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Modal Analysis of Fluid Flows: Applications and Outlook

Kunihiko Taira, Maziar S. Hemati, Steven L. Brunton, Yiyang Sun, Karthik Duraisamy, Shervin Bagheri, Scott T. M. Dawson, Chi-An Yeh

arXiv:1903.05750v2physics.flu-dyn

TL;DR

This paper surveys how modal analysis can provide physical insight into canonical aerodynamic flows and support their analysis, modeling, and control. It examines cylinder wakes, wall-bounded flows, airfoil wakes, and cavity flows, while highlighting data-collection choices, nonlinear extensions, and limits of linear modal representations.

  • Problem

    Canonical aerodynamic flows require techniques that reveal their dynamics and support analysis, modeling, and control across different flow settings.

  • Method

    The paper surveys applications of POD, DMD, resolvent analysis, sparse nonlinear identification, and related modal techniques across four canonical aerodynamic-flow examples.

  • Results

    Modal analyses capture dominant flow behavior with compact representations; for example, 6 modes capture 99.85% of fluctuation kinetic energy, while five modes capture over 95% of the initial-condition energy in a channel-flow reduced-order model.

  • Takeaways & Limitations

    Modal analysis provides complementary physical insights, but effective interpretation and reduced-order modeling depend on choosing data from relevant dynamical regimes and matching the representation to the flow.

  • Takeaways & Limitations

    Linear modal bases optimized for one dataset may require additional modes or recomputation when perturbations or actuation deform the flow, while noisy forcing can make DMD spectra appear marginally stable.

Abstract

from arXiv · show

We present applications of modal analysis techniques to study, model, and control canonical aerodynamic flows. To illustrate how modal analysis techniques can provide physical insights in a complementary manner, we selected four fundamental examples of cylinder wakes, wall-bounded flows, airfoil wakes, and cavity flows. We also offer brief discussions on the outlook for modal analysis techniques, in light of rapid developments in data science.

I. Introduction

Modal analysis addresses the challenge of interpreting increasingly complex, high-dimensional fluid-flow data by extracting underlying flow features and demonstrating their physical meaning. The paper surveys applications to canonical aerodynamic flows, while noting challenges for experimental datasets.

  • High-resolution simulations and measurements expose intricate flow physics but create vast datasets and high-dimensional nonlinear dynamics that require concise characterization.
  • Modal analysis extracts underlying flow features from flow-field data or flow-evolution operators to reveal common phenomena across diverse flows.
  • Modal decomposition also supports reduced-order modeling and control by identifying low-dimensional coordinate systems for dominant flow mechanisms.
  • The paper emphasizes interpreting modal outputs for physical insight rather than applying modal methods blindly.
  • Applications cover cylinder wakes, wall-bounded flows, airfoil wakes, and cavity flows, primarily through computational examples.
  • Experimental data require careful treatment because noise and bandwidth limitations pose challenges for data-driven modal analysis.

II. Cylinder wakes

Cylinder wakes provide a canonical setting for testing modal analysis because their recognizable shedding structures can represent complex flows compactly. The section demonstrates POD-based extraction of energetic structures and highlights limits when the flow is perturbed.

  • Cylinder wakes are fundamental bluff-body flows in which modal analysis has revealed dynamics and supported mode-based reduced-order flow control.
  • Persistent Kármán shedding across Reynolds numbers indicates that dominant wake structures can provide a low-dimensional representation of the flow.
  • A. Proper orthogonal decomposition: POD uses flow snapshots to extract spatial modes without requiring knowledge of the governing dynamics.
  • A. Proper orthogonal decomposition: At Re = 100, the first 2, 4, and 6 modes capture 94.84%, 98.68%, and 99.85% of fluctuation kinetic energy, respectively.
  • A. Proper orthogonal decomposition: With 8 modes, POD captures 99.97% of fluctuation kinetic energy, reducing the representation to a few spatial modes.
  • A. Proper orthogonal decomposition: The dominant POD modes reveal energetic spatial structures associated with the asymmetry of the Kármán wake.
  • A. Proper orthogonal decomposition: POD modes are optimal for the supplied data but may require augmentation or recomputation when perturbations or actuation deform the flow.

