Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations

Wang, Qiqi; Hu, Rui; Blonigan, Patrick

doi:10.1016/j.jcp.2014.03.002

Cited by 144 publications

(215 citation statements)

References 32 publications

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“…The reason for this is that we would need a rather long optimization horizon T to find a robust parameter combination that is independent of specific flow realizations. Unfortunately, the chaotic nature of turbulent flow fields makes long-time optimization using adjoint LES practically infeasible to date (see, e.g., Wang et al, 2014). However, the fact that we have only two parameters renders a parameter sweep computationally feasible.…”

Section: Parameter Sweepmentioning

confidence: 99%

Towards practical dynamic induction control of wind farms: analysis of optimally controlled wind-farm boundary layers and sinusoidal induction control of first-row turbines

Munters

Meyers

2018

Wind Energ. Sci.

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Abstract. Wake interactions between wind turbines in wind farms lead to reduced energy extraction in downstream rows. In recent work, optimization and large-eddy simulation were combined with the optimal dynamic induction control of wind farms to study the mitigation of these effects, showing potential power gains of up to 20 % (Munters and Meyers, 2017, Phil. Trans. R. Soc. A, 375, 20160100, https://doi.org/10.1098/rsta.2016.010). However, the computational cost associated with these optimal control simulations impedes the practical implementation of this approach. Furthermore, the resulting control signals optimally react to the specific instantaneous turbulent flow realizations in the simulations so that they cannot be simply used in general. The current work focuses on the detailed analysis of the optimization results of Munters and Meyers, with the aim to identify simplified control strategies that mimic the optimal control results and can be used in practice. The analysis shows that wind-farm controls are optimized in a parabolic manner with little upstream propagation of information. Moreover, turbines can be classified into first-row, intermediate-row, and last-row turbines based on their optimal control dynamics. At the moment, the control mechanisms for intermediate-row turbines remain unclear, but for first-row turbines we find that the optimal controls increase wake mixing through the periodic shedding of vortex rings. This behavior can be mimicked with a simple sinusoidal thrust control strategy for first-row turbines, resulting in robust power gains for turbines in the entrance region of the farm.

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Section: Parameter Sweepmentioning

confidence: 99%

Towards practical dynamic induction control of wind farms: analysis of optimally controlled wind-farm boundary layers and sinusoidal induction control of first-row turbines

Munters

Meyers

2018

Wind Energ. Sci.

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“…On the one hand, the discrete system being less chaotic than the actual flow makes sensitivity analysis more manageable. In particular, the cost of the two main adjoint-based sensitivity methods in the literature, namely the Least Squares Shadowing (LSS) 26 and the Ensemble Adjoint (EA) methods, increases with the number and/or magnitude of positive Lyapunov exponents in the discrete system. The computational cost of shadowing-based approaches, including the original LSS method, the Multiple Shooting Shadowing (MSS) method 2 and the Non-Intrusive LSS (NILSS) method, 3,17 is proportional to the number of positive Lyapunov exponents.…”

Section: Application To Chaotic Adjointsmentioning

confidence: 99%

“…Second, conventional sensitivity analysis methods break down for chaotic systems; 14 which compromises critical tasks in engineering such as flow control, design optimization, error estimation, data assimilation, and uncertainty quantification. While a number of sensitivity analysis methods have been proposed for chaotic systems, including Fokker-Planck methods, 23 FluctuationDissipation methods, 15 the Ensemble Adjoint (EA) method, 14 the Least Squares Shadowing (LSS) method 26 and the Non-Intrusive Least Squares Shadowing (NILSS) method, 17 they all come at a high computational cost. This is ultimately related to the positive portion of the Lyapunov spectrum of the system, and the cost of each method is sensitive to different aspects of it.…”

Section: Introductionmentioning

confidence: 99%

Lyapunov spectrum of scale-resolving turbulent simulations. Application to chaotic adjoints

Fernández

Wang

2017

23rd AIAA Computational Fluid Dynamics Conference

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We present an investigation of the Lyapunov spectrum of the chaotic, separated flow around the NACA 0012 airfoil at Reynolds number 2,400. The impact of the numerical discretization on the spectrum is investigated through time and space refinement studies. Numerical results show that the time discretization has a small impact on the Lyapunov exponents, whereas the spatial discretization can dramatically change them. In particular, the asymptotic Lyapunov spectrum for this wall-bounded flow is achieved with global CFL numbers as large as O(10 1 −10 2 ), whereas the system continues to become more and more chaotic as the mesh is refined even for meshes that are much finer than the best practice for this type of flows. Based on these results, we conclude the paper with a discussion on the feasibility of adjoint-based sensitivity analysis for chaotic flows.

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“…We applied a basic Euler method at each iteration step so it may be possible to reduce the number of iterations by using a higher order approach. Additionally the direction that we push our parameterisation vectors is not optimal, therefore it may also be possible to reduce the number of iterations by more accurately estimating this direction using a modified adjoint method (Wang et al, 2014), or by starting the optimisation with parameters defined by a fit to the high resolution statistics as in Achatz and Branstator (1999) and Frederiksen and Kepert (2006), rather than the low resolution climatology.…”

Section: F Vmentioning

confidence: 99%

“…Lea et al, 2000;Eyink et al, 2004) and approximations are required. Some attempts to solve this problem in a slightly different context include the methods of Abramov and Majda (2009) applied to climate response, who use the full non-linear model for the short time gradient estimate and a Gaussian model approximation for longer times, and Wang et al (2014) who uses a modified adjoint algorithm to stabilise the gradient estimation algorithm. Fortunately an estimate of ∂G/∂p does not need to be particularly accurate for the purposes of optimisation.…”

Section: Introductionmentioning

confidence: 99%

Optimisation of an idealised ocean model, stochastic parameterisation of sub-grid eddies

Cooper

Zanna

2015

Ocean Modelling

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An optimisation scheme is developed to accurately represent the sub-grid scale forcing of a high dimensional chaotic ocean system. Using a simple parameterisation scheme, the velocity components of a 30km resolution shallow water ocean model are optimised to have the same climatological mean and variance as that of a less viscous 7.5km resolution model. The 5 day lagcovariance is also optimised, leading to a more accurate estimate of the high resolution response to forcing using the low resolution model.The system considered is an idealised barotropic double gyre that is chaotic at both resolutions. Using the optimisation scheme, we find and apply the constant in time, but spatially varying, forcing term that is equal to the time integrated forcing of the submesoscale eddies. A linear stochastic term, independent of the large-scale flow, with no spatial correlation but a spatially varying amplitude and time scale is used to represent the transient eddies. The climatological mean, variance and 5 day lag-covariance of the velocity from a single high resolution integration is used to provide an optimisation target. No other high resolution statistics are required. Additional programming effort, for example to build a tangent linear or adjoint model, is not required either.The focus of this paper is on the optimisation scheme and the accuracy of the optimised flow. The method can be applied in future investigations into the physical processes that govern barotropic turbulence and it can perhaps be applied to help understand and correct biases in the mean and variance of a more realistic coarse or eddy-permitting ocean model. The method is complementary to current parameterisations and can be applied at the same time without modification.

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Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations

Cited by 144 publications

References 32 publications

Towards practical dynamic induction control of wind farms: analysis of optimally controlled wind-farm boundary layers and sinusoidal induction control of first-row turbines

Towards practical dynamic induction control of wind farms: analysis of optimally controlled wind-farm boundary layers and sinusoidal induction control of first-row turbines

Lyapunov spectrum of scale-resolving turbulent simulations. Application to chaotic adjoints

Optimisation of an idealised ocean model, stochastic parameterisation of sub-grid eddies

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