Universidade de São Paulo
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2010
Possible states in the flow around two circular
cylinders in tandem with separations in the
vicinity of the drag inversion spacing
PHYSICS OF FLUIDS, v.22, n.5, 2010
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Possible states in the flow around two circular cylinders in tandem
with separations in the vicinity of the drag inversion spacing
B. S. Carmo, J. R. Meneghini, and S. J. Sherwin
Citation: Phys. Fluids 22, 054101 (2010); doi: 10.1063/1.3420111
View online: http://dx.doi.org/10.1063/1.3420111
View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v22/i5
Published by the American Institute of Physics.
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PHYSICS OF FLUIDS 22, 054101 共2010兲
Possible states in the flow around two circular cylinders in tandem
with separations in the vicinity of the drag inversion spacing
B. S. Carmo,1,2,a兲 J. R. Meneghini,2,b兲 and S. J. Sherwin1,c兲
1
Department of Aeronautics, Imperial College, London SW7 2AZ, United Kingdom
Department of Mechanical Engineering, NDF, University of São Paulo, Poli, São Paulo 05508-900, Brazil
2
共Received 2 October 2009; accepted 29 March 2010; published online 4 May 2010兲
The possible states in the flow around two identical circular cylinders in tandem arrangements are
investigated for configurations in the vicinity of the drag inversion separation. By means of
numerical simulations, the hysteresis in the transition between the shedding regimes is studied and
the relationship between 共three-dimensional兲 secondary instabilities and shedding regime
determination is addressed. The differences observed in the behavior of two- and three-dimensional
flows are analyzed, and the regions of bistable flow are delimited. Very good agreement is found
between the proposed scenario and results available in the literature. © 2010 American Institute of
Physics. 关doi:10.1063/1.3420111兴
I. INTRODUCTION
In the external flow around solid bodies, it is a wellknown fact that the presence of other bodies in close proximity can change fundamental aspects of the flow, such as
fluid forces and transition thresholds. The effect of the presence of additional bodies in the fluid stream is called flow
interference. A particular type of flow interference which is
specially severe is the wake interference, which happens
when one body is immersed in the wake of another body.
In the case of bluff bodies, the most commonly applied
model to study wake interference is the flow around two
identical circular cylinders placed in tandem arrangements,
as illustrated in Fig. 1. It is known from experiments1,2 and
computations3,4 that different vortex shedding regimes can be
observed in the flow around this type of arrangement, depending on the center-to-center separation Lx. Adopting the
classification presented by Carmo et al.,5 illustrated in Fig. 2,
we see that three different shedding regimes are observed for
low Reynolds numbers. For very small separations, the shedding regime SG 共symmetric in the gap兲 is observed, as
shown in Fig. 2共a兲. In this regime, a pair of almost symmetric vortices is formed in the gap between the cylinders and
the root mean square 共rms兲 of the lift on the downstream
cylinder is very small. If the separation is gradually increased, the shedding regime eventually changes to AG 共alternating in the gap兲, in which regions of concentrated vorticity grow and decrease alternatively in time on each side of
the line that links the centers of the cylinders 关see Fig. 2共b兲兴.
This makes the rms of the lift coefficient on the downstream
cylinder increase. It is also important to highlight that the
drag on the downstream cylinder is usually negative for
shedding regimes SG and AG, as illustrated in Figs. 2共a兲 and
2共b兲. Finally, for larger separations, a complete vortex wake
is formed in the interstitial region, the rms of the lift on the
downstream cylinder increases significantly and the mean
a兲
Electronic mail: [email protected].
Electronic mail: [email protected].
c兲
Electronic mail: [email protected].
b兲
1070-6631/2010/22共5兲/054101/7/$30.00
drag on the downstream cylinder becomes positive, as shown
in Fig. 2共c兲. This shedding regime is called WG 共wake in the
gap兲. Since the transition between the shedding regimes AG
and WG is marked by the inversion of the drag coefficient on
the downstream cylinder, the transition from one to the other
is referred to as drag inversion, and in this paper we focus on
flows in the vicinity of such transition.
