Mixing Fronts in Chaotic Flows

TL;DR

This work addresses how concentration statistics in mixing fronts emerge from the interaction of macro-scale dispersion and micro-scale mixing in smooth chaotic flows. The authors develop a physically grounded closure by introducing an injection scale $s_i$ that balances dispersion and chaotic stretching, and they derive a parameter-free expression for the variance $\\sigma_c^2 \\\approx \\\frac{\\chi_0}{\\pi \\\gamma} Ei(-4\\pi s_B'/s_i)$ with $\\chi_0=2\\pi D (\\nabla \\bar{c})^2$, $s_B'= \\sqrt{\\kappa/\\sigma_\\gamma^2}$, and $s_i=4\\pi\\sqrt{D\\sigma_\\gamma^2/\\gamma^2}$. The theory, validated against direct numerical simulations in a 2D sine-flow, accurately captures the spectrum and variance across wide Peclet numbers and shows a Gaussian fluctuation distribution under forcing, simplifying the modeling of both conservative and reactive transport. The injection-scale closure provides a parameter-free tool to predict front statistics and could extend to porous media and microfluidic contexts, though extensions to 3D and time-dependent flows remain for future work. Overall, the paper offers a coherent framework that connects macroscale dispersion with microscale stirring to predict concentration statistics in mixing fronts.

Abstract

Mixing fronts develop at the interface of fluids with different solute concentrations leading to jointly evolving macroscopic and microscopic concentration gradients. Macroscopic gradients decay along the front through dispersion driven by microscopic velocity heterogeneity. The resulting macroscopic fluctuations are transmitted at the micro-scale through stretching and compression, where they are ultimately dissipated. While the elementary mechanisms leading to dispersion on the one hand and scalar mixing on the other hand are well understood, predicting how their coupling governs the evolution of concentration statistics within dispersing fronts remains a challenge. Here, we use theoretical derivations and numerical simulations of mixing fronts in chaotic flows to link the evolution of scalar variance to the microscale stirring and macroscale spreading properties of the flow. We argue that the transfer of energy between the macroscopic and microscopic scales operates at a characteristic length scale $s_i$, that we derive by comparing the characteristic times for scalar variance decay from dispersion and mixing. This leads to a closed expression for the concentration variance, which captures the results of numerical simulations with no fitting parameters for a broad range of Peclet numbers. For increasing stirring persistence, the numerical simulations deviate from the theory in the small Peclet range, which we explain qualitatively. These findings open a new avenue for predicting both conservative and reactive transport in mixing fronts, ranging from porous media flows to engineered flows in microfluidic devices.

Figures (8)

• Figure 1: Scalar front mixed and dispersed by a single-scale periodic chaotic flow (sine flow). Scalar concentrations (colorscale) obey Eq. \ref{['eq:ADE']} with $\kappa=5\cdot10^{-6}$ and $\boldsymbol{u}$ defined by Eq.\ref{['eq:sineflow']}, with $U=0.2$ ($\gamma\approx \sigma^2_\gamma\approx 0.05$), and $t=300$. The mixing scales appearing in mixing fronts are shown at real scale in the plot (from small to large). For the considered scenario, the Batchelor scale is $s_B\approx 0.01 s_v$, the modified Batchelor scale is $s_B'\approx 0.01 s_v$, the velocity scale is $s_v\equiv 1$, the injection scale is $s_i \approx 2.7 s_v$ and the dispersive scale $s_d \approx 1.41 s_v$.
• Figure 2: Sketch of the scalar spectrum of a freely dispersing front, with the dispersive spectrum at macroscale (Eq. \ref{['eq:spectrum_dispersive']}) and the Kraichnan spectrum at microscale (Eq. \ref{['eq:spectrum']}). $s_d$ is the dispersion scale, $s_i$ is the injection scale, $s_v$ is the velocity scale, $s_B'$ is the modified Batchelor scale and $s_B$ is the Batchelor scale. The shaded area determines the variability in concentration fluctuations $\sigma^2_c$
• Figure 3: a) Time evolution of the scalar variance b) Time convergence of the mean $<\sigma^2_c>$ over 100 realizations of the random phases for various $U$ at fixed $s_B'=5.3 \cdot 10^{-3}$.
• Figure 4: Concentration $c'$ for various amplitude $U$ for forced mixing $g\equiv \nabla\bar{c} =1$ (top and scalar decay $g=0$ (bottom). In the scalar decay case, concentrations are rescaled by the standard deviation. We used a fixed ratio $\gamma/\kappa=5\cdot10^5$ in all simulations.
• Figure 5: Comparison between power density spectrum obtained via numerical simulation (dots) and Eq. \ref{['eq:spectrum']} (lines) at fixed $U=0.1$ and various diffusivity (subplots b.1) and b.2)) and at fixed diffusivity $\kappa=10^{-5}$ for increasing flow persistence (subplots a.1) and a.2)). The columns show the same spectrum in log-log and log-lin scale. Note that we normalized $k$ by $1/s_B'$ in the right column to evidence the universal exponential cutoff. We used the theoretical value of $\chi_0$ obtained for a freely dispersing front (Eq. \ref{['eq:chi0']}) to normalize the spectra. Inset of b.1) shows the apparition of spurious frequencies at large flow persistence due to the periodicity of the sine-flow.