Flow Past Stationary and Elastically-Mounted Martin Grifﬁth (Swinburne · 2017-11-25 · GDR Contrôle des décollement Flow Past Stationary and Elastically-Mounted Circular Cylinders

GDR Contrôle des décollement

Flow Past Stationary and Elastically-Mounted Circular Cylinders in Tandem and Staggered Arrangements

Martin Griffith (Swinburne)David Lo Jacono (IMFT)John Sheridan (Monash)Justin Leontini (Swinburne)

/20 Problem definition2

øDU

St =fD

URe =

UD

⌫

U

øD

L T

Staggered

U

Side by side

U

Tandem

/20 Literature on FIXED cylinders3

See recent review of Zhou & Alam (2016)

W-T (1+2)W-T2W-T1

W-SGW-SD

P+WP-SSA

P-SSB

P-SSC

P-S2

1 2 3 4 5

1

2

3

4

P-SSA

T/D Zdravkovich (1987)Re < 105 P: proximity interference

W: Rear cylinder in the wakeT (tandem), S (staggered), SS (side by side)

Rich dynamic

L/D

/20

SLR

VIIS

VPEVPSE

SVSBB

SBB2SBB1

1 2 3 4 5

1

2

3

4

P-SSA

L/D

Sumner et al. (2000)800 < Re < 1900

Literature on FIXED cylinders3


W-T (1+2)W-T2W-T1

W-SGW-SD

P+WP-SSA

P-SSB

P-SSC

P-S2

1 2 3 4 5

1

2

3

4

P-SSA

T/D Zdravkovich (1987)Re < 105

Sumner et al. (2000)9 different regimes: SBB (single bluff bodies),BB (base bleed), SLR (shear layer reattach.),IS (induced separation), VPXX (vortex paring),SVS (clear vortex pattern from both), VI (vortex impingement)

/20

SLR

VIIS

VPEVPSE

SVSBB

SBB2SBB1

1 2 3 4 5

1

2

3

4

P-SSA

L/D

Sumner et al. (2000)800 < Re < 1900

Literature on FIXED cylinders3


W-T (1+2)W-T2W-T1

W-SGW-SD

P+WP-SSA

P-SSB

P-SSC

P-S2

1 2 3 4 5

1

2

3

4

P-SSA

T/D Zdravkovich (1987)Re < 105

S-Ib

S-II

T-II

S-Ia

T-I

1 2 3 4 5

1

2

3

4

P-SSA

L/D

T/D Hu & Zhou (2008)

Re = 7000

Hu & Zhou (2008)(focus after the near wake x/D=6)S: Single wakeT: Two wakes

/20 Literature on FIXED cylinders4

SLR

VIIS

VPEVPSE

SVSBB

SBB2SBB1

1 2 3 4 5

1

2

3

4

P-SSA

W-T (1+2)W-T2W-T1

W-SGW-SD

P+WP-SSA

P-SSB

P-SSC

P-S2

1 2 3 4 5

1

2

3

4

P-SSA

S-Ib

S-II

T-II

S-Ia

T-I

1 2 3 4 5

1

2

3

4

P-SSA

Good qualitative overlap considering Reynolds

/20 FIXED cylinders5

W-T (1+2)W-T2W-T1

W-SGW-SD

P+WP-SSA

P-SSB

P-SSC

P-S2

1 2 3 4 5

1

2

3

4

P-SSA

S-Ib

S-II

T-II

S-Ia

T-I

1 2 3 4 5

1

2

3

4

P-SSA

SLR

VIIS

VPEVPSE

SVSBB

SBB2SBB1

1 2 3 4 5

1

2

3

4

P-SSA

T/D

L/D

No gap flow (W-T2: wake interference; SLR: shear layer reattachment; SIb: single wake)

T/D < 0.4

Gap flow (W-SG: wake interference staggered; IS/VPE: induced separation/ pairing of vortex;S-II: unequal strength merging into single wake)

0.4 < T/D < 1.0

Temporally complex flow (P+W: proximity and wake interference;VPSE: vortex paring/splitting events)

1.0 < T/D < 1.3

Large gap: wake roll-up (outer SL interact)(P-S2: proximity in a staggered arrangment;SVS: separate vortex street)

