Turbulent diffusion is the transport of mass, heat, or momentum within a system due to random and chaotic time dependent motions. It occurs when turbulent fluid systems reach critical conditions in response to
shear flow The term shear flow is used in solid mechanics as well as in fluid dynamics. The expression ''shear flow'' is used to indicate:
* a shear stress over a distance in a thin-walled structure (in solid mechanics);Higdon, Ohlsen, Stiles and Weese (1960 ...
, which results from a combination of steep concentration gradients, density gradients, and high velocities. It occurs much more rapidly than
molecular diffusion
Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) o ...
and is therefore extremely important for problems concerning mixing and transport in systems dealing with
combustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combustion ...
,
contaminants
Contamination is the presence of a constituent, impurity, or some other undesirable element that spoils, corrupts, infects, makes unfit, or makes inferior a material, physical body, natural environment, workplace, etc.
Types of contamination
Wi ...
, dissolved oxygen, and solutions in industry. In these fields, turbulent diffusion acts as an excellent process for quickly reducing the concentrations of a species in a fluid or environment, in cases where this is needed for rapid mixing during processing, or rapid pollutant or contaminant reduction for safety.
However, it has been extremely difficult to develop a concrete and fully functional model that can be applied to the diffusion of a species in all turbulent systems due to the inability to characterize both an instantaneous and predicted fluid velocity simultaneously. In turbulent flow, this is a result of several characteristics such as unpredictability, rapid diffusivity, high levels of fluctuating vorticity, and dissipation of kinetic energy.
Applications
Atmospheric diffusion and pollutants
Atmospheric dispersion, or diffusion, studies how pollutants are mixed in the environment. There are many factors included in this modeling process, such as which level of atmosphere(s) the mixing is taking place, the stability of the environment and what type of contaminant and source is being mixed. The Eulerian and Lagrangian (discussed below) models have both been used to simulate atmospheric diffusion, and are important for a proper understanding of how pollutants react and mix in different environments. Both of these models take into account both vertical and horizontal wind, but additionally integrate
Fickian diffusion theory to account for turbulence. While these methods have to use ideal conditions and make numerous assumptions, at this point in time, it is difficult to better calculate the effects of turbulent diffusion on pollutants. Fickian diffusion theory and further advancements in research on atmospheric diffusion can be applied to model the effects that current emission rates of pollutants from various sources have on the atmosphere.
Turbulent diffusion flames
Using
planar laser-induced fluorescence
Planar laser-induced fluorescence (PLIF) is an optical diagnostic technique widely used for flow visualization and quantitative measurements. PLIF has been shown to be used for velocity, concentration, temperature and pressure measurements.
Wo ...
(PLIF) and
particle image velocimetry (PIV) processes, there has been on-going research on the effects of turbulent diffusion in flames. Main areas of study include combustion systems in gas burners used for power generation and chemical reactions in jet diffusion flames involving methane (CH
4), hydrogen (H
2) and nitrogen (N
2). Additionally, double-pulse Rayleigh temperature imaging has been used to correlate extinction and ignition sites with changes in temperature and the mixing of chemicals in flames.
Modeling
Eulerian approach
The Eulerian approach to turbulent diffusion focuses on an infinitesimal volume at a specific space and time in a fixed frame of reference, at which physical properties such as mass, momentum, and temperature are measured.
The model is useful because Eulerian statistics are consistently measurable and offer great application to chemical reactions. Similarly to molecular models, it must satisfy the same principles as the continuity equation below (where the advection of an element or species is balanced by its diffusion, generation by reaction, and addition from other sources or points) and the
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
:
where
= species concentration of interest,
= velocity
t= time,
= direction,
= molecular diffusion constant,
= rate of
generated reaction,
= rate of
generated by source.
Note that
is concentration per unit volume, and is not mixing ratio (
) in a background fluid.
If we consider an inert species (no reaction) with no sources and assume molecular diffusion to be negligible, only the advection terms on the left hand side of the equation survive. The solution to this model seems trivial at first, however we have ignored the random component of the velocity plus the average velocity in u
j= ū + u
j’ that is typically associated with turbulent behavior. In turn, the concentration solution for the Eulerian model must also have a random component c
j=
c+ c
j’. This results in a closure problem of infinite variables and equations and makes it impossible to solve for a definite c
i on the assumptions stated.
Fortunately there exists a closure approximation in introducing the concept of
eddy diffusivity and its statistical approximations for the random concentration and velocity components from turbulent mixing:
where K
jj is the eddy diffusivity.
[
Substituting into the first continuity equation and ignoring reactions, sources, and molecular diffusion results in the following differential equation considering only the turbulent diffusion approximation in eddy diffusion:
Unlike the molecular diffusion constant D, the eddy diffusivity is a matrix expression that may vary in space, and thus may not be taken outside the outer derivative.
]
Lagrangian approach
The Lagrangian model to turbulent diffusion uses a moving frame of reference to follow the trajectories and displacements of the species as they move and follows the statistics of each particle individually.[ Initially, the particle sits at a location x’ (x1, x2, x3) at time ''t''’. The motion of the particle is described by its probability of existing in a specific volume element at time ''t'', that is described by Ψ(x1, x2, x3, ''t'') dx1 dx2 dx3 = Ψ(x,''t'')dx which follows the probability density function (pdf) such that:
Where function ''Q'' is the probably density for particle transition.
The concentration of particles at a location x and time t can then be calculated by summing the probabilities of the number of particles observed as follows:
Which is then evaluated by returning to the pdf integral][
Thus, this approach is used to evaluate the position and velocity of particles relative to their neighbors and environment, and approximates the random concentrations and velocities associated with turbulent diffusion in the statistics of their motion.
]
Solutions
The resulting solution for solving the final equations listed above for both the Eulerian and Lagrangian models for analyzing the statistics of species in turbulent flow, both result in very similar expressions for calculating the average concentration at a location from a continuous source. Both solutions develop
Gaussian Plume
and are virtually identical under the assumption that the variances in the x,y,z directions are related to the eddy diffusivity: