On interference between two circular cylinders in staggered arrangement at high subcritical Reynolds numbers

*Corresponding author.

Journal of Wind Engineeringand Industrial Aerodynamics 80 (1999) 287—309

On interference between two circular cylinders in staggeredarrangement at high subcritical Reynolds numbers

Zhifu Gu*, Tianfeng SunDepartment of Mechanics and Engineering Science, State Key Laboratory for Turbulence Research,

Peking University, Beijing 100871, People+s Republic of China

Received 18 April 1998; received in revised form 15 June 1998; accepted 19 June 1998

Abstract

The results of a wind-tunnel investigation on the interference between two identical parallelcircular cylinders arranged in staggered configurations, carried out in a uniform smooth flow athigh subcritical Reynolds number, are presented. On the basis of the pressure measurementsand the flow visualization, three main flow patterns are classified and discussed. Close attentionis paid to the switching phenomenon of two different pressure patterns at critical angles. Thisswitching creates discontinuity of large lift forces on cylinders. With measurements of velocityprofiles and power spectral analysis in the gap between the two cylinders, the mechanism of theappearance of this extreme aerodynamic force is discussed. ( 1999 Elsevier Science Ltd.All rights reserved.

Keywords: Two circular cylinders; Staggered arrangements; Interference; Flow patterns

1. Introduction

In engineering practice, buildings and structures often appear in groups. Windloading on structures is one of the major factors which should be considered in design.It is well known that wind loading on members in a group may be quite different fromthat on isolated ones (e.g. see [1]). Because of the mutual interference, flow arounda group of structures may exhibit many interesting and unexpected phenomena. Thisgroup effect then attracts the attention of many engineers and researchers.

The circular cylinder is one of the popular forms used in engineering structures.Groups of circular cylinders exist in many different kinds of engineering, e.g. heatexchanger tube banks, cable bundles, reaction towers in the chemical industry, cooling

0167-6105/99/$ — see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 2 0 5 - 0

Fig. 1. Schematic diagram showing the arrangement of the two parallel cylinders and the definition of forcecoefficients.

towers, groups of chimneys, under-structure of offshore platforms, etc. Flows aroundtwo circular cylinders, which is the simplest case of group of structures, have beenstudied by many investigators. Reviews were given by Zdravkovich [2] and recentlyby Ohya et al. [3]. Many published results, however, were usually restricted to twocylinders in tandem or in a side-by-side arrangement at a Reynolds number of theorder of 104.

Two cylinders in staggered arrangements (see Fig. 1) are the cases most often foundin practice but seem to be the least studied. Hori [4] measured the pressure distribu-tions (also the vortex shedding frequency and wake velocity profiles) on one of the twocylinders at a Reynolds number Re"8]103 for gap ratios N/d equal to 1.2, 2.0 and3.0. He rotated the cylinders in such a way that the angle of incidence b varied from 0°to 180° and obtained results for both upstream and downstream cylinders. He foundthe lift and drag forces by integrating the pressure distributions. Suzuki et al. [5]measured the pressure distribution on the downstream cylinder at Re"2.3]105 forN/d from 1.05 to 3.85 and angle of incidence b"0°, 5°, 10° and 15°. Their resultsshowed the appearance of large negative pressures on the axis side of the wake and thesudden change of the lift forces for certain combinations of N/d and b. Zdravkovichand Priden [6] measured systematically the forces acting on the downstream cylinderwithin the range of ¸/d from 1—5, ¹/d from 1—3 at Re"6]104, and gave constantC

Dand C

Llines on the ¹/d and ¸/d plot. They also pointed out the phenomenon of

discontinuities or switching of the two pressure distributions which changed overintermittently at irregular time intervals. They called it the “gap-flow-switch” andgave an example in 1977 and another one in 1988. No more detailed investigationshave been carried out since then.

In recent years, the aerodynamic characteristics of two circular cylinders at highReynolds numbers, including the high subcritical and supercritical regions, have beenstudied systematically at the Peking University (see, e.g. [12,13]). The investigationstarted with the measurement of pressure distributions on the upstream and down-stream cylinders simultaneously. The results of interference on two cylinders at

288 Z. Gu, T. Sun/J. Wind Eng. Ind. Aerodyn. 80 (1999) 287–309

supercritical Reynolds number, have been reported by Gu [7] recently. This papergives the results of time-mean pressures and forces obtained for two cylinders ina staggered arrangement at high subcritical Reynolds number. With the results of flowvisualization, the flows are classified into three flow patterns. Particular attention ispaid to the study of how and why the discontinuities or switching processes of the twopressure distributions occur. With the aid of velocity measurement in the gap region,the mechanism of appearance of large negative pressure on the gap-side surface of thedownstream cylinder is discussed. It is shown that the switching processes may occuron both downstream and upstream cylinders at a certain angle of incidence of theflow, depending on the combination of gap ratio and Reynolds number.

