12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19 to 22, 2022

SPECTRAL PROPER ORTHOGONAL DECOMPOSITION OF THE
JUNCTION FLOW BEHIND A CANTILEVERED SQUARE CYLINDER

Ali Mohammadi1 , Chris Morton1 and Robert Martinuzzi1∗
1: Department of Mechanical and Manufacturing Engineering, University of Calgary
2500 University Dr. NW, Calgary, Alberta T2N 1N4, Canada.
∗ Corresponding author: rmartinu@ucalgary.ca

ABSTRACT

problem where a set of deterministic functions are sought
which best correlate with the original data and represent it
in a low-dimensional space (Sirovich, 1987; Berkooz et al.,1993). POD modes are energetically optimized, whichmakes them well suited to capture the coherent structures(Holmes et al., 2012). However, since there is no temporalconstraints on the modes, a single mode can contain contributions at significantly different temporal scales.
In direct Fourier transform (DFT) or dynamic mode decomposition (DMD), introduced by Rowley et al. (2009)
and Schmid (2010), the focus is on spanning the modespace based on fixed frequencies to identify coherent structures within narrow spectral bandwidths. In turbulent flows(e.g., shear layers or bluff body wakes), however, the coherent structures are naturally broad-band phenomena and thus
difficult to represent compactly within DMD mode space.
To bridge the gap between the energetically optimal
POD and the spectrally resolved DFT or DMD, Sieber et al.
(2016) introduced a time-domain filter operation applied on
the correlation matrix of POD to control the spectral bandwidth. The convolution operation of this SPOD results inbetter spectrally separated temporal modes. The SPOD filter length (2N f + 1) can be selected, such that the decomposition sweeps the possible outcomes between the limitingcases of POD and DFT. This spectral constraint permits abetter separation of phenomena that occur at multiple frequencies and energies (Sieber et al., 2016).
Kindree et al. (2018) reported several spectral frequencies in the wake of an h/d = 4 square cylinder (Re = 10600)
protruding a thin laminar boundary layer with a relative
thickness of δ /d = 0.21. Using POD, they associated thefs and 2 fs to the Kármán vortex shedding, fL ≈ 0.1 fs to thelow frequency signature from the free-end, and the fai =fs ± fL (i = 1 and 2) (named as accompanying frequencies)
to the interactions between the free-end and Kármán vortices. In addition, they reported frequency peaks around0.345 fs just above the ground plane, which were attributed
to primary HSV system instabilities (Baker, 1979). However, using POD, challenges emerged in separating dynamics in the junction region where the HSV, junction, andKármán vortices interacted. Specifically, a broadband spectral concentration around 0.4 fs was observed. The energy
was distributed over several modes containing significant
spectral energy contributions at other frequencies.
This study presents a further analysis of the PIV datain the junction region of the same square cylinder case. The
aim is to discuss the benefits of using SPOD to separate dynamics. The low-order representation using SPOD provides
a more interpretable basis to explain the physical phenom-

The wake of a cantilevered square cylinder of heightto-width ratio h/d = 4 protruding a thin laminar boundary
layer is investigated using time-resolved stereoscopic particle image velocimetry at a Reynolds number (Re) of 10600.
The PIV are processed using proper orthogonal decomposition (POD) and the time-domain spectral proper orthogonal
decomposition (SPOD). Spectral analysis in the wake region in the vicinity of the ground plane (z < d) show fluctuation energy concentration centered on the vortex shedding
frequency ( fs ), as well as weaker contributions at 0.1, 0.4,
0.9, 1.1, and 2 fs . It is shown that SPOD is significantly
more successful than POD in separating coherent contributions within narrow spectral bandwidths and resolving
spatial modes. Each of the identified frequencies is represented in a separate mode pair, which permits a better interpretation of the related dynamics. Notably, the improved
modal separation assisted in associating the 0.4 fs spectral
contribution to the interactions between the horseshoe andKármán vortices in the wall-body junction region.

