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.

Properties of a bidisperse particle–gas suspension Part 2. Viscous relaxation time small compared with collision time

1993 ◽  
Vol 247 ◽  
pp. 643-660 ◽  
Author(s):  
V. Kumaran ◽  
Donald L. Koch
Keyword(s):  
Relaxation Time ◽  
Vertical Direction ◽  
Stokes Number ◽  
Terminal Velocity ◽  
Distribution Functions ◽  
Mean Square ◽  
Collision Time ◽  
Gas Suspension ◽  
The Mean ◽  
Viscous Relaxation

The properties of a dilute bidisperse particle–gas suspension under low Reynolds number, high Stokes number conditions are studied in the limit τv [Lt ] τc, where τc is the time between successive collisions of a particle, and τv is the viscous relaxation time. In this limit, the particles relax close to their terminal velocity between successive collisions, and we use a perturbation analysis in the small parameter ε, which is proportional to τv/τc, about a base state in which all the particles settle at their terminal velocities. The mean velocities of the two species are O(ε) different from their terminal velocities, and the mean-square velocities are O(ε) smaller than the square of the terminal velocity. The distribution functions for the two species, which incorporate the first effects of collisions between particles settling at their terminal velocities, are derived. The velocity distribution is highly anisotropic in this limit, and the mean-square velocity in the vertical direction is twice that in the horizontal plane. The distribution function for each species is singular at its terminal velocity, and the distributions are non-zero in a finite region in velocity space between the two terminal velocities.

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.

New space-averaging procedure for inhomogeneous flow fields using balance equation formulations

2006 ◽  
Vol 220 (9) ◽  
pp. 1363-1374
Author(s):  
S Wattananusorn
Keyword(s):  
Compressible Flows ◽  
Incompressible Flows ◽  
Flow Fields ◽  
Coupling Factor ◽  
Balance Equations ◽  
Mean Velocity ◽  
New Space ◽  
Inhomogeneous Flow ◽  
The Mean ◽  
Dependent Flow

This paper features the possibility of averaging space-dependent flow fields using a coupling factor that links the equations of momentum and energy. The scheme is applied to the mean velocity, which is derived straightforwardly through the continuity equation. It creates a small imbalance, which can be eliminated later completely. Smaller discrepancies in the integration of systems of balance equations for inhomogeneous flow are the consequence. The procedure is verified on various flow patterns, and comparisons are made with other conventional methods and with some available experimental data. Despite investigating only numerical examples of incompressible flows here, the technique, in principle, is capable of dealing with compressible flows as well. Furthermore, the proposed method discards some variables required in other techniques while still providing useful and acceptable results for practical problems.

scholarly journals Parameterization of Mixed Layer and Deep-Ocean Mesoscales including Nonlinearity

2018 ◽  
Vol 48 (3) ◽  
pp. 555-572 ◽  
Author(s):  
V. M. Canuto ◽  
Y. Cheng ◽  
M. S. Dubovikov ◽  
A. M. Howard ◽  
A. Leboissetier
Keyword(s):  
Deep Ocean ◽  
Altimetry Data ◽  
Dynamical Equations ◽  
Mean Velocity ◽  
Ocean Stratification ◽  
Large Heat ◽  
Reference Experiment ◽  
The Mean ◽  
The Difference ◽  
Induced Velocity

AbstractIn 2011, Chelton et al. carried out a comprehensive census of mesoscales using altimetry data and reached the following conclusions: “essentially all of the observed mesoscale features are nonlinear” and “mesoscales do not move with the mean velocity but with their own drift velocity,” which is “the most germane of all the nonlinear metrics.” Accounting for these results in a mesoscale parameterization presents conceptual and practical challenges since linear analysis is no longer usable and one needs a model of nonlinearity. A mesoscale parameterization is presented that has the following features: 1) it is based on the solutions of the nonlinear mesoscale dynamical equations, 2) it describes arbitrary tracers, 3) it includes adiabatic (A) and diabatic (D) regimes, 4) the eddy-induced velocity is the sum of a Gent and McWilliams (GM) term plus a new term representing the difference between drift and mean velocities, 5) the new term lowers the transfer of mean potential energy to mesoscales, 6) the isopycnal slopes are not as flat as in the GM case, 7) deep-ocean stratification is enhanced compared to previous parameterizations where being more weakly stratified allowed a large heat uptake that is not observed, 8) the strength of the Deacon cell is reduced. The numerical results are from a stand-alone ocean code with Coordinated Ocean-Ice Reference Experiment I (CORE-I) normal-year forcing.