1.3 < T/D

/20

U⇤

Flow induced vibrations6

Borazjani & Sotiropoulos (2009)Tandem (T=0) at L=1.5 (1DOF & 2DOF)

Huera-Huarte & Gharib (2011) side by side (2DOF) Huera-Huarte & Bearman (2011) tandem (2DOF)Assi, Bearman & Menegheni (2010) tandem (2DOF)Carmo, Assi & Menegheni (2013) tandem (2DOF)Wang, Yang, Nguyen, Yu (2014) unequal D, tandemAlam et al. unequal D, tandemetc…

state1 state2 M⇣Y + (2⇡fN )2Y

⌘= Fy

U⇤ =U

DfN

Re = 200

m⇤ = 2.546

/20 Methodology7

Immersed boundary finite difference solver

✦ Results obtained using a sharp interface immersed boundary method✦ Fluid and structure are coupled via a Newmark-beta method✦ Each simulation run on a grid of 1024 x 512 cells (others at 2048 x 1024)✦ Geometries defined with 256 elements

Immerse complex geometry in

cartesian mesh

Identify ghost points: Internal points with external neighbours

Interpolate along normal enforcing boundary

condition to obtain ghost point value

/20 Validation (tandem)8

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2.0 4.0 6.0 8.0 10.0 12.0 14.0

frontrear

single

U⇤U⇤

U⇤ =U

DfN

Re = 200

m⇤ = 2.546L/D = 1.5

T/D = 0.0

/20 Parameter space9

0.0

1.0

2.0

3.0

4.0

5.0

0 2 4 6 8 10 12 14

L/D = 1.5

Static case

Tandem caseBorazjani & Sotiropoulos (2009)

Re=200, T/D = 0 U⇤

T/D

/20 Results10Towards isolated case

Higher amplitudes for the rear cylinder

spring less stiff, stronger effect of T/D

spring more stiff, lesser effect of T/D

/20 Frequency content11

a) T=0, U*=0 freq. slightly lower than isolatedb) T ~ 1.4, U*=0 complex/broadband content (QP or chaotic)c) for higher T values, recover isolated cylinderd) U*=0 and U*=14 (stiff/slack) fluid forcing dominante) U*=5, T=0 and T>~3 flow is periodic. Odd harmonics. In between, symmetry broken, even harmonic appears (QP, etc.)

f) difference in frequency between the two cylinders.

/20 Tentative summary12

T/D

U⇤Periodic & sync.P2 Periodic & sync.(period doubling)Pn Periodic & sync.

QP states

Chaotic states

/20

mode 3mode 2

Tentative summary13

Periodic and sync.

mode 2# mode 3#mode 1

gap flow dominated

wake pair dominated

desync region!

/20 Rigid case detail14

Page 16 of 29

f

FIV of tandem and staggered cylinders 17

Mode 1, T/D = 0.00

Gap flow, T/D = 1.00

Wake pair, T/D = 2.30


CL

CL

CL

CL

tUD

f D

U

800 900 1000 0.0 0.2 0.4 0.6 0.8 1.0

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

Figure 9. For stationary cylinders, U∗ = 0, plots of vorticity contours, time series of thecoefficient of lift of each cylinder and the corresponding spectra, red corresponding to theupstream cylinder and green the downstream, for cylinder offsets T/D = 0.0, 1.0, 2.3 and3.5.

Strouhal frequencies has been also been noted in Alam & Sakamoto (2005), a study offlow past staggered cylinders. For non-tandem and non-side-by-side arrangements, theyobserved, for a far greater Reynolds number than the current study, Re = 5.5 × 104,sometimes intermittently, different Strouhal frequencies for particular cases, describingthe flow as bi- or multistable.

This range of cross-stream offset 0.0 ! T/D ! 5.0 was also covered numerically in thework of Tong et al. (2015) for Re = 103, but traversed in terms of angle and pitch ratiobetween the two cylinders (see their figure 4). The description of the regimes providedhere is consistent with their figure 18 along a traverse at L/D = 1.5.

