scholarly journals Polydisperse streaming instability – I. Tightly coupled particles and the terminal velocity approximation

2020 ◽  
Vol 499 (3) ◽  
pp. 4223-4238
Author(s):  
Sijme-Jan Paardekooper ◽  
Colin P McNally ◽  
Francesco Lovascio
Keyword(s):  
Growth Rates ◽  
Time Scale ◽  
Stokes Number ◽  
Terminal Velocity ◽  
Streaming Instability ◽  
Dust Size Distribution ◽  
Dynamical Time ◽  
Tightly Coupled ◽  
Dust Component ◽  
Velocity Approximation

ABSTRACT We introduce a polydisperse version of the streaming instability (SI), where the dust component is treated as a continuum of sizes. We show that its behaviour is remarkably different from the monodisperse SI. We focus on tightly coupled particles in the terminal velocity approximation and show that unstable modes that grow exponentially on a dynamical time-scale exist. However, for dust to gas ratios much smaller than unity, they are confined to radial wavenumbers that are a factor $\sim 1/{\overline{\rm St}}$ larger than where the monodisperse SI growth rates peak. Here ${\overline{\rm St}}\ll 1$ is a suitable average Stokes number for the dust size distribution. For dust to gas ratios larger than unity, polydisperse modes that grow on a dynamical time-scale are found as well, similar as for the monodisperse SI and at similarly large wavenumbers. At smaller wavenumbers, where the classical monodisperse SI shows secular growth, no growing polydisperse modes are found under the terminal velocity approximation. Outside the region of validity for the terminal velocity approximation, we have found unstable epicyclic modes that grow on ∼104 dynamical time-scales.

scholarly journals Concentration and velocity statistics of inertial particles in upward and downward pipe flow

2017 ◽  
Vol 822 ◽  
pp. 640-663 ◽  
Author(s):  
J. L. G. Oliveira ◽  
C. W. M. van der Geld ◽  
J. G. M. Kuerten
Keyword(s):  
Liquid Velocity ◽  
Fluid Velocity ◽  
Three Dimensional ◽  
Stokes Number ◽  
Terminal Velocity ◽  
Point Particle ◽  
Inertial Particles ◽  
Carrier Fluid ◽  
Velocity Statistics ◽  
The Difference

Three-dimensional particle tracking velocimetry is applied to particle-laden turbulent pipe flows at a Reynolds number of 10 300, based on the bulk velocity and the pipe diameter, for developed fluid flow and not fully developed flow of inertial particles, which favours assessment of the radial migration of the inertial particles. Inertial particles with Stokes number ranging from 0.35 to 1.11, based on the particle relaxation time and the radial-dependent Kolmogorov time scale, and a ratio of the root-mean-square fluid velocity to the terminal velocity of order 1 have been used. Core peaking of the concentration of inertial particles in up-flow and wall peaking in down-flow have been found. The difference in mean particle and Eulerian mean liquid velocity is found to decrease to approximately zero near the wall in both flow directions. Although the carrier fluid has all of the characteristics of the corresponding turbulent single-phase flow, the Reynolds stress of the inertial particles is different near the wall in up-flow. These findings are explained from the preferential location of the inertial particles with the aid of direct numerical simulations with the point-particle approach.

Wind-generated gravity-capillary waves: laboratory measurements of temporal growth rates using microwave backscatter

1975 ◽  
Vol 70 (3) ◽  
pp. 417-436 ◽  
Author(s):  
T. R. Larson ◽  
J. W. Wright
Keyword(s):  
Growth Rates ◽  
Water Waves ◽  
Friction Velocity ◽  
Capillary Waves ◽  
Spectral Amplitude ◽  
Bragg Scattering ◽  
Laboratory Measurements ◽  
Wind Speeds ◽  
Tightly Coupled ◽  
Microwave Backscatter

