Coefficient turbulent diffusion

The energy of large and medium-size eddies can be characterized by the turbulent diffusion coefficient. A, m-/s. This parameter is similar to the parameter used by Richardson to describe turbulent diffusion of clouds in the atmosphere. Turbulent diffusion affects heat and mass transfer between different zones in the room, and thus affects temperature and contaminant distribution in the room (e.g., temperature and contaminant stratification along the room height—see Chapter 8). Also, the turbulent diffusion coefficient is used in local exhaust design (Section 7.6). 

Studies by Elterman show that turbulent diffusion coefficients in ventilated rooms outside jets and plumes can be described using the relationship-... 

The scalar flux is then proportional to a turbulent-diffusion coefficient ... 

Turbulent diffusion - The mixing of chemicals by turbulence, such that a turbulent diffusion coefficient can be defined separately from the temporal mean convection. 

Turbulent diffusion is not reaUy diffusion but the mixing of chemicals through turbulent eddies created by convection. Turbulent diffusion is thus a form of convection. Although it has the appearance of diffusion in the end (i.e., random mixing similar to diffusion), the causes of diffusion and turbulent diffusion are very different. Since the end products are similar, diffusion coefficients and turbulent diffusion coefficients are often simply added together. This process will be discussed in this chapter. 

B. MASS TRANSPORT EQUATION WITH TURBULENT DIFFUSION COEFFICIENTS... 

In this section, we will derive the most common equations for dealing with mass transport in a turbulent flow. Beginning with equation (2.14), we will take the temporal mean of the entire equation and eventually end up with an equation that incorporates turbulent diffusion coefficients. 

Therefore, let us consider the following thought process if the end result of turbulence, when visualized from sufficient distance, looks like diffusion with seemingly random fluctuations, then we should be able to identify the terms causing these fluctuations in equation (5.18). Once we have identified them, we will relate them to a turbulent diffusion coefficient that describes the diffusion caused by turbulent eddies. Looking over the terms in equation (5.18) from left to right, we see an unsteady term, three mean convective terms, the three unknown terms, the diffusive terms, and the source/sink rate terms. It is not hard to figure out which terms should be used to describe our turbulent diffusion. The unknown terms are the only possibility. 

Turbulent diffusion occurs because turbulent eddies are transporting mass, momentum, and energy over the eddy scale at the rotational velocity. This transport rate is generally orders of magnitude greater than the transport rate due to molecular motion. Thus, when a flow is turbulent, diffusion is normally ignored because e Z). The exception is very near the flow boundaries, where the eddy size (and turbulent diffusion coefficient) decreases to zero. 

Prandtl s mixing length hypothesis (Prandtl, 1925) was developed for momentum transport, instead of mass transport. The end result was a turbulent viscosity, instead of a turbulent diffusivity. However, because both turbulent viscosity and turbulent diffusion coefficient are properties of the flow field, they are related. Turbulent viscosity describes the transport of momentum by turbulence, and turbulent diffusivity describes the transport of mass by the same turbulence. Thus, turbulent viscosity is often related to turbulent diffusivity as... 

You have worked hard to study the internal dynamics of tetrachloroethene (PCE) and to calculate vertical turbulent diffusion coefficients in lakes. A friend of yours is more interested in the process of air-water exchange. One day, she sees some of your PCE data lying on your desk. She is very happy with the table below and... 

Illustrative Example 22.2 Vertical Turbulent Diffusion Coefficient in a Lake... 

Give two frequently used models for turbulent transport in the environment and explain how they relate to the turbulent diffusion coefficient. 

Which assumptions are necessary for determining vertical turbulent diffusion coefficients from repeated vertical temperature measurements made at a single location in the middle of a lake ... 

