Trajectories Of Inertial Particles Research Articles

Time reversibility and non-deterministic behaviour in oscillatorily sheared suspensions of non-interacting particles at high Reynolds numbers

Collections of inertial particles suspended in a viscous fluid and subjected to oscillatory shear have recently attracted much attention due to their relevance to a number of industrial applications and natural phenomena. It is known that, even at very low values of the flow Reynolds number, particle-to-particle interactions can lead to complex chaotic displacements despite the reversibility of the overarching fluid-dynamics (Stokes) equations. For high-Re flows, the loss of predictability after a finite time horizon is generically ascribed to the non-linear nature of the Navier–Stokes equations. Where the sources of nonlinearity are located exactly and how they influence the motion of particles, however, has not been clarified yet. We show that assuming particle interactions to be negligible, surprisingly, at high values of the Reynolds number the major source of non-deterministic behaviour comes from effects of stationary nature in the carrier flow. We report numerical simulations showing precisely how for geometries of finite extent such stationary effects emerge as the time-averaged non-linear response of the Navier–Stokes equations to the applied oscillatory forcing. They cause small deviations of the inertial particle's trajectory from the streamlines of the instantaneous oscillatory flow, which accumulate in time until the system behaviour becomes essentially non reversible.

Coherent Structures and Extreme Events in Rotating Multiphase Turbulent Flows

By using direct numerical simulations (DNS) at unprecedented resolution we study turbulence under rotation in the presence of simultaneous direct and inverse cascades. The accumulation of energy at large scale leads to the formation of vertical coherent regions with high vorticity oriented along the rotation axis. By seeding the flow with millions of inertial particles, we quantify -for the first time- the effects of those coherent vertical structures on the preferential concentration of light and heavy particles. Furthermore, we quantitatively show that extreme fluctuations, leading to deviations from a normal-distributed statistics, result from the entangled interaction of the vertical structures with the turbulent background. Finally, we present the first-ever measurement of the relative importance between Stokes drag, Coriolis force and centripetal forces along the trajectories of inertial particles. We discover that vortical coherent structures lead to unexpected diffusion properties for heavy and light particles in the directions parallel and perpendicular to the rotation axis.

Global stability of trajectories of inertial particles within domains of instability

Finite sized particles exhibit complex dynamics that differ from that of the underlying fluid flow. These dynamics such as chaotic motion, size dependent clustering and separation can have important consequences in many natural and engineered settings. Though fluid streamlines are global attractors for the inertial particles, regions of local instability can exist where the inertial particle can move away from the fluid streamlines. Identifying and manipulating the location of the so called stable and unstable regions in the fluid flow can find important applications in microfluidics. Research in the last two decades has identified analytical criteria that can partition the fluid domain into locally stable and unstable regions. In this paper, we identify two new mechanisms by which neutrally buoyant inertial particles could exhibit globally stable dynamics in the regions of the fluid flow that are thought to be locally unstable and demonstrate this with examples. The examples we use are restricted to the simpler case of time independent fluid flows.

Inertial Steady 2D Vector Field Topology

AbstractVector field topology is a powerful and matured tool for the study of the asymptotic behavior of tracer particles in steady flows. Yet, it does not capture the behavior of finite‐sized particles, because they develop inertia and do not move tangential to the flow. In this paper, we use the fact that the trajectories of inertial particles can be described as tangent curves of a higher dimensional vector field. Using this, we conduct a full classification of the first‐order critical points of this higher dimensional flow, and devise a method to their efficient extraction. Further, we interactively visualize the asymptotic behavior of finite‐sized particles by a glyph visualization that encodes the outcome of any initial condition of the governing ODE, i.e., for a varying initial position and/or initial velocity. With this, we present a first approach to extend traditional vector field topology to the inertial case.

Lagrangian coherent structures and inertial particle dynamics

In this work we investigate the dynamics of inertial particles using finite-time Lyapunov exponents (FTLE). In particular, we characterize the attractor and repeller structures underlying preferential concentration of inertial particles in terms of FTLE fields of the underlying carrier fluid. Inertial particles that are heavier than the ambient fluid (aerosols) attract onto ridges of the negative-time fluid FTLE. This negative-time FTLE ridge becomes a repeller for particles that are lighter than the carrier fluid (bubbles). We also examine the inertial FTLE (iFTLE) determined by the trajectories of inertial particles evolved using the Maxey-Riley equations with nonzero Stokes number and density ratio. Finally, we explore the low-pass filtering effect of Stokes number. These ideas are demonstrated on two-dimensional numerical simulations of the unsteady double-gyre flow.