Introduction
Turbulent flows feature a wide range of spatial andtemporal scales, which can represent spatially correlated
coherent motions and incoherent stochastic contributions.
For moderate to high Reynolds numbers (8000 < Re =U∞ d/ν < 150000, where U∞ and ν are the free-stream velocity and kinematic viscosity, respectively), the wake ofcantilevered cylinders is generally characterized by three
regions along the height: (a) wall-body junction, (b) midheight, and (c) free-end (Kawamura et al., 1984; Wanget al., 2006). The mid-height region is characterized by
the quasi-periodic Kármán vortex shedding, whereas theother two regions are characterized by high levels of streamwise vorticity. The wall-body junction region arises due
to junction and horseshoe vortices (HSV), which forms asa result of separation and roll-up of the incoming boundary layer (Ballio et al., 1998), and the free-end is associated with a region of low-frequency periodicity (Kindree
et al., 2018; Yauwenas et al., 2019). The interactions ofthese vortices with the Kármán vortices in the wake regionof low aspect ratio cantilevered cylinders (h/d < 10) giverise to inter-modal dynamics (Porteous et al., 2014). The
present work addresses the interaction of the junction flow
with the Kármán vortex shedding using time-domain SPOD
of Sieber et al. (2016).
POD, also known as singular value decomposition or
Karhunen-Loève expansion, is based on a maximization1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19 to 22, 20220

where u represents the total fluctuations. Upright boldsymbols indicate vectors and x and t represent position andtime, respectively. The POD and SPOD are used to determine the coherent contributions. For completeness, a brief
overview of these methods are given below. The POD isdefined as:ena and relation between different vortical structures anddynamics, especially between HSV and Kármán vortices.

Experimental Setup
Figure 1 presents the schematics of a cantilevered
h/d = 4 square cylinder protruding a thin laminar boundary layer, investigated experimentally at an inlet free-stream
velocity of U∞ = 14.3 m/s (corresponding to Re = 10600).
The square cylinder was machined aluminum mounted ona machined-flat rigid steel ground plate with a 4:1 ellipticalleading edge (minor axis matching the plate thickness). Theintersection of the cylinder central axis with the plate marks
the origin of the Cartesian coordinate system: x, y, andz denote streamwise, lateral, and spanwise directions withu, v, and w being the corresponding velocity components.
All coordinate variables and velocity components are nondimensionalized by d and U∞ , respectively. The laminarboundary layer developed naturally and reached a thicknessof δ /d = 0.21 (δ /h = 0.053) at the location of the obstaclewith the obstacle removed. The free stream turbulence wasmeasured to be less than 0.1%.

N

uc (x,t) = ∑ ai (t)φi (x)

(2)

i=1

where ai and φi are the temporal and spatial modal functions. At each PIV plane, the correlation, R, between two
snapshots, u(x) and v(x), is calculated through an innerproduct h., .i:hu(x), v(x)i =Zu(x)v(x)dxdy(3)

A

where A specifies the spatial region (each PIV plane) overwhich the correlation is integrated. The elements of R are
given by:zdRi, j =

3d
2d

d

x
y

Obstaclecross section(4)

R is an N ×N real symmetric positive-definite matrix; therefore, it’s eigenvectors, ai , are orthogonal to each other. ai
and their corresponding eigenvalues, λi , represent the temporal modes and their energies, respectively, and are bothsorted in a descending order. In addition, ai are scaled withthe energy of single modes.

h=4d



0
1 0
hu (x,ti ), u (x,t j )iN1(ai , a j ) = λi δi jNxd

Figure 1. Schematics of the experimental set-up in the test
section of the wind tunnel (not drawn to scale) with the coordinate system and related nomenclature.

TKE =

1 N
∑ λi
2 i=1

(5)

where (.,.) represents the scalar product and TKE is the total
kinetic energy of the fluctuations. Finally, the spatial modes0
are obtained from projecting each fluctuating snapshot (ui )onto its corresponding ai :The velocity fields were measured with time-resolved
stereoscopic particle image velocimetry (PIV). A LaVision
FlowMaster system was used for PIV measurements, whichwere run in a high inlet contraction (area ratio 36:1) opentest-section suction wind tunnel, described in details in Kindree et al. (2018), for which the free stream pressure gradient is zero. To ensure statistical convergence of the firstand second statistical moments and POD/ SPOD modes,
N ≈ 15000 snapshots were acquired in each plane.

