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Modal Analysis of Fluid Flows: An Overview

Kunihiko Taira, Steven L. Brunton, Scott T. M. Dawson, Clarence W. Rowley, Tim Colonius, Beverley J. McKeon, Oliver T. Schmidt, Stanislav Gordeyev, Vassilios Theofilis, Lawrence S. Ukeiley

arXiv:1702.01453v2physics.flu-dyn

TL;DR

Complex fluid flows contain important temporal and spatial features that researchers seek to extract and represent compactly. This overview organizes established modal-analysis techniques within a familiar linear-algebra framework to make them accessible for studying diverse fluid flows.

  • Problem

    Complex fluid flows contain diverse temporal and spatial features, creating a need to extract dominant features and represent high-dimensional data compactly.

  • Method

    The paper surveys data-based and operator-based modal-analysis techniques, grounding them in eigenvalue, singular-value, and pseudospectral analysis.

  • Results

    The overview presents established techniques including POD, Balanced POD, DMD, Koopman analysis, global stability analysis, and resolvent analysis.

  • Takeaways & Limitations

    Modal analysis provides established approaches for studying fluid flows and describing complex flow fields in lower-dimensional forms.

Abstract

from arXiv · show

Simple aerodynamic configurations under even modest conditions can exhibit complex flows with a wide range of temporal and spatial features. It has become common practice in the analysis of these flows to look for and extract physically important features, or modes, as a first step in the analysis. This step typically starts with a modal decomposition of an experimental or numerical dataset of the flow field, or of an operator relevant to the system. We describe herein some of the dominant techniques for accomplishing these modal decompositions and analyses that have seen a surge of activity in recent decades. For a non-expert, keeping track of recent developments can be daunting, and the intent of this document is to provide an introduction to modal analysis that is accessible to the larger fluid dynamics community. In particular, we present a brief overview of several of the well-established techniques and clearly lay the framework of these methods using familiar linear algebra. The modal analysis techniques covered in this paper include the proper orthogonal decomposition (POD), balanced proper orthogonal decomposition (Balanced POD), dynamic mode decomposition (DMD), Koopman analysis, global linear stability analysis, and resolvent analysis.

I. Introduction … A. Eigenvalue Decomposition

The paper introduces modal decomposition as a way to extract physically important flow features and represent complex, high-dimensional dynamics in low-dimensional form. It then establishes eigenvalue and singular value decompositions as foundations for the reviewed methods, emphasizing how eigenvalues characterize linear-system behavior.

  • I. Introduction: Dominant flow features such as von Kármán shedding, Kelvin–Helmholtz instability, and vortex pairing or merging can be extracted to describe complex flows in low-dimensional form.The expectation follows from their visual recognizability despite perturbations or parameter variations.
  • I. Introduction: At Re = 100 and α = 30°, a complex nonlinear separated flow over a flat-plate wing is well represented by two POD modes and the mean flow field.The dominant unsteady feature is a von Kármán vortex street in the wake.
  • I. Introduction: Modal decomposition extracts energetically and dynamically important flow features as spatial modes with characteristic energy, growth-rate, or frequency values.Modes can be obtained from flow-field data or governing equations.
  • I. Introduction: The overview presents eigenvalue and singular value decompositions, pseudospectral analysis, POD, Balanced POD, DMD, Koopman analysis, global stability analysis, and resolvent analysis.It provides overviews, descriptions, illustrative examples, and future outlooks for the methods, with an appendix on arranging flow data into vector and matrix forms.
  • II. Eigenvalue and Singular Value Decompositions: Eigenvalue decomposition applies to square matrices or operators with the same domain and range, whereas singular value decomposition applies to rectangular matrices whose domain and range need not coincide.Rectangular matrices cannot serve as propagators, although rectangular data matrices assembled from dynamical processes remain suitable for singular value decomposition.
  • A. Eigenvalue Decomposition: Eigenvectors identify directions preserved by an operator up to scalar multiplication, while eigenvalues determine whether the associated direction grows or shrinks.Eigenvalue decomposition requires a full set of n linearly independent eigenvectors; otherwise, a defective matrix uses Jordan form.
  • A. Eigenvalue Decomposition: The real and imaginary parts of λ_j represent growth or decay rate and frequency, respectively, for the state evolving in eigenvector direction v_j.A linear system is stable when Re(λ_j) ≤ 0 for all j.
  • A. Eigenvalue Decomposition: Pseudospectra address intermediate dynamics by relating eigenvalue sensitivity to perturbations with operator non-normality, while generalized eigenvalue problems account for mass matrices B.Recasting a generalized problem as standard is undesirable when B is noninvertible or its inversion causes ill-conditioning.