B. Dynamic mode decomposition

DMD separates cylinder-wake dynamics into spatial modes and temporal eigenvalues, providing interpretable information about oscillatory structures, frequencies, and growth or decay. Its relationship with POD is especially close for periodic wakes, but interpretation depends on mode properties, data selection, and measurement quality.

  • Dynamic mode decomposition: The first oscillatory DMD mode captures the top-down asymmetry associated with Kármán vortex shedding.The mode is shown through real, imaginary, magnitude, and phase representations.
  • Dynamic mode decomposition: Magnitude plots identify each mode’s active regions, while phase plots show relative phase between spatial regions.These representations help distill physical insights from the modal structures.
  • Dynamic mode decomposition: For the periodic cylinder wake, POD and DMD produce closely corresponding spatial modes, although DMD modes need not be orthogonal.POD modes are real-valued and orthogonal, whereas oscillatory DMD modes occur in complex-conjugate pairs.
  • Dynamic mode decomposition: DMD represents cylinder-wake dynamics through spatial modes and eigenvalues encoding temporal growth rates and frequencies.Each dynamic mode contains spatial information, while each eigenvalue supplies temporal information.
  • Dynamic mode decomposition: DMD amplitudes quantify the relative contribution of each mode to a system realization, with definitions based on initial or all snapshots.The cylinder-flow example uses amplitudes calculated from the initial snapshot.
  • Dynamic mode decomposition: Selecting snapshots by dynamical regime is essential for preserving interpretability, while experimental noise motivates noise-robust DMD variants.Using mixed-regime data can contaminate modal-analysis results, and measurement uncertainty can bias DMD.

C. Linear global stability analysis

Linear stability analysis formulates flow instabilities as an eigenvalue problem, while Galerkin projection and related reduced-order models represent cylinder-wake dynamics with few modal coordinates. These approaches identify instability mechanisms and enable compact dynamical predictions, but truncation, complexity, and changing flow conditions limit model utility.

  • C. Linear global stability analysis: Global stability analysis linearizes the Navier–Stokes equations about a steady base flow and solves for eigenvalues and spatial eigenvectors.The eigenvalue gives each instability’s growth or decay rate and frequency.
  • C. Linear global stability analysis: The cylinder wake becomes unstable at Recrit ≈46, where eigenvalues cross into the unstable complex plane and initiate von Kármán shedding through a Hopf bifurcation.Floquet analysis extends this framework to periodic base states and identifies three-dimensional modes A and B at ReA ≈189 and ReB ≈259.
  • 1. Galerkin modeling: Galerkin projection substitutes a POD expansion into the incompressible Navier–Stokes equations and projects the momentum dynamics onto orthogonal modes to obtain ODEs for POD coefficients.The resulting coefficients can be integrated to predict modal dynamics and reconstruct the full flow field.
  • 1. Galerkin modeling: Using four POD modes can produce unboundedly slow growth of wake oscillations, whereas six modes improve accuracy; models may also deviate over long times and across Reynolds numbers.A model that predicts short-duration dynamics well may lose accuracy over longer time frames.
  • 1. Galerkin modeling: Adding a shift mode representing the difference between the mean flow and steady equilibrium lets a Galerkin model capture transient growth from an unstable equilibrium to a shedding limit cycle with only 3 modes.Without the shift mode, this transient prediction is generally difficult.
  • 1. Galerkin modeling: Galerkin truncation can remove dynamically important low-energy modes, destabilize nonlinear terms, and become costly for multiscale flows because model complexity scales as O(r3).POD modes also deform with changing flow conditions and geometries, restricting transferability.

2. Sparse identification of nonlinear dynamics

SINDy identifies sparse nonlinear dynamics directly from time series of POD coefficients, providing interpretable reduced models alongside Galerkin projection. Its use of higher-order nonlinearities can improve representation of truncated effects and closely reproduce cylinder-wake dynamics.