It is known that the separation at which the drag inversion occurs depends on the initial conditions because, at least
for low Reynolds numbers, the flow is bistable in the vicinity
of the drag inversion point.6 For this reason, it is more appropriate to refer to a drag inversion range than to a drag
inversion point. For a fixed separation, this range is defined
in terms of Reynolds numbers. Likewise, for a fixed Reynolds number this range is defined in terms of center-tocenter distance. As far as the authors are aware, no study to
date has calculated the drag inversion range for low Reynolds numbers while taking into account the bistable nature
of the flow in this region 共in the work by Papaioannou et al.,7
the drag inversion separation was calculated only for increasing Reynolds numbers兲.
A recent study by Carmo et al.5 investigated the threedimensional instabilities observed in the nominally twodimensional time-periodic flow around two identical circular
cylinders placed in tandem in relation to the free stream.
These instabilities are known as secondary instabilities, since
they occur after the primary instability, which is the transition from steady flow to two-dimensional time-periodic flow.
The primary instability in the flow around two circular cylinders in tandem was investigated by Mizushima and
Suehiro.6 The results obtained by Carmo et al.5 were compared to those obtained for an isolated cylinder.8–10 A summary of the main results of that work are reproduced in Fig.
3, in which it can be seen that different modes appear in the
transition to three-dimensional flow in the wake for separations smaller than the drag inversion spacing. For such cases,
the three-dimensional structures appeared later in terms of
Reynolds number than for the flow around an isolated cylinder. It was shown that for configurations at shedding regime
22, 054101-1
© 2010 American Institute of Physics
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054101-2
Phys. Fluids 22, 054101 共2010兲
Carmo, Meneghini, and Sherwin
FIG. 1. Sketch of the flow around two circular cylinders in tandem
arrangement.
SG, the unstable mode at the onset of the secondary instability originated at the formation region, downstream of the
leeward cylinder. This mode, referred to as mode T1, has a
topology that breaks the spatial symmetry of the base flow,
and its physical mechanism appears to be associated with a
hyperbolic instability. For slightly larger separations, the
shedding regime changed to AG and a different unstable
mode, named mode T2, was observed. Mode T2 has its origin at the base of the downstream cylinder, upstream of the
vortex formation region. A centrifugal instability in this region seems to give rise to this mode. Like the single cylinder
mode A,8 mode T2 wake topology keeps the in-plane spatial
symmetry observed in the base flow. If the separation is increased a little more, but not so much as to change the base
flow shedding regime, a new unstable mode 共mode T3兲 is
initiated at the interstitial region. Like mode T1, mode T3
also breaks the spatial symmetry of the base flow. Some of
the mode attributes suggest that the underlying physical
mechanism is a cooperative elliptical instability. On the other
hand, if the separation was greater than the drag inversion
spacing 共shedding regime WG兲, the initial stages of the transition in the wake occurred in a similar way to that of the
isolated cylinder. The first three-dimensional instability,
mode A, arose earlier in Reynolds number terms when compared to the single cylinder case, and it is therefore concluded that the downstream cylinder has a destabilizing effect on the flow for separations larger than the drag inversion
spacing.
Although by Carmo et al.5 a full characterization of the
modes was presented and physical mechanisms were proposed to explain the instabilities, that paper did not address a
point of high practical interest in engineering, which is how
the onset of three-dimensional instabilities affect the drag
inversion. In the present paper, we investigate in detail how
the transition to three-dimensional flow affects the vortex
shedding regime, focusing on the vicinity of the drag inversion spacing and taking into account the hysteresis of the
regime transition. We obtain the possible flow states for
Reⱕ 500 by means of direct numerical simulations and use
the current results to explain previously published computational data.
1.5
CD, CL
1
0.5
0
-0.5
-1
-1.5
0
10
(a) Lx /D = 1.5 – regime SG
20
30
40
20
30
40
20
30
40
tU/D
1.5
CD, CL
1
0.5
0
-0.5
-1
-1.5
0
10
(b) Lx /D = 2.3 – regime AG
tU/D
1.5
CD, CL
1
0.5
0
-0.5
-1
-1.5
0
(c) Lx /D = 5 – regime WG
10
tU/D
FIG. 2. 共Left兲 Instantaneous vorticity contours illustrating the different shedding regimes observed in the flow around two circular cylinders in tandem
arrangements. Contours vary from ␻zD / U⬁ = −2.2 共light contours兲 to ␻zD / U⬁ = 2.2 共dark countours兲. 共Right兲 Drag coefficient 共gray solid line兲 and lift
coefficient 共black dashed line兲 time series for the downstream cylinder. Re= 200, two-dimensional simulations.
