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Influence of mass and damping ratios on VIVs of a cylinder with a downstream counterpart in cruciform arrangement
Journal of Fluids and Structures 28 (2012) 40–55
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Influence of mass and damping ratios on VIVs of a cylinder with a downstream counterpart in cruciform arrangement T. Nguyen a,n, M. Koide b, S. Yamada a, T. Takahashi a, M. Shirakashi a a b
Department of Mechanical Engineering, Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka, Niigata, Japan Faculty of Economics, Niigata Sangyo University, Karuigawa 4730, Kashiwazaki, Niigata, Japan
a r t i c l e in f o
abstract
Article history: Received 10 May 2010 Accepted 11 October 2011 Available online 21 November 2011
Previous wind tunnel experiments by the present authors showed that two kinds of longitudinal vortex induce cross-flow vibrations (LVIVs) on a circular cylinder accompanied by the vortex-synchronization (lock-in) by setting another cylinder downstream with proper gaps in cruciform arrangement, like the well-known Karman vortex induced vibration (KVIV) of a single cylinder. In this work, experiments on the three types of VIVs are carried out using a water tunnel to investigate influences of the mass ratio MR and the damping ratio z on the oscillation response. It is shown that the two LVIVs, i.e. the trailing- and the necklace-VIVs, occur in water flow over respective ranges of flow velocity U in spite of large differences in values of MR and z as compared with experiments in air flow. However, their influences on oscillation responses of the two LVIVs are largely different from each other, and also from the KVIV. They are compared by referring to the de-synchronization prediction based on the dimensional analysis and the equation on cylinder motion. The variation of behaviors among the three types of VIV are attributed to the differences among the three vortices in their threshold functions for synchronization and the vortex-synchronization functions which express the feedback effect of the cylinder oscillation on the alternating lift generated by the vortex. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Flow induced vibration Karman vortex Vortex induced vibration (VIV) Cruciform cylinders Longitudinal vortex
1. Introduction Vortex-induced vibration (VIV) of a cylinder in fluid flow is one of the most important problems in engineering and also one of the most fundamental but still challenging subjects in fluid mechanics. Among various types of VIVs, the transverse vibration of a spring supported rigid circular cylinder in uniform flow induced by the Karman vortex (KVIV) is the most prominent and simplest one from the view point of fluid mechanics, and the most commonly observed one in practical engineering. Hence, huge volume of research works have been devoted to clarify mechanism of KVIV, to establish methods for predicting behaviors, and, to develop techniques for preventing harmful effect of KVIV (Bearman, 1984; Blevins, 1990; Sarpkaya, 2004; Williamson and Govardhan, 2004). In air flow, where mass ratio MR ( ¼mass of oscillating structure/ displaced fluid mass) is in the order of hundreds, KVIV is regarded as essentially a resonant response occurring at the structure natural frequency fn. While, in water flow, where the mass ratio is considerably lower, say MR around the order of
Abbreviations: VIV, Vortex induced vibration; KVIV, Karman vortex induced vibration; LVIV, Longitudinal vortex induced vibration; TVIV, Trailing vortex induced vibration; NVIV, Necklace vortex induced vibration; VSF, Vortex-synchronization function n Corresponding author. Tel.: þ 0258 47 9730; fax: þ 0258 47 9770. E-mail address: [email protected] (T. Nguyen). 0889-9746/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfluidstructs.2011.10.006
T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
Nomenclature aspect ratio ( ¼L/d) blockage ratio [BR ¼(2dH d2)/H2] alternating lift coefficient, ( ¼ F vA =0:5rU 2 dLv ) alternating lift by vortex shedding and its amplitude H side length of square measuring section Fz alternating lift exerting on the upstream cylinder in z-direction L length of the cylinder (distance between the end plates) Lv spanwise length of vortex shedding region MR mass ratio, ¼4me/(pd2L) Re Reynolds number, ¼ rdU/m Sc Scruton number, ¼4pMRz SL, Su, SZ spectra of lift FL, velocity u and displacement Z St Strouhal number for vortex shedding, ¼ fvd/U U free flow velocity U0 free flow velocity at which fv0 ¼fn Z, ZA displacement of cylinder and its amplitude, pffiffiffi ZA ¼ 2Zrms c, cadd, ce structural, added and effective damping factors. (ce ¼cþ cadd) d diameter of the cylinders fn structure natural frequency fv vortex shedding frequency AR BR CL Fv, FvA
fz k m, madd, s t u
41
oscillation frequency of the upstream cylinder spring constant me structural, added and effective mass gap between the two cylinders time x-velocity component detected at a reference position
Greek letters
de z
m r f
ov , oz
effective logarithmic damping pffiffiffiffiffiffiffiffiffi factor damping ratio, ce =2 me k ¼ 2dpe viscosity of fluid density of fluid phase angle between Fv and Z angular frequencies of vortex shedding and cylinder oscillation
Subscript and superscript n
max rms 0 R
non-dimensional variable referred to d, fn and U values for maximum response root mean square value for stationary system resonance value predicted from desynchronization analysis
1–10, and the damping ratio z higher but resulting in a low combined mass-damping parameter MR z, the oscillation response is considerably different from the case in air flow as reported by Govardhan and Williamson (2000, 2004, 2006), Khalak and Williamson (1999), and Klamo et al. (2006). Farshidianfar and Zanganeh (2010) presented a modified wake oscillator model that can describe most of the behaviors of vortex-induced vibration for both high and low mass-damping ratio systems. A most recent review by Bearman (2011) presents comprehensive discussion for a rigid circular cylinder on the problems including two degree freedom motion, interference of tandem cylinders and the influence of Reynolds number on the maximum oscillation response. In earlier wind tunnel experiments by the present authors, it was found that KVIV is effectively suppressed by setting another cylinder downstream with a proper gap, i.e. the gap-to-diameter ratio s/do0.5, in cruciform arrangement (Bae et al., 2001). However, in the same arrangement, unexpected cross-flow vibrations occurred in higher velocity ranges accompanied by periodic shedding of longitudinal vortices with rotation axes parallel to the free stream (Shirakashi et al., 1994, 2001). Experiments on fixed cruciform system showed that the type of the longitudinal vortex depends on the value of s/d: (a) trailing vortex when s/do0.25 and (b) necklace vortex when 0.25os/do0.5, as shown in Fig. 1 (Takahashi et al., 1999). The two longitudinal vortices shed alternately from the upper and the lower sides of the upstream cylinder with frequencies considerably lower than that of Karman vortex, generating transverse exciting force on the upstream cylinder. A recent study for fixed system over wide ranges of
Fig. 1. Longitudinal vortices shedding from cruciform cylinder system (Koide et al., 2005). (a) Trailing vortex, 0o s/d o 0.25 and (b) necklace vortex, 0.25 os/d o 0.5.
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T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
experimental conditions showed that the two longitudinal vortices shed periodically in air flow over the ranges of 3.5 103 oReo9 104 and aspect ratio 7oAR o14 (blockage ratio 0.14oBR o0.26) with respective Strouhal numbers. However, one peculiar result observed in the smallest water tunnel experiment is the vanishing of necklace vortex at Re422 000, while trailing vortex persists over the whole measurement range (Koide et al., 2006; Nguyen et al., 2010). These two longitudinal-vortex-induced vibrations (LVIVs), i.e. trailing VIV (TVIV) and necklace VIV (NVIV), in air flow are regarded as a resonance response since they occur when the vortex shedding frequency fv coincides with the structure natural frequency fn. Another similar feature of these LVIVs to KVIV is that the resonance persists over a wide flow velocity range around U0, the velocity at which the natural vortex shedding frequency fv0 ¼fn, due to the synchronization of vortex shedding with the cylinder oscillation (lock-in phenomenon). In order to investigate the synchronization phenomenon of the vortices, the upstream cylinder was mechanically oscillated in a wind tunnel and the velocity u was detected by a hot wire probe set at a properly determined reference position, i.e. the position at which periodic change caused by each vortex was most definitely observed (Kumagai et al., 2002). The synchronization phenomenon was discerned by an abrupt increase in both the peak value of Su, the spectrum of u, and the cross-correlation between the cylinder displacement Z and u. Thus the threshold value was obtained showing that the synchronization velocity regions of the longitudinal vortices at an equal oscillation amplitude are considerably different from that of Karman vortex. The phase difference between Z and u are also obtained, which indicated that the phase of vortex shedding relative to the cylinder motion varies with velocity over the synchronization region. The objectives of this study is to investigate the behaviors of the three types of VIV in water flow, in which values of MR and z are largely different from those in wind tunnel experiments so far but the mass-damping factor MR z or Scruton number is comparable, and, to give physical understanding of their behaviors focusing on the influences of MR and z on the VIV responses. 2. Analysis 2.1. Phenomenon of vortex synchronization and simplification for analysis In this study, the phenomenon of VIVs simplified as shown in Fig. 2 is investigated experimentally. A rigid circular cylinder is fixed, or spring-supported for transverse motion, in uniform flow. Another rigid cylinder with an equal diameter is fixed downstream with a gap in cruciform arrangement. Both cylinders are spanned over essentially the whole width of measuring section, resulting in an equal length. It is assumed that the upstream cylinder is oscillating in the transverse (z) direction as given by Eq. (1) and that a periodic vortex shedding generates lift force Fv as Eq. (2) Z ¼ Z A sinð2pf z tÞ,
ð1Þ
F v ¼ F vA sinð2pf v t þ fÞ:
ð2Þ
In accordance to the many of research works so far (Khalak and Williamson, 1999; Williamson and Govardhan, 2004) the following simplification is introduced for the transverse force Fz exerted by the surrounding fluid in the direction of cylinder motion. The fluid force can be divided into the vortex force which exists when the body is at rest in the flow and the resistance force against the body motion which exists without flow. In fluid otherwise at rest, an oscillating cylinder experiences resistance force exerted by surrounding fluid, which is commonly assumed to consist of the inertia and viscous contributions. Therefore, the fluid force exerting on the oscillating cylinder in flow is assumed to be expressed as 2
F Z ¼ madd
d Z dZ þ Fv: cadd dt dt 2
ð3Þ
Fig. 2. Model of the cruciform cylinder system in uniform flow.