2. Experimental apparatus and data reduction

The experiments were conducted in a closed-return low-speed wind tunnel atthe Peking University. The tunnel has an open circular test section 2.25 m in dia-meter and 3.65 m long. The maximum speed is 50 m/s with a turbulent intensityaround 0.2%.

The circular cylinders tested were made from seamless steel tubes with a machine-finished surface. They were identical in size (157 mm in diameter and 1 m long). Therelevant aspect ratio was equal to 6.4 and the wind tunnel blockage was 8% of thecross-sectional area of the test section. Pressure taps were installed circumferentiallyevery 10° around the cylinders at the mid-span. The cylinders, with circular endplates of 1.5 m diameter at both ends, were mounted horizontally in two adjustableguide-slots which made the adjustment of the gap ratio of the cylinders easy. Thecylinder-group was capable of being rotated to any angle relative to the winddirection. This system was fixed on two columns which were rigidly attached to theground floor.

The measurement system for the surface pressure consisted of pressure transducers(PDCR-23d), a set of Scanivalves (SGM-48), two DC amplifiers (6M72), an A/Dconverter and a personal computer (IBM PC/XT). Two individual transducers wereused for each cylinder so that the pressure signals at the same azimuth angle h, (seeFig. 1) of both cylinders under test were recorded simultaneously. A typical run tookabout 60 s to complete the data-logging of all pressure taps for both cylinders. In thepresent test, only time-averaged pressures were measured in a low-turbulence uniformstream.

Fig. 1 is a schematic diagram showing the arrangement of the two parallel cylindersand the definition of force coefficients. Here d is the diameter of the cylinder, N thedistance between the centers of the two cylinders, and N/d the spacing ratio betweenthe two cylinders. The angle of wind direction b is defined as the angle between thedirection of the oncoming flow and the line connecting the centers of the twocylinders. As frequent reference will be made to the individual cylinders, the upstreamand the downstream cylinders are labelled A and B, respectively, in the followingdiscussion. The pressure coefficient C

p(h) is defined as C

p(h)"[ p(h)!p

=]/0.5o»2

=,

where h is the azimuth angle measured from the wind direction, positive clockwise,

Z. Gu, T. Sun/J. Wind Eng. Ind. Aerodyn. 80 (1999) 287–309 289

p(h) is the pressure measured on the surface of cylinder, and p=, o, »

=are respectively

the pressure, density and velocity of the oncoming flow. The Reynolds number Re isdefined as o»

=d/k, where k is the absolute coefficient of viscosity of air. In most cases,

the pressure measurements were conducted at a Reynolds number of 2.2]105.However, in some cases they were conducted at Re"3.3]105. No correction for theblockage effect has been made to the pressure data presented in this paper. It was feltthat the blockage effect was small, especially in the open test section, compared to thelarge flow interferences that were measured.

The drag and lift coefficients, denoted by CD

and CL, respectively, are defined

conventionally and are obtained by integrating appropriately the pressure distribu-tion around the circumference of the cylinder at mid span.

In some cases of interest, measurements of instantaneous velocity as a function oftime in the gap area between two cylinders were made by a hot-wire anemometer(TSI-1050). The analog output signals of the anemometer were linearized before beingrecorded on a tape recorder (TEAC-XR23). Later the signals on the tape wereanalyzed by a computer to obtain the real velocity profile, as well as the fluctuatingvelocity and its power spectrum.

3. Results and discussion

3.1. Single circular cylinder at high subcritical Reynolds number

It is well-known that Reynolds number plays an important role in the featuresof the flow around a circular cylinder. Recently, the variations with Reynoldsnumber in the range 101—107 of drag and base suction of circular cylinderswere discussed by Roshko [8], with an emphasis on the importance of the separatedlayers.

In order to perceive the effect of group interference, the pressure distribution fora single cylinder was measured first, and was then compared with the pressuredistribution for each of the two cylinders.

