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

DATA-DRIVEN MODEL FOR VORTEX LOCK-IN OF CIRCULARCYLINDERS IN AN ACOUSTICALLY-DRIVEN OSCILLATORY FLOW

Girish K. Jankee, Srikar Yadala, Eirik Æsøy, James R. Dawson & Nicholas A. WorthDepartment of Energy and Process Engineering
Norwegian University of Science and Technology
Trondheim N-7491, Norway
Girish.K.Jankee@ntnu.no

ABSTRACT( f0 ) and becomes half of the excitation frequency ( fe ). While
Barbi et al. (1986) reported a range of the lock-in envelope
between fe / f0 of 1 and 2 (second instability region), otherstudies such as Al-Mdallal et al. (2007) and Konstantinidis &Balabani (2007) have found this range to extend to 3 or the
existence of another lock-in regime around fe / f0 of 4 respectively. The amplitude of the acoustic forcing/oscillations has
been associated with the existence of these multiple lock-in
modes.

The concept of vortex lock-in for a single circular cylinder in an oscillating flow, induced through acoustic forcing, isrevisited over the second instability region. Multiple cylinder
diameters are investigated over a Reynolds number range between 500 and 7200. The lock-in behaviour is inspected, bothquantitatively and qualitatively, through hot-wire anemometry
and planar particle image velocimetry (PIV) measurements.
Initial results corroborate previous findings describing the frequency range over which lock-in is inherent for single cylinders. Additionally, the influence of the cylinder location in astanding wave (velocity node or pressure node) is observed toheavily influence the lock-in behaviour. A novel scaling which
describes the onset of lock-in is subsequently established. This
data-driven model demonstrates the necessity to account for
the Strouhal number in predicting the velocity fluctuations required to achieve lock-in over the second instability region.
Velocity fields also reveal the existence of bimodal shedding(symmetric or alternate) during lock-in, which can be associated to the non-dimensional groups present in the scaling.
Finally, the occurrence statistics of each shedding pattern isfound to be sensitive to the amplitude of the velocity fluctuations.

This lock-in behaviour can be further explored through ananalysis of the topological features of the structures present in
the wake of the cylinder. An extensive study by DetempleLaake & Eckelmann (1989) characterised vortical structures
across the second instability region as (i) being independentof fe / f0 , (ii) formed upon synchronisation, i.e., lock-in at
fs = 12 fe or (iii) occurring at fs = fe . Flow visualisation
revealed a kaleidoscope of patterns induced by the velocity
fluctuations, which included but is not limited to symmetric,
alternate (Kármán vortex street), vortex strings and isolated
vortex pairs. Identifying the type of vortical structures present
across the lock-in range is key in understanding the physicalmechanisms responsible for lock-in. However, this has provento be a complex task as is illustrated through the DNS resultsof Balachandar et al. (1997), where a cylinder had the same
shedding frequency at fe / f0 of 2 in both a steady and an oscillating flow, but the observed vortex dynamics were different.

INTRODUCTION

While multiple vortex patterns exist, a myriad of experimental and numerical studies have revealed that the perturbations across the flow direction during lock-in can be primarily associated to two types of vortex shedding: alternating and symmetric modes. Konstantinidis et al. (2005) notedthat the alternating mode corresponds to the 2S mode (classicalKármán vortex street), while the symmetric mode resembles
the 2P mode. The alternating mode generally occurs when theincoming flow is steady while Konstantinidis & Liang (2011)
have shown that in symmetric modes, the flow perturbations
promote the simultaneous formation of vortices in the shear
layer roll-up on both sides of the cylinder. This implies thatthe symmetric mode is synchronised with the oscillatory component. The type of mode present depends on the amplitudeof the flow perturbations. At low values, the alternating modeis predominant and even when the symmetric mode occurs,
the shed vortices are unstable and break down within a few
downstream diameters, giving rise to an anti-symmetric arrangement (Konstantinidis & Balabani, 2007).

