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Journal ArticleDOI

The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows

01 Jan 1993-Annual Review of Fluid Mechanics (Annual Reviews)-Vol. 25, Iss: 1, pp 539-575
TL;DR: The Navier-Stokes equations are well-known to be a good model for turbulence as discussed by the authors, and the results of well over a century of increasingly sophisticated experiments are available at our disposal.
Abstract: It has often been remarked that turbulence is a subject of great scientific and technological importance, and yet one of the least understood (e.g. McComb 1990). To an outsider this may seem strange, since the basic physical laws of fluid mechanics are well established, an excellent mathematical model is available in the Navier-Stokes equations, and the results of well over a century of increasingly sophisticated experiments are at our disposal. One major difficulty, of course, is that the governing equations are nonlinear and little is known about their solutions at high Reynolds number, even in simple geometries. Even mathematical questions as basic as existence and uniqueness are unsettled in three spatial dimensions (cf Temam 1988). A second problem, more important from the physical viewpoint, is that experiments and the available mathematical evidence all indicate that turbulence involves the interaction of many degrees of freedom over broad ranges of spatial and temporal scales. One of the problems of turbulence is to derive this complex picture from the simple laws of mass and momentum balance enshrined in the NavierStokes equations. It was to this that Ruelle & Takens (1971) contributed with their suggestion that turbulence might be a manifestation in physical

