversals when they exist. Such dynamics is substantially reproduced by a one dimensional stochastic model of an over-damped particle trapped in a two-well potential, forced by a colored noise. An extended model is also proposed that reproduces observed dynamics and trapping without potential barrier: the key ingredient is the ratio between the time scales of the noise correlation and the particle dynamics. A total agreement with experiments requires the introduction of spatially inhomogeneous fluctuations and a suited confinement strength.
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We use direct numerical simulations to calculate the joint probability density function of the relative distance $R$ and relative radial velocity component $V_R$ for a pair of heavy inertial particles suspended in homogeneous and isotropic turbulent flows. At small scales the distribution is scale invariant, with a scaling exponent that is related to the particle-particle correlation dimension in phase space, $D_2$. It was argued [1, 2] that the scale invariant part of the distribution has two asymptotic regimes: (1) $|V_R| ll R$ where the distribution depends solely on $R$; and (2) $|V_R| gg R$ where the distribution is a function of $|V_R|$ alone. The probability distributions in these two regimes are matched along a straight line $|V_R| = z^ast R$. Our simulations confirm that this is indeed correct. We further obtain $D_2$ and $z^ast$ as a function of the Stokes number, ${rm St}$. The former depends non-monotonically on ${rm St}$ with a minimum at about ${rm St} approx 0.7$ and the latter has only a weak dependence on ${rm St}$.
We use the Fokker-Planck equation and its moment equations to study the collective behavior of interacting particles in unsteady one-dimensional flows. Particles interact according to a long-range attractive and a short-range repulsive potential field known as Morse potential. We assume Stokesian drag force between particles and their carrier fluid, and find analytic single-peaked traveling solutions for the spatial density of particles in the catastrophic phase. In steady flow conditions the streaming velocity of particles is identical to their carrier fluid, but we show that particle streaming is asynchronous with an unsteady carrier fluid. Using linear perturbation analysis, the stability of traveling solutions is investigated in unsteady conditions. It is shown that the resulting dispersion relation is an integral equation of the Fredholm type, and yields two general families of stable modes: singular modes whose eigenvalues form a continuous spectrum, and a finite number of discrete global modes. Depending on the value of drag coefficient, stable modes can be over-damped, critically damped, or decaying oscillatory waves. The results of linear perturbation analysis are confirmed through the numerical solution of the fully nonlinear Fokker-Planck equation.
We show and explain how a long bead-spring chain, immersed in a homogeneous, isotropic turbulent flow, preferentially samples vortical flow structures. We begin with an elastic, extensible chain which is stretched out by the flow, up to inertial-range scales. This filamentary object, which is known to preferentially sample the circular coherent vortices of two-dimensional (2D) turbulence, is shown here to also preferentially sample the intense, tubular, vortex filaments of 3D turbulence. In the 2D case, the chain collapses into a tracer inside vortices. In 3D, on the contrary, the chain is extended even in vortical regions, which suggests that it follows axially-stretched tubular vortices by aligning with their axes. This physical picture is confirmed by examining the relative sampling behaviour of the individual beads, and by additional studies on an inextensible chain with adjustable bending-stiffness. A highly-flexible, inextensible chain also shows preferential sampling in 3D, provided it is longer than the dissipation scale, but not much longer than the vortex tubes. This is true also for 2D turbulence, where a long inextensible chain can occupy vortices by coiling into them. When the chain is made inflexible, however, coiling is prevented and the extent of preferential sampling in 2D is considerably reduced. In 3D, on the contrary, bending stiffness has no effect, because the chain does not need to coil in order to thread a vortex tube and align with its axis.
Phoresis, the drift of particles induced by scalar gradients in a flow, can result in an effective compressibility, bringing together or repelling particles from each other. Here, we ask whether this effect can affect the transport of particles in a turbulent flow. To this end, we study how the dispersion of a cloud of phoretic particles is modified when injected in the flow, together with a blob of scalar, whose effect is to transiently bring particles together, or push them away from the center of the blob. The resulting phoretic effect can be quantified by a single dimensionless number. Phenomenological considerations lead to simple predictions for the mean separation between particles, which are consistent with results of direct numerical simulations. Using the numerical results presented here, as well as those from previous studies, we discuss quantitatively the experimental consequences of this work and the possible impact of such phoretic mechanisms in natural systems.
We present an extensive numerical study of the time irreversibility of the dynamics of heavy inertial particles in three-dimensional, statistically homogeneous and isotropic turbulent flows. We show that the probability density function (PDF) of the increment, $W(tau)$, of a particles energy over a time-scale $tau$ is non-Gaussian, and skewed towards negative values. This implies that, on average, particles gain energy over a period of time that is longer than the duration over which they lose energy. We call this $textit{slow gain}$ and $textit{fast loss}$. We find that the third moment of $W(tau)$ scales as $tau^3$, for small values of $tau$. We show that the PDF of power-input $p$ is negatively skewed too; we use this skewness ${rm Ir}$ as a measure of the time-irreversibility and we demonstrate that it increases sharply with the Stokes number ${rm St}$, for small ${rm St}$; this increase slows down at ${rm St} simeq 1$. Furthermore, we obtain the PDFs of $t^+$ and $t^-$, the times over which $p$ has, respectively, positive or negative signs, i.e., the particle gains or loses energy. We obtain from these PDFs a direct and natural quantification of the the slow-gain and fast-loss of the particles, because these PDFs possess exponential tails, whence we infer the characteristic loss and gain times $t_{rm loss}$ and $t_{rm gain}$, respectively; and we obtain $t_{rm loss} < t_{rm gain}$, for all the cases we have considered. Finally, we show that the slow-gain in energy of the particles is equally likely in vortical or strain-dominated regions of the flow; in contrast, the fast-loss of energy occurs with greater probability in the latter than in the former.
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