The time-mean pressure distributions on a single cylinder at four typical Reynoldsnumbers, corresponding to the high subcritical (Re"2.2]105, Re"3.3]105), criti-cal (Re"3.9]105) and supercritical (Re"4.5]105) regions respectively, are shownin Fig. 2. For both Re"2.2]105 and Re"3.3]105, the pressure distributions onboth sides of the cylinder are quite symmetrical and show typical laminar separationsat about h"$80°. However, the values of base pressure, C

1", at these two Reynolds

numbers are quite different. The value of C1"

for Re"3.3]105 is reduced a great dealand the drag coefficient of the cylinder drops significantly. A hot wire probe wasplaced in the near wake behind the cylinder. The signals of the flow velocity weresampled and analyzed. A strong peak frequency was found in the power spectra,which corresponded to a Strouhal number S

5(defined as S

5"fd/»

=, where f is the

frequency of the dominant peak in the power spectrum of the velocity) of 0.19. Itsuggests that regular vortex shedding still occurs in the high subcritical region. Thevalues of drag coefficients C

D, for various Reynolds numbers, are shown in Fig. 3. It


Fig. 2. Time-mean pressure distributions on a single cylinder at four typical Reynolds numbers, corre-sponding to high subcritical (Re"2.2]105, Re"3.3]105), critical (Re"3.9]105) and supercritical(Re"4.5]105) regions.

seems that a Reynolds number of 3.3]105 is close to the beginning of the criticalregion. At the critical Reynolds number of 3.9]105, turbulent separation occurs onone side of the cylinder. The local minimum pressure coefficient C

1.*/reaches the

value of !3.0 and then rises with a small “step-like” curve, which indicates thata laminar separation “bubble” exists. On the other side of the cylinder, the flowmaintains laminar separation with a rather high base pressure C

1"of !0.5. At

Re"4.5]105, a typical supercritical flow pattern is present. The results of the presenttest agree well with the results given by Bearman [9] and Farell [10], in which moredetailed experiments on vortex shedding, as well as formation of separation bubbleswere reported and discussed, though the values of critical Reynolds number aresomewhat different from the present test, due to the different conditions of theexperiments, such as surface roughness of the cylinder and turbulence intensity inoncoming flow. As described by Bearman [9], there is a discontinuity recorded atRe"3.4]105 (rather than Re"3.9]105 in this experiment) in the plot of thevariation of base pressure coefficient with Reynolds number. He suggested that thiswas caused by a laminar bubble forming on only one side of the cylinder. Two highsubcritical Reynolds number Re, namely 2.2]105 and 3.3]105, were used for theexperiments on two circular cylinders in this paper.


Fig. 3. Variation of drag coefficients with Reynolds number for the single circular cylinder in the presenttest.

3.2. Classification of pressure patterns on two circular cylinders in staggeredarrangements: N/d"1.7

The case of N/d"1.7 and Re"2.2]105 is discussed first. The pressure distribu-tions, i.e. C

pversus h curves, for different b on cylinders A and B are shown in Fig. 4.

For b"0° the two cylinders are in tandem and all pressures on cylinder B arenegative. On the front part of cylinder B, around h"0°, the C

phas its minimum value

C1.*/

of !0.9 approximately, and is nearly equal to or slightly less than the backpressure coefficient C

18of cylinder A. There are two local peaks on either side of

cylinder B, where the local maximum values, C1.!9

, occur. The azimuth angles h., for

C1.!9

are in the ranges 65 to 70° and !65 to !70°, respectively. This result suggeststhat the flow within the region between the cylinders is somewhat “stagnant” and thatthe shear layers separating from cylinder A act directly on the upwind surface ofcylinder B.

For b"5°, all the values of Cp

for cylinder B are still negative. However, Cp, as

a function of h, has only one peak value which is located on the outer side, withh.

equal to 60°. On the other hand, Cp

attains a local minimum value near h"90°.The C

pvalues on the gap side and the rear (h from !100° to !180° and from 120° to


Fig. 4. Pressure distribution coefficients CP

versus h for different b, on cylinders A (pressure patterns IA)

and B (pressure patterns IB, II

Band III

B) for Re"2.2]105, N/d"1.7.