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Annu.
Rev.
Fluid Mech. 1993.25: 539-75
Copyrinht
0
1993
by
Annual
Reviews
Inc.
All
rights
reserved
THE PROPER ORTHOGONAL
DECOMPOSITION IN THE
ANALYSIS
OF
TURBULENT
FLOWS
Gal Berkooz, Philip Holmes, and John
L.
Lumley
Cornell University, Ithaca, New
York
14853
KEY
WORDS:
coherent structures, empirical eigenfunctions, modeling,
turbulence
1.
INTRODUCTION
1.1
The
Problems
of
Turbulence
It has often been remarked that turbulence
is
a subject of great scientific
and technological importance, and yet one of the least understood (e.g.
McComb
1990).
To
an outsider this may seem strange, since the basic
physical laws of fluid mechanics are well established, an excellent mathe-
matical model is available in the Navier-Stokes equations, and the results
of well over a century of increasingly sophisticated experiments are at our
disposal. One major difficulty,
of
course, is that the governing equations
are nonlinear and little is known about their solutions at high Reynolds
number, even in simple geometries. Even mathematical questions as basic
as existence and uniqueness are unsettled in three spatial dimensions (cf
Temam 1988).
A
second problem, more important from the physical
viewpoint, is that experiments and the available mathematical evidence all
indicate that turbulence involves the interaction of many degrees of free-
dom over broad ranges of spatial and temporal scales.
One of the problems of turbulence
is
to derive this complex picture from
the simple laws of mass and momentum balance enshrined in the Navier-
Stokes equations. It was to this that Ruelle
&
Takens (1971) contributed
with their suggestion that turbulence might be a manifestation in physical
539
0066-4189/93/0115-0539$02.00
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Annual Reviews
Annu. Rev. Fluid Mech. 1993.25:539-575. Downloaded from arjournals.annualreviews.org
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540
BERKOOZ, HOLMES & LUMLEY
space of a strange attractor in phase space. Since 1971 we have witnessed
great advances in dynamical-systems theory and many applications of it
to fluid mechanics, with, alas, mixed results in turbulence--despite the
attractive notion of using deterministic chaos in resolving the apparent
paradox of a deterministic model (Navier-Stokes) that exhibits apparently
random solutions. This is due not solely to the technical difficulties
involved: Proof of global existence and a finite-dimensional strange attrac-
tor for the 3-D equations in a general setting would be a great mathematical
achievement, but would probably be of little help to specific problems in,
say, turbomachinery. For a start, rigorous estimates of attractor dimension
(T6man 1988) indicate that any dynamical system which captures all
the relevant spatial scales will be of enormous dimension. Advances in such
areas will most probably nccessitate a dramatic reduction in complexity
by the removal of inessential degrees of freedom.
The first real evidence that this reduction in complexity might be possible
for fully developed turbulent flows came with the experimental discovery
of coherent structures around the outbreak of the second world war,
documented by J. T. C. Liu (1988). The existence of these structures
was probably first articulated by Liepmann (1952), and was thoroughly
exploited by Townsend (1956). Extensive experimental investigation did
not take place until after 1970, however (see Lumley 1989). Coherent
structures are organized spatial features which repeatedly appear (often in
flows dominated by local shear) and undergo a characteristic temporal life
cycle. The proper orthogonal decomposition, which forms the subject of
this review, offers a rational method for the extraction of such features.
Before we begin our discussion of it, a few more general observations on
turbulence studies are appropriate.
1.2
Experiments, Simulations, Analysis, and Understanding
In analytical studies of turbulence, two grand currents are clear: statistical
and deterministic. The former originates in the work of Reynolds (1894).
The latter is harder to pin down; linear stability theory is felt to have little
to do with turbulence. Nonlinear stability, however, and such things as
amplitude equations, definitely are relevant, so perhaps L. D. Landau
and J. T. Stuart should be credited with the beginnings of an analytical
nonstatistical approach. Lorenz’ work was certainly seminal. Over the past
twenty years a third stream has emerged and grown to a torrent which
threatens to carry everything in its path: computational fluid dynamics.
Both analytical approaches have drawbacks. Statistical methods, involv-
ing averaged quantities, immediately encounter closure problems (Monin
& Yaglom 1987), the resolution of which, even in sophisticated re-
normalization group theories (cf McComb 1990) usually requires use
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Annual Reviews
Annu. Rev. Fluid Mech. 1993.25:539-575. Downloaded from arjournals.annualreviews.org
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POD & TURBULENCE 541
empirical data (Tennekes & Lumley 1972). Nonetheless, they are intended
for and are used for fully developed turbulence. Analytical methods have
so far been unable to deal with the interaction of more than a few unstable
modes, usually in a weakly nonlinear context, and thus have been restricted
to studies of transition or pre-turbulence. Most of the dynamical systems
studies have been limited to this area. Computational fluid dynamics
bypasses the shortcomings of these methods by offering direct simulation
of the Navier-Stokes equations. However, unlike analysis, in which logical
deductions lead stepwise to an answer, simulation provides little under-
standing of the solutions it produces. It is more akin to an experimental
method, and no less valuable (or less confusing) for the immense quantity
of data it produces, especially at high spatial resolution.
Proper orthogonal decomposition (POD), while lacking the broad sweep
of the approaches mentioned above, nonetheless has something to offer
all three of these. 1. It is statistically based--extracting data from experi-
ments and simulations. 2. Its analytical foundations supply a clear under-
standing of its capabilities and limitations. 3. It permits the extraction,
from a turbulent field, of spatial and temporal structures judged essential
according to predetermined criteria and it provides a rigorous math-
ematical framework for their description. As such, it offers not only a tool
for the analysis and synthesis of data from experiment or simulation,
but also for the construction, from the Navier-Stokes equations, of low-
dimensional dynamical models for the interaction of these essential struc-
tures. Thus, coming full circle, we have a statistical technique that con-
tributes to deterministic dynamical analysis.
In Sections 3 and 4 we review applications of the proper orthogonal
decomposition, after developing its key features in Section 2. The latter is
necessarily mathematical in style and while space limitations preclude a
complete treatment, we include some of the new and lesser known results.
Proofs are omitted; see Berkooz (1991b, Chapter 2) for details. Section
explores relations to some other techniques used in turbulence studies
and Section 6 contains a concluding discussion. The remainder of this
introductory section contains an historical survey.
1.3 The Proper Orthoyonal Decomposition
The proper orthogonal decomposition is a procedure for extracting a basis
for a modal decomposition from an ensemble of signals. Its power lies in
the mathematical properties that suggest that it is the preferred basis to
use in many circumstances. The POD was introduced in the context of
turbulence by Lumley (1967, cf 1981). In other disciplines the same pro-
cedure goes by the names Karhunen-Lo~ve decomposition or principal
components analysis and it seems to have been independently rediscovered
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Annual Reviews
Annu. Rev. Fluid Mech. 1993.25:539-575. Downloaded from arjournals.annualreviews.org
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542
BERKOOZ~ HOLMES & LUMLEY
several times, cf Sirovich (1987). According to Lumley, quoting A.
Yaglom (personal communication), the POD was suggcsted independently
by several scientists, e.g. Kosambi (1943), Lo6ve (1945), Karhunen (1946),
Pougachev (1953), and Obukhov (1954). For use of the POD in other
disciplines see: Papoulis (1965)--random variables; Rosenfeld & Kak
(1982)--image processing; Algazi & Sakrison (1969)--signal analysis;
Andrews et al (1967)--data compression; Preisendorfer (1988)--ocean-
ography; and Gay & Ray (1986, 1988)--process identification and control
in chemical engineering. Introductory discussions of the method in the
context of fluid mechanical problems can also be found in Sirovich (1987,
1989, 1990) and Holmes (1990).
The attractiveness of the POD lies in the fact that it is a linear procedure.
The mathematical theory behind it is the spectral theory of compact, self-
adjoint operators. This robustness makes it a safe haven in the intimidating
world of nonlinearity; although this may not do the physical violence of
linearization methods, the linear nature of the POD is the source of its
limitations, as will emerge from what follows. Howcvcr, it should be made
clear that the POD makes no assumptions about the linearity of the
problem to which it is applied. In this respect it is as blind as Fourier
analysis, and as general.
2. FUNDAMENTALS OF THE PROPER
ORTHOGONAL DECOMPOSITION
2.1 The Eigenvalue Problem
For simplicity we introduce the proper orthogonal decomposition in the
context of scalar fields: (complex-valued) functions defined on a interval
f~ of the real line. The interval might be the width of the flow, or the
computational domain. We restrict ourselves to the space of functions
which are square integrable (or, in physical terms, fields with finite kinetic
energy) on this interval. We need an inner product (f, 9) ~nf(x)9*(x)dx,
and a norm Ilfll = (f, f)~/2. We start with an ensemble of realizations of
the function u(x), and ask which single (deterministic) function is most
similar to the members of u(x) on average? We need an averaging operation
(), which may be a time, space, ensemble, or phase average. We suppose
that the probabilistic structure of the ensemble is such that the average
and limiting operations can be interchanged (cf Lumley 1971). Mathema-
tically, the notion of "most similar" corresponds to seeking a function 4~
such that
max (l(u, ~)12)/(ff, ~O) = (l(u, ~b)l z)/(tk, (2.1)
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Annual Reviews
Annu. Rev. Fluid Mech. 1993.25:539-575. Downloaded from arjournals.annualreviews.org
by LEICESTER UNIVERSITY LIBRARY on 12/30/07. For personal use only.

