Higher order dynamic mode decomposition to identify
and extrapolate flow patterns
Soledad Le Clainche
a)
and Jose´ M. Vega
E.T.S.I. Aerona´utica y del Espacio, Universidad Polite´cnica de Madrid, 28040 Madrid, Spain
This article shows the capability of using a higher order dynamic mode decomposition (HODMD)
algorithm both to identify flow patterns and to extrapolate a transient solution to the attractor region.
Numerical simulations are carried out for the three-dimensional flow around a circular cylinder, and
both standard dynamic mode decomposition (DMD) and higher order DMD are applied to the non-
converged solution. The good performance of HODMD is proved, showing that this method guesses the
converged flow patterns from numerical simulations in the transitional region. The solution obtained
can be extrapolated to the attractor region. This fact sheds light on the capability of finding real flow
patterns in complex flows and, simultaneously, reducing the computational cost of the numerical
simulations or the required quantity of data collected in experiments. Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4997206]
I.
INTRODUCTION
Dynamic mode decomposition (DMD) is a technique
introduced by Schmid
23
that uses the Koopman linear oper-
ator
11
to calculate Fourier-like expansions for non-linear
dynamics. The strength of this technique mainly lies in its capa-
bility to reconstruct the original data set analyzed by means
of identifying its flow dynamics, which also allows to study
its main spatio-temporal flow patterns and physical insight.
Therefore, with such aim, DMD is applied to a set of K time
equispaced spatio-temporal snapshots relying in the follow-
ing Koopman assumption, which relates each snapshot v
k+1
(calculated at the time instant t
k +1
) with the previous snapshot
v
k
(calculated at time instant t
k
) by means of the Koopman
operator R, as
v
k+1
= R v
k
for k = 1, . . . , K − 1.
(1)
Based on this idea, after some calculations (detailed
below), it is possible to reconstruct the set of K time-
equispaced data in the sampled time interval t
1
≤ t ≤ t
1
+ T
using the following spatio-temporal expansion of modes:
M
v(x, t)
�
v
DMD
(x, t)
≡
\"1
a
m
u
m
(x) e
(δ
m
+iω
m
)t
m
=
1
for t
1
≤ t ≤ t
1
+ T ,
(2)
where the spatial fields u
m
(x) are known as DMD modes, and
a
m
, δ
m
, and ω
m
are their corresponding associated amplitudes,
growth rates, and frequencies. If the time interval (t
1
, t
1
+ T )
in which the data are first calculated, and next reconstructed, is
substituted by any interval more advanced in time (t
r
, t
r
+ T ),
with t
r
»
t
1
, the same expansion can be used to extrapolate
the original data to the attractor region, where the dynamics
are permanent δ
m
= 0 (saturated flow).
a)
Author to whom correspondence should be addressed: soledad.leclainche@
upm.es
The spatial and temporal complexity and dimension are
also relevant parameters in the previous expansion (2). On the
one hand, the number of expansion terms, M, can be referred
to as the spectral or temporal complexity, while the temporal
dimension, denoted by K, is dependent on the number of data
collected in time; thus the length of the time interval t
1
≤ t
≤ t
1
+ T , where T = K
·
∆t (time-equispaced data). On the other
hand, the length of the spatial vector x determines the spatial
dimension J, while the spatial complexity is determined by
the dimension of the subspace generated by the DMD modes,
as
N = dim (span{u
1
, . . . , u
M
}) ≤ min{M, J }. (3)
If the spatial and temporal complexities are equal, N
= M, conditions (1) and (2) are exact, meaning that the perfor-
mance of the standard DMD is optimal. The good performance
of DMD has been evidenced several times in the literature
either to study flow dynamics in numerical and experimen-
tal data
12,13,24
or to compute the global linear modes in linear
flows.