180°) have a practically constant value of Cp+!0.2. As the values of b increase, the

peak values of Cpon cylinder B also increase and may become positive. The minimum

pressure C1.*/

occurring at h"90° decreases monotonically. This type of pressuredistribution will be denoted by pattern I

Bfor future reference. Pattern I

Bwas observed

up to b"9.65°.However, at b equal to 9.7°, the pattern of pressure distribution on cylinder

B changes abruptly. The maximum pressure coefficient C1.!9

attains a value of unityat h

."30° and a large area of suction is developed on the gap-side surface, where the

C1.*/

is equal to a value of about !2.0. The back pressure C18

is nearly constant andequal to !0.30 for h from !150° to !180° and from 70° to 180°. This pressuredistribution will be denoted by pattern II

B, and the angle of b at which pattern II

Bfirst

appears will be denoted by CB1

. The change of pressure pattern from IB

to IIB, or vice

versa, at !B1

is not simply discontinuous, but may switch back and forth. A moredetailed discussion on the mechanism of generation of such a large suction on thegap-side surface of the downstream cylinder will be given later.

Pattern IIB

is present up to an angle b equal to 15°. The large area of suction thendisappears abruptly at b"16°, where the pressure distributions on both sides ofcylinder B appear to have a laminar-like separation. It should be mentioned here thatthe second sudden change in pressure distributions has not been reported previously.The location of h

1at which C

1.*/occurred are nearly equal to #70° and !50°. The

separation points h8

are located approximately at h"$80°. The pressure coeffi-cients C

18in the ranges !80°5h5!180° and 180°5h580° are nearly equal to

a constant value of !0.5. This type of pressure distribution is denoted by pattern IIIB

and the angle where IIB

changes to IIIB

is denoted by CB2

, which have not been


determined exactly here in this case. As the values of b further increase, the pressure oncylinder B maintains the pattern III

B, except that the angles of h

1, h

8and the

coefficient C18

shift. The h.

moves from 20° (b"16°) to 0° in the neighbourhood ofb"45°, and then becomes negative (located on the gap side) up to !20° in theside-by-side arrangement (b"90°).

When b increases from 0° to 90°, the pressure distribution of laminar-separationtype is retained on both sides of cylinder A, though the values of C

18are varying and

h.

moves toward the gap side gradually as b'45°. All these types of pressuredistributions on cylinder A will be denoted by pattern I

A. The general features of

IA

and IIIB

are about the same, though their details are quite different. At b"90°, theC

18on cylinder A is equal to !0.4 and on cylinder B, equal to !0.75. From

established results in the literature, biased flow results.

3.3. Flow visualization: classification of flow patterns N/d"1.5—2.0

In order to explain these interesting phenomena, flow visualizations were conduc-ted later in a smaller wind tunnel with the smoke-wire technique. The Reynoldsnumber used was 5.6]103 which was 50 times less than the Reynolds number used inthe pressure measurements.

It is believed that the aerodynamic characteristics of a single circular cylinder, suchas the vortex shedding frequency, the separation points, the shear layers and the wakestructures, are similar in the region of subcritical Reynolds number. The interferenceeffect between two cylinders, on the other hand, is mostly caused by the interactionsamong these characteristics. Therefore, if the experimental flow visualizations areconducted still in the region of subcritical Reynolds number, it is reasonable tosuppose that some valuable information will be obtained from the results to explainwhy the pressure patterns change abruptly on the downstream cylinder at certainangles of attack, and why the large suction areas are created on the gap-side surface ofdownstream cylinder, although the influence of the difference in Reynolds numbershould be considered.

Four smoke wires were used in the front and in the rear of each cylinder respective-ly. The pictures were taken with a time-delay, between the generation of smoke andthe flash of the camera, ranging from 60 to 100 ms. The experiments were conductedwith three different gap ratios for the two cylinders, namely N/d"1.5, 1.7 and 2.0. Asan example, several pictures for N/d"1.7 are shown in Fig. 5 and discussed here.

In general, three remarkably different pictures of the main flow patterns wereobtained, corresponding to the three pressure patterns on the downstream cylindermentioned in the previous section.

Due to the basic instability of the shear layers separated from the upstreamcylinder, two quite different pictures of the flow were obtained for N/d"1.7 andb"5°. However, in both pictures, only the separated shear layer from the gap side ofthe upstream cylinder may reach or touch the windward surface of the downstreamcylinder. It results in only one peak in the pressure distribution on the downstreamcylinder. In both cases, the downstream cylinder is submerged completely in the wakeof the upstream cylinder. The time-mean pressure distributions on the downstream


Fig. 5. Photograph of flow visualization with smoke-wire technique for two circular cylinders in staggeredarrangements (Re"5.6]103, N/d"1.7). (a) N/d"1.7, b"5°(I), Pattern I

B. (b) N/d"1.7, b"5°(II),

Pattern IB. (c) N/d"1.7, b"10°(I), Pattern I

B. (d) N/d"1.7, b"10°(II), Pattern II

B. (e) N/d"1.7,

b"15°(I), Pattern IIB. (f) N/d"1.7, b"15°(II), Pattern III

B. (g) N/d"1.7, b"25°(I), Pattern III

B.

