Annu.
Rev.
Fluid Mech. 1993.25: 539-75
Copyrinht
0
1993
by
Annual
Reviews
Inc.
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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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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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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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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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