5,7,20,22
However, when the spatial complexity is smaller than
the spectral complexity, N < M, the previous assumption is
not valid. Instead, it is necessary to introduce a higher order
dynamic mode decomposition (HODMD) approximation or
DMD-d algorithm. This higher order technique has been
recently introduced by Le Clainche and Vega,
16
as an exten-
sion of the classical DMD that is capable of providing highly
accurate results in cases in which the performance of the clas-
sical DMD is deteriorated or even fails. HODMD mixes the
ideas behind classical DMD with Takens’ delay embedding
theorem,
25
leading to a higher order Koopman assumption
that uses time-lagged snapshots, as
v
k+d
=
R
1
v
k
+
R
2
v
k+1
+
· · ·
+
R
d
v
k+d
−
1
for k = 1, . . . , K − d, (4)
and is used to calculate the DMD expansion proposed in
Eq. (2). Let us note that d ≥ 1 is tunable and, when d = 1
.
1
(i.e., applying DMD-1), this assumption exactly matches the
standard Koopman assumption presented in Eq. (1). In this
way, it is possible to relate DMD-d algorithm or HODMD
with classical DMD,
23
which in this article is defined as
DMD-1.
While the temporal complexity M is only dependent on
the (spectral) complexity of the real data (flow dynamics), both
the spatial and temporal dimensions, J and K, are dependent of
the type of data analyzed. For example, if DMD analysis is per-
formed to study the results of a three-dimensional numerical
simulation, the spatial dimension J is determined by the total
number of grid points defining the computational domain, as
J = n
x
× n
y
× n
z
, while the number of K equispaced temporal
data collected is a choice of the user, more related to the time
step of the numerical simulation and the computer’s storage
memory. As expected, the spatial dimension is decreased in the
case of two-dimensional numerical simulations (J = n
x
× n
y
)
and may be drastically decreased in the case of experiments.
For example, particle image velocimetry (PIV) experimental
measurements are usually performed on a small plane in which
the spatial dimension is J = n
x
× n
y
(smaller than in a numer-
ical simulation). Similarly, in hot wire (HW) measurements,
the spatial dimension J is equal to the number of HW probes
used in the experiment. Also, the temporal dimension K may
be determined by the type of experiment performed, since in
complex experiments, the number of data collected is restricted
by some external factors (i.e., wall confined flow with thermal
effects
15
).
A different issue is the calculation of the spatial complex-
ity N, in which, naturally, the spatial dimension J of the original
data and their temporal complexity M play a highly essen-
tial role [N ≤ min{M, J }, see Eq. (3)]. However, some other
external factors, such as noise or transient decaying dynamics,
may also affect this variable. Thus, it is necessary to analyze
carefully the parameter N and the sensitive factors that may
contribute to alter it, leading to a spatial complexity that is
smaller than the spectral complexity. In such cases, HODMD
must be used instead of the classical algorithm. It is possible
to mention three natural cases with larger spectral complexity
than spatial complexity, N < M:
(i)
When the spectral complexity M is very large but the
number J of collected data is limited, either due to
(a)
the number of involved differential equations (in
numerical simulations) or due to (b) the number of mea-
surements points coming from an experiment: Since
the spatial dimension J is small, N is small. Case (a) is
commonly found in non-linear dynamical systems and
becomes determined by the definition of the problem.
For example, in a periodic solution of the Lorenz equa-
tion or in a quasi-periodic solution of the Ginzburg-
Landau equation, one finds that N < M (Le Clainche
and Vega
16
). Case (b) is also usually found in sev-
eral types of experiments used for industrial purposes
such as the flight test
18
or in the analysis of magnetic
resonances.
(ii)
When the data are noisy: This is a case usually found in
experimental data. Even in simple flow dynamic cases
in which ideally N = M, the noise found in the spa-
tial structure defining the data needs to be calculated
first in order to be cleaned next. So, some of the spa-
tial modes are used to calculate this noise, leading to
the modification of the real calculations of spectral and
spatial complexities and, consequently, producing that
the effective value of N be smaller than M (Le Clainche
et al.
17
).