scholarly journals Wind Turbulence Statistics of the Atmospheric Inertial Sublayer under Near-Neutral Conditions

Atmosphere ◽  
2020 ◽  
Vol 11 (10) ◽  
pp. 1087
Author(s):  
Eslam Reda Lotfy ◽  
Zambri Harun
Keyword(s):  
Boundary Layer ◽  
Atmospheric Stability ◽  
Turbulent Velocity ◽  
Stability Conditions ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
Turbulent Statistics ◽  
The Mean ◽  
The Law ◽  
Velocity Variances

The inertial sublayer comprises a considerable and critical portion of the turbulent atmospheric boundary layer. The mean windward velocity profile is described comprehensively by the Monin–Obukhov similarity theory, which is equivalent to the logarithmic law of the wall in the wind tunnel boundary layer. Similar logarithmic relations have been recently proposed to correlate turbulent velocity variances with height based on Townsend’s attached-eddy theory. The theory is particularly valid for high Reynolds-number flows, for example, atmospheric flow. However, the correlations have not been thoroughly examined, and a well-established model cannot be reached for all turbulent variances similar to the law of the wall of the mean-velocity. Moreover, the effect of atmospheric thermal condition on Townsend’s model has not been determined. In this research, we examined a dataset of free wind flow under a near-neutral range of atmospheric stability conditions. The results of the mean velocity reproduce the law of the wall with a slope of 2.45 and intercept of −13.5. The turbulent velocity variances were fitted by logarithmic profiles consistent with those in the literature. The windward and crosswind velocity variances obtained the average slopes of −1.3 and −1.7, respectively. The slopes and intercepts generally increased away from the neutral state. Meanwhile, the vertical velocity and temperature variances reached the ground-level values of 1.6 and 7.8, respectively, under the neutral condition. The authors expect this article to be a groundwork for a general model on the vertical profiles of turbulent statistics under all atmospheric stability conditions.

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.

Investigations of Tripping Effect on the Friction Factor in Turbulent Pipe Flows

2009 ◽  
Vol 131 (7) ◽  
Author(s):  
A. Al-Salaymeh ◽  
O. A. Bayoumi
Keyword(s):  
Friction Factor ◽  
Reynolds Numbers ◽  
Original Data ◽  
Turbulent Pipe Flow ◽  
Wall Friction ◽  
Mean Flow ◽  
Mean Velocity ◽  
Fully Developed Turbulent Flow ◽  
The Mean ◽  
The Difference

Tripping devices are usually installed at the entrance of laboratory-scale pipe test sections to obtain a fully developed turbulent flow sooner. The tripping of laminar flow to induce turbulence can be carried out in different ways, such as using cylindrical wires, sand papers, well-organized tape elements, fences, etc. Claims of tripping effects have been made since the classical experiments of Nikuradse (1932, Gesetzmässigkeit der turbulenten Strömung in glatten Rohren, Forschungsheft 356, Ausgabe B, Vol. 3, VDI-Verlag, Berlin), which covered a significant range of Reynolds numbers. Nikuradse’s data have become the metric by which theories are established and have also been the subject of intense scrutiny. Several subsequent experiments reported friction factors as much as 5% lower than those measured by Nikuradse, and the authors of those reports attributed the difference to tripping effects, e.g., work of Durst et al. (2003, “Investigation of the Mean-Flow Scaling and Tripping Effect on Fully Developed Turbulent Pipe Flow,” J. Hydrodynam., 15(1), pp. 14–22). In the present study, measurements with and without ring tripping devices of different blocking areas of 10%, 20%, 30%, and 40% have been carried out to determine the effect of entrance condition on the developing flow field in pipes. Along with pressure drop measurements to compute the skin friction, both the Pitot tube and hot-wire anemometry measurements have been used to accurately determine the mean velocity profile over the working test section at different Reynolds numbers based on the mean velocity and pipe diameter in the range of 1.0×105–4.5×105. The results we obtained suggest that the tripping technique has an insignificant effect on the wall friction factor, in agreement with Nikuradse’s original data.

