Classification of the tripped cylinder wake and bi-stable phenomenon

International Journal of Heat and Fluid Flow 31 (2010) 545–560

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International Journal of Heat and Fluid Flow

journal homepage: www.elsevier .com/ locate/ i jhf f

Classification of the tripped cylinder wake and bi-stable phenomenon

Md. Mahbub Alam a, Y. Zhou a,*, J.M. Zhao b, O. Flamand c, O. Boujard c

a Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kongb School of Energy Science and Engineering, Harbin Institute of Technology, Chinac Centre Scientifique et Technique du Batiment, 11 rue Henri Picherit, 44000 Nantes, France

a r t i c l e i n f o

Article history:Received 27 May 2009Received in revised form 29 December 2009Accepted 18 February 2010Available online 20 March 2010

Keywords:TripwiresCylinder wakeBi-stable flowRain–wind-induced cable vibration

0142-727X/$ - see front matter � 2010 Published bydoi:10.1016/j.ijheatfluidflow.2010.02.018

* Corresponding author. Fax: +852 2365 4703.E-mail address: [email protected] (Y. Zhou)

a b s t r a c t

Experimental investigation is conducted in the wake of a circular cylinder with two tripwires attachedat azimuthal angle a = ±10� to 70� measured from the forward stagnation point at Reynolds numbers(Re) = 2.5 � 103–6 � 104. The tripwire diameter was 0.045d, where d is the cylinder diameter. Time-averaged drag (CD), fluctuating drag (CD,rms) and lift (CL,rms) were measured, along with Strouhal number(St) and the flow structure in the wake. The flow is classified as five regimes based on present measure-ments and also those in literature. In Regime A, a < 20�, the forces, St and St–Re relationship are all similarto their counterparts in a plain cylinder wake. In Regime B, 20� 6 a 6 35�, CD, CD,rms and CL,rms all declineconsiderably and St escalates with increasing a. In Regime C, 35� < a 6 40�, CD, CD,rms and CL,rms plummetrapidly. In Regime D, 40� < a < 45�, the flow may occur in two distinct stable modes, which switchesrandomly from one to another; accordingly, the instantaneous drag and St display drastic up-and-downjumps. Regime E occurs at a = 45–70�, where CD, CD,rms and CL,rms are large and St is small. This regime maybe further divided into two, viz. E1 (45� 6 a < 60�), where CD, CD,rms and CL,rms grow and St decreases withincreasing a, and E2 (60� 6 a < 70�), where the dependence of the parameters on a is less pronounced.The bi-stable phenomenon (Regime D) is given special attention and its possible connection to themechanism of well reported rain–wind-induced cable vibration is proposed.

� 2010 Published by Elsevier Inc.

1. Introduction

The imperfections of bluff body surfaces may be introduced byprotrusions, structural defects, contaminants and/or attachmentssuch as ice attached on power lines in the winter of cold areas,deposits on pipe lines in the sea, and rivulets running over a stayof cable-stayed bridges in rain and wind. The imperfections bringabout changes to the original flow and modify the boundary layer,including the turbulence level and flow separation, and subse-quently the wake flow structure. Naturally, the fluid forces onthe bluff body may be profoundly altered, resulting in possibly dra-matic effects.

Fage and Warsap (1929) are perhaps the first who studied theeffects of such imperfections on flow around a cylinder. In theirstudy, two small tripwires were placed symmetrically on the sur-face of a cylinder at a = ±65�, where a is the azimuthal angle be-tween a tripwire and the forward stagnation point. The tripwiresreduced time-averaged drag coefficient CD and altered the pressuredistribution on the cylinder surface. The observation lies in the factthat, given appropriate positioning, the tripwires may change thesubcritical flow regime to the transitional or critical regime. How-ever, they did not investigate the dependence of CD on a. James and

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.

Truong (1972) investigated experimentally the dependence of acylinder wake on the tripwire size and a. The range of Reynoldsnumber Re, based on the free-stream velocity U1 and cylinderdiameter d, was 104 to 105, the tripwire diameter was 0.006–0.063d, and a was from 0� to 180�. They found that an increasedtripwire diameter caused the transition of the boundary layer toturbulence at an earlier Re and/or a smaller a. Furthermore, thetripped flow depended much more on a than on the tripwire diam-eter. The diameter of tripwires must be quite large to change a verylow Re flow to a critical flow (Pearcey et al., 1982); for example, thechange occurs at a tripwire diameter of 0.003d and a = 65� forRe = 7.7 � 104 (Fage and Warsap, 1929), 0.025d and a = 50� for5 � 104 (Igarashi, 1986), 0.059d and a = 40� for 4 � 104 (Pearceyet al., 1982), and 0.063d and a = 35� for 3 � 104 (James and Truong,1972). Nebres and Batill (1993) studied the effect of a single wire of0.007–0.14d on the pressure distribution, Strouhal number(St � fsd/U1, where fs is the frequency of vortex shedding) and CD

at Re = 2 � 104–4 � 104 and a = 0–180�. The tripwire was foundto have a considerable effect on the wake at a = 20–70�, and bothSt and CD were highly sensitive to a. On the other hand, the trip-wire had no effect on the wake when positioned near the forwardstagnation point or the base region. Romberg and Popp (1998)investigated the effect of tripwires of 0.0125d in diameter ata = ±90� for Re = 103–105 on the instability behavior of one flexiblecylinder in a bundle of 18 and 24 cylinders. They noted that the

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Nomenclature

CD time-mean drag coefficientCD,rms fluctuating (rms) drag coefficientCL,rms fluctuating (rms) lift coefficientd diameter of the cylinderf frequencyfs vortex shedding frequencyRe Reynolds number, dU1/mSt Strouhal number, fsd/U1urms fluctuating streamwise velocityurms;max maximum urms

U1 free-stream velocityU time-averaged streamwise velocity

t timex, y coordinates along streamwise and lateral flow direc-

tions, respectivelyx0 the farthest x-coordinate of recirculation bubble (U=0)xurms;max x-coordinate of urms;max

* normalization by U1 and/or da azimuthal position of tripwire measured from the for-

ward stagnation pointh pressure tap position measured from the forward stag-

nation pointm kinematic viscosity of fluid

546 Md. Mahbub Alam et al. / International Journal of Heat and Fluid Flow 31 (2010) 545–560

placement of the tripwires enlarged the Ur range where the cylin-der was stable. Hover et al. (2001) applied two tripwires of diam-eter 0.0125d at a = ±70� to control vortex-induced vibration on anisolated cylinder. An early (premature) lock-in compared to a plaincylinder was observed, and the vibration amplitude was reducedmoderately.

Previous investigations mostly focused on the tripwire effect onSt and CD, paying little attention to fluctuating forces and thepossible dependence of the tripwire effect on Re. Furthermore,the variation of the flow structure around the cylinder with a hasnot been investigated. These aspects are important for the thor-ough understanding of the tripwire effect.

Alam et al. (2003a) used tripwires with a diameter of 0.081–0.13d, which were detached to the cylinder surface with a gap of0.008d, to reduce fluid forces acting on an isolated circular cylinderand two cylinders in either side-by-side or tandem arrangement(Re = 5.5 � 104). Two peaks were observed at St = 0.16 and 0.22,respectively, in the spectrum of the measured fluctuating lift signalon the isolated cylinder at a = ±41� to 44�, suggesting a bi-stableflow. This phenomenon was however not given any special atten-tion, let alone its possible impact upon or connection to engineer-ing applications.

One of important examples of tripwire-like surface protrusionin engineering is the formation of two water rivulets running overthe stay cable of cable-stayed bridges under light to moderatewind and rain, one on the lower windward side of the cable andthe other on the upper leeward side. The water rivulets are sus-pected to contribute to the so-called rain–wind-induced-vibration(RWIV) of the stay cables (Yamaguichi, 1990). The RWIV of the staycable cannot be simply ascribed to the well known vibration mech-anisms such as the vortex-excitation, classical galloping or wake-galloping (Hikami and Shiraishi, 1988; Matusmoto et al., 1995).Alam and Zhou (2007a) proposed that the RWIV could be causedby the lock-in of the circumferential rivulet oscillation, vortexshedding and cable vibration or the rivulet–vortex-induced vibra-tion (RVIV). This proposition was supported by theoretical analysisof Lemaitre (2006) and Lemaitre et al. (2006), who demonstratedthat the three dynamic processes could indeed be locked in,enhancing greatly the cable vibration. Nevertheless, this mecha-nism may not be the only one responsible for the RWIV (De Langreand Herman, 2006). Flamand (2007) proposed that the RWIV mightbe connected to a switch between the subcritical flow regime andthe critical, and the large change in drag between the two regimesmight be responsible for the RWIV. However, there has been so farno experimental or numerical evidence to substantiate thishypothesis. Since the tripwire may induce the change from thesubcritical flow regime to the critical, one naturally wonderswhether this hypothesis could be investigated by studying the trip-wire-perturbed wake, in particular when tripwires are placed at

the positions where the flow regime change occurs. One furtherwonders whether the bi-stable phenomenon reported by Alamet al. (2003a) could be connected to the RWIV.

