Branch/mode competition in the flow-induced vibration of a square cylinder

Texte intégral

(1)Open Archive Toulouse Archive Ouverte 2$7$2LVDQRSHQDFFHVVUHSRVLWRU\WKDWFROOHFWVWKHZRUNRI7RXORXVH UHVHDUFKHUVDQGPDNHVLWIUHHO\DYDLODEOHRYHUWKHZHEZKHUHSRVVLEOH. 7KLVLVSXEOLVKHU¶VYHUVLRQSXEOLVKHGLQKWWSRDWDRXQLYWRXORXVHIU. Official URL: KWWSVGRLRUJUVWD. To cite this version: =KDR-LVKHQJDQG1HPHV$QGUDVDQG/R-DFRQR'DYLGDQG 6KHULGDQ-RKQ%UDQFKPRGHFRPSHWLWLRQLQWKHIORZLQGXFHG YLEUDWLRQRIDVTXDUHF\OLQGHU

(2) 3KLORVRSKLFDO 7UDQVDFWLRQV$0DWKHPDWLFDO3K\VLFDODQG(QJLQHHULQJ 6FLHQFHV

(3) ,661;. $Q\FRUUHVSRQGHQFHFRQFHUQLQJWKLVVHUYLFHVKRXOGEHVHQW WRWKHUHSRVLWRU\DGPLQLVWUDWRUWHFKRDWDR#OLVWHVGLIILQSWRXORXVHIU.

(4)

(5)

(6) $*

(7) **")$*$*

(8) '**

(9)

(10) * * "

(11) . 0>-&;C#$59#<59A)69C'954#?=.%#0C#4&C14&?;<:-#0C';'#9%,CC. M^[F`JBT>dXBu M^hXbF|BJEY@q>wYhbQ|XEtxq~CwpJ XcxIo>BxXhf@_|MOAhH >IÎx

(12) 1ZsTIdR=T?h La>Z^']XvTIeSV>h+aig>uT

(13) JE|. '7#9=3'4=C5)C'%,#4-%#0C#4&C'95;7#%'C4*.4(:-4+C54#;,C !4-@':;-<AC"/%<59.#C C>;<:#1/#C 4;<-=>=C&'CB%#4.8?'C&';C1?.&';C&'C>25?;'CC!4.@':;.<BC &'C ?05>;'CC

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

(15) !@] 0=V7-] ?.+3(@5+N] P4.] O.H?] 0=V6-

(16) 7@-V+.-] W7*H(O6B@N] ] -.N+H6*.N] (] +>(NN] B0] 0=BXN] .Z39*6SA2] (] +BVE=.-] 6AO.K(+P6B@] *.OX..@] (] 0=BX] 06.>-] (@-] (] NOHV+OVH.] W6*H(P5@2]7@]H.NEB@N.]OB]P3(O]0>BX]06.>- ]]0.X]+(@BA5+(=] .Z(?E>.N]B0]O35N]O[E.]B0]E4.@B?.@B@](H.]WBHO.Z 6@-V+.-] W6*H(P5BAN] &] 2(=>BE9@2] (@-] (.HB 0=VOO.I ] ] 5N] B0].

(17) . . . . . )) $)# $ )

(18) )% )! )#%& )# &! ) "##)# %)(%)'! ) ) )$% ) #%! )%) # $ ) !##)% )#$! ) Ki?8nO_n?i?moS_nL?BP?X=cBBYvO= mniv;ovi?Q_n?i8;nOc_8mPnL8mmPK_PBO;8_nP]gYP;8oPc_mSa;OyOX8_= ]?;L8_O;8Y?_KQ_??iQ_K:?QaK:coL8gcn?_nO8Y?_?iK}L8iy?mtaKmcvj;?468a=8av_=?mOi8:X? gL?_c]?_c_ Sa gi8;nP;8X 8ggYP;8nSc_m 47 -_ p? X8no?i ;8m? ;8_ P_=v;? ;};YQ; B8oOKv? c_ mX?_=?i mnjw;nvi?mmv;L8mzSa=nvi:Qa? ncz?im;LP]a?}m8_=cU]9iQa? iPm?im8_= ;8a ?y?_ X?8=nc;8o8moicgLR;B8OXvi?mv;L8mnL?08;c]8.8iiczm&iP=K?;cYZ8gm?O_ 0L?v:OhvPncvm_8nvi?cB 8Yc_KmP=?Pnm?_KP_??iP_KO]gYO;8nPc_mL8m]coPy8o?=?|n?_mOy? Qay?mnPK8nPc_m S_ nL? X8mo L8YB ;?_ovj} nc :?no?i ;L8i8;n?iR? ]c=?X 8a= gi?=O;n Pnm c_m?o 8_= i?mgc_m? 'c]gi?L?_mOy? i?yP?zm c_ oNTm Y8iK? :c=} cBi?m?8i;L zciVL8y? :??_ KOy?_ 4 6 0L?n}g?m8a=i?KP]?mcBgL?_c]?a8=?g?_=c_PnL?]?;L9_P;8X8_=K?c]?uS;gicg?iq?m cBoL?mnjv;oxk@PPnL?BYczBP?Y=8_=BXvP=gicg?inP?m8a=OOOp?8YTK_]?_ocBnL?BYczF?X=8_= oL? mniw;nvi? 0L?v_=?iX}S_K ]c=?X Om nL?i?Bci? _cn c_X}LPKLX}_c_YP_?8i P_ _8nvk? :vo8Ymc ;c]giOm?m8_ ?|o?_mPy? _v]:?icBg8i8]?n?im *y?_mO]gYPBP?= ]c=?Xmmv;L8mp8ocB8mhv8i? ;idmmm?;rc_mY?_=?i:c=}Qa8vbRBci]CYczL8y?i?;?_oX}47:??_mLcz_oc?|LP:On]vXrgX? yP:i8nPc_8Y]c=?moL8nS_o?lAonci?y?8Y_?zi?KO]?mcBLPKL

