Possible states in the flow around two circular cylinders in tandem

1
Possible states in the flow around two circular
2
cylinders in tandem with separations in the vicinity
3
of the drag inversion spacing
4
Bruno S. Carmoa , Julio R. Meneghinib , Spencer J. Sherwina
a Department
5
6
b NDF,
of Aeronautics – Imperial College London, UK
Department of Mechanical Engineering, Poli, University of São Paulo, Brazil
Abstract
7
8
The possible states in the flow around two identical circular cylinders in
9
tandem arrangements are investigated for configurations in the vicinity of the
10
drag inversion separation. By means of numerical simulations, the hysteresis
11
in the transition between the shedding regimes is studied and the relationship
12
between (three-dimensional) secondary instabilities and shedding regime de-
13
termination is addressed. The differences observed in the behaviour of two-
14
and three-dimensional flows are analysed, and the regions of bi-stable flow
15
delimited. Very good agreement is found between the proposed scenario and
16
results available in the literature.
1
1
1
Introduction
2
In the external flow around solid bodies, it is a well-known fact that the presence of
3
other bodies in close proximity can change fundamental aspects of the flow, such as
4
fluid forces and transition thresholds. The effect of the presence of additional bodies
5
in the fluid stream is called flow interference. A particular type of flow interference
6
which is specially severe is the wake interference, which happens when one body is
7
immersed in the wake of another body.
8
In the case of bluff bodies, the most commonly applied model to study wake
9
interference is the flow around two identical circular cylinders placed in tandem
10
arrangements, as illustrated in figure 1. It is known from experiments (Zdravkovich,
11
1977; Igarashi, 1981) and computations (Mittal et al., 1997; Meneghini et al., 2001)
12
that different vortex shedding regimes can be observed in the flow around this type
13
of arrangement, depending on the centre-to-centre separation Lx . Adopting the
14
classification presented in Carmo et al. (2009), illustrated in figure 2, we see that
15
three different shedding regimes are observed for low Reynolds numbers. For very
16
small separations, the shedding regime SG (symmetric in the gap) is observed, as
17
shown in figure 2(a). In this regime, a pair of almost symmetric vortices is formed
18
in the gap between the cylinders and the root mean square (RMS) of the lift on
19
the downstream cylinder is very small. If the separation is gradually increased, the
20
shedding regime eventually changes to AG (alternating in the gap), in which regions
21
of concentrated vorticity grow and decrease alternatively in time on each side of the
22
line that links the centres of the cylinders (see figure 2(b)). This makes the RMS
23
of the lift coefficient on the downstream cylinder increase. It is also important to
2
Figure 1 – Sketch of the flow around two circular cylinders in tandem arrangement.
1
highlight that the drag on the downstream cylinder is usually negative for shedding
2
regimes SG and AG, as illustrated in figures 2(a)-(b). Finally, for larger separations,
3
a complete vortex wake is formed in the interstitial region, the RMS of the lift on the
4
downstream cylinder increases significantly and the mean drag on the downstream
5
cylinder becomes positive, as shown in figure 2(c). This shedding regime is called
6
WG (wake in the gap). Since the transition between the shedding regimes AG and
7
WG is marked by the inversion of the drag coefficient on the downstream cylinder,
8
the transition from one to the other is referred to as drag inversion and in this paper
9
we focus on flows in the vicinity of such transition.
10
It is known that the separation at which the drag inversion occurs depends on
11
the initial conditions because, at least for low Reynolds numbers, the flow is bi-
12
stable in the vicinity of the drag inversion point (Mizushima & Suehiro, 2005). For
13
this reason, it is more appropriate to refer to a drag inversion range than to a drag
14
inversion point. For a fixed separation, this range is defined in terms of Reynolds
15
numbers. Likewise, for a fixed Reynolds number this range is defined in terms of
16
centre-to-centre distance. As far as the authors are aware, no study to date has
17
calculated the drag inversion range for low Reynolds numbers while taking into
18
account the bi-stable nature of the flow in this region1 .
1
Papaioannou et al. (2006) calculated the drag inversion separation only for increasing Reynolds
numbers.