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054101-3
Phys. Fluids 22, 054101 共2010兲
Possible states in the flow around two circular cylinders
40
y
20
0
-20
-40
-20
0
x
20
40
FIG. 4. Mesh employed in the calculations of the flow around the configuration with Lx / D = 5.
FIG. 3. Variation in the critical Reynolds numbers 共a兲 and respective perturbation wavelengths 共b兲 with the center-to-center separation for modes T1
共䊏兲, T2 共䊊兲, T3 共〫兲, and A 共䉱兲. Mode T1 bifurcates from shedding regime
SG, modes T2 and T3 from shedding regime AG and mode A from shedding
regime WG.
II. NUMERICAL METHOD
The flows investigated in this paper were calculated using numerical simulations of the incompressible Navier–
Stokes equations, here written in nondimensional form,
1 2
⳵u
ⵜ u,
= − 共u · ⵜ兲u − ⵜp +
Re
⳵t
共1兲
ⵜ · u = 0.
共2兲
The cylinder diameter D is the reference length and the freestream speed U⬁ is the reference speed used in the nondimensionalization. u ⬅ 共u , v , w兲 is the velocity field, t is the
time, p is the static pressure, Re= ␳U⬁D / ␮ is the Reynolds
number, and ␮ is the dynamic viscosity of the fluid. The
pressure was assumed to be scaled by the constant density ␳.
The numerical solution of these equations was calculated
using a Spectral/hp discretization.11 The time integration
scheme adopted was a stiffly stable splitting scheme.12
Polynomials of degree 8 were used in the discretization
of the meshes for the two-dimensional simulations. The
meshes employed were the same as those used to obtain the
base flows by Carmo et al.;5 an example is shown in Fig. 4.
The boundary conditions were u = 1, v = 0 on the left, upper,
and lower boundaries of the mesh in the figure, ⳵u / ⳵x = 0,
⳵v / ⳵x = 0 on the right 共outflow兲 boundary, and u = v = 0 on the
cylinders’ walls. A high-order pressure boundary condition12
was employed on every boundary apart from the outflow
boundary, on which p = 0 was imposed.
The three-dimensional simulations were performed using a three-dimensional version of the Navier–Stokes solver
which uses a Spectral/hp element discretization in the xy
plane and Fourier modes in the spanwise direction.13 The
advantages of this approach is the high efficiency in the code
parallelization and that the meshes generated for the twodimensional simulations can be reused. In the threedimensional simulations, domains with spanwise lengths between 8D and 12D were employed, in order to comply with
the wavelength of the instability that was expected to arise.
Depending on the Reynolds number and spanwise length, 16
or 32 Fourier modes were used in the discretization in the
spanwise direction. Periodic boundary conditions were enforced on the planes at the boundaries perpendicular to the
cylinder axis.
In Sec. III we also present the critical Reynolds numbers
for the primary and secondary instabilities in the wake. The
data referring to the secondary instabilities were extracted by
Carmo et al.,5 but the data referring to the primary instabilities were calculated. To obtain the steady base flows, we
have employed the method presented by Tuckerman and
Barkley14 with the modification suggested by Blackburn.15
The stability analysis procedure was the same as that used by
Carmo et al.5
III. RESULTS AND DISCUSSION
A number of two- and three-dimensional simulations
were performed to investigate the boundaries of the drag
inversion range, fixing the geometric configuration and varying the Reynolds number. Each of the calculations was run
for at least 300 nondimensional time units for the twodimensional simulations and for at least 100 nondimensional
time units for the three-dimensional simulations. The mean
drag coefficient was used as the indicator of the shedding
regime 共AG or WG兲. Due to the hysteretic nature of the transition between these regimes, determining the upper 共lower兲
boundary requires that we start our flow simulations at a
lower 共upper兲 Reynolds number and increase 共reduce兲 its
value. The boundaries were defined taking the Reynolds
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054101-4
Carmo, Meneghini, and Sherwin
FIG. 5. Values of the drag coefficient of the downstream cylinder obtained
for the configuration with Lx / D = 3.8. The symbols in the graph correspond
to the different shedding regimes observed: 共䉱兲 WG, 共䊐兲 AG, and 共쎲兲
steady wake. The simulations were performed for increasing and decreasing
Reynolds number, so the hysteresis in the transition between the regimes AG
and WG could be outlined. The vorticity contours on the right hand side
illustrate each of the shedding regimes; dark contours mean positive vorticity and light contours mean negative vorticity.
number of the first calculation that showed a change in regime, with an uncertainty of ⌬Re= ⫾ 0.5.
