The effective long-time diffusivity for a passive scalar in a Gaussian model fluid flow

1978 ◽  
Vol 89 (2) ◽  
pp. 241-250 ◽  
Author(s):  
R. Phythian ◽  
W. D. Curtis
Keyword(s):  
Diffusion Processes ◽  
Fluid Velocity ◽  
Gaussian Model ◽  
Effective Diffusivity ◽  
Passive Scalar ◽  
Random Motion ◽  
Velocity Correlation ◽  
Molecular Diffusivity ◽  
Velocity Correlation Function ◽  
Long Time

The problem considered is the diffusion of a passive scalar in a ‘fluid’ in random motion when the fluid velocity field is Gaussian and statistically homogeneous, isotropic and stationary. A self-consistent expansion for the effective long-time diffusivity is obtained and the approximations derived from this series by retaining up to three terms are explicitly calculated for simple idealized forms of the velocity correlation function for which numerical simulations are available for comparison for zero molecular diffusivity. The dependence of the effective diffusivity on the molecular diffusivity is determined within this idealization. The results support Saffman's contention that the molecular and turbulent diffusion processes interfere destructively, in the sense that the total effective diffusivity about a fixed point is less than that which would be obtained if the two diffusion processes acted independently.

Path-integral methods for turbulent diffusion

1982 ◽  
Vol 123 ◽  
pp. 59-68 ◽  
Author(s):  
I. T. Drummond
Keyword(s):  
Path Integral ◽  
Effective Diffusion ◽  
Effective Diffusivity ◽  
Passive Scalar ◽  
Path Integral Representation ◽  
Integral Methods ◽  
Long Time ◽  
Negative Effect ◽  
Diffusion Function ◽  
Markovian Limit

We derive a path-integral representation for the effective diffusion function of a passive scalar field. We use it to calculate the long-time effective diffusivity in Gaussian turbulence in the near-Markovian limit. Our results confirm the negative effect of vorticity predicted by previous discussions. They also demonstrate that the helicity of the turbulence when present may be as important an influence as the vorticity.

On the spreading of a turbulent spot in the absence of a pressure gradient

1982 ◽  
Vol 123 ◽  
pp. 69-90 ◽  
Author(s):  
I. Wygnanski ◽  
M. Zilberman ◽  
Joseph H. Haritonidis
Keyword(s):  
Scalar Field ◽  
Effective Diffusion ◽  
Effective Diffusivity ◽  
Passive Scalar ◽  
Turbulent Spot ◽  
Path Integral Representation ◽  
Long Time ◽  
Negative Effect ◽  
Diffusion Function ◽  
Markovian Limit

We derive a path-integral representation for the effective diffusion function of a passive scalar field. We use it to calculate the long-time effective diffusivity in Gaussian turbulence in the near-Markovian limit. Our results confirm the negative effect of vorticity predicted by previous discussions. They also demonstrate that the helicity of the turbulence when present may be as important an influence as the vorticity.

Dispersion by random velocity fields

1975 ◽  
Vol 67 (1) ◽  
pp. 145-153 ◽  
Author(s):  
R. Phythian
Keyword(s):  
Correlation Function ◽  
Velocity Field ◽  
Velocity Fields ◽  
Random Motion ◽  
Velocity Correlation ◽  
Spectral Functions ◽  
Velocity Correlation Function ◽  
Lagrangian Velocity ◽  
Computer Calculations ◽  
Exact Computer

A simple approximation is proposed for the problem of the dispersion of marked particles in an incompressible fluid in random motion when the probability distribution of the velocity field is taken as Gaussian, homogeneous, isotropic, stationary and of zero mean. Approximations for the Lagrangian velocity correlation function and the dispersion are given and compared with exact computer calculations of Kraichnan. Agreement is found to be good except for time-independent velocity fields and singular wavenumber spectral functions.

Particle dispersion in isotropic turbulence under Stokes drag and Basset force with gravitational settling

1991 ◽  
Vol 225 ◽  
pp. 481-495 ◽  
Author(s):  
Renwei Mei ◽  
Ronald J. Adrian ◽  
Thomas J. Hanratty
Keyword(s):  
Isotropic Turbulence ◽  
Fluid Velocity ◽  
Stokes Number ◽  
Particle Dispersion ◽  
Velocity Correlation ◽  
Basset Force ◽  
Stokes Drag ◽  
High Particle ◽  
Velocity Correlation Function ◽  
Density Ratios

