Scalar variance derivation

The next step is to derive an expression for the scalar variance defined by... 

For a homogeneous scalar field with an isotropic filter, the conditional expected value of the scalar will have the property (+U,

transport equation can be derived for the residual scalar variance defined by11... 

Starting with the scalar transport equation, a transport equation for the inert-scalar variance was derived in Section 3.3 ((3.105), p. 85) ... 

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. 

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 scalar mixing studies and for infinite-rate reacting flows controlled by mixing, the variance of inert scalars is of interest since it is a measure of the local instantaneous departure of concentration from its local instantaneous mean value. For non-reactive flows the variance can be interpreted as a departure from locally perfect mixing. In this case the dissipation of concentration variance can be interpreted as mixing on the molecular scale. The scalar variance equation (1.462) can be derived from the scalar transport equation... 

The transport equation for the variance of an inert scalar (

and Reynolds averaging the resultant expression. This process leads to an unclosed term of the form 2[Pg.103]

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

CFD providers treat gas-phase combustion by using a mixture fraction model (Wang et al., 2006). The model is based on the solution of the transport equations for the fuel and oxidant mixture fractions as scalars and their variances. The combustion chemistry of the mixture fractions is modeled by using the equilibrium model through the minimization of the Gibbs free energy, which assumes that the chemistry is rapid enough to assure chemical equilibrium at the molecular level. Therefore, individual component concentrations for the species of interest are derived from the predicted mixture fraction distribution. 

---