14th International Symposium on Particle Image Velocimetry – ISPIV 2021
August 1–5, 2021

Application of SPOD analysis to PIV data obtained in
the wake of a circular cylinder undergoing vortex

induced vibrations
Christopher M. O’Neill1, Yannick Schubert2, Moritz Sieber2, Robert

Martinuzzi1, Chris Morton1∗

1Department of Mechanical Engineering, University of Calgary, Calgary, Canada
2Institut für Strömungsmechanik und Technische Akustik, Hermann-Föttinger-Institut, Technische

Universität Berlin, Berlin, Germany
∗ chris.morton@ucalgary.ca

Abstract
Vortex induced vibrations (VIV) of a circular cylinder have been investigated experimentally using a cyber-
physical apparatus with m∗ = 8, ζ = 0.005, and Re = 4000. This study considers the application of proper
orthogonal decomposition (POD) and spectral POD (SPOD) analysis to the wake dynamics of the low-mass-
ratio VIV of a circular cylinder in the lower branch at U∗ = 7.5. SPOD has been previously shown to better
separate frequency-centered modal dynamics, compared to POD. Coherent POD and SPOD modes were
compared and the newly separated third SPOD mode pair was found to have a periodicity characteristic of
vortex shedding and a peak in the temporal coefficient spectra at St = f D/U∞ = 0.2248. The literature has
identified that the wake dynamics within the lower branch are synchronized to the cylinder motion; however
the present study suggests that some hidden dynamics persist at the Strouhal frequency. Low order models
based on the first eight POD and SPOD modes were compared, and it was found that the filtering operation
in SPOD removes the uncorrelated stochastic energy component of the POD modes while producing a
comparable representation of the coherent deterministic part of the wake dynamics. Using SPOD to separate
the distinct frequency-centered dynamics into unique, interpretable mode pairs will simplify future efforts
to develop sparse dynamical models of the flow.

1 Introduction
Vortex-induced vibration of a circular cylinder is a fluid-structure interaction problem presenting engineering
challenges to the design of civil structures (e.g., cable-tensioned bridges) as well as mechanical engineering
devices (e.g., heat exchangers). Comprehensive reviews of VIV can be found in the works of Sarpkaya
(2004) and Williamson and Govardhan (2004). An elastically mounted cylinder in cross-flow is susceptible
to vortex induced vibrations when the vortex shedding frequency is near the natural frequency of the system.
The resonance of the structure and the associated ’lock-in’ phenomenon occur when the vortex shedding
frequency is synchronized to the vortex induced oscillations of the body. The structural dynamics of the
cylinder motion are given by:

mÿ+ cẏ+ ky = Fy (1)
where m,c,k are the structural mass, damping, and spring constant respectively. Fy is the transverse compo-
nent of the fluid force on the cylinder. The natural frequency of the system is defined as fN =

√
k/m. The

following section discusses the contributions of the cylinder wake dynamics to the amplitude and frequency
response of the system, and approaches in modeling of this fluid-structure interaction.

1.1 Cylinder VIV Wake Modes, Amplitude and Frequency Response
The VIV amplitude response (A∗ = A/D, where D is the cylinder diameter) for a low mass-damping sys-
tem (m∗ζ < 0.1, m∗ = 4m/ρπD2L, ζ = c/2

√
km) across a range of reduced velocities (U∗ = U∞/ fND) is



characterized by an initial branch, upper branch, lower branch, and a desynchronization region (Khalak and
Williamson, 1997). The topology of the wake for a given reduced velocity and response amplitude has been
characterized in controlled motion experiments and was classified into vortex shedding modes using phase-
averaging as described by Morse and Williamson (2009). The transitions between branches are somewhat
sensitive to m∗ζ; for consistency with the literature, the following description of the amplitude response
follows Khalak and Williamson (1997). A description of the amplitude response associated with the m∗ζ of
the apparatus used in this study can be found in Riches and Morton (2018).

In the initial branch (2 . U∗ . 4.75), with increasing U∗ the frequency of vortex shedding approaches
the natural frequency of the system causing an increase in the oscillation amplitude. The vortex shedding
frequency and the natural frequency are both present within the cylinder position spectrum. Following the
nomenclature of Morse and Williamson (2009), this branch is characterized by 2S-type vortex shedding in
which single counter-rotating vortices are shed alternately from the cylinder sides.