(19) 8]gXPnv=?yP:i8oScam 0LOmmow=}BeYYczmc_Jd] 478_=S_y?mnPK8n?mnL?Hczg8mo8_ ?Y8mr;8YZ}]cv_n?=mhv8i? ;idmmm?;oPc_;}YS_=?i zL?i?nL?BYczPmg?ig?_=O;vY8iocnL?Xc_K8|RmcBnL?;}YP_=?i8_=nL? ;}XQ_=?iPm;c_mni8O_?=nccm;OYX8o?8;icmmnL?CXczGKvi?1MPmm?o vgB8ZYmxa=?inL?:ic8=?i ;Y8mmcBSay?mnPK8nPc_mcBoL?BYvP=muv;ovi?Sao?i8;nPc_cB:XvDmX?_=?imnjw;ovi?mQa;icmm

(20) Icz 0LOmg8inP;vX8iK?c]?oj}Pmmvm;?gnP:Y?ncnzc]8S_gL?_c]?_8!3zL?i?nL?Bi?hv?_;}cB g?iPc=P; ycin?|mL?==SaK8a=nL?J?hv?_;}cBnL?:c=}cm;OXY8oPc_m}_;LicaO?#8a=nj9_my?im? K8YYcgQaK8_8?ifY8mnP;Samn8:O[Rn};8vm?=:} ;L8_K?mSanL?i?Y8qy?8_KX?cB8no8;WS_=v;?=:} oL?:c=}]cqc_i?mvZnO_KS_8?ic=}_8]O; Bei;?mQ_nL?m8]?=Oi?;nOc_8mnL?]coPc_57 0L?cB 8_ ?X8moP;8YY}]cv_n?= :c=} ;c_moi8Pa?=occm;OXY8o? 8;icmmp?mnj?8] =?g?_=m c_nL?:c=}]8mmSa cm;OYY8nOc_ oL? muw;nvi8X =8]gO_K <8_= oL? mgiPaKmoPE?mm :coL 8mmx]?= L?i? oc :? ;c_mo8_o p? CYvP= =?_mOo~ oL?WQa?]8oS; yOm;cmPn 8_= oL? S_BYcz mg??= $_v]:?i cB=O]?_mPc_Y?mmKidvgm;8_ :?=?BQ_?=:vno}gP;8XY}Bei 3mow=P?m nL? O_=?g?_=?_ng8i8]?n?imvm?=8i?"oL?]8mmi8nPc

(21) zL?i? OmnL?]8mmcBoL?CYvP= =PmgX8;?=:}oL? :c=}#oL? mojv;nvi8Y =8]gPaKi8oOczPp;c_mO=?i8nPc_ cB nL?8==?=]8mm ^%P_zLP;LOmp?8==?=]8mm#nL?i?=v;?=y?Xc>n~ ^%, 2`{,# 8_= oL? /?}_cX=m _v]:?i zL?i? , Pm nL? Y?_Kp cB nL? ;icmm m?;qc_ g?ig?_=P;vY8i oc oL? CYcz 0L? =PmgX8;?= BYvP= ]8mm Pm =?BOa?= 8m zL?i? ( Pm oL? K?c]?ni}m;icmm

(22) m?;sc_8X8i?88_=PmnL?X?_KnLcBp?:c=}P]]?im?=O_oL?CYvP=+ci oLOm mov=~oL?8aKY?cB8on8;VcBp?mhv8i?;idmmm?;oOc_zPnLi?mg?;oncnL?BYcz Pm8Ymc8y8iP8:Y? )?g?_=?_oy8iP8:X?m8i?_c_=O]?_mOc_8\R?=:}nL?X?_KoLm;8Y?, 8_=oL?_8nvi8XBi?hv?_;} Qaz8o?i `{.

(23) Independently, VIV and galloping have both received considerable attention. The former, in the context of slender bodies of circular cross section, has been actively researched, leading to multiple review articles [7–9]. The primary characteristic relevant to the current study is the definition of regimes of fluid–structure interaction, referred to as branches [10,11], that occur over U∗ ranges. Beginning with low flow velocity, the first branch of VIV is the initial branch, where the amplitude of oscillation increases with U∗ and the oscillation is modulated due to the influence of both the body’s natural frequency and the Strouhal frequency (the vortex shedding frequency of a stationary cylinder). Second is the upper branch, characterized by oscillations of large amplitude at a frequency around the body’s natural frequency, which appear to be unstable and chaotic [12–14]. Third is the lower branch, consisting of periodic and stable oscillations at amplitudes around 0.6D (with D the cylinder diameter) and a frequency around the natural frequency of the body. The vortex shedding and body oscillation are synchronized at the same frequency. Finally, the synchronization is lost and a desynchronized regime takes over, consisting of small oscillations at a fluctuating frequency around the Strouhal frequency. The FIV phenomenon of galloping has also received attention, investigations focusing on canonical configurations of a square cross section cylinder oriented with a flat face normal to the flow. While Den Hartog [15] first proposed a criterion for the onset of galloping of ice-covered cables, Parkinson & Smith [16] developed a very successful quasi-steady theory to predict the amplitude response of a square cylinder undergoing galloping. The theory is especially successful for relatively heavy bodies, typically when surrounded by air, where the galloping oscillation frequency is much lower than the vortex shedding frequency. Less attention has been paid to FIV of a square cylinder with variation of angle of attack. Our recent paper [2] experimentally investigated the influence of the angle of attack of a square cylinder with low mass-damping ratio on the body’s FIV response, mapping out the flow regimes as a function of α and U∗ . A new, previously unreported, higher branch (HB) of amplitude response was observed over a range of angles of attack, 10◦ < α < 22.5◦ , where the body oscillation amplitudes are considerably higher than those seen in the upper branch associated with VIV, but with an oscillation frequency locked onto approximately half of the Strouhal frequency. A numerical study by Zhao et al. [17], which allowed motion in both the cross-stream and streamwise directions, saw a similar high-amplitude response regime. A recent experimental study by Zhao et al. [6] refined and expanded our work [2], investigating representative angles of attack where the vibrations were dominated by galloping (α = 0◦ ), by VIV (α = 45◦ ), and an intermediate region where the HB has been observed (α = 20◦ ). This study investigates the same parameter range as [6], though, while the latter characterized the dynamics and wake vorticity production in the regimes across this α–U∗ range, this study examines the boundary regions between regimes and those featuring intermittent behaviour in oscillations using wavelet and recurrence plots. The following section details the experimental method. Section 3 presents the obtained experimental results and the dynamic responses of a freely vibrating square cylinder at α = 0◦ , 45◦ and 20◦ . Lastly, conclusions of this study are given in §4.. 2. Experimental method The experiments were conducted in the free-surface recirculating water channel of the Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Monash University. More details of this water channel facility and the experimental set-up can be found in [6]. The experimental set-up is shown in figure 2. The rigid square cylinder model used in this study was made from aluminium square cross section tubing with a side width of 24.6 mm and an immersed length of L = 620 mm, giving an aspect ratio range of 17.8 ≤ AR = L/H ≤ 25.2 across the α range. The displaced mass of water yields a minimum achievable mass ratio of m∗ = 2.64. Details of the air bearing system and measurement acquisitions can be found in [2]. An end conditioning platform was used to promote parallel vortex shedding. The platform height ensured that the cylinder was outside the boundary layer of the channel floor and provided a.

