momentum and heat transfer in a turbulent cylinder wake behind a streamwise oscillating cylinder

International Journal of Heat and Mass Transfer 48 (2005) 4062–4072

www.elsevier.com/locate/ijhmt

Momentum and heat transfer in a turbulent cylinderwake behind a streamwise oscillating cylinder

G. Xu, Y. Zhou *

Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received 22 March 2004; received in revised form 16 March 2005

Abstract

The work aims to understand the effect of an oscillating circular cylinder on momentum and heat transport in theturbulent wake of a downstream identical cylinder, which is slightly heated. The oscillating amplitude, A, was 0.79d (d isthe cylinder diameter) and the frequency was 0.17fs for L/d = 2.5 and 0.24fs for L/d = 4, where L and fs are the cylindercentre-to-centre spacing and the frequency of vortex shedding from the downstream cylinder, respectively, at A = 0.The velocity and temperature fields were measured using a three-wire probe at 10d behind the stationary cylinderand a Reynolds number of 5920 based on d and the free-stream velocity. It is found that the upstream cylinder oscil-lation modifies the frequency of vortex shedding from the stationary cylinder, which is locked on with one harmonic ofthe oscillating frequency. This harmonic frequency is nearest to and below the natural vortex-shedding frequency. Fur-thermore, the wake response to the oscillation depends on L/d in terms of the cross-stream distributions of mean veloc-ity, Reynolds stresses, temperature variance and heat fluxes; the turbulent Prandtl number decreases at L/d = 4 butincreases at L/d = 2.5. The observations are linked to whether the flow regime experiences a change under the upstreamcylinder oscillation.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Turbulent wake; Two tandem cylinder wake; Momentum and heat transfer; Streamwise oscillating cylinder

1. Introduction

The turbulent flow behind multiple slender cylindershas received significant attention in the past [1–3]. It is ofboth fundamental and practical significance to under-stand the momentum and heat transfer characteristicsof this flow. The simplest configuration of multiple cyl-inders is two in tandem, side-by-side and staggeredarrangements. Aerodynamic interference between twocylinders involves most of the generic flow features asso-

0017-9310/$ - see front matter � 2005 Elsevier Ltd. All rights reservdoi:10.1016/j.ijheatmasstransfer.2005.04.029

* Corresponding author. Tel.: +852 2766 6662.E-mail address: [email protected] (Y. Zhou).

ciated with multiple structures; the study of this flowmay provide insight into flow physics and heat transportaround more cylinders [4–7].

Using a laser-Doppler velocimeter, Kolar et al. [4]conducted ensemble-averaged measurements in the tur-bulent near wake of two side-by-side square cylindersat T* = 3 and Re = 23100, where, T is transverse spacingbetween the two cylinder axes, d is the characteristicheight of the cylinder, Re is Reynolds number basedon d and free-stream velocity U1. Unless otherwisestated, an asterisk in this paper denotes normalizationby either U1, d or temperature excess H0. They foundthat the inner vortices close to the flow centreline decayed

ed.

mailto:[email protected]

Flow

Movablethree-wire probe

L

d

y

xod

Oscillation

10d

Fig. 1. Schematic arrangement of measurements.

G. Xu, Y. Zhou / International Journal of Heat and Mass Transfer 48 (2005) 4062–4072 4063

more rapidly than the outer ones shed from the cylinderside near the free stream. Based on flow visualizationand particle image velocimetry (PIV) measurements,Sumner et al. [5] investigated the flow around two circu-lar cylinders of the same diameter in a staggeredarrangement with a centre-to-centre pitch ratio P* = 1–5 and an angle of incidence a = 0–90� for Re = 850–1900, and identified nine flow patterns depending on P

and a. Using a phase-averaging method [8], Zhouet al. [7] studied the momentum and heat transport inthe wake of two side-by-side circular cylinders atRe = 5800 and found that the flow structure at T* = 3was totally different from that at T* = 1.5, the two flowregimes showing distinct momentum and heat transportcharacteristics.

