Inhomogeneous turbulent mixing

A spectral model similar to (3.82) can be derived from (3.75) for the joint scalar dissipation rate eap defined by (3.139), p. 90. We will use these models in Section 3.4 to understand the importance of spectral transport in determining differential-diffusion effects. As we shall see in the next section, the spectral interpretation of scalar energy transport has important ramifications on the transport equations for one-point scalar statistics for inhomogeneous turbulent mixing. 

Owing to the complexity of multi-point descriptions, almost all scalar transport models for complex flows are based on one-point statistics. As shown in Section 2.1, one-point turbulence statistics are found by integrating over the velocity sample space. Likewise, 

Furthermore, space and time derivatives of mean quantities can be easily related to space and time derivatives of /u, (V, f x, t). For example, starting from (3.84), the time derivative of the scalar flux is given by 

In general, given /u, (V, f x, t), transport equations for one-point statistics can be easily derived. This is the approach used in transported PDF methods as discussed in Chapter 6. In this section, as in Section 2.2, we will employ Reynolds averaging to derive the one-point transport equations for turbulent reacting flows. 

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Table 3.2. The scalar statistics and unclosed quantities appearing in the transport equations for inhomogeneous turbulent mixing of an inert scalar. 

Obviously, a successful model for turbulent mixing must, at a minimum, be able to account for the time dependence of inhomogeneous turbulence.2 However, the situation... 

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. 

Implicit in the above discussion is an assumption that the aerosol is locally homogeneous over some region of space. In fact, aerosol particles that are present at a given time and location are derived from emissions that may have occurred at a variety of times and upwind locations. Spatial inhomogeneities tend to decrease over time because of turbulent mixing. In many studies it has been... 

An interpretation of these results is presented in terms of the competition between turbulent mixing and the inhomogeneities generated by the feeding of the reactor. The effects are illustrated (Pig. 1) on a realistic example, the simple kinetic model which has been used to give a near-quantitative description of the bistable region of the iodate-arsenous acid system [ 3,... 

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