Session EO: Turbulence: Control II

Model reduction and feedback control of the Blasius boundary-layer

Low-dimensional models of the transitional flat-plate boundary layer are considered for the design of feedback control. In particular, the recently introduced technique for approximating balanced truncation for very large systems using the method of snapshots is considered. This projection basis is computed from a composite snapshot set consisting of the impulse response of both the direct and adjoint linearized Blasius flow. The reduced-order model preserves flow states that are both controllable and observable and thus captures input-output characteristics of the flow, making it a natural projection basis for flow control. The error of the flow approximation obtained from balanced truncation is computed in terms of transfer function norms and compared to other commonly used methods for model reduction, such as Proper Orthogonal Decomposition (POD). The reduced-order model is then used to design a feedback control strategy such that the perturbation energy is minimized.   

Control and system identification of transition induced by free-stream turbulence

Control has been applied to disturbances in a laminar boundary-layer under a turbulent free-stream in a wind tunnel. A feed-back control system comprising of two units, each with four sensors (wall wires) and four actuators (intermittent suction through narrow holes), has been used. The suction through the holes is managed by fast solenoid valves and is turned on (with a delay) when a moment of low velocity is detected by the sensor straight upstream of the actuator. In each measurement sequence, the suction rate is constant, but it can be varied between runs. It is shown that the control system manages to inhibit the disturbance growth for a considerable distance downstream of the actuators. The disturbance structure with and without control The non-dimensional suction rate is one third of the one necessary if uniform suction is used to reach the same control effect. Linear system identification is used to evaluate which modifications of the control system that would be most beneficial: developed control logics, more advanced actuator technology or improved sensors.   

Optimal perturbation in a channel flow: adjoint-based and riccati-based control comparison

Active open-loop and closed-loop control of optimal instabilities amplified in a channel flow is investigated. The control is carried out at both the upper and the lower wall by blowing and suction. The state system considered is parabolic in time. We used both adjoint-based and Riccati-based control theories. In the adjoint-based method, we alternatively solve the state (forward time marching) and the adjoint (backward time marching) systems until convergence towards the optimal control. In the feedback control method, the control is the solution of a differential Riccati equation which marches in time. We show that the adjoint-based (open-loop) and the Riccati-based (closed-loop) control results are very similar. An analysis of the control robustness has been performed to take into account uncertainties coming from the state disturbances (such as surface irregularities, walls vibrations, acoustic vibrations, actuators inputs inaccuracies, etc ...).   

Robust $H_2$ Feedback Control of a Prototypical Fluid Flow System

Our ability to design efficient technologies involving complex fluid flows strongly depends on our ability to actively control these flows in real-time in an autonomous fashion, according to our needs. In this talk, we will describe the challenges one face in feedback control of multi-scale fluid flows, with wide range of spatial and temporal scales that are tightly coupled, using a spatially distributed network of sensors, actuators and controllers. A prototype system governed by partial differential equations with strong nonlinearity and solutions exhibiting saturating, periodic and soliton behavior will be considered in the talk. A $H_2$ optimal controller that works effectively in the presence of a class of disturbance uncertainty is designed and tested in the direct numerical simulations (DNS). Collocation spectral methods based on Fourier modes are used for the numerical computations. The solution of the resulting nonlinear matrix Riccati equations are obtained using the Eigenvalue and Schur decompositions, with and without balancing. $H_2$ norm is computed numerically using its time domain characterization in terms of gramians and avoiding costly improper integrals. $H_{\infty}$ norm is computed using an iteration procedure involving Hamiltonian matrix. The open and closed loop DNS results indicate various surprising things and nicely illustrate the intricacies involved in dealing with the full Navier-Stokes equations.   

Retrograde Estimation and Forecasting of Chaotic Systems. Part 1: Theoretical Foundations

Chaotic systems are characterized by long-term unpredictability. Previous methods designed to estimate and forecast such systems, such as extended Kalman filtering [a matrix-based approach] and 4Dvar [aka Moving-Horizon Estimation (MHE), a vector-based approach], are essentially based on the assumption that Gaussian uncertainties in the initial state estimate and Gaussian disturbances to the state and measurements lead to uncertainty on the state estimate at later times that is well described by a Gaussian model. This assumption is not valid in chaotic nonlinear systems. A new method is thus proposed which revisits past measurements in order to reconcile them with more recent measurements of the system. This new approach, which we refer to as Model Predictive Estimation (MPE), is a straightforward extension of 4Dvar/MHE, an operational algorithm recently adopted by the weather forecasting community. Our new method leverages backwards-in-time (aka,``retrograde") time marches of the system, a receding-horizon optimization framework, and adaptive adjustment of the optimization horizon based on the quality of the estimate at each iteration.   

