# Reynolds number effects on the bistable flows over a wavy circular cylinder

Kai<EMAIL_ADDRESS>Hongbo Zhu1  Yong Cao1  Dai Zhou1
<EMAIL_ADDRESS>1State Key Laboratory of Ocean Engineering, School of
Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong
University, Shanghai 200240, China 2Department of Mechanical and Aerospace
Engineering, Rutgers University, Piscataway, NJ 08854, USA

###### Abstract

The wake of wavy cylinder has been shown to exhibit bistability. Depending on
the initial condition, the final state of the wake can either develop into a
steady flow (state I), or periodic shedding (state II). In this paper, we
perform direct numerical simulations to reveal the Reynolds number effects on
these two wake states. With increasing Reynolds number, the steady vortical
structures in state I wake sways back and forth in the spanwise direction,
resulting in low-frequency fluctuations in drag forces, but not in lift. For
state II, the increase in Reynolds number is associated with the emergence of
another spectral peak in the lift coefficient. The secondary frequency is
associated with highly three-dimensional vortical structures in the wake. For
both states, the wakes transition to turblent flows at higher Reynolds
numbers, with the development of small-scale vortices. We further study the
streamwise gust flows over the wavy cylinder. The time-varying inflow velocity
results in a wide range of instantaneous Reynolds number spanning from the
absolutely unstable flow regime to the bistable regime. Depending on the
period of the inflow velocity variation, the wake perturbations grown at the
absolutely unstable flow regime can be damped out in state I wake, or grow
large enough to trigger the transition state II, resulting in loss of flow
control efficacy. The above analyses reveal novel flow physics of the bistable
states at unexplored Reynolds numbers, and showcase the complex transition
behavior between the two states in unsteady flows. The insights gained from
this study improve the understanding of the wake dynamics of the wavy
cylinder.

##  1 Introduction

Engineering structures featuring bluff bodies, such as deepwater riser, long-
span bridge cables are often plagued by the excessive fluid-induced forces and
vibrations, which increase the construction cost, and cause concerns of
fatigue failure. In order to alleviate these detrimental effects, a lot of
efforts have been devoted to the manipulation of flow behind bluff structures
(Choi et al., 2008; Kumar et al., 2008; Triantafyllou et al., 2016). In
particular, spanwise modification near the separation point, referred as 3D
forcing (Choi et al., 2008), has been proven effective for the reduction of
hydrodynamic forces on nominally two-dimensional bluff bodies. Typical
examples of this category include sinus axis and helical bumps (Owen et al.,
2001), small-size tabs (Yoon, 2005), spanwise blowing and suctions (Kim &
Choi, 2005), just to name a few.

Among the various realizations of the 3D forcing techniques, a circular
cylinder with straight axis and axially varying diameter (referred to as wavy
cylinder hereafter), has attracted a lot of attention due to its
omnidirectional geometry. Early experimental investigations by Ahmed & Bays-
Muchmore (1992) and Ahmed et al. (1993) have shown that pressure gradient
exists in the spanwise direction, leading to the formation of non-uniform
separation lines and the development of the three-dimensional wake. The
subsequent investigations discovered that such a three-dimensional wake is
associated with significant suppression of the Kármán vortex shedding, and
reduction in drag and lift forces(Lam & Lin, 2009; Xu et al., 2010; Lam & Lin,
2008; Lin et al., 2016; Jung & Yoon, 2014). Moreover, the spanwise-undulated
geometry is found to resemble the whiskers of harbor seals (Hanke et al.,
2010; Lyons et al., 2020; Murphy et al., 2021). Such shape has been shown to
exhibit superior hydrodynamic performance and to enhance the sensitivity of
the whisker even in turbid water (Beem & Triantafyllou, 2015). This finding
has inspired bio-mimicry innovations such as energy conserving flow sensors
(Beem et al., 2012).

Despite the favorable aerodynamic properties regarding the wavy cylinder as
described above, we have revealed in a series of studies that the flow control
efficacy of the wavy cylinder, even optimally designed to eliminate the Kármán
vortex shedding, can be lost in some occasions. For example, when the wavy
cylinder is yawed to the incoming flow, the Kármán vortex shedding that is
concealed in the non-yawed condition revives, leading to increased drag and
fluctuating lift (Zhang et al., 2018b). When flexibly mounted, our numerical
simulations found that the same wavy cylinder can also develop large-amplitude
vortex-induced vibrations similar to a straight cylinder (Zhang et al., 2017).
This observation is supported by the experimental study by Assi & Bearman
(2018). In addition, if the wavy cylinder is forced to oscillate in the
crossflow direction, the Kármán vortex shedding mode can revive, leading to
quasi-periodicity in the flow, as well as synchronization between the inherent
shedding frequency and the forcing frequency (Zhang et al., 2018a).