This work aims to study comprehensively fluid dynamics asso-ciated with the tripwire-perturbed cylinder wake. Specifically, CD,fluctuating drag and lift CD,rms and CL,rms (where subscript rms de-notes the root-mean-square value), and St are measured overRe = 2.5 � 103–6 � 104 and a = 10–70�, and compared with theircounterparts in a plain cylinder wake. Their dependence on Reand a is discussed in connection with the measured flow structureand flow characteristic parameters such as the vortex formationlength, recirculation bubble and wake width. A classification ofthe flow is proposed based on the measurements. The bi-stablephenomenon is examined in detail and connected to the RWIV.

2. Experimental details

2.1. Test facility and setup

Experiments were carried out in a closed circuit wind tunnelwith a 2.4-m-long square test section (0.6 m � 0.6 m). The bottomwall of the test section was inclined by about 1� to ensure a zeropressure gradient along the test section. The designed maximumoperating wind speed is 50 m/s and the longitudinal turbulenceintensity is approximately 0.5%. The wind speed was measuredusing a standard Pitot-static tube connected to an electronic mi-cro-manometer (Furness Control Limited, model FCO510). The testmodel was a smooth circular brass tube of d = 33.3 mm in diame-ter, which was mounted across the horizontal width of the test sec-tion, giving an aspect ratio of 18, considerably exceeding thethreshold West and Apelt (1993) proposed for the ‘long’ cylindercondition. The geometric blockage was approximately 5.5%. Thetwo-dimensionality of the flow has been checked by measuringthe frequency of vortex shedding from the cylinder at 0, ±4d, and±7d from the mid section of the cylinder. No difference was ob-served, suggesting a negligible end effect. Two brass tripwires witha diameter of 1.5 mm (0.045d) were attached symmetrically aboutthe forward stagnation point of the cylinder. The tripwires wereglued tightly to the cylinder surface, avoiding possible vibration.

Fig. 1 presents a schematic of experimental setup, including thedefinition of symbols. The angular position of the two tripwires,denoted by a, is measured from the forward stagnation point.The coordinate system is defined such that its origin is at the centerof the cylinder, with the x-axis along the free stream direction, they-axis along the lateral direction and the z-axis normal to both xand y, following the right-hand system. The free-stream velocityU1 was varied from 1.2 m/s to 29.3 m/s, corresponding toRe = 2.5 � 103–6 � 104 based on d. The a range, where CD drops

3d

2d

αα

Wind tunnel walls

d

x

y

Hotwire

Pressure tap

H = 0.6 m

0.5H

Fig. 1. Schematic of experimental setup.

Md. Mahbub Alam et al. / International Journal of Heat and Fluid Flow 31 (2010) 545–560 547

and St increases, compared with a plain cylinder, is 30–70�, irre-spective of single or two symmetrically positioned tripwires, givena diameter of 0.006–0.14d (James and Truong, 1972; Nebres andBatill, 1993; Alam et al., 2003a). The present tripwire diameter is0.045d, and a is varied from 10� to 70� to cover the entire rangeof drag reduction.

2.2. Force measurements

A three-component strain-gauge load cell (KYOWA Model LSM-B-500NSA1) characterized by excellent response, high resolutionand stiffness was installed at one end of the cylinder to measureinstantaneous integral lift and drag forces on the length of the cyl-inder exposed in the wind tunnel. To avoid the wind tunnel vibra-tion effect on the measurements, the load cell holder was mountedon an external rigid support detached from the wind tunnel. Staticcalibration was performed for the load cell along the lift and dragdirections, respectively, using dead weights. The load and outputvoltage relation of the load cell was linear. See Alam and Zhou(2007b) for more details of the load cell and force measurements.

Fig. 2. Mean and fluctuating drag coefficient (CD and CD,rms), fluctuating lift coefficient (CL

flow.

A pressure tap of 1 mm in diameter was made near the trailingstagnation point (h � 150� measured from front stagnation point)of the cylinder to measure the base pressure and connectedthrough a plastic tube to a semiconductor pressure transducer(LEEG SMP131) with a range of �3 to 3 kPa.

2.3. Hotwire measurements

A single tungsten wire of 5 lm in diameter, operated at an over-heating ratio of 1.8 on a constant temperature circuit, was placedat x* = 3 and y* = 2 (Fig. 1) to measure the streamwise velocity uand subsequently the predominant vortex shedding frequency inthe wake. Here superscript ‘*’ stands for normalization by d andU1. The hotwire signal was offset, amplified and then digitizedusing a 12-bit A/D board at a sampling frequency of 2.0 kHz. Thesampling duration of each hotwire signal was 10 s and at leastthree sample signals were obtained at each measurement location;the sampling duration corresponds to about 180–2520 vortexshedding cycles, depending on Re, and is considered to beadequately long. The power spectral density function, Eu, of u,

,rms), and Strouhal number (St) vs. Reynolds number (Re) for a plain cylinder in cross

0.233

42.5°

45°

60°Re = 6×104

0.278

0.180

0.154bi

trar

y)Re = 4×104

45°

60°

0.2210.289

0.180

0.153

Re = 2×104

Plain Cylinder

α = 10°

30°

40°

42.5°

45°

60°

0.197

0.197

0.210

0.228

0.256

0.180

0.156

log(

arbi

trar

y)

(b)(a)

(d)(c)

Plain Cylinder

α = 10°

30°

40°

42.5°

45°

60° Re = 2.5×103

0.200

0.200

0.209

0.205

0.200

0.164

0.159


was calculated using a fast Fourier transform algorithm based onan average of 20 runs, each consisting of 1024 (210) samples.Short-term Fourier transform (STFT) was used to analyze the hot-wire signal, exploring the time dependence of St.

2.4. PIV measurements and flow visualization

A Dantec PIV system (PIV2100) was used to measure flow in the(x, y) plane at mid-span of the cylinder. The cylinder surface andthe tunnel working section wall hit by the laser sheet were paintedblack to minimize reflection noises. The flow was seeded by smoke,generated from paraffin oil, with a particle size of about 1 lm indiameter, which was specially produced for PIV measurements.The PIV image size is 2048 � 2048 pixels, covering an area x* = 0–5 (166 mm) and y* = �2 to 2 (132 mm). In image processing, aninterrogation window of 64 � 64 pixels was used with 50% overlapin each direction. The ensuing in-plane velocity field consisted of63 � 63 vectors. The same number of spanwise vorticity datawas calculated from the velocity vectors. About 1800 images werecaptured for each test case to estimate the mean and fluctuatingflow fields, including the mean vorticity (x�) and mean and root-mean-square-streamwise velocities (U� and u�rms).

The same PIV system was used to visualize the flow in the (x, y)plane of mid-span of the cylinder. Smoke generated from paraffinoil was released from two 0.5-mm-in-diameter pinholes symmet-rically drilled at ±15�, respectively, from the nominal leading stag-nation point at the mid-span of the cylinder. The particle imageswere taken using a CCD camera (HiSense type 13, 4 M 8 bit,2048 � 2048 pixels). The Dantec FlowMap Processor (PIV2100type) was used to synchronize image taking and illumination.

0.1 10.1 1f*

Plain Cylinder

α = 10°

30°

40°

0.209

0.201

0.254

0.251

log(

ar

f*

Plain Cylinder

α = 10°

30°

40°

42.5°

0.199

0.199

0.228

0.260

Fig. 4. Power spectral density function of streamwise velocity u obtained from ahotwire placed at x* = 3 and y* = 2.

40.278

0.2154

3. Fluid forces and Strouhal numbers

3.1. Validation of measurements

The force and base pressure coefficients and St of a plain cylin-der were first measured. Being well documented in literature, thedata may serve as a validation for present measurements and alsoas a baseline for the analysis of the tripwire-perturbed wake. Fig. 2shows the dependence on Re of CD, CD,rms, CL,rms and St in a plain cyl-inder wake. The CD and St are about 1.15 and 0.2, respectively, forRe = 2.5 � 104–6.2 � 104, in agreement with previous reports (e.g.,Norberg, 1987; Nebres and Batill, 1993; Alam et al., 2003a). On theother hand, CD,rms and CL,rms, including those in literature, displayrelatively large scattering, which is ascribed to their high sensitiv-ity to experimental conditions such as turbulence intensity, block-

Fig. 3. Mean base pressure coefficient CPb and fluctuating base pressure coefficientCPb,rms vs. Re for a plain cylinder in cross flow.

age and aspect ratio (West and Apelt, 1997; Norberg, 2001, 2003).Present CD,rms and CL,rms are about 0.1 and 0.2, respectively, andchange little over Re = 2.5–4.0 � 104. With further increasing Re,they rise rapidly, reaching 0.28 and 0.55, respectively, atRe = 4.7 � 104 and then drop gradually to 0.1 and 0.3, respectively,at Re = 6.15 � 104. The rapid increase in CD,rms and CL,rms from

0.01 0.1 1

α =42.5°

6.1×10

5.1×104

4.0×104

3.1×104

6.2×10

2.0×104

1.6×104

1.1×104

6.2×103

0.2840.217

0.256

0.242

0.173

0.160

log(

arbi

trar

y)

Re = 2.5×103

f*

0.200

0.2890.223

0.222 0.285

6×10

3.0×104

Fig. 5. Dependence on Re of the power spectral density function of streamwisevelocity u measured at x* = 3 and y* = 2, a ¼ 42:5� .