When the reduced velocity is increased further above U∗ ≈ 4.75, the amplitude response of the cylinder
transitions to the upper branch, where the vortex shedding frequency becomes synchronized with the natural
frequency of the system and structural resonance occurs leading to large oscillations. The wake topology in
the upper branch is characterized by 2PO-type vortex shedding, which consists of pairs of counter-rotating
vortices. These pairs are shed alternately from the cylinder sides with the trailing vortex being significantly
weaker than the leading vortex (Morse and Williamson, 2009).

The lower branch occurs for 6.U∗. 9.5, and is characterized by a slightly-lower steady state amplitude
response compared to the upper branch. The wake topology in the lower branch is characterized by 2P-
type vortex shedding, where pairs of equally strong counter-rotating vortices shed from alternating sides
of the cylinder. While oscillation and shedding remain synchronized, the peak in the cylinder oscillation
spectra shifts to be slightly higher than the natural frequency. This synchronization of the natural and
oscillation frequencies is known as ‘lock-in’. According to Sarpkaya (2004), lock-in occurs when the motion
of the cylinder becomes a dominant factor in the interaction of the separated shear layers that causes vortex
shedding in the wake. The focus of this paper will be further analysis of the wake topology at U∗ = 7.5 in
the lower branch.

For reduced velocities above U∗ ≈ 9.5, the vortex shedding frequency is no longer synchronized with
the natural frequency, and the amplitude response of the system decays. In the cylinder position spectra, the
vortex shedding frequency and the natural frequency re-emerge as separate frequencies.

1.2 Further analysis of Cylinder VIV and Low Order Modeling
Beyond the phase-average based characterization of the topology of the wake by Morse and Williamson
(2009), VIV has been studied using several techniques including Proper Orthogonal Decomposition (POD)
Riches et al. (2018), Dynamic Mode Decomposition (DMD) Freire et al. (2015), and deep learning Raissi
et al. (2019), based on time-resolved Particle Image Velocimetry (PIV) data. In addition to characterizing the
flow topology, there is an interest in using these methods to develop physics-based models of the flow state.
POD has been used to develop low order models (LOM) of the flow state, such as described in Liberge
and Hamdouni (2010) where a 6-mode LOM was found to accurately reconstruct the velocity field for a
computational study of cylinder VIV at Re = 1690. Riches et al. (2018) applied POD to cylinder VIV in
the initial (Re = 3100) and upper (Re = 4100) branches and found that 6- and 7- mode LOMs, respectively,
represented the coherent part of the cylinder wake. Riches et al. (2018) also investigated the analytical
relationships between the POD modes in order to develop a model of the flow. As explored in Riches et al.
(2018), discerning which modes correspond to the coherent dynamics can be challenging. For example,
a temporal Gaussian filter was applied to data in a pre-processing step in order to separate low-frequency
and high frequency dynamics which could not be identified using POD alone. Spectral Proper Orthogonal
Decomposition (SPOD)(Sieber et al., 2016) involves a more robust filtering process, and has shown promise
in decoupling modal dynamics for bluff body wakes. SPOD recently was applied to the wake of a FX63-
137 airfoil with a Gurney flap, at Rec = 180× 103, and was found to produce a simpler wake model by
decoupling the interaction of the upstream vortex from the primary vortex shedding (Sieber et al., 2016).

1.3 Objective
The objective of this paper is to consider the application of SPOD to decomposition of the lower branch
of VIV, and illustrate the advantages SPOD compared to POD in extracting dynamically relevant modes.
This paper will first examine the effect of the SPOD filter length on the high energy modes, followed by



analyzing the cylinder wake. POD-based and SPOD-based LOM will be constructed and compared to
examine the benefits SPOD offers in terms of representing the flow state.

