Clarence W. Rowley

MODEL REDUCTION FOR FLUIDS, USING BALANCED PROPER ORTHOGONAL DECOMPOSITION

Many of the tools of dynamical systems and control theory have gone largely unused for fluids, because the governing equations are so dynamically complex, both high-dimensional and nonlinear. Model reduction involves finding low-dimensional models that approximate the full high-dimensional dynamics. This paper compares three different methods of model reduction: proper orthogonal decomposition (POD), balanced truncation, and a method called balanced POD. Balanced truncation produces better reduced-order models than POD, but is not computationally tractable for very large systems. Balanced POD is a tractable method for computing approximate balanced truncations, that has computational cost similar to that of POD. The method presented here is a variation of existing methods using empirical Gramians, and the main contributions of the present paper are a version of the method of snapshots that allows one to compute balancing transformations directly, without separate reduction of the Gramians; and an output p...

On self-sustained oscillations in two-dimensional compressible flow over rectangular cavities

Numerical simulations are used to investigate the resonant instabilities in two-dimensional flow past an open cavity. The compressible Navier–Stokes equations are solved directly (no turbulence model) for cavities with laminar boundary layers upstream. The computational domain is large enough to directly resolve a portion of the radiated acoustic field, which is shown to be in good visual agreement with schlieren photographs from experiments at several different Mach numbers. The results show a transition from a shear-layer mode, primarily for shorter cavities and lower Mach numbers, to a wake mode for longer cavities and higher Mach numbers. The shear-layer mode is characterized well by the acoustic feedback process described by Rossiter (1964), and disturbances in the shear layer compare well with predictions based on linear stability analysis of the Kelvin–Helmholtz mode. The wake mode is characterized instead by a large-scale vortex shedding with Strouhal number independent of Mach number. The wake mode oscillation is similar in many ways to that reported by Gharib & Roshko (1987) for incompressible flow with a laminar upstream boundary layer. Transition to wake mode occurs as the length and/or depth of the cavity becomes large compared to the upstream boundary-layer thickness, or as the Mach and/or Reynolds numbers are raised. Under these conditions, it is shown that the Kelvin–Helmholtz instability grows to sufficient strength that a strong recirculating flow is induced in the cavity. The resulting mean flow is similar to wake profiles that are absolutely unstable, and absolute instability may provide an explanation of the hydrodynamic feedback mechanism that leads to wake mode. Predictive criteria for the onset of shear-layer oscillations (from steady flow) and for the transition to wake mode are developed based on linear theory for amplification rates in the shear layer, and a simple model for the acoustic efficiency of edge scattering.

Maximum Power Point Tracking for Photovoltaic Optimization Using Ripple-Based Extremum Seeking Control

This study develops a maximum power point tracking algorithm that optimizes solar array performance and adapts to rapidly varying irradiance conditions. In particular, a novel extremum seeking (ES) controller that utilizes the natural inverter ripple is designed and tested on a simulated solar array with a grid-tied inverter. The new algorithm is benchmarked against the perturb and observe (PO) method using high-variance irradiance data gathered on a rooftop array experiment in Princeton, NJ. The ES controller achieves efficiencies exceeding 99% with transient rise-time to the maximum power point of less than 0.1 s. It is shown that voltage control is more stable than current control and allows for accurate tracking of faster irradiance transients. The limitations of current control are demonstrated in an example. Finally, the effect of capacitor size on the performance of ripple-based ES control is investigated.

Detection of Lagrangian coherent structures in three-dimensional turbulence

We use direct Lyapunov exponents (DLE) to identify Lagrangian coherent structures in two different three-dimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow. These results are compared with commonly used Eulerian criteria for coherent vortices. We find that, despite additional computational cost, the DLE method has several advantages over Eulerian methods, including greater detail and the ability to define structure boundaries without relying on a preselected threshold. As a further advantage, the DLE method requires no velocity derivatives, which are often too noisy to be useful in the study of a turbulent flow. We study the evolution of a single hairpin vortex into a packet of similar structures, and show that the birth of a secondary vortex corresponds to a loss of hyperbolicity of the Lagrangian coherent structures.

Locomotion of Articulated Bodies in a Perfect Fluid

AbstractThis paper is concerned with modeling the dynamics of N articulated solid bodies submerged in an ideal fluid. The model is used to analyze the locomotion of aquatic animals due to the coupling between their shape changes and the fluid dynamics in their environment. The equations of motion are obtained by making use of a two-stage reduction process which leads to significant mathematical and computational simplifications. The first reduction exploits particle relabeling symmetry: that is, the symmetry associated with the conservation of circulation for ideal, incompressible fluids. As a result, the equations of motion for the submerged solid bodies can be formulated without explicitly incorporating the fluid variables. This reduction by the fluid variables is a key difference with earlier methods, and it is appropriate since one is mainly interested in the location of the bodies, not the fluid particles. The second reduction is associated with the invariance of the dynamics under superimposed rigid motions. This invariance corresponds to the conservation of total momentum of the solid-fluid system. Due to this symmetry, the net locomotion of the solid system is realized as the sum of geometric and dynamic phases over the shape space consisting of allowable relative motions, or deformations, of the solids. In particular, reconstruction equations that govern the net locomotion at zero momentum, that is, the geometric phases, are obtained. As an illustrative example, a planar three-link mechanism is shown to propel and steer itself at zero momentum by periodically changing its shape. Two solutions are presented: one corresponds to a hydrodynamically decoupled mechanism and one is based on accurately computing the added inertias using a boundary element method. The hydrodynamically decoupled model produces smaller net motion than the more accurate model, indicating that it is important to consider the hydrodynamic interaction of the links.