The flow around two tandem cylinders has also at-tracted considerable attention. This flow may be classi-fied into three regimes [1,2]. For 1 < L* < 1.2–1.8,where L is the cylinder centre-to-centre spacing, freeshear layers separating from the upstream cylinder over-shoot the downstream one and then roll up into vortices.For 1.2–1.8 < L* < 3.4–3.8, the shear layers reattach onthe upstream side of the downstream cylinder. A vortexstreet is formed only behind the latter. For L* > 3.4–3.8,both cylinders generate vortices. Yiu et al. [9] measuredflow and temperature fields behind two stationary tan-dem cylinders with the downstream one slightly heated.Their measurements covered the three regimes, i.e.,L* = 1.3, 2.5 and 4.0. They found that vortical struc-tures, momentum and heat transport were stronglydependent on the flow regimes.

The initial conditions may have a profound effect ona flow. The effects of initial conditions on turbulentwakes may persist far downstream [10]. The equilibriumsimilarity analysis [11] indicated that the initial condi-tions and local Reynolds number effects dominated anaxisymmetric wake, and this dependence on upstreamconditions was asymptotic. Zhou and Antonia [12]examined the memory effects in turbulent plane wakesgenerated by circular, triangular and square cylindersand a screen of 50% solidity, respectively. Discernibledifferences in Reynolds stresses, fluctuating velocityspectra and root mean square vorticities were observedfor different generators even in the far wake. In engineer-ing, the structures such as offshore pilings and heatexchangers are frequently associated with flow-inducedoscillation. The structural oscillation alters the initialconditions and may affect the momentum and heattransport of the flow. In an experimental study of theflow structure and heat transfer over a heated cylinderoscillating at a small amplitude in the stream direction,Gau et al. [13] demonstrated that an oscillation fre-quency near twice the natural vortex-shedding frequencyenhanced heat transfer. Lai et al. [14] found that anoscillating cylinder could even lock in vortex sheddingfrom a neighbouring side-by-side arranged cylinder.

Mahir and Rockwell [15,16] investigated vortex forma-tion around two forced oscillating cylinders in thetandem and side-by-side arrangements. However, thereis a lack of information on how the cylinder oscillationwould alter the momentum and heat transport in theturbulent wake of two inline cylinders.

This work aims to understand how an upstreamoscillating cylinder influences the flow and temperaturefields in a turbulent cylinder wake. Two L* values, i.e.2.5 and 4, are investigated, covering two of the threeflow regimes in the absence of cylinder oscillation [2].The flow regime of very small L* is not investigateddue to a difficulty in carrying out oscillating-cylinderexperiments at small L*. The investigation is conductedat a fixed oscillation amplitude A* = 0.79 and frequencyf �e ¼ 0.025, resulting in a frequency ratio fe/fs = 0.17 at

L* = 2.5 and 0.24 at L* = 4, where the frequency fs ofvortex shedding from two stationary cylinders dependson L* [1,17]. A relatively low fe/fs is selected in view ofthe fact that past investigations are frequently conductedon the condition of synchronization, i.e. fe � fs, and thedata of a small fe/fs are scarce. More experimental de-tails are given in Section 2. The results are presentedin Sections 3–5 and concluded in Section 6.

2. Experimental conditions

Experiments were carried out in a close-circuit windtunnel with a 2.4 m long square working section of0.6 m · 0.6 m. The details of the tunnel were given in[7]. The wake was generated by two circular cylindersof the same diameter d = 0.0127 m, arranged in tandem(Fig. 1). The cylinders had an aspect ratio of 17 and werecantilever-supported in the horizontal mid-plane of theworking section, causing a blockage of 2.1%. In orderto reduce the end effects, circular plates with a diameterof 0.12 m and a thickness of 0.002 m, designed followingStansby�s recommendations [18], were mounted at bothends of the cylinders. The longitudinal spacing betweenthe cylinders, L*, is 2.5 and 4, respectively. The upstreamcylinder, made of aluminium, was forced to oscillate

0.001

0.01

0.1

1

10

100

0.001 0.01 0.1 1 10

f*=0.125

fe*=0.025

f*=0.149

E u

0.001

0.01

0.1

1

10

100

0.001 0.01 0.1 1 10

f*=0.125fe*=0.025

f*=0.149

E v

0.001

0.01

0.1

1

10

100

0.001 0.01 0.1 1 10

f*=0.125fe*=0.025

f*=0.149

f*=fd/U∞

E θ

a

b

c

Fig. 2. Spectra of u, v and h measured between the gap of thecylinders (x* = 10, and y* = 1.5) at L* = 2.5: (a) u; (b) v; (c) h.Solid line: oscillating upstream cylinder; dashed line: stationaryupstream cylinder.