  • 2. Sparse identification of nonlinear dynamics: SINDy discovers a low-order dynamical system from time-series data of POD coefficients by regressing the coefficient dynamics onto a library of basis functions.The state derivative is represented as a linear combination of functions θj(a).
  • 2. Sparse identification of nonlinear dynamics: The algorithm selects a sparse coefficient vector so that as few library terms as possible remain active, improving model interpretability.Known constraints, including energy conservation for quadratic nonlinearities in incompressible flows, can also be incorporated.
  • 2. Sparse identification of nonlinear dynamics: Unlike standard Galerkin projection, SINDy can include higher-order nonlinearities to account for effects of truncated POD terms.A cubic SINDy model nearly perfectly captures the true cylinder-wake dynamics and can also operate on sensor-based coordinates such as lift and drag.

III. Wall-bounded flows

Wall-bounded flows differ from oscillator-like flows because sufficiently large Reynolds numbers produce energetic structures across broad length and time scales. Modal and stability analyses therefore address both classical instability and transient growth associated with non-normal dynamics.

  • III. Wall-bounded flows: Wall-bounded shear flows can contain energetic structures across a broad range of length and time scales at sufficiently large Reynolds numbers.This contrasts with oscillator-type flows characterized by a single dominant frequency and length scale.
  • III. Wall-bounded flows: Laminar channel flow becomes linearly unstable at Re = 5 772, but transition to turbulence can occur at much lower Reynolds numbers than this modal-stability threshold.Linear dynamics can nevertheless produce substantial transient energy growth before perturbations decay.

A. Linearly stable laminar channel flow

The laminar channel example combines operator-based and data-driven modal analyses to connect stability, transient amplification, and reduced-order modeling. It shows that energy-based POD truncation can miss dynamically important components, whereas balanced POD better preserves the observed input-output dynamics.

  • Operator-based analysis: At Re = 2 000, kx = 0.25, and kz = 2, the channel system is asymptotically stable but non-normal, allowing transient energy growth and strong disturbance amplification.The analysis uses the Orr–Sommerfeld and Squire formulation about a parabolic laminar profile.
  • Operator-based analysis: The maximum single-frequency amplification occurs at wavespeed cr = 0.578, identified by the resolvent norm and associated forcing and response modes.Maximum finite-time energy growth occurs at τ = 58 and is characterized by singular vectors of exp(Lτ).
  • Data-driven analysis: DMD identifies only the eigenvalues active in the maximally growing trajectory, yet reconstructs that trajectory despite missing eigenvalues of the full operator.Eigenvalues near intersecting branches are especially sensitive to perturbations because of non-normality.
  • Reduced-order modeling: The leading POD modes are dominated by wall-normal vorticity, while the initial condition contains substantial wall-normal velocity, so three POD modes reconstruct the trajectory poorly.The first three POD modes represent less than 4% of the initial-condition energy; using five modes raises this measure above 95%.
  • Reduced-order modeling: Total energy content is not sufficient for choosing a projection basis because low-energy features can remain dynamically important.Balanced POD addresses this by using primal and adjoint modes to preserve system dynamics, and its three-mode model accurately captures the trajectory.
  • Flow control: Reduced-order models for transient-growth control must be tailored to the control objective because controller performance can be sensitive to ROM-generation parameters.Observer-based feedback cannot fully suppress transient growth in linearized flows that already exhibit it and can worsen worst-case growth.

B. Turbulent wall-bounded flows

For turbulent wall-bounded flows, modal analysis commonly uses mean-linearized operators and resolvent singular-value decompositions to extract structures and amplification mechanisms. Connections to spectral POD, state estimation, and colored-noise modeling extend these analyses toward reduced-complexity models and control.

  • Operator-based analysis: Mean-linearized operator analyses can provide substantial insight into turbulent-flow features even though they generally cannot predict exact trajectory evolution.The approach assumes spatial homogeneity in the streamwise and spanwise directions.
  • Resolvent analysis: Resolvent singular-value decomposition is a particularly useful operator-based approach for analyzing turbulent wall-bounded flows and their amplification mechanisms.The pseudospectrum can be more relevant than the spectrum for understanding typical fluid-flow instability and amplification behavior.
  • Connections to POD: Under uncorrelated response-mode coefficients, resolvent response modes coincide with spectral POD modes, linking operator-based and data-driven decompositions.This provides a direct connection between resolvent analysis and POD under the stated forcing assumption.
  • Modeling and control: Resolvent-based models have been applied to state estimation from limited measurements and to covariance completion for modeling nonlinear forcing as colored noise.These approaches are presented as reduced-complexity models that may guide future control investigations.