(h) N/d"1.7, b"25°(II), Pattern IIIB. (i) N/d"1.7, b"45°, Pattern III

B.

cylinder are the same and quite stable. Therefore, the pressure pattern IB

prevails ondownstream cylinder and there is no steady stagnation point on it. It should bementioned here that the averaged time for pressure measurement (about 60 s) is muchlonger than the exposure time (about 60 ms) for the pictures which were taken bymeans of smoke-wire technique.

Two pictures of different flow patterns which correspond to pressure patternsIB

and IIB, respectively, were obtained for b"10°. It is evident that in the case of


pattern IB, cylinder B is completely (for b"5°) or partially (for b"10° (I)) submerged

in the wake of cylinder A. The fluctuating amplitude of the smoke generated from thesecond wire is large. It means that the intensity of turbulence behind cylinder A islarge and coincides with the results of flow velocity measurements in the gap region(see Section 3.7). On the other hand, while the pattern II

Bprevails on cylinder B, as

indicated by Zdravkovich [11], the downstream cylinder is not submerged in thewake of the upstream cylinder but “displaces” the wake and in doing so “squeezes” thestreamlines on its gap side. However, from the pictures shown in Fig. 5, further preciseinformation can be found (see b"10° (II) and b"15° (I)). It is evident that the shearlayer separated from the gap side of cylinder A is not only seriously distorted, but alsoits high speed side reattaches onto the front part of cylinder B and then sweeps alongthe gap-side of cylinder B at rather high velocity. It is well-known that the reattach-ment of a shear layer on a surface usually causes the surface pressure to increase. Theterm “reattachment” normally means that the separated shear layer with its inner side(low-speed side) attaches onto the surface again. However, in this case, the separatedshear layer with its outer side (high-speed side) reattaches onto the surface of thedownstream cylinder. Thus, a much stronger surface suction in the area of thereattachment is expected. In combination with the results of the gap flow profilemeasurements presented in Section 3.7, the mechanism of the appearance of a largesuction area on the gap-side surface of the downstream cylinder, could be interpretedas the results of the shear layer separated from the gap-side surface of the upstreamcylinder with its high speed side “reattaching onto” and “sweeping along” the gap-sidesurface of the downstream cylinder, as well as the “passage” of the regular shedding athigh-frequency and high-speed (see also the power spectra of the velocity measured inthe gap region and presented in Section 3.7). Two pictures of flow patterns whichcorrespond to the pressure patterns II

Band III

Bwere also obtained at the same angle

of b"15° as shown in Fig. 5. When pattern IIIB

prevails, the shear layer separatedfrom cylinder A no longer reattaches onto cylinder B. Vortices are formed away fromthe gap side of the downstream cylinder. Thus, the suction pressure on the gap-sidesurface of cylinder B reduces greatly. Individual wake regions are formed behind eachcylinder, though the interference in the adjacent boundaries between the two regionsstill exists. The wake regions of the two cylinders become deformed (see the cases ofN/d"1.7, b"25° and N/d"1.7, b"45° in Fig. 5). This means that the gap flow,which comes from the oncoming flow, is established between the shear layers whichseparated from the gap-sides of both cylinders. It is evident that the wake width ofcylinder A is always narrower than that of cylinder B due to the biasing of the gap flowbetween the cylinders when pattern III

Bprevails.

For smaller and larger spacing ratios, i.e. N/d"1.5 and 2.0, similar flow visualiz-ation pictures are obtained, though the locations of C

B1and C

B2are different. The

values of CB1

and CB2

are also different from those obtained from the pressuremeasurements described previously, due to the Reynolds-number effect.

It is quite surprising that the changes of the flow patterns match quite well withthose of the pressure patterns on the downstream cylinders, though the values ofCB1

and CB2

are different in the results obtained from the flow visualization and fromthe pressure measurements. It suggests that the discontinuity of pressure changes on


Fig. 6. Sketches of the classification of three flow patterns of two circular cylinders in staggered arrange-ments (S: separation point; P: stagnation point; R: reattachment point; T: shear layer touch point).


the downstream cylinder is not sensitive to the difference in the Reynolds number, andpersists through the entire subcritical region though the details of the process may bedifferent.

Furthermore, to summarize the results of the flow visualizations and of the changesof pressure pattern which were discussed in the previous section, three quite differentflow patterns in essence can be classified and sketched as shown in Fig. 6.

