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

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.

💡 Research Summary

The paper investigates how design‑parameter uncertainty influences mixing performance in engineered injection‑extraction (EIE) systems that generate chaotic advection. Two representative chaotic flow fields are examined: the Rotated Potential Mixing (RPM) flow, which is low‑dimensional (2 or 4 hyper‑parameters controlling rotation angle and dwell time), and a quadrupole flow, which is high‑dimensional (16 hyper‑parameters describing pumping rates, well locations, hydraulic conductivities, and periodic re‑orientations). For each flow, a time‑varying global sensitivity analysis is performed using three established methods: Sobol variance‑based total‑effect indices, the Morris elementary‑effects method, and a modified activity‑score derived from the active‑subspace methodology.

The RPM flow is evaluated with both deterministic and stochastic rotations. The mixing metric is the fraction of the domain occupied by particles initially released in a small region (¯M), which grows monotonically due to numerical diffusion. The quadrupole flow’s metric is the maximal solute concentration (peak concentration). Sensitivity is computed at successive simulation times, allowing the authors to track how the importance of each design variable evolves as mixing proceeds.

Key methodological findings:

1. Sobol indices provide a full decomposition of first‑order and second‑order interactions but require N·(2n + 2) model evaluations, making them computationally expensive for the 16‑dimensional case.
2. The Morris method needs only N·(n + 1) evaluations; the authors demonstrate that as few as ten trajectories yield stable μ* (mean absolute elementary effect) and σ* (standard deviation) values. Morris is therefore the cheapest method while still capturing both main effects and non‑linearities.
3. The activity‑score approach reuses the Morris trajectories, computes gradients (or finite‑difference approximations) and rescales the scores to percentages of total output variance, delivering a quantitative interpretation without extra cost.

Results for the RPM flow show that the rotation angle Θ and the dwell time τ (or their means and variances in the stochastic version) dominate the early‑time mixing dynamics. All three methods agree on this ranking, and Sobol’s second‑order indices are essentially zero, indicating negligible interaction effects. As time progresses, the sensitivities of all parameters decline, reflecting the approach toward a saturated chaotic region.

For the quadrupole flow, only Morris and activity scores are applied due to the prohibitive cost of Sobol. Both methods identify the four pumping rates (k₁–k₄) and the set of well‑orientation angles (θ) as the most influential variables during the initial mixing stage. The sensitivity of these parameters varies markedly over time, suggesting that optimal control of pumping schedules and well re‑orientation can be used to target specific mixing times.

Convergence analysis confirms that Morris reaches stable estimates with roughly one‑quarter the number of model runs required by Sobol. Activity scores inherit this efficiency because they are built from the same sample set. Consequently, for high‑dimensional chaotic‑flow problems, the combination of Morris and activity scores offers a cost‑effective yet reliable sensitivity framework, while Sobol remains valuable for low‑dimensional cases where interaction effects are of particular interest.

Overall, the study provides a practical guideline for selecting global sensitivity analysis techniques in chaotic‑mixing applications, demonstrates how time‑varying sensitivities can inform design and operational decisions, and highlights the trade‑off between computational expense and insight depth across different methodological choices.