An analysis that includes the effects of Basset and gravitational forces is presented for the dispersion of particles experiencing Stokes drag in isotropic turbulence. The fluid velocity correlation function evaluated on the particle trajectory is obtained by using the independence approximation and the assumption of Gaussian velocity distributions for both the fluid and the particle, formulated by Pismen & Nir (1978). The dynamic equation for particle motion with the Basset force is Fourier transformed to the frequency domain where it can be solved exactly. It is found that the Basset force has virtually no influence on the structure of the fluid velocity fluctuations seen by the particles or on particle diffusivities. It does, however, affect the motion of the particle by increasing (reducing) the intensities of particle turbulence for particles with larger (smaller) inertia. The crossing of trajectories associated with the gravitational force tends to enhance the effect of the Basset force on the particle turbulence. An ordering of the terms in the particle equation of motion shows that the solution is valid for high particle/fluid density ratios and to 0(1) in the Stokes number.

Expressions for the effective diffusivity in materials with interphase boundaries

2004 ◽  
Vol 819 ◽  
Author(s):  
Irina V. Belova ◽  
Graeme E. Murch
Keyword(s):  
Polycrystalline Material ◽  
Effective Diffusivity ◽  
Poor Agreement ◽  
Two Phase ◽  
Interphase Boundaries ◽  
Long Time ◽  
Simulation Results ◽  
The Individual ◽  
Individual Grains ◽  
Good Agreement

AbstractWe address the problem of calculating the long-time-limit effective diffusivity in stable two- phase polycrystalline material. A phenomenological model is used where the high diffusivity interphase boundaries are treated as connected “coatings” of the individual grains. Derivation of expressions for the effective diffusivity with segregation is made along Maxwell lines. Monte Carlo simulation using lattice-based random walks is used to test the validity of the expressions. It is shown that for the case analysed the derived expressions for the effective diffusivity are in very good agreement with simulation results. The equivalent of the Hart equation is also derived. It is shown to be in poor agreement with simulation results.

Acetate and ammonium diffusivity in membrane-aerated biofilms: improving model predictions using experimental results

2005 ◽  
Vol 52 (7) ◽  
pp. 121-126 ◽  
Author(s):  
J.W. Shanahan ◽  
A.C. Cole ◽  
M.J. Semmens ◽  
T.M. LaPara
Keyword(s):  
Population Stratification ◽  
Fluid Velocity ◽  
Effective Diffusivity ◽  
Substantial Variation ◽  
Biomass Density ◽  
Bacterial Populations ◽  
Modified Model ◽  
Biofilm Reactors ◽  
Model Predictions ◽  
High Fluid

Membrane-aerated biofilm reactors (MABRs) are advantageous for wastewater treatment because of their ability to achieve both nitrification and denitrification in a single bioreactor. The stratification of membrane aerated biofilms, however, needs to be better understood so that MABRs can be properly designed and implemented. In this study, we present a modified multi-population model that accounts for variation in effective diffusivity in biofilms of variable biomass density. For biofilms grown at a low fluid velocity (2 cm s−1), the variation in effective diffusivity had a profound effect on the predicted stratification and activity of bacterial populations. For biofilms grown at a high fluid velocity (14 cm s−1), biomass density was relatively constant as a function of depth and thus there was less substantial variation in effective diffusivity; our modified model, therefore, predicted a population stratification that was similar to its original version under these conditions.

Grouted Waste Leach Tests: Pursuit of Mechanisms and Data for Long-Term Performance Assessment

1989 ◽  
Vol 176 ◽  
Author(s):  
R. Jeff Serne
Keyword(s):  
Diffusion Processes ◽  
Scale Up ◽  
Liquid Waste ◽  
Effective Diffusivity ◽  
Cementitious Materials ◽  
Solidified Waste ◽  
Effective Diffusivities ◽  
Long Term Performance ◽  
Term Performance

ABSTRACTAt Hanford, low-level liquid nuclear waste is being mixed with cementitious materials (grout) to form leach-resistant solid waste. Prior to grouting each liquid waste, an assessment must be performed to evaluate the long-term environmental impact. These predictions rely upon a diffusioncontrolled release model and short-term laboratory leach data on small grout samples. This paper describes size scale-up and inventory scale-up experiments that evaluate whether diffusion does in fact control the release of contaminants. The results of the volume scale-up test suggest that tests on grout cylinders between the sizes 3 cm dia. by 3 cm length and 30 cm dia. by 29 cm length yield comparable results. These data and other available literature suggest that extrapolation of leach results to large blocks of solidified waste is defensible.The inventory scale-up tests for 125I and selenium show a constant effective diffusivity regardless of the original inventory. This suggests that diffusion processes do in fact control the release of iodide and selenium from grout. The 99Tc effective diffusivities may not remain constant with changes in inventory.

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