4064 G. Xu, Y. Zhou / International Journal of Heat and Mass Transfer 48 (2005) 4062–4072

longitudinally by a computer-controlled motor system,which allowed the accurate adjustment of the oscillationfrequency and amplitude. The downstream cylinder,made of brass, was slightly heated electrically. The maxi-mum mean temperature of the heated wake was about0.8 �C above ambient, implying a negligible buoyancyeffect and heat may be considered to be a passive scalar.Unless otherwise stated, all temperatures in the paperare given with respect to ambient.

Experiments were conducted at Re = 5920 andx* = 10 with and without the upstream cylinder oscillat-ing (A* = 0.79). The coordinates x and y are defined inFig. 1. The oscillation frequency was fixed at f �

e ¼0.025, resulting in a frequency ratio of fe/fs = 0.17 forL* = 2.5 and 0.24 for L* = 4 since fs is dependent onL* [1,17]. The choice of such a low fe/fs ratio allowsthe oscillation effect to be investigated in the absenceof vortex shedding �locked-on� with fe.

A three-wire probe consisting of an X-wire and a coldwire was used to measure the longitudinal velocity (u),transverse velocity (v) and temperature (h) fluctuations.The cold wire was placed orthogonally to the plane ofthe X-wire and about 0.8 mm upstream of the X-wireintersection. The probe was mounted on a traversingmechanism controlled by a personal computer, whichallowed motion in three directions, with a resolutionof 0.1 mm and 0.01 mm in the streamwise and cross-stream directions, respectively. The sensing elements ofthe X-wire were made of 5 lm Pt–10%Rh wire approxi-mately 1 mm in length. For the cold wire, the sensingelement was made of 1.27 lm Pt–10%Rh wire, about1.2 mm in length. The temperature coefficient of the coldwire is 1.69 · 10�3 �C �1, determined using a 10 X plat-inum resistance thermometer operated in a Leed andNorthrup 8087 bridge with a resolution of 0.010 �C.The hot wires were operated on constant temperaturecircuits at an overheat ratio of 1.5. The cold wire wasoperated on a constant current (0.1 mA) circuit withoutput proportional to h. The probe was calibrated inthe free stream for velocity, yaw angle and temperatureusing a Pitot-static tube connected to a Furness micro-manometer and a thermocouple. A full velocity-yawangle calibration scheme [19] was used, with a ±20�yaw angle and a 3.0–11.0 m/s velocity range.

Velocity and temperature signals from the anemome-ter were passed through buck and gain circuits, low-passfiltered and digitised on a personal computer using a12 bit A/D converter at a sampling frequency of2fc = 3500 Hz, where fc is the cut-off frequency deter-mined by checking a spectral analyser. The samplingduration was 30 s. The velocity contamination of thecold wire signal was ignored because of the low operat-ing current (0.1 mA). No correction was applied to theinstantaneous velocity signals because of a small temper-ature excess, below 0.8 �C, at the measurementlocations.