B. Singular Value Decomposition (SVD)

The singular value decomposition generalizes eigendecomposition to rectangular matrices and provides optimal low-rank approximations. It also represents how matrices stretch and rotate vectors through singular values and paired left and right singular vectors.

  • B. Singular Value Decomposition (SVD): SVD generalizes eigendecomposition to rectangular matrices and supports dimensionality reduction through optimal low-rank matrix approximations.The decomposition has multiple uses and interpretations, particularly for dimensionality reduction.
  • B. Singular Value Decomposition (SVD): Geometrically, a rectangular matrix maps right singular-vector directions to left singular-vector directions, with singular values measuring the resulting stretching.The mapped unit sphere becomes an ellipse or ellipsoid whose semiaxes are characterized by the left singular vectors and singular values.
  • B. Singular Value Decomposition (SVD): The factorization uses unitary matrices U and V with diagonal Σ, whose nonnegative diagonal entries satisfy σ1 ≥ σ2 ≥ · · · ≥ σp ≥ 0.Here, p = min(m, n), and U and V contain the left and right singular vectors.
  • B. Singular Value Decomposition (SVD): For rectangular matrices, right singular vectors span the domain while left singular vectors span the range, allowing matrix effects to be represented by scalar singular values.This parallels eigenvalue decomposition but requires two basis-vector sets because the domain and range may differ.

C. Relationship between Eigenvalue and Singular Value Decompositions · D. Numerical Libraries for Eigenvalue and Singular Value Decompositions · E. Pseudospectra

Eigenvalue and singular value decompositions are closely connected and supported by standard numerical libraries. Pseudospectral analysis extends eigenvalue-based stability assessment by revealing perturbation sensitivity, non-normality, and transient amplification.

  • C. Relationship between Eigenvalue and Singular Value Decompositions: The singular vectors of A are orthonormal eigenvectors of AA* and A*A, while A’s nonzero singular values are square roots of their nonzero eigenvalues.Thus, an eigenvalue decomposition of AA* or A*A can recover the SVD quantities.
  • D. Numerical Libraries for Eigenvalue and Singular Value Decompositions: Standard numerical libraries provide readily available routines for eigenvalue and singular value decompositions.The paper lists MATLAB, LAPACK, ScaLAPACK, and ARPACK as examples.
  • D. Numerical Libraries for Eigenvalue and Singular Value Decompositions: MATLAB’s eig handles standard and generalized eigenvalue problems, while svd returns singular values and left and right singular vectors, including economy-sized SVD.MATLAB is described as user-friendly for small to moderate size modal decompositions.
  • D. Numerical Libraries for Eigenvalue and Singular Value Decompositions: LAPACK supplies Fortran 90 routines for basic eigenvalue and singular value decompositions, while ScaLAPACK targets dense and banded problems on parallel distributed-memory machines.These libraries address standard linear algebra and high-performance parallel computation, respectively.
  • D. Numerical Libraries for Eigenvalue and Singular Value Decompositions: ARPACK specializes in large-scale and generalized eigenvalue problems, also performs singular value decompositions, and supports serial and parallel computations.The library is written in FORTRAN 77.
  • E. Pseudospectra: Pseudospectral analysis measures eigenvalue-spectrum sensitivity to operator perturbations and complements eigenvalue stability analysis for transient and input–output dynamics.The paper later connects pseudospectra to resolvent analysis along the imaginary axis.
  • E. Pseudospectra: As δ →0, nearly linearly dependent eigenvectors make A ill-conditioned and produce noticeable transient growth despite stable eigenvalues and eventual decay to zero.The growth arises because nearly parallel eigenvectors require large coefficients during the transient.
  • E. Pseudospectra: Decreasing δ through 0.01, 0.001, and 0.0001 makes A increasingly non-normal, expands its ϵ-pseudospectra, and increases susceptibility to perturbations.Pseudospectral expansion can lower-bound transient amplification, with the Kreiss constant estimating transient behavior.