Annual Reviews
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Annu. Rev. Fluid Mech. 1993.25:539-575. Downloaded from arjournals.annualreviews.org
by LEICESTER UNIVERSITY LIBRARY on 12/30/07. For personal use only.

Citations
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Abstract: The description of coherent features of fluid flow is essential to our understanding of fluid-dynamical and transport processes. A method is introduced that is able to extract dynamic information from flow fields that are either generated by a (direct) numerical simulation or visualized/measured in a physical experiment. The extracted dynamic modes, which can be interpreted as a generalization of global stability modes, can be used to describe the underlying physical mechanisms captured in the data sequence or to project large-scale problems onto a dynamical system of significantly fewer degrees of freedom. The concentration on subdomains of the flow field where relevant dynamics is expected allows the dissection of a complex flow into regions of localized instability phenomena and further illustrates the flexibility of the method, as does the description of the dynamics within a spatial framework. Demonstrations of the method are presented consisting of a plane channel flow, flow over a two-dimensional cavity, wake flow behind a flexible membrane and a jet passing between two cylinders.

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Abstract: Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.

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Cites background or methods from "The Proper Orthogonal Decomposition..."

  • ...2, it is possible to obtain a lowrank approximation using dimensionality reduction techniques, such as the proper orthogonal decomposition (POD) (35, 37)....

    [...]

  • ...The first two most energetic POD modes capture a significant portion of the energy, and steady-state vortex shedding is a limit cycle in these coordinates....

    [...]

  • ...The POD (35, 37), provides a low-rank basis resulting in a hierarchy of orthonormal modes that, when truncated, capture the most energy of the original system for the given rank truncation....

    [...]

Journal ArticleDOI
TL;DR: In this article, a review of scale-invariance properties of high-Reynolds-number turbulence in the inertial range is presented, focusing on dynamic and similarity subgrid models and evaluating how well these models reproduce the true impact of the small scales on large scale physics and how they perform in numerical simulations.
Abstract: ▪ Abstract Relationships between small and large scales of motion in turbulent flows are of much interest in large-eddy simulation of turbulence, in which small scales are not explicitly resolved and must be modeled. This paper reviews models that are based on scale-invariance properties of high-Reynolds-number turbulence in the inertial range. The review starts with the Smagorinsky model, but the focus is on dynamic and similarity subgrid models and on evaluating how well these models reproduce the true impact of the small scales on large-scale physics and how they perform in numerical simulations. Various criteria to evaluate the model performance are discussed, including the so-called a posteriori and a priori studies based on direct numerical simulation and experimental data. Issues are addressed mainly in the context of canonical, incompressible flows, but extensions to scalar-transport, compressible, and reacting flows are also mentioned. Other recent modeling approaches are briefly introduced.

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TL;DR: 5. M. Green, J. Schwarz, and E. Witten, Superstring theory, and An interpretation of classical Yang-Mills theory, Cambridge Univ.
Abstract: 5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

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TL;DR: The intent of this document is to provide an introduction to modal analysis that is accessible to the larger fluid dynamics community and presents a brief overview of several of the well-established techniques.
Abstract: 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 flowfield, 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 [1–8]. For a nonexpert, 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.

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Cites background from "The Proper Orthogonal Decomposition..."

  • ...Excellent reviews on the POD can be found in [1,45], and chapter 3 of [46]....

    [...]