Understanding the characteristics of vortex shedding is ofparamount importance due to its ramifications in domains suchas the structural design of heat exchangers, offshore platforms
or bridges and instability control of flames (Sumner, 2010).
While many aspects of the physics of vortex shedding frombluff bodies have been well established, there are still many
scenarios where the impact of vortex shedding is less explored
and understood. In this paper, we investigate vortex shedding
off a single circular cylinder in the presence of acoustic forcing.
Disturbances can be added via controlled vibrations of thecylinder or through velocity fluctuations which can be generated by upstream flaps, active grids or by acoustically modulating the flowfield with electromagnetic speakers (Williamson,
1985; Barbi et al., 1986; Hall et al., 2003). In the latter
case, Blevins (1985) demonstrated that a sound field (standing wave) could alter the motion of particles and induce vortex
shedding at a frequency which fully correlated with the excitation frequency of the standing wave. Such a phenomenon
is termed vortex lock-in and occurs when the vortex shedding
frequency ( fs ) of the cylinder is shifted from its natural state

Moreover, both Ongoren & Rockwell (1988) and Konstantinidis et al. (2007) observed a bimodal behaviour during

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
2
4
(a)

(b)

3

Single cylinderconfiguration
(i)

5

1

x

Velocity
node case(ii)

Velocity
anti-node case

dc

z
y

Air flow
Air flow

Figure 1. (a) Schematic of the experimental setup (not to scale), with the following components: 1. Monacor KU-516 loudspeakers,
2. Test-section pipe, 3. Cylinder, 4. Laser sheet, 5. PIV camera, (b) Single cylinder at (i) velocity node location and (ii) velocity
anti-node location, with the hot wire probe positioned 4dc downstream

which there was a competition and switching between the alternating and the symmetric modes in the near wake. Konstantinidis et al. (2007) noted that this switching between themodes was random but the number of occurrence of each mode
was found to be a function of the excitation frequency, theamplitude of the perturbation as well as Reynolds number ofthe incoming flow. The existence of this bimodal behaviour
highlights the ambiguous nature of the flowfield during lockin and demonstrates that the physical mechanisms responsible for lock-in are yet to be fully understood. Furthermore,
although there is significant evidence which highlights a conjunction between the amplitude of oscillations, viz. velocity
fluctuations, and the vortex lock-in regime, the absence of any
scaling to capture this behaviour is remarkable. This article
primarily aims to establish such a scaling through a systematicparametric study by revisiting the concept of vortex lock-in
for a single circular cylinder in an oscillating flow. Additionally, the flow fields of the cylinders’ wakes are examined underconditions of lock-in with the aim of complementing existing
explanations on the mechanisms driving lock-in.

2.4, 4.0 and 6 mm. The amplitude of the velocity fluctuations
at the velocity anti-node was controlled through pressure measurements with Bruël and Kjær 1/4” condenser microphones
at four different axial locations and using the multiple microphone method of Seybert & Ross (1977). The forcing amplitude (u′ /U∞ ) was varied between 0 and 30 %. The cylinderbased Reynolds number (Re) of the flow was altered between
500 and 7200 to change the natural shedding frequency of thecylinder and thus, vary fe / f0 between a range of 1 to 2.5.
Firstly, hot-wire anemometry measurements were performed with a single-wire traversing hot wire probe of diameter 5 µm and prong spacing 1.25 mm which was operated
in constant temperature mode. The probe was aligned suchthat the wire was positioned at the center of the pipe and at adownstream distance of 4dc . Calibration of the hot-wire probe
was performed using a jet with a top hat velocity profile withvelocities ranging from 0-30 m/s. Following analysis of the
hot-wire measurements, planar PIV (both high speed and low
speed) was performed for selected cases to capture the dynamics of vortex shedding. These cases corresponded to fe / f0 of
0.7, 1.4 and 2.0 and u′ /U∞ of 7.5 and 15 % for the 1.7 mm
cylinder diameter. The high speed imaging setup (HS) consisted of a Phantom V2012 1MP camera fitted with a 200 mm
focal length lens and a Litron LDY300 high speed laser. 6000
image pairs were acquired at a rate of 3 kHz. The low speed
PIV setup (LS) consisted of a Lavision imager LX 16 MP camera, also fitted with a 200 mm focal length lens and extension
rings, and a Litron 200 mJ dual-pulse Nd:YAG laser. For thisconfiguration, the laser and camera were phase locked to theexcitation frequency signal at 12 equidistant phases. For eachphase, 200 image pairs were recorded. Vector fields were determined using Lavision Davis 10 software. For the HS cases,
an initial step with a window size of 64 × 64 px with an overlap of 50 % was applied followed by two passes of 24 × 24 px
with a 50 % overlap. This resulted in a resolution of 0.236 mm
per vector in both the x and y directions. For the LS cases, an
initial step with a window size of 64 × 64 px with an overlap of

