On the dispersion of small particles suspended in an isotropic turbulent fluid

1977 ◽  
Vol 83 (3) ◽  
pp. 529-546 ◽  
Author(s):  
M. W. Reeks
Keyword(s):  
Relaxation Time ◽  
Fluid Velocity ◽  
Fluid Motion ◽  
Particle Diffusion ◽  
Small Particles ◽  
Mean Velocity ◽  
Turbulent Fluid ◽  
A Value ◽  
Long Time ◽  
The Mean

A solution to the dispersion of small particles suspended in a turbulent fluid is presented, based on the approximation proposed by Phythian for the dispersion of fluid points in an incompressible random fluid. Motion is considered in a frame moving with the mean velocity of the fluid, the forces acting on the particle being taken as gravity and a fluid drag assumed linear in the particle velocity relative to that of the fluid. The probability distribution of the fluid velocity field in this frame is taken as Gaussian, homogeneous, isotropic, stationary and of zero mean. It is shown that, in the absence of gravity, the long-time particle diffusion coefficient is in general greater than that of the fluid, approaching with increasing particle relaxation time a value consistent with the particle being in an Eulerian frame of reference. The effect of gravity is consistent with Yudine's effect of crossing trajectories, reducing unequally the particle diffusion in directions normal to and parallel to the direction of the gravitational field. To characterize the effect of flow and gravity on particle diffusion it has been found useful to use a Froude number defined in terms of the turbulent intensity rather than the mean velocity. Depending upon the value of this number, it is found that the particle integral time scale may initially decrease with increasing particle relaxation time though it eventually rises and approaches the particle relaxation time. It is finally shown how this analysis may be extended to include the extra forces generated by the fluid and particle accelerations.

Properties of a bidisperse particle–gas suspension Part 1. Collision time small compared with viscous relaxation time

1993 ◽  
Vol 247 ◽  
pp. 623-641 ◽  
Author(s):  
V. Kumaran ◽  
Donald L. Koch
Keyword(s):  
Relaxation Time ◽  
Stokes Number ◽  
Distribution Functions ◽  
Gas Suspension ◽  
Balance Equations ◽  
Mean Velocity ◽  
The Mean ◽  
Viscous Relaxation ◽  
The Difference ◽  
Velocity Variances

The properties of a dilute bidisperse particle–gas suspension under low Reynolds number, high Stokes number conditions are studied in the limit τcτv using a perturbation analysis in the small parameter v, which is proportional to the ratio of timescales τc/τv. Here, τc is the time between successive collisions of a particle, and tv is the viscous relaxation time. The leading-order distribution functions for the two species are isotropic Gaussian distributions, and are identical to the molecular velocity distributions in a two-component gas at equilibrium. Balance equations are written for the mean and mean-square velocities, using a distribution function that is a small perturbation from the isotropic Gaussian. The collisional terms are calculated by performing an ensemble average over the relative configurations of the colliding particles, and the mean velocity and velocity variances are calculated correct to O(v2) by solving the balance equations. The difference in the mean velocities of the two species is O(v) smaller than the mean velocity of the suspension, and the fluctuating velocity is O(v½) smaller than the mean velocity.

scholarly journals Fluid motion between parallel planes. Dynamical stability

1946 ◽  
Vol 186 (1006) ◽  
pp. 391-409 ◽  
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Fluid Motion ◽  
Reynolds Numbers ◽  
Small Disturbance ◽  
Mean Velocity ◽  
High Reynolds Numbers ◽  
The Mean ◽  
Parabolic Flow ◽  
High Reynolds

If U is the velocity of the mean motion the following main results are obtained: 1. The region where U = c , c being the wave velocity, is the source where vibrations are generated; i.e. the slowly varying vibrations give rise to large rapidly varying vibrations in passing through the critical point. 2. Curved profiles admit a periodic motion at sufficiently high Reynolds numbers. 3. Parabolic flow is unstable at high Reynolds numbers; i.e. an infinitely small disturbance is sufficient to break up such flow. The critical Reynolds number is equal to R = U 0 h/v =6700, and the corresponding wavelength is about three times the width of the channel ( U 0 is the mean velocity at the axis, and h is the half-width of the channel).