Reduced-order models for control of fluids using the eigensystem realization algorithm

As sensors and flow control actuators become smaller, cheaper, and more pervasive, the use of feedback control to manipulate the details of fluid flows becomes increasingly attractive. One of the challenges is to develop mathematical models that describe the fluid physics relevant to the task at hand, while neglecting irrelevant details of the flow in order to remain computationally tractable. A number of techniques are presently used to develop such reduced-order models, such as proper orthogonal decomposition (POD), and approximate snapshot-based balanced truncation, also known as balanced POD. Each method has its strengths and weaknesses: for instance, POD models can behave unpredictably and perform poorly, but they can be computed directly from experimental data; approximate balanced truncation often produces vastly superior models to POD, but requires data from adjoint simulations, and thus cannot be applied to experimental data. In this article, we show that using the Eigensystem Realization Algorithm (ERA) (Juang and Pappa, J Guid Control Dyn 8(5):620–627, 1985) one can theoretically obtain exactly the same reduced-order models as by balanced POD. Moreover, the models can be obtained directly from experimental data, without the use of adjoint information. The algorithm can also substantially improve computational efficiency when forming reduced-order models from simulation data. If adjoint information is available, then balanced POD has some advantages over ERA: for instance, it produces modes that are useful for multiple purposes, and the method has been generalized to unstable systems. We also present a modified ERA procedure that produces modes without adjoint information, but for this procedure, the resulting models are not balanced, and do not perform as well in examples. We present a detailed comparison of the methods, and illustrate them on an example of the flow past an inclined flat plate at a low Reynolds number.

Turbulence, Coherent Structures, Dynamical Systems and Symmetry: Proper orthogonal decomposition

The proper orthogonal decomposition (POD) provides a basis for the modal decomposition of an ensemble of functions, such as data obtained in the course of experiments. Its properties suggest that it is the preferred basis to use in various applications. The most striking of these is optimality : it provides the most efficient way of capturing the dominant components of an infinite-dimensional process with only finitely many, and often surprisingly few, “modes.” The POD was introduced in the context of turbulence by Lumley in. In other disciplines the same procedure goes by the names: Karhunen–Loeve decomposition, principal components analysis, singular systems analysis, and singular value decomposition. The basis functions it yields are variously called: empirical eigenfunctions, empirical basis functions, and empirical orthogonal functions. According to Yaglom (see), the POD was introduced independently by numerous people at different times, including Kosambi, Loeve, Karhunen, Pougachev, and Obukhov. Lorenz, whose name we have already met in another context, suggested its use in weather prediction. The procedure has been used in various disciplines other than fluid mechanics, including random variables, image processing, signal analysis, data compression, process identification and control in chemical engineering, and oceanography. Computational packages based on the POD are now becoming available. In the bulk of these applications, the POD is used to analyse experimental data with a view to extracting dominant features and trends: coherent structures.

Modeling of transitional channel flow using balanced proper orthogonal decomposition

We study reduced-order models of three-dimensional perturbations in linearized channel flow using balanced proper orthogonal decomposition (BPOD). The models are obtained from three-dimensional simulations in physical space as opposed to the traditional single-wavenumber approach, and are therefore better able to capture the effects of localized disturbances or localized actuators. In order to assess the performance of the models, we consider the impulse response and frequency response, and variation of the Reynolds number as a model parameter. We show that the BPOD procedure yields models that capture the transient growth well at a low order, whereas standard POD does not capture the growth unless a considerably larger number of modes is included, and even then can be inaccurate. In the case of a localized actuator, we show that POD modes which are not energetically significant can be very important for capturing the energy growth. In addition, a comparison of the subspaces resulting from the two methods...

Reconstruction equations and the Karhunen—Loéve expansion for systems with symmetry

We present a method for applying the Karhunen–Loeve decomposition to systems with continuous symmetry. The techniques in this paper contribute to the general procedure of removing variables associated with the symmetry of a problem, and related ideas have been used in previous works both to identify coherent structures in solutions of PDEs, and to derive low-order models via Galerkin projection. The main result of this paper is to derive a simple and easily implementable set of reconstruction equationswhich close the system of ODEs produced by Galerkin projection. The geometric interpretation of the method closely parallels techniques used in geometric phases and reconstruction techniques in geometric mechanics. We apply the method to the Kuramoto–Sivashinsky equation and are able to derive accurate models of considerably lower dimension than are possible with the traditional Karhunen–Loeve expansion.

Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators

We present an estimator-based control design procedure for flow control, using reduced-order models of the governing equations linearized about a possibly unstable steady state. The reduced-order models are obtained using an approximate balanced truncation method that retains the most controllable and observable modes of the system. The original method is valid only for stable linear systems, and in this paper, we present an extension to unstable linear systems. The dynamics on the unstable subspace are represented by projecting the original equations onto the global unstable eigenmodes, assumed to be small in number. A snapshot-based algorithm is developed, using approximate balanced truncation, for obtaining a reduced-order model of the dynamics on the stable subspace. The proposed algorithm is used to study feedback control of two-dimensional flow over a flat plate at a low Reynolds number and at large angles of attack, where the natural flow is vortex shedding, though there also exists an unstable steady state. For control design, we derive reduced-order models valid in the neighbourhood of this unstable steady state. The actuation is modelled as a localized body force near the trailing edge of the flat plate, and the sensors are two velocity measurements in the near wake of the plate. A reduced-order Kalman filter is developed based on these models and is shown to accurately reconstruct the flow field from the sensor measurements, and the resulting estimator-based control is shown to stabilize the unstable steady state. For small perturbations of the steady state, the model accurately predicts the response of the full simulation. Furthermore, the resulting controller is even able to suppress the stable periodic vortex shedding, where the nonlinear effects are strong, thus implying a large domain of attraction of the stabilized steady state.