(24) . . . . .

(25) . . . . . .

(26) . % %# % % $ % %% % % %

(27) % %# % % %" %% !%. . . G5[vXBvvUv8>coAVv

(28) 5V:veH>v9qQIW:>]vF?>v>V: v-H>vV5cm]5QvB]>\m>V;>`vXBveH>v`q`c>TvIVv5K^ 5``mT>:vcXv8>v>\mIn5R>VcvcXvn59mmTv v#tv5V:vo5c>]v v#tvo>]>vT>5`m]>: oIcHvB^>>v:>95qvc>`c` v-H>v)>qVXR:`vVmT8>]vo5`vn5^J>:vIVvcH>v]5VG>vv v vv

(29) v -Xv[]XnJ:>v cJT>`>]I>`v 5V5QqÌ`vcH5cv95Vv]>n>5RvTX:>v9XT[>cIcIXVv5V:vNVc>]TIce>V9qvKVv cH> :qV695Rv `q`c>Tv cH>v 8X:qv X`9IQQ5cIXV`v o>]>v 5V5Qq`>:v oIcHv 9XVeNVmXm`v o5n>Q>cv k5V`BX_T` "0.v5V:v]>9m]]>V9>v 5V5QqÌ` v-H>v"0.v[]XnI:>`v c>T[X]5QQqv]>bQn>:v B]>\m>V9qv5V5Rq`J`v cX GJn>v NWÌGHcvNWcXvg>v:qV79`vXBvg>vnI8]5cJXV`veH]XmGHv cH>vcIT>vc]59>` v%VveHJ`v`fm:rveH>v"0. o5`v[>]BX]T>:vmÌVGvg>v9XT[Q>pv'X^R>cvo5n>Q>cv5cv`95Q>`v 2vvoJcHveH>v9>Vc]5QvB]>\m>V9q 5cv v#t v*>9m^^>V9>v5V5RqÌ`v9XT[R>T>Vlv"0.v8qv[]XnI:IVGv:qV795RvIV`JGHcvXBvcH>vj> `>]J>`vT5[[JVGv cH>v]>9m^^>V9>v XBvcH>v`q`c>TvoJgvcIT> v.HI`vJ`v 59HJ>n>:v8qv[QXccIVGv cH>vcIT> `>]J>`vXVcXv5v`\m5]>vT5e_JpvoH>]>vcH>v5p>`v]>[]>`>VcvcH>vc>T[X]5Qv9XT[XV>Vcv5V:v5v#>5nJÌ:> BmV9iXVvXBvg>v& VX^TvXBveH>v[H5av:ME>]>V9>vCZ]v>59HveO>v[XKWcvcXvqK>R:v5v`qTT>kJ9v8IV5]q `\m5]>vT5kIpv3.

(30) .

(31)

(32) -H>v9Q5``J9vG5QQX[KVGv]>`[XV`>vXBvcH>v`\m5]>v9qQIV:>]vI`v`HXoVvNVvBJGm]>vv5`v5vBmV9eIXVvXBv BX] eH>v 95`> v(Xc>vg5cvKWvBIGm]>v ]>[]>`>Vc`veH>vT>5VvXBvcH>vcX[vvXBvcH>v5T[QIcm:> [>5P`vVX]T5RJs>:v8qv$ v!R`Xv`HXoVv5]>v9XVcXm]`vXBv>V>]Gqv5`v5vBmW9hXVvXBvB]>\m>V9qv5V:v BZ]veH>v8X:qvX`9ISR5cIXV v-H>vVX]T5QIs>:vnI8]5eIXVvB]>\m>V9qvI`vGIn>Vv8qv -H>v9XVcXm^` ]>[]>`>Vcvg>vB]>\m>V9qv[Xo>]v`[>9e]mTvXBvcH>v9qRIV:>]v[X`JcIXVveLT>v`>]J>`v5cv>59HvT>5`m]>:v VX]T5RIt>:v8qvcH>vT5pJTmTv>V>]Grv5V:v`c59P>:vHX]JtXVc5RQqvcXv:J`[Q5qvcH>vB]>\m>V9qv]>`[XV`> 5`v 5v BmV9cLXVv XBv .H]>>v `qV9H]YVNt5cIXVv ]>GO>`v o>]>v J:>VeLDI>:v 8qv15Xv 24v oH>]> eH>v8X:qvTXcJXVv5V:vcH>vnX]c>pv`H>::IVGvB]>\m>V9I>`v`qV9H]XVJt>:v5cvvvvv5V:vv v-H>`> `qV9H]XVMt5cIXVv]>GJT>`vo>]>v`HXoVvcXvH5n>v9R>5]v[>_JX=J<ervoIcHvH5]TXVL@v9XT[XV>Vc`vnI`L8R> BZ]vg>vvvv5V:vvvv`qV9H]XVNt5cJXVv]>GJT>`vX8`>]n58Q>vIVvcH>vcXc5QvQIBcvBX]9>v5V:vnX]c>pvQJBcvBZ]9> `>>v2 v !`v>nJ:>VcvB^XTvBJGm]>vvcH>vBK^`cv`qV9H^XVJt5iXVv]>GIT>vX99m]`v5]XmV:v voH>]>vcH> 8X=q`vnJ8]5eJXVvB]>\m>V9qvT5c9H>`vcH>vnX]c>pv`H>::JVGvB^>\m>V9rvR>5:IVGvcXv5v`qV9H]XVJt5hXV 8>co>>Vvg>vcoXv>p[>9c>:vcXvX99m]voH>VvcH>v+e]XmH5QvVmT8>]v,dv/. u.