On the relationship between the non-local clustering mechanism and preferential concentration

‘Preferential concentration’ (Squires & Eaton,Phys. Fluids, vol. A3, 1991, pp. 1169–1178) refers to the clustering of inertial particles in the high strain, low-rotation regions of turbulence. The ‘centrifuge mechanism’ of Maxey (J. Fluid Mech., vol. 174, 1987, pp. 441–465) appears to explain this phenomenon. In a recent paper, Bragg & Collins (New J. Phys., vol. 16, 2014, 055013) showed that the centrifuge mechanism is dominant only in the regime$St\ll 1$, where$St$is the Stokes number based on the Kolmogorov time scale. Outside this regime, the centrifuge mechanism gives way to a non-local, path history symmetry breaking mechanism. However, despite the change in the clustering mechanism, the instantaneous particle positions continue to correlate with high strain, low-rotation regions of the turbulence. In this paper, we analyse the exact equation governing the radial distribution function and show how the non-local clustering mechanism is influenced by, but not dependent upon, the preferential sampling of the fluid velocity gradient tensor along the particle path histories in such a way as to generate a bias for clustering in high strain regions of the turbulence. We also show how the non-local mechanism still generates clustering, but without preferential concentration, in the limit where the time scales of the fluid velocity gradient tensor measured along the inertial particle trajectories approaches zero (such as white noise flows or for particles in turbulence settling under strong gravity). Finally, we use data from a direct numerical simulation of inertial particles suspended in Navier–Stokes turbulence to validate the arguments presented in this study.

Finite-Time Mass Separation for Comparative Visualizations of Inertial Particles

The visual analysis of flows with inertial particle trajectories is a challenging problem because time-dependent particle trajectories additionally depend on mass, which gives rise to an infinite number of possible trajectories passing through every point in space-time. This paper presents an approach to a comparative visualization of the inertial particles’ separation behavior. For this, we define the Finite-Time Mass Separation (FTMS), a scalar field that measures at each point in the domain how quickly inertial particles separate that were released from the same location but with slightly different mass. Extracting and visualizing the mass that induces the largest separation provides a simplified view on the critical masses. By using complementary coordinated views, we additionally visualize corresponding inertial particle trajectories in space-time by integral curves and surfaces. For a quantitative analysis, we plot Euclidean and arc length-based distances to a reference particle over time, which allows to observe the temporal evolution of separation events. We demonstrate our approach on a number of analytic and one real-world unsteady 2D field.

Subgrid-Scale Fluid Statistics along the Inertial Particle Trajectory in Isotropic Turbulence

Fluctuations of sub-grid scale (SGS) velocity and its influence on the motion of particles are important issues for large eddy simulation (LES) of gas-particle turbulence. We obtain the SGS statistics from the direct numerical simulation (DNS) data of homogeneous isotropic turbulent flow, which is filtered by the top filter in physical space. The result shows that the SGS kinetic energy seen by particles reach a minimum value when the Stokes number St closes to 1. Moreover, the turning point shifts towards a larger St as the filter width increases. Different from that observed in DNS, the Lagrangian integral time scale is larger than the Eulerian integral time scale in SGS. For small particles, the Lagrangian timescale of SGS fluid seen by particle is very close to that of the fluid itself. For particles which are large enough, it approaches the Eulerian timescale of SGS fluid. For the intermediate particles, the predicted curve of SGS fluid timescale seen by particle varies with St and its variation is non-monotonic.

Passive scalar characteristics along inertial particle trajectory in turbulent non-isothermal flows

The momentum and heat coupling between carrier fluid and particles are a complex and challenge topic in turbulent reactive gas-solid flow modeling. Most observations on this topic, either numerical or experimental, are based on Eulerian framework, which is not enough for developing the probability density function (PDF) model. In this paper, the instantous behavior and multi-particle statistics of passive scalar along inertial particle trajectory, in homogenous isotropic turbulence with a mean scalar gradient, are investigated by using the direct numerical simulation (DNS). The results show that St∼1.0 particles are easy to aggregate in high strain and low vorticity regions in the fluid field, where the scalar dissipation is usually much higher than the mean value, and that every time they move across the cliff structures, the scalar change is much more intensive. Anyway, the self-correlation of scalar along particle trajectory is significantly different from the velocities observed by particle, for which the prefer-concentration effect is evident. The mechanical-to-thermal time scale ratio averaged along the particles, p, is approximately two times smaller than that computed in the Eulerian frame r, and stays at nearly 1.77 with a weak dependence on particle inertia.