Retrograde Estimation and Forecasting of Chaotic Systems. Part 2: Application to Chaotic and Multiscale Model Problems

Application of the new algorithm for estimation and forecasting of chaotic systems, discussed in Part 1 of this presentation, to Lorenz, Kuramoto-Sivashinsky, and Navier-Stokes systems is discussed. For the Lorenz case, an interactive GUI has been developed to illustrate the various features and performance of the new strategy, which categorically outperforms existing strategies for the forecasting of this model. For the Kuramoto-Sivashinsky case, significant regularization is required to enable backwards-in-time marches of the state and forward-in-time marches of the adjoint; nonetheless, the retrograde algorithm is observed in numerical experiments to outperform existing methods. For the Navier-Stokes case, the regularization issue is less severe, and becomes even less of an issue as the Reynolds number is increased towards the Euler limit appropriate for many large-scale turbulent flows of interest (hurricanes, contaminant plumes, etc.).   

Retrograde Estimation and Forecasting of Chaotic Systems. Part 3: Adaptive Observation and Covariance Update Strategies

The variational (i.e., adjoint-based) strategies discussed in Parts 1 and 2 of this presentation do not attempt to model the variance or covariance of the estimation error, as done in the Kalman and extended Kalman filters. In the low-dimensional setting, when the system evolves slowly as compared with the computational model, it is straightforward to extend the idea of covariance modeling to the retrograde, or backwards-in-time, setting. The key idea of this extension is to ``undo'' the covariance update based on the linearization of the system around the previous estimator trajectory, then ``redo'' the covariance update based on the linearization of the system around the updated estimator trajectory. In the high-dimensional setting [as encountered when using high-fidelity computational models of fluid systems], low-rank approximations of the covariance may be used, as with Kalman filtering, in this retrograde framework. A consistent model of the spatial and temporal dynamics of the error covariance (that is, of the system uncertainty) may then be used to drive an adaptive observation algorithm in order to coordinate the motion of a swarm of mobile sensors, such as UAVs equipped with GPS and MEMS-based detectors of contaminants, in order to maximize the accuracy of the forecast.   

Efficient Derivative-Free Optimization

When optimizing the parameters affecting the evolution of a chaotic system, a peculiar challenge arises. The infinite-time average of the statistic of interest is only approximated by any finite-time simulation, with the truncation errors effectively decorrelated from one simulation to the next. Thus, the optimization surface is nonsmooth, and gradient-based optimization algorithms are ill suited. One of the most efficient derivative-free optimization algorithm available for such problems is the Surrogate Management Framework (SMF), which fits an interpolating function to the available data to identify regions of interest. The SMF is based on an underlying grid structure, with all function evaluations performed on this grid. Once discrete convergence is obtained, the grid is refined and the process repeated. In all previous SMF codes, Cartesian grids have been used. However, Cartesian grids are not nearly as uniform at packing or covering parameter space as various alternatives available from n-dimensional sphere packing theory. In the present talk, we show that, by leveraging such packings, the number of function evaluations required for convergence of the SMF algorithm is substantially reduced.   

Stability of a channel flow subject to blowing and suction in the form of a traveling wave

A recent study by Min et al. (JFM, 558, 2006) showed that drag in a channel could be sustained below that of a laminar flow when the flow was subjected to surface blowing and suction in the form of an upstream traveling wave. It was shown that a positive Reynolds shear stress, normally negative for a positive mean shear, was induced by the traveling wave, thus resulting in the mean drag reduction. In the present study, stability characteristics of a laminar channel flow subject to the same kind of blowing and suction have been investigated by using Floquet analysis and by examining maximum transient growth of initial energy. Unlike the behavior of the mean flow, linear stability characteristics were hardly affected by upstream-traveling waves, implying that the drag reduction was a direct consequence of traveling- wave induced flow fields. A more interesting behavior was found for cases with a downstream-traveling wave. The growth rate of the most unstable 2-D and 3-D disturbances significantly increased when the phase speed of the wave was around 40\% of the centerline velocity, while stabilization of 3-D disturbance was observed when the phase speed exceeded the centerline velocity. The destabilization appears to be caused by an otherwise stable Tollmien-Schlichting mode excited by the traveling wave. Detailed results with insights into a physical mechanism will be presented at the meeting.