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In most of literatures about KVIV, madd and cadd are assumed to be constant then effective mass me and effective damping factor ce defined by Eqs. (4), (5) are regarded as constants me ¼ m þ madd ,
ð4Þ
ce ¼ c þ cadd :
ð5Þ
Note that Eq. (3) with constant values of madd and cadd is not really true but an approximation to simplify the oscillation analysis. Since, the resistance against the cylinder motion should be proportional to the square of the relative flow velocity, and Fv generated by the synchronizing vortex accounts for all the fluid force, meaning that madd ¼0 and cadd ¼0, as suggested for the case of KVIV in Bearman (1984). The vortex shedding from a cylinder oscillating as Eq. (1) is determined by the flow condition, the system geometry and the oscillation motion, irrespective of the cause of cylinder oscillation. Hence, for geometrically similar systems shown in Fig. 2, i.e. for the systems with an equal aspect ratio AR and gap-to-diameter ratio s/d, the Strouhal number and the lift coefficient of the vortex force are expressed as follows, where ‘‘funct.’’ stands for a function in general f d f d St ¼ v ¼ funct: Re,Z nA , z , ð6Þ U U f d ¼ funct: Re,Z nA , z , U ð1=2ÞrU dLv F vA
CL ¼
2
ð7Þ
where Lv is the spanwise length of vortex shedding region and Lv ¼L for the Karman vortex. The two longitudinal vortices, as well as the Karman vortex, synchronize (lock-on) with cylinder oscillation when the oscillation amplitude is larger than their respective threshold values, ZAth. This criterion is expressed by dimensionless variables as f d Z nA 4Z nAth ¼ funct: Re, z : ð8Þ U The synchronization is defined as a phenomenon of flow in which the vortex shedding is driven by the cylinder oscillation. In consequence, the vortex shedding frequency and phase difference of the vortex force Fv from the cylinder displacement Z in turn are determined as below f v ¼ f z,
ð9Þ
f ¼ funct: Re,Z nA ,
f zd : U
ð10Þ
Since Eq. (7) under synchronization and Eqs. (9) and (10) express the manner in which the vortex shedding is driven by the cylinder oscillation they are referred to as ‘‘vortex-synchronization functions (VSFs)’’ henceforth in this paper. When the cylinder is steadily oscillating excited by the vortex force Fv, Eq. (9) always applies and Eqs. (6), (7) and (10) are rewritten as St ¼
f vd f d ¼ z , U U
on C L , f ¼ funct: Re,Z nA , nv , U
ð11Þ
ð12Þ
where
onv ¼
fn ¼
2pf v n ¼ fz, 2pf n
1 2p
sffiffiffiffiffiffiffi k : me
ð13Þ
ð14Þ
2.2. Equations for cylinder oscillation By applying Eqs. (1) and (2) to the equation of motion for cylinder oscillation with the assumption of constant me and ce, the following expressions are obtained for the oscillation response Z nA ¼
1 C L U n2 Lv 1 f , 2p3 M R d AR z
ð15aÞ
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T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
where fz is the amplitude magnification factor defined by 1 f z ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ð1ovn2 Þ2 þð2zonv Þ2
ð15bÞ
and tan f ¼
2zonv : 1onv2
ð16Þ
2.3. Desynchronization prediction as the reference When ZA oZAth, the synchronization does not occur and the oscillation behavior is essentially a forced vibration excited by the alternating lift generated by periodical vortex shedding. The frequency fv and the magnitude FvA of the vortex force are virtually equal to those for the same cylinder at rest, i.e. fv0 and FvA0, respectively. Then, the dimensionless frequency of the exciting force is given by
onv ¼
f v0 f d U ¼ St0 U n : ¼ v0 U f nd fn
ð17Þ
This desynchronization prediction leads that the maximum response occurs as the resonant oscillation at fv ¼fn, that is, at the velocity U0 at which fv0 ¼fn. Since, the resonance-frequency, -velocity and -amplitude at the maximum response are expressed in dimensionless forms as below n
f zR ¼ wnvR ¼ 1,
ð18Þ
U nR ¼ U n0 ¼ 1=St0 ,
ð19Þ
Z nAR ¼
1 C L0 Lv 1
p2 ScSt20 d AR
,
ð20Þ
If CL0, St0 and Lv/d are independent of Re and AR, Z nA ¼ 2onv2 zf z ðonv Þ: Z nAR
ð21Þ
Eq. (20) shows that the Scruton number Sc ¼4pMRz can be a single parameter to predict the maximum response of VIVs taking the both contributions of MR and z into account, if it were not for the synchronization. While, the damping ratio z n influences also the behavior of oscillation amplitude with ov, and then with flow velocity Un, as seen from Eq. (21). Thus n Eqs. (17), (20) and (21) give the desynchronization prediction of ZAScAR–Un curves for the VIVs in general. n n When ZA becomes larger than ZA th given by Eq. (8) for the vortex, the synchronization occurs and the VSFs (12) for the n vortex govern the values of lift coefficient CL and phase difference f. Consequently, the oscillation response, ZA and n n oz ( ¼ ov), together with CL and f should be obtained by solving the system of four equations, that is the two VSFs given as Eq. (12) and Eqs. (15a), (16). Note that the VSFs coupled with the threshold function (8) constitute an oscillator model for the VIV, which deviation of the oscillation response from the desynchronization prediction is caused by the feedback effect of cylinder motion on the vortex force Fv. Hence in this paper, the behavior of the three types of VIV are expressed by the above non-dimensional parameters and compared by normalizing with the respective desynchronization prediction to see the influence of mass and damping factor on the VIVs. 3. Experimental apparatus, measuring system and experimental conditions Outline of the experimental apparatus is shown in Fig. 3. The measuring section has 100 mm square cross section and 500 mm length. The diameter d of the cylinders is 10 mm and d¼15 mm is also tested for comparison. Although the blockage ratio BR [ (2dH d2)/H2] in this work is significantly higher, say BR ¼0.19 for d ¼10 mm and 0.28 for d¼15 mm, than the limitation of usual wind/water tunnel experiments, earlier wind tunnel experiments confirmed that the effect of BR on the longitudinal vortex shedding is insignificant at these values of BR (Nguyen et al., 2010). The upstream cylinder is rigidly fixed, or supported by twin-plate-springs at the both ends outside the measuring section so as to make the motion pure cross-flow mode, while the downstream cylinder is mounted rigidly on a traversing device to adjust the gap s. End plates are attached to the upstream cylinder to remove influence of flow through slots where the upstream cylinder is passing (Shirakashi et al., 1985). Measurement instrument set up is also illustrated in Fig. 3(a). An electromagnetic flow meter is set to the pipe downstream the measuring section to measure the flow rate. The velocity profile of the free flow in the measuring section has 95% uniform region both in horizontal and vertical direction. The turbulence level obtained by a hot film probe at the center of the measuring section is less than 2.8% over the experimental range of Re. The free flow velocity U is determined
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Fig. 3. (a) Water tunnel, supporting structure and measuring system and (b) cross section of the measuring section (viewed from downstream).
as the uniform value in the profile, which is obtained from the flow rate divided by the cross-sectional area of the measuring section multiplying a correction factor. This factor keeps a constant value of 1.05 over experimental range of Reynolds number. A hot-film probe is set at respective reference positions for each of the three vortices in near wake where the periodic vortex shedding is most definitely detected in the velocity u (Koide et al., 2005). A laser displacement sensor is used to measure the displacement Z of the upstream cylinder. In this paper, the vibration amplitude is expressed by the root-mean-square value of Z. Fluctuating lift force loading on the fixed upstream cylinder is measured by a load cell. The natural frequency fn and the effective logarithmic damping factor de are determined from free damping oscillation in the water tunnel at U ¼0, then the effective mass me is calculated by the formulame ¼ k=ð2pf n Þ2 . As seen in Fig. 3(b), the parts of cylinder and support outside the measuring section are in water at rest. Hence, madd and de estimated in this way include their contribution, and there is no rational way to separate it from those attributed to the cylinder in the measuring section. Therefore, me and ce obtained in water flow at U ¼0 is regarded as the structure mass and damping in this paper.
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Table 1 Characteristics of the elastically supported cylinder system.
d (mm) L (mm) k (N/m) fn (Hz) z (–) MR (–) Sc (–)
LVIVs in air flow (Shirakashi et al., 2001)
KVIV in air flow (Koide et al., 2002)
3 types of VIV in water flow
26 318 652 14.5–15.5 0.0014–0.003 360 6.4–14.4
26 318 623 16.94 0.16 10 3 264 16.4
10, 15 98 111 4.5–5.13 0.024–0.082 6–30 2.2–15.6
Fig. 4. Spectra of lift on fixed upstream cylinder: (a) Karman vortex (d ¼10 mm, U ¼1.4 m/s); (b) trailing vortex (d¼ 10 mm, s/d¼ 0.08, U¼ 1.3 m/s); (c) necklace vortex (d ¼10 mm, s/d ¼ 0.3, U ¼1.9 m/s); and (d) vanishing of necklace vortex at a higher flow velocity (d ¼ 10 mm, s/d ¼0.3, U¼ 2.8 m/s).
Experiments on the trailing vortex are carried out at the gap-to-diameter ratio s/d ¼0.08, and on the necklace vortex s/d ¼0.28 or 0.3, since at these values the trailing and the necklace vortex shedding were most clearly observed in the wind tunnel experiments (Koide et al., 2005; Shirakashi et al., 2001). Experimental conditions are summarized in Table 1, where the conditions for wind tunnel experiment in Refs. Shirakashi et al. (2001) and Koide et al. (2002) are compared. 4. Experimental results 4.1. Vortex shedding from fixed system Examples of spectrum SL of lift on fixed cylinder caused by the Karman vortex and the longitudinal vortices are shown in Fig. 4. The peaks in SL for the three vortices are observed at respective frequencies as seen in Fig. 4 (a), (b) and (c). For each vortex, a peak at the equal frequency is observed in the corresponding velocity spectrum Su, assuring that the three vortices generate alternating lift forces. It is peculiar that the periodic shedding of necklace vortex seems to vanish beyond a certain velocity (or Reynolds number) as seen in Fig. 4(d). The normalized vortex shedding frequency, Strouhal number St0 and the lift coefficient CLrms0 obtained from the amplitude of lift force are plotted against the Reynolds number in Fig. 5, compared with the earlier wind tunnel experiments of Shirakashi et al. (2001). For Karman vortex shedding from a fixed single cylinder, St0, CLrms0–Re curves
T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
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Fig. 5. Strouhal number and lift coefficient versus Reynolds number for fixed system: (a) Karman vortex (single cylinder); (b) trailing vortex (s/d¼ 0.08); and (c) necklace vortex (s/d¼ 0.28 or 0.3).
show good agreement between data in water and air flows. The values of St0, CLrms0 for the longitudinal vortices in water flow agree fairly well with those in air flow, and are slightly dependent on Re as seen in Fig. 5(b) and (c). The similarity of the two longitudinal vortices shedding from fixed system is well assured except the vanishing of necklace vortex for Reynolds number range of Re 420 000. In Fig. 5(c), the St0 ¼ 0 plots at Re 422 000 does not mean steady vortex but vanishing of necklace vortex peak in SL as seen in Fig. 4(d). The values of St0 and CL0 in Fig. 5 indicated by the horizontal lines are determined as respective constant values based on the data in the figure for the later use in the desynchronization prediction. Based on the dynamic similarity theory, the peculiar behavior of the necklace vortex in the high Re region can be attributed to the slight defects of geometrical similarity and/or difference in characteristics of the approaching flow. The investigation about universality of longitudinal vortices (Nguyen et al., 2010) showed that the necklace vortex is definitely observed to shed in a wind tunnel experiment with BR ¼26% (AR ¼7.1) when 3.5 103 oRe o9 104, and that it vanishes at around Re ¼2 104 in water tunnel experiment with BR ¼ 19% (AR ¼10) and 28% (AR ¼6.8) like the results in this work. Therefore, influence of the aspect ratio or blockage ratio on the behavior of vortex shedding is excluded from the potential causes for the necklace vortex vanishing. Since the end plates are not strictly geometrically similar, an experiment with the upstream cylinder without end plates fixed directly to the side walls of the measuring section was conducted. The result showed the necklace vortex vanishing at the same velocity as Fig. 5. Hence, the effect of end plates is also excluded. Therefore, the potential causes remaining at present are the differences in uniformity and turbulence of approaching flow. 4.2. Effective mass and damping The effective mass and the logarithmic damping factor were obtained from free damping oscillation of the single cylinder after giving an initial impact. The effect of surrounding air is examined first by carrying out the experiment in a vacuum box. No
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T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
Fig. 6. Effective mass and damping of single cylinder obtained by free damping oscillation in water tunnel, normalized by their counterparts in air at rest.