(33) (a). 2.4 1:1. 1:3. 1:5. 2.0 1.6 A*10 1.2 0.8 0.4 0 (b). 0. 3. –0.5 –1.0. 2 f *y. –1.5 –2.0. 1. –2.5 –3.0. 0 2. 4. 6. 8. 10. 12. 14 U*. 16. 18. 20. 22. 24. 26. Figure 3. The normalized amplitude response and the logarithmic-scale frequency power spectral density (PSD) contours as a function of the reduced velocity for α = 0◦ . Note that the frequency PSD contours here are constructed by stacking the frequency PSD based on short-time Fourier transforms (STFT) at each U∗ [6]. The 1 : 1, 1 : 3 and 1 : 5 synchronization regimes are highlighted with blue shading in (a), and their boundaries are illustrated by the vertical dashed lines in (b). The dotted-dashed slope line represents the Strouhal number St 0.131 (measured in the fixed cylinder case). (Online version in colour.). The location of the other synchronization regimes can also be understood by considering the behaviour of the body’s oscillation frequency and the vortex shedding frequency with increasing U∗ . Figure 3b shows that over the entire range of U∗ tested, the body’s primary oscillation frequency fy∗ remains essentially constant. Only in the synchronization regimes does it vary noticeably, and even then only by a small amount. Figure 3 shows the synchronization regimes (1 : 1 regime over 5.8 ≤ U∗ ≤ 6.2, 1 : 3 regime over 11.2 ≤ U∗ ≤ 15.4, and 1 : 5 regime over 21.5 ≤ U∗ ≤ 22.5) occurring at equally spaced increments U∗ = 6, U∗ = 14 and U∗ = 22 during which nonlinear synchronization causes the body’s oscillation and the vortex shedding to shift from their natural values and synchronize at the values reported in figure 3. To demonstrate the temporal evolution of the frequency content for the body vibration across these synchronization regimes, figures 4 and 5 show time series of the normalized body displacement along with the frequency energy contours based on CWT at various U∗ values. At U∗ = 4.0, where the body oscillations are non-periodic, the vibration frequency is dominated by the Strouhal frequency but with some unstable behaviour over time. As the reduced velocity is increased to U∗ = 5.0, the body vibration appears to be more periodic, with the vibration frequency modulated by the Strouhal frequency and the natural frequency of the system. At U∗ = 6.0, where the 1 : 1 synchronization is observed, the cylinder vibration frequency appears to be.

(34) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . !9-)-9%9.9"%)! 79! -1*96#)9-' !#/9 9%#95.9/9(1"69#)69%#/%1)-9 -9%"9

(35) 9 .94 )%1-94 2-9&)9 9 %.9/ .9"%/-9.9"%*! 79%69'%-/%$9)%!9/-9#3.) 9'%-/%#9 /97,%9%59 4 %.698909-9.9#%)! 79.!9 #"94)-%#9#9%%3+9. . ./#7/7

(36) 75/7/7 &$3&7'7//&3/7'&.,#67'54,7/7(,' 067.7.2##7 %'3$/75/7,,*3&67'%('&&/.7//7,7!&/,% //&/75/7,#0 4$67%375",7 ('5,7 3,/,7 /7 7 $'/7 &7 /5&7 07 7 7&7 7 7 ,%.7 5 $7 /7 4,0'&7 +3&67 .7 '% &/7 67 7 ,*3&67 $'5,7 0&7 7 .'&-67 +3&67 '%('&&/7 )(,.7 &/,%//&0$67 5/7 ,'&7 &' .7 '4,7 1 %7 .7 '%('&&/7 '7,7 ,+3&67.

(37) . . . . . . . . . . . . . (R\<r<oN<rdCuL<^dm\/VO<:5WO`:<m:OrhW/5<\<û0Wd_JNuLvL<Dm<j}<^9<_<mJ5d_ud~mr3/r<:d_#+)/u /mOdr/WBHm8_vO`}<:Dod\GJ~m< '`WO`< <mrOdÔ_5dWdm ]e=s we1n;s xM= M1n]eaP> sa6MneaQ1wQea 1s Ps Qc7n=1s=; 4w xM= 6e]4Pa1zea eE wM= enw= YQFw 1a; K1YXeiPaK n=sXw Pa se]= 1i=qQe;Q6 4=M1Qen Pa wM= 4e; es7PYX1xQeas 1s s==a Sc wM= 1=X=w si=6xn1 "s y= n=;6=; =Ye6Pw Ps sxPXX EnxM=n Sc6n=1s=; we Yg1w=;ScxM=sa6MneaP1xQean=KQ]=xM=6XPa;=nQ4n1xPeaPsMQKMXi=qPe;P61a;=MQ4Pws we 6e]iea=axs xM= EQnsw 1a; wM= wMTn; M1n]eaP>s PxM ae 1i=nPe;P6Qw en Qaw=q]Qww=a6 in=s=awPawM= sa6MneaP1xQea n1aK= "w wM==a; eEwMPssa6MneaQ2wPean=KQea xM= P4n1xPea En=k=a6 61a 4= s==a we 6eax1Qa 1 ]e;= 6e]i=wT{ea M=n=4 wM= MPKMY i=nQe;P6 sa6MneaQ1xPea 4@e]=s asx14X= 1a; 1 Xe=nFn?k=a6 i1Pn PwM 4ne1;41a; aePs= Ps in=s=aw1swM=enw=sM=;;SaK1a;K1YXeiPaKEen6=s;=sa6MneaQ=4=Een=n=K1PaQaKi=qPe;P6Qx

(38) "w 1MPKM=nn=;6=; =Ye6Qw MQY= xM=;e]Pa1awP4n1xPeaEn=k=a6n=]1Pas]6M Ye=n wM1ab xM=EQFxMM1n]eaP>Ps6easQsw=axY in=s=aw