Particle dynamics in the channel flow of a turbulent particle–gas suspension at high Stokes number. Part 2. Comparison of fluctuating force simulations and experiments

2011 ◽  
Vol 687 ◽  
pp. 41-71 ◽  
Author(s):  
Partha S. Goswami ◽  
V. Kumaran
Keyword(s):  
Relaxation Time ◽  
Particle Velocity ◽  
Volume Fraction ◽  
Quantitative Agreement ◽  
Mean Square ◽  
Mass Loading ◽  
Mean Velocity ◽  
Velocity Fluctuations ◽  
The Mean ◽  
Viscous Relaxation

AbstractThe particle and fluid velocity fluctuations in a turbulent gas–particle suspension are studied experimentally using two-dimensional particle image velocimetry with the objective of comparing the experiments with the predictions of fluctuating force simulations. Since the fluctuating force simulations employ force distributions which do not incorporate the modification of fluid turbulence due to the particles, it is of importance to quantify the turbulence modification in the experiments. For experiments carried out at a low volume fraction of $9. 15\ensuremath{\times} 1{0}^{\ensuremath{-} 5} $ (mass loading is 0.19), where the viscous relaxation time is small compared with the time between collisions, it is found that the gas-phase turbulence is not significantly modified by the presence of particles. Owing to this, quantitative agreement is obtained between the results of experiments and fluctuating force simulations for the mean velocity and the root mean square of the fluctuating velocity, provided that the polydispersity in the particle size is incorporated in the simulations. This is because the polydispersity results in a variation in the terminal velocity of the particles which could induce collisions and generate fluctuations; this mechanism is absent if all of the particles are of equal size. It is found that there is some variation in the particle mean velocity very close to the wall depending on the wall-collision model used in the simulations, and agreement with experiments is obtained only when the tangential wall–particle coefficient of restitution is 0.7. The mean particle velocity is in quantitative agreement for locations more than 10 wall units from the wall of the channel. However, there are systematic differences between the simulations and theory for the particle concentrations, possibly due to inadequate control over the particle feeding at the entrance. The particle velocity distributions are compared both at the centre of the channel and near the wall, and the shape of the distribution function near the wall obtained in experiments is accurately predicted by the simulations. At the centre, there is some discrepancy between simulations and experiment for the distribution of the fluctuating velocity in the flow direction, where the simulations predict a bi-modal distribution whereas only a single maximum is observed in the experiments, although both distributions are skewed towards negative fluctuating velocities. At a much higher particle mass loading of 1.7, where the time between collisions is smaller than the viscous relaxation time, there is a significant increase in the turbulent velocity fluctuations by ${\ensuremath{\sim} }1$–2 orders of magnitude. Therefore, it becomes necessary to incorporate the modified fluid-phase intensity in the fluctuating force simulation; with this modification, the mean and mean-square fluctuating velocities are within 20–30 % of the experimental values.

Gravity waves over a non-uniform flow

1969 ◽  
Vol 39 (4) ◽  
pp. 817-829 ◽  
Author(s):  
H. G. M. Velthuizen ◽  
L. Van Wijngaarden
Keyword(s):  
Velocity Distribution ◽  
Gravity Waves ◽  
Propagation Velocity ◽  
Fluid Velocity ◽  
Wall Layer ◽  
Mean Flow ◽  
Viscous Effects ◽  
Critical Height ◽  
Mean Velocity ◽  
The Mean

This paper is concerned with the propagation of small amplitude gravity waves over a flow with non-uniform velocity distribution. For such a flow Burns derived a relation for the velocity of propagation in terms of the velocity distribution of the mean flow. This result is derived here in another way and some of its implications are discussed. It is shown that one of these is hardly acceptable physically. Burns's result holds only when a real value of the propagation velocity is assumed; the mentioned difficulties vanish if complex values are allowed for, implying damping or growth of the waves. Viscous effects which are the cause of damping or growth are important in the wall layer near the bottom and also at the critical depth, which is present when the wave speed is between zero and the fluid velocity at the free surface.In § 2 the basic equations for the present problem are given. In § 3 exchange of momentum and energy between wave and primary flow is discussed. This is analogous to what happens at the critical height in a wind flow over wind-driven gravity waves. In § 4 the viscous effects at the bottom are included in the analysis and the complex equation for the propagation velocity is derived. Finally in § 5 illustrations of the theory are given for long waves over running flow and for the flow along a ship advancing in a wavy sea. In these examples a negative curvature of the mean velocity profile is shown to have a stabilizing effect.

