Differential diffusion

Differential diffusion occurs when the molecular diffusivities of the scalar fields are not the same. For the simplest case of two inert scalars, this implies F / and y 2 1 (see (3.140)). In homogeneous turbulence, one effect of differential diffusion is to de-correlate the scalars. This occurs first at the diffusive scales, and then backscatters to larger scales until the energy-containing scales de-correlate. Thus, one of the principal difficulties of modeling differential diffusion is the need to account for this length-scale dependence. 

Scalar correlation at the diffusive scales can be measured by the scalar-gradient correlation function  

Likewise, scalar correlation at the energy-containing scales is measured by the scalar correlation function  

The Reynolds-number dependence of differential-diffusion effects on gajg is distinctly different than on pap, and can be best understood by looking at scalars in homogeneous, stationary turbulence with and without uniform mean scalar gradients. 

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

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Pulsed field gradient NMR has become a standard method for measurement of diffusion rates. Stilbs [272] and others have exploited in particular the FT version for the study of mixtures. An added advantage of PFG-NMR is that it can be employed to simplify complex NMR spectra. This simplification is achieved by attenuation of resonances based on the differential diffusion properties of components present in the mixture. 

Note that in the special case of size-independent growth, this term can be expressed as a closed function of the moments, i.e., G,t(c) = G(c)mk. Note also that when deriving Eq. (102) we have neglected the size-dependence of This is justified in turbulent flows and, in any case, to do otherwise would require a micromixing model that accounts for differential diffusion (Fox, 2003). 

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. 

Again, this is strictly only true at sufficiently high Reynolds number where differential-diffusion effects become negligible. 

At present, there exists no completely general RANS model for differential diffusion. Note, however, that because it solves (4.37) directly, the linear-eddy model discussed in Section 4.3 can describe differential diffusion (Kerstein 1990 Kerstein et al. 1995). Likewise, the laminar flamelet model discussed in Section 5.7 can be applied to describe differential diffusion in flames (Pitsch and Peters 1998). Here, in order to understand the underlying physics, we will restrict our attention to a multi-variate version of the SR model for inert scalars (Fox 1999). 

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-... 

For differential diffusion, yap>. Thus, the dissipation transfer rate,... 

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.,

[Pg.156]

In order to illustrate how the multi-variate SR model works, we consider a case with constant Re>. = 90 and Schmidt number pair Sc = (1, 1/8). If we assume that the scalar fields are initially uncorrelated (i.e., pup 0) = 0), then the model can be used to predict the transient behavior of the correlation coefficients (e.g., pap(i)). Plots of the correlation coefficients without (cb = 0) and with backscatter (Cb = 1) are shown in Figs. 4.14 and 4.15, respectively. As expected from (3.183), the scalar-gradient correlation coefficient gap(t) approaches l/yap = 0.629 for large t in both figures. On the other hand, the steady-state value of scalar correlation pap depends on the value of Cb. For the case with no backscatter, the effects of differential diffusion are confined to the small scales (i.e., (), / h and s)d) and, because these scales contain a relatively small amount of the scalar energy, the steady-state value of pap is close to unity. In contrast, for the case with backscatter, de-correlation is transported back to the large scales, resulting in a lower steady-state value for p p. 

As discussed in Section 3.4, differential-diffusion effects will decrease with increasing Reynolds number. A single molecular mixing time scale T[Pg.168]

Ignoring the conditional fluctuations and differential-diffusion effects, the transport equation for Q(f x, t) becomes... 

We have used Fick s law of diffusion with separate molecular diffusivities for each species. However, most PDF models for molecular mixing do not include differential-diffusion effects. 

In order to go beyond the simple description of mixing contained in the IEM model, it is possible to formulate a Fokker-Planck equation for scalar mixing that includes the effects of differential diffusion (Fox 1999).83 Originally, the FP model was developed as an extension of the IEM model for a single scalar (Fox 1992). At high Reynolds numbers,84 the conditional scalar Laplacian can be related to the conditional scalar dissipation rate by (Pope 2000)... 

Rank(Cg) = Ns. In this case, the allowable region will be Ns-dimensional for all time. This can occur, for example, when differential diffusion is important, or may be due to how the scalar fields are initialized at time zero. 

Property (iii) applies in the absence of differential-diffusion effects. In this limit, the FP model becomes... 

Before leaving the FP model, it is of interest to consider particular limiting cases wherein the form of (e 0) is relatively simple. For example, in many non-premixed flows without differential diffusion, the composition vector is related to the mixture fraction by a linear transformation 107... 

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... 

Thus, as noted by Yeung andPope (1993), since the molecular diffusivities do not appear on the right-hand side, molecular differential diffusion affects the coherency only indirectly, i.e., through inter-scale transfer processes which propagate incoherency from small scales to large scales. The choice of the model for the scalar transfer spectra thus completely determines the long-time behavior of pap in the absence of mean scalar gradients. 

The Lagrangian spectral relaxation model for differential diffusion in homogeneous turbulence. Physics of Fluids 11, 1550-1571. 

Linear-eddy modelling of turbulent transport. Part 3. Mixing and differential diffusion in round jets. Journal of Fluid Mechanics 216, 411 —4-35. 

Kronenburg, A. and R. W. Bilger (1997). Modeling of differential diffusion effects in nonpremixed nonreacting turbulent flow. Physics of Fluids 9, 1435-1447. 

Nilsen, V. and G. Kosaly (1997). Differentially diffusing scalars in turbulence. Physics of Fluids 9, 3386-3397. 

Differential diffusion in turbulent reacting flows. Combustion and Flame 117, 493-513. 

Pitsch, H. and N. Peters (1998). A consistent flamelet formulation for non-premixed combustion considering differential diffusion effects. Combustion and Flame 144,... 

Saylor, J. R. and K. R. Sreenivasan (1998). Differential diffusion in low Reynolds number water jets. Physics of Fluids 10, 1135-1146. 

Multi-scalar triadic interactions in differential diffusion with and without mean scalar gradients. Journal of Fluid Mechanics 321, 235-278. 

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