The above observations hint that Kármán vortex shedding mode is inherent to
the wavy cylinder wake. Indeed, in our latest work (Zhang et al., 2020), the
bistability in the wake of the wavy cylinder was unveiled. It was shown that,
even in the fixed configuration, the final state of the wavy cylinder wake
depends on the initial condition. As depicted in figure 1, for weakly
perturbed initial condition (such as uniform initial condition), the wake
settles to the fixed point featuring steady flow with low drag and zero lift
(state I). On the other hand, if the initial condition is highly perturbed,
the wake finally arrives at the limit cycle state featuring Kármán vortex
shedding (state II), indicating the loss of flow control efficacy. However,
our previous work was limited to low Reynolds number of $Re\approx 150$. The
wake dynamics of the wavy cylinder at higher Reynolds numbers is not
systematically explored.

Figure 1: Schematic diagram of the bi-stable states of the wavy cylinder wake
at $Re=150$. State I represents the steady wake, and state II represents the
oscillatory wake. Reproduced from Zhang et al. (2020) with permission obtained
from AIP Publishing.

In this paper, we extend our previous works to study the Reynolds number
effects ($Re$ = 30 – 300) on the bistable flows over the wavy cylinder. The
aim is to discover new flow phenomena in both wake states in the unexplored
$Re$ regimes. In addition, we simulate time-varying flows over the wavy
cylinder to reveal the complex transition behaviors between the two wake
states. The insights gained from this study should improve the current
understanding of the wake dynamics of the wavy cylinder, and potentially aid
its use in engineering applications. In what follows, we present the
computational setup in §2. The results are discussed in §3, where we unveil
some unique flow phenomena with increasing Reynolds number, and shed light
upon the transition between the two states in streamwise gust flows. We
conclude this paper by summarizing the findings in §4.

##  2 Computational setup

The geometry of the wavy cylinder is schematically depicted in figure 2. The
diameter of the wavy cylinder varies sinusoidally along the spanwise direction
$z$ according to

$D(z)=D_{m}+a\cos(2\pi z/\lambda),$ (1)

where $D_{m}$ is the averaged diameter, $a$ and $\lambda$ are the geometric
amplitude and wavelength, respectively. In the current paper, we assign
$a=0.175D_{m}$ and $\lambda=2.5D_{m}$. This set of parameters has been shown
by Lam & Lin (2009) and Zhang et al. (2020) to exhibit satisfactory flow
control efficacy. The wavy cylinder is subjected to a uniform incoming flow
$U_{\infty}$ in the $x$ direction. For non-dimensionalization, we normalize
the spatial variables by the averaged diameter $D_{m}$, velocity by
$U_{\infty}$, time by $D_{m}/U_{\infty}$, and frequency by $U_{\infty}/D_{m}$.
The Reynolds number, defined as $Re\equiv U_{\infty}D_{m}/\nu$, where $\nu$ is
the kinematic viscosity of the fluid, is varied from 30 to 300.

Figure 2: ($a$) Schematic view of problem setup and $(b)$ mesh design.

The flow is governed by the incompressible Navier-Stokes equations, which are
solved by the direct numerical simulations using the open-source software
OpenFOAM. Both the time and space are discretized with second-order accurate
schemes. The wavy cylinder is placed in the center of a circular computational
domain of $30D_{m}$ in radius. Periodic boundary condition is specified at the
spanwise ends of the domain. It is noted that at higher Reynolds numbers, the
flows at the two spanwise boundaries may not be periodic. Thus, we use three
wavelengths ($7.5D_{m}$ in spanwise direction) in the spanwise direction to
alleviate the unphysical boundary condition effects on the wake dynamics. The
cylinder surface is treated as no-slip wall. A uniform flow condition with
freestream velocity $U_{\infty}$ is specified at the inlet. In order to
achieve the state I wake, we initialize the flow with uniform velocity. For
state II wake, the flow is initialized with the saturated unsteady flow at
$Re=100$, at which the wake is absolutely unstable. The flow statistics are
collected after the unphysical transient initial flows are flushed out from
the computational domain.

An O-type mesh with resolution $N_{c}\times N_{r}\times N_{z}=200\times
180\times 180$ (where $N_{c}$, $N_{r}$ and $N_{z}$ represent the grids in the
circumferential, radial and spanwise directions) is used for the domain
discretization. The mesh is concentrated in the vicinity of the cylinder to
better resolve the near wake. The nondimensional time-step is set to be
$\Delta t=0.01$. Mesh dependency test is omitted here since it has been
carefully done in our previous study (Zhang et al., 2020).

