Scaling properties of differential molecular diffusion effects in turbulence

Abstract

Scaling properties of a normalized concentration difference in a turbulent flow containing two scalars of unequal diffusivity are determined by similarity analysis and numerical simulation. Similarity hypotheses applied to the power spectrum of the normalized concentration difference, termed the differential diffusion, yield predicted dependences of the variance of the differential diffusion on the turbulence Reynolds number (Re) and on the Schmidt numbers (Sc) of the scalars. In particular, the variance is found to be proportional to Re−1/2. This and other predictions are supported by numerical simulations of multiple scalar mixing using a one-dimensional stochastic mixing model. The analysis and numerical results indicate fundamental distinctions between the physical mechanisms governing the scalar spectral cascade and those governing spectral transfer of the differential diffusion. The relationships of predicted scalings to passive mixing measurements that have been reported and to behaviors expected in reacting flow are noted.

A multiple mapping conditioning model for differential diffusion

“…This Re −1/2 scaling is consistent with the theory developed by Kerstein et al 22 and later corroborated by the DNS of Nilsen and Kosály. 28 Reynolds number dependence is also analysed by Fox 6 for the study of differential diffusion in forced homogeneous isotropic turbulence and scales as z 2 ∼ Re −0.3 which is close to the −1/2 theoretical value discussed above.…”

“…Whereas much previous literature 21,28,6,19,22 has chosen to quantify differential diffusion in terms of the difference of two passive scalars, we instead quantify differential diffusion in terms of the ratio of scalar dissipation timescales. The advantage is that this quantity is stationary in time.…”

“…Following on from Bilger's 20 observation of differential diffusion effects in methane diffusion flames, Bilger and Dibble 21 introduce a differential diffusion variable that is the difference between two mixture fractions (or passive scalars). That quantity is subsequently used by Kerstein et al 22 in their analysis of the Reynolds number scaling of differential diffusion, and in a series of DNS studies by Yeung and co-workers, [23][24][25][26] Jaberi et al 27 and Nilsen and Kosály. 28,29 Experimental studies of differential diffusion are reported for both reacting and non-reacting flows.…”

The Lagrangian spectral relaxation model for differential diffusion in homogeneous turbulence

“…The Reynolds-number exponent ͑0.3͒ predicted by the LSR model is less than the value ͑0.5͒ that might be expected based on classical spectral arguments. 17 The difference, however, can be attributed to backscatter: if c d ϭ0 ͑no backscatter͒ all decorrelation would be confined to the dissipation range which scales like Re Ϫ1/2 . With backscatter, decorrelation ''leaks'' back to larger scales which decrease more slowly with an increasing Reynolds number, thereby decreasing the exponent.…”

“…The linear-eddy model ͑LEM͒ 16 is an example of such a model, and it has been applied to study the scaling properties of differential diffusion with respect to the Reynolds and Schmidt numbers. 17 Scalar spectral transport models are another option; 18 however, the computational requirements of such models are generally considered to be too large to be useful for practical applications. Modifications of existing ''equilibrium'' mixing models-like the conditional moment closure ͑CMC͒ 12,13 -have thus received particular attention in the recent literature.…”