scholarly journals How much does turbulence change the pebble isolation mass for planet formation?

2018 ◽  
Vol 615 ◽  
pp. A110 ◽  
Author(s):  
S. Ataiee ◽  
C. Baruteau ◽  
Y. Alibert ◽  
W. Benz
Keyword(s):  
Aspect Ratio ◽  
Planet Formation ◽  
Turbulent Viscosity ◽  
Analytical Calculation ◽  
Radial Pressure ◽  
Stokes Number ◽  
High Viscosity ◽  
Necessary Condition ◽  
Pressure Maximum ◽  
Hydrodynamical Simulations

Context. When a planet becomes massive enough, it gradually carves a partial gap around its orbit in the protoplanetary disk. A pressure maximum can be formed outside the gap where solids that are loosely coupled to the gas, typically in the pebble size range, can be trapped. The minimum planet mass for building such a trap, which is called the pebble isolation mass (PIM), is important for two reasons: it marks the end of planetary growth by pebble accretion, and the trapped dust forms a ring that may be observed with millimetre observations. Aims. We study the effect of disk turbulence on the PIM and find its dependence on the gas turbulent viscosity, aspect ratio, and particles Stokes number. Methods. By means of 2D gas hydrodynamical simulations, we found the minimum planet mass to form a radial pressure maximum beyond the orbit of the planet, which is the necessary condition to trap pebbles. We then carried out 2D gas plus dust hydrodynamical simulations to examine how dust turbulent diffusion impacts particles trapping at the pressure maximum. We finally provide a semi-analytical calculation of the PIM based on comparing the radial drift velocity of solids and the root mean square turbulent velocity fluctuations around the pressure maximum. Results. From our results of gas simulations, we provide an expression for the PIM vs. disk aspect ratio and turbulent viscosity. Our gas plus dust simulations show that the effective PIM can be nearly an order of magnitude larger in high-viscosity disks because turbulence diffuse particles out of the pressure maximum. This is quantified by our semi-analytical calculation, which gives an explicit dependence of the PIM with Stokes number of particles. Conclusions. Disk turbulence can significantly alter the PIM, depending on the level of turbulence in regions of planet formation.

scholarly journals Influences of protoplanet-induced three-dimensional gas flow on pebble accretion

2020 ◽  
Vol 643 ◽  
pp. A21
Author(s):  
Ayumu Kuwahara ◽  
Hiroyuki Kurokawa
Keyword(s):  
Gas Flow ◽  
Planet Formation ◽  
Early Stage ◽  
Three Dimensional ◽  
Stokes Number ◽  
Flow Shear ◽  
Outer Planets ◽  
Symmetric Structure ◽  
Hydrodynamical Simulation ◽  
Hydrodynamical Simulations

Context. Pebble accretion is among the major theories of planet formation. Aerodynamically small particles, called pebbles, are highly affected by the gas flow. A growing planet embedded in a protoplanetary disk induces three-dimensional (3D) gas flow. In our previous work, Paper I, we focused on the shear regime of pebble accretion and investigated the influence of planet-induced gas flow on pebble accretion. In Paper I, we found that pebble accretion is inefficient in the planet-induced gas flow compared to that of the unperturbed flow, particularly when St ≲ 10−3, where St is the Stokes number. Aims. Following on the findings of Paper I, we investigate the influence of planet-induced gas flow on pebble accretion. We did not consider the headwind of the gas in Paper I. Here, we extend our study to the headwind regime of pebble accretion. Methods. Assuming a nonisothermal, inviscid sub-Keplerian gas disk, we performed 3D hydrodynamical simulations on the spherical polar grid hosting a planet with the dimensionless mass, m = RBondi∕H, located at its center, where RBondi and H are the Bondi radius and the disk scale height, respectively. We then numerically integrated the equation of motion for pebbles in 3D using hydrodynamical simulation data. Results. We first divided the planet-induced gas flow into two regimes: flow-shear and flow-headwind. In the flow-shear regime, where the planet-induced gas flow has a vertically rotational symmetric structure, we find that the outcome is identical to what we obtained in Paper I. In the flow-headwind regime, the strong headwind of the gas breaks the symmetric structure of the planet-induced gas flow. In the flow-headwind regime, we find that the trajectories of pebbles with St ≲ 10−3 in the planet-induced gas flow differ significantly from those of the unperturbed flow. The recycling flow, where gas from the disk enters the gravitational sphere at low latitudes and exits at high latitudes, gathers pebbles around the planet. We derive the flow transition mass analytically, mt, flow, which discriminates between the flow-headwind and flow-shear regimes. From the relation between m, mt, flow and mt, peb, where mt, peb is the transition mass of the accretion regime of pebbles, we classify the results obtained in both Paper I and this study into four groups. In particular, only when the Stokes gas drag law is adopted and m < min(mt, peb, mt, flow), where the accretion and flow regime are both in the headwind regime, the accretion probability of pebbles with St ≲ 10−3 is enhanced in the planet-induced gas flow compared to that of the unperturbed flow. Conclusions. Combining our results with the spacial variety of turbulence strength and pebble size in a disk, we conclude that the planet-induced gas flow still allows for pebble accretion in the early stage of planet formation. The suppression of pebble accretion due to the planet-induced gas flow occurs only in the late stage of planet formation, more specifically, in the inner region of the disk. This may be helpful for explaining the distribution of exoplanets and the architecture of the Solar System, both of which have small inner and large outer planets.

