We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space–time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic Tools in Turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time, while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg–Landau equation and a turbulent jet.

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Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

Dynamic mode decomposition (DMD) represents an effective means for capturing the essential features of numerically or experimentally generated flow fields. In order to achieve a desirable tradeoff between the quality of approximation and the number of modes that are used to approximate the given fields, we develop a sparsity-promoting variant of the standard DMD algorithm. Sparsity is induced by regularizing the least-squares deviation between the matrix of snapshots and the linear combination of DMD modes with an additional term that penalizes the l1-norm of the vector of DMD amplitudes. The globally optimal solution of the resulting regularized convex optimization problem is computed using the alternating direction method of multipliers, an algorithm well-suited for large problems. Several examples of flow fields resulting from numerical simulations and physical experiments are used to illustrate the effectiveness of the developed method.

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Sparsity-promoting dynamic mode decomposition

Dynamic mode decomposition (DMD) is an Arnoldi-like method based on the Koopman operator. It analyzes empirical data, typically generated by nonlinear dynamics, and computes eigenvalues and eigenmodes of an approximate linear model. Without explicit knowledge of the dynamical operator, it extracts frequencies, growth rates, and spatial structures for each mode. We show that expansion in DMD modes is unique under certain conditions. When constructing mode-based reduced-order models of partial differential equations, subtracting a mean from the data set is typically necessary to satisfy boundary conditions. Subtracting the mean of the data exactly reduces DMD to the temporal discrete Fourier transform (DFT); this is restrictive and generally undesirable. On the other hand, subtracting an equilibrium point generally preserves the DMD spectrum and modes. Next, we introduce an “optimized” DMD that computes an arbitrary number of dynamical modes from a data set. Compared to DMD, optimized DMD is superior at calculating physically relevant frequencies, and is less numerically sensitive. We test these decomposition methods on data from a two-dimensional cylinder fluid flow at a Reynolds number of 60. Time-varying modes computed from the DMD variants yield low projection errors.

http://cwrowley.princeton.edu/papers/ChenDMD10.pdf

Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses

This article reviews linear instability analysis of flows over or through complex two-dimensional (2D) and 3D geometries. In the three decades since it first appeared in the literature, global instability analysis, based on the solution of the multidimensional eigenvalue and/or initial value problem, is continuously broadening both in scope and in depth. To date it has dealt successfully with a wide range of applications arising in aerospace engineering, physiological flows, food processing, and nuclear-reactor safety. In recent years, nonmodal analysis has complemented the more traditional modal approach and increased knowledge of flow instability physics. Recent highlights delivered by the application of either modal or nonmodal global analysis are briefly discussed. A conscious effort is made to demystify both the tools currently utilized and the jargon employed to describe them, demonstrating the simplicity of the analysis. Hopefully this will provide new impulses for the creation of next-generation algorithms capable of coping with the main open research areas in which step-change progress can be expected by the application of the theory: instability analysis of fully inhomogeneous, 3D flows and control thereof.

Global Linear Instability

Closed-Loop Turbulence Control: Progress and Challenges

A general theoretical formalism is developed to assess how base-flow modifications may alter the stability properties of flows studied in a global approach of linear stability theory. It also comprises a systematic approach to the passive control of globally unstable flows by the use of small control devices. This formalism is based on a sensitivity analysis of any global eigenvalue to base-flow modifications. The base-flow modifications investigated are either arbitrary or specific ones induced by a steady force. This leads to a definition of the so-called sensitivity to base-flow modifications and sensitivity to a steady force. These sensitivity analyses are applied to the unstable global modes responsible for the onset of vortex shedding in the wake of a cylinder for Reynolds numbers in the range 47≤Re≤80. First, it is demonstrated how the sensitivity to arbitrary base-flow modifications may be used to identify regions and properties of the base flow that contribute to the onset of vortex shedding. Secondly, the sensitivity to a steady force determines the regions of the flow where a steady force acting on the base flow stabilizes the unstable global modes. Upon modelling the presence of a control device by a steady force acting on the base flow, these predictions are then extensively compared with the experimental results of Strykowski & Sreenivasan (J. Fluid Mech., vol. 218, 1990, p. 71). A physical interpretation of the suppression of vortex shedding by use of a control cylinder is proposed in the light of the sensitivity analysis.

Sensitivity analysis and passive control of cylinder flow

This review article addresses the dynamics and control of low-frequency unsteadiness, as observed in some aerodynamic applications. It presents a coherent and rigorous linearized approach, which enables both to describe the dynamics of commonly encountered open-flows and to design open-loop and closed-loop control strategies, in view of suppressing or delaying instabilities. The approach is global in the sense that both cross-stream and streamwise directions are discretized in the evolution operator. New light will therefore be shed on the streamwise properties of open-flows. In the case of oscillator flows, the unsteadiness is due to the existence of unstable global modes, i.e., unstable eigenfunctions of the linearized Navier-Stokes operator. The influence of nonlinearities on the dynamics is studied by deriving nonlinear amplitude equations, which accurately describe the dynamics of the flow in the vicinity of the bifurcation threshold. These equations also enable us to analyze the mean flow induced by the nonlinearities as well as the stability properties of this flow. The open-loop control of unsteadiness is then studied by a sensitivity analysis of the eigenvalues with respect to base-flow modifications. With this approach, we manage to a priori identify regions of the flow where a small control cylinder suppresses unsteadiness. Then, a closed-loop control approach was implemented for the case of an unstable open-cavity flow. We have combined model reduction techniques and optimal control theory to stabilize the unstable eigenvalues. Various reduced-order-models based on global modes, proper orthogonal decomposition modes, and balanced modes were tested and evaluated according to their ability to reproduce the input-output behavior between the actuator and the sensor. Finally, we consider the case of noise-amplifiers, such as boundary-layer flows and jets, which are stable when viewed in a global framework. The importance of the singular value decomposition of the global resolvent will be highlighted in order to understand the frequency selection process in such flows. © 2010 by ASME.