2 Methods
Experiments were carried out in a water tunnel facility at the University of Calgary. A cyber-physical
apparatus was employed to study VIV of a circular cylinder, as described in Riches and Morton (2018).
Following the works of Hover et al. (1997), Mackowski and Williamson (2011), and Derakhshandeh et al.
(2015), the apparatus operates by real-time simulation of the differential equation of the cylinder’s motion,
given in Equation 1. The transverse component of the force on the cylinder (Fy) is taken as the input, and
the structural mass, damping, and spring constant are applied in software to compute the desired kinematics.
The cylinder is then forced to follow the desired kinematics using a motorized traverse. The transverse force
on the cylinder was measured using an ATI Mini40 force transducer with a resolution of 0.005 [N]. The
traverse was driven using a ClearPath digital brushless servo motor (MCVC-2341P) and the position was
measured using an Omron optical encoder with a position resolution of 0.0375 [mm]. The block diagram of
Equation 1 was used to compute the kinematics, and is shown below in Figure 1b.

Figure 1: (a) Experimental apparatus placed in a section of the University of Calgary water channel, with
PIV imaging setup shown. (b) Cyber-physical apparatus path generation algorithm and system controller.
Additional details on this system can be found in Riches (2018). Figure 1 is adapted from Riches and Morton
(2018).

The parameters investigated in this study correspond to m∗ = 8, ζ = 0.005, Re = 4000, and U∗ = 7.5.
The free-stream velocity was set to fix the Re, while U∗ was set by altering the natural frequency of the
system. The structural natural frequency of the system is defined in air. The natural frequency and damping
ratio were confirmed based on the system accurately reproducing an oscillating free decay after release from
an initial displacement, as shown in Riches and Morton (2018). Good agreement was found between the am-
plitude and frequency response of the cyber-physical apparatus with data from Govardhan and Williamson
(2006) and Klamo et al. (2006), as discussed in Riches and Morton (2018).

Time-resolved two-component planar PIV measurements were obtained for a plane located at the mid-
span of the cylinder, as shown in Figure 1a. Images were captured using a high speed Phantom Miro M340
digital camera focused on the mid-span plane illuminated by a laser sheet generated by a Photonics 20mJ
Nd:YLF high repetition rate pulsed laser with a wavelength of l = 527 [nm]. PIV images were processed
using LaVision DaVis 8.3 software. A total of N = 4767 vector fields were obtained at a sample frequency
of 24 Hz, spanning approximately 290 oscillation periods of the cylinder. Velocity vector uncertainties were
calculated using the correlation statistics method of Wieneke (2015), as employed within DaVis 8.3 and
were an average of 2.2% in the free stream and 4.0% in the wake.

2.1 Proper orthogonal decomposition
The velocity field u(x, t) is split into mean ū(x) and fluctuating u′(x, t) components and then the fluctuating
component is decomposed into a finite sum of spatial eigenfunctions φi(x) multiplied by a time-dependent



temporal modal coefficients ai(t) following the snapshot POD method of Sirovich (1987):

u(x, t) = ū(x)+u′(x, t) = ū(x)+
N

∑
i=1

φi(x)ai(t) (2)

To compute the correlation between snapshots, the L2 inner product is first defined using 〈,〉. Applying the
inner product to some arbitrary velocity fields p(x) and q(x) gives:

〈p(x),q(x)〉=
∫

Ω

p(x)q(x)dΩ (3)

where Ω defines the spatial volume where the correlation is integrated. If velocity fields p(x) and q(x)
are orthonormal, then 〈p(x),q(x)〉= δi, j. Using the inner product, the elements of the snapshot correlation
matrix Ri, j can be computed according to:

Ri, j =
1
N
〈u′(x, ti),u′(x, t j)〉 (4)

The temporal coefficients ai(t) and mode energies λi are determined from the eigenvectors and eigenvalues
of the correlation matrix:

Rai(t) = λiai(t) (5)
The spatial modes φi(x) are computed from the projection of the snapshots onto the temporal coefficients:

φi(x) =
1

Nλi

N

∑
j=1

ai(t j)u′(x, t j) (6)

The spatial modes are orthonormal by construction and therefore they satisfy the condition:

〈φi(x),φ j(x)〉= δi, j (7)

The process of decomposing the fluctuating field using POD produces orthogonal modes and coefficients
such that a minimal number of modes contain the maximum amount of energy within the field; however,
the modes may contain energy at multiple frequencies and therefore may not represent a specific dynamical
structure in the wake. Once a decomposition of the flow has been performed, a low-order model of the field
u′LOM(x, t) can be created using an expansion of the first Nt POD modes associated with coherent dynamics
in Equation 2, thus yielding the coherent part of the fluctuating velocity field (Rowley (2005); Noack et al.
(2011); Bourgeois et al. (2013)). This low order model can be found according to:

u′LOM(x, t) =
Nt

∑
k=1

φi(x)ai(t), (8)

