Scalar covariance

As discussed in Chapter 5, the complexity of the chemical source term restricts the applicability of closures based on second- and higher-order moments of the scalars. Nevertheless, it is instructive to derive the scalar covariance equation for two scalars pa and pp with molecular-diffusion coefficients ra and I, respectively. Starting from (1.28), p. 16, the transport equation for ( p a p p) can be found following the same steps that were used for the Reynolds stresses. This process yields34 

Reynolds averaging of (3.135) leads to terms of the form ( p pV2cp a), which can be rewritten as in (3.92). The final covariance transport equation can be expressed as 

Spatial transport of scalar covariance is described by the triple-correlation term u,(p a(p p), and the molecular-transport term defined by 

The dissipation term35 in (3.136) is written as the product of the joint scalar dissipation rate eap defined by 

In high-Reynolds-number turbulent flows, is negligible, and thus the triple-correlation 

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Note that evaluating the correlation functions at r = 0 yields the corresponding one-point statistics. For example, Rap(0, t) is equal to the scalar covariance W,/prp). 

Similar relationships exist for the scalar flux energy spectrum and the scalar covariance energy spectrum. 

The derivation of the transport equation for g xg,p is analogous to that used to derive the transport equation for the scalar covariance. The resultant expression is... 

The physics of this term can be understood using the scalar cospectrum Eap(ic, t), which, similar to the scalar spectrum, represents the fraction of the scalar covariance at wavenum-... 

Note that in (3.163) the characteristic time scale for scalar-covariance dissipation is... 

In homogeneous turbulence, the governing equations for the scalar covariance, (3.137), and the joint scalar dissipation rate, (3.166), reduce, respectively, to... 

With mean scalar gradients, the scalar covariances and joint dissipation rates attain steady-state values found by setting the right-hand sides of (3.179) and (3.180) equal to zero. This yields... 

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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The case of uniform mean scalar gradients was introduced in Section 3.4, where Gia (see (3.176)) denotes the ith component of the gradient of (< In this section, we will assume that the mean scalar gradients are collinear so that GiaGip = GiaGia = G,pG,p = G2. The scalar covariance production term then reduces to V p = 2rTG2. In the absence of differential diffusion, the two scalars will become perfectly correlated in all wavenumber bands, i.e.,

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Or, to put it another way, the simplest hrst-order moment closure is to assume that all scalar covariances are zero ... 

The failure of first-order moment closures for the treatment of mixing-sensitive reactions has led to the exploration of higher-order moment closures (Dutta and Tarbell 1989 Heeb and Brodkey 1990 Shenoy and Toor 1990). The simplest closures in this category attempt to relate the covariances of reactive scalars to the variance of the mixture fraction (I 2). The latter can be found by solving the inert-scalar-variance transport equation ((3.105), p. 85) along with the transport equation for (f). For example, for the one-step reaction in (5.54) the unknown scalar covariance can be approximated by... 

Fikewise, the reacting-scalar covariances can be found from... 

The higher-order conditional scalar moments are defined in terms of the conditional fluctuations. For example, the conditional scalar covariances are defined by... 

The only new unclosed term that appears in the composition PDF transport equation is (Ui ijy>. The exact form of this term will depend on the flow. However, if the velocity and scalar fields are Gaussian, then the scalar-conditioned velocity can be expressed in terms of the scalar flux and the scalar covariance matrix ... 

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. 

Likewise, the time-evolution equation for the scalar-covariance spectrum Eap , t) can be written as... 

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. 

The scalar-scalar transfer function Saoi(K q) appearing in the final term on the right-hand side of (A.6) denotes the contribution of scalar mode q to the scalar-variance transfer spectrum at k. (See Yeung (1994) and Yeung (1996) for examples of these functions extracted from DNS.) Similarly, the scalar-covariance transfer spectrum can be decomposed using... 

Consistent with these definitions, DNS data (Yeung 1994 Yeung 1996) show that Taa is always positive, while Taa is always negative. Analogous definitions and remarks hold for the scalar-covariance transfer function, i.e., for Tap and T p. 

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. 

Note that evaluating the correlation functions at r = 0 yields the corresponding one-point statistics. For example, Rap(0, t) is equal to the scalar covariance Like the velocity spatial correlation function discussed in Section 2.1, the scalar spatial correlation function provides length-scale information about the underlying scalar field. For a homogeneous, isotropic scalar field, the spatial correlation function will depend onlyonr = r, i.e., R,p(r, t). The scalar integral scale and the scalar Taylor microscale... 

Note that in (3.163) the characteristic time scale for scalar-covariance dissipation is eap/Waf p)d- We show in Chapter 4 that eap and (

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