[Paper Review] Models and measures of mixing and effective diffusion

Zhi Lin, Katarína Boďová+1 more|arXiv (Cornell University)|Nov 5, 2010

Fluid Dynamics and Turbulent Flows3 references3 citations

TL;DR

This paper resolves quantitative inconsistencies among different measures of effective diffusivity in mixing by proposing a generalized Dispersion-Diffusion Theory (DDT), which unifies particle dispersion and scalar variance suppression. By extending Batchelor’s 1949 theory to inhomogeneous, anisotropic flows, DDT accurately predicts mixing efficiency without requiring scale separation, yielding a uniformly valid $\mathcal{E}_p \lesssim \text{Pe}^1$ scaling at high Péclet numbers.

ABSTRACT

Mixing a passive scalar field by stirring can be measured in a variety of ways including tracer particle dispersion, via the flux-gradient relationship, or by suppression of scalar concentration variations in the presence of inhomogeneous sources and sinks. The mixing efficiency or efficacy of a particular flow is often expressed in terms of enhanced diffusivity and quantified as an effective diffusion coefficient. In this work we compare and contrast several notions of effective diffusivity. We thoroughly examine the fundamental case of a steady sinusoidal shear flow mixing a scalar sustained by a steady sinusoidal source-sink distribution to explore apparent quantitative inconsistencies among the measures. Ultimately the conflicts are attributed to the noncommutative asymptotic limits of large P$\acute{ ext{e}}$clet number and large length-scale separation. We then propose another approach, a generalization of Batchelor's 1949 theory of diffusion in homogeneous turbulence, that helps unify the particle dispersion and concentration variance suppression measures.

Motivation & Objective

• To identify and resolve apparent contradictions in effective diffusivity measures derived from different physical settings: particle dispersion, flux-gradient, and variance suppression.
• To analyze the fundamental case of a steady sinusoidal shear flow with a monochromatic source-sink distribution to expose inconsistencies arising from noncommutative asymptotic limits (large Péclet number and large length-scale separation).
• To develop a unified theoretical framework that reconciles particle dispersion and concentration variance suppression in mixing, especially in the absence of scale separation.
• To propose a generalization of Batchelor’s 1949 theory of diffusion in homogeneous turbulence to inhomogeneous and anisotropic flows, enabling accurate prediction of scalar variance reduction.
• To assess the validity and limitations of the flux-gradient model in steady-state mixing with sources and sinks, particularly when scale separation is absent.

Proposed method

• Derive the effective diffusivity tensor $\mathbf{K}^{\text{eff}}$ from long-time particle dispersion statistics using the covariance of particle displacements: $K_{ij}^{\text{eff}} = \frac{1}{2}\lim_{t\to\infty}\frac{d}{dt}\mathds{E}[(X_i(t)-X_i(0))(X_j(t)-X_j(0))]$.
• Apply homogenization theory to the advection-diffusion equation with a source-sink term, showing its breakdown when length-scale separation $r = l_d/l_u$ is not pristine.
• Introduce the Dispersion-Diffusion Theory (DDT) as a generalization of Batchelor’s 1949 theory, modeling scalar concentration via an integral solution resembling a diffusion equation with time-dependent effective diffusivity.
• Use the dispersion relation $\mathds{E}[(X_i(t)-X_i(0))(X_j(t)-X_j(0))] \sim (2\kappa t + U^2 t^2)\delta_{ij}$ to capture both molecular diffusion and advective dispersion in the particle covariance.
• Derive the effective mixing efficiency $\mathcal{E}_p$ from the DDT approximation, showing $\mathcal{E}_p \sim r\,\text{Pe}$ for $\text{Pe} \gg 1 \gtrsim r$, saturating rigorous upper bounds.
• Compare predictions of DDT with those from flux-gradient and homogenization-based models, demonstrating consistency in the absence of scale separation and resolving non-commutative limit conflicts.

Experimental results

Research questions

• RQ1 Why do different measures of effective diffusivity—particle dispersion, flux-gradient, and variance suppression—yield conflicting scalings in the high Péclet number limit?
• RQ2 What causes the apparent inconsistency between effective diffusivity predictions when the limits of large Péclet number and large length-scale separation are taken in different orders?
• RQ3 Can a unified theoretical framework be constructed that accurately predicts scalar variance suppression using only information from particle dispersion, even without scale separation?
• RQ4 How does the generalized Dispersion-Diffusion Theory (DDT) reconcile the particle dispersion and concentration variance suppression measures in inhomogeneous, anisotropic flows?
• RQ5 Is the flux-gradient model, which assumes infinite scale separation, physically relevant for systems with comparable source-sink and stirring length scales?

Key findings

• The apparent contradiction in effective diffusivity measures arises from the noncommutativity of the large Péclet number and large length-scale separation limits, not from mathematical inconsistency.
• Homogenization theory fails to describe scalar mixing when the source-sink distribution has a length scale comparable to the stirring scale ($r \lesssim 1$), invalidating the assumption of pristine scale separation.
• The proposed Dispersion-Diffusion Theory (DDT) successfully unifies particle dispersion and variance suppression by generalizing Batchelor’s 1949 theory to inhomogeneous flows, yielding a uniformly valid effective diffusivity.
• For high Péclet numbers and $r \lesssim 1$, the DDT predicts scalar variance suppression scaling as $\mathcal{E}_p \sim r\,\text{Pe}$, which saturates the rigorous upper bound and matches physical intuition.
• The DDT framework respects the upper bound on mixing efficiency and provides a consistent prediction of $\mathcal{E}_p \lesssim \text{Pe}^1$ at high Péclet numbers, even in the absence of scale separation.
• The flux-gradient model, which assumes infinite scale separation, cannot be reconciled with DDT or variance suppression measures in systems with comparable source-sink and stirring scales, raising questions about its physical relevance in such settings.