3
1.5
CD, CL
1
0.5
0
-0.5
-1
-1.5
0
(a) Lx {D
10
20
30
40
20
30
40
20
30
40
tU∞/D
1.5 – regime SG
1.5
CD, CL
1
0.5
0
-0.5
-1
-1.5
0
(b) Lx {D
10
tU∞/D
2.3 – regime AG
1.5
CD, CL
1
0.5
0
-0.5
-1
-1.5
0
(c) Lx {D
10
tU∞/D
5 – regime WG
Figure 2 – Left – Instantaneous vorticity contours illustrating the different shedding
regimes observed in the flow around two circular cylinders in tandem arrangements.
Contours vary from ωz D{U8 2.2 (light contours) to ωz D{U8 2.2 (dark countours). Right – Drag coefficient (grey solid line) and lift coefficient (black dashed line)
time series for the downstream cylinder. Re 200, two-dimensional simulations.
1
A recent study by Carmo et al. (2009) investigated the three-dimensional insta-
2
bilities observed in the nominally two-dimensional time-periodic flow around two
3
identical circular cylinders placed in tandem in relation to the free-stream. These
4
instabilities are known as secondary instabilities, since they occur after the primary
4
1
instability, which is the transition from steady flow to two-dimensional time-periodic
2
flow (the primary instability in the flow around two circular cylinders in tandem was
3
investigated by Mizushima & Suehiro, 2005). The results obtained by Carmo et al.
4
(2009) were compared to those obtained for an isolated cylinder (Williamson, 1988,
5
1996; Barkley & Henderson, 1996). A summary of the main results of that work
6
are reproduced in figure 3, in which it can be seen that different modes appear in
7
the transition to three-dimensional flow in the wake for separations smaller than the
8
drag inversion spacing. For such cases, the three-dimensional structures appeared
9
later in terms of Reynolds number than for the flow around an isolated cylinder. It
10
was shown that for configurations at shedding regime SG, the unstable mode at the
11
onset of the secondary instability originated at the formation region, downstream
12
of the leeward cylinder. This mode, referred to as mode T1, has a topology that
13
breaks the spatial symmetry of the base flow, and its physical mechanism appears
14
to be associated to a hyperbolic instability. For slightly larger separations, the shed-
15
ding regime changed to AG and a different unstable mode, named mode T2, was
16
observed. Mode T2 has its origin at the base of the downstream cylinder, upstream
17
of the vortex formation region. A centrifugal instability in this region seems to give
18
rise to this mode. Like the single cylinder mode A (Williamson, 1988), mode T2
19
wake topology keeps the in-plane spatial symmetry observed in the base flow. If
20
the separation is increased a little more, but not so much as to change the base
21
flow shedding regime, a new unstable mode (mode T3) is initiated at the interstitial
22
region. Like mode T1, mode T3 also breaks the spatial symmetry of the base flow.
23
Some of the mode attributes suggest that the underlying physical mechanism is a
5
1
cooperative elliptical instability. On the other hand, if the separation was greater
2
than the drag inversion spacing (shedding regime WG), the initial stages of the tran-
3
sition in the wake occurred in a similar way to that of the isolated cylinder. The
4
first instability, mode A, arose earlier in Reynolds number terms when compared to
5
the single cylinder case, and it is therefore concluded that the downstream cylinder
6
has a destabilising effect on the flow for separations larger than the drag inversion
7
spacing.
8
Although in Carmo et al. (2009) a full characterisation of the modes was pre-
9
sented and physical mechanisms were proposed to explain the instabilities, that
10
paper did not address a point of high practical interest in engineering, which is how
11
the onset of three-dimensional instabilities affect the drag inversion. In the present
12
paper, we investigate in detail how the transition to three-dimensional flow affects
13
the vortex shedding regime, focusing on the vicinity of the drag inversion spacing
14
and taking into account the hysteresis of the regime transition. We obtain the pos-
15
sible flow states for Re
16
current results to explain previously published computational data.