C. Spatially developing flows

Spatially developing boundary layers are globally stable but locally convectively unstable, so external forcing generates disturbances that amplify downstream. Modal methods reveal this spatial evolution, while BPOD addresses upstream input sensitivity that leading POD modes often miss.

  • Flow physics: A flat-plate boundary layer is globally asymptotically stable but locally convectively unstable under a parallel-flow approximation.Disturbances grow as they are transported downstream by the mean flow.
  • Flow physics: External perturbations continuously enter the boundary layer, trigger Tollmien–Schlichting waves or streamwise vortices, and can amplify during downstream propagation.The disturbances may induce transition if they exceed a sufficient amplitude threshold.
  • Reduced-order modeling: BPOD modes account for sensitivity to upstream forcing, whereas leading POD modes emphasize energetic structures farther downstream and provide little upstream spatial support.This difference makes small accurate Galerkin models of input-output dynamics difficult to obtain from leading POD modes alone.
  • Data-driven limitations: DMD assigns marginally stable temporal eigenvalues to continuously noise-driven data even when the system has a damped global spectrum.The modes can still convey spatial inherent dynamics corresponding to spatial stability analysis.
  • DMD interpretation: In periodically forced flat-plate flow, the first DMD mode represents a Tollmien–Schlichting wave at the forcing frequency, while the third represents a 2ω wave generated by nonlinear interactions.The fundamental mode decays immediately downstream of forcing, then grows between branches I and II before peaking farther downstream.

IV. Airfoil wakes

Modal analysis of airfoil wakes identifies coherent structures, instability modes, and norm-dependent physical features, while supporting analysis of separation, noise, and wake dynamics.

  • Snapshot duration, temporal resolution, and spatial-domain selection determine whether data-based analyses resolve shear-layer or wake structures.Restricting the domain or using spatial weighting can bring separation-bubble structures into low-rank POD modes, while DMD separates structures by frequency.
  • For turbulent NACA 0012 flow at Re = 408 000, kinetic-energy and pressure norms identify similar tonal-noise structures but different second-pair modes.The pressure norm reveals spanwise harmonic structures, whereas the kinetic-energy norm reveals streamwise structures unrelated to tonal-noise generation.
  • Biglobal stability analysis examines unstable equilibrium flows over NACA 0012 airfoils using spanwise-periodic perturbations and eigenmodes.The formulation uses q′(x) = q̂(x, y) exp(iωt + iβz).
  • Floquet analysis of periodic NACA 4415 flow at Re = 500 and α = 20° reveals three-dimensional instabilities at β = 3 and β = 11.The short-wavelength β = 11 instability is stronger.

C. Flow control

Modal analysis guides airfoil flow control by identifying forcing parameters and instability structures linked to separation, aerodynamic performance, and tip-vortex attenuation.

  • Resolvent analysis guides periodic forcing for separation control by analyzing turbulent mean flows around separated airfoils.
  • The modal mixing metric M(β, ω) combines response modal Reynolds stresses, resolvent gain, and spatial weighting over the separation bubble.It quantifies momentum mixing and evaluates actuation choices through spanwise wavenumber β and frequency ω.
  • Agreement between M(β, ω) and controlled-flow performance indicates that the metric predicts aerodynamic-performance enhancement across the actuation parameter space.
  • Tip-vortex attenuation is motivated by the prolonged vortex presence, which creates a safety hazard for aircraft operations.
  • Global stability modes can define trailing-edge perturbations that modify the wake and attenuate tip-vortex circulation.The first and fifth instability modes are used to introduce the perturbations.

V. Cavity flows

Cavity-flow modal analysis reveals Rossiter-related and other instability structures, identifies amplification mechanisms, and informs control while requiring attention to spanwise and sidewall effects.