significant difference was observed over the range of absolute pressure between the atmospheric pressure and 0.01 MPa, which shows that influence of the air is negligible. Then, the same procedure was carried out in the water tunnel with U from 0 to 3 m/s under the condition of small terminal oscillation. The results in Fig. 6 show that the additional mass is virtually constant, while additional damping increases significantly with U when Un 410. In this paper, me and ce are regarded to be constant, and values obtained by the free damping oscillation in still water are used for the desynchronization prediction. 4.3. Oscillation behavior of the three VIVs An example of behaviors of the three VIVs in water flow, i.e. vibration amplitude Zrms and fz versus flow velocity U, is presented in Fig. 7(a) compared with that in air flow as Fig. 7(b). The fv0 U curves are for the vortex shedding frequencies obtained from measurements for the respective fixed counterparts. It is noted that the oscillation frequencies in air flow for the three VIVs are always virtually equal to fn. The three VIVs are clearly observed in water flow in ways similar to their respective counterparts in air flow, in spite of large differences in the fluid property, dimension and structure parameters between the two experiments. It is noted that when the oscillation amplitude is small in air flow, the spectrum of displacement, Sz, has a peak at fn, in addition to fv, due to the selective resonance which commonly occurs for a low-damped system exerted by alternating force having broadly distributing spectrum as shown in Fig. 4. In Fig. 7, and in the following figures in this paper, such data for fz are not plotted. In water flow, two peaks sometimes appeared in Sz even when the oscillation amplitude was considerable. In such cases, the frequency of the higher peak are taken as fz and plotted in the figures. The amplitude of NVIV in water keeps around the highest level up to the maximum velocity in Fig. 7(a), making the higher border velocity of NVIV unclear. This may be associated with the vanishing of necklace vortex expressed by the broken line for fv0–U. The oscillation of NVIV is not stable but very intermittent in the velocity range where the necklace vortex shedding is unclear. 4.3.1. Vibration induced by Karman vortex (KVIV) In order to compare results of water tunnel experiment on KVIV with largely varying values of MR and z in Table 1, data n of Zrms–U are normalized into ZrmsSc–Un since in Eq. (20) Lv =L ¼ Lv =dAR ¼ 1 for the Karman vortex. Fig. 8(a) shows the
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49
Fig. 7. Behaviors of the three VIVs: (a) in water flow, d ¼10 mm, MR ¼15, fn ¼4.88 Hz, de ¼ 0.2; (b) in air flow, d¼ 26 mm, MR ¼264 (KVIV) and 360 (LVIVs), fn E15 Hz, de E0.01 (Kato et al., 2006).
normalized experimental data for MR ¼ 6, 15 in water flow and MR ¼264 in air flow (Koide et al., 2002) when the Scruton number’s values are about 6–7. The normalized oscillation frequency is plotted in Fig. 8(b). The curves in Fig. 8(a) and (b) are for the desynchronization prediction obtained from Eqs. (17), (20) and (21) using St0 ¼0.21 and CLrms0 ¼0.4 as shown in Fig. 5(a). n The desynchronization prediction gives reasonable values for the maximum response, i.e. the amplitude Zrms max and n the velocity Umax, when Sc is comparable but MR and z are largely different. It also well predicts the initial velocity of KVIV and the tendency that the Un range expands wider, especially towards higher velocity, when z is larger. However, the measured oscillation amplitudes are, generally speaking, considerably larger than the desynchronization prediction, suggesting that the lift coefficient under synchronization may be larger than the value for the stationary cylinder. The n n synchronization is clearly observed in fz in Fig. 8(b), that is fz is constant over the Un range where oscillation amplitude is n large. As reported in many literatures, fz in this region deviates higher than unity when MR is smaller, or, z is larger (Khalak and Williamson, 1999; Williamson and Govardhan, 2007). 4.3.2. Vibration induced by trailing vortex (TVIV) n Behavior of TVIV is investigated by normalizing in a similar manner as the preceding section for KVIV. In Fig. 9, ZrmsScAR n n and fz are plotted against U for systems at which MR ¼6, 15 in water flow and MR ¼ 360 in air flow. The curves are their desynchronization predictions for St0 ¼ 0.08 and CLrms0 ¼ 0.22 indicated in Fig. 5(b), and Lv/d¼6.2 from Shirakashi et al. (2001). The normalized vibration response of TVIV in the initial stage, say 6oUn o10, collapses on a single curve for all the systems, consequently the deviation from the desynchronization prediction being larger when z is smaller. Although the curve deviates rapidly from the measurement in the higher Un range, the maximum amplitude for MR ¼360 and 15 is fairly well predicted by Eq. (20). The TVIV range extends in the higher Un for MR ¼15 in water flow as compared with MR ¼360 in n air flow. However, this tendency is not coherent for MR ¼ 6. The non-dimensional oscillation frequency fz in large TVIV region increases with Un with a gradient which becomes steeper with decreasing MR, as seen in Fig. 9(b). By comparing Fig. 9 with Fig. 8, it is seen that the normalized behaviors of TVIV is similar to that of KVIV, including the influence of the damping ratio, z. 4.3.3. Vibration induced by necklace vortex (NVIV) Fig. 10 shows the normalized behavior of NVIV for the same systems as in the preceding two sections. Here, St0 ¼0.04 and CLrms0 ¼0.33 as indicated by the horizontal lines in Fig. 5(c), and Lv/d¼5.0 (Shirakashi et al., 2001) are used for the n desynchronization prediction. Compared with the other two VIVs, ZrmsScAR–Un plots among the systems with different values of MR are largely scattering and its influence seems incoherent. The maximum amplitude by the desynchronization prediction is much larger than the measured. This is because Eq. (20) gives several-times-diameter maximum amplitude
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Fig. 8. Oscillation behavior of KVIV compared with the desynchronization prediction.
Fig. 9. Oscillation behavior of TVIV compared with the desynchronization prediction.
when St0 ¼0.04 for a system with Sc around 10, to which the assumptions for the desynchronization do not apply. It is also unnatural that the oscillation for MR ¼6 (in water) decays at Un 425 while it holds much higher maximum value up to Un E30 for MR ¼360 (in air). This may be attributed to the necklace-vortex-vanishing observed for the fixed system as shown in Fig. 4(d) and Fig. 5(c). The large vibration in Fig. 10(a) for MR ¼15 appearing at high values of Un, say Un 435, is
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51
Fig. 10. Oscillation behavior of NVIV compared with the desynchronization prediction.
not sinusoidal but modulating intermittently. Hence, the maximum amplitude and the range of NVIV cannot be properly determined from the present data for low values of MR. 4.4. Behavior of the maximum response Analysis in Section 2 showed that the VIV behaviors of equal Sc systems are governed by the damping ratio z if the synchronization did not occur. Therefore, the maximum response and the velocity range of VIVs against z are investigated in this section. n n n The maximum response velocity Unmax, the maximum amplitude Zrms max and fz max, the value of fz at Un ¼Unmax, are normalized by their respective desynchronization prediction values and plotted against z in Fig. 11. A feature common to the three VIVs is that the normalized oscillation frequency of maximum amplitude shown in Fig. 11(c) is equal to unity when z is smaller than 0.007. The desynchronization equations give good prediction, generally speaking, for the maximum response of KVIV over the measurement range of z. The normalized maximum response velocity of KVIV, St0Unmax, is slightly higher than unity for the lowest value of z, as seen in Fig. 11(a) and approaches to unity with increasing z. The maximum amplitude of KVIV is estimated fairly well by the desynchronization prediction, Eq. (20), as Fig. 11(b). Note that the ordinate in the figure is log-scale. The maximum response frequency deviates slightly higher with z 40.01. The maximum response behavior of TVIV in Fig. 11 is similar, roughly speaking, to that of KVIV described above. Two significant differences are: the normalized maximum response velocity St0Unmax of TVIV in Fig. 11(a) is constant but significantly larger than unity irrespective of z. There seem tendencies of decreasing maximum amplitude and increasing maximum response frequency with increasing z as seen in Fig. 11(b) and(c). Although the data of NVIV are scattered in Fig. 11, there observed definite difference in its behavior from those of the other two VIVs. The most prominent is the low value of the maximum response as observed in Fig. 11(b), due to the high St0 value of NV, as suggested in Section 4.3.3. 4.5. Velocity range of VIVs n
The regions of three VIVs defined by Zrms 40.02 are presented in the z–UnSt0 plane in Fig. 12. The value UnSt0 ¼1 on the ordinate means the resonance velocity by the desynchronization prediction. It is seen that the velocity range of KVIV thus normalized extends from a value slightly lower than unity to around 1.4 when z is lower than around 0.01 and expands wider with increasing z especially to the higher velocity region. This behavior of KVIV range is in agreement with many
52
T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
Fig. 11. Maximum response of 3 VIVs: J, KVIV; m, TVIV; and , NVIV. (a) Normalized velocity at which the resonance occurs; (b) normalized maximum amplitude; and (c) normalized oscillation frequency at which the resonance occurs.
Fig. 12. Response regions of VIVs for Sc¼ 2–16 on UnSt0 versus z plane (limited by border lines and background color patterns).
other references such as Williamson and Govardhan (2000, 2004) where they observed dependence on the mass ratio MR but not on z. The tendency of velocity range of TVIV versus z is similar to that of KVIV, but considerably wider even after normalizing by St0, towards both higher and lower velocity, in contrast to KVIV. For NVIV, the velocity range is considerably narrower (in relative measure) and shifted to the range much lower than UnSt0 ¼1. When z 40.01, the higher border value of Un was not determined due to the intermittent oscillation as mentioned in the preceding section.
T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
53
n
Fig. 13. Lift phase f and ratio of amplitude versus ov.