In Illustrative Example 19.4, the dissolution of a non-aqueous-phase liquid (NAPL) into groundwater was discussed. Here we consider a similar (although somewhat hypothetical) case. Assume that a mixture of chlorinated solvents totally covers the flat bottom of a small pond (maximum depth zmax = 4 m, surface area Asurface = 104 m2) forming a dense non-aqueous-phase liquid (DNAPL). The DNAPL is contaminated by benzene which dissolves into the water column and is vertically transported by turbulent diffusion. The pond is horizontally well mixed. The vertical turbulent diffusion coefficient is , = 0.1 cm2s l and approximately constant over the whole water column. 

Currents in rivers and streams are turbulent. Turbulent mixing can be described by the Fickian laws (Eqs.18-6 and 18-14) and by empirical turbulent diffusion coefficients Ea, where a stands for x, y, z (Chapter 22). The main source of turbulence is the friction between the water and the river bed. It can be expected that increasing roughness of the river leads to increasing turbulence, much in the same way as a large roughness causes the mean flow u to become slow (see the effect on Eq. 24-4 if the friction coefficient f increases). In fact, turbulence in rivers can be scaled by the shear velocity, u, defined in Eq. 24-5. 

Figure 24.4 Mixing processes in a river. Ey and E, are the turbulent diffusion coefficients in the lateral and vertical direction, respectively h0 is the maximum depth. Longitudinal dispersion, djs, results from the variation of velocity in a given cross section of the river. A pollutant added to the river in cross section A-B mixes vertically and laterally into the whole river cross-section.

Like Ez, the lateral turbulent diffusion coefficient Ey is also characterized by u ... 

Another explanation of the lithium gap in the Hyades could be found in terms of turbulent diffusion and nuclear destruction. Turbulence is definitely needed to explain the lithium abundance decrease in G stars. If this turbulence is due to the shear flow instability induced by meridional circulation (Baglin, Morel, Schatzman 1985, Zahn 1983), turbulence should also occur in F stars, which rotate more rapidly than G stars. Fig. 2 shows a comparison between the turbulent diffusion coefficient needed for lithium nuclear destruction and the one induced by turbulence. Li should indeed be destroyed in F stars This effect gives an alternative scenario to account for the Li gap in the Hyades. The fact that Li is normal in the hottest observed F stars could be due to their slow rotation. 

The diffusion in a velocity field with a wide velocity spectrum supposedly describing turbulence is considered in the spirit of cascade-renormalization ideas. For the latter case of isotropic turbulence, we construct an ordinary differential equation for the turbulent diffusion coefficient. 

Here r = v(X)X, i.e., by order of magnitude it becomes the turbulent diffusion coefficient. The ratio r/D0 is the Peclet number—the analogue of the Reynolds number. 

Thus, in this paper we have obtained an exact solution of the diffusion equation for one-dimensional motion of an incompressible fluid, and determined the effective diffusion coefficient. We have constructed an approximate theory of turbulent diffusion as a cascade process of motion interaction on different scales. We have obtained an expression for the turbulent diffusion coefficient with the correct transformation properties under time reversal. 

It is interesting to note that in a study by A. P. Mirabel and A. S. Monin, written after Ya.B. s paper, the expression for the turbulent diffusion coefficient in a two-dimensional turbulent field differs from the one obtained by Ya.B. by a factor which is equal to some logarithmic power of the ratio of the mixing scale to the energy-supply scale. 

The second limit is named the cognition limit and arises from the less controlled assumptions concerning the complicated and ill acquainted phenomena involved in the process. Considering the interface as an equilibrium Gibbs interface and introducing the turbulent flow from the turbulent diffusion coefficient are two famous examples which illustrate this class of cognition limits. 

D = vertical turbulent diffusion coefficient R = production rate... 

Empirical parameters governing atmospheric dispersal pervade the literature on this subject. Like most cases of turbulent transport, elimination of a disposable coefficient in one place leads to a reappearance of one somewhere else. The present work uses an experimentally determined turbulent diffusion coefficient, D, in Equation 19. Near the ground and near the inversion base we must assign a height (z) dependence to the diffusion coefficient. 

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