Maziar S. Hemati

Dynamic mode decomposition for large and streaming datasets

We formulate a low-storage method for performing dynamic mode decomposition that can be updated inexpensively as new data become available; this formulation allows dynamical information to be extracted from large datasets and data streams. We present two algorithms: the first is mathematically equivalent to a standard “batch-processed” formulation; the second introduces a compression step that maintains computational efficiency, while enhancing the ability to isolate pertinent dynamical information from noisy measurements. Both algorithms reliably capture dominant fluid dynamic behaviors, as demonstrated on cylinder wake data collected from both direct numerical simulations and particle image velocimetry experiments.

De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets

The dynamic mode decomposition (DMD)—a popular method for performing data-driven Koopman spectral analysis—has gained increased popularity for extracting dynamically meaningful spatiotemporal descriptions of fluid flows from snapshot measurements. Often times, DMD descriptions can be used for predictive purposes as well, which enables informed decision-making based on DMD model forecasts. Despite its widespread use and utility, DMD can fail to yield accurate dynamical descriptions when the measured snapshot data are imprecise due to, e.g., sensor noise. Here, we express DMD as a two-stage algorithm in order to isolate a source of systematic error. We show that DMD’s first stage, a subspace projection step, systematically introduces bias errors by processing snapshots asymmetrically. To remove this systematic error, we propose utilizing an augmented snapshot matrix in a subspace projection step, as in problems of total least-squares, in order to account for the error present in all snapshots. The resulting unbiased and noise-aware total DMD (TDMD) formulation reduces to standard DMD in the absence of snapshot errors, while the two-stage perspective generalizes the de-biasing framework to other related methods as well. TDMD’s performance is demonstrated in numerical and experimental fluids examples. In particular, in the analysis of time-resolved particle image velocimetry data for a separated flow, TDMD outperforms standard DMD by providing dynamical interpretations that are consistent with alternative analysis techniques. Further, TDMD extracts modes that reveal detailed spatial structures missed by standard DMD.

Improving separation control with noise-robust variants of dynamic mode decomposition

Flow separation can lead to degraded performance in many engineered systems, which has led to sustained interest in developing strategies for suppressing and controlling flow separation. Separation control strategies based on open-loop forcing via synthetic jets have demonstrated a relative degree of success in various studies; however, many of these studies have relied upon trial-and-error “tuning” of a synthetic jet’s operating parameters for satisfactory performance with respect to a particular flow configuration. Subsequent work has focused on improving the general understanding of fluid flow separation from a dynamical systems perspective, with the aim of isolating key mechanisms that can be exploited for more systematic controller designs. Numerical studies have shown that dynamically dominant flow characteristics, identified by the dynamic mode decomposition (DMD), can be used to guide the design of open-loop separation control strategies. While these approaches have proven valuable for dynamical analyses in numerics, standard formulations of DMD have recently been shown to possess systematic errors that can lead to misleading results when the data are corrupted by some degree of measurement noise (e.g., sensor noise in experimental studies). Here, we make use of DMD to synthesize time-resolved particle image velocimetry (TR-PIV) data from a canonical separation experiment in an effort to inform the design of open-loop separation control strategies; to this end, we make use of a noise-aware version of DMD—introduced in Hemati et al. (2015)—to assess the impact of measurement noise on the conclusions drawn for informing open-loop controller design. Additionally, we extend the noise-aware framework to formulate a noise-robust version of the streaming DMD algorithm presented in Hemati et al. (2014). Dynamic characterizations afforded by DMD-based techniques are then used to inform open-loop separation control strategies that are tested in experiments. We find that open-loop forcing at a frequency associated with the dominant DMD mode reduces the mean height of the separation bubble, suggesting that DMD-based techniques may provide a systematic means of designing open-loop control strategies aimed at suppressing flow separation.

Wake sensing for aircraft formation flight

It is well established that flying aircraft in formation can lead to improved aerodynamic efficiency. However, successfully doing so is predicated on having knowledge of the lead aircraft’s wake position. Here, a wake-sensing strategy for estimating the wake position and strength in a two-aircraft formation is explored in a simplified proof-of-concept setting. The wake estimator synthesizes wing-distributed pressure measurements, taken on the trailing aircraft, by making use of an augmented lifting-line model in conjunction with both Kalman-type and particle filters. Simple aerodynamic models are introduced in constructing the filter to enable fundamental wake-sensing challenges to be identified and reconciled. The various estimation algorithms are tested in a vortex lattice simulation environment, thus allowing the effects of modeling error to be analyzed. It is found that biases in the position estimates no longer arise if a particle filter is used in place of the Kalman-type filters. Filter divergence ...

Parameter-Varying Aerodynamics Models for Aggressive Pitching-Response Prediction

Current low-dimensional aerodynamic-modeling capabilities are greatly challenged in the face of aggressive flight maneuvers, such as rapid pitching motions that lead to aerodynamic stall. Nonlinearities associated with leading-edge vortex development and flow separation push existing real-time-capable aerodynamics models beyond their predictive limits, which puts reliable real-time flight simulation and control out of reach. In the present development, a push toward realizing real-time-capable models with enhanced predictive performance for flight operations has been made by considering the simpler problem of modeling an aggressively pitching airfoil in a low-dimensional manner. A parameter-varying model, composed of three coupled quasi-linear sub-models, is proposed to approximate the lift, drag, and pitching-moment response of an airfoil to arbitrarily prescribed aggressive ramp–hold pitching kinematics. An output-error-minimization strategy is used to identify the low-dimensional quasi-linear parameter...