5. Discussion Although a few experimental data on ZAth for Karman vortex are reported so far, they do not correlate well with each other (Koide et al., 2002), may be due to the differences in its definition and way to determine the synchronization from experiment. On the longitudinal vortices, the only existing data for ZAth is those by the present authors (Shirakashi et al., 2001) obtained through wind tunnel experiment. However, in spite of the fact that the data on ZAth for the three vortices n are still insufficient, it seems rational that the minimum value of ZAth appears at fz ¼fv0 and considerably low, say less than around 0.01–0.03 for the three vortices. Thus, in the VIVs with significant amplitude the synchronization occurs practically always since the oscillation frequency fz is equal to the vortex shedding frequency fv as described in Section 2.2. n Concerning the vortex-synchronization functions (VSFs) of the Karman vortex, there are many data for CL(ZA) as Eq. (7), but they do not so well agree maybe due to the dependence on factors such as Reynolds number, aspect ratio, fzd/U ratio, etc. Only a limited number of reports on VSF of f are now available on the Karman vortex, and no experiment has been carried out in the case of the longitudinal vortices. Hence, it is worthwhile to discuss the influence of z on vortex-induced vibrations in general, based on the existing data of the threshold function and VSF of f for KVIV. Fig. 13 shows the desynchronization prediction for f and ZA/ZAR for Scruton number Sc¼10 by Eqs. (13) and (14) as typical examples of air and water flows, i.e. z ¼0.00227 (MR ¼ 360) and 0.0794 (MR ¼10), respectively. Note that the abscissa is proportional to the velocity outside the synchronization region, but not in the synchronization region. In Fig. 13(b), the threshold curve ZAth/ZAR of Karman vortex estimated from Koide et al. n (2002) is added using approximation ov ¼fv0/fn. The result on VSF of f for Karman vortex by Carberry et al. (2001) is reproduced in Fig. 14, obtained by PIV measurement in water flow with a fixed velocity around a cylinder oscillated mechanically with a constant amplitude. In order to apply the threshold amplitude by Koide et al. (2002) to Fig. 13(b) and f(fz) in Fig. 14 to Fig. 13(a), it is assumed that their dependence on Re as suggested in Eqs. (8) and (10) are insignificant supported by experimental results as shown n n in Fig. 5. For the VSF of f, dependence on ZA in Eq. (10) is also neglected since Sc is constant meaning that ZA will not be largely different. Based on these assumptions, f in Fig. 14 becomes applicable by converting the relationship into f(fz/fv0). Thus the influences of z on VIVs observed in the experimental results are explained as below. (1) Intersections of ZA/ZAR and ZAth/ZAR curves in Fig. 13(b) (A, B for z ¼0.00227 and A’, B’ for z ¼0.0794) give the outside n limit of synchronization velocity range. Therefore, the VIV range of ov and of Un in turn, can be larger when z is larger.
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T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
Fig. 14. Lift phase, flift, and amplitude of the lift coefficient CL versus frequency ratio, fe/f0 ¼fz/fv0 in forced oscillation experiment (Carberry et al., 2001), U ¼0.09 m/s, d ¼ 25.4 mm, ZA/d ¼0.5, Re ¼2.3 103, f0 ¼ 0.7484: , lift phase; K, CL.
(2) In the synchronization VIV, the point P (Un,f) moves between A–B or A’–B’ from left to right along the solid or chaindot curve in Fig. 13(a). However, it does not mean that all the points in these sections of the curve are traced, since at the same time f is governed by the VSF as given by Fig. 14 for the case of Karman vortex, where the point (fz/fv0, f) moves from right to left with increasing flow velocity U. n (3) In the case of VIVs with small z, ov E1 between A and B, meaning that the oscillation frequency is virtually fixed at fn in agreement with the experimental result as shown in Section 4.3.1 Fig. 14 shows that f o0 when fz/fv0 is larger than a certain critical value, say [fz/fv0]crit ¼ 0.82. The negative f never occurs in VIVs since it would cause energy transfer from the cylinder to the fluid. Therefore, for the KVIV in air flow with increasing U, fz(¼ fn)/fv0 41 when velocity is low and hence the synchronization does not occur. When the velocity attains above the value corresponding to [fz/fv0]crit, the synchronization occurs if the amplitude is larger than the threshold value and the point P(Un,f) drops somewhere on A–B curve as the solution of the equation system composed of Eqs. (12) (15a, b) and (16). (4) In the case of synchronization KVIV, Fig. 14 suggests that the value of f changes from a value near to zero (A or A0 in Fig. 13(a) at the lowest) to 1601. Corresponding change of f is indicated as C, C0 in Fig. 13(a). The normalized frequency n n fz ( ¼ ov) is nearly equal to unity over the range of synchronization KVIV in spite of the jump of f when z is low as n n indicated by A–C. The deviation of fz becomes larger as A’–C’ when z is higher, and constant value of fz in water flow in the higher velocity region, so-called lower branch response, is explained as the result of constant plateau of f E1601 for fz/fv0 o0.8 in Fig. 14 (see also Eq. (16)). n n (5) The resonance response can be attained only when f ¼ p/2 (and hence fz ¼ ov ¼1) is stably attained under the restraint of the VSFs, which will lead the desynchronization prediction for the maximum response reasonable. For KVIV, this condition may not be satisfied as suggested by Fig. 14. Although the above discussion is based on KVIV data, the same consideration will be applied for the two LVIVs if the threshold function and VSF of f are obtained. Hence, the different deviating ways of experimental results from desynchronization prediction among the three VIVs are attributed to the difference of these functions among them. 6. Conclusion In this work, experiments were carried out in water flow over the ranges of Reynolds number Re ¼3 103–4 104 and Scruton number Sc¼2–16, which are comparable with the previous wind tunnel experiments, in order to investigate influences of the mass ratio MR and damping ratio z. The two LVIVs are confirmed to occur in water flow in spite of the large differences in the dimension and the mass ratio and/or damping factor as compared with the earlier wind tunnel experiments. However, behaviors of the two LVIVs in water flow deviates from their respective counterparts in air flow, in different manners as compared with the case of KVIV. The results are compared and discussed by referring to the desynchronization prediction. Analysis for the influences of MR and z based on the dimensional analysis and the equation of cylinder motion showed that the desynchronization oscillation behavior of VIVs with varying velocity is determined by z, while its maximum amplitude of VIVs without synchronization is given by Sc taking both the contributions of MR and z into account. If the threshold function of the vortex is given, the desynchronization prediction can give the range of synchronization VIV. The behavior of synchronization VIVs with varying velocity can be obtained by solving the simultaneous equation systems
T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
55
composed of the two VSFs and two equations obtained from the equation of cylinder motion. Nevertheless, the desynchronization prediction can elucidate influences of z such as the oscillation frequency higher than the structural natural frequency and the broadening of the VIV velocity range. Therefore, what is most required for oscillation prediction is establishment of these functions for the three vortices. The vanishing of necklace vortex at high Re and the unstable intermittent nature of NVIV in higher velocity range are also left for further experiments. References Bae, H.M., Baranyi, L., Koide, M., Takahashi, T., Shirakashi, M., 2001. Suppression of Karman vortex excitation of a circular cylinder by a second cylinder set downstream in cruciform arrangement. Journal of Computational and Applied Mechanics 2 (2), 175–188. Blevins, R.D., 1990. Flow-Induced Vibration. Van Nostrand Reinhold, New York. Bearman, P.W., 1984. Vortex shedding from oscillating bluff bodies. Annual Reviews of Fluid Mechanics 16, 195–222. Bearman, P.W., 2011. Circular cylinder wakes and vortex-induced vibrations. Journal of Fluids and Structures 27, 648–658. Carberry, J., Sheridan, J., Rockwell, D., 2001. Forces and wake modes of an oscillating cylinder. Journal of Fluids and Structures 15, 523–532. Farshidianfar, A., Zanganeh, H., 2010. A modified wake oscillator model for vortex-induced vibration of circular cylinders for a wide range of massdamping ratio. Journal of Fluids and Structures 26, 430–441. Govardhan, R., Williamson, C.H.K., 2000. Modes of vortex formation and frequency response of a freely vibrating cylinder. Journal of Fluid Mechanics 420, 85–130. Govardhan, R., Williamson, C.H.K., 2004. Critical mass in vortex-induced vibration of a cylinder. European Journal of Mechanics. B/Fluids 23, 17–27. Govardhan, R.N., Williamson, C.H.K., 2006. Defining the ‘‘modified Griffin plot’’ in vortex-induced vibration: Revealing the effect of Reynolds number using controlled damping. Journal of Fluid Mechanics 561, 147–180. Kato, N., Koide, M., Takahashi, T., Shirakashi, M., 2006. Influence of cross-sectional configuration on the longitudinal vortex excitation of the upstream cylinder cruciform two-cylinder system. Journal of Fluid Science and Technology 1 (2), 126–137. Khalak, A., Williamson, C.H.K., 1999. Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. Journal of Fluids and Structures 13, 813–815. Klamo, J.T., Leonard, A., Roshko, A., 2006. The effects of damping on the amplitude and frequency response of a freely vibrating cylinder in cross-flow. Journal of Fluids and Structures 22, 845–856. Koide, M., Takahashi, T., Shirakashi, M., 2005. Experimental study on universality of longitudinal vortices shedding periodically from crisscross circular cylinder system in uniform flow. In: Proceedings of International Conference on Jets, Wakes and Separated Flows. Mie, Japan, pp. 433–438 (in Japanese). Koide, M., Ootani, K., Yamada, S., Takahashi, T., Shirakashi, M., 2006. Vortex excitation caused by longitudinal vortices shedding from cruciform cylinder system in water flow. JSME International Journal B-49 (4), 1043–1048. Koide, M., Tomida, S., Takahashi, T., Shirakashi, M., 2002. Influence of cross-sectional configuration on the synchronization of Karman vortex shedding with the cylinder oscillation. JSME International Journal 45 (2), 249–258. Kumagai, I., Takahashi, T., Shirakashi, M., 2002. Influence of oscillation of upstream circular cylinder on longitudinal vortex excitation of criss-cross circular cylinders. Transactions of the Japan Society of Mechanical Engineers 68–669 (B), 99–106 (Japanese). Nguyen, T., Koide, M., Takahashi, T., Shirakashi, M., 2010. University of longitudinal vortices shedding from a cruciform two circular cylinder system in uniform flow. Journal of Fluid Science and Technology 5 (3), 603–616. Shirakashi, M., Takahashi, T., Kumagai, I., Matsumoto, T., 2001. Vortex-induced vibration of the upstream cylinder of a two-cylinder system in cruciform arrangement. Journal of Computational and Applied Mechanics 2 (1), 103–122. Shirakashi, M., Ishida, Y., Wakiya, S., 1985. Higher velocity resonance of circular cylinder in cross flow. Transactions of ASME Journal of Fluids Engineering 107, 392–396. Sarpkaya, T., 2004. A critical review of the intrinsic nature of vortex-induced vibrations. Journal of Fluids and Structures 19, 389–447. Takahashi, T., Baranyi, L., Shirakashi, M., 1999. Configuration and frequency of longitudinal vortices shedding from two circular cylinders in cruciform arrangement. Journal of the Visualization Society of Japan 19 (75), 328–336. Williamson, C.H.K., Govardhan, R., 2007. A brief review of recent results in vortex-induced vibrations. Journal of Wind Engineerng and Industrial Aerodynamics doi:10.1016/j.jweia.2007.06.019. Williamson, C.H.K., Govardhan, R., 2004. Vortex-induced vibrations. Annual Reviews of Fluid Mechanics doi:10.1146/annuarev.Fluid.36.050802.122128.