Figure 9 plots details of four cases, picking out the various regimes of behaviourclassified in figure 6 and further described in figures 7 and 8. The flows for offsetT/D = 0.0 corresponds to mode 1 where the front cylinder does not shed vortices andonly one vortex street is present. Offset T/D = 1.0 however corresponds to the gap-dominated flow regime and the flow is more complex with stronger contributions from allharmonics of the primary frequency present in the lift coefficient signal of each cylinder.Despite the gap flow, the cylinders still largely shed vortices as one body. In both casesthe vortex-formation region is larger than for the single, isolated cylinder, resulting in alower vortex-shedding frequency. For T/D = 2.3, the flow is now wake-pair dominated.A periodic beating is evident in the time series of the cylinder oscillation, indicative ofthe difference in primary vortex shedding frequency. For the case of T/D = 3.50, the

Page 17 of 29

mode 1

gap flow

wake pair

/20 Rigid case detail14

Page 16 of 29

f


Mode 1, T/D = 0.00




CL

CL

CL

CL

tUD

f D

U

800 900 1000 0.0 0.2 0.4 0.6 0.8 1.0

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

Figure 9. For stationary cylinders, U∗ = 0, plots of vorticity contours, time series of thecoefficient of lift of each cylinder and the corresponding spectra, red corresponding to theupstream cylinder and green the downstream, for cylinder offsets T/D = 0.0, 1.0, 2.3 and3.5.

Strouhal frequencies has been also been noted in Alam & Sakamoto (2005), a study offlow past staggered cylinders. For non-tandem and non-side-by-side arrangements, theyobserved, for a far greater Reynolds number than the current study, Re = 5.5 × 104,sometimes intermittently, different Strouhal frequencies for particular cases, describingthe flow as bi- or multistable.

This range of cross-stream offset 0.0 ! T/D ! 5.0 was also covered numerically in thework of Tong et al. (2015) for Re = 103, but traversed in terms of angle and pitch ratiobetween the two cylinders (see their figure 4). The description of the regimes providedhere is consistent with their figure 18 along a traverse at L/D = 1.5.

Figure 9 plots details of four cases, picking out the various regimes of behaviourclassified in figure 6 and further described in figures 7 and 8. The flows for offsetT/D = 0.0 corresponds to mode 1 where the front cylinder does not shed vortices andonly one vortex street is present. Offset T/D = 1.0 however corresponds to the gap-dominated flow regime and the flow is more complex with stronger contributions from allharmonics of the primary frequency present in the lift coefficient signal of each cylinder.Despite the gap flow, the cylinders still largely shed vortices as one body. In both casesthe vortex-formation region is larger than for the single, isolated cylinder, resulting in alower vortex-shedding frequency. For T/D = 2.3, the flow is now wake-pair dominated.A periodic beating is evident in the time series of the cylinder oscillation, indicative ofthe difference in primary vortex shedding frequency. For the case of T/D = 3.50, the

Page 17 of 29

Page 16 of 29

18 Martin D. Griffith, David Lo Jacono, John Sheridan and Justin S. Leontini







CL

CL

CL

CL

CL

CL

tUD

f D

U

800 900 1000 0.0 0.2 0.4 0.6 0.8 1.0

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

Figure 10. For stationary cylinders, U∗ = 0, plots of vorticity contours, time series of thecoefficient of lift of each cylinder and the corresponding spectra, red corresponding to theupstream cylinder and green the downstream, for cylinder offsets T/D = 1.40, 1.43, 1.47, 1.51,1.54 and 1.60.

frequency difference persists, but is much smaller, resulting in a weaker, longer-periodbeating in the signal as the interaction between the wakes lessens as the cylinders aremoved further apart.