3. Vortex-shedding frequencies

Fig. 2 presents the u-, v- and h-spectra, Eu, Ev and Eh,at L* = 2.5. The spectra in the absence of oscillation dis-play a pronounced peak at f �

s ¼ 0.149, indicating thedominant vortex frequency. The u-spectrum (Fig. 2a)shows a peak at the second harmonic of f �

s , i.e.,f* = 0.298. With the upstream cylinder oscillating, apeak at f �

e ¼ 0.025 appears in the u and v-spectrum, as

0.0001

0.01

1

100

0.001 0.01 0.1 1 10

f*=0.1496fe

*2fe*

fs*=fe*=0.025

E u

0.0001

0.01

1

100

0.001 0.01 0.1 1 10

f*=0.1496fe

*2fe*

fs*=fe*=0.025

f*=fd/U∞

E v

a

b

Fig. 3. Spectra of u and v and measured between the gap of thecylinders (x* = �1.25, y* = 1.5) at L* = 2.5: (a) u; (b) v. Solidline: oscillating upstream cylinder; dashed line: stationaryupstream cylinder.


expected. Furthermore, a number of peaks at the har-monics of f �

e ,up to 5f �e , are discernible in both the u-

and v-spectra, among which the one at f � ¼ 0.125 ¼5f �

e is most pronounced, probably resulting from vortexshedding locked on with the fifth harmonic of f �

e . Thephenomenon that the natural fs is modified or lockedon with the cylinder oscillating frequency has beenwidely reported in the literature. The extent to which fscan be modified depends on the oscillation amplitude[20]; larger amplitude corresponds to a greater maxi-mum modification of fs. In the present context, the dif-ferent harmonics of the oscillation may compete witheach other to lock on fs. The first harmonic is supposedto have the largest oscillation amplitude. However, thisharmonic could not lock on fs because of a large devia-tion between them. Apparently, the harmonic that has aminimum deviation from and also lower than the natu-ral fs (the harmonic that is higher than fs should havesmaller oscillation amplitude) has the greatest potentialto lock on fs. As a result, fs is locked on with 5fe. It isnoted that, with a change in f �

e , the modified vortex-shedding frequency may vary but is always smaller thanf �s . The result suggests an effective method to modifyvortex shedding from a cylinder by oscillating an up-stream cylinder and adjusting its oscillating frequency,whose harmonic may dictate the vortex-sheddingfrequency behind the downstream cylinder.

The h-spectrum, Eh (Fig. 2c), displays a pronouncedpeak at f �

s ¼ 0.149 in the absence of oscillation and anumber of peaks at the harmonics of f �

e once the oscil-lation is introduced. The peak at the fifth harmonicf * = 0.125 is apparently connected to the dominant vor-tices, while others are probably linked to the oscillatingshear layers separated from the upstream heated cylin-der. This can be verified by examining the spectra ofhot-wire signals measured between the cylinders.

In order to understand the relationship between thevortical structures generated by the two cylinders, anX-wire was placed at x* = �1.25 and y* = 1.5 to detectpossible vortex shedding from the upstream cylinder.Because of large flow fluctuations between the cylinders,the u and v components derived from the X-wire werenot so accurate as at x/d = 10. This is however not crit-ical since the signals are used only to determine the dom-inant frequencies. The u- and v- spectra, Eu and Ev (Fig.3), display a sharp peak at f * � 0.149 and a broad peakover f * � 0.4–2 for the stationary upstream cylinder.The former peak occurs at the same frequency as thedominant vortices measured behind the downstream cyl-inder and the latter is ascribed to the instability of shearlayers associated with the upstream cylinder. It has beenestablished that the vortex street is not formed in the gapof two cylinders at L* = 2.5 [2]. In fact, the co-existenceof the two peaks indicates an incomplete formation ofvortices generated by the upstream cylinder [1,17]. Oncethe cylinder is oscillated, Eu and Ev exhibit one pro-

nounced peak at f �e and a few minor ones at the harmon-

ics of f �e , up to 6f �

e , some of which coincide with thoseobserved in Eh. Note that the broad peak overf * � 0.4–2 now disappears and, meanwhile, the peakat f * � 0.149 appears significantly more pronounced,suggesting a complete formation of vortices betweenthe cylinders and enhancement of vortex shedding bythe oscillation. This is supported by the laser-inducedfluorescence (LIF) flow visualization (Fig. 4) conductedat Re = 5920 for L* = 2.5. Two pin holes of 0.5 mmdiameter were drilled at ±45� from the leading stagna-tion point at the mid span of the two cylinders. Smoke,generated from Paraffin oil, of a particle size of about1 lm in diameter was released from the pin hole, thusproviding seeding for flow. A Dantec standardPIV2100 system was used to capture the flow images.More details of the LIF measurements were given in[21]. Vortices are evident between the gap of vorticeswith the upstream cylinder oscillating (Fig. 4b) but arenot observed without (Fig. 4a). It is evident that the up-stream cylinder oscillation leads to a change in the flowregime at L* = 2.5 from one-cylinder shedding to ‘‘co-shedding’’ from both cylinders.