Data-Based Modal Decomposition Methods · III. Proper Orthogonal Decomposition (POD) · A. Description

Data-based modal decomposition methods use only flow-field output data, without requiring knowledge of the dynamics. POD optimally represents flow data with a minimal orthogonal basis, while its variants and implementation choices address computational scale, frequency content, and dynamical relevance.

  • Data-Based Modal Decomposition Methods: Data-based methods, including POD, Balanced POD, and DMD, require only numerical or experimental flow-field output data, not the system dynamics.These methods decompose flow-field datasets into physically meaningful modes.
  • III. Proper Orthogonal Decomposition (POD): POD optimizes the mean-square representation of flow data, providing an objective decomposition into a minimal number of basis functions.It was introduced to extract coherent structures from turbulent flows and is also known as the Karhunen-Loève procedure.
  • III. Proper Orthogonal Decomposition (POD): POD represents fluctuations using orthogonal spatial modes, temporal coefficients, and energy levels ordered by each mode’s relative captured energy.For velocity data, the eigenvalues correspond to captured kinetic energy.
  • III. Proper Orthogonal Decomposition (POD): The method of snapshots replaces the large n×n covariance eigenproblem with an m×m problem when m ≪n, reducing computational and memory requirements.It recovers the same dominant spatial modes and is presently the most widely used POD method in fluid mechanics.
  • III. Proper Orthogonal Decomposition (POD): POD modes are optimal in the L2 sense, minimizing both mean-square truncation error and the number of modes required for a specified error.This fastest-convergent property explains POD’s extensive use in flow analysis and related data-processing tasks.
  • III. Proper Orthogonal Decomposition (POD): POD’s orthogonal minimal-dimensional basis supports reduced-order modeling, while high-order modes can represent incoherent noise when noise is weaker than the signal.Galerkin projection uses POD modes to reduce high-dimensional discretizations to ordinary differential equations for temporal coefficients.
  • III. Proper Orthogonal Decomposition (POD): POD has limitations: it ranks modes by energy rather than dynamical importance, spatial-mode coefficients can mix frequencies, and higher-order correlations are ignored.Spectral POD addresses frequency mixing, while Balanced POD and DMD address dynamical importance.

B. Illustrative Examples

The section illustrates POD through turbulent separated flow over a NACA 0012 airfoil and open-cavity flow experiments. These examples show dominant coherent structures and mode similarity across varied Mach numbers.

  • NACA 0012 airfoil: POD was applied to a three-dimensional LES velocity field for incompressible turbulent separated flow over a NACA 0012 airfoil at Re = 23, 000 and α = 9◦.The wake exhibited large-scale vortical structures associated with von Kármán shedding.
  • Open-cavity flow: In open-cavity flow experiments, mode 1 contained between 15 and 20 percent of the energy, while the first 7 modes represented 50 percent.Snapshot POD was applied to two-component PIV data spanning free stream Mach numbers from 0.2 through 0.73.
  • NACA 0012 airfoil: The airfoil analysis visualized instantaneous and time-averaged streamwise velocity fields alongside the four most dominant POD modes.These fields were shown on a spanwise slice.
  • Open-cavity flow: The cavity modes had similar wavelengths across 4 free stream Mach numbers, and orthogonality checks quantitatively verified their similarity.The similarity implied the underlying turbulence had the same structure across the investigated Mach-number range despite mean-flow differences.

C. Outlook

POD remains a foundational approach for extracting coherent structures in unsteady fluid flows, while numerous variants address shortcomings of standard POD. Its orthogonal modes also support low-order modeling and closed-loop flow control through Galerkin projection.

  • C. Outlook: POD is the bedrock of modal decomposition techniques for extracting coherent structures in unsteady fluid flows.
  • C. Outlook: Variants including Balanced POD, Split POD, Sequential POD, Temporal POD, and Joint POD have emerged to address shortcomings of standard POD.
  • C. Outlook: POD mode orthogonality enables low-order, sparse models and supports Galerkin-projection reduced-order modeling and model-based closed-loop flow control.