New feature of the solution of a Timoshenko beam carrying the moving mass particle

The paper deals with the problem of vibrations of a Timoshenko beam loaded by a travelling mass particle. Such problems occur in a vehicle/track interaction or a power collector in railways. Increasing speed involves wave phenomena with significant increase of amplitudes. The travelling speed approaches critical values. The moving point mass attached to a structure in some cases can exceed the mass of the structure, i.e. a string or a beam, locally engaged in vibrations. In the literature, the travelling inertial load is often replaced by massless forces or oscillators. Classical solution of the motion equation may involve discussion concerning the contribution of the Dirac delta term, multiplied by the acceleration of the beam in a moving point in the differential equation. Although the solution scheme is classical and successfully applied to numerous problems, in the paper the Lagrange equation of the second kind applied to the problem allows us to obtain the final solution with new features, not reported in the literature. In the case of a string or the Timoshenko beam, the inertial particle trajectory exhibits discontinuity and this phenomenon can be demonstrated or proved mathematically in a particular case. In practice, large jumps of the travelling inertial load is observed.

Scalar Statistics along Inertial Particle Trajectory in Isotropic Turbulence

The statistics of a passive scalar along inertial particle trajectory in homogeneous isotropic turbulence with a mean scalar gradient is investigated by using direct numerical simulation. We are interested in the influence of particle inertia on such statistics, which is crucial for further understanding and development of models in non-isothermal gas-particle flows. The results show that the scalar variance along particle trajectory decreases with the increasing particle inertia firstly; when the particle's Stokes number St is less than 1.0, it reaches the minimal value when St is around 1.0, then it increases if St increases further. However, the scalar dissipation rate along the particle trajectory shows completely contrasting behavior in comparison with the scalar variance. The mechanical-to-thermal time scale ratios averaged along particle, 〈r〉p, are approximately two times smaller than that computed in the Eulerian frame r, and stay at nearly 1.77 with a weak dependence on particle inertia. In addition, the correlations between scalar dissipation and now structure characteristics along particle trajectories, such as strain and vorticity, are also computed, and they reach their maximum and minimum, 0.31 and 0.25, respectively, when St is around 1.0.

Can We Trust the Dust? Evidence of Dust Segregation in Molecular Clouds

Maps of estimated dust column density in molecular clouds are usually assumed to reliably trace the total gas column density structure. In this work we present results showing a clear discrepancy between the dust and the gas distribution in the Taurus molecular cloud complex. We compute the power spectrum of a 2MASS extinction map of the Taurus region and find it is much shallower than the power spectrum of a 13CO map of the same region previously analyzed. This discrepancy may be explained as the effect of grain growth on the grain extinction efficiency. However, this would require a wide range of maximum grain sizes, which is ruled out based on constraints from the extinction curve and the available grain models. We show that major effects due to CO formation and depletion are also ruled out. Our result may therefore suggest the existence of intrinsic spatial fluctuations of the dust to gas ratio, with amplitude increasing toward smaller scales. Preliminary results of numerical simulations of trajectories of inertial particles in turbulent flows illustrate how the process of clustering of dust grains by the cloud turbulence may lead to observable effects. However, these results cannot be directly applied to large scale supersonic and magnetized turbulence at present.

Inertial particle approximation to solutions of the ShallowWater Equations on the rotating spherical Earth

The present study assesses the relationship between solutions of the mechanical problem of particle motion on the surface of a rotating sphere subject only to the gravitation force (called inertial particle motion) and the fluid dynamical problem there, described by the shallow water equations (SWE). Trajectories of fluid parcels advected by a time-dependent velocity field subject to the SWE on the sphere are computed numerically and compared to analytical formulae of inertial particle trajectories. In addition, the free surface height of an ensemble of noninteracting particles is estimated within the classical mechanics framework and compared to computed height of the SWE. The comparison between solutions of the two systems shows very good qualitative as well as quantitative agreement for times of several inertial periods in the following basic low-energy cases: inertial particle oscillations in mid-latitudes (corresponding to inertial waves in fluid dynamics) and inertial motion near the equator. Moreover, for realistic values of the reduced gravity (gH of 1 to 100 m2 s−2) and for time interval of 1–2 d the periods of the trajectories of fluid parcels nearly coincide with those of inertial particles. These results are obtained for a wide range of initial velocity fields and they imply that, at least for time intervals considered, the Coriolis force dominates the motion even after the pressure gradient forces become sufficiently large to affect the motion. They also highlight the fact that fluid parcels of non-linear inertialwaves are subject to the same westward drift as inertial particles and provide an explanation for existence of so called ‘inertial peak’ in the internal oceanic wave spectrum.