Turbulence eddy diffusion coefficient

There are a number of criticisms to this approach. First, the model is incomplete, since once growth begins it continues without limit. Nonlinear saturation and interactions with predators would be needed to stop this. The diffusion coefficient D certainly does not originate from the Brownian motion of the organisms, since this would be irrelevant to these processes above, say, on the millimeter scale. It is rather a turbulent eddy-diffusion coefficient aimed to... 

Eq. 7.2, where is then the eddy diffusion coefficient (Taylor and Spencer 1990). The height of the turbulent zone, within the atmospheric boundary layer, is orders of magnitude greater than that of the laminar flow layer, and dispersion of contaminant vapors in the turbulent zone is relatively rapid. 

Figure 5.7 gives some relationships for eddy diffusion coefficient profiles under different conditions that will be handy in applications of turbulent diffusive transport. 

We will apply equation (5.20) to solve for the concentration profile of suspended sediment in a river, with some simplifying assumptions. Suspended sediment is generally considered similar to a solute, in that it is a scalar quantity in equation (5.20), except that it has a settling velocity. We will also change our notation, in that the bars over the temporal mean values will be dropped. This is a common protocol in turbulent transport and will be followed here for conformity. Thus, if an eddy diffusion coefficient, e, is in the transport equation,... 

It is interesting to compare equations (6.32) and (6.33) with those for a fully developed laminar flow, equations (6.29) and (6.30). In Example 5.1, we showed that eddy diffusion coefficient in a turbulent boundary layer was linearly dependent on distance from the wall and on the wall shear velocity. If we replace the diffusion coefficient in equation (6.30) with an eddy diffusion coefficient that is proportional to hu, we get... 

Interfacial transfer of chemicals provides an interesting twist to our chemical fate and transport investigations. Even though the flow is generally turbulent in both phases, there is no turbulence across the interface in the diffusive sublayer, and the problem becomes one of the rate of diffusion. In addition, temporal mean turbulence quantities, such as eddy diffusion coefficient, are less helpful to us now. The unsteady character of turbulence near the diffusive sublayer is crucial to understanding and characterizing interfacial transport processes. 

The eddy diffusion coefficients that we introduced in Chapter 5 were steady quantities, using mean turbulence quantities (e.g., the temporal mean of u C). This temporal mean character of eddy diffusion coefficients can be misleading in determining the thickness of a diffusive boundary layer because of the importance of unsteady characteristics. We will review some conceptual theories of mass transfer that have been put forward to describe the interaction of the diffusive boundary layer and turbulence. 

Turbulent flow means that, superimposed on the large-scale flow field (e.g., the Gulf Stream), we find random velocity components along the flow (longitudinal turbulence) as well as perpendicular to the flow (transversal turbulence). The effect of the turbulent velocity component on the transport of a dissolved substance can be described by an expression which has the same form as Fick s first law (Eq. 18-6), where the molecular diffusion coefficient is replaced by the so-called turbulent or eddy diffusion coefficient, E. For instance, for transport along the x-axis ... 

The coefficient Ex is called the turbulent (or eddy) diffusion coefficient it has the same dimension as the molecular diffusion coefficient [L2 1]. The index x indicates the coordinate axis along which the transport occurs. Note that the turbulentjliffusion coefficient can be interpreted as the product of a mean transport distance Lx times a mean velocity v = (Aa At) l Egex, as found in the random walk model, Eq. 18-7. 

For turbulence it is convenient to describe particle flux in terms of an eddy diffusion coefficient, similar to a molecular diffusion coefficient. Unlike a molecular diffusion coefficient, however, the eddy diffusion coefficient is not constant for a given temperature and particle mobility, but decreases as the eddy approaches a surface. As particles are moved closer and closer to a surface by turbulence, the magnitude of their fluctuations to and from that surface diminishes, finally reaching a point where molecular diffusion predominates. As a result, in turbulent deposition, turbulence establishes a uniform aerosol concentration that extends to somewhere within the viscous sublayer. Then molecular diffusion or particle inertia transports the particles the rest of the way to the surface. 