17
2
¤ 500 by means of direct numerical simulations and use the
Numerical method
The flows investigated in this paper were calculated using numerical simulations of
the incompressible Navier–Stokes equations, here written in non-dimensional form:
Bu pu.∇qu ∇p
Bt
∇.u 0.
6
1 2
∇ u,
Re
(1)
(2)
500
450
400
Recr
350
300
250
200
150
100
0
1
2
3
4
5
6
7
8
9 10 11
7
8
9 10 11
Lx/D
(a)
10.0
9.0
8.0
λcr /D
7.0
6.0
5.0
4.0
3.0
2.0
0
1
2
3
4
5
6
Lx/D
(b)
Figure 3 – Variation of the critical Reynolds numbers (a) and respective perturbation
wavelengths (b) with the centre-to-centre separation for modes T1 (), T2 (), T3 ()
and A (N). Mode T1 bifurcates from shedding regime SG, modes T2 and T3 from
shedding regime AG and mode A from shedding regime WG.
1
2
The cylinder diameter D is the reference length and the free-stream speed U8 is the
reference speed used in the non-dimensionalisation. u pu, v, wq is the velocity field,
ρU8D{µ is the Reynolds number and µ
3
t is the time, p is the static pressure, Re
4
is the dynamic viscosity of the fluid. The pressure was assumed to be scaled by the
7
1
constant density ρ. The numerical solution of these equations was calculated using
2
a Spectral/hp discretisation as presented in Karniadakis & Sherwin (2005). The
3
time integration scheme adopted was the stiffly stable splitting scheme presented by
4
Karniadakis et al. (1991).
5
Polynomials of degree 8 were used in the discretisation of the meshes for the
6
two-dimensional simulations. The meshes employed were the same as those used to
7
obtain the base flows in Carmo et al. (2009); an example is shown in figure 4. The
8
boundary conditions were u
9
10
11
12
1, v 0 on the left, upper and lower boundaries
of the mesh in the figure, B u{B x
and u v
0, Bv{Bx 0 on the right (outflow) boundary
0 on the cylinders’ walls. The high-order pressure boundary condition
described in Karniadakis et al. (1991) was employed on every boundary apart from
the outflow boundary, on which p 0 was imposed.
13
The three-dimensional simulations were performed using a three-dimensional ver-
14
sion of the Navier-Stokes solver which uses a Spectral/hp element discretization in
15
the xy plane and Fourier modes in the spanwise direction (Karniadakis, 1990). The
16
advantages of this approach is the high efficiency in the code parallelisation and that
17
the meshes generated for the two-dimensional simulations can be re-used. In the
18
three-dimensional simulations, domains with spanwise lengths between 8D and 12D
19
were employed, in order to comply with the wavelength of the instability that was
20
expected to arise. Depending on the Reynolds number and spanwise length, 16 or 32
21
Fourier modes were used in the discretisation in the spanwise direction and periodic
22
boundary conditions enforced on the planes at the boundaries perpendicular to the
23
cylinder axis.
8
Figure 4 – Mesh employed in the calculations of the flow around the configuration
with Lx {D 5.
1
In the next section we also present the critical Reynolds numbers for the primary
2
and secondary instabilities in the wake. The data referring to the secondary instabil-
3
ities were extracted from Carmo et al. (2009), but the data referring to the primary
4
instabilities were calculated. To obtain the steady base flows, we have employed the
5
method presented by Tuckerman & Barkley (2000) with the modification suggested
6
by Blackburn (2002). The stability analysis procedure was the same as that used in
7
Carmo et al. (2009).
8
3
9
A number of two- and three-dimensional simulations were performed to investigate
10
the boundaries of the drag inversion range, fixing the geometric configuration and
Results and discussion
9
1
varying the Reynolds number. Each of the calculations was run for at least 300
2
non-dimensional time units for the two-dimensional simulations and for at least
3
100 non-dimensional time units for the three-dimensional simulations. The mean
4
drag coefficient was used as the indicator of the shedding regime (AG or WG).