(iii)
When the data analyzed come from a transitory stage:
This case may be represented by the Stuart-Landau
equation
1
and is a case usually found in non-converged
numerical simulations, or in the transitory stage of an
experiment. Even though the converged dynamics may
be represented by an equal value of the spatial and tem-
poral complexities (N = M), when DMD analysis is
performed in a transitory region, some of the dynam-
ics describing the flow are decaying (δ
m
< 0). Thus,
as in the previous item, some of the spatial modes are
used to calculate these decaying modes, modifying the
actual values of the spectral and spatial complexities
and, consequently, leading to N < M.
Cases (i) and (ii) are related to interpolation (reconstruc-
tion of the original data) and are already explained in detail
in the literature.
16,17
Case (iii), which is more related to pat-
tern identification and extrapolation, is still an open topic that
will be addressed in this article. If it were possible to calculate
the main dynamics describing the flow in the transient region
and to construct the DMD expansion proposed in Eq. (2),
it would be (in turn) possible to extrapolate the solution to
the attractor region by only retaining the non-decaying DMD
modes.
In this article, the transient regime of the three-
dimensional flow around a circular cylinder is analyzed using
HODMD, with the aim to show the good performance of
this technique both to calculate the real flow patterns in non-
converged solutions and its high potential to be used as a
reduced order model for extrapolation. The latter is useful
either to reduce the computational cost in numerical simu-
lations or to reduce the time and the number of data col-
lected in experimental measurements, which sometimes, due
to their high complexity, are subject to external, out of control
restrictions.
15
The article is organized as follows. The above men-
tioned extension of standard DMD, called HODMD, is briefly
described in Sec. II. The cylinder wake test problem will be
considered in Sec. III, where the numerical method to simulate
the flow around the cylinder will also be described. The main
results of the paper are given in Sec. IV, where both classi-
cal DMD and higher order DMD will be applied to guess the
dynamics on the attractor from results computed in the tran-
sient region, comparing both the relevant involved frequencies
and the flow patterns. The paper ends with some concluding
remarks, in Sec. V.
II.
HIGHER ORDER DYNAMIC MODE
DECOMPOSITION ALGORITHM
The algorithm presented in this section is explained more
in detail in Ref. 16. As anticipated, we consider a data set of
K equispaced snapshots, collected in a snapshot matrix V
K
(whose columns are the snapshots, from the first to the Kth
V
K
1
1
1
R
1
snapshot) in the following way:
1
= [v
1
, . . . , v
k
],
(5)
where v
k
is the vector representing the spatio-temporal data
(snapshot) collected at time instant t
k
. The HODMD algo-
rithm is applied to this snapshot matrix to study the main flow
dynamics and to represent the spatio-temporal snapshot as the
following DMD expansion [same as in Eq. (2)]:
The remaining of this step, in turn, considers two cases,
depending on the value of d:
(a)
If d = 1: DMD-1 algorithm.
The standard higher order Koopman assumption
(10) is used in the dimension reduced snapshot as
vˆ
k+1
� R
ˆ
1
vˆ
k
, for k = 1, . . . , K − 1.
(11)
In the matrix form,
K
K
−
1
M
v(x, t)
�
v
DMD
(x, t)
≡
\"1
a
m
u
m
(x) e
(δ
m
+iω
m
)t
m
=
1
for t
1
≤ t ≤ t
1
+ T .
(6)
HODMD, using the DMD-d algorithm, considers two
V
ˆ
2
� R
ˆ
V
ˆ
1
,
with R
ˆ
= U
T
RU. (12)
Let us note that this equation is similar to the classical
Koopman assumption presented in Eq. (3), but applied
to the dimension reduced snapshot. Thus, the DMD-1
algorithm is similar to classical DMD.
23
main steps:
1.