Re = 4 � 104 to 4.7 � 104 coincides with Norberg’s (2001) observa-tion, which is attributed to a significant reduction in the vortex for-mation length, resulting from transition to turbulence in theseparated shear layer (Norberg, 2003). In contrast, the change inthe vortex formation length does not affect St appreciably.

Fig. 3 presents time-averaged base pressure coefficient (CPb) androot-mean-square base pressure coefficient (CPb,rms). The �CPb isnearly a constant, about 1.2, for the range of Re examined, in agree-ment with Norberg’s (1987) and Alam et al’s (2003a,b) reports. Ingeneral, CPb,rms correlates well with CL,rms or CD,rms. CPb,rms is about0.2 for Re = 2.5–4 � 104 and then increases steadily to about 0.3from Re = 4 � 104 to 4.7 � 104, which coincides with the Re-rangewhere CD,rms and CL,rms grow. A further increasing Re leads to adeclining CPb,rms. The above comparison provides a validation forpresent measurements.

3.2. Strouhal number and bi-stable flow

Fig. 4 shows the power spectral density function Eu of stream-wise velocity u obtained from the hotwire at Re = 2.5 � 103,

0.1

0.2

0.3

0.1

0.2

0.3

0.1

0.2

0.3

0.2

0.3

0.2

0.3

0.1

0.2

0.3

0 0.2 0.4

0.1

0.2

(a)

(b) α = 10°

(d) α = 40°

(e) α = 42.5°

(f) α = 45°

Time (s)

(g) α = 60°

(c) α = 30°

f *

f *

f *

f *

f *

f *

f *

Fig. 6. Time–frequency spectrogram of the hotwire signal at different a, Re = 4 � 104.

2 � 104, 4 � 104 and 6 � 104 for the range of a = 10–60�. The samelog scale (arbitrary) is used in each plot for the convenience ofcomparison. At Re = 2.5 � 103, a single pronounced peak is ob-served in Eu for a = 10–60�. This peak is insensitive to a in its mag-nitude and occurs at approximately the same frequency fora < 42.5� (Fig. 4a). At a higher Re (=2 � 104), Eu displays again a sin-gle peak (Fig. 4b). However, St increases appreciably from a = 10�to 42.5�. At Re = 4 � 104, the peak height decays from a = 10� to42.50� (Fig. 4c). Interestingly, at a = 42.5� two peaks occur atf* = 0.221 and 0.289, respectively, where asterisk denotes normali-zation by U1 and d. Similar observation is made at Re = 6 � 104

(Fig. 4d). The occurrence of two peaks apparently depends on Re,as highlighted in Fig. 5, which displays the evolution of Eu ata = 42.5� with Re. It is evident that two peaks occur for Re P3 � 104, but not for Re < 3 � 104. Bearman (1969) observed twopeaks at St = 0.2 and 0.3, respectively, in the transition regime ofa plain cylinder wake (Re = 3.55 � 105), which corresponded tothe separated (subcritical) and single bubble flows, respectively.He pointed out that the shedding process was non-stationary andthe physical phenomena associated with the two peaks could not

0.20.40.60.8

0.20.40.60.8

0.20.40.60.8

0.20.40.60.8

0.20.40.60.8

0.20.40.60.8

0.6 0.8 1

0.20.40.60.8

Eui/E

ui,m

Eui/E

ui,m

Eui/E

ui,m

Eui/E

ui,m

Eui/E

ui,m

Eui/E

ui,m

Eui/E

ui,m

The instantaneous spectrum magnitude Eui is normalized by its maximum Eui,m.


exist simultaneously. The existence of two peaks indicates a bi-sta-ble nature of the flow.

The Fourier transform produces averaged spectral coefficientsthat are independent of time and is useful to identify dominant fre-quencies in a signal. When the Fourier spectrum of the signaldisplays more than one peak, the temporal dependence of the

(b)

(a) T

T

Fig. 7. Schematic and smoke flow visualization of the bi-stable state: (a) Mode I: the bounII: early separation from the tripwire.

0 10 20 30 40 50 60 70 80.12

0.16

0.20

0.24

0.28

0.32

Re = 5×104

St

α

0.12

0.16

0.20

0.24

0.28

0.32

St

Re = 2.5×103

0.12

0.16

0.20

0.24

0.28

0.32

Re = 3×104

St

B D

E1 E2

E C A

(a)

(c)

(e)

Fig. 8. St–a relation

frequencies, where the peaks occur, i.e., the time–frequency plot,is valuable for understanding flow physics (Farge, 1992; Alamand Sakamoto, 2005; Alam and Zhou, 2008). Time–frequency anal-ysis using short-time Fourier transform (STFT) identifies the distri-bution of power as a function of the frequency at every instant forthe signal analyzed and is especially suitable for understanding a

ripwire

ripwire α = 42.5°, Re = 4×104

α = 42.5°, Re = 4×104

dary layer reattaches the cylinder surface and separation is postponed and (b) Mode

0

Re = 4×104

Re = 2×104

0 10 20 30 40 50 60 70 80

Re = 6×104

α

B D

E1 E2

E CA

(b)

(d)

(f)

at various Re.


non-stationary signal, providing more detailed information on thefrequency component of the signal than simple spectral analysis.Fig. 6 presents the time–frequency analysis of the u signal atRe = 4 � 104 for various a. It is well known that in STFT analysisthe frequency and temporal resolutions go against each other. Alarge window size improves the frequency resolution but reducesthe temporal resolution, and vice versa. In present STFT analysis,Gaussian window function is used. The width of the window istuned to be about 0.06 s to achieve a compromise between fre-quency and temporal resolutions. In order to explore the intermit-tency of a signal at any time, the transient spectrum is normalizedto its maximum, thus providing the dominant frequency of everyinstant and facilitating the examination of intermittency. Ata = 10� (Fig. 6b), the spectral energy is concentrated at aboutf* = St = 0.2 throughout the duration examined, just like the caseof a plain cylinder (Fig. 6a). At a = 30� (Fig. 6c) and 40� (Fig. 6d),the concentration of the spectral energy shifts to f* � 0.23 and0.26, respectively, consistent with the pronounced peak inFig. 4c. The frequency about which the energy is concentrated isindependent of time at a = 30� (Fig. 6c) but appears fluctuatingabout its average at a = 40� (Fig. 6d), which is attributed to the ad-vent of the bi-stable state. The observation agrees well with thesharp peak at a = 10� or 30� and the broad peak at a = 40� inFig. 4c. At a = 42.5� (Fig. 6e), the energy concentration occursintermittently at f* = St � 0.28 or 0.22. The observation suggeststhat two stable states corresponding to the two St occur intermit-tently, switching randomly from one to the other. Apparently,there exist two stable states of flow. Each state lasts for

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 00.4

0.8

1.2

1.6

2.0

2.4

CD

R e = 3 × 1 0 4

4 × 1 0 4

5 × 1 0 4

6 × 1 0 4

α

0 10 20 30 40 50 60 70 800.00

0.04

0.08

0.12

0.16

0.20

0.24

(b)

α

0.0

0.1

0.2

0.3

0.4

0.5Re = 3×104

4×104

5×104

6×104

(a)

(a)

(b)

(c)

B D

E1 E2

E C A

CD

,rm

sC

L,r

ms

Fig. 9. Dependence on a of: (a) time-averaged drag, CD, (b) fluctuating lift, CL,rms, (c)fluctuating drag, CD,rms at various Re.

0.03–0.10 s. The duration of St = 0.22 accounts for about 58% ofthe total time and the other for 42%.

In order to gain a better picture of the bi-stable flow, smokeflow visualization was conducted immediately downstream ofthe tripwire at a = 42.5� (Re = 4 � 104). With one half of the cylin-der in shadow, only flow over one side of the cylinder is captured.Two modes of flow separation are identified, which are illustratedin Fig. 7. In Mode I, the boundary layer separating from the trip-wire, as indicated by white color in Fig. 7a, reattaches on the cylin-der surface. In Mode II, on the other hand, the boundary layerseparating from the tripwire does not reattach, as is evident inFig. 7b, that is, the presence of the tripwire may have triggeredearly flow separation under the current flow condition, whichmay correspond to a wide wake or a reduced vortex shedding fre-quency. As such, we may assert that Modes I and II correspond tof* � 0.28 and 0.22, respectively. At a = 45� (Fig. 6f), the switchingbetween two flow states at a = 42.5� disappears, though the fluctu-ation of the frequency (f* � 0.18) about which the spectral energyis concentrated is still discernible. At a = 60� (Fig. 6g), the time–fre-quency spectrogram resembles that at a = 10� (Fig. 6b) except forthe occurrence of the spectral energy concentration at f* = St �0.15.

