Time-varying sensitivity analysis for mixing in chaotic flows: a comparison study

TL;DR

The paper tackles the challenge of understanding how design parameters influence mixing in chaotic flows by performing time-varying sensitivity analyses on two flow models with different dimensionalities: the RPM flow (2 or 4 parameters) and the quadrupole flow (16 parameters). It systematically compares Sobol indices, Morris scores, and an active-subspace-based activity score, assessing their convergence, interactions, and temporal evolution of parameter importance. Across the RPM and quadrupole cases, the methods yield broadly consistent rankings, with Morris offering substantial computational savings, and ASM providing a path to surrogate modeling when feasible. The study demonstrates that parameter influence on mixing can shift over time and depends on problem dimensionality, informing practical design and optimization of chaotic advection systems in applications like groundwater remediation.

Abstract

Engineered injection and extraction systems that create chaotic advection are promising procedures for enhancing mixing between two species. Mixing efficiencies vary considerably, so carefully selecting the design parameters, like pumping rates, well locations, or operation times, is crucial. While numerous studies investigate the conditions required to achieve chaotic flow, sensitivity analyses addressing its impact on mixing have rarely been performed. However, selecting a suitable sensitivity analysis method depends on the underlying system and is often restricted by the computational cost, especially when considering complex, high-dimensional models. Moreover, the most appropriate metric to quantify mixing (e.g., plume area, peak concentration) can also be system-specific. We perform a time-varying sensitivity analysis on the mixing enhancement of two chaotic flow fields with different complexities. The rotated potential mixing (RPM) flow is parametrized using two or four hyperparameters, while the quadrupole flow utilizes 16 hyperparameters. We compare three global sensitivity analysis methods: Sobol indices, Morris scores, and a modification of the activity scores. We evaluate the temporal evolution of the sensitivity of the design parameters, compare the performance of the three methods, and highlight their potential in analyzing parameter interactions. The analysis of the RPM flow shows comparable sensitivities for all methods. Additionally, our numerical experiments show that Morris is the cheapest method, needing at most four times fewer model evaluations than Sobol to reach convergence. This motivates us to only use the computationally cheaper but as reliable Morris and activity scores on the 16-dimensional model, yielding again consistent results.

Figures (26)

• Figure 1: Visualization for the source (blue) and sink (red) location for the RPM flow for the non-randomized and the randomized case.
• Figure 2: Visualization of the Quadrupole model, including the boundary condition and the conductivity field. The locations of the wells depend on the hyperparameter for $r$ and $\theta$ as depicted in the top right corner.
• Figure 3: Sensitivity rankings for the non-randomized RPM flow with $\Theta \in [0, \pi]$ and $\tau \in [0, 1]$ in \ref{['fig:sSens_ranking_bigintervals']} and $\Theta \in [0.4\pi, 0.6\pi]$ and $\tau \in [0.4, 0.6]$ in \ref{['fig:sSens_ranking_smallintervals']} over time $t$. The lower the score, the higher the parameter is ranked in sensitivity. Differences between the rankings of the different methods are highlighted with gray boxes.
• Figure 4: Results of the sensitivity analysis for the non-randomized RPM flow with $\Theta \in [0, \pi]$ and $\tau \in [0, 1]$ in \ref{['fig:sSens_bigintervals']} and $\Theta \in [0.4\pi, 0.6\pi]$ and $\tau \in [0.4, 0.6]$ in \ref{['fig:sSens_smallintervals']} over time $t$. We show the first, second, and total-order Sobol indices $S$ with respect to the left y-axis with solid lines and the Morris activity scores $\hat{\mu}^{2*}$ as well as the active subspace activity scores $\hat{\alpha}$ with respect to the right y-axis with dashed lines. For the results of Sobol we also show a $95~\%$ confidence interval computed using bootstrapping.
• Figure 5: Relationship between $\hat{\mu}^*$ and $\hat{\sigma}$ over time $t \geq 1$ for the non-randomized RPM flow with $\Theta \in [0, \pi]$ and $\tau \in [0, 1]$ in \ref{['fig:sSens_relationship_big']} and $\Theta \in [0.4\pi, 0.6\pi]$ and $\tau \in [0.4, 0.6]$ in \ref{['fig:sSens_relationship_small']}. The solid line indicates $\frac{\hat{\sigma}}{\hat{\mu}^*} = 1$, the dashed line indicates $\frac{\hat{\sigma}}{\hat{\mu}^*} = 0.5$ and the dash-dotted line indicates $\frac{\hat{\sigma}}{\hat{\mu}^*} = 0.1$.