Dynamics of a Small Spherical Particle in Steady Two-Dimensional Vortex Flows

1994 ◽  
Vol 47 (6S) ◽  
pp. S61-S69 ◽  
Author(s):  
Juan C. Lasheras ◽  
Kek-Kiong Tio
Keyword(s):  
Spherical Particle ◽  
Mass Density ◽  
Fluid Velocity ◽  
Stokes Number ◽  
Viscous Drag ◽  
Particle Inertia ◽  
Small Perturbations ◽  
Rankine Vortex ◽  
Small Spherical Particle ◽  
Limiting Case

The equation of motion of a small spherical particle in an isolated Rankine vortex is analyzed using an asymptotic scheme valid for the limiting case of small Stokes number St. The effects of particle inertia and added mass, gravity, the acceleration of the fluid, viscous drag, and the Basset history force are taken into consideration. For the case of an isolated Rankine vortex, the analysis shows that in the region where the fluid velocity is large enough, the viscous drag constrains the particle to move with a velocity equal to that of the fluid plus a perturbation of order St. This perturbative term incorporates the effects of gravity, the density difference between the particle and the fluid, and the local acceleration of the fluid. In the region where the fluid velocity is small, the particle moves with a velocity equal to the sum of the fluid velocity and the rising/settling velocity of the particle in still fluid, the effects of particle inertia and fluid acceleration appearing as small perturbations. Throughout the whole region of the flow, the effect of the Basset force always appears at higher order than the other forces acting on the particle and may, consequently, be neglected. The analysis also shows that a particle with a mass density greater than that of the fluid always escapes from the central region of the vortex, but a buoyant particle may be trapped by the equilibrium point located there.

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.

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.

Dispersion resulting from flow through spatially periodic porous media

1980 ◽  
Vol 297 (1430) ◽  
pp. 81-133 ◽  
Keyword(s):  
Porous Media ◽  
Velocity Vector ◽  
Tracer Particle ◽  
Fluid Velocity ◽  
Complex Structure ◽  
Molecular Diffusivity ◽  
Spatially Periodic ◽  
Flow Through ◽  
Limiting Case ◽  
Interstitial Velocity

A rigorous theory of dispersion in both granular and sintered spatially-periodic porous media is presented, utilizing concepts originating from Brownian motion theory. A precise prescription is derived for calculating both the Darcy-scale interstitial velocity vector v* and dispersivity dyadic D* of a tracer particle. These are expressed in terms of the local fluid velocity vector field v at each point within the interstices of a unit cell of the spatially periodic array and, for the dispersivity, the molecular diffusivity of the tracer particle through the fluid. Though the theory is complete, numerical results are not yet available owing to the complex structure of the local interstitial velocity field v. However, as an illustrative exercise, the theory is shown to correctly reduce in an appropriate limiting case to the well-known Taylor-Aris results for dispersion in circular capillaries.

An Investigation of Crossing Trajectory Effects in Turbulent Shear Flow (Keynote)

2005 ◽  
Author(s):  
Lionel Thomas ◽  
Benoiˆt Oesterle´
Keyword(s):  
Shear Flow ◽  
Isotropic Turbulence ◽  
Fluid Velocity ◽  
Stokes Number ◽  
Turbulent Shear ◽  
Inertial Particles ◽  
Turbulent Shear Flow ◽  
Velocity Magnitude ◽  
Dispersed Particles ◽  
Lagrangian Tracking

The dispersion of small inertial particles moving in a homogeneous, hypothetically stationary, shear flow is investigated using both theoretical analysis and numerical simulation, under one-way coupling approximation. In the theoretical approach, the previous studies are extended to the case of homogeneous shear flow with a corresponding anisotropic spectrum. As it is impossible to obtain a closed theoretical solution without some drastic simplifications, the motion of dispersed particles is also investigated using kinematic simulation where random Fourier modes are generated according to a prescribed anisotropic spectrum with a superimposed linear mean fluid velocity profile. The combined effects of particle Stokes number and dimensionless drift velocity (magnitude and direction) are investigated by computing the statistics from Lagrangian tracking of a large number of particles in many flow field realizations, and comparison is made between the observed effects in shear flow and in isotropic turbulence.

Recent Advances in the Knowledge of Transonic Air Flow

1956 ◽  
Vol 60 (544) ◽  
pp. 241-252 ◽  
Author(s):  
C. H. E. Warren
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Layers ◽  
Velocity Potential ◽  
Fluid Velocity ◽  
Linear Partial Differential Equation ◽  
Inviscid Flow ◽  
Stream Velocity ◽  
Linear Boundary ◽  
Small Perturbations

The most powerful theoretical tool in the solution of the aerodynamic problems of aircraft is the theory of small perturbations, which states that if a wing is thin (or a body slender), and if the incidence is small, then in inviscid flow the fluid velocity at any point can be treated as a small perturbation from the stream velocity. The backbone of our knowledge of the aerodynamics of aircraft is provided by this theory, to which the effects of thick wings and large incidences, and the effect of viscosity, introducing as it does the concept of boundary layers, can be added as additional or correction effects. It is known that at subsonic and again at supersonic speeds, the theory of small perturbations is a linear theory; that is, the assumptions implicit in it lead to a linear partial differential equation for the velocity potential, with linear boundary conditions.