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Dynamics and Control of Global Instabilities in Open-Flows: A Linearized Approach

This article deals with the linear dynamics of a transitional boundary layer subject to two-dimensional Tollmien–Schlichting instabilities. We consider a flat plate including the leading edge, characterized by a Reynolds number based on the length of the plate equal to Re = 6 × 105, inducing a displacement thickness-based Reynolds number of 1,332 at the end of the plate. The global linearized Navier–Stokes equations only display stable eigenvalues, and the associated eigen-vectors are known to poorly represent the dynamics of such open flows. Therefore, we resort to an input–output approach by considering the singular value decomposition of the global resolvent. We then obtain a series of singular values, an associated orthonormal basis representing the forcing (the so-called optimal forcings) as well as an orthonormal basis representing the response (the so-called optimal responses). The objective of this paper is to analyze these spatial structures and to closely relate their spatial downstream evolution to the Orr and Tollmien–Schlichting mechanisms. Analysis of the spatio-frequential diagrams shows that the optimal forcings and responses are respectively localized, for all frequencies, near the upstream neutral point (branch I) and the downstream neutral point (branch II). It is also shown that the spatial growth of the dominant optimal response favorably compares with the eN method in regions where the dominant optimal forcing is small. Moreover, thanks to an energetic input–output approach, it is shown that this spatial growth is solely due to intrinsic amplifying mechanisms related to the Orr and Tollmien–Schlichting mechanisms, while the spatial growth due to the externally supplied power by the dominant optimal forcing is negligible even in regions where the dominant optimal forcing is strong. The energetic input–output approach also yields a general method to assess the strength of the instability in amplifier flows. It is based on a ratio comparing two quantities of same physical dimension, the mean-fluctuating kinetic energy flux of the dominant optimal response across some boundary and the supplied mean external power by the dominant optimal forcing. For the present boundary-layer flow, we have computed this amplification parameter for each frequency and discussed the results with respect to the Orr and Tollmien–Schlichting mechanisms.

Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer

The stability behaviour of a low-Reynolds-number recirculation flow developing in a curved channel is investigated using a global formulation of hydrodynamic stability theory. Both the resonator and amplifier dynamics are investigated. The resonator dynamics, which results from the ability of the flow to self-sustain perturbations, is studied through a modal stability analysis. In agreement with the literature, the flow becomes globally unstable via a three-dimensional stationary mode. The amplifier dynamics, which is characterized by the ability of the flow to exhibit large transient amplifications of initial perturbations, is studied by looking for the two- and three-dimensional initial perturbations that maximize the energy gain over a given time horizon. The optimal initial two-dimensional perturbations have the form of wave packets localized in the upstream part of the recirculation bubble. It is shown that they are first amplified while travelling downstream along the shear layer of the recirculation bubble and then decay when leaving the recirculation bubble. Maximal energy gain is thus achieved for a time horizon approximately corresponding to the propagation of the wave packet along the whole recirculation bubble. The resonator and amplifier dynamics are associated with different types of structures in the flow: three-dimensional steady structures for the resonator dynamics and nearly two-dimensional unsteady structures for the amplifier dynamics. A comparison of the strength of the two dynamics is proposed. The transient energetic growth of the two-dimensional unsteady perturbations is large at moderate time, compared to the very weak exponential growth of the three-dimensional stationary mode. This suggests that, as soon as there is noise in the system, the amplifier dynamics dominates the resonator dynamics, thus explaining the appearance of unsteadiness rather than the emergence of stationary structures in similar experimental flows.

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Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework

Non-modal analysis determines the potential for energy amplification in stable flows. The latter is quantified in the frequency domain by the singular values of the resolvent operator. The present work extends previous analysis on the effect of base-flow modifications on flow stability by considering the sensitivity of the flow non-modal behavior. Using a variational technique, we derive an analytical expression for the gradient of a singular value with respect to base-flow modifications and show how it depends on the singular vectors of the resolvent operator, also denoted as the optimal forcing and optimal response of the flow. As an application, we examine zero-pressure-gradient boundary layers where the different instability mechanisms of wall-bounded shear flows are all at work. The effect of the component-type non-normality of the linearized Navier-Stokes operator, which concentrates the optimal forcing and response on different components, is first studied in the case of a parallel boundary layer. The effect of the convectivetype non-normality of the linearized Navier-Stokes operator, which separates the spatial support of the structures of the optimal forcing and response, is studied in the case of a spatially evolving boundary layer. The results clearly indicate that base-flow modifications have a strong impact on the Tollmien-Schlichting (TS) instability mechanism whereas the amplification of stream-wise streaks is a very robust process. This is explained by simply examining the expression for the gradient of the resolvent norm. It is shown that the sensitive region of the Lift-up (LU) instability spreads out all over the flat plate and even upstream of it, whereas it reduces to the region comprised between branch I and branch II for the TS waves.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S002211201100382X

Olivier Marquet

Papers

Sensitivity analysis and passive control of cylinder flow

Dynamics and Control of Global Instabilities in Open-Flows: A Linearized Approach

Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer

Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework

Effect of base-flow variation in noise amplifiers: the flat-plate boundary layer