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Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs
Influence of mass and damping ratios on VIVs of a cylinder with a downstream counterpart in cruciform arrangement T. Nguyen a,n, M. Koide b, S. Yamada a, T. Takahashi a, M. Shirakashi a a b
Department of Mechanical Engineering, Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka, Niigata, Japan Faculty of Economics, Niigata Sangyo University, Karuigawa 4730, Kashiwazaki, Niigata, Japan
a r t i c l e in f o
abstract
Article history: Received 10 May 2010 Accepted 11 October 2011 Available online 21 November 2011
Previous wind tunnel experiments by the present authors showed that two kinds of longitudinal vortex induce cross-flow vibrations (LVIVs) on a circular cylinder accompanied by the vortex-synchronization (lock-in) by setting another cylinder downstream with proper gaps in cruciform arrangement, like the well-known Karman vortex induced vibration (KVIV) of a single cylinder. In this work, experiments on the three types of VIVs are carried out using a water tunnel to investigate influences of the mass ratio MR and the damping ratio z on the oscillation response. It is shown that the two LVIVs, i.e. the trailing- and the necklace-VIVs, occur in water flow over respective ranges of flow velocity U in spite of large differences in values of MR and z as compared with experiments in air flow. However, their influences on oscillation responses of the two LVIVs are largely different from each other, and also from the KVIV. They are compared by referring to the de-synchronization prediction based on the dimensional analysis and the equation on cylinder motion. The variation of behaviors among the three types of VIV are attributed to the differences among the three vortices in their threshold functions for synchronization and the vortex-synchronization functions which express the feedback effect of the cylinder oscillation on the alternating lift generated by the vortex. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Flow induced vibration Karman vortex Vortex induced vibration (VIV) Cruciform cylinders Longitudinal vortex
1. Introduction Vortex-induced vibration (VIV) of a cylinder in fluid flow is one of the most important problems in engineering and also one of the most fundamental but still challenging subjects in fluid mechanics. Among various types of VIVs, the transverse vibration of a spring supported rigid circular cylinder in uniform flow induced by the Karman vortex (KVIV) is the most prominent and simplest one from the view point of fluid mechanics, and the most commonly observed one in practical engineering. Hence, huge volume of research works have been devoted to clarify mechanism of KVIV, to establish methods for predicting behaviors, and, to develop techniques for preventing harmful effect of KVIV (Bearman, 1984; Blevins, 1990; Sarpkaya, 2004; Williamson and Govardhan, 2004). In air flow, where mass ratio MR ( ¼mass of oscillating structure/ displaced fluid mass) is in the order of hundreds, KVIV is regarded as essentially a resonant response occurring at the structure natural frequency fn. While, in water flow, where the mass ratio is considerably lower, say MR around the order of
Abbreviations: VIV, Vortex induced vibration; KVIV, Karman vortex induced vibration; LVIV, Longitudinal vortex induced vibration; TVIV, Trailing vortex induced vibration; NVIV, Necklace vortex induced vibration; VSF, Vortex-synchronization function n Corresponding author. Tel.: þ 0258 47 9730; fax: þ 0258 47 9770. E-mail address: [email protected] (T. Nguyen). 0889-9746/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfluidstructs.2011.10.006
T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
Nomenclature aspect ratio ( ¼L/d) blockage ratio [BR ¼(2dH d2)/H2] alternating lift coefficient, ( ¼ F vA =0:5rU 2 dLv ) alternating lift by vortex shedding and its amplitude H side length of square measuring section Fz alternating lift exerting on the upstream cylinder in z-direction L length of the cylinder (distance between the end plates) Lv spanwise length of vortex shedding region MR mass ratio, ¼4me/(pd2L) Re Reynolds number, ¼ rdU/m Sc Scruton number, ¼4pMRz SL, Su, SZ spectra of lift FL, velocity u and displacement Z St Strouhal number for vortex shedding, ¼ fvd/U U free flow velocity U0 free flow velocity at which fv0 ¼fn Z, ZA displacement of cylinder and its amplitude, pffiffiffi ZA ¼ 2Zrms c, cadd, ce structural, added and effective damping factors. (ce ¼cþ cadd) d diameter of the cylinders fn structure natural frequency fv vortex shedding frequency AR BR CL Fv, FvA
fz k m, madd, s t u
41
oscillation frequency of the upstream cylinder spring constant me structural, added and effective mass gap between the two cylinders time x-velocity component detected at a reference position
Greek letters
de z
m r f
ov , oz
effective logarithmic damping pffiffiffiffiffiffiffiffiffi factor damping ratio, ce =2 me k ¼ 2dpe viscosity of fluid density of fluid phase angle between Fv and Z angular frequencies of vortex shedding and cylinder oscillation
Subscript and superscript n
max rms 0 R
non-dimensional variable referred to d, fn and U values for maximum response root mean square value for stationary system resonance value predicted from desynchronization analysis
1–10, and the damping ratio z higher but resulting in a low combined mass-damping parameter MR z, the oscillation response is considerably different from the case in air flow as reported by Govardhan and Williamson (2000, 2004, 2006), Khalak and Williamson (1999), and Klamo et al. (2006). Farshidianfar and Zanganeh (2010) presented a modified wake oscillator model that can describe most of the behaviors of vortex-induced vibration for both high and low mass-damping ratio systems. A most recent review by Bearman (2011) presents comprehensive discussion for a rigid circular cylinder on the problems including two degree freedom motion, interference of tandem cylinders and the influence of Reynolds number on the maximum oscillation response. In earlier wind tunnel experiments by the present authors, it was found that KVIV is effectively suppressed by setting another cylinder downstream with a proper gap, i.e. the gap-to-diameter ratio s/do0.5, in cruciform arrangement (Bae et al., 2001). However, in the same arrangement, unexpected cross-flow vibrations occurred in higher velocity ranges accompanied by periodic shedding of longitudinal vortices with rotation axes parallel to the free stream (Shirakashi et al., 1994, 2001). Experiments on fixed cruciform system showed that the type of the longitudinal vortex depends on the value of s/d: (a) trailing vortex when s/do0.25 and (b) necklace vortex when 0.25os/do0.5, as shown in Fig. 1 (Takahashi et al., 1999). The two longitudinal vortices shed alternately from the upper and the lower sides of the upstream cylinder with frequencies considerably lower than that of Karman vortex, generating transverse exciting force on the upstream cylinder. A recent study for fixed system over wide ranges of
Fig. 1. Longitudinal vortices shedding from cruciform cylinder system (Koide et al., 2005). (a) Trailing vortex, 0o s/d o 0.25 and (b) necklace vortex, 0.25 os/d o 0.5.
42
T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
experimental conditions showed that the two longitudinal vortices shed periodically in air flow over the ranges of 3.5 103 oReo9 104 and aspect ratio 7oAR o14 (blockage ratio 0.14oBR o0.26) with respective Strouhal numbers. However, one peculiar result observed in the smallest water tunnel experiment is the vanishing of necklace vortex at Re422 000, while trailing vortex persists over the whole measurement range (Koide et al., 2006; Nguyen et al., 2010). These two longitudinal-vortex-induced vibrations (LVIVs), i.e. trailing VIV (TVIV) and necklace VIV (NVIV), in air flow are regarded as a resonance response since they occur when the vortex shedding frequency fv coincides with the structure natural frequency fn. Another similar feature of these LVIVs to KVIV is that the resonance persists over a wide flow velocity range around U0, the velocity at which the natural vortex shedding frequency fv0 ¼fn, due to the synchronization of vortex shedding with the cylinder oscillation (lock-in phenomenon). In order to investigate the synchronization phenomenon of the vortices, the upstream cylinder was mechanically oscillated in a wind tunnel and the velocity u was detected by a hot wire probe set at a properly determined reference position, i.e. the position at which periodic change caused by each vortex was most definitely observed (Kumagai et al., 2002). The synchronization phenomenon was discerned by an abrupt increase in both the peak value of Su, the spectrum of u, and the cross-correlation between the cylinder displacement Z and u. Thus the threshold value was obtained showing that the synchronization velocity regions of the longitudinal vortices at an equal oscillation amplitude are considerably different from that of Karman vortex. The phase difference between Z and u are also obtained, which indicated that the phase of vortex shedding relative to the cylinder motion varies with velocity over the synchronization region. The objectives of this study is to investigate the behaviors of the three types of VIV in water flow, in which values of MR and z are largely different from those in wind tunnel experiments so far but the mass-damping factor MR z or Scruton number is comparable, and, to give physical understanding of their behaviors focusing on the influences of MR and z on the VIV responses. 2. Analysis 2.1. Phenomenon of vortex synchronization and simplification for analysis In this study, the phenomenon of VIVs simplified as shown in Fig. 2 is investigated experimentally. A rigid circular cylinder is fixed, or spring-supported for transverse motion, in uniform flow. Another rigid cylinder with an equal diameter is fixed downstream with a gap in cruciform arrangement. Both cylinders are spanned over essentially the whole width of measuring section, resulting in an equal length. It is assumed that the upstream cylinder is oscillating in the transverse (z) direction as given by Eq. (1) and that a periodic vortex shedding generates lift force Fv as Eq. (2) Z ¼ Z A sinð2pf z tÞ,
ð1Þ
F v ¼ F vA sinð2pf v t þ fÞ:
ð2Þ
In accordance to the many of research works so far (Khalak and Williamson, 1999; Williamson and Govardhan, 2004) the following simplification is introduced for the transverse force Fz exerted by the surrounding fluid in the direction of cylinder motion. The fluid force can be divided into the vortex force which exists when the body is at rest in the flow and the resistance force against the body motion which exists without flow. In fluid otherwise at rest, an oscillating cylinder experiences resistance force exerted by surrounding fluid, which is commonly assumed to consist of the inertia and viscous contributions. Therefore, the fluid force exerting on the oscillating cylinder in flow is assumed to be expressed as 2
F Z ¼ madd
d Z dZ þ Fv: cadd dt dt 2
ð3Þ
Fig. 2. Model of the cruciform cylinder system in uniform flow.