5
Due to the hysteretic nature of the transition between these regimes, determining
6
the upper (lower) boundary requires that we start our flow simulations at a lower
7
(upper) Reynolds number and increase (reduce) its value. The boundaries were
8
defined taking the Reynolds number of the first calculation that showed a change of
9
regime, with an uncertainty of ∆Re
0.5.
10
The primary instability was investigated by means of global linear stability anal-
11
ysis of the solutions of the steady Navier-Stokes equations. The critical Reynolds
12
number for each of the configurations was obtained using a search algorithm that
13
employed Newton’s method. The resolution in Reynolds number of this algorithm
14
was 1, i.e. only integer Reynolds numbers were tested, and the Reynolds number
15
was considered to be the critical one if the real part of the eigenvalue resulting from
16
the stability calculations had modulus smaller than 0.001.
17
Although is usually said that regime AG is associated with a negative mean drag
18
on the downstream cylinder, this is not always the case for low Reynolds numbers.
19
Figure 5 shows values of mean drag coefficient obtained for decreasing Reynolds
20
numbers for the configuration with Lx {D
3.8 and vorticity contours illustrating
21
the three different wakes observed (steady flow, shedding regime AG and shedding
22
regime WG). The discontinuities associated with the two changes of regime are clear
23
in the graph. It can be seen that the drag coefficient is small, but not negative, for
10
Figure 5 – Values of the drag coefficient of the downstream cylinder obtained for
decreasing Reynolds numbers, Lx {D 3.8. The symbols in the graph correspond to
the different shedding regimes observed: N – WG, – AG, – steady wake. 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.
1
regime AG. For that reason, it was impossible to define a general threshold value for
2
the drag coefficient which would indicate the change of shedding regime; in order to
3
find the thresholds of shedding regime transition it was necessary to examine each
4
configuration individually, checking the flow field contours and drag coefficient time
5
histories.
6
7
Figure 6 displays the results of the calculations on a Re vs. Lx {D map. The
curves showing the variation of the critical Reynolds number with the centre-to-
11
1
centre distance for the modes A and T3, obtained by Carmo et al. (2009), and the
2
critical Reynolds numbers for the primary instability are also plotted on the map.
3
The four regions of bi-stable flow are marked in shades of grey. The bottom one
4
is located under the curve of critical Reynolds numbers for the primary instability,
5
and corresponds to a region in the parameter space where only steady flow or two-
6
dimensional flow with shedding regime WG are possible. The second grey region
7
from the bottom is located between the curve of critical Reynolds numbers for
8
mode A and the curve of critical Reynolds numbers for the primary instability.
9
Hence only two-dimensional time-periodic flows are possible in this region, but the
10
vortex shedding regime can be either AG (2d-AG) or WG (2d-WG), depending on
11
the initial conditions. The third region of bi-stable flow is between the curves of
12
critical Reynolds numbers for mode A and mode T3. In this region, two-dimensional
13
flows at regime AG (2d-AG) and three-dimensional flows at regime WG (3d-WG)
14
are possible. Lastly, the fourth top region of bi-stable flow is situated above the
15
mode T3 critical Reynolds number curve. The flow in this region is always three-
16
dimensional and, depending on the initial conditions, the vortex shedding regime
17
can be AG (3d-AG) or WG (3d-WG). Plots of vorticity iso-surfaces, obtained by
18
means of three-dimensional simulations, are shown in figure 7. These plots illustrate
19
each of the states observed and discussed previously.
20
To help to understand how to interpret the map in figure 6, let us describe
21
two examples of change of state. Suppose we have a flow around the configuration
22
23
with Lx {D
3.8 and Re 175.
According to the map in figure 6, this flow will
be three-dimensional with shedding regime WG (3d-WG). If the Reynolds number
12
Figure 6 – Map of Reynolds number against centre-to-centre separation showing the
possible vortex shedding regimes and the variation of the critical Reynolds numbers
of modes T3 and A with Lx {D, in the neighbourhood of the drag inversion range.
Symbols are: – mode T3 critical Reynolds numbers, N – mode A critical Reynolds
numbers, – primary instability critical Reynolds numbers, – 2-d transition from
shedding regime WG to steady flow, – 2-d vortex shedding transition from WG
to AG, – 3-d vortex shedding transition from WG to AG, – 2-d vortex shedding
transition from AG to WG, – 3-d vortex shedding transition from AG to WG.