IV. Balanced Proper Orthogonal Decomposition (Balanced POD)

Balanced POD approximates balanced truncation to identify flow modes that reflect both how readily states are excited by specified inputs and how strongly they affect specified outputs. It uses direct and adjoint information to produce two mode sets for reduced-order modeling and analysis of coherent structures.

  • Inputs, outputs, and modes: Balanced POD extracts balancing and adjoint modes for specified inputs, such as disturbances or actuation, and outputs, such as sensor measurements or modeled quantities.The outputs may include amplitudes of POD modes.
  • Inputs, outputs, and modes: Balanced POD approximates balanced truncation by balancing controllability, which measures ease of excitation, and observability, which measures influence on future outputs.Reduced-order models seek to retain states that are both highly controllable and highly observable.
  • Relation to POD: Unlike POD’s single orthogonal, energy-ranked mode set, Balanced POD produces two mode sets while using similar singular value decomposition computations to describe coherent structures.The two procedures are closely related, but Balanced POD distinguishes balancing modes from adjoint modes.
  • Required data: Balanced POD requires linear responses to impulsive inputs and adjoint impulse responses, whose simulations reveal state observability and make the method effective for non-normal systems.The dataset contains one forward time series per input and one adjoint response per output.

A. Description · B. Illustrative Example

Balanced POD uses forward and adjoint linear-system snapshots to construct balancing and adjoint modes ordered by Hankel singular values, providing a controllability–observability balancing transform. It is especially useful for non-normal flows and feedback control, as illustrated by stabilization of an unstable flat-plate-wing wake at Re = 100.

  • A. Description: Balanced POD takes snapshots from linearized forward and companion adjoint simulations to produce balancing and adjoint modes.The inputs are two snapshot sets obtained from linearized forward and adjoint simulations.
  • A. Description: The resulting modes are ranked by Hankel singular values and form a coordinate transform balancing controllability and observability.The balancing and adjoint modes correspond to columns of the transforms and are ordered by Σ = diag(σ1, σ2, . . . , σm).
  • A. Description: For snapshot-based Balanced POD, empirical Gramians are approximated from forward and adjoint data matrices, whose cross-product is factorized by SVD.For MIMO systems, stacked forward and adjoint solutions give W_c ≈ XX^T and W_o ≈ ZZ^T, while the SVD is applied to Z^T X.
  • Strengths and Weaknesses: Balanced POD captures highly observable, small-energy perturbations in non-normal systems with large transient growth and typically outperforms POD there.POD ranks modes by energy content, whereas Balanced POD can capture perturbations identified through adjoint modes.
  • Strengths and Weaknesses: Balanced POD provides an input-output model suitable for feedback control, while adjoint snapshots make experimental application difficult; ERA avoids adjoint simulations when only the model is needed.Both forward and adjoint simulations should be based on linear dynamics, although extensions to nonlinear systems exist.
  • Control of Wake Behind a Flat-Plate Wing: At Re = 100, Balanced POD analyzed and controlled the unsteady wake behind a flat-plate wing using linearized forward and adjoint simulations.The balancing modes resembled traditional POD modes, while adjoint modes identified flow regions capable of triggering large downstream perturbations.
  • Control of Wake Behind a Flat-Plate Wing: For the unstable flat-plate wake at Re = 100 and α = 35°, a Balanced POD reduced-order model was used to compare baseline and feedback-controlled lift histories.The figure shows baseline lift as a dashed line and controlled cases with different feedback-initiation times as solid lines.

C. Outlook · V. Dynamic Mode Decomposition (DMD) · A. Description

The paper outlines DMD as a data-driven decomposition of time-resolved flow data into modes with characteristic frequencies and growth or decay rates. It also describes DMD’s snapshot-based linear-algebra formulation, extensions, and practical strengths and limitations, while identifying future directions for Balanced POD.