##  3 Results

###  3.1 Reynolds number effects on bistable flows

Figure 3: ($a$) Time-averaged drag coefficient, ($b$) rms lift coefficient,
and ($c$) shedding frequency of the wavy cylinder. The shaded area indicates
the bistable regime.

We first present an overview of the bistable flows from the perspective of
aerodynamic forces of the wavy cylinder. The time-averaged drag coefficients
($\overline{C_{D}}$), root-mean-squared (rms) lift coefficients
($C_{L}^{\prime}$), and the dominant shedding frequency $St$ are shown in
figure 3. For $Re\lesssim 110$, the flow over wavy cylinder exhibits similar
dynamic features with that of the two-dimensional cylinder (Zhang et al.,
2020). The wake remains steady for $Re$ below 60, and becomes absolutely
unstable with increasing Reynolds number. The bistability commences at
$Re=120$, as evidenced by the bifurcations in the $\overline{C_{D}}$-$Re$ and
$C_{L}^{\prime}$-$Re$ curves. The drag and lift coefficients in state I are
significantly smaller than that in state II. No dominant frequency is detected
in the lift coefficients for state I wakes, because the lift is either exactly
zero for $Re=120$ – 220 (note that this does not mean the wake is absolutely
steady), or feature small random variations at higher $Re$.

The representative vortical structures in the bistable regime are shown in
figure 4, with a preview of the Reynolds number effects. For state I, the
wakes at $Re=120-150$ are absolutely steady. The vortical structures
visualized by $Q$ criteria feature a tongue-like structure that is located
within each geometric wavelength. In between $Re=160$ and 220, these tongue-
like structures sways back and forth with low frequency, resulting in
unsteadiness in the drag coefficients, but not in lift. For $Re$ beyond 230,
the vortex sheets breaks down in downstream, populating the far wake with
small-scale vortices. A more detailed analysis of the $Re$ effects on state I
wake is presented in §3.1.1.

Figure 4: Summary of Reynolds-number effects on the bistable flows over the
wavy cylinder. The vortical structures are visualized by isosurfaces of
$Q=0.1$ for state I, and $Q=0.5$ for state II.

For state II, the wake is primarily dominated by periodic vortex shedding. The
spanwise vortex tubes are more or less associated with waviness that is
dictated by the geometric wavelength. With increasing Reynolds number, the
three dimensionality enhances in the wake. This is also reflected in the lift
spectra as increasing numbers of dominant frequencies, as will be discussed in
§3.1.2.

####  3.1.1 Reynolds number effects on state I

Figure 5: Time histories of drag and lift coefficients of state I. $(a)$
$Re=150$, $(b)$ $Re=160$, $(c)$ $Re=200$ and $(d)$ $Re=300$.

In state I, the large-scale Kármán vortex shedding is suppressed in the wake.
However, with increasing Reynolds number, the wake can still exhibit
unsteadiness. We show the drag and lift coefficients of the wavy cylinder in
state I wake at different Reynolds numbers in figure 5. At $Re=150$, both drag
and lift coefficients are constant after the initial transient are flushed
out, suggesting the complete steadiness of the wake. As $Re$ increases to 160,
while the lift coefficient remains at zero, the drag coefficient exhibit
periodic variation beyond $t\approx 500$. It is noted that such phenomenon is
associated with even smaller time-averaged drag than that in the steady flow
regime. Such phenomenon is also observed with increasing Reynolds number,
although the time at which the unsteadiness in drag commences is much earlier
than that at $Re=160$. At higher Reynolds numbers, both drag and lift exhibit
random unsteadiness, suggesting the emergence of small-scale vortices and the
transition to turbulent flow.

Figure 6: Unsteady flow for the state I at $Re=160$. $(a)$ Drag coefficient
and $(b)$ vortical structures visualized by isosurfaces of $Q=0.1$. The three
time instants correspond to the black dots in ($a$).