2.2 Spectral Proper Orthogonal Decomposition
Spectral POD, as described in Sieber et al. (2016), is an alternative decomposition method to POD. Begin-
ning with the snapshot correlation matrix defined in Equation 4, a low-pass filtering is performed along the
diagonal to form a filtered correlation matrix, which has elements Si, j given by:

Si, j =
N f

∑
k=−N f

gkRi+k, j+k (9)

where N f is the filter length, and gk is an entry of the 1-D filter kernel g of length 2N f +1. Sieber et al. (2016)
showed that in the case of a box-filter with equal coefficients given by gk = 1/(2N f +1), that with N f = 0,
SPOD is equivalent to POD. Further with N f = N, SPOD is equivalent to the discrete fourier transform
(DFT). After the filtering, remaining steps in the decomposition proceed as described above in POD, yielding
a set of spatial modes and temporal coefficients. Based on Sieber et al. (2016), a Gaussian filter kernel is used
in the following results, due to preferable properties of smooth roll off, except for N f = 0 and N f = 4767,
where a box filter was used to recover POD and the DFT. Mode pairing was performed as described in
Sieber et al. (2016), including identification of the primary frequency for the temporal coefficient of each
mode pair.



3 Results

3.1 SPOD Filter Length
Investigation of the dynamical features of the flow was first carried out using a POD-based approach similar
to that of Riches et al. (2018). This corresponds to a filter length of N f = 0 in Equation 9. The choice of
filter length 0 < N f < N represents the trade off between the spectrally agnostic POD and spectrally pure
DFT. In a practical sense, as the filter length is increased from zero (POD), the spectral bandwidth of the
mode temporal coefficients is constrained. The resulting spatial modes are affected through the projection
operation, and energy outside the mode spectral bandwidth is redistributed to other modes.

The filter length parameter N f determines the number of 2D PIV vector fields averaged by the filter
and thus proper selection depends on both the sample rate and the frequency of the underlying dynamics.
The filter length can be examined with reference to the number of 2D PIV vector fields for one oscillation
of the cylinder (Nosc). The cylinder oscillation frequency fosc = 1.753 [Hz] (St = foscD/U∞ = 0.15) was
determined based on spectral analysis of the cylinder position signal, see Figure 2. The number of vector
fields capturing a given oscillation cycle is:

Nosc =
fs

fosc
=

24
1.753

= 13.69≈ 14 (10)

The effects of the filter length on the spatial modes and temporal coefficient spectra was examined over a
wide range of N f . For brevity only N f = [0,4Nosc,5Nosc,N] are shown in Figure 3. Figure 3(i) shows the
energy distribution of mode pairs (y-axis) and their associated frequencies (x-axis). The size and color of
the markers represents the correlation between the modes in the mode pair; pairs with large yellow markers
are well correlated. Figure 3(ii,iv,vi,viii) shows the u-component of the first four most energetic POD spatial
mode pairs, or the equivalent SPOD mode pair. Figure 3(iii,v,vii,ix) shows the temporal coefficient PSDF
corresponding to the mode pairs in Figure 3(ii,iv,vi,viii).

The spatial mode in Figure 3a(ii) shows a periodicity that is characteristic of vortex shedding, and the
corresponding temporal coefficient spectrum for the pair shown in Figure 3a(iii) has a peak at the St =
f D/U∞ = 0.15, which corresponds to the oscillation frequency of the cylinder. Additional energy content in
this pair is spread across multiple frequencies, including a secondary peak at f D/U∞ ≈ 0.2. The structures
in the spatial mode in Figure 3a(iv) show half the spatial wavelength compared to Figure 3a(ii) and the PSDF
of the temporal coefficient in Figure 3a(v) shows a peak at St = f D/U∞ = 0.30; therefore, this mode pair
corresponds to the second harmonic of the vortex shedding at the oscillation frequency. The second harmonic
mode pair has energy content spread across multiple frequencies, including a secondary peak at f D/U∞ ≈
0.2. The spatial modes in Figure 3a(vi) and (viii) contain large scale features and the corresponding temporal
coefficient spectrum contains a broad range of scales, including broad peaks at f D/U∞ ≈ 0.1 and f D/U∞ ≈
0.2.