Fig. 4. Photographs of flow visualization at L/d = 2.5 andRe = 5920: (a) without cylinder oscillation; (b) with cylinderoscillation (fe/fs = 0.17, A/d = 0.79).

0.001

0.01

0.1

1

10

100

0.001 0.01 0.1 1 10

a

b

c

f*=0.1124fef *e=* 0.025

E u

0.001

0.01

0.1

1

10

100

0.001 0.01 0.1 1 10

f =0.1124fe*

fe*=0.025

E v

0.001

0.01

0.1

1

10

100

0.001 0.01 0.1 1 10

4fe*3fe

*

fe*=0.025

f*=fd/U∞

E θ

*

Fig. 5. Spectra of u, v and h measured behind the cylinders(x* = 10 and y* = 1.5) at L* = 4: (a) u; (b) v; (c) h. Solid line:oscillating upstream cylinder; dashed line: stationary upstreamcylinder.


At L* = 4, Eu, Ev and Eh at x* = 10 (Fig. 5) all showone pronounced peak at f �

s ¼ 0.112 in the absence ofoscillation, apparently resulting from the dominant vor-tices in the flow. Once the upstream cylinder is oscil-lated, one strong peak occurs at f �

e ¼ 0.025 andmeanwhile the vortex-shedding frequency is reduced to0.1, locked on with the fourth harmonic of f �

e . As inthe case of L* = 2.5, minor peaks at the other harmonicsof f �

e are also discernible in the spectra.Eu and Ev (Fig. 6) measured between the stationary

cylinders appear qualitatively the same as their counter-parts behind the downstream cylinder, exhibiting onepronounced peak at f �

s ¼ 0.118. The observation con-forms to previous reports that the frequencies of vortexshedding from the two cylinders are identical [17,22]. Inthe presence of the upstream cylinder oscillation, how-ever, both Eu and Ev display their most pronouncedpeak at 5f �

e , implying that vortex shedding from theoscillating cylinder is locked on with the fifth harmonic

of f �e , deviating from vortex shedding from the

downstream cylinder, which is locked on with 4f �e

(Fig. 5). The deviation is not well understood at thisstage.

It is noteworthy that the vortex-shedding frequencybehind the downstream cylinder drops as a result ofthe upstream cylinder oscillation, irrespective of the L*

value, changing from 0.149 to 0.125 at L* = 2.5 andfrom 0.112 to 0.1 at L* = 4.

0.0001

0.01

1

100

0.001 0.01 0.1 1 10

3fe* 5fe

*

2fe*

fs*=fe*=0.025

E u

0.0001

0.01

1

100

0.001 0.01 0.1 1 10

b

a

5fe*

2fe*

fs*=fe*=0.025

f*=fd/U∞

E v

Fig. 6. Spectra of u and v measured between the gap of thecylinders (x* = �1.25, y* = 1.5) at L* = 4: (a) u; (b) v. Solid line:oscillating upstream cylinder; dashed line: stationary upstreamcylinder.

0.7

0.8

0.9

1.0

-6 -2 2 6

U*

0

0.2

0.4

0.6

0.8

1.0

-6 -2 2 6

y/d

Θ*

a

b

Fig. 7. Cross-flow distributions of (a) mean velocity U � and (b)temperature H� at L* = 2.5 and x* = 10. Open symbol:stationary upstream cylinder; solid symbol: oscillating upstreamcylinder.