Collisional sheet flows of sediment driven by a turbulent fluid

1998 ◽  
Vol 370 ◽  
pp. 29-52 ◽  
Author(s):  
JAMES T. JENKINS ◽  
DANIEL M. HANES
Keyword(s):  
Particle Velocity ◽  
Particle Concentration ◽  
Fluid Velocity ◽  
Packed Bed ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Volume Flux ◽  
Velocity Fluctuations ◽  
Turbulent Fluid ◽  
The Mean

We consider a sheet flow in which heavy grains near a packed bed interact with a unidirectional turbulent shear flow of a fluid. We focus on sheet flows in which the particles are supported by their collisional interactions rather than by the velocity fluctuations of the turbulent fluid and introduce what we believe to be the simplest theory for the collisional regime that captures its essential features.We employ a relatively simple model of the turbulent shearing of the fluid and use kinetic theory for the collisional grain flow to predict profiles of the mean fluid velocity, the mean particle velocity, the particle concentration, and the strength of the particle velocity fluctuations within the sheet. These profiles are obtained as solutions to the equations of balance of fluid and particle momentum and particle fluctuation energy over a range of Shields parameters between 0.5 and 2.5. We compare the predicted thickness of the concentrated region and the predicted features of the profile of the mean fluid velocity with those measured by Sumer et al. (1996). In addition, we calculate the volume flux of particles in the sheet as a function of Shields parameter.Finally, we apply the theory to sand grains in air for the conditions of a sandstorm and calculate profiles of particle concentration, velocity, and local volume flux.

scholarly journals Measurements of the average properties of a suspension of bubbles rising in a vertical channel

2001 ◽  
Vol 429 ◽  
pp. 307-342 ◽  
Author(s):  
ROBERTO ZENIT ◽  
DONALD L. KOCH ◽  
ASHOK S. SANGANI
Keyword(s):  
Volume Fraction ◽  
Fluid Velocity ◽  
Vertical Channel ◽  
Bubble Velocity ◽  
Large Reynolds Number ◽  
Mean Velocity ◽  
Velocity Fluctuations ◽  
Velocity Variance ◽  
The Mean ◽  
Bubble Collision

Experiments were performed in a vertical channel to study the behaviour of a monodisperse bubble suspension for which the dual limit of large Reynolds number and small Weber number was satisfied. Measurements of the liquid-phase velocity fluctuations were obtained with a hot-wire anemometer. The gas volume fraction, bubble velocity, bubble velocity fluctuations and bubble collision rate were measured using a dual impedance probe. Digital image analysis was performed to quantify the small polydispersity of the bubbles as well as the bubble shape.A rapid decrease in bubble velocity with bubble concentration in very dilute suspensions is attributed to the effects of bubble–wall collisions. The more gradual subsequent hindering of bubble motion is in qualitative agreement with the predictions of Spelt & Sangani (1998) for the effects of potential-flow bubble–bubble interactions on the mean velocity. The ratio of the bubble velocity variance to the square of the mean is O(0.1). For these conditions Spelt & Sangani predict that the homogeneous suspension will be unstable and clustering into horizontal rafts will take place. Evidence for bubble clustering is obtained by analysis of video images. The fluid velocity variance is larger than would be expected for a homogeneous suspension and the fluid velocity frequency spectrum indicates the presence of velocity fluctuations that are slow compared with the time for the passage of an individual bubble. These observations provide further evidence for bubble clustering.

The mean value of the fluctuations in pressure and pressure gradient in a turbulent fluid

1938 ◽  
Vol 34 (4) ◽  
pp. 534-539
Author(s):  
A. E. Green
Keyword(s):  
Unit Volume ◽  
Isotropic Turbulence ◽  
Fluid Motion ◽  
Mean Value ◽  
Mean Square ◽  
Turbulent Fluid ◽  
Dissipation Of Energy ◽  
The Mean ◽  
Rate Of Dissipation ◽  
Image Position

1. Taylor has shown (1) that two characteristic lengthsλ and λη may be defined for turbulent fluid motion. The length λ, which is connected with the dissipation of energy, is, for isotropic turbulence, given bywhere is the mean rate of dissipation of energy per unit volume and represents the mean square value of any component of velocity. The length λη can be defined in terms of thuswhere For isotropic turbulence Taylor assumed thatwhere B is a constant. Since the turbulence is isotropic,and so, from (1), (2), (3) and (4) we have

scholarly journals VI. Experiments upon the heart of the dog with reference to the maximum volume of blood sent out by the left ventricle in a single beat and the influence of variations in venous pressure, arterial pressure, and pulse-rate upon the work done by the heart

1884 ◽  
Vol 175 ◽  
pp. 139-160 ◽  
Keyword(s):  
Venous Pressure ◽  
Uncertain Data ◽  
Mean Velocity ◽  
Long Time ◽  
Cause Of Failure ◽  
The Mean ◽  
The Right ◽  
The University ◽  
Short Time ◽  
Work Done

