An ensemble-variational approach for open-loop flow control

Abstract

The design of effective control strategies for unsteady flows governed by complex, nonlinear dynamics remains a central challenge in fluid mechanics. Adjoint-based optimisation methods, while efficient for high-dimensional problems, require the derivation and implementation of adjoint equations and can exhibit numerical sensitivities in systems that are not smoothly differentiable. This work proposes an ensemble-variational (EnVar) framework as a non-intrusive alternative, in which cost-function gradients are approximated through a finite ensemble of perturbed control vectors, requiring no modification of the forward solver. The present approach is tailored to address high-dimensional control problems, here considering the case where the actuator is represented as a spatially distributed forcing field. The methodology is assessed on two-dimensional open-cavity flows across Reynolds regimes spanning from quasi-periodic to chaotic dynamics. In the quasi-periodic regime, the EnVar framework recovers control strategies consistent with adjoint-based optimisation, achieving significant reductions in kinetic energy fluctuations and driving the flow toward a periodic limit cycle. In the chaotic regime, the framework remains effective in estimating gradient information and mitigating flow fluctuations. Taken together, these results demonstrate that the EnVar method constitutes a computationally efficient, parallelisable, and non-intrusive alternative to adjoint-based approaches for high-dimensional flow optimisation across a broad range of unsteady and chaotic regimes.