Galerkin Method for Nonlinear Dynamics

Noack, Bernd R.; Schlegel, Michael; Morzyński, Marek; Tadmor, Gilead

doi:10.1007/978-3-7091-0758-4_3

Cited by 29 publications

(30 citation statements)

References 49 publications

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“…We purposefully investigate a regime at Re = 2 × 10 4 , where the flow evolves in a chaotic manner (Auteri et al 2002;Peng, Shiau & Hwang 2003, see also the animation of the vorticity field in the supplementary movie is available at https://doi.org/10.1017/jfm.2020.707). The chaotic nature of the problem ensures that the frequency spectrum of velocity fluctuations is continuous and energy transfers are scattered in modal space, rather than being highly organised as for periodic flows (Noack et al 2011). The domain is defined by the non-dimensional Cartesian coordinates x = (x, y) and the velocity vector u(t, x) is defined by the components u(t, x) and v(t, x)).…”

Section: Resultsmentioning

confidence: 99%

“…first 40 time units and then saturates on a value that is approximately two orders of magnitude larger than what observed in DNS. The over-prediction occurs because the truncation of the small scales in the ansatz (2.2) leads to a significant imbalance of the production-dissipation budget within the model (Noack, Papas & Monkewitz 2005;Noack et al 2008Noack et al , 2011Balajewicz et al 2013). A qualitatively similar behaviour, if not worse, is then necessarily observed for the model sparsified with the greedy approach, since neglecting weak interactions alone does not cure the original dissipation problems.…”

Section: Energy Interactions Identified By the Regression And Conservmentioning

confidence: 94%

“…The right-hand side of (2.7) is composed of three terms describing energy transfers between the hierarchy of modes. The first two describe variations of energy due to production/dissipation arising from interactions with the mean flow and from viscous effects (Noack et al 2011). The third term defines variations of energy arising from inviscid nonlinear interactions between triads of modes.…”

Section: Reduced-order Modellingmentioning

confidence: 99%

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Thel₁-based sparsification of energy interactions in unsteady lid-driven cavity flow

2020

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Section: Resultsmentioning

confidence: 99%

Section: Energy Interactions Identified By the Regression And Conservmentioning

confidence: 94%

Section: Reduced-order Modellingmentioning

confidence: 99%

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Thel₁-based sparsification of energy interactions in unsteady lid-driven cavity flow

2020

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“…To alleviate the computational concerns, control strategies are built on low-order dynamical models obtained via model reduction (Protas 2004; Pinier et al. 2007; Barbagallo, Sipp & Schmid 2009; Noack, Morzynski & Tadmor 2011) or with the use of system identification techniques (Huang & Kim 2008; Bagheri, Brandt & Henningson 2009; Semeraro et al. 2011; Illingworth, Morgans & Rowley 2012; Brunton, Proctor & Kutz 2016).…”

Section: Introductionmentioning

confidence: 99%

Cluster-based feedback control of turbulent post-stall separated flows

et al. 2019

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We propose a novel model-free self-learning cluster-based control strategy for general nonlinear feedback flow control technique, benchmarked for high-fidelity simulations of post-stall separated flows over an airfoil. The present approach partitions the flow trajectories (force measurements) into clusters, which correspond to characteristic coarsegrained phases in a low-dimensional feature space. A feedback control law is then sought for each cluster state through iterative evaluation and downhill simplex search to minimize power consumption in flight. Unsupervised clustering of the flow trajectories for in-situ learning and optimization of coarse-grained control laws are implemented in an automated manner as key enablers. Re-routing the flow trajectories, the optimized control laws shift the cluster populations to the aerodynamically favorable states. Utilizing limited number of sensor measurements for both clustering and optimization, these feedback laws were determined in only O(10) iterations. The objective of the present work is not necessarily to suppress flow separation but to minimize the desired cost function to achieve enhanced aerodynamic performance. The present control approach is applied to the control of two and three-dimensional separated flows over a NACA 0012 airfoil with large-eddy simulations at an angle of attack of 9 • , Reynolds number Re = 23, 000 and free-stream Mach number M ∞ = 0.3. The optimized control laws effectively minimize the flight power consumption enabling the flows to reach a low-drag state. The present work aims to address the challenges associated with adaptive feedback control design for turbulent separated flows at moderate Reynolds number.

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“…Physics-based nonlinear ROMs usually rely on Galerkin projection of the full Navier-Stokes equations (as opposed to the linearized version around the fixed point), most often using POD modes. However, such models typically require fine tuning and calibration because of instabilities stemming from the structure of the expansion and its truncation (Noack et al 2011). While designing nonlinear ROMs is a challenge of fundamental interest by itself, it is however not immediately clear that their enhanced fidelity can be made profitable for closed-loop control applications.…”

Section: Introductionmentioning

confidence: 99%

Linear iterative method for closed-loop control of quasiperiodic flows

et al. 2019

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This work proposes a feedback-loop strategy to suppress intrinsic oscillations of resonating flows in the fully nonlinear regime. The frequency response of the flow is obtained from the resolvent operator about the mean flow, extending the framework initially introduced by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to study receptivity mechanisms in turbulent flows. Using this linear time-invariant model of the nonlinear flow, modern control methods such as structured ${\mathcal{H}}_{\infty }$-synthesis can be used to design a controller. The approach is successful in damping self-sustained oscillations associated with specific eigenmodes of the mean-flow spectrum. Despite excellent performance, the linear controller is however unable to completely suppress flow oscillations, and the controlled flow is effectively attracted towards a new dynamical equilibrium. This new attractor is characterized by a different mean flow, which can in turn be used to design a second controller. The method can then be iterated on subsequent mean flows, until the coupled system eventually converges to the base flow. An intuitive parallel can be drawn with Newton’s iteration: at each step, a linearized model of the flow response to a perturbation of the input is sought, and a new linear controller is designed, aiming at further reducing the fluctuations. The method is illustrated on the well-known case of two-dimensional incompressible open-cavity flow at Reynolds number $Re=7500$, where the fully developed flow is initially quasiperiodic (2-torus state). The base flow is reached after five iterations. The present work demonstrates that nonlinear control problems may be solved without resorting to nonlinear reduced-order models. It also shows that physically relevant linear models can be systematically derived for nonlinear flows, without resorting to black-box identification from input–output data; the key ingredient being frequency-domain models based on the linearized Navier–Stokes equations about the mean flow. Applicability to amplifier flows and turbulent dynamics has, however, yet to be investigated.

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Galerkin Method for Nonlinear Dynamics

Cited by 29 publications

References 49 publications

Thel₁-based sparsification of energy interactions in unsteady lid-driven cavity flow

Thel₁-based sparsification of energy interactions in unsteady lid-driven cavity flow

Cluster-based feedback control of turbulent post-stall separated flows

Linear iterative method for closed-loop control of quasiperiodic flows

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Galerkin Method for Nonlinear Dynamics

Cited by 29 publications

References 49 publications

Thel1-based sparsification of energy interactions in unsteady lid-driven cavity flow

Thel1-based sparsification of energy interactions in unsteady lid-driven cavity flow

Cluster-based feedback control of turbulent post-stall separated flows

Linear iterative method for closed-loop control of quasiperiodic flows

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Thel₁-based sparsification of energy interactions in unsteady lid-driven cavity flow

Thel₁-based sparsification of energy interactions in unsteady lid-driven cavity flow