(39) "wwMPs 4ea;1qn=KPeawM=enw= sM=;;PaK1a;4e;es6QVX1wPeaPX[Paw=n]Pww=awY&e6U4qQ=FYs==aPaxM=$,1ssMenw4ns|eE i=nPe;P6PxMe==nPwPsasw14Y=1a;kP6UXn==nxswe;=sa6MneaP=;K1XXeiQaK "w QcxM=sa6MqeaQ2xQean=KQeawM=Fa;1]=aw1XFn?k=a62a;wM=FPFxMM1n]eaT>6e]iea=axs 4=6e]=sx14X=n=sYxPaKQasx14X= Q4n1wPea1]iXPx;=s %ne]xM=14e=n=sX|wM=I=k=a6 wS]=1a1XsPs41s=;ea$,ineQ;=s1aQasPKMwSawexM=sw14PXPweFwM=sa7MpfaT1xPea4=x==a wM=P4n1xPea En=k=a6 6e]iea=axs1a;xM=Pn 4=M1Pen e=n wP]=

(40) Zw1Xse MPKMYPKMxswM1wxM= wn1asQwPean=KP]=s1n=s=asPxQ=weaePt1a;61an=sXwPa]e;=sPw6MPaKPa4ea;1nn=KPeas .

(41) *aXPU=wM=61s=1w xM=61t1w Psaewss6=iwQ4X=weK1XXeiQaK "s=iY1Qa=;4xM= l1sQsw1xQ6yAneE -.K1XXeiPaKsMeX;eaXe66q M=a! 1w wM= 41t 1aKX= eF.

(42) (a). 0.8. desynchronized. 0.6 A*10 0.4 0.2 I. II. III IV V. 0 (b). 0. 3. –0.5 –1.0. 2 f *y. –1.5 –2.0. 1. –2.5 –3.0. 0 2. 4. 6. 8. 10. 12. 14. 16. U*. Figure 6. The amplitude and frequency responses as a function of U∗ for α = 45◦ . Five regimes (I–V) are identified within the VIV lock-in region. In (b), the dotted-dashed slope line represents the Strouhal number St 0.176. Note that the frequency PSD contours here are constructed based on STFT analysis [6]. (Online version in colour.) attack (here 45◦ ), where C¯ y is the mean transverse lift coefficient. This does occur for α = 0◦ , but it does not hold for α = 45◦ . However, the α = 45◦ case, like any bluff body that causes alternate vortex shedding, is susceptible to VIV, and this phenomenon dominates the FIV response of this case. Figure 6 shows the variation of the amplitude of oscillation as a function of U∗ , as well as the variation of the frequency content of the time–displacement series. Note that five regimes (I–V) were identified within the VIV lock region by Zhao et al. [6], based on an overview examination of the fluid force coefficients and the corresponding phases, and the wake mode. It shows that the response type is quite varied, with large-amplitude oscillations for 3 < U∗ < 7.5 (the region designated as the upper branch by Nemes et al. [2]) consisting of five regimes, the boundaries of which are marked by the dashed vertical lines on figure 6. The characteristics of each of these regimes, as well as the initial branch and synchronization regions, are highlighted below, in order of increasing U∗ . Summarizing the regimes, for U∗ < 3, the flow is in the initial branch as for a canonical cylinder case. The oscillations are small, comprising Strouhal and natural frequencies, resulting in a modulated response seen in the wavelet analysis in figure 7 for U∗ = 2.6. Between U∗ = 3 and 4.4, the flow follows the upper branch behaviour of a circular cylinder, and the first of a series of synchronized regimes, where the vortex shedding and the body oscillation occur at the same frequency. In this first regime (I), vortex shedding occurs in the 2S mode [2,6], similar to the classic Kármán vortex street. The amplitude of oscillation in this regime is almost.

(43) . . .

(44) !". . . . . . . . . . . . . @. . . . .

(45) @. . . . /,:2@. - -$"$--%-" ,-+#-$ %- -*&-&-!'+-"+-&("$- $--

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

(47) . . . . . . . . . . . . +# #++$+ * +

(48) ) !+ #

(49) $+ ++($+$+!%

(50) )+!)+

(51) $%!#+ # ++ + $+ ' !%#+' %#+!+

(52) $% + +&!+ +'"#++

(53) %!+. . SccS4*I*c5Pc$cSI$AP5T6DAcSDc$cS47I)c)5PWA'ScP]A'4JDA6`*)cJ*27>*c c!4*cDAP*ScD.cT46Pc I*26>*c F'XJPc '<DRc SDc S4*c GD5ASc [4*J*c S4*c &D)]Pc A$TXJ$<c .J*HX*A']c 5Pc *HX$<cSDc S4*c TIDX4$=c .J*HX*B']

(54) c!4*I*c6Pc=5SS<*c)6P'*@5&<*c)6..*J*B'*c7CcS4*c.<D[c&*S[-AcT46PcI*28?*c$A)cS4*cGJ*Z6DXPc DA*cc4D[*Z*IcS4*J*c6Pc$c)5PS6C'Sc'4%C2*c6AcS4*cDP'6<=$WDBcI*PGDAP*c$PcP4D[Bc8Bc.62XJ*Pcc%C)c

(55) c AcS4*c<$SS*Ic$c>D)*cP[5S'46A2c&*S[**AcS4*cS[DcJ*PGDAP*Pc6Pc*Z5)*BSc.DJc c5)*BS6.6*)c&]c$c PX))*Ac.J*HX*A']cP[6S'4c$B)c>D)X=$S*)cI*PGDBP*c6BcDP'6==$S5DAc.J*HX*A']cD''XJI5A2c$Sc Kcbc

(56) cBc T4*c[$Z*<*ScG<DScS4*c&D)]cDP'6=<$T6DBcP46.SPc/LE>cS4*cP]A'4IDA6`*)cB$SXN$<c.J*HX*A']cSDcP<:24S=]c $&DZ*cS4*c TJDX4$<c1+HX*A']c$<DA2c[6S4c$Ac$PPD'6$S*)cP>$<<c=DPPcD.cG*J6D)6'6S^ c 52XI*ccP4D[PcT4$ScS4*c$>G<6TX)*cD.cDP'5=<$S6DAc.5JPSc6B'I*$P*PcS4*Ac)*'J*$P*Pc[6S4c6A(I*$P6B2c 9AcS46PcJ*26>*c[6S4c$cG*$;cDP'6<=$S6DAc$IDYB)c