EXPERIMENTAL SETUP
The experimental setup is shown in figure 1. The testsection was made from a 560 mm long acrylic pipe with a diameter of 35 mm and the flow into the test rig was monitored
and regulated using an Alicat mass flow controller with a maximum range of 1000 slpm and accuracy of ±2.5 slpm. A TTi40 MHz waveform generator was used to create a sinusoidal
waveform for driving the loudspeakers. The flow was forced
acoustically at frequencies of 730, 1460 and 2190 Hz, whichcorresponded to standing wave modes of the pipe, while theamplitude was varied by adjusting the input voltage. At 730
Hz, a velocity node and anti-node of the acoustic mode werelocated at 150 mm and 270 mm from the pipe exit respectively(see figure 1(b)). These corresponded to the cylinder locations
studied in this paper. The cylinder diameters, dc , were 1.7, 2.0,

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

Figure 2. (a) Contour plot of the power spectra for an example case showcasing peak detection of dominant frequencies at lock-in.
Plot of the vortex shedding frequency of the 2 mm cylinder, fs , against the excitation frequency, fe , both normalised by its natural
shedding frequency, f0 , at (b) velocity anti-node and (c) velocity node, as the amplitude of the velocity fluctuations at the velocity
anti-node is varied.
50 % was applied followed by two passes of 32 × 32 px witha 75 % overlap. This resulted in a resolution of 0.046 mm pervector in both the x and y directions.

flow velocity and vortex shedding occurs solely at f0 . This isevident in figure 2(c), where the amplitude of the velocity fluctuations is always defined at the velocity anti-node. Therefore,
while this parameter increases, the corresponding amplitude atthe velocity node is consistently abated, resulting in a cylinder
positioned at the node not experiencing lock-in. This confirmsthat the intensity of the velocity fluctuations is an importantparameter in dictating the lock-in behaviour of the cylinder.
Additional insight into lock-in can be achieved byanalysing the frequency ratio ( fe / f0 ) corresponding to the onset of lock-in ( fs / f0 = 0.5), for each amplitude of the velocity fluctuations at the velocity anti-node. From figure 3(a), theanalysis of lock-in onset points for multiple cylinder diameters
demonstrates a reasonable collapse and suggests the possibility to derive a universal scaling. As a first attempt, it is possible
to try and fit the data points according to the equation below:

RESULTS
Scaling of vortex lock-in
Hot-wire anemometry data was first analysed to determine the dominant frequencies present in the flow. The natural shedding frequency of the cylinder was first detected ateach Re through identification of the most energetic peak in the
power spectra, with the flow unforced. This also showed that
the Strouhal number for the cylinders tested varied between
0.18 and 0.25. In the presence of acoustic forcing, two peaks
emerge from the power spectra, corresponding to fe and fs(figure 2(a)). By tracking the evolution of fs with Re, the lockin characteristics of the cylinder can be quantified as displayedin figure 2, where the 2 mm cylinder was located at (b) the velocity anti-node and (c) the velocity node. Figure 2(b) shows
the variation of fs / f0 with fe / f0 as the forcing amplitude is increased, with the dashed line representing a mean fitting curvefrom Barbi et al. (1986). Initial observations demonstrate thatthe lock-in behaviour of the single cylinder at the velocity antinode corroborates well with the limits identified by Barbi et al.
(1986). The lock-in envelope ranges from fe / f0 of 1 to 2.2,
similar to the findings of Barbi et al. (1986) and Al-Mdallal
et al. (2007). However, the fe / f0 range tested in this study
is not deemed exhaustive enough to assess the occurrence ofmultiple lock-in modes, as ascertained by Konstantinidis &Balabani (2007). Note that for brevity, only the 2 mm caseis shown in figure 2 but all dc tested exhibited comparable behaviour.
Figures 2(b) and 2(c) also highlight the influence of thecylinder location on the lock-in characteristics, in the presenceof a standing wave. At a velocity node, lock-in is nonexistent. Such a trend arises since in a standing wave, the velocity
fluctuations are at their apogee at the velocity anti-node andzero at the node. Thus at a node, while pressure oscillations
are large, the cylinder is only exposed to the mean incoming

 n
fe
u′
=A
.
U∞
f0

(1)

Following non-linear regression, the values of constants
A and n were found to be 0.23 and -2.68, respectively, with ther.m.s error of the fit equivalent to 0.13. Such a moderate value
in the r.m.s error implies that equation 1 does not fully capture
the non-dimensional groups relevant in describing the relation
between u′ /U∞ and fe / f0 . Therefore, a similar methodology
as the one employed by Jankee & Ganapathisubramani (2021)
is used to fit the scaling parameters to dimensional variables.
The values of u′ corresponding to the onset of lock-in are fitted against dimensional parameters defined in equation 2, after
which the non-dimensional groups are formed using the coefficients obtained through the non-linear regression procedure(table 1). This method avoids any bias as fitting the scaling
parameters against predefined non-dimensional groups, suchas fe / f0 and u′ /U∞ only, limits the results to these assumed
groups.

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

Figure 3. (a) Experimentally acquired data points describing the onset of lock-in for multiple values of fe / f0 , prior to scaling, and (b)Comparison of u′ /U∞ corresponding to the onset of lock-in between experimental data (u′ /U∞ )expt and predictions by the data-driven
model (u′ /U∞ )mod .

u′ = F ( f0 , fe ,U∞ , dc ) = λ f0 a fe bU∞ c dc d ,model is shown in equation 4 and a subsequent comparison
between predicted values and experimentally measured valuesof u′ /U∞ displays good agreement (figure 3(b)). This novel
data-driven model reveals that in addition to fe / f0 , the valueof u′ /U∞ at which lock-in begins also depends on the Strouhal
number. This is because the Strouhal number is not a constantvalue, but rather, it denotes a range of values which dependon the diameter of the cylinder. Using equation 4, it becomes
possible to predict the velocity fluctuations required to induce
lock-in given the cylinder diameter, the freestream velocity,
the natural shedding frequency of the cylinder at that velocity
and the excitation frequency.

(2)where λ is a constant and a, b, c and d are exponents
obtained from non-linear regression.

Table 1. Fitted coefficients and normalised residual for thescaling of u′ corresponding to the onset of lock-in and f0 . The
95 % confidence interval limits (CI95 ) for the fitted parameters
are also provided in the brackets.