Determination of the generation cross section of a second harmonic on a small spherical particle

1975 ◽  
Vol 22 (3) ◽  
pp. 404-406
Author(s):  
Yu. A. Bykovskii ◽  
I. E. Nakhutin ◽  
�. A. Manykin ◽  
Yu. G. Rubezhnyi ◽  
P. P. Polu�ktov
Keyword(s):  
Spherical Particle ◽  
Cross Section ◽  
Second Harmonic ◽  
Small Spherical Particle

Apparatus for continuous, fast, and precise measurements of position and velocity of a small spherical particle

1983 ◽  
Vol 54 (12) ◽  
pp. 1643-1647 ◽  
Author(s):  
T. S. Venkataraman ◽  
William W. Eidson ◽  
Leonard D. Cohen ◽  
James D. Farina ◽  
Charles Acquista
Keyword(s):  
Spherical Particle ◽  
Small Spherical Particle

On the dynamics of buoyant and heavy particles in a periodic Stuart vortex flow

1993 ◽  
Vol 254 ◽  
pp. 671-699 ◽  
Author(s):  
Kek-Kiong Tio ◽  
Amable Liñán ◽  
Juan C. Lasheras ◽  
Alfonso M. Gañán-Calvo
Keyword(s):  
Dynamical System ◽  
Computational Study ◽  
Mass Density ◽  
Order Term ◽  
Equilibrium Points ◽  
Density Ratio ◽  
Viscous Drag ◽  
Perturbation Scheme ◽  
Heavy Particles ◽  
Drag Forces

In this paper, we study the dynamics of small, spherical, rigid particles in a spatially periodic array of Stuart vortices given by a steady-state solution to the two-dimensional incompressible Euler equation. In the limiting case of dominant viscous drag forces, the motion of the particles is studied analytically by using a perturbation scheme. This approach consists of the analysis of the leading-order term in the expansion of the ‘particle path function’ Φ, which is equal to the stream function evaluated at the instantaneous particle position. It is shown that heavy particles which remain suspended against gravity all move in a periodic asymptotic trajectory located above the vortices, while buoyant particles may be trapped by the stable equilibrium points located within the vortices. In addition, a linear map for Φ is derived to describe the short-term evolution of particles moving near the boundary of a vortex. Next, the assumption of dominant viscous drag forces is relaxed, and linear stability analyses are carried out to investigate the equilibrium points of the five-parameter dynamical system governing the motion of the particles. The five parameters are the free-stream Reynolds number, the Stokes number, the fluid-to-particle mass density ratio, the distribution of vorticity in the flow, and a gravitational parameter. For heavy particles, the equilibrium points, when they exist, are found to be unstable. In the case of buoyant particles, a pair of stable and unstable equilibrium points exist simultaneously, and undergo a saddle-node bifurcation when a certain parameter of the dynamical system is varied. Finally, a computational study is also carried out by integrating the dynamical system numerically. It is found that the analytical and computational results are in agreement, provided the viscous drag forces are large. The computational study covers a more general regime in which the viscous drag forces are not necessarily dominant, and the effects of the various parametric inputs on the dynamics of buoyant particles are investigated.

scholarly journals An Improved Multiscale Model for Dilute Turbulent Gas-Particle Flows Based on the Equilibration of Energy Concept

2005 ◽  
Author(s):  
Ying Xu ◽  
Shankar Subramaniam
Keyword(s):  
Fluid Phase ◽  
Multiscale Model ◽  
Stokes Number ◽  
Fluid Turbulence ◽  
Energy Concept ◽  
Particle Flows ◽  
Homogeneous Particle ◽  
Limiting Case ◽  
Physical Phenomena ◽  
Simple Flow

The objective of this study is to assess, and possibly improve, models for turbulent particle-laden flows. We begin by understanding the behavior of two existing models—one proposed by Simonin [Von Karman Institute of Fluid Dynamics Lecture Series, 1996], and the other by Ahmadi [Int. J. Multiphase Flow, 1990]—in the limiting case of statistically homogeneous particle-laden turbulent flow. The decay of particle and fluid phase turbulent kinetic energy (TKE) are compared with direct numerical simulation results. Even this simple flow poses a significant challenge to current models which have difficulties in reproducing important physical phenomena, such as the variation of TKE decay with particle Stokes number. Some of these problems can be traced to the model for the interphase TKE transfer timescale. A new model for the interphase transfer timescale is proposed that accounts for the interaction of particles with a range of fluid turbulence scales. A new multiphase turbulence model—the Equilibration of Energy Model (EEM)—is proposed, that incorporates this multiscale interphase transfer concept. The particle and fluid TKE evolution predicted by this new EEM model correctly reproduce the trends with particle Stokes number.

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