Fig. 8 illustrates the relationship between St and a for differentRe. The five regions marked by A–E on the top of Fig. 8 denote dis-tinct variations in St with a and will be discussed later. Three dis-tinct types of the St–a relationship can be identified. (1) At a low Re(62.5 � 103), St is insensitive to a at a < 42.5�, with about the samevalue (0.20) as in a plain cylinder wake, but drops rapidly betweena = 42.5� and 45�, and displays little variation for a > 45�, asillustrated in Fig. 8a. (2) For 2.5 � 103 < Re 6 2 � 104 (Fig. 8b), Stgrows significantly from about 0.20 to 0.26 at a = 42.5�, and thendrops quickly to 0.18 from a = 42.5� to 45�, and finally follows amonotonic decrease to a constant below 0.16. (3) For Re > 2 � 104

(Fig. 8c–f), two distinct St values are observed for 40� < a < 45�due to the bi-stable flow phenomenon; the St–a relationshipis otherwise similar to that in (2). The a range, where Stdeclines rapidly, is independent of Re for the Re-range examined,similarly to the observation by Nebres and Batill (1993) for

0 10 20 30 40 50 60 70 800.0

0.1

0.2

0.3

0.4

0.5

(b)

α

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

(a)

-CP

b

Re = 4×104

5×104

6×104

CP

b,rm

s

B D

E1 E2

E C A

Fig. 10. Dependence on a of: (a) negative base pressure coefficient �CPb and (b)fluctuating base pressure coefficient CPb,rms.

70°

α

t*

60°

50°

42.5°

40°

30°

20°

10°

0 30

Fig. 11. Time traces of the pressure signal at different a. Re = 4 � 104.


Re = 2 � 104–4 � 104. Nebres and Batill however observed a slightdependence of this a range on Re, if Re < 2 � 104, and did not reportthe bi-stable phenomenon.

St is about the same as in a plain cylinder wake in Regime A,exhibits a steady rise with increasing a in Regime B, reaches themaximum in Regime C, and drops rapidly in Regime D. In Regime

100

-0.5

0

0.5

Ruu

(a) α = 10°

-0.5

0

0.5

-0.5

0

0.5

-0.5

0

0.5

-0.5

0

0.5

Ruu

(f) α = 50°

-0.5

0

0.5

(g) α = 60°

Ruu

-0.5

0

0.5

(h) α = 70°

Ruu

t*100

-0.5

0

0.5

(b) α = 20°

(c) α = 30°

Ruu

(d) α = 40°

Ruu

(e) α = 42.5°

Ruu

Fig. 12. Autocorrelation coefficient Ruu of the

E, St continues its drop, albeit slowly, in E1 but remains unchangedin E2, with a value substantially smaller than in a plain cylinderwake.

3.3. Forces and base pressure

The extent to which tripwires could modify the boundary layerand hence flow separation, wake and fluid dynamic parameters isdependent on both Re and a. Fig. 9 shows the dependence of CD,CL,rms and CD,rms on a at a number of Re. The plain cylinder data isincluded, as indicated by a = 0�. Evidently, given the same a, CD issmaller for a higher Re. Note a difference between Re P 4 � 104

and 3 � 104. As a increases, the former is characterized by a re-duced CD, compared with a plain cylinder, but the latter by a con-siderably increased CD for a 6 35�, as observed by Nebres and Batill(1993) at Re = 3 � 104 and a 6 30�. The observation suggests thatthe presence of the tripwires may act to postpone flow separationat a 6 35� only if Re exceeds a certain level say 4 � 104 for presentexperimental conditions. This is possibly due to the fact that at alow Re the tripwire-generated turbulence in the boundary layeris inadequate to postpone flow separation and to increase pressureon the rear surface. As such, a postponed boundary layer with ahigher turbulence is connected to a larger pressure in the rear sur-face (Zdravkovich, 1997).

Given the same Re, CD drops gradually with increasing a. But thedrop to the minimum from a = 35� to 40� is rather rapid, reachingalmost 50% that of a plain cylinder at Re = 3 � 104. Then CD recoversrapidly to a level well above that of a plain cylinder. In this paper,the angle at which the minimum CD occurs is referred to as the crit-ical a. Using a single tripwire of 0.09d in diameter at Re = 3 � 104,

101 102

101 102

hotwire signal at different a. Re = 4 � 104.


Nebres and Batill (1993) measured a maximum drag reduction ata = 40� of only 30% that of the plain cylinder. The difference sug-gests that CD on a cylinder is more effectively suppressed withtwo tripwires used than with one. From a = 45� to 60�, the trip-wires act as ‘edges’, forcibly separating and deflecting the bound-ary layers towards the free stream. Furthermore, the transversedistance between the ‘edges’ grows with a, leading to an increasedwake width. As a consequence, CD grows quickly, exceeding theplain cylinder value for a > 50�.

The dependence of CL,rms and CD,rms on a (Fig. 9b and c) displaysa variation similar to CD. With increasing a, both CD,rms and CL,rms

decrease monotonically to the critical a (�40�), the maximum de-crease reaching about 60% that of the plain cylinder at Re = 6 � 104.With two symmetrically arranged tripwires of 0.08–0.12d in diam-eter, Alam et al. (2003a) managed to reduce CD, CL,rms and CD,rms by67%, 87% and 61%, respectively (a = 40�), obtaining slightly largerreductions than the present. The larger reduction obtained in theirinvestigation could be ascribed to two factors. (i) Their tripwireswere separated by 0.008d from the cylinder surface, which maybe more effective to generate turbulence in the boundary layers.(ii) They measured forces on the cylinder only, without consideringforces on the tripwire.

The essence for the use of tripwires to change CD, CD,rms andCL,rms lies in the fact that the tripwire-produced protrusion altersthe boundary layer around the cylinder and the alteration dependsstrongly on a. As a matter of fact, the sharp fall in CD from a = 35� to40�, similarly to the crisis region in a plain cylinder wake, resultsfrom a change from the laminar to turbulent boundary layer

Laminar reatta

turbulent sep

Laminar reattachment and

separation

Turbulent reattachment and

separation

No reattachment

Mode I

Mode II

(a) (b)

(d)

Fig. 13. Sketch of flow separation: (a)

(Schewe, 1983; Almosnino and McAlister, 1984; Zdravkovich,1997).

It is of relevance to compare the tripwire diameter d0 with theboundary layer thickness d. At a given location, d varies inverselywith Re1/2 (Zdravkovich, 1997). On a plain cylinder at h = 60�, d/dwas estimated to be 0.5, 0.16, 0.05, and 0.02 at Re = 102, 103, 104

and 105, respectively (Zdravkovich, 1997). The present d0 = 0.045dand d0/d is then estimated to be 0.4–2 for Re = 2.5 � 103–6 � 104.Nebres and Batill (1993) obtained d/d = 0.017–0.007 at h = 45� forRe = 1 � 104–4 � 104 and then d0/d = 2.64–6.42, that is, the tripwiremay protrude out of the boundary layer at small h or be immersedat large h, depending on Re. With a tripwire attached to the cylin-der, the boundary layer is modified and may develop over the trip-wire, which can be laminar or turbulent, depending on a and Re.

CD is closely linked to CPb; a more pronouncedly negative CPb

corresponds to a larger CD and vice versa. Naturally, CPb,rms is con-nected to CD,rms and CL,rms, and provides an indication on thestrength of Karman vortex shedding. Indeed, the dependences of�CPb and CPb,rms on a (Fig. 10) exhibit almost the same variationas CD and CD,rms, respectively. As such, with a increasing, �CPb re-duces significantly for 20� < a < 35� and drastically for35� < a < 40�. Nebres and Batill (1993) measured circumferentialtime-averaged pressure distribution around the cylinder with atripwire of 0.09d at Re = 3 � 104 and observed that �CPb startedto drop between a = 20� and 30� and continued its drop untila = 42�. They ascribed the drop in �CPb to the occurrence of transi-tion to turbulence in the boundary layer. It is worth noting that�CPb and CPb,rms are not sensitive to Re, for the range examined,

chment but

aration

Turbulent reattachment and

separation

No reattachment

(c)

(e)

Regime A, (b) B, (c) C, (d) D, (e) E.


for a plain cylinder (Fig. 3) but are in the presence of tripwires atleast for a < 40� or >50� (Fig. 10). This is because the tripwire itselfacts as a turbulence generator in the boundary layer.