1
is gradually increased to 300, the flow will then be located inside the upper grey
2
region in the map, in which the regimes 3d-AG and 3d-WG are possible. Since
3
the previous state of the flow was 3d-WG and the change of Reynolds number was
4
gradual, the flow will keep the same state at Re
5
further increased to 500, the flow will now be in a region of the parameter space
6
where only 3d-AG flows are possible, so the flow will then change to this state. If
7
the Reynolds number is then gradually changed back to 300, the flow will keep the
8
state 3d-AG. The same ratiocination is valid if the separation between the cylinders
13
300.
If the Reynolds number is
(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
Figure 7 – Plots of vorticity iso-surfaces illustrating the different states observed in
the vicinity of the drag inversion separation, three-dimensional simulations. Translucent surfaces represent iso-surfaces of |ωz |. Solid light grey and dark grey surfaces
represent iso-surfaces of negative and positive ωx respectively.
1
is changed instead of the Reynolds number – the flow will always retain its previous
2
state when entering a grey region coming from a white region.
3
It may also happen that the flow at a certain state is taken gradually to a region
4
in which two states are possible, but none of them is the initial state of the flow.
5
In this case the flow will assume the possible state that retains the shedding regime
6
of the initial state. For example, suppose we have a flow around the configuration
14
1
with Lx {D
3.2 at Re 250 with regime 3d-AG; the flow will be in the upper
2
grey region of figure 6. If the Reynolds number is gradually decreased to 150, the
3
flow will then be in the dark grey region immediately below, in which two states are
4
possible, 2d-AG and 3d-WG. However, none of them is equal to the initial state of
5
the flow. So the flow will change to the state that keeps the shedding regime, i.e.
6
the flow will change to 2d-AG.
7
A point worthy of note is that the boundaries of the drag inversion range have
8
different orientations depending on whether the flow is two-dimensional or three-
9
dimensional: the boundaries have a negative slope for two-dimensional flows and
10
a positive slope for three-dimensional flows. This means that once the flow is un-
11
stable to three-dimensional perturbations, the dependence of the shedding regime
12
on the Reynolds number is inverted. It was shown in Carmo & Meneghini (2006)
13
that in two-dimensional flows, an increase in the Reynolds number makes the forma-
14
tion length shorter. This occurs because at higher Reynolds numbers the spanwise
15
vorticity in the shear layers separate from top and bottom of the cylinder wall is
16
stronger, and this stronger vorticity facilitates the interaction between these shear
17
layers. A shorter formation length favours shedding regime WG. In contrast, when
18
the flows were three-dimensional the results obtained by Carmo & Meneghini (2006)
19
showed longer formation lengths, owing to the fact that three-dimensional diffusion
20
and spanwise de-correlation weakened the interaction between the opposite shear
21
layers. The DPIV measurements carried out by Noca et al. (1998) showed that, for
22
the flow around an isolated cylinder, the formation length increases with Reynolds
23
number for p300 Re
1500q, indicating that the three-dimensional effects prevail
15
1
over the two-dimensional ones in this Reynolds number range. The results in figure 6
2
demonstrate that this is also true for the flow around two circular cylinders in tan-
3
dem. This was also one of the conclusions drawn by Papaioannou et al. (2006), who
4
deduced that the variation of the single cylinder formation length and variation of
5
the tandem arrangement drag inversion separation with Reynolds number appeared
6
to be consistent in both two- and three-dimensional simulation results.
