Eddy models

At present, there exists no completely general RANS model for differential diffusion. Note, however, that because it solves (4.37) directly, the linear-eddy model discussed in Section 4.3 can describe differential diffusion (Kerstein 1990 Kerstein et al. 1995). Likewise, the laminar flamelet model discussed in Section 5.7 can be applied to describe differential diffusion in flames (Pitsch and Peters 1998). Here, in order to understand the underlying physics, we will restrict our attention to a multi-variate version of the SR model for inert scalars (Fox 1999). 

The PDF of an inert scalar is unchanged by the first two steps, but approaches the well mixed condition during step (3).108 The overall rate of mixing will be determined by the slowest step in the process. In general, this will be step (1). Note also that, except in the linear-eddy model (Kerstein 1988), interactions between Lagrangian fluid particles are not accounted for in step (1). This limits the applicability of most mechanistic models to cases where a small volume of fluid is mixed into a much larger volume (i.e., where interactions between fluid particles will be minimal). 

As discussed in Section 4.3, the linear-eddy model solves a one-dimensional reaction-diffusion equation for all length scales. Inertial-range fluid-particle interactions are accounted for by a random rearrangement process. This leads to significant computational inefficiency since step (3) is not the rate-controlling step. Simplifications have thus been introduced to avoid this problem (Baldyga and Bourne 1989). 

Of all of the methods reviewed thus far in this book, only DNS and the linear-eddy model require no closure for the molecular-diffusion term or the chemical source term in the scalar transport equation. However, we have seen that both methods are computationally expensive for three-dimensional inhomogeneous flows of practical interest. For all of the other methods, closures are needed for either scalar mixing or the chemical source term. For example, classical micromixing models treat chemical reactions exactly, but the fluid dynamics are overly simplified. The extension to multi-scalar presumed PDFs comes the closest to providing a flexible model for inhomogeneous turbulent reacting flows. Nevertheless, the presumed form of the joint scalar PDF in terms of a finite collection of delta functions may be inadequate for complex chemistry. The next step - computing the shape of the joint scalar PDF from its transport equation - comprises transported PDF methods and is discussed in detail in the next chapter. Some of the properties of transported PDF methods are listed here. 

Desjardin, P. E. and S. H. Frankel (1996). Assessment of turbulent combustion submodels using the linear eddy model. Combustion and Flame 104, 343-357. 

Kerstein, A. R. (1988). A linear-eddy model of turbulent scalar transport and mixing. 

Linear-eddy modeling of turbulent transport. II Application to shear layer mixing. 

Linear-eddy modelling of turbulent transport. Part 3. Mixing and differential diffusion in round jets. Journal of Fluid Mechanics 216, 411 —4-35. 

McMurtry, P. A., S. Menon, and A. R. Kerstein (1993). Linear eddy modeling of turbulent combustion. Energy and Fuels 7, 817-826. 

Frankel, S.H., C.K. Madnia, P. A. McMurtry, and P. Givi. 1993. Binary scalar mixing and reaction in homogeneous turbulence Some linear eddy model results. 

Zimberg, M. J., S. H. Prankel, J. P. Gore, and Y. R. Sivathanu. 1998. A study of coupled turbulence, soot chemistry and radiation effects using the linear eddy model. Combustion Flame 113 454-69. 

Rough River Flow and Large-Eddy Model... 

The above expression is very general and includes both the case of stagnant waters (u = 0, e.g., Eq. 20-24) as well as situations in which the water flow-induced turbulence dominates the exchange velocity relative to the influence of the wind. Obviously, as wind speed changes, for a given river the situation may switch between current-dominated and wind-dominated regimes. Another factor which influences the shape of the empirical function / of Eq. 20-31 is the typical size of the turbulent structures (the eddies) relative to the water depth. This leads to two different models, the small-eddy and the large-eddy model, respectively (Fig. 20.8 and Box 20.3). 

The small-eddy model applies, if the nondimensional roughness parameter d fulfills ... 

If the roughness parameter d increases beyond 136, the turbulent eddies become larger and begin to feel the limited vertical extension of the water column. As a result, the exponent n in Eq. 1 steadily increases until it reaches a new constant value at 1/2. At this point the eddies have reached the full depth of the river and are thus able to transport water fast between surface and bottom. This is the situation described by the large-eddy model of O Connor and Dobbins (1958). We formulate it in the general form of Eq. 1 ... 

The constant in Eq. (5) turns out to be of order 1. This yields the large-eddy model by O Connor and Dobbins... 

Figure 20.8 Depending on the roughness of the river bed, the production of turbulence leads either to (a) eddies which are much smaller than the river depth h, or to (b) large eddies which are able to transport dissolved chemicals fast from and to the water surface. Both situations can be described by two different models for air-water exchange (a) the small-eddy model by Lamont and Scott (1970), and (b) the large-eddy model by O Connor and Dobbins (1958). See Box 20.3 for details.

This river stretch would be treated with the large-eddy model without bubble enhancement. 

To calculate vPCE a/w we note that for both discharge regimes the nondimensional roughness parameter d (Eq. 20-36) is larger than 136. Thus, we use the large-eddy model (Eq. 20-35) ... 

Air-water exchange deserves closer inspection. On one hand, regarding the depth and size of the river the small-eddy model (Eq. 20-32) may be appropriate to calculate v of chloroform. Since the Mississippi River is wide and its waters flow rather slowly, we can, on the other hand estimate air-water exchange from wind speed while noting that the Henry s law coefficient of chloroform indicates a water side-controlled process. The reader is invited to compare the two approaches. 

The first group of models ( eddy models) assumes that the liquid renewal is due to small-scale eddies of the turbulent field. These models are based on idealized eddy structures of turbulence in the bubble vicinity. Lamont and Scott [1] have assumed that the small scales of turbulent motion, which extend from the smallest viscous motions to the iner-... 

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