T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
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In most of literatures about KVIV, madd and cadd are assumed to be constant then effective mass me and effective damping factor ce defined by Eqs. (4), (5) are regarded as constants me ¼ m þ madd ,
ð4Þ
ce ¼ c þ cadd :
ð5Þ
Note that Eq. (3) with constant values of madd and cadd is not really true but an approximation to simplify the oscillation analysis. Since, the resistance against the cylinder motion should be proportional to the square of the relative flow velocity, and Fv generated by the synchronizing vortex accounts for all the fluid force, meaning that madd ¼0 and cadd ¼0, as suggested for the case of KVIV in Bearman (1984). The vortex shedding from a cylinder oscillating as Eq. (1) is determined by the flow condition, the system geometry and the oscillation motion, irrespective of the cause of cylinder oscillation. Hence, for geometrically similar systems shown in Fig. 2, i.e. for the systems with an equal aspect ratio AR and gap-to-diameter ratio s/d, the Strouhal number and the lift coefficient of the vortex force are expressed as follows, where ‘‘funct.’’ stands for a function in general f d f d St ¼ v ¼ funct: Re,Z nA , z , ð6Þ U U f d ¼ funct: Re,Z nA , z , U ð1=2ÞrU dLv F vA
CL ¼
2
ð7Þ
where Lv is the spanwise length of vortex shedding region and Lv ¼L for the Karman vortex. The two longitudinal vortices, as well as the Karman vortex, synchronize (lock-on) with cylinder oscillation when the oscillation amplitude is larger than their respective threshold values, ZAth. This criterion is expressed by dimensionless variables as f d Z nA 4Z nAth ¼ funct: Re, z : ð8Þ U The synchronization is defined as a phenomenon of flow in which the vortex shedding is driven by the cylinder oscillation. In consequence, the vortex shedding frequency and phase difference of the vortex force Fv from the cylinder displacement Z in turn are determined as below f v ¼ f z,
ð9Þ
f ¼ funct: Re,Z nA ,
f zd : U
ð10Þ
Since Eq. (7) under synchronization and Eqs. (9) and (10) express the manner in which the vortex shedding is driven by the cylinder oscillation they are referred to as ‘‘vortex-synchronization functions (VSFs)’’ henceforth in this paper. When the cylinder is steadily oscillating excited by the vortex force Fv, Eq. (9) always applies and Eqs. (6), (7) and (10) are rewritten as St ¼
f vd f d ¼ z , U U
on C L , f ¼ funct: Re,Z nA , nv , U
ð11Þ
ð12Þ
where
onv ¼
fn ¼
2pf v n ¼ fz, 2pf n
1 2p
sffiffiffiffiffiffiffi k : me
ð13Þ
ð14Þ
2.2. Equations for cylinder oscillation By applying Eqs. (1) and (2) to the equation of motion for cylinder oscillation with the assumption of constant me and ce, the following expressions are obtained for the oscillation response Z nA ¼
1 C L U n2 Lv 1 f , 2p3 M R d AR z
ð15aÞ
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T. Nguyen et al. / Journal of Fluids and Structures 28 (2012) 40–55
where fz is the amplitude magnification factor defined by 1 f z ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ð1ovn2 Þ2 þð2zonv Þ2
ð15bÞ
and tan f ¼
2zonv : 1onv2
ð16Þ
2.3. Desynchronization prediction as the reference When ZA oZAth, the synchronization does not occur and the oscillation behavior is essentially a forced vibration excited by the alternating lift generated by periodical vortex shedding. The frequency fv and the magnitude FvA of the vortex force are virtually equal to those for the same cylinder at rest, i.e. fv0 and FvA0, respectively. Then, the dimensionless frequency of the exciting force is given by
onv ¼
f v0 f d U ¼ St0 U n : ¼ v0 U f nd fn
ð17Þ
This desynchronization prediction leads that the maximum response occurs as the resonant oscillation at fv ¼fn, that is, at the velocity U0 at which fv0 ¼fn. Since, the resonance-frequency, -velocity and -amplitude at the maximum response are expressed in dimensionless forms as below n
f zR ¼ wnvR ¼ 1,
ð18Þ
U nR ¼ U n0 ¼ 1=St0 ,
ð19Þ
Z nAR ¼
1 C L0 Lv 1
p2 ScSt20 d AR
,
ð20Þ
If CL0, St0 and Lv/d are independent of Re and AR, Z nA ¼ 2onv2 zf z ðonv Þ: Z nAR
ð21Þ
Eq. (20) shows that the Scruton number Sc ¼4pMRz can be a single parameter to predict the maximum response of VIVs taking the both contributions of MR and z into account, if it were not for the synchronization. While, the damping ratio z n influences also the behavior of oscillation amplitude with ov, and then with flow velocity Un, as seen from Eq. (21). Thus n Eqs. (17), (20) and (21) give the desynchronization prediction of ZAScAR–Un curves for the VIVs in general. n n When ZA becomes larger than ZA th given by Eq. (8) for the vortex, the synchronization occurs and the VSFs (12) for the n vortex govern the values of lift coefficient CL and phase difference f. Consequently, the oscillation response, ZA and n n oz ( ¼ ov), together with CL and f should be obtained by solving the system of four equations, that is the two VSFs given as Eq. (12) and Eqs. (15a), (16). Note that the VSFs coupled with the threshold function (8) constitute an oscillator model for the VIV, which deviation of the oscillation response from the desynchronization prediction is caused by the feedback effect of cylinder motion on the vortex force Fv. Hence in this paper, the behavior of the three types of VIV are expressed by the above non-dimensional parameters and compared by normalizing with the respective desynchronization prediction to see the influence of mass and damping factor on the VIVs. 3. Experimental apparatus, measuring system and experimental conditions Outline of the experimental apparatus is shown in Fig. 3. The measuring section has 100 mm square cross section and 500 mm length. The diameter d of the cylinders is 10 mm and d¼15 mm is also tested for comparison. Although the blockage ratio BR [ (2dH d2)/H2] in this work is significantly higher, say BR ¼0.19 for d ¼10 mm and 0.28 for d¼15 mm, than the limitation of usual wind/water tunnel experiments, earlier wind tunnel experiments confirmed that the effect of BR on the longitudinal vortex shedding is insignificant at these values of BR (Nguyen et al., 2010). The upstream cylinder is rigidly fixed, or supported by twin-plate-springs at the both ends outside the measuring section so as to make the motion pure cross-flow mode, while the downstream cylinder is mounted rigidly on a traversing device to adjust the gap s. End plates are attached to the upstream cylinder to remove influence of flow through slots where the upstream cylinder is passing (Shirakashi et al., 1985). Measurement instrument set up is also illustrated in Fig. 3(a). An electromagnetic flow meter is set to the pipe downstream the measuring section to measure the flow rate. The velocity profile of the free flow in the measuring section has 95% uniform region both in horizontal and vertical direction. The turbulence level obtained by a hot film probe at the center of the measuring section is less than 2.8% over the experimental range of Re. The free flow velocity U is determined
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Fig. 3. (a) Water tunnel, supporting structure and measuring system and (b) cross section of the measuring section (viewed from downstream).
as the uniform value in the profile, which is obtained from the flow rate divided by the cross-sectional area of the measuring section multiplying a correction factor. This factor keeps a constant value of 1.05 over experimental range of Reynolds number. A hot-film probe is set at respective reference positions for each of the three vortices in near wake where the periodic vortex shedding is most definitely detected in the velocity u (Koide et al., 2005). A laser displacement sensor is used to measure the displacement Z of the upstream cylinder. In this paper, the vibration amplitude is expressed by the root-mean-square value of Z. Fluctuating lift force loading on the fixed upstream cylinder is measured by a load cell. The natural frequency fn and the effective logarithmic damping factor de are determined from free damping oscillation in the water tunnel at U ¼0, then the effective mass me is calculated by the formulame ¼ k=ð2pf n Þ2 . As seen in Fig. 3(b), the parts of cylinder and support outside the measuring section are in water at rest. Hence, madd and de estimated in this way include their contribution, and there is no rational way to separate it from those attributed to the cylinder in the measuring section. Therefore, me and ce obtained in water flow at U ¼0 is regarded as the structure mass and damping in this paper.
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Table 1 Characteristics of the elastically supported cylinder system.
d (mm) L (mm) k (N/m) fn (Hz) z (–) MR (–) Sc (–)
LVIVs in air flow (Shirakashi et al., 2001)
KVIV in air flow (Koide et al., 2002)
3 types of VIV in water flow
26 318 652 14.5–15.5 0.0014–0.003 360 6.4–14.4
26 318 623 16.94 0.16 10 3 264 16.4
10, 15 98 111 4.5–5.13 0.024–0.082 6–30 2.2–15.6
Fig. 4. Spectra of lift on fixed upstream cylinder: (a) Karman vortex (d ¼10 mm, U ¼1.4 m/s); (b) trailing vortex (d¼ 10 mm, s/d¼ 0.08, U¼ 1.3 m/s); (c) necklace vortex (d ¼10 mm, s/d ¼ 0.3, U ¼1.9 m/s); and (d) vanishing of necklace vortex at a higher flow velocity (d ¼ 10 mm, s/d ¼0.3, U¼ 2.8 m/s).
Experiments on the trailing vortex are carried out at the gap-to-diameter ratio s/d ¼0.08, and on the necklace vortex s/d ¼0.28 or 0.3, since at these values the trailing and the necklace vortex shedding were most clearly observed in the wind tunnel experiments (Koide et al., 2005; Shirakashi et al., 2001). Experimental conditions are summarized in Table 1, where the conditions for wind tunnel experiment in Refs. Shirakashi et al. (2001) and Koide et al. (2002) are compared. 4. Experimental results 4.1. Vortex shedding from fixed system Examples of spectrum SL of lift on fixed cylinder caused by the Karman vortex and the longitudinal vortices are shown in Fig. 4. The peaks in SL for the three vortices are observed at respective frequencies as seen in Fig. 4 (a), (b) and (c). For each vortex, a peak at the equal frequency is observed in the corresponding velocity spectrum Su, assuring that the three vortices generate alternating lift forces. It is peculiar that the periodic shedding of necklace vortex seems to vanish beyond a certain velocity (or Reynolds number) as seen in Fig. 4(d). The normalized vortex shedding frequency, Strouhal number St0 and the lift coefficient CLrms0 obtained from the amplitude of lift force are plotted against the Reynolds number in Fig. 5, compared with the earlier wind tunnel experiments of Shirakashi et al. (2001). For Karman vortex shedding from a fixed single cylinder, St0, CLrms0–Re curves
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Fig. 5. Strouhal number and lift coefficient versus Reynolds number for fixed system: (a) Karman vortex (single cylinder); (b) trailing vortex (s/d¼ 0.08); and (c) necklace vortex (s/d¼ 0.28 or 0.3).
show good agreement between data in water and air flows. The values of St0, CLrms0 for the longitudinal vortices in water flow agree fairly well with those in air flow, and are slightly dependent on Re as seen in Fig. 5(b) and (c). The similarity of the two longitudinal vortices shedding from fixed system is well assured except the vanishing of necklace vortex for Reynolds number range of Re 420 000. In Fig. 5(c), the St0 ¼ 0 plots at Re 422 000 does not mean steady vortex but vanishing of necklace vortex peak in SL as seen in Fig. 4(d). The values of St0 and CL0 in Fig. 5 indicated by the horizontal lines are determined as respective constant values based on the data in the figure for the later use in the desynchronization prediction. Based on the dynamic similarity theory, the peculiar behavior of the necklace vortex in the high Re region can be attributed to the slight defects of geometrical similarity and/or difference in characteristics of the approaching flow. The investigation about universality of longitudinal vortices (Nguyen et al., 2010) showed that the necklace vortex is definitely observed to shed in a wind tunnel experiment with BR ¼26% (AR ¼7.1) when 3.5 103 oRe o9 104, and that it vanishes at around Re ¼2 104 in water tunnel experiment with BR ¼ 19% (AR ¼10) and 28% (AR ¼6.8) like the results in this work. Therefore, influence of the aspect ratio or blockage ratio on the behavior of vortex shedding is excluded from the potential causes for the necklace vortex vanishing. Since the end plates are not strictly geometrically similar, an experiment with the upstream cylinder without end plates fixed directly to the side walls of the measuring section was conducted. The result showed the necklace vortex vanishing at the same velocity as Fig. 5. Hence, the effect of end plates is also excluded. Therefore, the potential causes remaining at present are the differences in uniformity and turbulence of approaching flow. 4.2. Effective mass and damping The effective mass and the logarithmic damping factor were obtained from free damping oscillation of the single cylinder after giving an initial impact. The effect of surrounding air is examined first by carrying out the experiment in a vacuum box. No
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Fig. 6. Effective mass and damping of single cylinder obtained by free damping oscillation in water tunnel, normalized by their counterparts in air at rest.