(57)

(58) c !4*cJ*)X'*)cZ*<D'5S]c c>$I;PcT4*c&*32cD.c$c.DXIT4cJ*26>*c#c)6PS6A'Sc.ID>c S4*c DT4*IPc6BcS4*c4624$>G=6SX)*cXGG*Jc&J$A'4cJ*26DAc$Pc6Sc6PcADScG*L6D)6' c62XI*ccP4D[PcS4$Sc[49<*c T4*c1*HX*A']c 'DBS*BScD.c V*c &D)]c DP'5<<$S6DAcI*>$5APc )D>6B$S*)c&]c$c P6A2<*c 'D>GDB*ASc $Sc$c Z$=X*cP<624T=]c$&DZ*cS4*c SJDX4$<c.J*HX*A'_cS4*c[$Z*<*SPcP4D[c4624*Jc/M*HX*A(6*P

(59) cD>G%J7B2c T4*c[$Z*=*ScG=DSc&*S[**Ac c

(60) c$A)cccS4*cDP(6<<$S6DAPc$GG*$IcSDc2Dc.JD>c$c>D)X<$S*)c DP'6=<$T:DBcJ*PGDBP*c8C'J*$P8B2c$B)c)*'J,$P6B2c.L*HX*A']cDZ*Jc c']'<*PcSDc$c4624*J.L*HX*A']c 'D>GDB*BSc$A)c<*PPcDJ2$A6`*)c>D)X<$S6DB

(61) c!4*c'$XP*cD.cS46PcJ*PGDBP*c5PcXB'=*$Jc$=S4DX24cc 4]GDS4*P7`*)cS4$Sc6Sc[$Pc)X*cSDcS4*cP]PS*>Pc6A'J*$P*)cPXP'*GS5&5=6S]cSDc*\S*O$=cAD6P* c *28>*c#c&*25APc$Sc c$A)c'DAT8AX*PcSDc

(62) c$.S*Jc[46'4cS4*c&D)]cDP'6=<$T6DBc$A)c ZDIS*\c P4*))8A2c&*'D>*c)*P]B'4IDA5a*) c 6;*c I*26>*c J*26?*c #c'DBP6PUQcD.c P]B'4IDB9a*)c DP'6=<$T6DBPc$Sc$c.I*HX*B']cP<624T<]c$&DZ*cS4*c TJDY4$<c.I*HX*B'^

(63) c 5A$<=_c$Sc

(64) c$PcP4D[Ac5Ac02XI*ccS4*c&D)]cDP(6<<$T6DBc$B)cS4*cZDJS*\cP4*))8A2c&*26Bc SDc)*P]B'4JEA6`*

(65) c"4*cZDJS*\cP4*))6A2cF'XIPc$Sc$cGI6>$I]c/J*HX*B']c'<DP*cSDc&XScP=524S<]c&*<D[c.

(66) the Strouhal frequency for the fixed body; however, the spectra are broadband, indicating that the flow is not strictly periodic. The amplitude of oscillation is only a weak negative function of U∗ , as shown in figure 6. For higher U∗ , there is an evident desynchronization in the time series with frequency components varying over a broadband region.. (c) Sub-harmonic VIV response at α = 20◦ As demonstrated by Nemes et al. [2], while the case at α = 0◦ is dominated by galloping response, and the case at α = 45◦ is dominated by VIV response, there exists a transition between the two FIV response phenomena with variation of the angle of attack. The FIV response regime map in the U∗ –α parameter space presented in [2] shows a new response regime labelled the ‘higher’ branch, as it is characterized by highly periodic oscillations with amplitudes larger than those present in the upper branch (UB). Later, the study of [6], by examining the representative angle of attack α = 20◦ , provided further details of the vibration dynamics (i.e. the structural vibration response and the fluid forcing) and revealed that a new 2(2S) wake mode was associated with a sub-harmonic synchronization in the HB. This section presents a follow-up analysis using CWT to gain a proper understanding of the temporal behaviour of the frequency response in various response regimes for the case of α = 20◦ . Also, a recurrence analysis is given to shed light on dynamical states of the structural vibration. Figure 9 shows the amplitude and frequency responses as a function of U∗ for the α = 20◦ case. As can be seen, five response regimes are identifiable: an initial branch (IB), a UB, a HB, and two desynchronization regions. Similar to the response typical of VIV, an initial branch is encountered for low reduced velocities (i.e. U∗ < 4.2), where the body’s oscillation is characterized by low amplitudes with the oscillation frequency modulated by the Strouhal frequency and fnw . The UB and HB regimes, where the body oscillations are highly periodic, occur over the reduced velocity ranges 4.4 ≤ U∗ ≤ 6.4 and 7.9 ≤ U∗ ≤ 9.4, which are highlighted with blue shading. The UB regime is associated with a 1 : 1 synchronization with a 2S vortex shedding mode consisting of two single opposite-signed vortices shed per body oscillation cycle, while the HB regime is associated with a sub-harmonic synchronization (or designated by 1 : 2 synchronization hereby) with 2(2S) vortex shedding mode consisting of two cycles of two single opposite-signed vortices shed per body oscillation cycle (see [6]). Interestingly, the HB regime is bounded by two intermittency regions (IR-I and IR-II highlighted with grey shading) in transition to the desynchronization regions (DR-I and DR-II) at lower and higher reduced velocities. It should be noted that DR-I occurs unexpectedly over the range of 6.4 < U∗ < 7.4 immediately after the UB regime, which is distinctly different from the VIV response of the cases of α = 45◦ and a circular cylinder. On the other hand, DR-II at high reduced velocities (i.e. U∗ > 10.3) appears to be similar to that seen in the case of α = 45◦ , where the body vibration become desynchronized with the vortex shedding and the vibration frequency exhibits broadband noise around a dominant component slightly below the Strouhal frequency. To examine the temporal behaviour of the body vibration in the response regimes, figure 10 shows time series of the normalized body displacement and its frequency content based on CWT at various reduced velocities. At U∗ = 3.1 from the IB regime, where the body vibration frequency is modulated generally by the Strouhal frequency and fnw , the frequency content appears to be unstable over time, resulting in disordered body oscillations, which is different from the cases of α = 45◦ and a circular cylinder. At a higher reduced velocity U∗ = 5.1 in the UB regime, the frequency content is clearly stable over time, with the dominant component close to fy∗ = 1, resulting in highly periodic body oscillations. Further, at U∗ = 7.8 from the IR-I region, while highly periodic large-amplitude (HB) body oscillations with two harmonic frequency components (i.e. the first and second harmonics) occur for a long time period, desynchronized oscillations with much lower amplitudes are also intermittently encountered. When the HB regime is reached at U∗ = 8.0, the harmonic components are consistently stable over time. The further higher reduced velocity case of U∗ = 10.1 sees intermittent switching behaviour, where highly periodic.