φi (x) =0
1 N
ai (t j )u (x,t j )∑Nλi j=1(6)

The POD spatial modes are orthonormal by construction:
hφi , φ j i = δi j .
In the SPOD method proposed by Sieber et al. (2016),
the aim is to enforce diagonal similarity of the correlation
matrix. This is achieved by operating a low-pass filter along
the diagonals of R. The elements of the filtered correlation
matrix, S, are given by:

Analytical Methodology
Following Hussain (1983), the velocity field, u(x,t), is
subject to a triple decomposition, into a mean, U(x), and00
coherent and incoherent contributions (uc (x,t) and u (x,t),
respectively):00u(x,t) = U(x) + uc (x,t) + u (x,t)|{z}u0 (x,t)Nf

Si, j =

∑ gk Ri+k, j+k(7)k=−N fwhere g is a symmetric finite impulse response filter witha length of 2N f + 1. S converges to a symmetric Toeplitz
matrix if the filter length extends over the entire time series.

(1)2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19 to 22, 2022
tial distributions are obtained. However, this comes at thepenalty of retaining less TKE with the first modes. Here,N f = 6.4/ fs was selected to provide the optimum balance
between spectral separation and not losing significant portions of TKE. Following Holmes et al. (2012), a symmetric/ antisymmetric split was carried out on the dataset prior
to performing the decomposition to accelerate the convergence in terms of PIV snapshots needed.
Figure 4 illustrates the u and v components of selected
spatial modes obtained using POD (columns 1 and 2) and
SPOD (columns 4 and 5). To the right of each set, the
PSDF of the corresponding temporal modal coefficients areshown. For POD, rows 1 to 3 present the first, third, andfifth most energetic antisymmetric modes (φa1 , φa3 , and
φa5 , respectively) and rows 4 to 6 present the first three mostenergetic symmetric modes (φs1 to φs3 , respectively). For
better visualization, the SPOD modes are presented basedon their resemblance to the presented POD modes (withrespect to both spatial distribution and spectral behavior).
In both antisymmetric and symmetric fields, only the firstmode of a harmonic pair is shown for brevity.
For POD, the first two antisymmetric modes (φa1 and
φa2 ) are associated to Kármán vortex shedding, with a sharp
frequency peak at fs = 120 Hz (St = fs d/U∞ = 0.11). The
other frequencies ( fh , fai , and 2 fs ) can also be seen in the
PSDF. However, these frequencies either appear in several
modes or together in the same modes, which makes the interpretation of the spatial modes difficult. For instance, thePOD modes most closely corresponding to fai are the (φa5 ,φa6 ) mode pair. However, energetic contents corresponding
to these frequencies can also be seen in the spectra of φa1 to
φa4 . On the other hand, the temporal spectra of φs2 showsan accumulation of energy around both fL and fh as well as
the sharp spectral peak of 2 fs .
By applying a filter to the correlation matrix, SPOD
allows separation of these frequencies and provides a moreinterpretable basis. For instance, in the antisymmetric field,
fs and fai are clearly separated: the (φa1 , φa2 ) mode paircorresponds to fs and (φa3 , φa4 ) corresponds to fai . The fh
is also better resolved with SPOD. Compared to the spectraof (φa3 , φa4 ) POD mode pair, the spectra of (φa5 , φa6 ) SPOD
mode pair contains significantly less energetic contents oneither sides of fh . Accordingly, the spatial distribution isbetter resolved and the coherent motion associated with fh
can be differentiated from that of the shed structures.
The better resolved spatial distribution of the (φa5 , φa6 )SPOD mode pair allows for a comparison with the oil-filmvisualization. Maximum magnitude of the spatial Eigenfunction for the u component are aligned along the dark
pigment lines, indicated by green arrows in Fig. 2(c). Thisis consistent with the energetic contents observed around fh
in the spectra of Fig. 3. Preliminary analysis of SPOD results at higher planes (z < 0.5) show similar patterns, where
the fh gradually weakens with height such that at z = 0.5, it
is no longer detected. Hence, the modes associated with fh
are expressed in the region close to the ground plane where
the HSV downstream extensions are in the proximity of theshed vortices. This suggests that the fh is related to the perturbation of the downstream extensions of the HSV system.
The energetic contents around fh in the spectra of the(φs5 , φs6 ) SPOD mode pair is more clearly separated compared to the POD modes. The frequency and spatial distribution of this mode pair resemble the ones of the (φa5 ,φa6 ) SPOD mode pair, suggesting a connection with theprimary HSV system instabilities. However, when recon-

The rest of the procedure of SPOD is the same as for
POD. The temporal modal coefficients remain orthogonal,
but the spatial modes are no longer orthogonal. The TKE
is still represented with the same definition, however, lessenergy will be recovered compared to POD by the firstmodes. However, similar to POD, if all SPOD modes areused for reconstruction, the flow field is recovered. For a
more comprehensive discussion on SPOD, readers are referred to Sieber et al. (2016).