significant difference was observed over the range of absolute pressure between the atmospheric pressure and 0.01 MPa, which shows that influence of the air is negligible. Then, the same procedure was carried out in the water tunnel with U from 0 to 3 m/s under the condition of small terminal oscillation. The results in Fig. 6 show that the additional mass is virtually constant, while additional damping increases significantly with U when Un 410. In this paper, me and ce are regarded to be constant, and values obtained by the free damping oscillation in still water are used for the desynchronization prediction. 4.3. Oscillation behavior of the three VIVs An example of behaviors of the three VIVs in water flow, i.e. vibration amplitude Zrms and fz versus flow velocity U, is presented in Fig. 7(a) compared with that in air flow as Fig. 7(b). The fv0 U curves are for the vortex shedding frequencies obtained from measurements for the respective fixed counterparts. It is noted that the oscillation frequencies in air flow for the three VIVs are always virtually equal to fn. The three VIVs are clearly observed in water flow in ways similar to their respective counterparts in air flow, in spite of large differences in the fluid property, dimension and structure parameters between the two experiments. It is noted that when the oscillation amplitude is small in air flow, the spectrum of displacement, Sz, has a peak at fn, in addition to fv, due to the selective resonance which commonly occurs for a low-damped system exerted by alternating force having broadly distributing spectrum as shown in Fig. 4. In Fig. 7, and in the following figures in this paper, such data for fz are not plotted. In water flow, two peaks sometimes appeared in Sz even when the oscillation amplitude was considerable. In such cases, the frequency of the higher peak are taken as fz and plotted in the figures. The amplitude of NVIV in water keeps around the highest level up to the maximum velocity in Fig. 7(a), making the higher border velocity of NVIV unclear. This may be associated with the vanishing of necklace vortex expressed by the broken line for fv0–U. The oscillation of NVIV is not stable but very intermittent in the velocity range where the necklace vortex shedding is unclear. 4.3.1. Vibration induced by Karman vortex (KVIV) In order to compare results of water tunnel experiment on KVIV with largely varying values of MR and z in Table 1, data n of Zrms–U are normalized into ZrmsSc–Un since in Eq. (20) Lv =L ¼ Lv =dAR ¼ 1 for the Karman vortex. Fig. 8(a) shows the
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Fig. 7. Behaviors of the three VIVs: (a) in water flow, d ¼10 mm, MR ¼15, fn ¼4.88 Hz, de ¼ 0.2; (b) in air flow, d¼ 26 mm, MR ¼264 (KVIV) and 360 (LVIVs), fn E15 Hz, de E0.01 (Kato et al., 2006).
normalized experimental data for MR ¼ 6, 15 in water flow and MR ¼264 in air flow (Koide et al., 2002) when the Scruton number’s values are about 6–7. The normalized oscillation frequency is plotted in Fig. 8(b). The curves in Fig. 8(a) and (b) are for the desynchronization prediction obtained from Eqs. (17), (20) and (21) using St0 ¼0.21 and CLrms0 ¼0.4 as shown in Fig. 5(a). n The desynchronization prediction gives reasonable values for the maximum response, i.e. the amplitude Zrms max and n the velocity Umax, when Sc is comparable but MR and z are largely different. It also well predicts the initial velocity of KVIV and the tendency that the Un range expands wider, especially towards higher velocity, when z is larger. However, the measured oscillation amplitudes are, generally speaking, considerably larger than the desynchronization prediction, suggesting that the lift coefficient under synchronization may be larger than the value for the stationary cylinder. The n n synchronization is clearly observed in fz in Fig. 8(b), that is fz is constant over the Un range where oscillation amplitude is n large. As reported in many literatures, fz in this region deviates higher than unity when MR is smaller, or, z is larger (Khalak and Williamson, 1999; Williamson and Govardhan, 2007). 4.3.2. Vibration induced by trailing vortex (TVIV) n Behavior of TVIV is investigated by normalizing in a similar manner as the preceding section for KVIV. In Fig. 9, ZrmsScAR n n and fz are plotted against U for systems at which MR ¼6, 15 in water flow and MR ¼ 360 in air flow. The curves are their desynchronization predictions for St0 ¼ 0.08 and CLrms0 ¼ 0.22 indicated in Fig. 5(b), and Lv/d¼6.2 from Shirakashi et al. (2001). The normalized vibration response of TVIV in the initial stage, say 6oUn o10, collapses on a single curve for all the systems, consequently the deviation from the desynchronization prediction being larger when z is smaller. Although the curve deviates rapidly from the measurement in the higher Un range, the maximum amplitude for MR ¼360 and 15 is fairly well predicted by Eq. (20). The TVIV range extends in the higher Un for MR ¼15 in water flow as compared with MR ¼360 in n air flow. However, this tendency is not coherent for MR ¼ 6. The non-dimensional oscillation frequency fz in large TVIV region increases with Un with a gradient which becomes steeper with decreasing MR, as seen in Fig. 9(b). By comparing Fig. 9 with Fig. 8, it is seen that the normalized behaviors of TVIV is similar to that of KVIV, including the influence of the damping ratio, z. 4.3.3. Vibration induced by necklace vortex (NVIV) Fig. 10 shows the normalized behavior of NVIV for the same systems as in the preceding two sections. Here, St0 ¼0.04 and CLrms0 ¼0.33 as indicated by the horizontal lines in Fig. 5(c), and Lv/d¼5.0 (Shirakashi et al., 2001) are used for the n desynchronization prediction. Compared with the other two VIVs, ZrmsScAR–Un plots among the systems with different values of MR are largely scattering and its influence seems incoherent. The maximum amplitude by the desynchronization prediction is much larger than the measured. This is because Eq. (20) gives several-times-diameter maximum amplitude
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Fig. 8. Oscillation behavior of KVIV compared with the desynchronization prediction.
Fig. 9. Oscillation behavior of TVIV compared with the desynchronization prediction.
when St0 ¼0.04 for a system with Sc around 10, to which the assumptions for the desynchronization do not apply. It is also unnatural that the oscillation for MR ¼6 (in water) decays at Un 425 while it holds much higher maximum value up to Un E30 for MR ¼360 (in air). This may be attributed to the necklace-vortex-vanishing observed for the fixed system as shown in Fig. 4(d) and Fig. 5(c). The large vibration in Fig. 10(a) for MR ¼15 appearing at high values of Un, say Un 435, is
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Fig. 10. Oscillation behavior of NVIV compared with the desynchronization prediction.
not sinusoidal but modulating intermittently. Hence, the maximum amplitude and the range of NVIV cannot be properly determined from the present data for low values of MR. 4.4. Behavior of the maximum response Analysis in Section 2 showed that the VIV behaviors of equal Sc systems are governed by the damping ratio z if the synchronization did not occur. Therefore, the maximum response and the velocity range of VIVs against z are investigated in this section. n n n The maximum response velocity Unmax, the maximum amplitude Zrms max and fz max, the value of fz at Un ¼Unmax, are normalized by their respective desynchronization prediction values and plotted against z in Fig. 11. A feature common to the three VIVs is that the normalized oscillation frequency of maximum amplitude shown in Fig. 11(c) is equal to unity when z is smaller than 0.007. The desynchronization equations give good prediction, generally speaking, for the maximum response of KVIV over the measurement range of z. The normalized maximum response velocity of KVIV, St0Unmax, is slightly higher than unity for the lowest value of z, as seen in Fig. 11(a) and approaches to unity with increasing z. The maximum amplitude of KVIV is estimated fairly well by the desynchronization prediction, Eq. (20), as Fig. 11(b). Note that the ordinate in the figure is log-scale. The maximum response frequency deviates slightly higher with z 40.01. The maximum response behavior of TVIV in Fig. 11 is similar, roughly speaking, to that of KVIV described above. Two significant differences are: the normalized maximum response velocity St0Unmax of TVIV in Fig. 11(a) is constant but significantly larger than unity irrespective of z. There seem tendencies of decreasing maximum amplitude and increasing maximum response frequency with increasing z as seen in Fig. 11(b) and(c). Although the data of NVIV are scattered in Fig. 11, there observed definite difference in its behavior from those of the other two VIVs. The most prominent is the low value of the maximum response as observed in Fig. 11(b), due to the high St0 value of NV, as suggested in Section 4.3.3. 4.5. Velocity range of VIVs n
The regions of three VIVs defined by Zrms 40.02 are presented in the z–UnSt0 plane in Fig. 12. The value UnSt0 ¼1 on the ordinate means the resonance velocity by the desynchronization prediction. It is seen that the velocity range of KVIV thus normalized extends from a value slightly lower than unity to around 1.4 when z is lower than around 0.01 and expands wider with increasing z especially to the higher velocity region. This behavior of KVIV range is in agreement with many
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Fig. 11. Maximum response of 3 VIVs: J, KVIV; m, TVIV; and , NVIV. (a) Normalized velocity at which the resonance occurs; (b) normalized maximum amplitude; and (c) normalized oscillation frequency at which the resonance occurs.
Fig. 12. Response regions of VIVs for Sc¼ 2–16 on UnSt0 versus z plane (limited by border lines and background color patterns).
other references such as Williamson and Govardhan (2000, 2004) where they observed dependence on the mass ratio MR but not on z. The tendency of velocity range of TVIV versus z is similar to that of KVIV, but considerably wider even after normalizing by St0, towards both higher and lower velocity, in contrast to KVIV. For NVIV, the velocity range is considerably narrower (in relative measure) and shifted to the range much lower than UnSt0 ¼1. When z 40.01, the higher border value of Un was not determined due to the intermittent oscillation as mentioned in the preceding section.
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n
Fig. 13. Lift phase f and ratio of amplitude versus ov.