Turbulent agglomeration. Far turbulent agglomeration two cases should be considered. First, if the inertia of the aerosol particles is approximately the same as that of the medium, the particles will move about with the same velocities as associated air parcels and can be characterized by a turbulence or eddy diffusion coefficient DT. This coefficient can have a value 104 to 106 times greater than aerosol diffusion coefficients. Turbulent agglomeration processes can be treated in a manner similar to conventional coagulation except that the larger diffusion coefficients are used. 

For a moderate wind speed of 2 m s-1, the eddy diffusion coefficient is usually 0.05 to 0.2 m2 s-1 just above a plant canopy. Under these conditions, Kj might be about 2 m2 s-1 at 30 m above the canopy and can exceed 5 m2 s-1 at or above 300 m, where turbulent mixing is even greater. By comparison, D1W is 2.4 x 10-5 m2 s-1 and Dcch is 1.5 x 10-5 m2 s-1 in air at 20°C (Appendix I). Thus Kj is 104 to 105 times larger in the turbulent air above the canopy than these > s. The random motion of air packets is indeed much more effective than the random thermal motion of molecules in moving H2O and CO2. 

CO2 concentrations in the air can vary over a wide range within different plant communities. For a corn crop exposed to a low wind speed (below 0.3 m s-1 at the top of the canopy), for a rapidly growing plant community, or for other dense vegetation where the eddy diffusion coefficient may be relatively small, the CO2 mole fraction in the turbulent air within the plant stand can be 200 pmol mol-1 during a sunny day. On the other hand, for sparse desert vegetation, especially on windy or overcast days, generally does not decrease even 2 pmol mol-1 from the value at the top of the canopy. 

Suppose that Jw, above some plant canopy reaches a peak value equivalent to 1.0 mm of water hour-1 during the daytime when the air temperature is 30°C and is 0.10 mm hour-1 at night when Tta is 20°C. Assume that during the daytime the relative humidity decreases by 20% across the first 30 m of turbulent air and that the eddy diffusion coefficient halves at night because of a reduced ambient wind speed compared to during the daytime. 

Eddy Diffusion Coefficients. The eddy diffusivities, Kh (x,y,z,t) and Kv (x,y,z,t), which depend on the turbulent structure of the atmosphere, are two of the more elusive quantities that must be estimated. They are not established through direct measurement they must be calculated from observed data. Most of the data that have been acquired to determine Kr (or Kh) have been limited to the surface layer (79) few data are available for conditions under which an elevated inversion was present. As a result, relatively little guidance is available in the literature that can be used to estimate these parameters. 

The parameter D is usually called a turbulent (or eddy) diffusion coefficient when it arises from fluid turbulence its value varies enormously from one situation to another, depending on the intensity of turbulence and on whether the environmental medium is air or water. The diagram in Fig. 1-6 shows the Fickian mass flux arising from a concentration gradient in a smoke plume. 

In Chapter 7 we define mass transfer coefficients for binary and multicomponent systems. In subsequent chapters we develop mass transfer models to determine these coefficients. Many different models have been proposed over the years. The oldest and simplest model is the film model this is the most useful model for describing multicomponent mass transfer (Chapter 8). Empirical methods are also considered. Following our discussions of film theory, we describe the so-called surface renewal or penetration models of mass transfer (Chapter 9) and go on to develop turbulent eddy diffusivity based models (Chapter 10). Simultaneous mass and energy transport is considered in Chapter 11. 

A comparison of the interactive film models that use the Chilton-Colburn analogy to obtain the heat and mass transfer coefficients with the turbulent eddy diffusivity models. 