Results and Discussion
Figure 2(a),(b) shows the schematics of a typical laminar HSV system at (a) y = 0 (symmetry plane) and (b)x − y plane close to the base plate. Figure 2(c) shows themean surface flow patterns from oil-film visualization experiments of Kindree et al. (2018) on the ground plane.
The incoming boundary layer initially separates along L1 ,the foremost point of which is the separation saddle point,
S. The topology of the HSV system consists of large primary (PV) and corner (CV) vortices and small secondary
vortices (SV). The interaction of the CV and obstacle generally gives rise to a corner junction vortex (JV ) and the interactions of the CV and PV with the wall give rise to small
counter-rotating vortices near the wall (BAV1 and BAV2 , respectively). The dashed lines, L1 , L2 , and L3 , are separationlines associated with SV, BAV2 and BAV1 , respectively, andL4 is related to JV . As the PV and CV stretch and extend
laterally from the symmetry plane, the trajectory of theircores follow L2 and L3 .
The HSV system is linked to significant unsteady dynamics. Baker (1979, 1991) categorized the behavior of
HSV systems into four regimes based on the boundary layer
displacement thickness (δ ∗ ): (i) non-oscillatory, (ii) low
frequency secondary oscillations associated with the entire HSV system ( f ∗ δ ∗ /U = 0.01; f ∗ being the dimensional oscillation frequency), (iii) high frequency primaryoscillations associated with the instability of PV (0.07 <f ∗ (δ ∗ d)0.5 /U∞ < 0.09), and (iv) turbulent. The present
laminar boundary layer case corresponds to the third category. In the wake, frequency peaks around f = 0.345 (corresponding to f ∗ (δ ∗ d)0.5 /U∞ = 0.093) were reported in thevelocity fluctuation spectra in regions corresponding to theleft corner of the PIV plain in Fig. 2(c), which were attributed to primary oscillation instability of the HSV system(Kindree et al., 2018).
Figure 3 shows the power spectral density function(PSDF) of fluctuating velocity components (u0 and v0 ) at
various locations at z = 0.11, selected strategically basedon an initial grid search. All frequencies reported earlier byKindree et al. (2018) can be seen in these spectra ( fai cannot
be easily discerned). In addition, a broader band accumulation of energy around fh = 0.4 fs is observed at (3.8, 0.8),(4.1, 1.1), and (4.8, 1.6) points, all of which lie along thepigment line indicated by the green arrows (between L2 andL3 lines) in Fig. 2(c). Although this hump of energy is difficult to discern due to its proximity to fs , its significancewill be discussed in considering the SPOD decomposition.
To provide a clearer interpretation of the dynamics associated with each frequency, an illustrative comparison of
results obtained from the POD and SPOD of the velocity
field data is provided for the z = 0.11 plane in the cylinder wake. In the case of SPOD, several filter lengths (N f =0.8/ fs to 25.6/ fs ) were considered. As N f increases, better
separation of frequencies in PSDF and more refined spa-

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19 to 22, 2022

flow
(a)

(c)

4

SV PV CV

3
2

L4

1

BAV2 BAV1 JV

L2

y 0

L3

-1

L4

(b) S

L3

-2

-3

L2

L1

-4

L1

-3

-2

0

-1

x

1

3

2

6

5

4

Figure 2. Left side: schematics of a typical laminar horse-shoe votex (HSV) system: (a) side view and (b) top view (sketch
inspired by earlier studies: Baker (1979); Simpson (2001); Lin et al. (2002); Kirkil & Constantinescu (2012); Martinuzzi& Tropea (1993); Seal et al. (1997); Ballio et al. (1998), and not drawn to scale. Vortices are named following Kirkil &Constantinescu (2012)). Right side: oil-film visualization of the flow patterns on the ground plate overlaid with the PIV
window (red dashed-lines). Figure is reprinted with permission from Kindree et al. (2018).
(1.3, 0.5)

0(2.5, 0.0)

(3.8, 0.8)

(4.1, 1.1)