scholarly journals Do nuclear rings in barred galaxies form at the shear minimum of the rotation curve?

2020 ◽  
Vol 494 (4) ◽  
pp. 6030-6035 ◽  
Author(s):  
Mattia C Sormani ◽  
Zhi Li
Keyword(s):  
Rotation Curve ◽  
Turbulent Viscosity ◽  
Necessary Condition ◽  
Scale Free ◽  
Barred Galaxies ◽  
Acoustic Instability ◽  
Central Regions ◽  
Hydrodynamical Simulations ◽  
Spurious Result ◽  
Self Gravity

ABSTRACT It has been recently suggested that (i) nuclear rings in barred galaxies (including our own Milky Way) form at the radius where the shear parameter of the rotation curve reaches a minimum; and (ii) the acoustic instability of Montenegro et al. is responsible for driving the turbulence and angular momentum transport in the central regions of barred galaxies. Here, we test these suggestions by running simple hydrodynamical simulations in a logarithmic barred potential. Since the rotation curve of this potential is scale free, the shear minimum theory predicts that no ring should form. We find that in contrast to this prediction, a ring does form in the simulation, with morphology consistent with that of nuclear rings in real barred galaxies. This proves that the presence of a shear-minimum is not a necessary condition for the formation of a ring. We also find that perturbations that are predicted to be acoustically unstable wind up and eventually propagate off to infinity, so that the system is actually stable. We conclude that (i) the shear-minimum theory is an unlikely mechanism for the formation of nuclear rings in barred galaxies; and (ii) the acoustic instability is a spurious result and may not be able to drive turbulence in the interstellar medium, at least for the case without self-gravity. The question of the role of turbulent viscosity remains open.

scholarly journals Evolution of α Centauri b’s protoplanetary disc

2020 ◽  
Vol 496 (2) ◽  
pp. 2436-2447 ◽  
Author(s):  
Rebecca G Martin ◽  
Jack J Lissauer ◽  
Billy Quarles
Keyword(s):  
Steady State ◽  
Aspect Ratio ◽  
Orbital Period ◽  
Circular Orbit ◽  
Planet Formation ◽  
Eccentric Orbit ◽  
Collision Velocity ◽  
Small Disc ◽  
Hydrodynamical Simulations ◽  
Binary Orbit

ABSTRACT With hydrodynamical simulations we examine the evolution of a protoplanetary disc around α Centauri B including the effect of the eccentric orbit binary companion α Centauri A. The initially circular orbit disc undergoes two types of eccentricity growth. First, the eccentricity oscillates on the orbital period of the binary, Porb, due to the eccentricity of the binary orbit. Secondly, for a sufficiently small disc aspect ratio, the disc undergoes global forced eccentricity oscillations on a time-scale of around $20\, P_{\rm orb}$. These oscillations damp out through viscous dissipation leaving a quasi-steady eccentricity profile for the disc that oscillates only on the binary orbital period. The time-averaged global eccentricity is in the range 0.05–0.1, with no precession in the steady state. The periastrons of the gas particles are aligned to one another. The higher the disc viscosity, the higher the disc eccentricity. With N-body simulations we examine the evolution of a disc of planetesimals that forms with the orbital properties of the quasi-steady protoplanetary disc. We find that the average magnitude of the eccentricity of particles increases and their periastrons become misaligned to each other once they decouple from the gas disc. The low planetesimal collision velocity required for planet formation suggests that for planet formation to have occurred in a disc of planetesimals formed from a protoplanetary disc around α Centauri B, said disc’s viscosity must be have been small and planet formation must have occurred at orbital radii smaller than about $2.5\, \rm au$. Planet formation may be easier with the presence of gas.

scholarly journals Pebble-isolation mass: Scaling law and implications for the formation of super-Earths and gas giants