Figure 10 focuses on cases where there is a significant difference between the primaryfrequencies, spanning the range 1.3 ! T/D ! 1.9, the same range as shown in the insetin figure 8. This range encompasses the transition from gap-pair dominated to wake-pairdominated flow. The inset shows a frequency content strongly dependent on cross-streamoffset. Data have been obtained at a high resolution of T/D, and figure 10 presents sixexample flows in this region. Figure 10 shows the complexity of the wake and the subtle

Page 18 of 29

P2

P3

P4

mode 1

gap flow

wake pair

/20 Tandem case 15Borazjani & Sotiropoulos (2009)Tandem (T=0) at L=1.5 (1DOF)

mode 1 mode 2 mode 3


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 10 12 14

0.0

0.5

1.0

1.5

0 2 4 6 8 10 12 14

U∗ U∗

CL

rm

s

CD

rm

s

A∗ M

AX

γ

front

rear

single

front

rear

single

front

rear

single

−2π

−3π2

−π

−π

2

0

mode1

mode2

mode3

Figure 11. For T/D = 0.0 and L/D = 1.5, the variation with U∗ of the maximum displacementof the each cylinder, A∗

MAX , the phase difference between the cylinder oscillations, γ, and theroot-mean-square values of the drag and lift coefficients. Solid symbols represent results fromthe current study, while hollow ones are from Borazjani & Sotiropoulos (2009).

Sotiropoulos (2009) and δt = 0.004 in the current. This difference is mostly due to thehigher grid resolution used in the current study. To examine the effect of timestep, wehave also run both grid domain sizes using timesteps of δt = 0.004 and 0.002, to examineany sensitivity to temporal resolution. We found no significant effect resulting from thischange in timestep. We ran further tests at lower grid resolution and larger timestep; theresults for A∗

MAX returned by the M8 grid for a timestep up to δt = 0.0125 were greaterthan those returned for the higher resolution grid. The loose-coupling scheme outlinedin Borazjani & Sotiropoulos (2009) was also implemented in the current code, but hadnegligible effect on the results. In short, it is not completely clear why this discrepancyexists. We note that the single cylinder results from the current study match very closelywith the numerical results from Leontini et al. (2006) (see Griffith et al. (2016)) whichemployed a highly accurate spectral-element method.

Figure 12 plots Lissajous curves of lift coefficient and displacement across the U∗ range.These curves show a strong similarity to those of figure 8 of Borazjani & Sotiropoulos(2009). The main differences here are that the orbits are much closer to symmetric aroundY = 0 and CL = 0 in the current results, (which may account for some of the differencesbetween the two studies) and also the more meandering paths of the cases for U∗ = 5and 6, indicative of some quasi-periodicity.

It is shown below that these cases, U∗ = 5 and 6, occur in a distinct shedding regimewhere vortices formed at the rear of the front cylinder are forced through the gap between

Page 20 of 29

mode 1mode 3mode 2

/20 Tandem case 15Borazjani & Sotiropoulos (2009)Tandem (T=0) at L=1.5 (1DOF)

mode 1 mode 2 mode 3


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 10 12 14

0.0

0.5

1.0

1.5

0 2 4 6 8 10 12 14

U∗ U∗

CL

rm

s

CD

rm

s

A∗ M

AX

γ

front

rear

single

front

rear

single

front

rear

single

−2π

−3π2

−π

−π

2

0

mode1

mode2

mode3

Figure 11. For T/D = 0.0 and L/D = 1.5, the variation with U∗ of the maximum displacementof the each cylinder, A∗

MAX , the phase difference between the cylinder oscillations, γ, and theroot-mean-square values of the drag and lift coefficients. Solid symbols represent results fromthe current study, while hollow ones are from Borazjani & Sotiropoulos (2009).

Sotiropoulos (2009) and δt = 0.004 in the current. This difference is mostly due to thehigher grid resolution used in the current study. To examine the effect of timestep, wehave also run both grid domain sizes using timesteps of δt = 0.004 and 0.002, to examineany sensitivity to temporal resolution. We found no significant effect resulting from thischange in timestep. We ran further tests at lower grid resolution and larger timestep; theresults for A∗

MAX returned by the M8 grid for a timestep up to δt = 0.0125 were greaterthan those returned for the higher resolution grid. The loose-coupling scheme outlinedin Borazjani & Sotiropoulos (2009) was also implemented in the current code, but hadnegligible effect on the results. In short, it is not completely clear why this discrepancyexists. We note that the single cylinder results from the current study match very closelywith the numerical results from Leontini et al. (2006) (see Griffith et al. (2016)) whichemployed a highly accurate spectral-element method.