As the filter length is increased in the range 0 ≤ N f ≤ 4767, the trend towards decreasing spectral
bandwidth can be observed by comparing the temporal coefficient spectra in Figures 3(iii,v,vii,ix). For
example in Figure 3b(v), most of the mode energy in the temporal coefficient spectra is contained in the
range approximately 0.26≤ f D/U∞ ≤ 0.35. Increasing the filter length from N f = 56 to N f = 70, in Figure
3c(v) most of the energy is now contained within 0.28≤ f D/U∞ ≤ 0.33. Thus as the filter length increases,
the temporal coefficients for the first two mode pairs are constrained to progressively narrower frequency
bands without significant energy loss as shown by comparison of Figures 3(i).

Within the temporal coefficient spectra, the secondary peak observed in Figure 3a(v) and 3a(vi) at
f D/U∞ ≈ 0.2 is absent for N f > 56. The broad peak in Figure 3a(ix) is significantly weakened for N f = 56
and almost absent for N f = 70, suggesting N f ≥ 70 is required to separate the third mode pair from the
fourth mode pair. Comparing Figure 3a-c(vii), the peak at f D/U∞ ≈ 0.2 is well defined for N f ≥ 56. In-
creasing the filter length above N f = 70 further constrained the frequency content of the third mode pair
and progressively reduced the energy of the mode pair. The frequency bandwidth of the third mode pair
is broader than the first two mode pairs, thus further constraining the bandwidth of this mode would only
exclude relevant dynamics and their energy content. Higher filter lengths such as N f = 112 offered no clear
benefit in separating the dynamics associated with the fourth mode pair. Thus N f = 70 was chosen based on
the trade off between increasing energy losses of the third mode pair and frequency separation of the fourth
mode pair.



Figure 2: Power Spectral Density Function (PSDF) for the cylinder displacement signal. The dotted line is
located at f = 1.753 Hz, and corresponds to the cylinder oscillation frequency ( fosc).

3.2 Analysis of Wake Dynamics for SPOD with N f = 70

Following selection of an appropriate filter length (N f = 70) for separating the relevant dynamics, it is
important to consider what new fundamental physical insights have been established. For this purpose,
Figure 4 shows a direct comparison of the spatial modes and temporal coefficient spectra between POD (a)
and SPOD (b). From left to right, each row of Figure 4 consists of the u- and v-component of the spatial
modes for the first and second modes in the pair, followed by the temporal coefficient spectra. As previously
discussed, the mode pairs (1,2) and (3,4) in both POD and SPOD have spatial modes and temporal coefficient
spectra characteristic of the oscillation frequency and its second harmonic, respectively. The POD mode
pairs are contaminated with low and high frequency content, with multiple peaks visible in the spectra for
all four mode pairs in Figure 4a. Moreover, the third and fourth mode pairs (5,6 and 7,8) are difficult to
interpret and contains a mixture of low frequency content with broad local maxima located at f D/U∞ ≈ 0.1
and f D/U∞ ≈ 0.2. In contrast, the SPOD mode pair (5,6) has a spatial wavelength and mode shape typical
of a vortex shedding phenomena, and a well defined peak in the frequency domain at St = f D/U∞ = 0.2248,
which is close to the expected Strouhal number of ∼ 0.21 for a stationary cylinder at Re = 4000 (Norberg,
2003). These results suggest that SPOD modes (5,6) represent a natural vortex shedding process which is
not synchronized to the cylinder oscillation, and that this process may persist within the lower branch. The
SPOD mode pair (7,8) has a longer spatial wavelength with a peak in the frequency domain at f D/U∞ =
0.0753, which corresponds closely to the beat frequency ( f D/U∞ = 0.2248−0.1508 = 0.074) between the
cylinder oscillation represented in SPOD modes (1,2) and the natural vortex shedding represented in SPOD
modes (5,6).

