Theoretical and Computational Fluid Dynamics

, Volume 31, Issue 5–6, pp 579–593 | Cite as

Cluster-based control of a separating flow over a smoothly contoured ramp

  • Eurika Kaiser
  • Bernd R. Noack
  • Andreas Spohn
  • Louis N. Cattafesta
  • Marek Morzyński
Original Article


The ability to manipulate and control fluid flows is of great importance in many scientific and engineering applications. The proposed closed-loop control framework addresses a key issue of model-based control: The actuation effect often results from slow dynamics of strongly nonlinear interactions which the flow reveals at timescales much longer than the prediction horizon of any model. Hence, we employ a probabilistic approach based on a cluster-based discretization of the Liouville equation for the evolution of the probability distribution. The proposed methodology frames high-dimensional, nonlinear dynamics into low-dimensional, probabilistic, linear dynamics which considerably simplifies the optimal control problem while preserving nonlinear actuation mechanisms. The data-driven approach builds upon a state space discretization using a clustering algorithm which groups kinematically similar flow states into a low number of clusters. The temporal evolution of the probability distribution on this set of clusters is then described by a control-dependent Markov model. This Markov model can be used as predictor for the ergodic probability distribution for a particular control law. This probability distribution approximates the long-term behavior of the original system on which basis the optimal control law is determined. We examine how the approach can be used to improve the open-loop actuation in a separating flow dominated by Kelvin–Helmholtz shedding. For this purpose, the feature space, in which the model is learned, and the admissible control inputs are tailored to strongly oscillatory flows.


Flow control Markov model Cluster analysis Liouville equation Flow separation Feedback control 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentUniversity of WashingtonSeattleUSA
  2. 2.Institut PPRIMEUPR 3346 CNRS – Université de Poitiers – ENSMAFuturoscope ChasseneuilFrance
  3. 3.LIMSI-CNRS, UPR 3251Orsay CedexFrance
  4. 4.Technische Universität BraunschweigBraunschweigGermany
  5. 5.Florida Center for Advanced Aero-PropulsionFlorida State UniversityTallahasseeUSA
  6. 6.Chair of Virtual EngineeringPoznan University of TechnologyPoznańPoland

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