Figure 12 plots Lissajous curves of lift coefficient and displacement across the U∗ range.These curves show a strong similarity to those of figure 8 of Borazjani & Sotiropoulos(2009). The main differences here are that the orbits are much closer to symmetric aroundY = 0 and CL = 0 in the current results, (which may account for some of the differencesbetween the two studies) and also the more meandering paths of the cases for U∗ = 5and 6, indicative of some quasi-periodicity.

It is shown below that these cases, U∗ = 5 and 6, occur in a distinct shedding regimewhere vortices formed at the rear of the front cylinder are forced through the gap between

Page 20 of 29

Page 21 of 29

mode 1mode 3mode 2

/20 Tandem case16

Page 22 of 29

2Sr rear shedding

2Pf front shedding (out of phase)

2P shedding (pi/2 phase)

/20 Staggered cases — small offset17

T/D

U⇤


Mode 2, U∗ = 5.0, T/D = 0.0

Mode 21, U∗ = 5.0, T/D = 0.4

Mode 3, U∗ = 8.0, T/D = 0.0

Mode 31, U∗ = 8.0, T/D = 0.4

Y

D

Y

D

Y

D

Y

D

tUD

f D

U

800 900 1000 0.0 0.2 0.4 0.6 0.8 1.0

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

Figure 14. For U∗ = 5.0 and 8.0, and T/D = 0.0 and 0.4, plots of vorticity contours, timeseries of the displacement of each cylinder and the corresponding spectra for the displacements,red corresponding to the upstream cylinder and green the downstream. These four cases giveexamples, top to bottom, of the modes 2, 21, 3 and 31, as shown on figure 6.

of the cylinders. These are shear layers and vortices formed on different cylinders, havingdifferent formation areas and exposure to the oncoming freestream. There are no pairsof alternating vortices being shed from one cylinder at a given frequency; there areseveral frequencies possible, tied to vortex and vorticity formation on both sides of bothcylinders. Therefore, this region features both periodic flows and flows characterised byquasiperiodicity and disordered vortex shedding.

From figure 15, the common feature of the cases shown is the gap flow. In all cases,the positive vorticity generated from the upstream cylinder completely or substantiallypasses through the gap between the two cylinders. Following on from this is an interactionor pairing of the shear layers and vortices on the inside sides of the two cylinders;together with the elastic-mountings of the cylinders, the combination of several unrelatedfrequencies results in the quasiperiodic flows seen for the reduced velocity cases U∗ = 4.0,5.0 and 10.0 from figure 15. The case shown for U∗ = 5.0 exhibits a strong intermittencyindicative of a chaotic flow.

A distinction needs to be drawn between these gap flow dominated cases and thetandem modes 2 and 3. The tandem modes also feature strong gap flow, but the gap ispresent due to the elasticity of the cylinder mountings (U∗) - and can therefore changesides - rather than the initial position of the cylinders (T/D). Therefore, the “Gap flowdominated” region can extend over the entire range of U∗.

Increasing the cross-stream offset, the region marked “Wake pair dominated” on

Page 24 of 29

mode 2mode 3

mode 2#mode 3#

mode 3 mode 3#

Asymmetric versions of mode 2 and 3

2Pf

Pf+Sr

2P

P+S

/20 Staggered cases18

T/D

mode 2mode 3mode 1

Gap flow


U∗ = 0.0, T/D = 1.00

U∗ = 4.0, T/D = 1.00

U∗ = 5.0, T/D = 1.00

U∗ = 8.0, T/D = 1.00

U∗ = 10.0, T/D = 1.00

U∗ = 14.0, T/D = 1.00

CL

Y

D

Y

D

Y

D

Y

D

Y

D

tUD

f D

U

800 900 1000 0.0 0.2 0.4 0.6 0.8 1.0

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

1

1e-04

1e-08

1

0

-1

Figure 15. For T/D = 1.0, plots of vorticity contours, time series of the displacement of eachcylinder and the corresponding spectra for the displacements, red corresponding to the upstreamcylinder and green the downstream, for reduced velocities, U∗ = 0.0, 4.0, 5.0, 8.0, 10.0 and 14.0.These six cases give examples of the gap flow dominated region of figure 6.