Singular value decomposition (SVD) is applied to the
snapshot matrix V
K
, leading to a representation of the
spatio-temporal data as an expansion of spatial and tem-
poral modes, U and T, respectively, and singular values
Σ, written in a matrix form as
Then, SVD is applied to the snapshot matrix V
ˆ
K −1
to perform its pseudo-inverse, and the linear operator
R
ˆ
is calculated. This operator contains the dynamics
of the system. So, its eigenvectors and eigenvalues
yield the modes u
m
(x), growth rates δ
m
, and fre-
quencies ω
m
appearing in the DMD expansion (6).
The mode amplitudes a
m
are calculated upon least
V
K
T
K
K
1
�
U Σ T
,
≡
U V
ˆ
1
, with
V
ˆ
1
=
Σ
T
T
(
≡
U
T
V
K
).
squares fitting. A second tolerance ε is set in order to
(7)
A dimension reduction of the spatial terms is carried
out in this step, retaining N (=spatial complexity) spa-
tial modes. The value of N is selected to ensure that the
SVD approximation satisfies a certain root mean square
(RMS) error (selected by the user), as
retain the M most relevant DMD modes, exhibiting
the largest amplitudes a
m
. This parameter will deter-
mine the spectral complexity of the DMD expansion
(2), calculated for a specific accuracy ε (tunable by
the user).
(b)
If d > 1: DMD-d algorithm.
EE (N ) ≡
2
N +1
σ
2
+
· · ·
+ σ
2
2
≤ ε
1
,
(8)
The higher order Koopman assumption is used as
in Eq. (10). This equation is written as the following
1
+
· · ·
+ σ
R
where R ≤ min{J, K } is the rank of the snapshot matrix,
σ
i
are the singular values (contained in matrix Σ) sorted
in decreasing order, and ε
1
is a parameter tuneable by the
user to control the error of the approximation. In the case
modified Koopman equation:
v˜
k+1
� R
˜
v˜
k
, (13)
where the modified snapshots v˜
k
and the modified
Koopman matrix R
˜
are
of noisy data or unconverged solutions, this parameter
could be compared to the uncertainty contained in the
data.
17
Equation (7) implies that the size of the snapshots
v˜
k
≡
vˆ
k
vˆ
k+1
. . .
,
0
I
0
. . .
0 0
0 0 I . . .
0
0
R
˜
≡
. . . . . . . . . . . . . . . . . .
.
0 0 0 . . .
I
0
v
ˆ
k+d
−
2
v
k
is reduced to vˆ
k
in the following way:
v
k
= U vˆ
k
.
(9)
v
ˆ
k+d
−
1
R
ˆ
1
R
ˆ
2
R
ˆ
3
. . .
R
ˆ
d
−
1
R
ˆ
d
(14)
When the data collected are too noisy, as in the case of an
experiment, a higher order singular value decomposition
(HOSVD)
4,10,26
is used instead of SVD at this step.
17
The
HOSVD algorithm performs a SVD in each one of the
spatial directions. Thus, in this way, it is possible to better
clean the data from noise or to remove non-permanent
modes (δ
m
< 0), which somehow could be considered as
noise (snapshots reduction).
2. The DMD-d algorithm (with d ≥ 1) is applied to the tem-
poral modes V
ˆ
K
(the reduced snapshot matrix) obtained
in the previous step. Therefore, the higher order Koopman
assumption presented in Eq. (4) is used in the reduced
snapshots vˆ
k
(contained in a reduced linear manifold)
as
v
ˆ
k+d
�
R
ˆ
1
v
ˆ
k
+
R
ˆ
2
v
ˆ
k+1
+
· · ·
+
R
ˆ
d
v
ˆ
k+d
−
1
for k = 1,
· · ·
, K − d.
(10)
The modified Koopman operator R
-
is calculated as
in the previous case, and its eigenvectors and eigen-
values yield the modes u
m
(x), growth rates δ
m
, and
frequencies ω
m
of the DMD expansion (6). Again, the
mode amplitudes a
m
are calculated upon least squares
fitting, and a second tolerance ε is set in order to
retain the M most relevant DMD modes, exhibiting
the largest amplitudes a
m
. This parameter will deter-
mine the spectral complexity of the DMD expansion
(2), calculated for a specific accuracy ε (tunable by
the user).