7
Using figure 6, we can draw comparisons between the current results and data
8
from earlier research that employed three-dimensional numerical simulations. Deng
9
et al. (2006) performed three-dimensional simulations of the flow around diverse
10
11
12
13
14
15
16
17
tandem configurations at Re
220, using a virtual boundary method.
served three-dimensional flow for all configurations with Lx {D
Lx {D
¥
They ob-
4, whereas for
2 the flow remained two-dimensional. These results did not depend on the
initial conditions. This is entirely consistent with the map in figure 6: for Re
the configuration Lx {D
220,
2 is in a zone where only two-dimensional flow in the
AG regime is possible, and the configurations with Lx {D
¥ 4 are in a zone where
only three-dimensional flows in the WG regime are possible. When analysing the
results of Deng et al. (2006) for Lx {D
3.5, it should be borne in mind that the
18
spanwise length of the domain used in their computations was 8D. They observed
19
that the emergence of three-dimensional structures depended on the initial condi-
20
tions: three-dimensional flow was seen to occur if the initial flow field was in the
21
WG regime, whereas any three-dimensional perturbations died out if a flow in the
22
AG regime was used as initial condition. In the map in figure 6, it can be seen
23
that the the flow around the configuration with Lx {D
16
3.5 at Re 220 is in a
1
2
3
region where the possible flows are three-dimensional, either in the AG or in the
WG regimes. However, figure 3(b) shows that, for Lx {D
at the onset pRe cr
3.5, mode T3 wavelength
217q is λz {D 9.97. Therefore, the calculations performed by
4
Deng et al. (2006) were unable to capture mode T3 instability because the spanwise
5
length of the domain they used was too short. Deng et al. (2006) also tried to find
6
the Reynolds number for which the flow around the configuration Lx {D
3.5 at
7
regime AG would become unstable to three-dimensional perturbations. They ran a
8
series of simulations increasing the Reynolds number in steps of 10, using the final
9
solution of each simulation as the initial condition for the next one. They observed
250, and the wavelength and sym-
10
that the flow became three-dimensional for Re
11
metry of the three-dimensional structures were similar to those of mode A. However,
12
our results show that this transition occurs at Re
13
should be mode T3. Again, this difference can be explained by the short spanwise
14
length of their calculations. We performed additional two-dimensional calculations
15
and found that the upper Re limit of the two-dimensional drag inversion range for
16
17
configuration Lx {D
3.5 is Re 240.
217, and the unstable mode
We therefore assume that from Re
240
to Re 250 there was a shedding regime change to WG in the simulations of Deng
18
et al. (2006), and that this was accompanied by the appearance of mode A structures
19
in the flow.
20
21
22
23
In another study that used three-dimensional numerical simulations, Papaioannou et al. (2006) observed regime AG for 250 ¤ Re
around configurations with Lx {D
¤ 500 in simulations of the flow
¤ 3.5, whereas regime WG was observed in simu-
lations of the flow around configurations with Lx {D
17
¥ 3.8.
The current results are
1
2
3
4
5
mostly in line with this; it can be seen in the map in figure 6 that three-dimensional
flow in the AG regime is possible for Lx {D
¤ 3.5 for 250 ¤ Re ¤ 500 and three-
dimensional flow in the WG regime is possible for Lx {D
¥ 3.8 for 250 ¤ Re ¤ 475.
The only disagreement between our results and those from Papaioannou et al. (2006)
is in the regime observed for Lx {D
3.8 at Re 500. For this case, the map in fig-
6
ure 6 indicates that only three-dimensional flow in the AG regime is possible, while
7
Papaioannou et al. report that they found regime WG at the same conditions. A
8
possible reason for this discrepancy is the number of non-dimensional time units
9
for which the flow equations are integrated. We have used 100 non-dimensional
10
time units for all calculations, and we observed that for some of the cases, the
11
change of regime only happened after the equations were integrated for 40 or 50
12
non-dimensional time units. Papaioannou et al. do not report the time length of
13
their calculations. Another possible reason has to do with the size of the domain.
14
The mesh used in Papaioannou et al. (2006) is significantly smaller than that used
15
for the current results.
16
4
17
In this paper, a thorough investigation of the possible flow states in the drag in-
18
version range of the flow around two circular cylinders in tandem at low Reynolds
19
numbers was presented. For the first time, the regions of bi-stable flow were carefully
20
identified, taking into account the hysteresis of the shedding regimes and the influ-
21
ence of the secondary instabilities. The presence of three-dimensional flow structures
22
was observed to induce notable changes in the response of the flow to the variation
Conclusion
18
1
of Reynolds number and cylinder separation. Results available in the literature were
2
reviewed in the light of the new data and almost all the observations made by other
3
authors were consistent with the current findings. We believe that the analysis pre-
4
sented helps to improve the understanding of flows with wake interference, and can
5
be very useful for future investigations of other aspects of such flows.