(4.8, 1.6)

PSDF of u′

10

fL

fs

fL

2fs

fh fs

2fs

fh fs

2fs

fh fs

2fs

fh fs

2fs

−3

10
−6

10
−9

(a)

(b)

(c)

(d)

(e)

10
0

PSDF of v ′

10

fL

fh fs

2fs

fs

fL

fh fs

2fs

fh fs

2fs

−3

10
−6

10
−9

(f )

(g)

(h)

(i)

(j)

10
−3

10

−2

10

10
fd/U ∞

−1

0

10

−3

10

−2

10

10
fd/U ∞

−1

0

10

−3

10

−2

10

−1

10
fd/U ∞

0

10

−3

10

−2

10

10
fd/U ∞

−1

0

10

−3

10

−2

10

10
fd/U ∞−1

0

10

Figure 3. Power spectral density functions (PSDF) of u and v fluctuating velocity components (first and second rows, respectively). (a),( f ) at (1.3, 0.5); (b),(g) at (2.5, 0.0); (c),(h) at (3.8, 0.8); (d),(i) at (4.1, 1.1); and (e),( j) at (4.8, 1.6).

structing the velocity fields using these mode pairs, the vortical structures appears convective with the antisymmetricpair, but not with the symmetric one. This suggests thateach mode pair captures different aspects of interaction between the primary vortices of the HSV system and Kármánvortices in the junction region (z ≤ 1).
The other spectral peaks and related dynamics (i.e., fL
and 2 fs ) in the symmetric field is also better resolved using
SPOD. With POD, except φs1 , which shows a clear spectral
peak around fL , the spectra of next five modes (φs2 to φs6 )exhibit energetic contents around multiple frequencies (i.e.,fL , fh , fs , and 2 fs ; φs4 to φs6 are not shown for brevity).
φs4 and φs5 show energetic contents around fL , fs , and 2 fs ,while φs6 shows energetic contents around fL and fs . With
SPOD, however, energetic contributions at fL , fh , and 2 fs

appear in separate mode pairs: (φs1 , φs2 ) contain energetic
contributions at fL , (φs3 , φs4 ) at 2 fs , and (φs5 , φs6 ) at fh .
The low frequency fL has been associated with theflapping instability of the separated shear layers from thefree-end (Kindree et al., 2018; Yauwenas et al., 2019). At
the z = 0.11 plane, this flapping motion is mainly represented by the (φs1 , φs2 ) SPOD mode pair and accounts for
2.8% of TKE. The u component appears most energetic(representing 2.1% of TKE vs. 0.5% and 1% with the v
and w components, respectively). The spatial modal function for u indicates that high contributions due to this mode
are concentrated along the edge of the recirculation region- i.e., along the shear layer. The SPOD analysis of higher
planes shows that the relative TKE contribution related to
fL increases with height. The w component becomes more4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19 to 22, 2022240.7% of TKE