The growth rates of wind-induced water waves at fixed fetch were measured in a laboratory wave tank using microwave backscatter. The technique strongly filters out all wavenumber component pairs except for a narrow window at the resonant Bragg scattering conditions. For these waves the spectral amplitude was measured as a function of the time after a fixed wind was abruptly started. The radars were aligned to respond to waves travelling in the downwind direction at wavelengths of 0·7-7 cm. Wind speeds ranged from 0·5 to 15 m/s. Fetches of 1·0, 3·0 and 8·4 m were used. In every case, the spectral amplitude initially grew at a single exponential rate β over several orders of magnitude, and then abruptly ceased growing. No dependence of the growth rate on fetch was observed. For all wavelengths and wind speeds the data can be fitted by \[ \beta (k,u_{*},{\rm fetch})=f(k)\,u^n_{*}, \] with n = 1·484 ± 0·027. Here u* is the friction velocity obtained from vertical profiles of mean horizontal velocity. For each wind speed, f(k) had a relative maximum near k = kn ≃ 3·6 cm−1. Rough estimates of β/2ω, where ω is the water wave frequency, and of the wind stress supported by short waves indicate that the observed growth rates are qualitatively very large. These waves are tightly coupled to the wind, and play a significant role in the transfer of momentum from wind to water.

scholarly journals Growth after the streaming instability

2019 ◽  
Vol 624 ◽  
pp. A114 ◽  
Author(s):  
Beibei Liu ◽  
Chris W. Ormel ◽  
Anders Johansen
Keyword(s):  
Dynamical Evolution ◽  
Bimodal Distribution ◽  
Stokes Number ◽  
Volume Density ◽  
Mass Star ◽  
Type I ◽  
Solar Mass ◽  
Streaming Instability ◽  
Two Component ◽  
Self Gravity

Context. Streaming instability is a key mechanism in planet formation, clustering pebbles into planetesimals with the help of self-gravity. It is triggered at a particular disk location where the local volume density of solids exceeds that of the gas. After their formation, planetesimals can grow into protoplanets by feeding from other planetesimals in the birth ring as well as by accreting inwardly drifting pebbles from the outer disk. Aims. We aim to investigate the growth of planetesimals into protoplanets at a single location through streaming instability. For a solar-mass star, we test the conditions under which super-Earths are able to form within the lifetime of the gaseous disk. Methods. We modified the Mercury N-body code to trace the growth and dynamical evolution of a swarm of planetesimals at a distance of 2.7 AU from the star. The code simulates gravitational interactions and collisions among planetesimals, gas drag, type I torque, and pebble accretion. Three distributions of planetesimal sizes were investigated: (i) a mono-dispersed population of 400 km radius planetesimals, (ii) a poly-dispersed population of planetesimals from 200 km up to 1000 km, (iii) a bimodal distribution with a single runaway body and a swarm of smaller, 100 km size planetesimals. Results. The mono-dispersed population of 400 km size planetesimals cannot form protoplanets of a mass greater than that of the Earth. Their eccentricities and inclinations are quickly excited, which suppresses both planetesimal accretion and pebble accretion. Planets can form from the poly-dispersed and bimodal distributions. In these circumstances, it is the two-component nature that damps the random velocity of the large embryo through the dynamical friction of small planetesimals, allowing the embryo to accrete pebbles efficiently when it approaches 10−2 M⊕. Accounting for migration, close-in super-Earth planets form. Super-Earth planets are likely to form when the pebble mass flux is higher, the disk turbulence is lower, or the Stokes number of the pebbles is higher. Conclusions. For the single site planetesimal formation scenario, a two-component mass distribution with a large embryo and small planetesimals promotes planet growth, first by planetesimal accretion and then by pebble accretion of the most massive protoplanet. Planetesimal formation at single locations such as ice lines naturally leads to super-Earth planets by the combined mechanisms of planetesimal accretion and pebble accretion.

scholarly journals Streaming instability of multiple particle species in protoplanetary disks

2018 ◽  
Vol 618 ◽  
pp. A75 ◽  
Author(s):  
Noemi Schaffer ◽  
Chao-Chin Yang ◽  
Anders Johansen
Keyword(s):  
Particle Size ◽  
Size Distribution ◽  
Protoplanetary Disks ◽  
Single Species ◽  
Stokes Number ◽  
Dust Particles ◽  
Particle Sizes ◽  
Particle Species ◽  
Streaming Instability ◽  
Large Particles