The primary instability was investigated by means of
global linear stability analysis of the solutions of the steady
Navier–Stokes equations. The critical Reynolds number for
each of the configurations was obtained using a search algorithm that employed Newton’s method. The resolution in
Reynolds number of this algorithm was 1, i.e., only integer
Reynolds numbers were tested, and the Reynolds number
was considered to be the critical one if the real part of the
eigenvalue resulting from the stability calculations had
modulus smaller than 0.001.
Although is usually said that regime AG is associated
with a negative mean drag on the downstream cylinder, this
is not always the case for low Reynolds numbers. Figure 5
shows values of mean drag coefficient obtained for the configuration with Lx / D = 3.8 and vorticity contours illustrating
the three different wakes observed 共steady flow, shedding
regime AG and shedding regime WG兲. The calculations were
performed with increasing and decreasing Reynolds numbers. The discontinuities associated with the changes in regime are clear in the graph, as well as the hysteresis in the
transition between the regimes AG and WG. It can be seen
that, for regime AG, the drag coefficient is small, but not
negative. For that reason, it was impossible to define a general threshold value for the drag coefficient which would
indicate the change in shedding regime; in order to find the
thresholds of shedding regime transition it was necessary to
examine each configuration individually, checking the flow
field contours and drag coefficient time histories.
Figure 6 displays the results of the calculations on a Re
versus Lx / D map. The curves showing the variation in the
Phys. Fluids 22, 054101 共2010兲
FIG. 6. Map of Reynolds number against center-to-center separation showing the possible vortex shedding regimes and the variation in the critical
Reynolds numbers of modes T3 and A with Lx / D, in the neighborhood of
the drag inversion range. Symbols are 共〫兲 mode T3 critical Reynolds numbers, 共䉱兲 mode A critical Reynolds numbers, 共⫻兲 primary instability critical
Reynolds numbers, 共⽧兲 two-dimensional transition from shedding regime
WG to steady flow, 共䊏兲 two-dimensional vortex shedding transition from
WG to AG, 共䊐兲 three-dimensional vortex shedding transition from WG to
AG, 共쎲兲 two-dimensional vortex shedding transition from AG to WG, and
共䊊兲 three-dimensional vortex shedding transition from AG to WG. Dashed
lines are employed for data obtained from stability calculations, while solid
lines are used for data calculated by means of direct numerical simulations.
critical Reynolds number with the center-to-center distance
for the modes A and T3, obtained by Carmo et al.,5 and the
critical Reynolds numbers for the primary instability are also
plotted on the map. The four regions of bistable flow are
marked in shades of gray. The bottom one is located under
the curve of critical Reynolds numbers for the primary instability and corresponds to a region in the parameter space
where only steady flow or two-dimensional flow with shedding regime WG are possible. The second gray region from
the bottom is located between the curve of critical Reynolds
numbers for mode A and the curve of critical Reynolds numbers for the primary instability. Hence only two-dimensional
time-periodic flows are possible in this region, but the vortex
shedding regime can be either AG 共2d-AG兲 or WG 共2d-WG兲,
depending on the initial conditions. The third region of
bistable flow is between the curves of critical Reynolds numbers for mode A and mode T3. In this region, twodimensional flows at regime AG 共2d-AG兲 and threedimensional flows at regime WG 共3d-WG兲 are possible.
Lastly, the fourth top region of bistable flow is situated above
the mode T3 critical Reynolds number curve. The flow in
this region is always three dimensional and, depending on
the initial conditions, the vortex shedding regime can be AG
共3d-AG兲 or WG 共3d-WG兲. Plots of vorticity isosurfaces, obtained by means of three-dimensional simulations, are shown
in Fig. 7. These plots illustrate each of the states observed
and discussed previously.