4. Time-averaged velocity and temperature

and Reynolds stresses

Fig. 7 compares the lateral distributions of the meanvelocity U � and temperature H� at L* = 2.5 (x* = 10)with and without the upstream cylinder oscillating.The experimental uncertainties in U � and H� are esti-mated to be 2% and 8%, respectively. The latter is lar-gely caused by the temperature drift in the cold-wiremeasurement. Both U � and H� are reasonably symmet-rical about the flow centreline. The cylinder oscillationcauses an appreciable reduction in the maximum veloc-ity defect but appears to have a negligible effect on H�.Figs. 8 and 9 present the cross-flow distributions ofthe Reynolds stresses ðu�2; v�2; u�v�Þ, temperature vari-ance ðh�2Þ, and heat fluxes ðu�h� and v�h�Þ. The experi-mental uncertainties for u�2; v�2; u�v�; h�2; u�h� and v�h�

are estimated to be about 5%, 5%, 5%, 10%, 8% and8%, respectively. The distributions of u�2; v�2 and h�2

are symmetrical about the flow centerline. So is u�h�.On the other hand, u�v� and v�h� are anti-symmetrical.As a result of oscillation, the u�2 distribution grows

wider, and the maximum v�2; h�2; u�v� and v�h� allincrease appreciably.

The observations are linked to the earlier finding thatthe upstream cylinder oscillation changes the flow fromthe reattachment regime to the co-shedding regime (Sec-tion 3). This change in the initial condition from theshear layer reattachment to the vortex impingement onthe downstream cylinder enhances the cross-flow fluctu-ation, which is supported by the increased v�2 (Fig. 8b).As a consequence, more high-speed fluid is entrainedinto the wake and more slow fluid is transferred towardsthe free stream. The increased mixing causes a growingwidth in u�2 (Fig. 8a) and increased h�2 near the center-line (Fig. 8c), and also an enhancement of the momen-tum and heat transport (Fig. 9a and c). Note that thecross-flow distribution of u�h� (Fig. 9b) in the presenceof oscillation is totally different from that in the absenceof oscillation. This is not unexpected in view of two dif-ferent flow regimes associated with the two cases.

It is worthwhile pointing out that the u�2 distributionexhibits single peak, regardless of oscillation, whereas its

0

0.005

0.010

0.015

0.020

0.025

-6 -2 2 6

a

b

c

u*2

0

0.01

0.02

0.03

0.04

-6 -2 2 6

v*

0

0.1

0.2

0.3

0.4

-6 -2 2 6

y/d

θ*2

2

Fig. 8. Cross-flow distributions of velocity and temperature

variances at L* = 2.5 and x* = 10: (a) u�2; (b) v�2; (c) h�2. Opensymbol: stationary upstream cylinder; solid symbol: oscillatingupstream cylinder.

-0.0050

-0.0025

0

0.0025

0.0050

-6 -2 2 6

u*v*

-0.015

-0.010

-0.005

0

0.005

0.010

-6 -2 2 6

u*θ*

-0.04

-0.02

0

0.02

0.04

-6 -2 2 6

y/d

v*θ*

a

b

c

Fig. 9. Cross-flow distributions of Reynolds shear stress andheat fluxes at L* = 2.5 and x* = 10: (a) u�v�; (b) u�h�; (c) v�h�.Open symbol: stationary upstream cylinder; solid symbol:oscillating upstream cylinder.


counterpart behind an isolated cylinder shows a twin-peak distribution at the same x* [23]. Two causes maybe responsible for the observed difference, that is, vorti-ces are presently weak or lateral spacing between cross-stream vortices is small, compared with a single cylindercase. The wake half-width based on the U � distributionis 1.45d and 1.73d for the two cases of the non-oscilla-tion and oscillation, respectively, significantly largerthan 0.81d for a single cylinder wake at Re = 5830 [8].Furthermore, the peaks in u�v� and v�h� occur at

y/d � ±2, while their counterparts behind a single cylin-der occur at y/d � ±1 [8]. The observations point tounequivocally increased lateral spacing between cross-stream vortices in the wake of two tandem cylinders.Itmay be therefore inferred that the vortices are presentlyrather weak in strength, contributing to the single peaku�2 distribution.