3.3 Comparison of Low Order Models for POD and SPOD
A low order model of the flow field dynamics was constructed from the first four mode pairs for both POD
and SPOD decompositions of the wake, as described by Equation 8. Figure 5 shows the Reynolds stresses
from the raw PIV images, and compares the residuals of the two low order models. The Reynolds stress
residuals for the two LOMs are similar overall; however, the SPOD-based LOM has slightly higher u′u′ and
v′v′ residuals in the near wake (approximately −1.8 < y/D < 0.4 and x/D < 0.4), compared to the POD
LOM. Based on the lower energy of the SPOD modes compared to the POD modes, it is expected that the
residuals will be slightly higher in the frequency constrained SPOD LOM, for the same number of modes
reconstructed. Figure 6 shows the vorticity contours of the two LOM for two vector fields in a representative
shedding cycle. The two LOM show remarkably similar representations of the coherent dynamics in the
flow. The SPOD LOM thus offers an advantage compared to the POD LOM because each of the four
mode pairs in the SPOD LOM is associated with a distinct frequency-centered dynamic. Essentially, SPOD
has properly separated these four dynamics and removed the uncorrelated stochastic contributions seen in
the POD modes, leaving a deterministic coherent representation of the flow, which is desirable from the
perspective of further modeling.

4 Conclusions
This study considers the application of SPOD analysis to the wake dynamics of the low-mass-ratio VIV of a
circular cylinder in the lower branch, at U∗= 7.5. The effect of the SPOD filter length on the coherent modes



Figure 3: Comparison of SPOD filter lengths N f = [0,56,70,4767] in (a-d). Subplots: (i) Mode pair energy
content as a function of the Strouhal number. Circle size and color represent the correlation of the temporal
coefficients of the mode pair. (ii,iv,vi,viii) Spatial modes (u-component) for the first 3 most energetic mode
pairs from POD, and the same modes in SPOD. (iii,v,vii,ix) Power Spectrum Density Functions (PSDF)
of the temporal coefficients for the mode pairs in (ii,iv,vi,viii). The dotted lines at f D/U∞ = [0.15,0.3]
correspond to the cylinder oscillation frequency and second harmonic.

was examined, and a filter length of N f = 70 was chosen to properly separate the four distinct frequency-
centered dynamics within the first eight modes, while retaining as much energy as possible. The coherent
POD and SPOD modes were compared and the newly separated third SPOD mode pair was found to have a
periodicity characteristic of vortex shedding and a peak in the temporal coefficient spectra at St = f D/U∞ =
0.2248. The fourth SPOD mode pair was found to correspond to the beat frequency between the first and
third SPOD mode pairs. While the literature has identified that the wake dynamics within the lower branch
are synchronized to the cylinder motion, the present study suggests that some hidden dynamics persist at the
Strouhal frequency that were revealed only through application of SPOD analysis. Low order models based



Figure 4: Spatial modes and temporal coefficient spectra for the first three coherent mode pairs identified
using POD (a) and SPOD (b). The dashed lines in the spectra are located at f D/U∞ = 0.15,0.30 and
correspond to the cylinder oscillation frequency and second harmonic, respectively.

on the first eight POD and SPOD modes were compared, and it was found that the filtering operation in
SPOD removes the uncorrelated stochastic energy contributions seen in the POD modes while producing a
comparable representation of the coherent deterministic part of the wake dynamics. Using SPOD to separate
the distinct frequency-centered dynamics into unique, interpretable mode pairs will simplify future efforts
to develop sparse dynamical models of the flow.



Figure 5: Comparison of Reynolds stresses for the two LOM based on the POD and SPOD decompositions
of the wake. Raw data is shown in (a), and the residuals for the POD and SPOD LOM are shown in (b).

Figure 6: Vorticity contours for the raw PIV velocity fields (left column) and 8-mode low order models
(LOM) for POD (N f = 0, center column) and SPOD (N f = 70, right column). The contours shown are taken
from a representative shedding cycle, for brevity two vector fields are omitted between the shown contours.

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	Introduction
	Cylinder VIV Wake Modes, Amplitude and Frequency Response 
	Further analysis of Cylinder VIV and Low Order Modeling
	Objective

	Methods
	Proper orthogonal decomposition
	Spectral Proper Orthogonal Decomposition

	Results
	SPOD Filter Length
	Analysis of Wake Dynamics for SPOD with Nf=70
	Comparison of Low Order Models for POD and SPOD

	Conclusions