To help understand how to interpret the map in Fig. 6, let
us describe two examples of change in state. Suppose we
have a flow around the configuration with Lx / D = 3.8 and
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054101-5
Possible states in the flow around two circular cylinders
Phys. Fluids 22, 054101 共2010兲
(a) Lx /D = 3.5, Re = 400: 3d-AG
(b) Lx /D = 3.5, Re = 350: 3d-WG
(c) Lx /D = 3.7, Re = 75: 2d-AG
(d) Lx /D = 3.7, Re = 85: 2d-WG
FIG. 7. Plots of vorticity isosurfaces illustrating the different states observed in the vicinity of the drag inversion separation, three-dimensional simulations.
Translucent surfaces represent isosurfaces of 兩␻z兩. Solid light gray and dark gray surfaces represent isosurfaces of negative and positive ␻x, respectively.
Re= 175. According to the map in Fig. 6, this flow will be
three dimensional with shedding regime WG 共3d-WG兲. If the
Reynolds number is gradually increased to 300, the flow will
then be located inside the upper gray region in the map, in
which the regimes 3d-AG and 3d-WG are possible. Since the
previous state of the flow was 3d-WG and the change in
Reynolds number was gradual, the flow will keep the same
state at Re= 300. If the Reynolds number is further increased
to 500, the flow will now be in a region of the parameter
space where only 3d-AG flows are possible, so the flow will
then change to this state. If the Reynolds number is then
gradually changed back to 300, the flow will keep the state
3d-AG. The same reasoning is valid if the separation between the cylinders is changed instead of the Reynolds
number—the flow will always retain its previous state when
entering a gray region coming from a white region.
It may also happen that the flow at a certain state is taken
gradually to a region in which two states are possible, but
none of them is the initial state of the flow. In this case the
flow will assume the possible state that retains the shedding
regime of the initial state. For example, suppose we have a
flow around the configuration with Lx / D = 3.2 at Re= 250
with regime 3d-AG; the flow will be in the upper gray region
of Fig. 6. If the Reynolds number is gradually decreased to
150, the flow will then be in the dark gray region immediately below, in which two states are possible, 2d-AG and
3d-WG. However, none of them is equal to the initial state of
the flow. So the flow will change to the state that keeps the
shedding regime, i.e., the flow will change to 2d-AG.
A point worthy of note is that the boundaries of the drag
inversion range have different orientations depending on
whether the flow is two dimensional or three dimensional:
the boundaries have a negative slope for two-dimensional
flows and a positive slope for three-dimensional flows. This
means that once the flow is unstable to three-dimensional
perturbations, the dependence of the shedding regime on the
Reynolds number is inverted. It has been shown by Carmo
and Meneghini16 that in two-dimensional flows, an increase
in the Reynolds number makes the formation length shorter.
This occurs because at higher Reynolds numbers the spanwise vorticity in the shear layers that separate from the top
and the bottom of the cylinder wall is stronger, and this
stronger vorticity facilitates the interaction between these
shear layers. A shorter formation length favors shedding regime WG. In contrast, when the flows were three dimensional the results obtained by Carmo and Meneghini16
showed longer formation lengths, owing to the fact that
three-dimensional diffusion and spanwise decorrelation
weakened the interaction between the opposite shear layers.
DPIV measurements17 showed that, for the flow around an
isolated cylinder, the formation length increases with Reynolds number for 共300⬍ Re⬍ 1500兲, indicating that the
three-dimensional effects prevail over the two-dimensional
ones in this Reynolds number range. The results in Fig. 6
demonstrate that this is also true for the flow around two
circular cylinders in tandem. This was also one of the conclusions drawn by Papaioannou et al.,7 who deduced that the
variation in the single cylinder formation length and varia-
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054101-6
Phys. Fluids 22, 054101 共2010兲
Carmo, Meneghini, and Sherwin
tion in the tandem arrangement drag inversion separation
with Reynolds number appeared to be consistent in both twoand three-dimensional simulation results.
Using Fig. 6, we can draw comparisons between the current results and data from earlier research that employed
three-dimensional numerical simulations. In Ref. 18 threedimensional simulations of the flow around diverse tandem
configurations at Re= 220 were performed using a virtual
boundary method. The authors of that paper observed threedimensional flow for all configurations with Lx / D ⱖ 4,
whereas for Lx / D = 2 the flow remained two dimensional.