Figs. 10–12 present the cross-flow distributions of

U �;H�; u�2; v�2; h�2; u�v�; u�h� and v�h� at L* = 4, respec-tively. A number of points are noteworthy. Firstly,U �;H�; h�2; u�v� and v�h� are qualitatively similar to

0.6

0.7

0.8

0.9

1.0

-6 -2 2 6

a

b

U*

0

0.2

0.4

0.6

0.8

1.0

-6 -2 2 6

y/d

Θ*

Fig. 10. Cross-flow distributions of mean velocity U � (a) andtemperature H� (b) at L* = 4 and x* = 10. Open symbol:stationary upstream cylinder; solid symbol: oscillating upstreamcylinder.

0

0.005

0.010

0.015

0.020

0.025

-6 -2 2 6

u*2

0

0.01

0.02

0.03

a

b

c

-6 -2 2 6

v*θ*

2

0

0.05

0.10

0.15

0.20

0.25

-6 -2 2 6

y/d

2

Fig. 11. Cross-flow distributions of velocity and temperature

variance at L* = 4.0 and x* = 10: (a) u�2; (b) v�2; (c) h�2. Opensymbol: stationary upstream cylinder; solid symbol: oscillatingupstream cylinder.


those at L* = 2.5. However, unlike their counterparts atL� ¼ 2.5; u�2 and v�2 tend to show a twin-peak distribu-tion, which is probably due to increased lateral spacingbetween cross-stream vortices. Indeed, the mean velocityhalf-width increases slightly from 1.45d at L* = 2.5 to1.52d at L* = 4. Furthermore, compared with the caseof L* = 2.5, the maximum velocity defect is appreciablylarger, but v�2 and h�2 are smaller. The difference is againconnected to the distinct initial conditions. Secondly, theupstream cylinder oscillation produces an effect very dif-ferent from that at L* = 2.5. U � is essentially unchangedin the presence of the oscillation. But there is an appre-ciable decrease in the magnitude of v�2; u�v�; u�h� andv�h�, in marked contrast with the case at L* = 2.5, wherethe quantities all arise in magnitude. The difference inthe quantities at L* = 2.5 is largely attributed to thechange in the flow regime from the reattachment tothe co-shedding once the upstream cylinder is forcedto oscillate. At L* = 4, however, the flow regime remainsthe same with or without the upstream cylinder oscillat-ing; therefore, the oscillation effect on the wake isexpected to be different from the case at L* = 2.5. As

noted in Eu and Ev (Figs. 5 and 6), the predominant vor-tex frequency is 4fe between the cylinders but jumps to5fe behind the downstream cylinder. Presumably, theinteraction between the two types of vortices of differentfrequencies may accelerate the decay of the vortices be-hind the downstream cylinder. As a result, v�2 is consid-erably reduced and subsequently both momentum ðu�v�Þand heat transport ðv�h�Þ impair. It is noted that thedistribution of u�h� is qualitatively unchanged by the

-0.010

-0.005

0

0.005

0.010

-6 -2 2 6

u*v*

-0.02

-0.01

0

-6 -2 2 6

u*θ*

-0.04

-0.02

0

0.02

0.04

-6 -2 2 6

y/d

v*θ*

a

b

c

Fig. 12. Cross-flow distributions of Reynolds shear stress andheat fluxes at L* = 4.0 and x* = 10: (a) u�v�; (b) u�h�; (c) v�h�.Open symbol: stationary upstream cylinder; solid symbol:oscillating upstream cylinder.

0

0.25

0.50

0.75

-3.0 -1.5 0 1.5 3.0

y/d

Prt

Fig. 13. Cross-flow distributions of turbulent Prandtl numberPrt at x* = 10. Circle: L* = 4; square: 2.5. Open symbol:stationary upstream cylinder; solid symbol: oscillating upstreamcylinder.


oscillation. This is reasonable since the flow regime isunaffected at L* = 4 by the oscillation.