The benefit of this method lies in the resolution of
the eigenvalue problem for the modified Koopman
matrix R
˜
, which enforces the dynamic contained in
the delayed snapshot matrix to evoke the same solu-
tion. So, either the noise or non-permanent devices are
naturally removed from the solution.
σ
z
III.
THE THREE-DIMENSIONAL CYLINDER WAKE
Let us now consider a classical fluid mechanics
case, namely, the incompressible three-dimensional cylinder
wake.
28
For large aspect ratio cylinders, the spatio-temporal
structure of the wake only depends on the Reynolds num-
ber, defined as Re = U
∞
D/ν, where U
∞
is the incoming free
stream velocity, D is the cylinder diameter, and ν is the kine-
matic viscosity. At small Re, the wake is two-dimensional and
steady, but at Re � 46, it exhibits a primary Hopf bifurca-
tion,
9,21
which still produces a two-dimensional but unsteady
(in fact, periodic) von Karman vortex street flow. This peri-
odic flow remains orbitally stable up to Re � 190, where it
suffers a secondary bifurcation (Floquet multiplier = 1) to
three-dimensional periodic oblique waves, as repeatedly found
experimentally in large aspect ratio containers.
8,27
This thresh-
old was (more precisely) calculated numerically via Floquet
analysis by Barkley and Henderson,
2
as (critical Reynolds
and Henderson.
2
In order to reduce the computational cost, the
three-dimensional numerical simulation was initialized with
this two-dimensional converged solution. The boundary con-
ditions (b.c.) set in the cylinder surface was no-slip. In the inlet
surface, it was enforced uniform flow (u
x
= U
∞
= 1, u
y
= 0),
while at the top and bottom parts of the domain and the out-
let surface, the b.c. used were standard outflow conditions (p
= 0, ∂
x
u
x
= 0, ∂
x
u
y
= 0).
14
A second test case was carried
out setting uniform b.c. (as in the inlet surface) in the top and
bottom surfaces, as in Barkley and Henderson
2
and the results
obtained were similar. This fact proves that the boundaries of
the computational domain are sufficiently far from the cylinder
studied, minimizing their effects over the solution presented.
Finally, in the three-dimensional expansion in the spanwise
direction, the b.c. were set to periodic (Fourier expansion),
with a period L
z
= 4, which is consistent with the expected
period (15) for this value of the Reynolds number. The time
☞
3
number) Re
c
= 188.5 ± 1.0, by imposing periodicity in the step in the numerical simulations was set to 5
·
10
, in order
span direction, with wavenumber β (associated spatial period
L
z
= 2π/ β), whose critical value at Re = Re
c
was found to
to maintain the Courant-Friedrichs-Lewy condition (CFL) in a
value smaller than 1. The numerical code was evolved in time
be L
c
= 3.96 ± 0.02. For Re ≥ Re
c
, Barkley and Hender-
during 1325 time units, saving snapshots (each containing the
1132 end-points of the 1132 elements) every ∆t = 0.5 time
son estimated the frequency of the spanwise pattern as that
associated with the most unstable mode, known as mode A,
which has been studied in more detail by Blackburn et al.
3
In particular, for Re = 220 (the case that will be consid-
ered below), the experimental spatial period (estimated by
Barkley and Henderson
2
from the experimental visualiza-
tions of Williamson
27
) and temporal frequency (measured by
Williamson
27
),
L
z
= 4.01, f
exp
� 0.185 (or ω
exp
= 2π/f
exp
� 1.1624),
(15)
can be taken as reference. For this value of L
z
, the Floquet
analysis in Barkley and Henderson
2
yielded the frequency
f � 0.20 Hz (or ω � 1.2566), which compares reason-
ably well with its experimental counterpart [see Eq. (15)].