  • C. Outlook: Balanced POD research is extending toward nonlinear and unstable systems, harmonically forced data, more efficient algorithms, and applications beyond simulations.Current directions include analytic treatment of impulse-response tails and randomized methods; adjoint-simulation requirements still limit experimental use.
  • V. Dynamic Mode Decomposition (DMD): DMD decomposes time-resolved data into modes, each with a characteristic oscillation frequency and growth or decay rate.It uses the eigendecomposition of a best-fit linear operator approximating the dynamics in the data.
  • V. Dynamic Mode Decomposition (DMD): DMD combines data-driven spatial and temporal structure extraction, is rooted in linear algebra, and requires no governing equations or a priori dynamical knowledge.This enables applications beyond fluid dynamics and supports substantial algorithmic development.
  • A. Description: DMD begins with equally time-spaced snapshot pairs arranged as columns of X and X#, then approximates their relationship through a best-fit linear operator.The inputs are snapshot pairs from experiments or simulations, commonly obtained from a time series.
  • A. Description: The operator A = X#X+ defines DMD eigenvalues and modes through its eigendecomposition; when m ≪n, reduced-SVD algorithms avoid explicitly computing A.Projected modes can be obtained from the reduced SVD representation.
  • A. Description: DMD variants address real-time computation, sparse representations, external inputs, and nonlinear systems, although nonlinear applications require sufficiently rich snapshot measurements.Examples include optimized DMD, sparsity-promoting DMD, input-aware extensions, and augmented measurements such as squared velocities.
  • Strengths and Weaknesses: DMD can isolate frequency-specific structures and accommodate diverse data, but mode relevance is subjective because modes are nonorthogonal and lack an objective importance ranking.Unlike POD modes, DMD modes have no single correct ranking of eigenvalue importance.
  • Strengths and Weaknesses: DMD typically needs time-resolved data and can be unreliable for nonlinear, noisy, intermittent, or traveling-wave dynamics unless appropriate extensions or measurements are used.Robust sensor-noise algorithms, multi-resolution and time-delay variants, and nonlinear measurement enrichment address some limitations.

B. Illustrative Example … A. Description

The paper illustrates DMD on complex three-dimensional flows and then frames Koopman analysis as an operator-theoretic approach that represents nonlinear dynamics through infinite-dimensional linear dynamics. It also identifies challenges in selecting observables, finding eigenfunctions, and constructing useful finite-dimensional approximations.

  • Jet in Crossflow: DMD applied to 251 snapshots of a three-dimensional jet-in-crossflow DNS identified modes with distinct frequencies and amplitudes, including modes at dimensionless frequencies 0.0175 and 0.141.The modes were visualized using streamwise-velocity contours, while the flow snapshot was shown with the λ2 criterion.
  • Canonical Separated Flow with Control: In a separated turbulent flow over a finite-thickness plate at Re = 100, 000, POD and DMD were used to analyze wake dynamics modified by synthetic-jet actuation.The fundamental and secondary DMD modes corresponded to the actuation frequency 4.40 and its superharmonic.
  • C. Outlook: DMD’s connections to the Koopman operator support modeling and control of nonlinear systems, but selecting observables for accurate finite-dimensional Koopman approximations remains an open question.The paper notes that algorithmic improvements remain necessary.
  • Operator-Based Modal Analysis Methods: Operator-based modal analysis focuses on the state-dynamics operator rather than only flow data, covering Koopman, global linear stability, and resolvent analyses.Koopman analysis treats finite-dimensional nonlinear dynamics as infinite-dimensional linear dynamics, while global linear and resolvent analyses reveal stability and input characteristics.
  • VI. Koopman Analysis: Koopman analysis offers an alternative perspective on complex systems by describing their dynamics with an infinite-dimensional linear operator acting on observable functions.The framework originated with Koopman’s treatment of Hamiltonian systems and has become central to data-driven characterization of complex systems.
  • VI. Koopman Analysis: Koopman analysis provides modes, eigenvalues governing modal dynamics, and eigenfunctions serving as intrinsic observables; DMD modes form a subset of Koopman modes.It can provide an equation-free route to coherent structures and dynamics, which are related to POD modes in fluids.
  • A. Description: Given a nonlinear dynamical system and snapshot data of observables, Koopman analysis produces a modal decomposition and a linear dynamical system describing modal evolution.DMD typically approximates the Koopman mode decomposition through linear regression, with nonlinear measurement functions potentially improving the approximation.
  • Strengths and Weaknesses: Koopman methods enable linear techniques and control tools for nonlinear systems, but infinite-dimensional operators, low-order approximations, and Koopman eigenfunctions remain difficult to obtain.Without suitable eigenfunctions, a finite-dimensional Koopman-invariant subspace may be unavailable; integrating control theory with Koopman analysis also remains incomplete.