figure 6 covers the parameter space entirely for U∗ > 2.0. In this region, instead ofbeing determined by the interaction of vorticity in the gap, the flow is dominated bypairing of vortices shed from one cylinder. Figure 16 presents flows for the same reducedvelocities in figure 15, but for a cross-stream offset of T/D = 3.0. All of the flows arein the “Wake pair dominated” region. Although mixing of vortex streets can occur inthe far wake (see the vorticity snapshots for U∗ = 4.0 and 5.0) the cylinder oscillationsare determined by the pairing and interaction of the vortices formed on each cylinder.This distinguishes the categorization from the T-I and T-II regimes described by Hu& Zhou (2008) and used by Tong et al. (2015), which delineate the flow according to

Page 25 of 29

Gap flow

Upstream positive vorticitypasses through the gap and

interact with rear SL.Leads to complicated dynamics

/20Staggered cases — wake interaction19

T/D

mode 2mode 1

Gap flow

wake pair


U∗ = 0.0, T/D = 3.00

U∗ = 4.0, T/D = 3.00

U∗ = 5.0, T/D = 3.00

U∗ = 8.0, T/D = 3.00

U∗ = 10.0, T/D = 3.00

U∗ = 14.0, T/D = 3.00

CL

Y

D

Y

D

Y

D

Y

D

Y

D

tUD

f D

U

800 900 1000 0.0 0.2 0.4 0.6 0.8 1.0

1

1e-04

1e-08

2

0

-2

1

1e-04

1e-08

2

0

-2

1

1e-04

1e-08

2

0

-2

1

1e-04

1e-08

2

0

-2

1

1e-04

1e-08

2

0

-2

1

1e-04

1e-08

1

0

-1

Figure 16. For T/D = 3.0, plots of vorticity contours, time series of the displacement of eachcylinder and the corresponding spectra for the displacements, red corresponding to the upstreamcylinder and green the downstream, for reduced velocities, U∗ = 0.0, 4.0, 5.0, 8.0, 10.0 and 14.0.These six cases give examples of the wake pair dominated region of figure 6.

vortex interactions at 6 cylinder diameters downstream. The “Wake flow dominated”categorization refers to the forcing on the cylinder and hence on the near-wake. In thecases shown figure 16, the vorticity topology in the near wake is clearly defined by thewake pairs shed from each single cylinder.

For all six cases, the vortices appear organized. However, only the cases for reducedvelocity U∗ = 4.0 and 5.0 are classified as periodic, as shown on figure 5. For these twocases, the cylinders oscillate at the same primary frequency, with significant amplitude.For reduced velocity U∗ = 0.0, 8.0, 10.0 and 14.0, the cylinders either strictly do not,or only barely, vibrate, maintaining transverse separation between the cylinders. In each

Page 26 of 29

Periodic for small range of U*Hardly any oscillations for higher U*

/20 Conclusions20

Two identical elastically mounted cylinders placed in tandem or staggered fashion at Reynolds number of 200.The streamwise separation is held constant: L/D = 1.5.

Gap flow is essential for understanding the results for static cylinders. Same is true for vibrating cylinders as the gap flow changes direction.

Unlike isolated vibrating cylinder, matching frequency at the natural frequency does NOT lead to synchronisation (QP and chaotic flow)

For static configuration (U* =0), three base modes were observed (as expected from lit.): no gap, gap pair, wake pair. We show evidence of rich dynamics between gap pair/wake pair (around T/D=1.5) with Pn, QP, chaotic states (similar to driven configurations).

For the tandem configuration (T/D=0), three modes are observed: mode 1 (no oscillation and vortices shed from the rear cylinder), mode 2 (large amplitude, out of phase motions, 2Pf wake), mode 3 (largest amplitude, rear cylinder chasing the front cylinder, 2P wake). Mode 2 & 3 produces periodic large gap flow.

Flow-induced vibration of two cylinders in tandem and staggered arrangementsGriffith, Lo Jacono, Sheridan & Leontini, Journal of Fluid Mechanics, 833, 98-130, 2017

Documents

Flow Past Stationary and Elastically-Mounted Martin Grifﬁth (Swinburne · 2017-11-25 · GDR Contrôle des décollement Flow Past Stationary and Elastically-Mounted Circular Cylinders