2018 ◽  
Vol 612 ◽  
pp. A30 ◽  
Author(s):  
Bertram Bitsch ◽  
Alessandro Morbidelli ◽  
Anders Johansen ◽  
Elena Lega ◽  
Michiel Lambrechts ◽  
...  
Keyword(s):  
Turbulent Viscosity ◽  
Scaling Law ◽  
Stokes Number ◽  
Giant Planets ◽  
Mass Scaling ◽  
The Core ◽  
Pressure Bump ◽  
Hydrodynamical Simulations ◽  
Gas Accretion ◽  
Migration Pathways

The growth of a planetary core by pebble accretion stops at the so-called pebble isolation mass, when the core generates a pressure bump that traps drifting pebbles outside its orbit. The value of the pebble isolation mass is crucial in determining the final planet mass. If the isolation mass is very low, gas accretion is protracted and the planet remains at a few Earth masses with a mainly solid composition. For higher values of the pebble isolation mass, the planet might be able to accrete gas from the protoplanetary disc and grow into a gas giant. Previous works have determined a scaling of the pebble isolation mass with cube of the disc aspect ratio. Here, we expand on previous measurements and explore the dependency of the pebble isolation mass on all relevant parameters of the protoplanetary disc. We use 3D hydrodynamical simulations to measure the pebble isolation mass and derive a simple scaling law that captures the dependence on the local disc structure and the turbulent viscosity parameter α. We find that small pebbles, coupled to the gas, with Stokes number τf < 0.005 can drift through the partial gap at pebble isolation mass. However, as the planetary mass increases, particles must be decreasingly smaller to penetrate the pressure bump. Turbulent diffusion of particles, however, can lead to an increase of the pebble isolation mass by a factor of two, depending on the strength of the background viscosity and on the pebble size. We finally explore the implications of the new scaling law of the pebble isolation mass on the formation of planetary systems by numerically integrating the growth and migration pathways of planets in evolving protoplanetary discs. Compared to models neglecting the dependence of the pebble isolation mass on the α-viscosity, our models including this effect result in higher core masses for giant planets. These higher core masses are more similar to the core masses of the giant planets in the solar system.

Modified equations for hydraulic calculation of thermally insulated oil pipelines for the case of a power-law fluid

2021 ◽  
pp. 388-395
Author(s):  
Марат Замирович Ямилев ◽  
Азат Маратович Масагутов ◽  
Александр Константинович Николаев ◽  
Владимир Викторович Пшенин ◽  
Наталья Алексеевна Зарипова ◽  
...  
Keyword(s):  
Power Law ◽  
Analytical Calculation ◽  
High Viscosity ◽  
Hydraulic Calculation ◽  
Flow Index ◽  
Pipeline System ◽  
Power Law Fluid ◽  
Simplified Approach ◽  
Oil Pipeline ◽  
Oil Pipelines

Теплогидравлический расчет неизотермических трубопроводов является наиболее важным гидравлическим расчетом в рамках решения задач обеспечения надежности и безопасности работы нефтепроводной системы. Для практических расчетов применяются формулы Дарси - Вейсбаха и Лейбензона. При этом в ряде случаев (короткие теплоизолированные участки, поверхностный обогрев нефтепроводов) можно использовать упрощенный подход к расчету, пренебрегая изменением температуры или учитывая температурные поправки. В настоящее время формулы для аналитического расчета движения высоковязких нефтей в форме уравнения Лейбензона получены только для ньютоновской и вязкопластичной жидкостей. Для степенной жидкости соответствующие зависимости отсутствуют, расчет ведется с использованием формулы Дарси - Вейсбаха. Целью настоящей статьи является представление формулы Дарси - Вейсбаха для изотермических течений степенной жидкости в форме уравнения Лейбензона. Данное представление позволит упростить процедуру проведения аналитических выкладок. В результате получены модифицированные уравнения Лейбензона для определения потери напора на участке нефтепровода в диапазоне индекса течения от 0,5 до 1,25. В указанном диапазоне относительное отклонение от результатов расчетов с использованием классических формул Метцнера - Рида и Ирвина не превышает 2 %. The thermal-hydraulic calculation of non-isothermal pipelines is the most important hydraulic calculation in the framework of solving the problems of ensuring the reliability and safety of the oil pipeline system. For practical calculations, the Darcy - Weisbach and Leibenson formulas are used. Moreover, in a number of cases (short heat-insulated sections, surface heating of oil pipelines), a simplified approach to the calculation can be used, neglecting temperature changes or taking into account temperature corrections. At present, formulas for the analytical calculation of the motion of high-viscosity oils in the form of the Leibenson equation have been obtained only for Newtonian and viscoplastic fluids. For a power-law fluid, there are no corresponding dependences; the calculation is carried out using the Darcy - Weisbach formula. The purpose of this article is to present the Darcy - Weisbach formula for isothermal flows of a power-law fluid in the Leibenzon form, which will simplify the procedure for performing analytical calculations. The modified Leibenzon equations are obtained to determine the head loss in the oil pipeline section in the range of the flow index from 0.5 to 1.25. In the specified range, the relative deviation from the results of calculations using the classical Metzner - Reed and Irwin formulas does not exceed 2 %.