B. Illustrative Example

The example shows how suitable nonlinear measurements can transform a simple nonlinear system into a linear Koopman representation. This representation enables analytical future-state prediction while illustrating challenges in extending Koopman analysis to complex fluid systems.

  • The illustrative system is a two-variable nonlinear dynamical system with a single fixed point at the origin.
  • For λ ≪µ < 0, the origin is stable, and trajectories rapidly approach a slow manifold before converging to the origin.
  • A nonlinear coordinate transformation using measurements y makes the nonlinear dynamics linear and defines a Koopman-invariant observable subspace.
  • The restricted Koopman operator becomes a 3 × 3 matrix K, allowing analytical prediction through y(t) = exp(Kt)y(0).
  • Applying Koopman analysis to complex systems remains difficult because discovering coordinate transformations that linearize the problem is challenging.

C. Outlook · VII. Global Linear Stability Analysis · A. Description

The outlook emphasizes richer and better-identified observables for Koopman analysis, while global linear stability analysis characterizes small perturbations by solving eigenvalue problems about exact steady or time-periodic base flows. Its matrix-based and time-stepping implementations provide modal growth rates and frequencies but can be computationally expensive and remain inherently linear.

  • C. Outlook: Koopman analysis increasingly seeks richer nonlinear observables and principled sparse selection because DMD’s linear observables may inadequately represent fluid dynamics.Recent methods also construct Koopman eigenfunctions using regularized advection.
  • VII. Global Linear Stability Analysis: Global linear stability analysis converts linearized Navier–Stokes equations about an exact steady or unsteady base flow into an eigenvalue problem for multidimensional disturbances.Global modes describe disturbances on base flows varying in two or three spatial directions.
  • A. Description: The method takes a steady or time-periodic laminar base flow, inhomogeneous in two or three spatial directions, at specified Reynolds and Mach numbers.The base flow may be stable or unstable.
  • A. Description: Global stability analysis outputs spatial mode patterns with associated growth rates and frequencies for small-amplitude perturbations, rather than relying on flow-field snapshots.The perturbation decomposition assumes ||q′||/||q0|| ≪1, and quadratic perturbation terms are neglected.
  • Matrix-Based Approach: The matrix-based approach solves a generalized eigenvalue problem whose operators depend on the base state, wavenumber, Reynolds number, and Mach number.For large operators, matrix-free matrix-vector operations can determine eigenvalues without storing the operators.
  • Time-Stepping Approach: The time-stepping approach computes linear operators without storing matrices, but numerical discretization can introduce spurious modes requiring extraction of physically meaningful modes.High computational effort motivates sparse high-order discretization and exploiting flow symmetry.
  • Strengths and Weaknesses: Global analysis determines unstable modes and, when only stable modes exist, transient-growth analysis can assess short-horizon energy amplification; however, the framework is inherently linear and base-flow-dependent.Exact governing-equation solutions are required for stability interpretation, whereas modes about mean turbulent flows may instead identify coherent-structure frequencies and spatial patterns.

B. Illustrative Examples … A. Description

The paper illustrates modal analysis through global-stability studies of separated wings and finite-span cavities, then develops resolvent analysis as a forcing–response framework for coherent structures and flow control. The outlook emphasizes extensions to hypersonic, three-dimensional, parabolized, adjoint, and fully turbulent-flow settings, while noting computational and forcing-information limitations.