624
0.7% of TKE

10

y

φa5

10
2

2

0

0

−2
6

10

2
4
0.3% of TKE

y

φs1

2

2

0

0

−2

10

−2

10

6

2
4
0.9% of TKE

2

2

0

0

−2
6

2fs

10

−2

10

2

0

0

fh 2fs

10

−2

−2

10

4

x

6

2

4

x

6

6

2
4
0.4% of TKE

2

2

0

0

−2

0

6

2fs

−5

10

−2

2

0

0

−2
6

6

10

10

10

−2

10

−5

6

10

10

2

0

0

−2

10

−5

6

10

−2

10

fh

2
4
0.6% of TKE

2

2

0

0

−5

6

10

−2

−2

10

−5

−2
2

4

x

6

2

4

x

6

0

0

0

2fs
10

0

0

−2
2
4
0.6% of TKE

0

0

6

10

0

0

fL

2
4
0.1% of TKE

2−2

f d/U ∞

−5−22
4
0.4% of TKE0

fa2

fa1 fa224
0.2% of TKE

2

0

10

−2
2
4
1.4% of TKE

020% of TKE

y2
4
0.9% of TKE

10

fs

−2
2

−2

0

fL
10

−2

0

10

10

−5

6
10

2

0

0

fL

2
4
0.8% of TKE

0

−2

0

fh

10

−2
2
4
1.2% of TKE

φa1

25% of TKE
10

−5

6

10

2

0

fL

10

y

−5

6

10

10

−2
2
4
1.9% of TKE

φs2

10

2

0

fs

10

y

−2

−2
2
4
3.5% of TKE

φs3

fs

−5

6

6

0.7% of TKE−224
0.4% of TKE

1.3% of TKE

−2

10

6

−5

1.7% of TKE

0

−2

10

10

fh

y

0

−2
2
4
0.3% of TKE

y

2

0

0

y

y

φa3

10
2

10

0

fs

0.6% of TKE

10

2

PSDF

0

10

−2

10

1.1% of TKE

6

y

2
4
1.3% of TKE

y

6

2

0

φa5

2
4
1.5% of TKE−2

2.9% of TKE

−2

−5

φa3

−2

10

1.4% of TKE

0

φs1

0

fs

3.8% of TKE

2

φs5

2

v
13.6% of TKE
102.9% of TKE

y

φa1

10

u
6.2% of TKE

PSDF

0

φs3

v
16.9% of TKE

2.1% of TKE

u7.6% of TKE0

f d/ U ∞Figure 4. Decomposition of the velocity field. Spatial modal functions for u and v, and PSDF from POD (left three columns)
and SPOD with a filter length of N f = 6.4/ fs (right three columns). For POD: φa1 , φa3 , and φa5 represent the first, third, andfifth most energetic antisymmetric and φs1 , φs2 , and φs3 represent the first, second, and third symmetric modes. For SPOD: themodes most similar to the selected POD modes with regards to their spatial distributions and spectral patterns are shown (i.e.,
φa1 , φa5 , and φa3 for the antisymmetric modes, and φs1 , φs5 , and φs3 for the symmetric modes).

Conclusion

significant towards the mid-span height due to the intensified downwash effects. Hence, the observed behavior of thismode is consistent with the shear layer flapping instability.

A comparative study is done on the ability of POD
and SPOD techniques in separating dynamics and providing a more interpretable basis to study them in the wakeof an h/d = 4 cantilevered square cylinder. It was shownthat SPOD can effectively (i) separate vortical structures
within narrow spectral bandwidth without significant losses
of TKE presentation, (ii) pair similar modes, and (iii) deliver more clearly resolved phase relationships between
temporal modes, which allow a clearer representation of
the flow dynamics and nonlinear interactions. In particular to the presented case, SPOD successfully separated abroad band energy distribution around fh = 0.4 fs , which
was not possible with POD. The comparison of the morerefined related spatial modes and the oil-film visualization
flow patterns on the ground plate assisted in associating thisfrequency to perturbations of the downstream extensions of
the HSV due to interactions with the Kármán vortices.

Figure 5 shows the phase portraits of select temporalmodes using POD (first row) and SPOD (second row). The
trajectories obtained in the phase space with SPOD show
a clearer relationship between temporal modes: in aa1 vs.
aa2 , the cyclical relationship between the two legs of a harmonic pair is better resolved; in aa1 vs. aa4 (also in aa1
vs. aa3 ), a clearer bi-modal behavior, which corresponds to
fa1 and fa2 , can be identified. In considering aa1 vs. as1 ,the parabolic relationship between the shedding strengthand the slow base flow variation, described by Noack et al.
(2003), is more clearly rendered. Finally, in aa1 vs. as3 andaa1 vs. as4 , clear Lissajou portraits are seen, which indicatethat the phase dynamics of the 2 fs −harmonic are enslaved
to the fundamental harmonic (Bourgeois et al., 2013).

5

POD
aa1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19 to 22, 2022
2

2

1

1

1

1

1

0.5

0.5

0.5

0

0

0

0

0

−1

−1

−0.5

−0.5

−0.5

SPOD
aa1

−2
−2

0

2

−2
−2

0

2

−1
−2

0

2

−1
−2

0

2

−1
−2

2

2

1

1

1

1

1

0.5

0.5

0.5

0

0

0

0

0

−1

−1

−0.5

−0.5

−0.5

−2
−2

−2
−2

0

aa2

2

0

2

−1
−2

0

aa4

as1

2

−1
−2

0

as3

2

−1
−2

0

2

0

2

as4

Figure 5. Phase portrait of select temporal modes obtained from POD (top row) and SPOD (bottom row): aa1 vs. aa2 , aa4 ,
as1 , as3 , and as4 . Every second point is plotted.

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