scholarly journals Evaluation of mean velocity and mean speed for test ventilated room from RANS and LES CFD modeling

2019 ◽  
Vol 85 ◽  
pp. 02004 ◽  
Author(s):  
Nikolay Ivanov ◽  
Marina Zasimova ◽  
Evgueni Smirnov ◽  
Detelin Markov
Keyword(s):  
Kinetic Energy ◽  
Cfd Modeling ◽  
Navier Stokes ◽  
Velocity Data ◽  
Mean Velocity ◽  
Energy Field ◽  
Eddy Simulation ◽  
Large Eddy ◽  
The Mean ◽  
The Difference

The paper presents and discusses data for the ventilation airflow in an isothermal room corresponding to the Nielsen et al. (1978) test computed with Large Eddy Simulation (LES) and Reynolds-Averaged Navier-Stokes (RANS) approaches. As LES computations provide directly both the speed and velocity components data, the difference between the mean speed and mean velocity values is computed and discussed. For the RANS computations that give the mean velocity data only, application of the velocity-to-speed conversion procedure based on the turbulence kinetic energy field provided by a turbulence model resulted in accurate mean speed evaluation.

scholarly journals Flow in a commercial steel pipe

2008 ◽  
Vol 595 ◽  
pp. 323-339 ◽  
Author(s):  
L. I. LANGELANDSVIK ◽  
G. J. KUNKEL ◽  
A. J. SMITS
Keyword(s):  
Pressure Drop ◽  
Flow Rate ◽  
Roughness Height ◽  
Steel Pipe ◽  
Mean Flow ◽  
Mean Velocity ◽  
Flow Measurements ◽  
Commercial Steel ◽  
The Mean ◽  
The Difference

Mean flow measurements are obtained in a commercial steel pipe with krms/D = 1/26 000, where krms is the roughness height and D the pipe diameter, covering the smooth, transitionally rough, and fully rough regimes. The results indicate a transition from smooth to rough flow that is much more abrupt than the Colebrook transitional roughness function suggests. The equivalent sandgrain roughness was found to be 1.6 times the r.m.s. roughness height, in sharp contrast to the value of 3.0 to 5.0 that is commonly used. The difference amounts to a reduction in pressure drop for a given flow rate of at least 13% in the fully rough regime. The mean velocity profiles support Townsend's similarity hypothesis for flow over rough surfaces.

Implementation of a Mixing Length Turbulence Formulation Into the Dynamic Wake Meandering Model

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
Rolf-Erik Keck ◽  
Dick Veldkamp ◽  
Helge Aagaard Madsen ◽  
Gunner Larsen
Keyword(s):  
Cross Section ◽  
Turbulent Mixing ◽  
Eddy Viscosity ◽  
High Fidelity ◽  
Mixing Length ◽  
Two Dimensional ◽  
Wake Region ◽  
Mean Velocity ◽  
The Mean ◽  
The Difference

The work presented in this paper focuses on improving the description of wake evolution due to turbulent mixing in the dynamic wake meandering (DWM) model. From wake investigations performed with high-fidelity actuator line simulations carried out in ELLIPSYS3D, it is seen that the current DWM description, where the eddy viscosity is assumed to be constant in each cross-section of the wake, is insufficient. Instead, a two-dimensional eddy viscosity formulation is proposed to model the shear layer generated turbulence in the wake, based on the classical mixing length model. The performance of the modified DWM model is verified by comparing the mean wake velocity distribution with a set of ELLIPSYS3D actuator line calculations. The standard error (defined as the standard deviation of the difference between the mean velocity field of the DWM and the actuator line model), in the wake region extending from 3 to 12 diameters behind the rotor, is reduced by 27% by using the new eddy viscosity formulation.

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