Numerical Simulation of Localized Bulging in an Inflated Hyperelastic Tube with Fixed Ends

2020 ◽  
pp. 2050118
Author(s):  
Zehui Lin ◽  
Linan Li ◽  
Yang Ye
Keyword(s):  
Axial Stress ◽  
Necessary Condition ◽  
Theoretical Studies ◽  
Inflation Pressure ◽  
Material Models ◽  
Rubber Material ◽  
Bifurcation Phenomenon ◽  
A Value ◽  
Pressure Maximum ◽  
Almost All

When a hyperelastic tube is inflated, the inflation pressure has a maximum for almost all rubber material models, but has no maximum for commonly used arterial models. It is generally believed that the pressure having a maximum is a necessary condition for localized bulging to occur, and therefore aneurysms cannot be modeled as a mechanical bifurcation phenomenon. However, recent theoretical studies have shown that if the axial stretch is fixed during inflation, localized bulging may still occur even if a pressure maximum does not exist in uniform inflation. In this paper, numerical simulations are conducted to confirm this theoretical prediction. It is also demonstrated that if the axial pre-stretch is not sufficiently large, unloading near the two ends can reduce the axial stress to a value close to zero and Euler-type buckling then occurs.

Flow Prediction in Rotating Ducts Using Coriolis-Modified Turbulence Models

1980 ◽  
Vol 102 (4) ◽  
pp. 456-461 ◽  
Author(s):  
J. H. G. Howard ◽  
S. V. Patankar ◽  
R. M. Bordynuik
Keyword(s):  
Shear Stress ◽  
Aspect Ratio ◽  
Wall Shear Stress ◽  
Coriolis Force ◽  
Rectangular Cross Section ◽  
Turbulent Viscosity ◽  
Side Wall ◽  
Turbulence Models ◽  
Measured Velocity ◽  
Wall Shear

A parabolic numerical analysis procedure has been used to predict the flow in a straight, radial rotating channel of rectangular cross-section, chosen as a simple model of an impeller passage. A two equation turbulence model was employed, with alternative modifications, to include the influence of Coriolis force on turbulent kinetic energy. Alternative Coriolis force terms were evaluated by comparisons in a high-aspect-ratio duct with measured velocity, wall shear stress and turbulent viscosity. Secondary velocity predictions were checked with data from a low-aspect-ratio duct where the Coriolis modification of turbulence was found less influential than the secondary flow in the modification of side wall shear stress.

scholarly journals Influences of protoplanet-induced three-dimensional gas flow on pebble accretion

2020 ◽  
Vol 633 ◽  
pp. A81 ◽  
Author(s):  
Ayumu Kuwahara ◽  
Hiroyuki Kurokawa
Keyword(s):  
Shear Flow ◽  
Gas Flow ◽  
Gravitational Interaction ◽  
Three Dimensional ◽  
Stokes Number ◽  
Thermal Mass ◽  
Hydrodynamical Simulation ◽  
Wide Range ◽  
Accretion Model ◽  
Hydrodynamical Simulations

Context. The pebble accretion model has the potential to explain the formation of various types of planets. The main difference between this and the planetesimal accretion model is that pebbles not only experience the gravitational interaction with the growing planet but also a gas drag force from the surrounding protoplanetary disk gas. Aims. A growing planet embedded in a disk induces three-dimensional (3D) gas flow, which may influence pebble accretion. However, so far the conventional pebble accretion model has only been discussed in the unperturbed (sub-)Keplerian shear flow. In this study, we investigate the influence of 3D planet-induced gas flow on pebble accretion. Methods. Assuming a nonisothermal, inviscid gas disk, we perform 3D hydrodynamical simulations on the spherical polar grid, which has a planet located at its center. We then numerically integrate the equation of motion of pebbles in 3D using hydrodynamical simulation data. Results. We find that the trajectories of pebbles in the planet-induced gas flow differ significantly from those in the unperturbed shear flow for a wide range of investigated pebble sizes (St = 10−3–100, where St is the Stokes number). The horseshoe flow and outflow of the gas alter the motion of the pebbles, which leads to a reduction of the width of the accretion window, wacc, and the accretion cross section, Aacc. On the other hand, the changes in trajectories also cause an increase in the relative velocity of pebbles to the planet, which offsets the reduction of wacc and Aacc. As a consequence, in the Stokes regime, the accretion probability of pebbles, Pacc, in the planet-induced gas flow is comparable to that in the unperturbed shear flow except when the Stokes number is small, St ~ 10−3, in 2D accretion, or when the thermal mass of the planet is small, m = 0.03, in 3D accretion. In contrast, in the Epstein regime, Pacc in the planet-induced gas flow becomes smaller than that in the shear flow in the Stokes regime in both 2D and 3D accretion, regardless of assumed St and m. Conclusions. Our results combined with the spacial variety of turbulence strength and pebble size in a disk, suggest that the 3D planet-induced gas flow may be helpful to explain the distribution of exoplanets and the architecture of the Solar System.