5. Discussion Although a few experimental data on ZAth for Karman vortex are reported so far, they do not correlate well with each other (Koide et al., 2002), may be due to the differences in its definition and way to determine the synchronization from experiment. On the longitudinal vortices, the only existing data for ZAth is those by the present authors (Shirakashi et al., 2001) obtained through wind tunnel experiment. However, in spite of the fact that the data on ZAth for the three vortices n are still insufficient, it seems rational that the minimum value of ZAth appears at fz ¼fv0 and considerably low, say less than around 0.01–0.03 for the three vortices. Thus, in the VIVs with significant amplitude the synchronization occurs practically always since the oscillation frequency fz is equal to the vortex shedding frequency fv as described in Section 2.2. n Concerning the vortex-synchronization functions (VSFs) of the Karman vortex, there are many data for CL(ZA) as Eq. (7), but they do not so well agree maybe due to the dependence on factors such as Reynolds number, aspect ratio, fzd/U ratio, etc. Only a limited number of reports on VSF of f are now available on the Karman vortex, and no experiment has been carried out in the case of the longitudinal vortices. Hence, it is worthwhile to discuss the influence of z on vortex-induced vibrations in general, based on the existing data of the threshold function and VSF of f for KVIV. Fig. 13 shows the desynchronization prediction for f and ZA/ZAR for Scruton number Sc¼10 by Eqs. (13) and (14) as typical examples of air and water flows, i.e. z ¼0.00227 (MR ¼ 360) and 0.0794 (MR ¼10), respectively. Note that the abscissa is proportional to the velocity outside the synchronization region, but not in the synchronization region. In Fig. 13(b), the threshold curve ZAth/ZAR of Karman vortex estimated from Koide et al. n (2002) is added using approximation ov ¼fv0/fn. The result on VSF of f for Karman vortex by Carberry et al. (2001) is reproduced in Fig. 14, obtained by PIV measurement in water flow with a fixed velocity around a cylinder oscillated mechanically with a constant amplitude. In order to apply the threshold amplitude by Koide et al. (2002) to Fig. 13(b) and f(fz) in Fig. 14 to Fig. 13(a), it is assumed that their dependence on Re as suggested in Eqs. (8) and (10) are insignificant supported by experimental results as shown n n in Fig. 5. For the VSF of f, dependence on ZA in Eq. (10) is also neglected since Sc is constant meaning that ZA will not be largely different. Based on these assumptions, f in Fig. 14 becomes applicable by converting the relationship into f(fz/fv0). Thus the influences of z on VIVs observed in the experimental results are explained as below. (1) Intersections of ZA/ZAR and ZAth/ZAR curves in Fig. 13(b) (A, B for z ¼0.00227 and A’, B’ for z ¼0.0794) give the outside n limit of synchronization velocity range. Therefore, the VIV range of ov and of Un in turn, can be larger when z is larger.
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Fig. 14. Lift phase, flift, and amplitude of the lift coefficient CL versus frequency ratio, fe/f0 ¼fz/fv0 in forced oscillation experiment (Carberry et al., 2001), U ¼0.09 m/s, d ¼ 25.4 mm, ZA/d ¼0.5, Re ¼2.3 103, f0 ¼ 0.7484: , lift phase; K, CL.
(2) In the synchronization VIV, the point P (Un,f) moves between A–B or A’–B’ from left to right along the solid or chaindot curve in Fig. 13(a). However, it does not mean that all the points in these sections of the curve are traced, since at the same time f is governed by the VSF as given by Fig. 14 for the case of Karman vortex, where the point (fz/fv0, f) moves from right to left with increasing flow velocity U. n (3) In the case of VIVs with small z, ov E1 between A and B, meaning that the oscillation frequency is virtually fixed at fn in agreement with the experimental result as shown in Section 4.3.1 Fig. 14 shows that f o0 when fz/fv0 is larger than a certain critical value, say [fz/fv0]crit ¼ 0.82. The negative f never occurs in VIVs since it would cause energy transfer from the cylinder to the fluid. Therefore, for the KVIV in air flow with increasing U, fz(¼ fn)/fv0 41 when velocity is low and hence the synchronization does not occur. When the velocity attains above the value corresponding to [fz/fv0]crit, the synchronization occurs if the amplitude is larger than the threshold value and the point P(Un,f) drops somewhere on A–B curve as the solution of the equation system composed of Eqs. (12) (15a, b) and (16). (4) In the case of synchronization KVIV, Fig. 14 suggests that the value of f changes from a value near to zero (A or A0 in Fig. 13(a) at the lowest) to 1601. Corresponding change of f is indicated as C, C0 in Fig. 13(a). The normalized frequency n n fz ( ¼ ov) is nearly equal to unity over the range of synchronization KVIV in spite of the jump of f when z is low as n n indicated by A–C. The deviation of fz becomes larger as A’–C’ when z is higher, and constant value of fz in water flow in the higher velocity region, so-called lower branch response, is explained as the result of constant plateau of f E1601 for fz/fv0 o0.8 in Fig. 14 (see also Eq. (16)). n n (5) The resonance response can be attained only when f ¼ p/2 (and hence fz ¼ ov ¼1) is stably attained under the restraint of the VSFs, which will lead the desynchronization prediction for the maximum response reasonable. For KVIV, this condition may not be satisfied as suggested by Fig. 14. Although the above discussion is based on KVIV data, the same consideration will be applied for the two LVIVs if the threshold function and VSF of f are obtained. Hence, the different deviating ways of experimental results from desynchronization prediction among the three VIVs are attributed to the difference of these functions among them. 6. Conclusion In this work, experiments were carried out in water flow over the ranges of Reynolds number Re ¼3 103–4 104 and Scruton number Sc¼2–16, which are comparable with the previous wind tunnel experiments, in order to investigate influences of the mass ratio MR and damping ratio z. The two LVIVs are confirmed to occur in water flow in spite of the large differences in the dimension and the mass ratio and/or damping factor as compared with the earlier wind tunnel experiments. However, behaviors of the two LVIVs in water flow deviates from their respective counterparts in air flow, in different manners as compared with the case of KVIV. The results are compared and discussed by referring to the desynchronization prediction. Analysis for the influences of MR and z based on the dimensional analysis and the equation of cylinder motion showed that the desynchronization oscillation behavior of VIVs with varying velocity is determined by z, while its maximum amplitude of VIVs without synchronization is given by Sc taking both the contributions of MR and z into account. If the threshold function of the vortex is given, the desynchronization prediction can give the range of synchronization VIV. The behavior of synchronization VIVs with varying velocity can be obtained by solving the simultaneous equation systems
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composed of the two VSFs and two equations obtained from the equation of cylinder motion. Nevertheless, the desynchronization prediction can elucidate influences of z such as the oscillation frequency higher than the structural natural frequency and the broadening of the VIV velocity range. Therefore, what is most required for oscillation prediction is establishment of these functions for the three vortices. The vanishing of necklace vortex at high Re and the unstable intermittent nature of NVIV in higher velocity range are also left for further experiments. References Bae, H.M., Baranyi, L., Koide, M., Takahashi, T., Shirakashi, M., 2001. Suppression of Karman vortex excitation of a circular cylinder by a second cylinder set downstream in cruciform arrangement. Journal of Computational and Applied Mechanics 2 (2), 175–188. Blevins, R.D., 1990. Flow-Induced Vibration. Van Nostrand Reinhold, New York. Bearman, P.W., 1984. Vortex shedding from oscillating bluff bodies. Annual Reviews of Fluid Mechanics 16, 195–222. Bearman, P.W., 2011. Circular cylinder wakes and vortex-induced vibrations. Journal of Fluids and Structures 27, 648–658. Carberry, J., Sheridan, J., Rockwell, D., 2001. Forces and wake modes of an oscillating cylinder. Journal of Fluids and Structures 15, 523–532. Farshidianfar, A., Zanganeh, H., 2010. A modified wake oscillator model for vortex-induced vibration of circular cylinders for a wide range of massdamping ratio. Journal of Fluids and Structures 26, 430–441. Govardhan, R., Williamson, C.H.K., 2000. Modes of vortex formation and frequency response of a freely vibrating cylinder. Journal of Fluid Mechanics 420, 85–130. Govardhan, R., Williamson, C.H.K., 2004. Critical mass in vortex-induced vibration of a cylinder. European Journal of Mechanics. B/Fluids 23, 17–27. Govardhan, R.N., Williamson, C.H.K., 2006. Defining the ‘‘modified Griffin plot’’ in vortex-induced vibration: Revealing the effect of Reynolds number using controlled damping. Journal of Fluid Mechanics 561, 147–180. Kato, N., Koide, M., Takahashi, T., Shirakashi, M., 2006. Influence of cross-sectional configuration on the longitudinal vortex excitation of the upstream cylinder cruciform two-cylinder system. Journal of Fluid Science and Technology 1 (2), 126–137. Khalak, A., Williamson, C.H.K., 1999. Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. Journal of Fluids and Structures 13, 813–815. Klamo, J.T., Leonard, A., Roshko, A., 2006. The effects of damping on the amplitude and frequency response of a freely vibrating cylinder in cross-flow. Journal of Fluids and Structures 22, 845–856. Koide, M., Takahashi, T., Shirakashi, M., 2005. Experimental study on universality of longitudinal vortices shedding periodically from crisscross circular cylinder system in uniform flow. In: Proceedings of International Conference on Jets, Wakes and Separated Flows. Mie, Japan, pp. 433–438 (in Japanese). Koide, M., Ootani, K., Yamada, S., Takahashi, T., Shirakashi, M., 2006. Vortex excitation caused by longitudinal vortices shedding from cruciform cylinder system in water flow. JSME International Journal B-49 (4), 1043–1048. Koide, M., Tomida, S., Takahashi, T., Shirakashi, M., 2002. Influence of cross-sectional configuration on the synchronization of Karman vortex shedding with the cylinder oscillation. JSME International Journal 45 (2), 249–258. Kumagai, I., Takahashi, T., Shirakashi, M., 2002. Influence of oscillation of upstream circular cylinder on longitudinal vortex excitation of criss-cross circular cylinders. Transactions of the Japan Society of Mechanical Engineers 68–669 (B), 99–106 (Japanese). Nguyen, T., Koide, M., Takahashi, T., Shirakashi, M., 2010. University of longitudinal vortices shedding from a cruciform two circular cylinder system in uniform flow. Journal of Fluid Science and Technology 5 (3), 603–616. Shirakashi, M., Takahashi, T., Kumagai, I., Matsumoto, T., 2001. Vortex-induced vibration of the upstream cylinder of a two-cylinder system in cruciform arrangement. Journal of Computational and Applied Mechanics 2 (1), 103–122. Shirakashi, M., Ishida, Y., Wakiya, S., 1985. Higher velocity resonance of circular cylinder in cross flow. Transactions of ASME Journal of Fluids Engineering 107, 392–396. Sarpkaya, T., 2004. A critical review of the intrinsic nature of vortex-induced vibrations. Journal of Fluids and Structures 19, 389–447. Takahashi, T., Baranyi, L., Shirakashi, M., 1999. Configuration and frequency of longitudinal vortices shedding from two circular cylinders in cruciform arrangement. Journal of the Visualization Society of Japan 19 (75), 328–336. Williamson, C.H.K., Govardhan, R., 2007. A brief review of recent results in vortex-induced vibrations. Journal of Wind Engineerng and Industrial Aerodynamics doi:10.1016/j.jweia.2007.06.019. Williamson, C.H.K., Govardhan, R., 2004. Vortex-induced vibrations. Annual Reviews of Fluid Mechanics doi:10.1146/annuarev.Fluid.36.050802.122128.