(CI95 )

λ

a

b

0.34

2.82

−2.57 0.75(±0.50) (±0.16)cdεr.m.s0.25

0.01(±0.57) (±0.55) (±0.51)

 −2.57 
f0 dc 0.25
u′fe
= 0.34
.
U∞
f0
U∞

(−)

Flowfield during vortex lock-in
While the scaling in equation 4 provides a reliable approach to determine the onset of lock-in, the composition of
the non-dimensional groups also assist in interpreting the flow
field during lock-in. Firstly, the emergence of the Strouhal
number can be associated to the vortex shedding with a steadyflow, which usually takes the form of the classical Kármán vortex street (2S mode). Secondly, at the point of lock-in, since
fs = 12 fe , equation 4 can be rearranged to include the actual
shedding frequency of the cylinder, fs :Following the Buckingham π theorem, multiple nondimensional groups can be formed out of these five dimensional parameters. Hence, grouping these parameters can bedone in numerous different ways and the chosen groups are
based upon sensible assumptions. Using the coefficients in table 1, it is reasonable to combine the velocity perturbations andthe mean incoming flow velocity as u′ /U∞ . Further, the excitation frequency and natural shedding frequency can be coupledas ( fe / f0 )−2.57 . Using these choices, the scaling equation simplifies to: −2.57
u′fe= 0.34f0 0.25U∞ −0.25 dc 0.25 .
U∞
f0

(4)





u′
2 fs −2.57 f0 dc 0.25
= 0.34
.
U∞
f0
U∞

(5)

The rearranged equation 5 reveals that at the point of lockin, the shedding frequency ( fs ) is also an important parameter
which can be linked to the vortex patterns formed when theoscillating component of the flow becomes substantial. Although it may not be instinctive to expect the shedding pattern
to change from an instant to another, figures 4(a) and 4(b) confirms that it is possible to achieve different modes of shedding,i.c. 2S and 2P, at the same phase of the excitation cycle. Figures 4(a) and 4(b) depict the most dominant modes observed(3)This indicates the emergence of another non-dimensionalfrequency term in comparison to equation 1. Since the coefficients are the same, the remaining dimensional parameters
can be grouped together to form the cylinder based Strouhal
number, as ( f0 dc /U∞ )0.25 . The final form of this data-driven

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
enhances the symmetric nature of the vortex roll-up and shedding. Additionally, the percentage of snapshots with 2P mode
initially increases with fe / f0 up to a value of 1.5, after which aslump is observed at fe / f0 of 2.0. Konstantinidis et al. (2007)
explained that a vortex must have achieved a certain criticalstrength before shedding. In an acoustically-driven oscillatory
flow, sound energy is fed into the shedding process by superposition and this leads to an increase in circulation. When fe / f0
is less than 2.0, the vortex shedding frequency, fs , is less thanf0 , i.e. the vortex has more time to develop. Thus, each vortex accumulates more vorticity than a single vortex can handle,
thereby leading to a redistribution into several vortices which
are shed as pairs in the 2P mode. This also occurs when theamplitude of the perturbation is higher and justifies the higher
percentage of snapshots with the symmetric mode. At fe / f0
of 2.0, our findings show that there is minimal influence of theamplitude of perturbations on the probability of occurrence ofthe symmetric mode. Rather, the alternating mode is prevalentat fe / f0 of 2.0, similar to the observations of Konstantinidis &Balabani (2006).
Since the vortex roll-up and shedding is not phase-locked
to the imposed perturbation, an analysis of the time-averaged
flow features is deemed more sensible. In particular, the wakeproperties behind the cylinder at various lock-in locations isinspected. Distribution of the mean and r.m.s velocity values along the wake centerline for selected cases are shownin figures 5(b) and 5(c). The general trend in the mean velocity profile is the existence of a peak velocity deficit value
downstream of the cylinder, after which the velocity increases
asymptotically. The results show that the time-averaged wakestructure is modified in a systematic manner as the fe / f0 isvaried across the lock-in range. Griffin (1995) defined the formation length of the recirculation bubble as the location of thepeak in velocity fluctuations. Both the peak in velocity deficit
and the peak in velocity fluctuations follow the same trend. As
fe / f0 increases, the location of the peak moves closer to thecylinder which not only implies that the recirculation bubble
is shortening, but also that vortex roll-up and shedding occurs
closer to the cylinder. Analysis of the r.m.s profiles further indicates that higher amplitudes of perturbations result in higher
ur.m.s /U∞ and shorter wakes.