With this background in mind, we shall use the HODMD
method to analyze the cylinder at Re = 220 using numeri-
cally generated data via a computational fluid dynamics (CFD)
tool.
The numerical code used is Nek5000,
19
an open source
CFD code that uses spectral elements as spatial discretization.
The three-dimensional incompressible Navier-Stokes equa-
tions has been solved, in a Cartesian reference frame aligned
with the streamwise, vertical, and spanwise directions, x, y, and
z, respectively. The dimensions of the computational domain
coincide with those used in Barkley and Henderson,
2
namely,
x∈ [−15, 25], y∈ [−20, 20], and z∈ [0, 4], respectively; the
axis of the cylinder (whose nondimensional diameter is D
= 1) is the straight line x = y = 0. The mesh contains 1132 ele-
ments and was generated using the open source code Gmsh.
6
Finally, each element was discretized using Gauss-Lobatto-
Legendre points of order p = 5. A grid convergence study has
been performed first, in a two dimensional mesh, in order to
set the polynomial order to 5. Then, FFT has been applied
to several points in the computational domain, finding in
all cases that the dominant frequency was f
2d
� 0.20 (ω
2d
= 1.2566), similar to the frequency f calculated by Barkley
units, meaning that we have stored a total number of 2650
snapshots. However, discarding the transient 0 ≤ t ≤ 575,
only the last 1500 snapshots (750 time units) will be used
below. Because the considered value of the Reynolds num-
ber is not too far from its threshold value for the transition to
three-dimensional dynamics, the considered snapshots are not
expected to be fully converged to the final periodic attractor,
which is found after evolving the code in time during 2900 time
units. The dominant frequency calculated in the attractor was
ω
num
= 1.1711, comparable to the spanwise velocity obtained
in the experiment (15). Figure 1 shows the time evolution of
the residuals in the three-dimensional numerical simulations
in a representative point of the computational domain (cylin-
der wake). As seen, the fact that the Navier-Stokes equations
are initialized with the converged solution of the two dimen-
sional case is reflected in this velocity evolution, in which it is
possible to identify the growth of the spanwise mode.
FIG. 1. Evolution of spanwise velocity in the three-dimensional numerical
simulations, taken in the cylinder wake at (x, y, z) = (2, 0, 0) [the cylinder is
located at (0, 0, z)].
IV.
PATTERN FORMATION AND EXTRAPOLATION
TO THE ATTRACTOR REGION
We perform various sets of HODMD computations,
restricting the data to the plane y = 0 and using the three veloc-
ity components [to somewhat mimic stereo particle image
velocimetry (PIV) data] and considering both the near-field
and far-field in the streamwise direction, namely,
y = 0-near-field : 0.5 ≤ x ≤ 4, y = 0, 0 ≤ z ≤ 4, (16)
y = 0-far-field : 4 ≤ x ≤ 25, y = 0, 0 ≤ z ≤ 4, (17)
and two sets of tolerances, namely,
rough: ε
1
= 10
−2
, ε = 2
·
10
−2
more precise: ε
1
= 10
−4
, ε = 3
·
10
−3
.
(18)
Additional computations have been performed using (a) only
the in-plane velocities in the y = 0-near field data and (b) the
whole three-dimensional data in the near field. As expected,
those results (omitted here for the sake of brevity) are slightly
worse in case (a) and slightly better in case (b), compared with
the results that will be presented in this section.
In all applications below, the extrapolation properties of
both DMD-1 and DMD-d will be evidenced. The analyses have
been performed in two different sets of data, using the rough
and more precise tolerances. In the first case, we collect a set
of 500 snapshots, starting at time 575, sufficiently far from
the attractor (time � 2900) to prove the benefits of extrapolat-
ing instead of integrating Navier-Stokes equations, in terms of
reduction of computational time. In the second case, we collect
a set of 1000 snapshots, starting at time 875 in order to show
that, when the distance to the attractor is reduced, DMD-d is
capable of giving a better prediction of the flow patterns of the
attractor.