4. Flow classification

Before the discussion of flow classification, it is useful to exam-ine turbulence characteristics in the pressure signal at h � 150�,which may be best manifested with the predominating fluctuationat St filtered out. Fig. 11 shows the time traces of the high-pass fil-tered pressure signals with a cut-off frequency of 200 Hz, corre-sponding to St = 0.33, at different a. The pressure signals appearfluctuating rather uniform at a = 10� and substantially more irreg-ular at a P 20�, suggesting perhaps laminar and turbulent separa-tions, respectively. The duration of the fluctuation of relativelysmall amplitude gets prolonged from a = 20� to 42.5�, possibly be-cause the transition to turbulence in the boundary layer moves to-wards the tripwire with increasing a. From a = 50� to 70�, theuniform fluctuations tend to return because the laminar boundarylayer separated from the tripwire does not reattach. The turbulentcharacteristics of the near wake may be reflected in the autocorre-lation coefficient Ruu (Fig. 12) of the hotwire-measured u at x* = 3and y* = 2 (no filtering was performed). The occurrence of the firstvalley in Ruu is getting smaller in t* from a = 10� to 40�, in consis-tence with the corresponding increase in St (Fig. 8d). Ruu ata = 30� (Regime B) and 40� (Regime C) displays a noticeable dispar-ity from those at a < 20� (Regime A). Firstly, the former decaysmuch more rapidly with increasing t*. Secondly, its periodical var-iation is considerably weakened in particular at a = 40�, implyingan increased turbulence in the shear layers due to turbulent sepa-ration (to be discussed more later). Ruu regains the periodical

0.14

0.18

0.22

0.26

0.30(c)

St

α = 40

0 1 2 3 4 5 60.14

0.18

0.22

0.26

0.30

St

4/10Re

(e)α = 45

506070

0.14

0.18

0.22

0.26

0.30

Plain Cylinder

α = 10

St

(a)

Fig. 14. Dependence of St on Re: (a) R

variation at a P 42.5� (Regimes D and E). At a = 42.5�, Ruu showsperiodicity since flow in Mode II is similar to that in Regime E, nei-ther with the boundary layer, separated from the tripwire,reattached.

The flow may be classified as five regimes, as marked at the topof Figs. 8–10, based on the dependence on a of St, CD, CD,rms, CL,rms,CPb and CPb,rms, along with the measured pressure and velocity sig-nals. The sketches of flow patterns in Fig. 13 are based on not onlythe measurements of St, forces, base pressure and near wake butalso previous reports. Nebres and Batill (1993) conducted hotwiremeasurements in the boundary layer over a cylinder attached witha single tripwire to gain insight into the boundary layer transition,and Alam et al. (2003a) performed the surface pressure measure-ment and surface oil-flow visualization on a cylinder attached withtwo tripwires to identify the boundary layer reattachment andseparation.

Regime A or the laminar reattachment and separation regime(Fig. 13a) occurs at a < 20�, where the cylinder boundary layer reat-taches behind the tripwire and the final separation remains lami-nar; the flow separation point is affected perhaps only veryslightly. At small a, the laminar boundary layer developed beforeand over the tripwire may separate and reattach on the cylindersurface without transition into turbulence. One may wonder howthis could be possible. At small a, the separation between theboundary layer, separated from the tripwire, and the cylinder sur-face is converging and the separated boundary layer may smoothlyreattach without transition. However, at large a, this separation isdiverging and the boundary layer may experience transition beforeor when reattaching on the cylinder surface.

Regime B or the laminar reattachment but turbulent separationregime (20� 6 a 6 35�, Fig. 13b) is characterized by a boundarylayer that, once separating from the tripwire, reattaches on the

7

0 1 2 3 4 5 6 74/10Re

(d)α = 42.5

(b)α = 20

253035

egime A; (b) B; (c) C; (d) D; (e) E.


cylinder surface. Transition to turbulence in the reattached bound-ary layer occurs, resulting in a turbulent separation with the sepa-ration point shifting downstream with increasing a. For a larger a,St rises but CD, CD,rms, CL,rms, �CPb and CPb,rms all decrease rather rap-idly, below their counterpart in a plain cylinder wake. As shown inFigs. 8–10, there is a gradual and continuous variation in St, CD,CD,rms, CL,rms, �CPb and CPb,rms with increasing a; in fact, the variationis rather mild in Regime A but rapid in Regime B. In the Re-rangeexamined, the boundary layer over a plain cylinder is laminarbut turbulent, once separated. In Regime A, although flow separa-tion is laminar, the transition occurs in the separated boundary la-ter and shifts toward the separation point with increasing a(Fig. 13a). Similarly, the transition in the reattached boundary layermoves toward the reattachment point (Fig. 13b). Since the fluctu-ating pressure on the cylinder after-body surface contributes mostto CL,rms, the difference between the two regimes is evident in CL,rms

(Fig. 9b), which drops significantly from Regime A to B. The differ-ence between the two regimes can be seen also by comparing the

0.1 0.2 0.3 0.4 0.5 0.6

0.7

0.8

0.9

1

0.7

0.8

0.91

-1

0

1

0

-

0.1 0.2 0.3 0.4 0.5 0.6

0.7

0.80.9

1

0.7

0.8

0.9

1

-1

0

1

0

(a)

(c) α = 30°

y∗

y∗

y∗

x∗

Plain Cylinder

Regime B

0 0.1 0.2 0.3 0.4

1

0.5

0.9

0.6

0.7

0.8

0.5

0.6

0.9

0.7

0.8

1

-1

0

1

(g) α = 60°

Regime E: E2

(e) α = 42.5°

Regime D

0.1 0.2 0.5 0.6 0.7

0.8

0.9

1

0.8

0.9

1

0.3 0.4

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

-1

0

1

0

-

y∗

Fig. 15. Iso-contours of time-averaged streamwise ve

pressure signals (Fig. 11) and the recirculation bubble sizes ata = 10� and 30� (Fig. 15).

In Regime C or the turbulent reattachment and separation re-gime (35� < a 6 40�, Fig. 13c), the transition to turbulence reachesthe reattachment point, generating a turbulent reattachment. As aresult, CD, CD,rms, CL,rms, �CPb and CPb,rms all drop drastically and Streaches the maximum. In Regime D or the bi-stable flow regime(Fig. 13d, 40� < a < 45�), the boundary layer is in a critical conditionand, after separating from the tripwire, can reattach on thecylinder surface (Mode I) or can remain separated without reat-tachment (Mode II). The two states occur intermittently. Conse-quently, two distinct St values are observed. In Mode I, theboundary layer reattaching on the cylinder surface produces a nar-row wake with a high St; in Mode II, a wide wake with a lower St isgenerated. This regime is a transition from the crisis (Regime C) toan early separation regime, i.e., Regime E (Fig. 13e). Regime E or theearly separated flow regime corresponds to 45� 6 a < 70�, wherethe boundary layer separating from the tripwire does not reattach

0 0.1 0.2 0.3 0.4 0.5

0.6

0.70.8

0.91

0.60.7

0.80.9

1

0 1 2 3 4 5

1

0

1

(b) α = 10°

x∗

Regime A

0.5

0 0.1 0.2 0.3 0.4

0.5

0.70.8

0.91

0.60.7

0.80.9

1

0.6

0 1 2 3 4 5

-1

0

1

(f) α = 45°

Regime E: E1

0.1 0.2 0.3 0.4 0.50.6

0.7

0.80.9

1

0.70.8

10.9

0 1 2 3 4 5

1

0

1

0

(d) α = 40°

Regime C

locity U� in different flow regimes, Re = 4 � 104.


on the cylinder surface, thus separating early compared to a plaincylinder and resulting in a wide wake. With a increasing, the wakegrows in width, reaching a maximum at a � 60� and approaching aconstant for a > 60�. Accordingly, CD, CD,rms and CL,rms all rise froma = 45� to 60� and tend to approach a constant for a > 60�. RegimeE may be further divided into two: E1 (45� 6 a < 60�) and E2

(60� 6 a < 70�). Whilst CD, CD,rms and CL,rms rise rapidly with increas-ing a in E1, their change is insignificant in E2.

The St–Re relationship is also distinct from one regime to an-other (Fig. 14). In Regime A (a < 20�, Fig. 14a), St is essentially thesame as in a plain cylinder wake and independent of Re. St riseswith increasing Re monotonically in Regime B (20� 6 a 6 35�,Fig. 14b) but non-monotonically for Regime C (35� < a 6 40�,Fig. 14c). In Regime D (40� < a < 45�), after an initial drop up toRe = 1 � 104, St increases rapidly with Re, similarly to the transitionregime of a plain cylinder in Re at about 3 � 105 (Higuchi et al.,1989; Farrel and Blessmann, 1983; Schewe, 1983; Almosnino andMcAlister, 1984). Beyond a ‘critical’ Re (2 � 104) the bi-stable flowleads to two different St. The higher St reaches its maximum ofabout 0.29 at Re = 2 � 104. Igarashi (1986) observed a slightly high-er St (0.3–0.35). His tripwire diameter was smaller (0.008–0.016d)than the present (0.045d). A larger tripwire diameter should corre-spond to a smaller maximum St because of increased additionalbluffness to the cylinder. Interestingly, a further reduction in thetripwire diameter to say 0.0d, i.e., a plain cylinder in the critical re-gime, the maximum St rises to 0.42 (Paranthoen et al., 1999). St inRegime E (45� 6 a < 70�) is insensitive to Re as in Regime A, thoughmuch lower.