The radial drift and diffusion of dust particles in protoplanetary disks affect both the opacity and temperature of such disks, as well as the location and timing of planetesimal formation. In this paper, we present results of numerical simulations of particle-gas dynamics in protoplanetary disks that include dust grains with various size distributions. We have considered three scenarios in terms of particle size ranges, one where the Stokes number τs = 10−1−100, one where τs = 10−4−10−1, and finally one where τs = 10−3−100. Moreover, we considered both discrete and continuous distributions in particle size. In accordance with previous works we find in our multispecies simulations that different particle sizes interact via the gas and as a result their dynamics changes compared to the single-species case. The larger species trigger the streaming instability and create turbulence that drives the diffusion of the solid materials. We measured the radial equilibrium velocity of the system and find that the radial drift velocity of the large particles is reduced in the multispecies simulations and that the small particle species move on average outwards. We also varied the steepness of the size distribution, such that the exponent of the solid number density distribution, dN∕da ∝ a−q, is either q = 3 or q = 4. Overall, we find that the steepness of the size distribution and the discrete versus continuous approach have little impact on the results. The level of diffusion and drift rates are mainly dictated by the range of particle sizes. We measured the scale height of the particles and observe that small grains are stirred up well above the sedimented midplane layer where the large particles reside. Our measured diffusion and drift parameters can be used in coagulation models for planet formation as well as to understand relative mixing of the components of primitive meteorites (matrix, chondrules and CAIs) prior to inclusion in their parent bodies.

scholarly journals A general-purpose time-step criterion for simulations with gravity

2020 ◽  
Vol 495 (4) ◽  
pp. 4306-4313 ◽  
Author(s):  
Michael Y Grudić ◽  
Philip F Hopkins
Keyword(s):  
Time Scale ◽  
General Purpose ◽  
Gravitational Acceleration ◽  
Time Step ◽  
Simulation Code ◽  
Time Stepping ◽  
Dynamical Time ◽  
Adaptive Time Step ◽  
Order Of Magnitude ◽  
True Time

ABSTRACT We describe a new adaptive time-step criterion for integrating gravitational motion, which uses the tidal tensor to estimate the local dynamical time-scale and scales the time-step proportionally. This provides a better candidate for a truly general-purpose gravitational time-step criterion than the usual prescription derived from the gravitational acceleration, which does not respect the equivalence principle, breaks down when $\boldsymbol {a}=0$, and does not obey the same dimensional scaling as the true time-scale of orbital motion. We implement the tidal time-step criterion in the simulation code gizmo, and examine controlled tests of collisionless galaxy and star cluster models, as well as galaxy merger simulations. The tidal criterion estimates the dynamical time faithfully, and generally provides a more efficient time-stepping scheme compared to an acceleration criterion. Specifically, the tidal criterion achieves order-of-magnitude smaller energy errors for the same number of force evaluations in potentials with inner profiles shallower than ρ ∝ r−1 (i.e. where $\boldsymbol {a}\rightarrow 0$), such as star clusters and cored galaxies. For a given problem these advantages must be weighed against the additional overhead of computing the tidal tensor on-the-fly, but in many cases this overhead is small.

Motion of Descending Solid Particles and Local Flow Around the Particles in Downward Turbulent Water Duct Flow (Keynote)

2011 ◽  
Author(s):  
Yoshimichi Hagiwara ◽  
Hideto Fujii ◽  
Katsutoshi Sakurai ◽  
Takashi Kuroda ◽  
Atsuhide Kitagawa
Keyword(s):  
Reynolds Number ◽  
Time Scale ◽  
High Speed ◽  
Fluid Velocity ◽  
Stokes Number ◽  
Solid Particles ◽  
Duct Flow ◽  
Local Flow ◽  
Particle Reynolds Number ◽  
Turbulent Water