5. Turbulent Prandtl number

The turbulent Prandtl number, Prt, defined by [24]

Prt ¼uv=ðoU=oyÞvh=ðoH=oyÞ

ð1Þ

is given in Fig. 13. The mean velocity and temperaturegradients, oU=oy and oH=oy, were obtained from aleast-square sixth-order polynomial fit to U and H. Thismethod of determining Prt is similar to that used by [25].The resulting uncertainty in Prt was estimated from theuncertainties in uv; vh and oH=oU and was found to beabout 17%. Prt is not shown near y* = 0, where a singu-lar point occurs. Prt is in the order of 1 and declinestowards y* = 0, suggesting that heat transport is pro-gressively more effective than momentum transporttowards the wake centreline.

Without the oscillation, Prt is significantly larger atL* = 4 than at L* = 2.5, which correspond to the co-shedding regime and the reattachment regime, respec-tively. Examining U �;H�; u�v� and v�h� (Figs. 7, 9, 10and 12) unveils that, while the difference inoU �=oy; oH�=oy and v�h� is relatively small betweenthe two regimes, u�v� in the reattachment regime is onlyabout one fourth that in the co-shedding regime. Theobservation suggests a substantially more effectivemomentum transport in the co-shedding regime thanin the reattachment regime, which accounts for thehigher Prt at L* = 4.

The oscillation effect is opposite for the two cases; Prtdrops at L* = 4 but goes up at L* = 2.5. At L* = 2.5, theupstream cylinder oscillation gives rise to a switch fromthe flow reattachment regime to the co-shedding regime.This regime change is responsible for rising Prt. In fact,it can be inferred from Figs. 7 and 9 that the oscillationcauses little variation in oU �=oy or oH�=oy and an in-crease in the v�h� magnitude by only 17%. However,u�v� climbs more than 100% in magnitude because ofthe regime change, overwhelming other effects. AtL* = 4, with the flow regime unchanged, the oscillationhas essentially no influence on U � and H� or oU �=oyand oH�=oy, but reduces u�v� by about 40% and v�h�


by 30%, thus resulting in a decline in Prt. It may be con-cluded that the momentum transport is more liable tothe oscillation effect than the heat transport.

6. Conclusions

The turbulent cylinder wake behind a longitudinallyoscillating cylinder has been investigated. The investiga-tion is conducted at large oscillating amplitude(A* = 0.79) and a low frequency ratio (fe/fs = 0.17 forL* = 2.5 and 0.24 for L* = 4), leading to followingconclusions:

1. For all L* investigated, the frequency of vortex shed-ding from the stationary cylinder is found to be mod-ified by the upstream cylinder oscillation; it is lockedon with one harmonic of the oscillating frequency(fe), which is 5fe at L* = 2.5 and 4fe at L* = 4. Thisharmonic frequency is nearest to and below that ofnatural vortex shedding. In the co-shedding flowregime behind two stationary cylinders, the predom-inant vortex frequency in the gap of the cylinders isidentical to that behind the downstream cylinder.However, the upstream cylinder oscillation maycause the two frequencies to slightly deviate fromeach other, which has yet to be better understoodin future investigations.

2. The oscillation effect on the flow depends on L*.While the flow regime at L* = 4 remains unchangedby the upstream cylinder oscillation, that atL* = 2.5 is altered from the reattachment to the co-shedding regime. As a result, the oscillation gives riseto a decrease in the velocity defect at L* = 2.5 but nochange at L* = 4. Furthermore, v�2; u�v�; u�h� andv�h� is weakened in strength at L* = 4, but increasedat L* = 2.5.

3. The upstream cylinder oscillation has an appreciableeffect on turbulent Prandtl number Prt. It is foundthat the momentum transport is significantly moreeffective in the co-shedding regime than in the reat-tachment regime, resulting in a larger Prt at L* = 4than at L* = 2.5. Therefore, with the upstream cylin-der oscillating, the flow regime change from reattach-ment to co-shedding at L* = 2.5 is associated with arise in Prt. On the other hand, at L* = 4 the oscilla-tion does not change the flow regime and adverselyaffects momentum transport more than heat trans-port, contributing to a smaller Prt.

Acknowledgements

Authors wish to acknowledge support given to themby the Central Research Grant of The Hong Kong Poly-technic University through Grant G-YW74.

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momentum and heat transfer in a turbulent cylinder wake behind a streamwise oscillating cylinder

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