Scalar correlation 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. 

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. 

---