(67) (a). HB. 1.2 1.0 UB DR-I. 0.8 A*10. 0.6. DR-II. IR-I. 0.4. 1:1. 0.2. IR-II 1:2. IB. 0 (b). 0. 3. –0.5 –1.0. 2 f *y. –1.5 –2.0. 1. –2.5 –3.0. 0 2. 4. 6. 8. 10. 12. 14. 16. U*. Figure 9. The amplitude response and the logarithmic-scale frequency PSD contours as a function of the reduced velocity for the case of α = 20◦ . The 1 : 1 and 1 : 2 synchronization regimes are highlighted with blue shading in (a). Two intermittency regions (IR-I and IR-II) are highlighted with grey shading. In (b), the vertical dashed lines illustrate the boundaries of the regimes. The dotted-dashed line represents the Strouhal number St 0.176. Note that the frequency PSD contours here are constructed based on STFT analysis [6]. (Online version in colour.). oscillations are interrupted by oscillations of disorder, which is indicative of a transition to the desynchronization regime at high reduced velocity. To further demonstrate how the dynamical state of the body vibration behaves across the regimes of interest from the UB regime to the IR-II region, figure 11 presents the corresponding recurrence plots (RPs) based on the body displacement signal for the selected representative U∗ values. Recurrence plots were first introduced by Eckmann et al. [18] to visually analyse recurring patterns in time series of dynamical systems. As then, they have been employed to identify chaos in nonlinear dynamical systems in a large variety of scientific areas, from physics, to finance and economics, and to biological systems. The theory and construction method of RPs used for the present study can be found in [20]. As illustrated in figure 11a, for the case of U∗ = 5.1 from the UB regime, where the body vibration is highly periodic, the RP is characterized by diagonal oriented periodic chequerboard structures. These structures are symmetric about the main (45◦ ) diagonal (also known as the line of identity (LOI)). According to [20], the equally spaced diagonal lines parallel to the LOI are indicative of highly periodic recurrent dynamics with a single dominant frequency, noting that the horizontal distance between two consecutive diagonal lines depicts the vibration period. For the U∗ = 7.8 from the IR-I region (figure 11b), periodic recurring patterns are present for 300 < τ < 315; however, the diagonal lines become slightly curvy, indicating that the evolution of the dynamical states is similar but at different rates (e.g. with multiple.

(68) . . . . . . . . . . . . . . . . . L. . . L. . L. . . .$!$..%.# -.,".$ &. ..+%.&.! ',.!,.%(!$.

(69) $.. . %. * !)$.* )$.!. ..*!$..)!.. )&9E&5$IL $64865&5A@ L E;B,&;L ";6E5%L L L A+&IL &H+-#-AL @64&L-@62"B&%L #2"$1L %6A@L G-B,L F"<I.5*L@.K&@L G+-$+L ";&L "AA;-#EA&%L A6L A,&L $+"6A.$L #E<@D L @L A+&L @I@A&4L &F62F&@L (E;B,&<L A+&L !L @&&@L @64&L +6;/K65A"2L $E>F&%L3/5&@L"5%L A+&.;L 4.<<6;&%L $6E5A&<8"<A@L (6<L =LL L /5%-$"A/5*L B+"AL@64&L@A"A&@L%6L56AL$+"5*&L6;L$+"5*&L@36G3IL(7;L@64&LB.4& L6;LA+&L L$"@&L(>64L A+&L L ;&*-4&LB+&L!L&H,.#-A@L+-*,3IL 8&;-6%.$L 8"AA&?@LG.B+L$E;F&%L%-"*65"2L 30&@L/5%.$"A.5*L A+&L &H-@A&5$&L 6(L +"<465.'L(<&:E&5$-&@ L -5"23JL (7;L A+&L L $"@&L C&L!L$6<;&@865%.5*L A6L A+&L L;&*/65L-@L$+"<"$A&</K&%L#IL8&<.6%-$L8"AA&?@L/5A&>4/AA&5A2IL.5A&<;E8A&%L#IL$+"6A/$L#E;@A@L& * L.

(70) (a) 0.5 y*. t. 0.5 30. (b) 1.0 0.5 0 y* 0.5 1.0 330. 25. 325. 20. 320. 15. t 315. 10. 310. 5. 305. 0. 0. 0. 5. 10. 15 t. 20. 25. (c) y*. t. 300. 30. 300 305 310 315 320 325 330 t. (d) 1 0 1. 1 y* 0 1. 30. 30. 25. 25. 20. 20. 15. t. 15. 10. 10. 5. 5. 0. 0 0. 5. 10. 15 t. 20. 25. 30. 0. 5. 10. 15 t. 20. 25. 30. Figure 11. Recurrence plots (lower) of the time series of the normalized body displacement (upper) for the case of α = 20◦ showing different dynamical states in different response regimes: (a) periodic state at U∗ = 5.1 (UB), (b) periodic and transient chaotic states at U∗ = 7.8 (IR-I), (c) periodic state at U∗ = 8.0 (HB) and (d) periodic and transient chaotic states at U∗ = 10.1 (IR-II). (Online version in colour.). around τ = 20). The above RPs show that there exist transient chaos and abrupt changes in the intermittency region IR-I and IR-II, which are consistent with the analysis using CWT.. 4. Conclusion This paper presents a CWT-based analysis of time series of the FIV of a square cross section cylinder at three representative angles of attack, α = 0◦ , 45◦ and 20◦ . These are associated with different mechanisms of fluid–structure interaction. The use of the CWT technique provides an insight into the temporal evolution of the frequency content, and reveals the intermittency.

