Reduced-order models for closed-loop wake control

Lower Order Description and Reconstruction of Sparse Scanning Lidar Measurements of Wind Turbine Inflow Using Proper Orthogonal Decomposition

Preview measurements of the inflow by turbine-mounted lidar systems can be used to optimise wind turbine performance or alleviate structural loads. However, nacelle-mounted lidars suffer data losses due to unfavourable environmental conditions and laser beam obstruction by the rotating blades. Here, we apply proper orthogonal decomposition (POD) to the simulated line-of-sight wind speed measurements of a turbine-mounted scanning lidar obtained from two large eddy simulations. This work aimed at identifying the dominant POD modes that can be used to subsequently derive a reduced-order representation of the turbine inflow. Secondly, we reconstructed the data points lost due to blade passage by using Gappy-POD. We found that only a few modes are required to capture the dynamics of the wind field parameters commonly used for lidar-assisted wind turbine control, such as the effective wind speed, vertical shear and directional misalignment. By evaluating turbine-relevant metrics in the time and frequency domain, we found that a ten-mode reconstruction could accurately describe most spatio-temporal variations in the inflow. Furthermore, a modal interpretation is presented by direct comparison with these wind field parameters. We found that the Gappy-POD method performs substantially better than spatial interpolation techniques, accurately reconstructing up to even 50% of missing data. A POD-based wind field reconstruction offers a trade-off between wind field reconstruction techniques requiring flow assumptions and more complex physics-based representations, offers dimensional reduction and can overcome the blade passage limitation of nacelle-mounted lidar systems.

A minimization principle for the description of modes associated with finite-time instabilities

We introduce a minimization formulation for the determination of a finite-dimensional, time-dependent, orthonormal basis that captures directions of the phase space associated with transient instabilities. While these instabilities have finite lifetime, they can play a crucial role either by altering the system dynamics through the activation of other instabilities or by creating sudden nonlinear energy transfers that lead to extreme responses. However, their essentially transient character makes their description a particularly challenging task. We develop a minimization framework that focuses on the optimal approximation of the system dynamics in the neighbourhood of the system state. This minimization formulation results in differential equations that evolve a time-dependent basis so that it optimally approximates the most unstable directions. We demonstrate the capability of the method for two families of problems: (i) linear systems, including the advection-diffusion operator in a strongly non-normal regime as well as the Orr-Sommerfeld/Squire operator, and (ii) nonlinear problems, including a low-dimensional system with transient instabilities and the vertical jet in cross-flow. We demonstrate that the time-dependent subspace captures the strongly transient non-normal energy growth (in the short-time regime), while for longer times the modes capture the expected asymptotic behaviour.

Attractor local dimensionality, nonlinear energy transfers and finite-time instabilities in unstable dynamical systems with applications to two-dimensional fluid flows

We examine the geometry of the finite-dimensional attractor associated with fluid flows described by Navier–Stokes equations and relate its nonlinear dimensionality to energy exchanges between dynamical components (modes) of the flow. Specifically, we use a stochastic framework based on the dynamically orthogonal equations to perform efficient order-reduction and describe the stochastic attractor in the reduced-order phase space in terms of the associated probability measure. We introduce the notion of local fractal dimensionality to describe the geometry of the attractor and we establish a connection with the number of positive finite-time Lyapunov exponents. Subsequently, we illustrate in specific fluid flows that the low dimensionality of the stochastic attractor is caused by the synergistic activity of linearly unstable and stable modes as well as the action of the quadratic terms. In particular, we illustrate the connection of the low-dimensionality of the attractor with the circulation of energy: (i) from the mean flow to the unstable modes (due to their linearly unstable character), (ii) from the unstable modes to the stable ones (due to a nonlinear energy transfer mechanism) and (iii) from the stable modes back to the mean (due to the linearly stable character of these modes).