  • Large-Scale Separation Cells on Stalled Airfoils: Global stability analysis explains large-scale separation cells on stalled wings by combining primary and secondary eigenvalue and transient-growth analyses.The studies examine massively separated, spanwise-homogeneous laminar flows over spanwise-periodic wings with varying thickness and camber.
  • Finite-Span Open Cavity Flow: Trig global stability analysis identifies the leading traveling shear-layer mode causing transition in a finite-span open cavity, alongside stationary and traveling centrifugal modes.The time-stepper approach resolves the critical conditions and spatial characteristics of these distinct eigenmode classes.
  • C. Outlook: Global-stability methods are extending toward hypersonic-transition prediction, parabolized three-dimensional base flows, and adjoint analyses of receptivity, sensitivity, and control.Hypersonic base flows may be computed using continuum or Direct Simulation Monte Carlo methods, while parabolized formulations address flows varying strongly in two directions and weakly in the third.
  • VIII. Resolvent Analysis: Resolvent analysis relates input forcing to flow output and complements eigenvalue and global-stability analyses by treating forced linear dynamics.For incompressible Navier–Stokes flow, nonlinear terms provide the input and the divergence-free velocity field is the output.
  • A. Description: At a specified frequency, singular-value decomposition of the resolvent yields optimal forcing and response modes, with the largest singular value representing squared gain.The formulation requires a base velocity profile and can exploit spatial homogeneity through Fourier decomposition.
  • Strengths and Weaknesses: Fully inhomogeneous flows remain an unmet computational challenge, and predicting energetic observed modes requires information about the nonlinear forcing, not amplification alone.Resolvent modes are efficient for one-dimensional base flows, but two-dimensional base flows cost significantly more; observations depend on amplification and forcing weight.
  • A. Description: Resolvent analysis can use turbulent mean flows as base states and interpret coherent structures as responses sustained by nonlinear or background-turbulence forcing.This makes it useful for convectively unstable flows, such as turbulent jets, that lack intrinsic unstable global modes.

B. Illustrative Examples · C. Outlook · IX. Concluding Remarks

The illustrative examples show how resolvent analysis identifies dominant structures and physically meaningful amplification in turbulent channel flow and jets. The outlook and conclusions emphasize parametric sensitivity, connections to other modal methods, and the use of extracted modes for reduced-order modeling, flow control, and aerodynamic design.

  • B. Illustrative Examples: In turbulent channel flow, the leading resolvent response modes include left- and right-going pairs representative of very-large-scale motions.The modes use ±kζ wavenumbers, with velocity-component magnitudes (|u|, |v|, |w|) = (1, 0.05, 0.17).
  • B. Illustrative Examples: In a turbulent jet, the leading resolvent mode dominates the gain over a broad Strouhal-number interval.A sharp peak near St ≈ 0.4 is associated with acoustic resonance in the potential core, while a broader peak corresponds to the preferred mode.
  • C. Outlook: Resolvent analysis offers a promising basis for parametric sensitivity studies and dynamical modeling through identification and weighting of nonlinear forcing interactions.Forcing weights can be obtained or approximated from full-field snapshots or snapshot projections.
  • C. Outlook: Resolvent modes can efficiently represent exact coherent Navier–Stokes solutions, supporting approximate continuation and searches for new exact solutions.The paper also notes formal connections among resolvent analysis, Koopman analysis, DMD, and exact nonlinear traveling-wave solutions.
  • IX. Concluding Remarks: The paper surveys POD, Balanced POD, DMD, Koopman analysis, global linear stability analysis, and resolvent analysis as modal techniques for examining fluid flows.These techniques extract modal structures that illuminate different aspects of flow fields while sharing some similarities.
  • IX. Concluding Remarks: Extracted modes can support reduced-order models that capture flow dynamics at significantly lower computational cost.The paper identifies closed-loop flow control and aerodynamic design as applications of such models.

A. Appendix

The appendix explains how to arrange flow-field snapshots into a data matrix for modal decomposition. It covers structured and more general grids, vector-data stacking, mode visualization, and the resulting tall, skinny matrix used by POD, Balanced POD, and DMD.

  • Data-matrix construction: Flow-field snapshots are organized into a data matrix X using uniformly spaced grids and constant time steps, with scaling or preprocessing for general grids.The example uses a two-dimensional velocity field q = (u, v).
  • Snapshot organization: At each time instance, the velocity data are stacked into a column vector x(t) using the spatial coordinates of a collocated setup.The coordinates are indexed by i = 1, . . . , nξ and j = 1, . . . , nη.
  • Vector-data representation: For a two-dimensional vector field, stacking yields a column vector of size n = 2nxny, while scalar and three-dimensional data require corresponding component adjustments.Unstacking supports visualization of the decomposition modes after analysis.
  • Modal decomposition: The resulting fluid-flow data matrix is generally tall and skinny, with n ≫m, and supports POD, Balanced POD, and DMD decompositions.The matrix is formed once the column data vectors are available at the desired time levels.
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