In the five chapters that make up Part II (Chapters 7-11) we consider the estimation of rates of mass and energy transport in multicomponent systems. Multicomponent mass transfer coefficients are defined in Chapter 1, Chapter 8 develops the multicomponent film model, Chapter 9 describes unsteady-state diffusion models, and Chapter 10 considers models based on turbulent eddy diffusion. Chapter 11 shows how the additional complication of simultaneous mass and energy transfer may be handled. 

Small particles in a turbulent gas dilfuse from one point to another as a result of the eddy motion. The eddy diffusion coefficient of the particles will in general differ from that of the carrier gas. An expression for the particle eddy diffusivity can be derived for a Stokesian particle, neglecting the Brownian motion. In carrying out the analysis, it is assumed that the turbulence is homogeneous and that there is no mean gas velocity. The statistical properties of the system do not change with time. Essentially what we have is a stationary, uniform turbulence in a large box. This is an approximate representation of the core of a turbulent pipe flow, if we move with the mean velocity of the flow. 

Numerical simulation of the eddy diffusion of particles in the turbulent core of a pipe flow indicates that for particles smaller than about 170 rm, particle and gas eddy diffusion coefficients are about the same (Uijltewaal, 1995). The studies were made for three Reynolds numbers 5500,18,3(X), and 42,000 with particles of about unit density and a pipe diameter of 5 cm. Hence for the usual ranges of interest in aero.sol dynamics, particle and gas eddy diffusion coefficients can be a.ssumed equal in the turbulent core. However, the viscou.s sublayer near the wall of a turbulent pipe flow alters the situation as discussed in the next section. 

The value of D depends upon the conditions of transport. For the often-used case of diffusion under nonturbulent conditions, molecular diffusion prevails, and D is a molecular diffusion coeffi-cient. f As such, its value depends on temperature, pressure, relative size of molecules involved, and in some cases, whether all molecules, including /, are polar. If turbulent conditions prevail, we have an eddy diffusion coefficient, usually designated by the symbol 8. The models used in this chapter do not involve the turbulent case, largely because it lacks a firm basis for estimation, i.e., degrees of turbulence are not easily evaluated. 

As indicated, the flux may be expressed either in units of molecules/m2 s or in units of kg/m2 s. Here, p and n are the density and number density of air, respectively, and K is called the eddy diffusion coefficient. This quantity must be treated as a tensor because atmospheric diffusion is highly anisotropic due to gravitational constraints on the vertical motion and large-scale variations in the turbulence field. Eddy diffusivity is a property of the flowing medium and not specific to the tracer. Contrary to molecular diffusion, the gradient is applied to the mixing ratio and not to number density, and the eddy diffusion coefficient is independent of the type of trace substance considered. In fact, aerosol particles and trace gases are expected to disperse with similar velocities. 

It turns out that turbulent diffusion can be described with Fick s laws of diffusion that were introduced in the previous section, except that the molecular diffusion coefficient is to be replaced by an eddy or turbulent diffusivity E. In contrast to molecular diffusivities, eddy dififusivities are dependent only on the phase motion and are thus identical for the transport of different chemicals and even for the transport of heat. What part of the movement of a turbulent fluid is considered to contribute to mean advective motion and what is random fluctuation (and therefore interpreted as turbulent diffusion) depends on the spatial and temporal scale of the system under investigation. This implies that eddy diffusion coefficients are scale dependent, increasing with system size. Eddy diffusivities in the ocean and atmosphere are typically anisotropic, having much large values in the horizontal than in the vertical dimension. One reason is that the horizontal extension of these spheres is much larger than their vertical extension, which is limited to approximately 10 km. The density stratification of large water bodies further limits turbulence in the vertical dimension, as does a temperature inversion in the atmosphere. Eddy diffusivities in water bodies and the atmosphere can be empirically determined with the help of tracer compounds. These are naturally occurring or deliberately released compounds with well-estabhshed sources and sinks. Their concentrations are easily measured so that their dispersion can be observed readily. 

---