Controlling three-dimensional falling liquid films: from weighted-residual models to direct numerical simulation
Abstract
We consider the problem of controlling a three-dimensional falling liquid film using same-fluid blowing and suction through the substrate. Building upon recent results for the corresponding two-dimensional problem, we employ a reduced-dimensional weighted-residual model for the film height and downstream flux as the basis for controller design. Linear stability analysis is used to estimate the number of actuators required to stabilise the film, while a linear-quadratic regulator (LQR) framework is used to construct feedback controls for the linearised system. The resulting gain matrix is then combined with observations from direct numerical simulation (DNS) of the three-dimensional Navier-Stokes equations, allowing controls derived from the reduced-dimensional model to be applied directly to the full flow. Through systematic numerical tests, we demonstrate that these controls successfully stabilise the flat-film solution across a physically relevant range of Reynolds numbers and domain configurations. We further show that both the number and placement of actuators play a crucial role in control performance, with spanwise instabilities requiring additional actuator coverage in the cross-stream direction. Although the extension from two to three dimensions substantially increases the cost of controller construction, this cost is incurred entirely offline during the computation of the gain matrix, making the overall framework computationally tractable. Our results demonstrate that LQR feedback controls derived from a three-dimensional weighted-residual model retain predictive value when applied to fully resolved DNS, extending previous two-dimensional control methodologies and helping bridge the gap between theoretical control design and experimentally relevant falling-film flows.
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BibTeX
@misc{holroyd2026controlling,
title = {{Controlling three-dimensional falling liquid films: from weighted-residual models to direct numerical simulation}},
author = {Oscar A. Holroyd and Susana N. Gomes and Radu Cimpeanu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31377},
eprint = {2609.31377},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31377},
}
RIS
TY - UNPB
TI - Controlling three-dimensional falling liquid films: from weighted-residual models to direct numerical simulation
AU - Oscar A. Holroyd
AU - Susana N. Gomes
AU - Radu Cimpeanu
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31377
UR - https://arxiv.org/abs/2609.31377
AB - We consider the problem of controlling a three-dimensional falling liquid film using same-fluid blowing and suction through the substrate. Building upon recent results for the corresponding two-dimensional problem, we employ a reduced-dimensional weighted-residual model for the film height and downstream flux as the basis for controller design. Linear stability analysis is used to estimate the number of actuators required to stabilise the film, while a linear-quadratic regulator (LQR) framework is used to construct feedback controls for the linearised system. The resulting gain matrix is then combined with observations from direct numerical simulation (DNS) of the three-dimensional Navier-Stokes equations, allowing controls derived from the reduced-dimensional model to be applied directly to the full flow. Through systematic numerical tests, we demonstrate that these controls successfully stabilise the flat-film solution across a physically relevant range of Reynolds numbers and domain configurations. We further show that both the number and placement of actuators play a crucial role in control performance, with spanwise instabilities requiring additional actuator coverage in the cross-stream direction. Although the extension from two to three dimensions substantially increases the cost of controller construction, this cost is incurred entirely offline during the computation of the gain matrix, making the overall framework computationally tractable. Our results demonstrate that LQR feedback controls derived from a three-dimensional weighted-residual model retain predictive value when applied to fully resolved DNS, extending previous two-dimensional control methodologies and helping bridge the gap between theoretical control design and experimentally relevant falling-film flows.
ER -