(71) behaviour and transitions of the FIV response regimes for the three α cases. For the α = 0◦ case that exhibits a classic galloping response, highly periodic body oscillations and stable dynamics were encountered in the 1 : 1, 1 : 3 and 1 : 5 synchronization regions over the U∗ range investigated. At the boundary of these regions the time series exhibits intermittent switching of the syncrhonization. For the α = 45◦ case, which exhibited a VIV-dominated response, the CWT analysis reveals intermittent mode competition in the body oscillation time trace throughout the synchronization region at the branch boundary values. Finally, for the α = 20◦ case of asymmetric orientation, intermittent vibration and competition appeared in the initial branch (U∗ < 4.2). As U∗ was increased to the upper branch regime (over 4.4 ≤ U∗ ≤ 6.4) associated with a 1 : 1 synchronization, highly periodic body vibration was observed with stable dynamics over time. Stable dynamics was also found in the high branch associated with a 1 : 2 sub-harmonic synchronization. Outside the HB regime, the periodic body oscillations contain intermittency, IR-I and IR-II, and a transition to two subsequent desynchronization regions. Complementary recurrence plots show that periodic states are interrupted by chaotic bursts in the IR-I and IR-II regions. Data accessibility. The datasets supporting this article have been uploaded to the online repository https://doi.org/10.6084/m9.figshare.5991232 [21]. Authors’ contributions. J.Z. designed and performed the experiments, participated in data analysis and drafted the manuscript. A.N. participated in data analysis and drafted the manuscript. D.L.J. and J.S. participated in data analysis and reviewed the manuscript. All the authors gave their final approval for publication. Competing interests. We declare we have no competing interests. Funding. The financial support from Australian Research Council Discovery Project grant no. DP110102141 and the Centre National de la Recherche Scientifique (CNRS) grant no. PICS161793 (under the Projet International de Coopération Scientifique) is gratefully acknowledged. Acknowledgements. The authors thank Nat DeRose, Mark Symonds and Hugh Venables for their help in the construction of the experimental apparatus.. References 1. Wang Z, Du L, Zhao J, Sun X. 2017 Structural response and energy extraction of a fully passive flapping foil. J. Fluids Struct. 72, 96–113. (doi:10.1016/j.jfluidstructs.2017.05.002) 2. Nemes A, Zhao J, Lo Jacono D, Sheridan J. 2012 The interaction between flow-induced vibration mechanisms of a square cylinder with varying angles of attack. J. Fluid Mech. 710, 102–130. (doi:10.1017/jfm.2012.353) 3. Blevins RD 1990 Flow-induced vibration, 2nd edn. Malabar, FL: Krieger. 4. Naudascher E, Rockwell D. 2005 Flow-induced vibrations: an engineering guide. New York, NY: Dover. 5. Païdoussis MP, Price S, De Langre E. 2010 Fluid–structure interactions: cross-flow-induced instabilities. Cambridge, UK: Cambridge University Press. 6. Zhao J, Leontini JS, Lo Jacono D, Sheridan J. 2014 Fluid–structure interaction of a square cylinder at different angles of attack. J. Fluid Mech. 747, 688–721. (doi:10.1017/jfm.2014.167) 7. Bearman PW. 1984 Vortex shedding from oscillating bluff bodies. Annu. Rev. Fluid Mech. 16, 195–222. (doi:10.1146/annurev.fl.16.010184.001211) 8. Sarpkaya T. 2004 A critical review of the intrinsic nature of vortex-induced vibrations. J. Fluids Struct. 19, 389–447. (doi:10.1016/j.jfluidstructs.2004.02.005) 9. Williamson CHK, Govardhan R. 2004 Vortex-induced vibration. Annu. Rev. Fluid Mech. 36, 413–455. (doi:10.1146/annurev.fluid.36.050802.122128) 10. Khalak A, Williamson CHK. 1996 Dynamics of a hydroelastic cylinder with very low mass and damping. J. Fluids Struct. 10, 455–472. (doi:10.1006/jfls.1996.0031) 11. Govardhan R, Williamson CHK. 2000 Modes of vortex formation and frequency response of a freely vibrating cylinder. J. Fluid Mech. 420, 85–130. (doi:10.1017/S0022112000001233) 12. Hover FS, Techet AH, Triantafyllou MS. 1998 Forces on oscillating uniform and tapered cylinders in crossflow. J. Fluid Mech. 363, 97–114. (doi:10.1017/S0022112098001074) 13. Morse TL, Williamson CHK. 2009 Prediction of vortex-induced vibration response by employing controlled motion. J. Fluid Mech. 634, 5–39. (doi:10.1017/S0022112009990516).

(72) 14. Zhao J, Leontini JS, Lo Jacono D, Sheridan J. 2014 Chaotic vortex induced vibrations. Phys. Fluids 26, 121702. (doi:10.1063/1.4904975) 15. Den Hartog JP. 1932 Transmission line vibration due to sleet. Trans. Am. Inst. Electr. Eng. 51, 1074–1076. (doi:10.1109/T-AIEE.1932.5056223) 16. Parkinson GV, Smith JD. 1964 The square prism as an aeroelastic non-linear oscillator. Q. J. Mech. Appl. Math. 17, 225–239. (doi:10.1093/qjmam/17.2.225) 17. Zhao M, Cheng L, Zhou T. 2013 Numerical simulation of vortex-induced vibration of a square cylinder at low Reynolds number. Phys. Fluids 25, 023603. (doi:10.1063/1.4792351) 18. Eckmann J, Kamphorst SO, Ruelle D. 1987 Recurrence plots of dynamical systems. Europhys. Lett. 4, 973. (doi:10.1209/0295-5075/4/9/004) 19. Lighthill J. 1986 Fundamentals concerning wave loading on offshore structures. J. Fluid Mech. 173, 667–681. (doi:10.1017/S0022112086001313) 20. Marwan N, Romano MC, Thiel M, Kurths J. 2007 Recurrence plots for the analysis of complex systems. Phys. Rep. 438, 237–329. (doi:10.1016/j.physrep.2006.11.001) 21. Zhao J, Nemes A, Lo Jacono D, Sheridan J. 2018 Branch/mode competition in the flowinduced vibration of a square cylinder. Dryad Digital Repository. (doi:10.6084/m9.figshare. 5991232).

(73)