Let us start with the first application, in which both DMD-
1 and DMD-250 are applied to the set of 500 snapshots. The
value d = 250 has been selected after some calibration (looking
for consistency and robustness in the results presented
17
). As in
the remaining applications of the HODMD method described
above, this value of d can be decreased by 50 without signif-
icant changes in the performance of the method. In this first
application of the method, the rough tolerances are used in the
near field. The plots of the damping rates and mode amplitudes
vs. the retained frequencies are given in Fig. 2. The following
can be observed in these plots:
•
With these tolerances, DMD-250 identifies two kinds
of modes: transient (δ < 0) and permanent (δ � 0)
modes. Since the ω = 0 (mean flow) and ten pairs of
complex conjugate modes (plotted with blue circles)
exhibit a damping rate of ∼2
·
10
☞ 3
and thus are use-
ful for extrapolation to estimate the periodic attractor,
these modes will be called the permanent modes below.
There is a small gap that separates permanent modes
from the remaining 19 pairs of modes to be called
transient modes below (candidates to be eliminated in
extrapolation), plotted with blue crosses.
Besides, the permanent modes include the experimen-
tally measured fundamental frequency with a relative
error ∼10
−3
, and their second, third, and fourth harmon-
ics. In addition, somehow unexpectedly, DMD-250
also identifies a 1/3-subharmonic of the experimen-
tal frequency. Moreover, the exact commensurability
relations for these harmonics and subharmonics hold
within a maximum relative error of ∼7
·
10
☞ 3
.
•
The performance of DMD-1 could be seen as qualita-
tively similar, since it produces a similar gap between
permanent and transient modes. However, DMD-1 only
identifies the mean flow and four pairs of complex con-
jugate modes (of 240 modes calculated in total) includ-
ing the experimental frequency (with a relative error of
∼5
·
10
☞3
), its harmonic, and the 1/3-subharmonic mode
previously presented.
The results above raise the question on whether the 1/3-
subharmonic identified above by DMD-250 is the relevant
fundamental frequency (though with a much smaller mode
amplitude than the experimentally detected frequency), and
thus, strictly speaking, the period of the attractor is three times
larger than assumed. This issue is first analyzed considering
the precise tolerances in Eq. (18), which gives the results plot-
ted in Fig. 3 (a second analysis will be done below by using
data from the fully converged attractor). As can be seen in the
following:
•
Though the gap between permanent and transient
modes is similar than in Fig. 2, the amount (namely,
15) of permanent modes identified by DMD-250
is larger, their growth rate is smaller (∼10
☞3
), and
the amount of transient modes is larger. DMD-250
identifies as permanent modes the mean flow and
7 pairs of complex conjugate modes, which include
the “fundamental frequency” (with a relative error of
∼1.9
·
10
☞3
) plus two harmonics of the fundamental
FIG. 2. The y = 0-near field defined in
Eq. (16) with the rough tolerances in
Eq. (18), considering the three veloc-
ity components: damping rates (left)
and mode amplitudes (right) vs. the
retained frequencies obtained via DMD-
250 (blue symbols) and DMD-1 (red
symbols) applied in a set of 500 snap-
shots starting at time 575. Those modes
exhibiting the smaller damping rates
(δ � 0) are plotted with circles, while
the remaining modes are plotted with
crosses. The experimental frequencies
±ω
exp
[see Eq. (15)] are indicated with
vertical lines.
![FIG. 12. The mode associated with the mean flow [top, plotting, from left to right, � (ux ), � (uy ), and � (uz )] and the mode associated with the experimental frequency [bottom, plotting, from left to right, � (ux ), 5(ux ), � (uy ), 5(uy ),� (uz ), 5(uz )].](/figures/fig-12-the-mode-associated-with-the-mean-flow-top-plotting-3ipxtatg.webp)