These results did not depend on the initial conditions. This is
entirely consistent with the map in Fig. 6: for Re= 220, the
configuration Lx / D = 2 is in a zone where only twodimensional flow in the AG regime is possible, and the configurations with Lx / D ⱖ 4 are in a zone where only threedimensional flows in the WG regime are possible. When
analyzing the results of Deng et al.18 for Lx / D = 3.5, it should
be borne in mind that the spanwise length of the domain used
in their computations was 8D. They observed that the emergence of three-dimensional structures depended on the initial
conditions: three-dimensional flow was seen to occur if the
initial flow field was in the WG regime, whereas any threedimensional perturbations died out if a flow in the AG regime was used as initial condition. In the map in Fig. 6, it
can be seen that the flow around the configuration with
Lx / D = 3.5 at Re= 220 is in a region where the possible flows
are three dimensional, either in the AG or in the WG regimes.
However, Fig. 3共b兲 shows that, for Lx / D = 3.5, mode T3
wavelength at the onset 共Recr = 217兲 is ␭z / D = 9.97. Therefore, the calculations performed by Deng et al.18 were unable
to capture mode T3 instability because the spanwise length
of the domain they used was too short. Deng et al.18 also
tried to find the Reynolds number for which the flow around
the configuration Lx / D = 3.5 at regime AG would become
unstable to three-dimensional perturbations. They ran a series of simulations increasing the Reynolds number in steps
of 10, using the final solution of each simulation as the initial
condition for the next one. They observed that the flow became three dimensional for Re= 250, and the wavelength and
symmetry of the three-dimensional structures were similar to
those of mode A. However, our results show that this transition occurs at Re= 217, and the unstable mode should be
mode T3. Again, this difference can be explained by the
short spanwise length of their calculations. We performed
additional two-dimensional calculations and found that the
upper Re limit of the two-dimensional drag inversion range
for configuration Lx / D = 3.5 is Re= 240. We therefore assume
that from Re= 240 to Re= 250 there was a shedding regime
change to WG in the simulations of Deng et al.,18 and that
this was accompanied by the appearance of mode A structures in the flow.
In another study that used three-dimensional numerical
simulations, Papaioannou et al.7 observed regime AG for
250ⱕ Reⱕ 500 in simulations of the flow around configurations with Lx / D ⱕ 3.5, whereas regime WG was observed in
simulations of the flow around configurations with Lx / D
ⱖ 3.8. The current results are mostly in line with this; it can
be seen in the map in Fig. 6 that three-dimensional flow in
the AG regime is possible for Lx / D ⱕ 3.5 for 250ⱕ Re
ⱕ 500 and three-dimensional flow in the WG regime is possible for Lx / D ⱖ 3.8 for 250ⱕ Reⱕ 475. The only disagreement between our results and those from Papaioannou et al.7
is in the regime observed for Lx / D = 3.8 at Re= 500. For this
case, the map in Fig. 6 indicates that only three-dimensional
flow in the AG regime is possible, while in Papaioannou
et al.7 it is reported that regime WG is found at the same
conditions. A possible reason for this discrepancy is the number of nondimensional time units for which the flow equations are integrated. We have used 100 nondimensional time
units for all calculations, and we observed that for some of
the cases, the change in regime only happened after the equations were integrated for 40 or 50 nondimensional time units.
In Papaioannou et al.7 the time length of their calculations is
not reported. Another possible reason has to do with the size
of the domain. The mesh used by Papaioannou et al.7 is
significantly smaller than that used for the current results.
IV. CONCLUSION
In this paper, a thorough investigation of the possible
flow states in the drag inversion range of the flow around two
circular cylinders in tandem at low Reynolds numbers was
presented. For the first time, the regions of bistable flow
were carefully identified, taking into account the hysteresis
of the shedding regimes and the influence of the secondary
instabilities. The presence of three-dimensional flow structures was observed to induce notable changes in the response
of the flow to the variation of Reynolds number and cylinder
separation. Results available in the literature were reviewed
in the light of the new data and almost all the observations
made by other authors were consistent with the current findings. We believe that the analysis presented helps to improve
the understanding of flows with wake interference and can be
very useful for future investigations of other aspects of such
flows.
ACKNOWLEDGMENTS
B.S.C. thankfully acknowledges CAPES-Brazil for financial support during his Ph.D. at Imperial College London.
J.R.M. wishes to acknowledge FAPESP, CNPq, Finep, and
Petrobras for financial support. S.J.S. would like to acknowledge financial support under an EPSRC-GB advanced research fellowship.
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