Scalar covariance model

The SR model introduced in Section 4.6 describes length-scale effects and contains an explicit dependence on Sc. In this section, we extend the SR model to describe differential diffusion (Fox 1999). The key extension is the inclusion of a model for the scalar covariance (4> aft p) and the joint scalar dissipation rate. In homogeneous turbulence, the covari-... 

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

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Generate a mixing model that predicts the correct joint scalar PDF shape for a given scalar covariance matrix, including the asymptotic collapse to a Gaussian form. 

Couple it with a model for the joint scalar dissipation rate that predicts the correct scalar covariance matrix, including the effect of the initial scalar length-scale distribution. 

At first glance, an extension of the IEM model would appear also to predict the correct scalar covariance ... 

The extension of the SR model to differential diffusion is outlined in Section 4.7. In an analogous fashion, the LSR model can be used to model scalars with different molecular diffusivities (Fox 1999). The principal changes are the introduction of the conditional scalar covariances in each wavenumber band 4> a4> p) and the conditional joint scalar dissipation rate matrix (e). For example, for a two-scalar problem, the LSR model involves three covariance components (/2, W V/A) and and three joint dissipation... 

For higher-order reactions, a model must be provided to close the covariance source terms. One possible approach to develop such a model is to extend the FP model to account for scalar fluctuations in each wavenumber band (instead of only accounting for fluctuations in In any case, correctly accounting for the spectral distribution of the scalar covariance chemical source term is a key requirement for extending the LSR model to reacting scalars. 

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. 

P(j+i)j for / = 1— 1 to be independent of Sc. This is the assumption employed in the SR model, but it can be validated (and modified) using DNS data for the scalar spectrum and the scalar-scalar transfer function. The linearity assumption discussed earlier implies that the rate constants will be unchanged (for the same Reynolds and Schmidt numbers) when they are computed using the scalar-covariance transfer spectrum. 

In refs (Kim,2004 Kim, 2005) we take one step further estimating corrections to the Gaussian effective potential for the U(l) scalar electrodynamics where it represents the standard static GL effective model of superconductivity. Although it was found that, in the covariant pure (f)4 theory in 3 + 1 dimensions,corrections to the GEP are not large (Stancu,1990), we do not expect them to be negligible in three dimensions for high Tc superconductivity, where the system is strongly correlated. 

In Section 3.3, the general transport equations for the means, (3.88), and covariances, (3.136), of 0 are derived. These equations contain a number of unclosed terms that must be modeled. For high-Reynolds-number flows, we have seen that simple models are available for the turbulent transport terms (e.g., the gradient-diffusion model for the scalar fluxes). Invoking these models,134 the transport equations become... 

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). 

The prior estimates of X and P allow the n 1 Kalman gain K to be calculated through Eqn.(5), where R(k) is the covariance of the kth measurem it. It is worth nothing that the elements of K(k) are computed as the optimal filter weights producing the minimum variance fit of the model to the data. Moreover, the invoted quantity in Eqn.(5) is scalar, so only the reciprocal must be calculated. 

Calculations of the relativistic optical potential using this model and the set of covariants in eq. (4.9) with the 7172 term, make it clear that the PS invariant was indeed responsible for the low energy behavior of the scalar and vector optical potentials. Replacement of C p = 7 72 with... 

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