The most important factor to be determined before calculating the work done by the heart is the quantity of blood forced from the ventricles at each systole. Most of the efforts to determine this quantity have been based either upon faulty observations upon the dead heart, or upon the uncertain data obtained by estimating the mean velocity of the stream of blood in the aorta. Professor Martin accordingly suggested to us that we should attempt to measure it directly on the isolated Dog’s heart. The work thus undertaken was carried on during the greater part of the university session, 1881-82, and the results obtained are given in the following pages. The method of isolating the heart was essentially that described in Professor Martin’s paper (Phil. Trans., 1883, p. 663). In the course of this work many unexpected difficulties arose, necessitating changes in the apparatus and the method of operating, and preventing us for a long time from obtaining any successful results. In our experiments it was necessary not only that the heart should live and beat, but that it should be in the best possible physiological condition, and any marked pulmonary œdema made an experiment nearly valueless. This most frequent cause of failure was mainly owing to the fact that, on account of the large quantity of blood required for an experiment, we were obliged to use Calf’s blood obtained from the butcher; very often this blood, as Professor Martin states in his paper, will bring about œdema of the lungs in a short time; large quantities of exuded serum pour out of the tracheal cannula, the air-passages in the lungs become choked up with liquid, and the circulation from the right to the left side of the heart is greatly impeded. We have succeeded, however, in making a considerable number of experiments in which all the conditions were favourable, the œdema of the lungs not occurring to any marked extent until after many observations had been made.

An electron microscope study of evaporating small particles: the Kelvin equation for liquid lead and the mean surface energy of solid silver

1970 ◽  
Vol 318 (1535) ◽  
pp. 507-522 ◽  
Keyword(s):  
Electron Microscope ◽  
Surface Energy ◽  
Microscope Study ◽  
Electron Microscope Study ◽  
Liquid Lead ◽  
Small Particles ◽  
Kelvin Equation ◽  
A Value ◽  
Carbon Substrates ◽  
The Mean

An electron microscope study of evaporating liquid lead particles, formed on carbon substrates by nucleation from the vapour, has led to a confirmation of the Kelvin equation which relates equilibrium vapour pressure to surface curvature. Observations of the same kind made on particles of evaporating solid silver have produced evaporation curves similar to those for lead. Interpretation of these curves in terms of the Kelvin equation for solids has yielded a value for the mean surface energy of silver of 1.20 ± 0.06 J m -2 (1200 ± 60 erg cm -2 ) at 1005 ± 25 K.

THE EFFECT OF ALTERATION OF THYROXINE BINDING CAPACITY ON THE DIALYZABLE AND ABSOLUTE FRACTIONS OF TRIIODOTHYRONINE IN CIRCULATION

1973 ◽  
Vol 72 (2) ◽  
pp. 265-271 ◽  
Author(s):  
J. H. Dussault ◽  
D. A. Fisher ◽  
J. T. Nicoloff ◽  
V. V. Row ◽  
R. Volpe
Keyword(s):  
Correlation Coefficient ◽  
Inverse Relationship ◽  
Binding Capacity ◽  
Hormone Concentration ◽  
Thyroxine Binding Globulin ◽  
Serum Thyroxine ◽  
A Value ◽  
The Mean ◽  
Male Subjects ◽  
Female Subjects

ABSTRACT In order to determine the effect of alterations in binding capacity of thyroxine binding globulin (TBG) on triiodothyronine (T3) metabolism, studies were conducted in 10 patients with idiopathically low (7 subjects) or elevated (3 subjects) TBG levels and 10 subjects given norethandrolone (7 male subjects) or oestrogen (3 female subjects). Measurements of serum thyroxine (T4) concentration, maximal T4 binding capacity, serum T3 concentration and per cent dialyzable T3 were conducted. Serum T3 was measured both by chemical and radioimmunoassay methods. In patients with idiopathically low TBG, the mean serum T4 concentration was low (2.4 μg/100 ml), the mean serum T3 level low (55 ng/100 ml), the mean per cent dialyzable T3 increased (0.52%), and the calculated free T3 concentration normal (186 pg/100 ml). In patients with idiopathically high TBG levels the mean T4 concentration was high (10.3 μg/100 ml), the mean T3 level slightly elevated (127 ng/100 ml), the% dialyzable T3 low (0.10%) and the calculated free T3 concentration low normal (123 pg/100 ml). The correlation coefficient between the per cent dialyzable T3 and maximal TBG binding capacity in the 20 subjects was 0.68, a value significant at the P < 0.01 level. Thus, alterations in binding capacity of TBG seem to influence T3 and T4 metabolism similarly; the inverse relationship between the % of dialyzable hormone and total hormone concentration tends to keep the absolue levels of free hormones stable.

Sign in / Sign up

Export Citation Format

Share Document