Convective Heat Exchange and Friction in a Thin Laminar-to-Turbulent Boundary Layer on the Impermeable Lateral Surfaces of Blunted Cones Featuring a Low Aspect Ratio

2021 ◽  
pp. 25-37
Author(s):  
V.V. Gorskiy ◽  
A.G. Savvina
Keyword(s):  
Boundary Layer ◽  
Aspect Ratio ◽  
Turbulent Boundary Layer ◽  
Convective Heat ◽  
Turbulent Viscosity ◽  
Aircraft Design ◽  
Effective Length ◽  
Low Aspect Ratio ◽  
Viscosity Models ◽  
Semi Empirical

In order to provide a high-quality solution to the problem of computing convective heat transfer parameters in a laminar-to-turbulent boundary layer, it is necessary to numerically integrate differential equations describing that layer, completed by semi-empirical turbulent viscosity models, said models having been tested by comparing their output to the results of experimental investigations where the gas dynamics of a gas flow around a body is correctly simulated. Developing relatively simple yet adequately accurate computation methods becomes crucial for practical applications. To date, the effective length method, being simple yet apparently boasting an acceptable accuracy, has become the most widespread technique for solving this problem in aircraft design and aerospace technology. However, this statement is not correct for large Reynolds numbers on a hemisphere. Under these conditions, semi-empirical apparent turbulent viscosity models provide significantly better matches to experimental data. The paper analyses the feasibility of using a similar approach for the lateral surface of a blunted cone featuring a low aspect ratio. We describe a new efficient approach to solving this problem, demonstrating a high accuracy and maximum simplicity when used in practice. We check the results of systematic computations using our method against comparable data obtained via the most frequently cited approaches to solving this problem

scholarly journals Radial Pressure in the Solar Nebula as Affecting the Motions of Planetesimals

1971 ◽  
Vol 13 ◽  
pp. 355-361 ◽  
Author(s):  
Fred L. Whipple
Keyword(s):  
Pressure Gradient ◽  
Planet Formation ◽  
Radial Pressure ◽  
Particle Growth ◽  
Mean Free Path ◽  
Asteroid Belt ◽  
Positive Pressure ◽  
Density Gradients ◽  
Gravitational Acceleration ◽  
Radial Temperature

In the classical rotating Laplacian-type nebula, pressure gradients can develop radially to the protosun because of central radiation, particle ejection, and magnetic-field expansion or because of radial temperature or total gas density gradients. Except for the last two effects, the acting central acceleration for the gas is reduced from the gravitational value; the pressure gradient in the gas caused by temperature or density gradients may either add to or subtract from the gravitational acceleration, depending on the sense of the pressure gradient. Planetesimals in the nebula may thus experience tangential accelerations (+ or —) with respect to the gas because of the differential radial accelerations acting on the particles and the gas. As a consequence, the planetesimals may spiral outward or inward with respect to the protosun. The present paper deals with growing planetesimals and a range of drag laws depending on the Reynolds number and on the ratio of particle size to mean free path.Particles spiral in the direction of positive pressure gradient, thus being concentrated toward toroidal concentrations of gas. The effect increases with decreasing rates of particle growth, i.e., with increasing time scales of planet formation by accretion. In the outer regions, where evidence suggests that comets were formed and Uranus and Neptune were so accumulated, the effect of the pressure gradient is to clear the forming comets from those regions. The large mass of Neptune may have developed because of this effect, perhaps Neptune’s solar distance was reduced from Bode’s “law,” and perhaps no comet belt exists beyond Neptune. In the asteroid belt, on a slow time scale, the effect may have spiraled planetesimals toward Mars and Jupiter, thus contributing to the lack of planet formation in this region.

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