(a)

-2 -1 0 1 20.0

0.5

1.0

1.5

*U

*y

Plain Cylinder

α = 10

30

40

42.5

60

(b)

* 0U =

*x

*y

o** xx 0=

Fig. 16. (a) Schematic of the definition of streamwise velocity distribution along y*,(b) lateral distribution of the time-averaged streamwise velocity U� at x� ¼ x�0 fordifferent tripwire positions (Re = 4 � 104).

5. The near wake

Forces, St and flow field are closely linked with each other.Therefore, it is important to examine flow field to understand thor-oughly the observations made earlier. The near wake characteristicparameters, including the wake width, vortex formation length,recirculation bubble, separation angle (the angle between shearlayer at separation and free-stream velocity) and momentum defi-cit of the flow may be investigated through the iso-contours oftime-averaged and rms streamwise velocities, i.e., U� and u�rms.Fig. 15 presents the iso-contours of U� in different regimes alongwith those of a plain cylinder wake. The recirculation bubble size,enclosed by U� = 0, provides a measure for the strength of recircu-lation. The recirculation bubble in Regime A is slightly greater in lat-eral width than in a plain cylinder wake, but the separation angle isa little smaller. The observation is attributed to the fact that theplacement of a tripwire close to the forward stagnation point makesthe shear layer narrow, widening the recirculation bubble. With anincreased a in Regime B, the separation angle becomes even smal-ler. So does the recirculation bubble size, due to a postponed sepa-ration of the turbulent shear layer. The streamwise length of thebubble is shortest in this regime, implying a minimum flow rever-sal. In Regime C, the separation angle is the smallest, coupled withthe narrowest recirculation bubble, which is ascribed to the turbu-lent reattachment and postponed separation. Regime D is similar toRegime C, albeit with a wider shear layer that is linked with the bi-stable flow. In Regime E, the recirculation bubble increases signifi-cantly in both lateral and streamwise extents. The streamwiseextent however retreats from Sub-regime E1–E2.

The lateral distribution of U� may provide information on thedeficit of streamwise momentum, which is directly connected toCD. Choose the most downstream point x� ¼ x�0 of the U� = 0 con-tour (Fig. 16a) to compare the lateral distributions of U� betweendifferent regimes. As shown in Fig. 16b, the velocity deficit regionin Regime A (a = 10�) is slightly larger than in the plain cylinderwake. With increasing jy�j, U� recovers from 0 to 1.0 rapidly in Re-

gimes C (a = 40�) and D (a = 42.5�) and slowly in Regime E (a = 60�).Regimes D and E are characterized by the narrowest and widestvelocity deficit (U� < 1.0) regions and, accordingly, the smallestand largest CD, respectively.

Fig. 17 shows the iso-contours of the fluctuating streamwisevelocity u�rms. The two symmetrical concentrations occur at the rol-lup position of the two shear layers. Following Bloor (1964), the x*-coordinate of the concentration is defined as the vortex formationlength, L�f , which is closely connected to CD, CD,rms and CL,rms. A shortL�f corresponds to large CD, CD,rms and CL,rms or vice versa (Bloor,1964; Griffin, 1971; Zdravkovich, 1997). Another importantparameter is the wake width, w�, defined as the lateral separationbetween the free shear layers in the wake (Roshko, 1954; Bala-chandar et al., 1997). An alternative definition for this width isthe lateral separation between the two maximum concentrationsof u�rms (Griffin and Ramberg, 1974; Ramberg, 1983). This paperadopts the latter definition. This width has a great influence onthe forces. Table 1 summarizes L�f , w� and the maximum (u�rms;max)of u�rms, extracted from the u�rms-contours, in different regimes. Thefluctuating wake and the fluctuating forces on the cylinder arestrongly coupled. With a increasing from Regime A to E, w� first re-treats and then grows, reaching the minimum and the maximumin Regimes D (a = 42.5�) and E2 (a = 60�), respectively. Interest-ingly, when the reattached boundary layer changes from laminarto turbulent separation (Regimes A–B) with increasing a, w�

shrinks very slightly; on the other hand, when the laminar reat-tachment (Regime B) changes to the turbulent (Regime C) with afurther increase in a, w� reduces from 1.2 to 0.9. At the same U1,the smaller the wake width, the higher is St (Griffin and Ramberg,1974; Bearman and Trueman, 1972). This is fully consistent withthe measured forces (Fig. 9) and St (Fig. 8). Based on their measure-ment of U at x* = 0.25, Nebres and Batill (1993) showed that thewake width defined as the lateral separation between the pointsof the maximum velocity in the two shear layers decreased froma = 0� to about 42�, and then increased, reaching its maximum ata = 70� before decreasing again as a approached 180�. This defini-tion of w* is in line with what Roshko (1954) and Balachandar et al.(1997) proposed. Their observation is very similar to ours where w�

is estimated from the u�rms-contours. L�f does not follow the samevariation as w�. The largest and smallest L�f are observed in RegimeC (a = 40�) and Sub-regime E2 (a = 60�), respectively. Note that thepresence of tripwires prolongs L�f , compared to the plain cylinder,

(g) α = 60°

Regime E: E2

0.375

0.375

0.325

0.3 0.275

0.275

0.3

0.325

0.25

0.25

0.35

0.35

0.30.2750.25

0.375

0.375

-1

0

1

0.35

0.35

0.325 0.3

0.30.3250.25

0.25

0.225

0.225

0.2

0.2

0.250.2

0.375

0.375

0 1 2 3 4 5

-1

0

1

(a) (b) α = 10°

y∗

y∗

y∗

Regime A

Regime B

Plain Cylinder

0.350.30.250.2

0.4

0.4

0.4

0.4

0.35

0.30.25

0.2

0.350.3

0.250.2

-1

0

1

0.25 0.3 0.35

0.4

0.4

0.35

0.30.25

0.2

0.35

0.30.25

0.2

0.2

0.4

0.4

-1

0

1

(c) α = 30°

Regime E: E1

0.45 0.4

0.35

0.30.25

0.35

0.4

0.350.3

0.25

0.35

0.30.25

0.2

0.2

0.4

0.45

-1

0

1

(f) α = 45°(e) α = 42.5°

Regime D

0.260.20.24

0.28

0.280.22

0.3

0.3

-1

0

1

x∗

x∗

0.30.35

0.35

0.3

0.3

0.40.45

0.4

0.25

0.25

0.25

0.35

0.4

0.45

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 50 1 2 3 4 5

0 1 2 3 4 5

-1

0

1

Regime C

(d) α = 40°

y∗

0.45

0.44

0.44

0.45

0.45

0.44

Fig. 17. Iso-contours of u�rms in different flow regimes, Re = 4 � 104.

Table 1Formation length L�f , wake width w� and peak magnitude (u�rms;max) of u�rms at differentregimes (here a = 0 denotes the case of plain cylinder).

Regimes – A B C D E1 E2

a (�) 0 10 30 40 42.5 45 60L�f 2.0 2.5 2.4 2.8 2.5 2.7 1.7

w* 1.0 1.3 1.2 0.90 0.70 1.3 1.4u�rms;max 0.375 0.45 0.44 0.45 0.30 0.45 0.375


except in Sub-regime E2. The change in L�f is small from Regimes Ato B but is large, increasing from 2.4 to 2.8, from Regimes B to C.The results point to that the change in the nature of flow reattach-ment has a more pronounced effect on the near wake than that inthe nature of flow separation.

With the tripwires placed symmetrically about the leading stag-nation point, CL was zero for the a range examined, thus not pre-sented. In addition, the iso-contours of U� (Fig. 15) and u�rms

(Fig. 17) are all symmetric about the wake centerline, suggestingthat flow separation and reattachment are statistically symmetric.