To understand the unique unsteady flow at $Re=160$ – 220 where the drag is
unsteady but the lift is not, we plot the vortical structures of state I wake
at $Re=160$ in figure 6. As the drag coefficient varies periodically with
time, the tongue-like vortical structure within each wavelength sways back and
forth along the spanwise direction in sync. Each time the vortical structures
are positioned near the section of minimum diameter, the drag coefficient
reaches maximum. The minimum drag is reached when the vortical structures are
positioned towards either sides. Thus, one cycle of vortical motion
corresponds to two cycles in the drag variation. Since the sway motions of the
vortical structures are symmetric with respect to the $x$-$z$ plane, such
unsteady flow does not induced variation in the lift coefficients. Compared
with the Kármán vortex shedding, the frequency of the vortical motion is much
lower. A summary of the period of the sway motion of the vortical structures
is shown in table 1. With increasing Reynolds number, the sway period
decreases initially, and saturates at $T_{sway}\approx 80$ for higher $Re$.

Table 1: Summary of the periods of the sway motion of the vortical structures $Re$ | 160 | 170 | 180 | 190 | 200 | 210 | 220
---|---|---|---|---|---|---|---
$T_{sway}$ | 94.1 | 86.3 | 81.4 | 80.5 | 81.1 | 81.7 | 81.8

####  3.1.2 Reynolds number effects on state II

Figure 7: ($a$) Lift spectrum and ($b$) time history of lift coefficients for
state II.

The lift coefficients and their spectra of the state II wakes are shown in
figure 7 for some representative Reynolds numbers. At $Re=150$, the lift
coefficient exhibit purely periodic oscillations dominated by a single
frequency at $f=0.185$. With increasing Reynolds number, the lift coefficient
becomes quasi-periodic, and an additional frequency emerges at $f=0.244$,
although its spectral amplitude is much weaker. At higher Reynolds number,
multiple peaks are observed in the lift spectrum, suggesting the flow is
becoming turbulent.

Figure 8: Dynamic modes for the state II wake at $Re=200$. $(a)$ $f_{1}=0.185$
and ($b$) $f_{2}=0.244$. The two modes are visualized using the same contour
level of $Q$ criterion.

We use dynamic mode decomposition (Schmid, 2010, 2022) to reveal the coherent
structures associated with the two prominent frequencies at $Re=200$. As shown
in figure 8, for the primary shedding frequency $f_{1}=0.185$, the dynamic
mode features spanwise coherent vortical structures with slight waviness. Such
structure resembles the typical two-dimensional Kármán vortex shedding mode,
and is the dominant dynamic mode for the single-frequency unsteady flows at
lower Reynolds numbers. For $f_{2}=0.244$, the modal structure appears more
complex as it involves both spanwise and streamwise vortical structures. In
addition, the vortical structures in this mode are also observed in the far
wake, while the mode associated with $f_{1}$ is only strong in the near wake.
Nevertheless, the effects of the second mode on the lift of the wavy cylinder
is much smaller than the previous one.

###  3.2 Wake dynamics in time-varying inflow

We further study the aerodynamics of the wavy cylinder in streamwise gust
flows. In order to save computational time, the spanwise length of the
computational model is reduced to a single wavelength of $2.5D_{m}$ in these
simulations. A time-varying inflow velocity is specified at the inlet as

$U_{in}=[\operatorname{erf}(0.02(t-300))+1]\cdot 0.35\cdot\sin(2\pi t/T)+1.5,$
(2)

where $T$ denotes the period of the sinusoidal velocity variation. With the
Gaussian error function in equation (2), the inlet velocity initially stay
fixed, resulting in a constant Reynolds number of 150 up to $t=300$. This
allows the wake to develop into the steady state. Beyond $t=300$, the velocity
transitions to the sinusoidally varying part smoothly. The maximum
instantaneous Reynolds number is 220, at which the flow exhibits bistability,
and the minimum 80, at which the wake is absolutely unstable.

We assess the drag and lift coefficients of the wavy cylinder under different
period $T$ in figure 9. For $T=6$, which is of the same scale with the natural
shedding period $T_{n}=5.55$ for the state II wake at $Re=150$, the dynamic
inflow induces a quasi-periodic lift coefficient. This is typical for flows
over streamwise oscillating cylinders (Leontini et al., 2011; Kim & Choi,
2019), which is a sister problem of the current varying velocity over a fixed
cylinder. This suggests that, despite the initially steady flow, the unsteady
shedding state in the wavy cylinder’s wake revives, and interacts with the
velocity excitation, giving rise to the quasi-periodic flow behavior. This
observation is in accordance with our previous findings regarding the wake
response of wavy cylinder under external forcings (Zhang et al., 2018a, 2020).