6
References
7
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability
8
analysis of the wake of a circular cylinder. Journal of Fluid Mechanics 322, 215–
9
241.
10
11
Blackburn, H. M. 2002 Three-dimensional instability and state selection in an
oscillatory axisymmetric swirling flow. Physics of Fluids 14 (11), 3983–3996.
12
Carmo, B. S. & Meneghini, J. R. 2006 Numerical investigation of the flow
13
around two circular cylinders in tandem. Journal of Fluids and Structures 22,
14
979–988.
15
Carmo, B. S., Meneghini, J. R. & Sherwin, S. J. 2009 Secondary instabilities
16
in the flow around two circular cylinders in tandem. Accepted for publication in
17
the Journal of Fluid Mechanics .
18
Deng, J., Ren, A. L., Zou, J. F. & Shao, X. M. 2006 Three-dimensional flow
19
around two circular cylinders in tandem arrangement. Fluid Dynamics Research
20
38, 386–404.
19
1
2
Igarashi, T. 1981 Characteristics of the flow around two circular cylinders arranged in tandem. Bulletin of JSME 24 (188), 323–331.
3
Karniadakis, G. E. 1990 Spectral Element–Fourier methods for incompressible
4
turbulent flows. Computer Methods in Applied Mechanics and Engineering 80,
5
367–380.
6
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting
7
methods for the incompressible Navier–Stokes equations. Journal of Computa-
8
tional Physics 97, 414–443.
9
10
Karniadakis, G. E. & Sherwin, S. J. 2005 Spectral/hp Element Methods for
Computational Fluid Dynamics, 2nd edn. Oxford University Press.
11
Meneghini, J. R., Saltara, F., Siqueira, C. L. R. & Ferrari Jr, J. A.
12
2001 Numerical simulation of flow interference between two circular cylinders in
13
tandem and side-by-side arrangements. Journal of Fluids and Structures 15, 327–
14
350.
15
Mittal, S., Kumar, V. & Raghuvanshi, A. 1997 Unsteady incompressible flows
16
past two cylinders in tandem and staggered arrangements. International Journal
17
for Numerical Methods in Fluids 25, 1315–1344.
18
19
20
21
Mizushima, J. & Suehiro, N. 2005 Instability and transition of flow past two
tandem circular cylinders. Physics of Fluids 17 (10), 104107.
Noca, F., Park, H. G. & Gharib, M. 1998 Vortex formation length of a circular
cylinder p300
Re 4000q using DPIV. In Proceedings of Bluff Body Wakes
20
1
and Vortex-Induced Vibration, Washington, DC (ed. P. W. Bearman & C. H. K.
2
Williamson), p. 46.
3
Papaioannou, G. V., Yue, D. K. P., Triantafyllou, M. S. & Karni-
4
adakis, G. E. 2006 Three-dimensionality effects in flow around two tandem
5
cylinders. Journal of Fluid Mechanics 558, 387–413.
6
Tuckerman, L. S. & Barkley, D. 2000 Bifurcation analysis for timesteppers.
7
In Numerical methods for bifurcation problems and large-scale dynamical systems
8
(ed. E. Doedel & L. S. Tuckerman), IMA volumes in Mathematics and its appli-
9
cations, vol. 119, pp. 543–556. New York: Springer Verlag.
10
11
12
13
Williamson, C. H. K. 1988 The existence of two stages in the transition to threedimensionality of a cylinder wake. Physics of Fluids 31 (11), 3165–3168.
Williamson, C. H. K. 1996 Three-dimensional wake transition. Journal of Fluid
Mechanics 328, 345–407.
351
Zdravkovich, M. M. 1977 Review of flow interference between two circular cylin-
352
ders in various arrangements. ASME Journal of Fluids Engineering 99, 618–633.
21