A flight simulator for agile fighter aircraft and nonlinear aerodynamics

Physics-based reduced-order models have been developed that can accurately and efficiently simulate key aspects of aircraft flight operations including aerodynamics, aeroelasticity, and control-surface dynamics at subsonic, transonic, and supersonic flight speeds and rapidly changing, nonlinear post-stall conditions. The modeling technology enables flight simulations and virtual flight testing of agile (highly maneuverable) fighter aircraft. Order reduction is effected by transforming from physical space to modal space using the method of proper orthogonal decomposition. Modal models constructed with a relatively small set of data from high-fidelity, computationally intensive CFD simulations (where flow properties are computed in physical space) are capable of accurately predicting the flight dynamics for a wide range of aggressive aircraft maneuvers in simulations that are significantly faster than real time. Model accuracy is demonstrated through comparisons with data from CFD simulations of an open-source fighter aircraft with and without wing stores (modeled after an F-16) and an F-16 aircraft with articulating control surfaces. Model evaluations include both rigid and flexible versions of the aircraft.

Dynamic mode shaping for fluid flow control: New strategies for transient growth suppression

Sub-critical transition to turbulence is often attributed to transient energy growth that arises from non-normality of the linearized Navier-Stokes operator. Here, we introduce a new dynamic mode shaping perspective for transient growth suppression that focuses on using feedback control to shape the spectral properties of the linearized flow. Specifically, we propose a dynamic mode matching strategy that can be used to reduce non-normality and transient growth. We also propose a dynamic mode orthogonalization strategy that can be used to eliminate non-normality and fully suppress transient growth. Further, we formulate dynamic mode shaping strategies that aim to handle some of the practical challenges inherent to fluid flow control applications, namely high-dimensionality, nonlinearity, and uncertainty. Dynamic mode shaping methods are demonstrated on a number of simple illustrative examples that show the utility of this new perspective for transient growth suppression. The methods and perspectives introduced here will serve as a foundation for realizing effective flow control in the future.

Identifying dynamic modes of separated flow subject to ZNMF-based control from surface pressure measurements

Flow Subject to ZNMF-based Control from Surface Pressure Measurements Eric Deem∗, Louis Cattafesta† Mechanical Engineering, Florida State University, Tallahassee, FL, 32310 Hao Zhang‡, Clarence Rowley § Mechanical and Aerospace Engineering, Princeton University , Princeton, NJ, 08544 Maziar Hemati¶ Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN, 55455 Francois Cadieux‖, and Rajat Mittal ∗∗ Mechanical Engineering, Johns Hopkins University , Baltimore, MD, 21218

Learning wake regimes from snapshot data

Fluid wakes are often categorized by visual inspection according to the number and grouping of vortices shed per cycle (e.g., 2S, 2P, P+S). While such categorizations have proven useful for describing and comparing wakes, the criterion excludes features that are essential to a wake’s evolution (i.e., the relative positions and strengths of the shed vortices). For example, not all 2P wakes exhibit the same dynamics; thus, the evolution of wake patterns among 2P wakes can be markedly distinct. Here, we explore the notion of labeling wakes according their dynamics based on empirical snapshot data. Snapshots of the velocity field are reduced to a representative feature vector, which is then processed using machine learning techniques tailored to the task of wake regime learning. The wake regime learning framework is evaluated on an idealized 2P wake model, which can be configured to simulate different (known) dynamical regimes. A simple version of the wake regime learning framework successfully discriminates between dynamically distinct 2P wakes. The results presented here suggest that the wake regime learning perspective may facilitate the development of a new dynamics-based wake labeling convention in the future.

Unsteady aerodynamic response modeling: A parameter-varying approach

Current low-dimensional aerodynamic modeling capabilities are greatly challenged in the face of aggressive flight maneuvers, such as rapid pitching motions that lead to aerodynamic stall. Nonlinearities associated with leading-edge vortex development and flow separation push existing real-time-capable aerodynamics models beyond their predictive limits. The inability to accurately predict the aerodynamic response of an aircraft to sharp maneuvers makes flight simulation for pilot training unrealistic and, thus, ineffective at adequately preparing pilots to safely handle compromising flight scenarios. Inaccurate low-dimensional models also put practical approaches for aerodynamic optimization and control out of reach. In the present development, we make a push toward realizing real-time-capable models with enhanced predictive performance for flight operations by considering the simpler problem of modeling an aggressively pitching airfoil in a low-dimensional manner. We propose a parameter-varying model, composed of three coupled quasi-linear sub-models, to approximate the response of an airfoil to arbitrarily prescribed aggressive ramp-hold pitching kinematics. An output error minimization strategy is used to identify the lowdimensional quasi-linear parameter-varying sub-models from input-output data gathered from low-Reynolds number (Re = 100) direct numerical fluid dynamics simulations. The resulting models have noteworthy predictive capabilities for arbitrary ramp-hold pitching maneuvers spanning a broad range of operating points, thus making the models especially useful for aerodynamic optimization and real-time control and simulation.