6. Discussion: bi-stable flow phenomenon and RWIV

The bi-stable flow phenomenon in Regime D has barely been gi-ven any attention previously, let alone its possible impact uponengineering applications. As described in Introduction, under lightto moderate wind and rain, two circumferentially oscillating waterrivulets may occur, running along the surface of stay cables ofcable-stayed bridges; their important role in the RWIV phenome-non has been confirmed (e.g., Ohshima, 1987; Hikami and Shirai-shi, 1988; Flamand, 1995; Bosdogianni and Olivari, 1996). Thecircumferential oscillation effectively changes the position of therivulets, thus modifying the boundary layer and its separation.The rivulet oscillation frequency is much lower than the vortexshedding frequency; their ratio is 1 to 8–20 (Gu and Du, 2005).As such, the rivulets could be considered quite steady, compared

0 1 2 3 4 5 6 70

15

30

45

60

[St(α=42.5°) - St(α=60°)]/St(α=42.5°)[St(α=40°) - St(α=60°)]/St(α=40°)

4/10Re

0.00

0.04

0.08

0.12

0.16

St(α=42.5°) - St(α=60°)St(α=40°) - St(α=60°)ΔSt (42.5°, 60°)

ΔSt (40°, 60°)

Stρ (42.5°, 60°)

Stρ (40°, 60°)

ΔSt

Stρ

(a)

(b)

Fig. 19. Dependence of difference in St on Re: (a) DSt(a1, a2) = |St(a1) � St(a2)|, (b)qst (a1, a2) =|St(a1) � St(a2)|/ St(a1).


to vortex shedding. Subsequently, a trip wire may be used to a cer-tain degree to simulate the rivulet, and the present tripwire-per-turbed wake may provide insight into the mechanisms of theRWIV.

It has been previously reported that, with the movement of astay cable, the rivulet on the leeward side of the cable may oscillatecircumferentially from 30–47� to 50–68�, depending on Re (Hikamiand Shiraishi, 1988). Consider a rivulet moving from a = 40� to 60�,which corresponds to Regimes C and E, respectively, in the trip-wire-perturbed wake. The flow in Regime C is very different fromthat in Regime E. It is worthwhile quantifying the variation withthe a change in terms of St and CD, which are two importantparameters concerning RWIV. Let us define

Dgða1;a2Þ ¼ jgða1Þ � gða2Þj ð1Þ

and

qgða1;a2Þ ¼ jgða1Þ � gða2Þj=gða1Þ; ð2Þ

where g may stand for St or CD, a1 and a2 correspond to the angularpositions of the rivulet in Regimes C and E, respectively. Apparently,Dg(a1, a2) andqg(a1, a2) provide a measure for the actual and rela-tive differences in g, respectively, as the rivulet moves from a1 to a2.

Fig. 18 presents the dependence of DCD(a1, a2) and qCD(a1, a2) on

Re for a1 = 40� or 42.5� and a2 = 60�. DCD(40�, 60�) (or qCD(40�, 60�))

is always larger than DCD(42.5�, 60�) (or qCD(42.5�, 60�)), that is, the

rivulet oscillation of the larger amplitude produces a greater fluctu-ation of forces. While qCD

(40�, 60�) is almost constant, about 120%for the Re-range examined, qCD

(42.5�, 60�) is somewhat smaller atRe < 4 � 104 and reaches about 100% at Re P 4 � 104.

Fig. 19 shows the dependence of DSt(a1, a2) and qSt(a1, a2) on Refor a1 = 40� or 42.5� and a2 = 60�. The variation in qSt(a1, a2) is lessthan 35% at Re < 3 � 104 but reaches about 38–45% at Re P 3 � 104,where DSt(a1, a2) or qSt(a1, a2) is less dependent on Re. The obser-vation suggests that the perturbation of tripwires on the wake ismore effective at Re P 3 � 104, and is consistent with the reportsthat the RWIV is prone to occurrence at Re = 4 � 104–1.5 � 105

(Cosentino et al., 2003a) or 5.5 � 104–1.2 � 05 (Hikami and Shirai-shi, 1988).

The bi-stable flow is associated with two flow states/regimesand two frequencies, switching randomly from one to the other.

1 2 3 4 5 6 7

(b)

(a)

[CD

(α=60°) - CD

(α=42.5°)]/CD

(α=42.5°)[C

D(α=60°) - C

D(α=40°)]/C

D(α=40°)

4/10Re

0.6

0.8

1.0

1.2

1.4

1.6

CD

(α=60°) - CD

(α=42.5°)C

D(α=60°) - C

D(α=40°)

ΔCD (42.5°, 60°)

ΔCD (40°, 60°)

ΔCD

DC

ρ

DCρ (42.5°, 60°)

DCρ (40°, 60°)

1.4

1.2

1.0

0.8

0.6

0.4

0.2

Fig. 18. Dependence of difference in CD on Re: (a) DCD(a1, a2) = |CD(a1) � CD(a2)| and(b) qCD

(a1, a2) = |CD(a1) � CD(a2)|/CD(a1).

Naturally, the fluid forces associated with the two states of the floware quite different (Fig. 9). For the stationary cylinder, the a rangeof Regime D is limited to 40–45� and the switch between the flowstates is random. In engineering, the cable is flexible and thisswitch, associated with a large change in instantaneous drag,may provide initial excitation for the cable to oscillate. When syn-chronized with the natural frequency of the cable, which is muchlower than the vortex shedding frequency, the switch may betuned to quasi-periodical, resulting in a violent cable vibration.The bi-stable flow occurs in a restricted range of Re P 3 � 104,which is consistent with the limited range of wind speeds, 5–15 m/s, or the limited Re-range where RWIV occurs (Hikami andShiraishi, 1988; Cosentino et al., 2003a).

7. Conclusions

The effect of tripwires on the forces, St and wake of a circularcylinder has been examined at Re = 2.5 � 103–6 � 104. Two trip-wires of 0.045d in diameter were attached symmetrically on thecylinder surface at a = ±10� to 70�. The investigation leads to thefollowing conclusions:

The tripwire-perturbed wake may be classified into five re-gimes. In Regime A (a < 20�), the cylinder boundary layers separat-ing from the tripwires reattach on the cylinder surface. A laminarseparation of the reattached shear layer occurs, and the flow sepa-ration point is affected perhaps only very slightly. As a result, theeffect of tripwires on the wake, forces and St is small. In RegimeB (20� 6 a 6 35�), the boundary layer separating from the tripwirereattaches on the cylinder surface; transition to turbulence occursin the reattached boundary layer, resulting in postponed separa-tion from the cylinder. With increasing a, the transition to turbu-lence shifts upstream. Meanwhile, St increases and CD, CD,rms,CL,rms, �CPb and CPb,rms all decrease rather rapidly. In Regime C(35� < a 6 40�), the transition to turbulence occurs now at reat-tachment point, and CD, CD,rms, CL,rms, �CPb and CPb,rms all drop dras-tically. In Regime D (40� < a < 45�), the boundary layer separatingfrom the tripwire may reattach the cylinder surface (Mode I) or re-main separated without reattachment (Mode II). Modes I and II areconnected to a high and low St; respectively. Both modes are stableand occur intermittently and randomly, generating a bi-stableflow. Regime E (45� 6 a < 70�) is characterized by boundary layersthat, once separating from the tripwire, will not reattach on the


cylinder surface. With a increasing, CD, CD,rms and CL,rms all rise froma = 45� to 60� and tend to approach a constant for a > 60�. This re-gime may be further divided into two: E1 (45� 6 a < 60�) where CD,CD,rms and CL,rms increase and St decrease with increasing a, and E2

(60� 6 a < 70�) where the change of these force coefficients with ais insignificant.

The wake characteristic parameters, including the vortex for-mation length, wake width, recirculation bubble size and separa-tion angle, are linked to the boundary layer behavior and aredistinct from one regime to another. Compared to a plain cylinderwake, L�f is prolonged in all regimes, except Sub-regime E2, and islargest in Regime C. With increasing a from Regime A to E, w* firstdecreases and then increases, reaching the minimum and the max-imum in Regimes D and E2, respectively. While a change from thelaminar to turbulent separation from the cylinder (Regimes A–B)has a small effect on L�f and w*, that from the laminar reattachment(Regime B) to the turbulent (Regime C) causes a significant elonga-tion in L�f and retreat in w*. The early separation without reattach-ment (Regime E) leads to a large growth in w*. The lateral width ofthe recirculation bubble retreats gradually up to Regime C and thengrows from D to E2. While the retreat results from postponed sep-aration and small separation angle, the growth is connected toearly separation and large separation angle.

The Re–St relationship is distinct from one regime to another. InRegime A, St is insensitive to Re for the Re-range examined, simi-larly to a plain cylinder wake. This is not unexpected because bothcases are characterized by a laminar flow separation. However, Stdisplays a significant variation with increasing Re in Regimes Band C, which are characterized by turbulent flow separation. Inthe two regimes, transition to turbulence occurs in the reattachedboundary layer and shifts towards the tripwire with increasing Re,resulting in a postponed separation and subsequently an increasein St. In Regime E, laminar separation occurs, though earlier com-pared with a plain cylinder wake or Regime A. Consequently, St re-mains almost constant.

It is proposed that the bi-stable flow in Regime D may have aconnection to the RWIV phenomenon observed in cable-stayedbridges. The two water rivulets formed on a stay cable in thesimultaneous presence of rain and wind may act in a way as thepresent trip wires. At the right wind speed and yaw angle ofthe stay cable, the flow around the cable and rivulets may fall inRegime D. As a matter of fact, the quasi-periodical circumferentialoscillation of the rivulets (Hikami and Shiraishi, 1988; Cosentinoet al., 2003b) may forcefully synchronize the switch of flow be-tween Regimes C and E or between the subcritical and the criticalflow regimes, which occurs otherwise randomly. Noting the largedifference (90–105%) in CD between Regime C and E, the quasi-periodical switch, in particular, if synchronized with the fluid–structure system frequency, could be adequate to excite RWIV. Itis worth highlighting that the bi-stable flow occurs in a restrictedrange of Re P 3 � 104, which is consistent with the Re-range ofRWIV, as observed by Cosentino et al. (2003a) and Hikami andShiraishi (1988).

