Scalar variance model

The terms involving y in the SR model equations correspond to the fraction of the scalar-variance production that falls into a particular wavenumber band. In principle, yn could be found from the scalar-flux spectrum (Fox 1999). Instead, it is convenient to use a self-similarity hypothesis that states that for Sc = 1 at spectral equilibrium the fraction of scalar variance that lies in a particular wavenumber band will be independent of V. Applying this hypothesis to (4.103)-(4.106) yields 31... 

Thus the SR model yields the standard scalar-variance transport equation for homogeneous flow ... 

Like the scalar variance, (4.102), in the multi-variate SR model the scalar covariance is divided into finite wavenumber bands (

[Pg.155]

These variables are governed by exactly the same model equations (e.g., (4.103)) as the scalar variances (inter-scale transfer at scales larger than the dissipation scale thus conserves scalar correlation), except for the dissipation range (e.g., (4.106)), where... 

As done below for two examples, expressions can also be derived for the scalar variance starting from the model equations. For the homogeneous flow under consideration, micromixing controls the variance decay rate, and thus y can be chosen to agree with a particular model for the scalar dissipation rate. For inhomogeneous flows, the definitions of G and M(n) must be modified to avoid spurious dissipation (Fox 1998). We will discuss the extension of the model to inhomogeneous flows after looking at two simple examples. 

On the other hand, the form of the scalar-variance transport equation will depend on Gs and For example, applying the model to the mixture fraction 6 in the absence of... 

There is no information on the instantaneous scalar dissipation rate and its coupling to the turbulence field. A transported PDF micromixing model is required to determine the effect of molecular diffusion on both the shape of the PDF and the rate of scalar-variance decay. 

Since the scalar dissipation rate controls the time dependence of die scalar variance, a good model for die scalar dissipation rate is crucial for predicting the form of the scalar PDF at a particular time t. 

Gap is the corresponding scalar-covariance source term, and Tap is the scalar-covariance transfer spectrum. In the following, we will relate the SR model for the scalar variance to (A.2) however, analogous expressions can be derived for the scalar covariance from (A.4) by following the same procedure. 

In the present version of the SR model, the fractions y, and yn are assumed to be time-independent functions of Rei and Sc. Likewise, the scalar-variance source term Va is closed with a gradient-diffusion model. The SR model could thus be further refined (with increased computational expense) by including an explicit model for the scalar-flux spectrum. 

By far, the most widely employed models for reactive flow processes are based on Reynolds-averaged Navier Stokes (RANS) equations. As discussed earlier in Chapter 3, Reynolds averaging decomposes the instantaneous value of any variable into a mean and fluctuating component. In addition to the closure equations described in Chapter 3, for reactive processes, closure of the time-averaged scalar field equations requires models for (1) scalar flux, (2) scalar variance, (3) dissipation of scalar variance, and (4) reaction rate. Details of these equations are described in the following section. Broadly, any closure approach can be classified either as a phenomenological, non-PDF (probability density function) or as a PDF-based approach. These are also discussed in detail in the following section. 

This equation contains three new terms, namely flux of scalar variance, production of variance and dissipation of scalar variance, which require further modeling to close the equation. The flux terms are usually closed by invoking the gradient diffusion model (with turbulent Schmidt number, aj, of about 0.7). This modeled form is already incorporated in Eq. (5.21). The variance production term is modeled by invoking an analogy with turbulence energy production (Spalding, 1971) ... 

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