The Stokes number, the ratio of the particle time scale to flow time scale, is a promising quantity for estimating changes in statistics of turbulence due to particles. First, we explored the Stokes numbers in some recent studies. Secondly, we discussed the results of our direct numerical simulation for turbulent flow with a high-density particle in a vertical duct. In the discussion, we defined the particle Reynolds number from the mean fluid velocity in the near-particle region at any time. We evaluated a new local Stokes number for the particle. It is found that the Stokes number is effective for the prediction of the distance between the particle center and one wall. Finally, we carried out experiments for turbulent water flow with aluminum balls of 1 mm in diameter in a vertical channel. The motions of aluminum balls and tracer particles in the flow were captured with a high-speed video camera. We found that the experimental results for the time changes in the wall-normal distance of the ball and the particle Reynolds number for the ball are similar to the predicted results.

scholarly journals Dynamics of dusty vortices – I. Extensions and limitations of the terminal velocity approximation

2019 ◽  
Vol 488 (4) ◽  
pp. 5290-5299 ◽  
Author(s):  
Francesco Lovascio ◽  
Sijme-Jan Paardekooper
Keyword(s):  
Fluid Model ◽  
Velocity Model ◽  
Terminal Velocity ◽  
Viscous Stress Tensor ◽  
Spatial Convergence ◽  
Two Fluid Model ◽  
Numerical Solver ◽  
The Stability ◽  
Velocity Approximation ◽  
Protoplanetary Discs

ABSTRACT Motivated by the stability of dust laden vortices, in this paper we study the terminal velocity approximation equations for a gas coupled to a pressureless dust fluid and present a numerical solver for the equations embedded in the FARGO3D hydrodynamics code. We show that for protoplanetary discs it is possible to use the barycentre velocity in the viscous stress tensor, making it trivial to simulate viscous dusty protoplanetary discs with this model. We also show that the terminal velocity model breaks down around shocks, becoming incompatible with the two-fluid model it is derived from. Finally we produce a set of test cases for numerical schemes and demonstrate the performance of our code on these tests. Our implementation embedded in FARGO3D using an unconditionally stable explicit integrator is fast, and exhibits the desired second-order spatial convergence for smooth problems.

scholarly journals End Products of Cometary Evolution: Cometary Origin of Earth-Crossing Bodies of Asteroidal Appearance

1989 ◽  
Vol 116 (1) ◽  
pp. 537-556 ◽  
Author(s):  
G.W. Wetherill
Keyword(s):  
Time Scale ◽  
Large Fraction ◽  
Volatile Component ◽  
Cometary Nucleus ◽  
Short Period ◽  
Deep Earth ◽  
Dynamical Time ◽  
Time Required ◽  
Almost All ◽  
Cometary Origin

AbstractBecause there is no necessary connection between the time required to remove the volatile component of a cometary nucleus by solar heating (physical lifetime) and the dynamical lifetime of a comet, it is possible that a comet may evolve into an observable object of asteroidal appearance. Almost all comets have dynamical lifetimes much shorter than their physical lifetimes and in these cases complete loss of volatiles will not occur. Mechanisms do exist, however, whereby a small but significant fraction of comets will have longer dynamical lifetimes, permitting them to evolve first into Jupiter-family short period comets and then into comets with relatively safe decoupled orbits interior to Jupiter’s orbit. Observed Jupiter-family objects of asteroidal appearance (e.g., 1983SA) are much more likely to be of cometary rather than asteroidal origin. “Decoupling” is facilitated by several mechanisms: perturbations by the terrestrial planets, perturbations by Jupiter and the other giant planets (including resonant perturbations) and non-gravitational orbital changes caused by the loss of gas and dust from the comet. The dynamical time scale for decoupling is probably 105–106 years and almost all decoupled comets are likely to be of asteroidal appearance. Once decoupled, the orbits of the resulting Apollo-Amor objects will evolve on a longer (107–108 year) time scale, and the orbital evidence for these objects having originally been comets rather than asteroids will nearly disappear. Statistically, however, a large fraction of the bodies in deep Earth-crossing orbits with semi-major axes ≳ 2.2 AU are likely to be cometary objects in orbits that have not yet diffused into the steady state distribution. For plausible values of the relevant parameters, estimates can be made of the number of cometary Apollo-Amor “asteroids,” the observed number of Earthcrossing active and inactive short period comets, and the production rate of short period comets. These estimates are compatible with other theoretical and observational inferences that suggest the presence of a significant population of Apollo objects that formerly were active comets.

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