Acknowledgements

YZ wishes to acknowledge support given to him by the ResearchGrants Council of the Government of the HKSAR through Grant Pol-yU 5334/06E. Messrs Huang J.F., Bei H.L. and Wang X.W.’s assis-tance in experiments is gratefully acknowledged.

References

Alam, M.M., Sakamoto, H., 2005. Investigation of Strouhal frequencies of twostaggered bluff bodies and detection of multistable flow by wavelets. J. FluidsStruct. 20, 425–449.

Alam, M.M., Zhou, Y., 2007a. Turbulent wake of an inclined cylinder with waterrunning. J. Fluid Mech. 589, 261–303.

Alam, M.M., Zhou, Y., 2007b. Dependence of Strouhal number, drag and lift on theratio of cylinder diameters in a two-tandem cylinder wake. In: The Proceedingsof the 16th Australasian Fluid Mechanics Conference, Gold Coast, Australia, pp.750–757.

Alam, M.M., Zhou, Y., 2008. Strouhal numbers, forces and flow structures aroundtwo tandem cylinders of different diameters. J. Fluids Struct. 24, 505–526.

Alam, M.M., Sakamoto, H., Moriya, M., 2003a. Reduction of fluid forces acting on asingle circular cylinder and two circular cylinders by using tripping rods. J.Fluids Struct. 18, 347–366.

Alam, M.M., Moriya, M., Takai, K., Sakamoto, H., 2003b. Fluctuating fluid forcesacting on two circular cylinders in a tandem arrangement. J. Wind Eng. Ind.Aerodyn. 91, 139–154.

Almosnino, D., McAlister, K.W., 1984. Water tunnel study of transition flow aroundcircular cylinders. National Aeronautics and Space Administration, NASA,85879.

Balachandar, S., Mittal, R., Najjar, F.M., 1997. Properties of the mean recirculationregion in the wakes of two-dimensional bluff bodies. J. Fluid Mech. 351, 167–199.

Bearman, P.W., Trueman, D.M., 1972. An investigation of the flow aroundrectangular cylinders. Aeronaut. Quart. 23, 229–237.

Bearman, P.W., 1969. On vortex shedding from a circular cylinder in the criticalReynolds number régime. J. Fluid Mech. 37, 577–585.

Bloor, S.M., 1964. The transition to turbulence in the wake of a circular cylinder. J.Fluid Mech. 19, 290–309.

Bosdogianni, A., Olivari, D., 1996. Wind- and rain-induced oscillations of cables ofstayed bridges. J. Wind Eng. Ind. Aerodyn. 64, 171–185.

Cosentino, N., Flamand, O., Ceccoli, C., 2003a. Rain–wind induced vibration ofinclined stay cables. Part I: Experimental investigation and physicalexplanation. Wind Struct. 6, 471–484.

Cosentino, N., Flamand, O., Ceccoli, C., 2003b. Rain–wind induced vibration ofinclined stay cables. Part II: Mechanical modeling and parametercharacterization. Wind Struct. 6, 485–498.

De Langre, E., Herman, P., 2006. Private communication.Fage, A., Warsap, J.H., 1929. The effects of turbulence and surface roughness on the

drag of a circular cylinder. Aero. Res. Comm. R.M. No. 1283.Farge, M., 1992. Wavelet transforms and their applications to turbulence. Annu.

Rev. Fluid Mech. 24, 395–457.Farrel, C., Blessmann, J., 1983. On critical flow around smooth circular cylinders. J.

Fluid Mech. 136, 375–391.Flamand, O., 2007. Private communication.Flamand, O., 1995. Rain–wind induced vibration of cables. J. Wind Eng. Ind.

Aerodyn. 57, 353–362.Griffin, O.M., Ramberg, S.E., 1974. The vortex street wakes of a vibrating cylinders. J.

Fluid Mech. 66, 729–738.Griffin, O.M., 1971. The unsteady wake of an oscillating cylinder at low Reynolds

number. J. Appl. Mech. 38, 523–532.Gu, M., Du, X., 2005. Experimental investigation of rain–wind-induced vibration of

cables in cable-stayed bridges and its mitigation. J. Wind Eng. Ind. Aerodyn. 93,79–95.

Higuchi, H., Kim, H.J., Farell, C., 1989. On flow separation and reattachment around acircular cylinder at critical Reynolds numbers. J. Fluid Mech. 200, 149–171.

Hikami, Y., Shiraishi, N., 1988. Rain–wind induced vibration of cables in cablestayed bridges. J. Wind Eng. Ind. Aerodyn. 29, 409–418.

Hover, F.S., Tvedt, H., Triantafyllou, M.S., 2001. Vortex-induced vibrations of acylinder with tripping wires. J. Fluid Mech. 448, 175–195.

Igarashi, T., 1986. Effect of tripping wires on the flow around a circular cylindernormal to an airstream. Bull. Jpn. Soc. Mech. Eng. 29, 2917–2924.

James, D.F., Truong, Q.S., 1972. Wind load on cylinder with spanwise protrusion.Proc. ASCE, J. Eng. Mech. Div. 98, 1573–1589.

Lemaitre, C., Alam, M.M., Hémon, P., De Langre, E., Zhou, Y., 2006. Rainwater rivuletson a cable subject to wind. C.R. Mécan. 334, 158–163.

Lemaitre, C., 2006. Dynamics of Rainwater Film on a Stay Cable Subject to Wind.PhD Thesis, Ecole Polytechnique, France.

Matusmoto, M., Saitoh, T., Kitazawa, M., Shiraito, H., Nishizaki, T., 1995. Responsecharacteristics of rain–wind induced vibration of stay-cables of cable-stayedbridges. J. Wind Eng. Ind. Aerodyn. 57, 323–333.

Nebres, J., Batill, S., 1993. Flow about a circular cylinder with a single large-scalesurface perturbation. Exp. Fluids 15, 369–379.

Norberg, C., 2003. Fluctuating lift on a circular cylinder: review and newmeasurements. J. Fluids Struct. 17, 57–96.

Norberg, C., 1987. Effects of Reynolds number and a low-intensity freestreamturbulence on the flow around a circular cylinder. Publication 87/2, Departmentof Applied Thermodynamics and Fluid Mechanics, Chalmers University ofTechnology, Sweden.

Norberg, C., 2001. Flow around a circular cylinder: aspects of fluctuating lift. J.Fluids Struct. 15, 459–469.

Ohshima, K., 1987. Aerodynamic stability of the cables of a cable-stayed bridgesubject to rain (a case study of the Ajigawa bridge). In: Proceedings of US-JapanJoint Seminar on Natural Resources, pp. 324–336.

Paranthoen, P., Browne, L.W.B., Masson, S.L., Dumouchel, F., Lecordier, J.C., 1999.Characteristics of the near wake of a cylinder at low Reynolds numbers. Eur. J.Mech. B/Fluids 18, 659–674.

Pearcey, H.H., Cash, R.F., Salter, I.J., 1982. Flow past circular cylinders: simulation offull-scale flows at model scale. NMI Rep. 131, 1–54.


Ramberg, S.E., 1983. The effects of yaw and finite length upon the vortex wakes ofstationary and vibrating circular cylinders. J. Fluid Mech. 128, 81–107.

Romberg, O., Popp, K., 1998. The influence of trip-wires on the fluid-damping-controlled instability of a flexible tube in a bundle. J. Fluids Struct. 12,17–32.

Roshko, A., 1954. On the drag and shedding frequency of two-dimensional bluffbodies. NACA Tech. Note No. 3169.

Schewe, G., 1983. On the force fluctuations acting on a circular cylinder in cross-flow from subcritical up to transcritical Reynolds numbers. J. Fluid Mech. 133,265–285.

West, G.S., Apelt, C.J., 1993. Measurements of fluctuating pressures and forces on acircular cylinder in the Reynolds number range 104 to 2.5 � 105. J. Fluids Struct.7, 227–244.

West, G.S., Apelt, C.J., 1997. Fluctuating lift and drag forces on finite length of acircular cylinder in the subcritical Reynolds number range. J. Fluids Struct. 11,135–158.

Yamaguichi, H., 1990. Analytical study on growth mechanism of rain vibration ofcables. J. Wind Eng. Ind. Aerodyn. 33, 73–80.

Zdravkovich, M.M., 1997. Flow Around Circular Cylinders. Vol. 1: Fundamentals.Oxford Science Publications.

Classification of the tripped cylinder wake and bi-stable phenomenon

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