Figure 4. Contour plot of the normalised vorticity for instantaneous snapshots within the same phase of an actuation cycle
denoting the occurrence of bimodal behaviour with (a) symmetric mode with a 2P vortex pattern and (b) alternating modewith a 2S vortex pattern. (c) Number of occurence of the symmetric mode across all phases as fe / f0 and u′ /U∞ are varied.

CONCLUSION
A parametric study was conducted in order to investigate the concept of vortex lock-in for circular cylinders in an
oscillatory flow. Cylinders of various diameters were positioned at the velocity node and velocity anti-node in a standing
wave, and hot-wire anemometry was performed to determine
the dominant frequencies present in the flowfield. Planar PIV
measurements were subsequently carried out for selected casesin order to better understand the physical mechanisms driving
lock-in.
The results demonstrated good agreement with previous
findings of Barbi et al. (1986) and the significance of the velocity perturbations was also established by comparing the shedding frequency at the velocity anti-node and velocity node.
This led to the derivation of a novel scaling based on the experimental data, which encapsulates both the fluidic properties of the flow and the geometrical properties of the cylinder.
This newly established data-driven model allows the prediction of the flow perturbations required to trigger lock-in, given
the cylinder diameter, excitation frequency, steady mean velocity and the corresponding natural shedding frequency are
known. The model and experimental data showed good agree-in the flowfield at the point of lock-in. These were the symmetric mode, characterised by the 2P shedding pattern, andthe alternating mode, characterised by the 2S mode. It was observed that these two modes competed against each other andthe switch from one mode to another was random across allphases and all cases. Such a bimodal nature of the shedding
is consistent with the findings of Konstantinidis et al. (2007).
However, this also implies that the low speed PIV data is extraneous since phase locking is only meaningful for those conditions where the vortex roll-up or vortex shedding is synchronised with the imposed perturbation. However, a statisticalanalysis of the vortex shedding patterns present in all instantaneous snapshots is still valid.
Hence, the probability of occurrence of these two types ofvortex shedding pattern was computed across all the cases (figure 4(c)). Firstly, a general trend is that the percentage of snapshots with the symmetric mode is consistently higher at a large
value of u′ /U∞ compared to a lower value, across the whole
range of fe / f0 . This is expected since the induced perturbation
is purely symmetric and any increase in amplitude will amplify
the energy transfer between the flow and the cylinder, which

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 5. (a) Contour plot of the time-averaged streamwise velocity for fe / f0 of 0.7 and u′ /U∞ of 0.075. The pink dotted line
represents the location across which the profiles are taken. Note that for brevity, only one case is shown here. (b) Distribution of
the mean streamwise velocity along the wake centerline for various combinations of fe / f0 and u′ /U∞ . (c) Distribution of the r.m.s
fluctuation of the streamwise velocity across the wake centerline for various combinations of fe / f0 and u′ /U∞ .

ment with an r.m.s error of only 0.01. The scaling establishedthe Strouhal number of the cylinder as an important parameter, even at the point of lock-in. The non-dimensional parameters established by the scaling were also used to provide an
explanation for the bimodal nature of the vortex shedding, as
observed from PIV measurements. Statistical analysis of thevortex shedding patterns was carried out and the time-averaged
wake properties were assessed, with any sensitivity on the flow
perturbations and/or fe / f0 discussed.

from single and tandem cylinders in the presence of appliedsound. Journal of Fluids and Structures 18 (6), 741–758.
Jankee, Girish K & Ganapathisubramani, Bharathram 2021
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Acknowledgement
The authors acknowledge funding from the ResearchCouncil of Norway (FRIPRO), project 299946.

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