For $T=200$, which is much higher than natural shedding period, the lift
coefficient remains low, and exhibits a pulsing behavior. During each cycle of
velocity variation, the lift starts to fluctuate when the velocity is about to
reach minimum. Such oscillations are due to the absolute instability of the
wake as the instantaneous Reynolds number drops below $Re\approx 120$. As the
velocity increases from its minimum value, the fluctuation in the lift
coefficient grows drastically. However, since the perturbation in the flow
does not have time grow strong enough to trigger the transition from state I
to state II, the oscillatory lift coefficient peaks before the velocity
reaches the maximum value, and then damps out at high instantaneous Reynolds
numbers, at which the wake stays at state I.

Figure 9: Drag and lift coefficients of the wavy cylinder under time-varying
inflow velocity. $(a)$ $T=6$, $(b)$ $T=200$ and $(c)$ $T=1000$. The gray solid
lines are the force coefficients. The black dashed lines denote the variations
of the inflow velocity, which results in a range of instantaneous Reynolds
number from 80 to 220.

Given long period of $T=1000$, the perturbation in the flow is allowed to grow
strong enough to trigger the transition to state II, as shown in figure
9($c$). The fluctuations in the lift coefficient are almost in phase with the
inlet velocity, and are much higher than that in the $T=200$ case. In
addition, as the velocity reaches maximum, oscillations also become noticeable
in the drag coefficient. The wake vortical structures for the $T=200$ and
$T=1000$ cases are shown in figure 10. In the former, the flow is featured by
the typical state I wake with weak undulations. For the latter, the wake is
unsteady throughout one velocity variation cycle, and the vortical structures
appear stronger with increasing instantaneous inlet velocity. While an
exhaustive investigation of the effect of the streamwise gust period $T$ on
the wake behavior is out of scope for this study, the above analysis showcases
the complex transition behaviors between the bistable states in an unsteady
flow setting, and calls for caution in the use of wavy cylinder as a flow
control device.

Figure 10: Wake structures in the wake of the wavy cylinder under time-varying
inflow velocity. $(a)$ $T=200$ and $(b)$ $T=1000$. The vortical structures are
visualized by isosurfaces of $|\bm{\omega}|$ (transparent) and $Q$ (solid).

##  4 Conclusions

We have carried out direct numerical simulations to reveal the Reynolds number
effects on the wake dynamics of a wavy circular cylinder. Depending on the
initial condition, the wake of the wavy cylinder can settle into two distinct
states, i.e., one featuring low drag and minimum lift fluctuation (state I),
and another featuring unsteady vortex shedding with higher drag and lift
(state II). For state I, the wake is characterized by steady tongue-like
vortical structures at low $Re$. With increasing Reynolds number, such tongue-
like structures sway back and forth in the spanwise direction. The sway motion
is related to low-frequency variation in the drag force, but does not affect
the lift. For state II, a transition from single-frequency vortex shedding to
dual-frequency is observed with increasing Reynolds number. The emergence of
the second peak is associated with highly three-dimensional vortical
structures. The wakes at both states transitions to turbulent flows with the
emergence of small-scale vortices at higher Reynolds numbers.

In addition, we have also simulated steamwise gust flows over the wavy
cylinder to shed light on the transition behaviors between the two wake
states. The time-varying inflow velocity results in a wide range of
instantaneous Reynolds number spanning from the absolutely unstable flow
regime to the bistable regime. For gust period that is of the same scale with
the natural shedding period of the state II wake, the unsteady shedding state
revives despite the initially steady flow, and leads to quasi-periodic flow
behavior. With much higher gust period, the wake perturbations grow in the
absolutely unsteady flow regime as the instantaneous Reynolds number drops
below the threshold of bistability. If the perturbations do not grow strong
enough in this regime, they are damped out in the state I wake as the
instantaneous $Re$ enters the bistable regime, However, given long enough gust
period, the perturbations can grow significantly strong to trigger the
transitions from state I to state II, giving rise to unsteady flow over the
entire gust period.

The above analysis have revealed novel flow physics regarding the bistable
states of the wavy cylinder at unexplored Reynolds numbers, and showcased the
complex transition behavior between the two states in an unsteady flow
setting. The insights gained from this study improve the understanding of the
wake dynamics of the wavy cylinder, and could potentially guide its use in
engineering applications.

## Acknowledgments

KZ is grateful for the computing resources at Amarel cluster provided through
the Office of Advanced Research Computing (OARC) at Rutgers University, on
which some of the simulations were carried out. HZ acknowledges the financial
support from Natural Science Foundation of China (grant no. 52101322). DZ
thanks the support from Program for Intergovernmental International S&T
Cooperation Projects of Shanghai Municipality, China (grant no. 22160710200)
and the Oceanic Interdisciplinary Program of Shanghai Jiao Tong University
(grant no. SL2020PT201).

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