References
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TL;DR: This chapter discusses the concept of a Random Variable, the meaning of Probability, and the axioms of probability in terms of Markov Chains and Queueing Theory.
Abstract: Part 1 Probability and Random Variables 1 The Meaning of Probability 2 The Axioms of Probability 3 Repeated Trials 4 The Concept of a Random Variable 5 Functions of One Random Variable 6 Two Random Variables 7 Sequences of Random Variables 8 Statistics Part 2 Stochastic Processes 9 General Concepts 10 Random Walk and Other Applications 11 Spectral Representation 12 Spectral Estimation 13 Mean Square Estimation 14 Entropy 15 Markov Chains 16 Markov Processes and Queueing Theory

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TL;DR: A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.
Abstract: Office hours: MWF, immediately after class or early afternoon (time TBA). We will cover the mathematical foundations of probability theory. The basic terminology and concepts of probability theory include: random experiments, sample or outcome spaces (discrete and continuous case), events and their algebra, probability measures, conditional probability A First Course in Probability (8th ed.) by S. Ross. This is a lively text that covers the basic ideas of probability theory including those needed in statistics. Theoretical concepts are introduced via interesting concrete examples. In 394 I will begin my lectures with the basics of probability theory in Chapter 2. However, your first assignment is to review Chapter 1, which treats elementary counting methods. They are used in applications in Chapter 2. I expect to cover Chapters 2-5 plus portions of 6 and 7. You are encouraged to read ahead. In lectures I will not be able to cover every topic and example in Ross, and conversely, I may cover some topics/examples in lectures that are not treated in Ross. You will be responsible for all material in my lectures, assigned reading, and homework, including supplementary handouts if any.

10,221 citations


"The Proper Orthogonal Decomposition..." refers background or methods in this paper

  • ...[See the formula for the conditional expectation of joint normal variables in Feller (1957).] Using (5....

    [...]

  • ...Armbruster et al’s (1988) proof of structural stability of such heteroclinic cycles in O(2)-symmetric systems provides a mathematical foundation and does much to explain their robustness and persistence over a range of parameters and with different truncations (cf Armbruster et al 1989, Holmes 1991)....

    [...]

  • ...In this section we comment on the connection between the POD and certain other analysis techniques. We start by describing the connection between the POD and linear stochastic estimation, as applied by Adrian and coworkers in Adrian (1979), Adrian & Moin (1988), Adrian et (1987), and Moin et al (1987)....

    [...]

  • ...The best studied low-dimensional model is that developed by Aubry et al (1988) for the wall region of a turbulent boundary layer....

    [...]

  • ...of coherent structures around the outbreak of the second world war, documented by J. T. C. Liu (1988). The existence of these structures was probably first articulated by Liepmann (1952), and was thoroughly exploited by Townsend (1956)....

    [...]

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TL;DR: In this paper, the authors used the representations of the noise currents given in Section 2.8 to derive some statistical properties of I(t) and its zeros and maxima.
Abstract: In this section we use the representations of the noise currents given in section 2.8 to derive some statistical properties of I(t). The first six sections are concerned with the probability distribution of I(t) and of its zeros and maxima. Sections 3.7 and 3.8 are concerned with the statistical properties of the envelope of I(t). Fluctuations of integrals involving I2(t) are discussed in section 3.9. The probability distribution of a sine wave plus a noise current is given in 3.10 and in 3.11 an alternative method of deriving the results of Part III is mentioned. Prof. Uhlenbeck has pointed out that much of the material in this Part is closely connected with the theory of Markoff processes. Also S. Chandrasekhar has written a review of a class of physical problems which is related, in a general way, to the present subject.22

5,806 citations


"The Proper Orthogonal Decomposition..." refers methods in this paper

  • ...In the following treatment, we adapt Lumley's application of the shot-noise decomposition (Rice 1944; Lumley 1971, 1981)....

    [...]

Journal ArticleDOI

5,359 citations


"The Proper Orthogonal Decomposition..." refers background or methods in this paper

  • ...The additional symmetry imposed on the ensemble of flows by artificial addition of images of flows under symmetry group elements, as advocated by Sirovich (1987), may therefore obscure the true nature a particular system....

    [...]

  • ...This approach has been advocated by Sirovich (1987) and applied in many studies (e.g. Sirovich & Park 1990)....

    [...]

  • ...XnE MZTHOD OF S}qA]~S~OXS This method was proposed by Sirovich (1987)....

    [...]

  • ...Introductory discussions of the method in the context of fluid mechanical problems can also be found in Sirovich (1987) (1989) (1990) and Holmes (1990)....

    [...]

  • ...Berkooz (1991b) gives an argument for the equivalence of the method of snapshots to the original formulation of the eigenvalue problem, as well as a linear independence condition omitted by Sirovich (1987)....

    [...]