  • Open-cavity flow forms a Kelvin–Helmholtz-driven feedback loop that generates large vortical structures, pressure fluctuations, acoustic waves, and Rossiter modes.
  • POD of compressible cavity flows identifies energetic structures concentrated in the shear-layer region.
  • DMD applied to linearized cavity flow extracts global stability modes and identifies Kelvin–Helmholtz instabilities associated with flow oscillations.For L/D = 1 and Re = 4 500, unstable modes occupy a branch with λr > 0.
  • Biglobal stability uses spanwise-periodic perturbations to study two-dimensional Rossiter-related modes at β = 0 and three-dimensional instabilities at β > 0.
  • For L/D = 6 and ReD = 502, compressible-cavity stability analysis examines how Mach number and spanwise wavelength affect three-dimensional instability properties.
  • Resolvent analysis identifies optimal forcing and response modes through amplification gain, with a maximum around StD ≈ 0.15 for the specified cavity flow.The gain peak need not coincide with the leading global-stability frequency, a difference attributed to operator non-normality.
  • Sidewall effects alter cavity-flow global characteristics, requiring full three-dimensional simulations and triglobal analysis when spanwise periodicity is insufficient.

C. Flow control

Modal-analysis insights support open- and closed-loop cavity-flow control by selecting three-dimensional actuation patterns and constructing reduced-order feedback models.

  • Two-dimensional leading-edge control does not simultaneously suppress all resonant tones, whereas three-dimensional actuation reduces amplitudes across all resonant tones.
  • Biglobal stability and resolvent analysis can select actuator spanwise wavelength λ or wavenumber β = 2π/λ for three-dimensional cavity control.
  • Spatial modes enable reduced-order cavity-flow models for feedback control by projecting state dynamics onto an appropriate modal basis.
  • A closed-loop incompressible cavity-flow controller uses wall-normal blowing/suction as actuation and integrated wall-normal shear stress near the cavity edges as output.The setup targets suppression of flow unsteadiness at ReD = 7 500.
  • Global, POD, and BPOD modes are ordered respectively by growth rate, energy content, and Hankel singular values for control design.

D. Landing gear well

Modal analysis identifies physically relevant structures in complex flows and supports reduced-order modeling, sparse computation, and control. The landing-gear-well example links transverse velocity POD modes to vortical structures responsible for acoustic tones.

  • D. Landing gear well: Velocity POD modes in a turbulent landing-gear-well flow correlate with vortical structures responsible for generating acoustic tones.The extracted dominant modes reveal shear-layer structures within the highly complex flow.
  • VI. Outlook: Modal methods are fundamentally linear, whereas many flows evolve on low-dimensional manifolds rather than low-dimensional linear subspaces.Nonlinear dimensionality-reduction methods are proposed as potential alternatives for more accurate and efficient reduced-order models.
  • VI. Outlook: Sparse measurements and randomized linear algebra exploit low-dimensional or low-rank structure to reduce sensing and computational costs.Compressed sensing supports compact representations and flow reconstruction, while randomized projections accelerate matrix decompositions.
  • VI. Outlook: Reduced-order models support design, optimization, estimation, and control, but nonlinear projected dynamics can retain online costs scaling as O(n).Hyper-reduction or sparse sampling is therefore needed when nonlinear dynamics remain high-dimensional.

1. Dynamical systems models

The paper frames modal analysis as a route to predictive models for design, optimization, estimation, and control, while surveying data-driven modeling and stabilization challenges. It emphasizes that truncated or nonlinear reduced-order models require care to remain efficient and dynamically reliable.

  • 2. Closure models and stabilization: Truncating modes to energetic coherent structures can neglect fine-scale structures that still influence reduced-order-model dynamics and stability.Stabilization strategies include energy-based inner products, symmetry transformations, and least-squares Petrov-Galerkin projection.
  • 2. Closure models and stabilization: Nonlinear reduced-order models can retain high-dimensional online evaluations, motivating hyper-reduction and sparse-sampling techniques.Without acceleration, projected nonlinear dynamics may scale as O(n) despite a reduced state dimension r.
  • 1. Dynamical systems models: The survey applies modal analysis to study, model, and control canonical aerodynamic flows and discusses data-science developments for large, sparse, interpretable models.The outlook connects these developments with analysis of high-dimensional flows with complex dynamics.
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