For the first pattern (b(CB1

), the downstream cylinder is completely or partially“submerged” in the wake of the upstream cylinder and the pressure pattern I

Bon

cylinder B results. The shear layer separated from the gap side of the upstreamcylinder touches (but does not reattach onto) the windward surface of the down-stream cylinder, whereas the shear layer separated from the outer side of the upstreamcylinder cannot reach the downstream cylinder any longer. Thus, there is no steadystagnation point on the downstream cylinder. This kind of flow pattern, with respectto the downstream cylinder, could be classified as wake interference. In the secondpattern (C

B1(b(C

B2), the shear layer separated from the gap side of the upstream

cylinder reattaches onto and then separates again from the gap-side surface of thedownstream cylinder. The location of the separation point on the gap-side of thedownstream cylinder (h

8"150°) is rather far back. Therefore, a large suction area is

created and pressure pattern IIB

on cylinder B results. The oncoming flow, which isadjacent to the outer side of the shear layer separated from the gap side of theupstream cylinder, reaches the windward surface of the downstream cylinder directly.A steady stagnation point, with a value of unity on cylinder B, is established. On thisoccasion, since the shear layers separated from the upstream cylinder play an impor-tant role in the interference effect this pattern is called shear layer interference. The lastone (b'C

B2), is denoted by neighbourhood interference because only the wake

shapes of both cylinders are deformed. The individual wake regions of each cylinder,as well as the gap flow which comes from the oncoming flow, are established.

On the basis of this understanding of the main characteristics of the three flowpatterns for two circular cylinders in staggered arrangements, the following sectionsdiscuss the feature of this unique switching process in several ways.

3.4. Effect of Reynolds number: N/d"1.7

For higher Reynolds number Re"3.3]105 and the same gap ratio N/d"1.7, thepressure distributions on cylinders A and B are given in Fig. 7. For b"0°, the C

pon

the front part of cylinder B is equal to the C18

of cylinder A. It is similar to the case ofRe"2.2]105, although the magnitudes increase to a value of about !0.75. For anangle b ranging from 0° to 9.75°, patterns I

Band I

Aexist on the cylinders B and A,

respectively. However if b is kept equal to 9.75° and some kind of disturbance isintroduced into the stream, the pressure pattern I

Bon cylinder B may change to

pattern IIB. At the same time, the C

pdistribution on the gap side of cylinder A also

changes abruptly around h"90° to a minimum pressure coefficient C1.*/

of !3.1,which is the typical value for a turbulent separation. On the outer side of cylinder A,the C

pdistribution remains of a type similar to that for a laminar separation. The

cylinder A has the same C18

value of !0.5 as for cylinder B. Such a pressure


Fig. 7. Pressure coefficients Cp

versus h for different b, on cylinders A (pressure patterns IA

and IIA) and

B (pressure patterns IB, II

Band III

B) for Re"3.3]105, N/d"1.7.

distribution on cylinder A will be denoted by pattern IIA

and the angle at whichpattern II

Astarts to appear by C

A1(C

A1may not be equal to C

B1in other cases in

general).Similar to the previous case of Re"2.2]105, a large suction area on the gap side of

cylinder B disappears when the angle CB2

is somewhere between 23° to 24°. Thenpattern III

Bprevails as the angle b increase up to 90°. For cylinder A, the pressure

pattern IIA

persists to a much larger value of b, say up to 58° and at b"60°, thepressure distributions return to pattern I

A. This b angle will be denoted by !

A2.

Nevertheless, the process of disappearance of the large suction area on the gap side ofcylinder A seems to proceed gradually in contrast with that of !

B2. Thus, no definite

value of CA2

can be found in the test. For b'60°, pattern IA

prevails. At b"90°, theC

18of both cylinders are unstable although their averaged values seem to be the same.

No biased flow was expected or observed.

3.5. Effects of smaller gap ratio: N/d"1.5

Now the case of N/d"1.5 and Re"2.2]105 will be considered. The pressuredistributions on cylinders A and B for various b’s are shown in Fig. 8. Three differentpressure patterns I

B, II

Band III

Bappear on cylinder B as they do in the case of

N/d"1.7. Now the CB1

is between 8.5° and 8.7°, and CB2

is equal to 16.5° and theC

p!h curve on the gap-side surface reveals that the switching process is going on.

For cylinder A, pattern IA

prevails for the angle b ranging from 0° upto 25.2°. Atb"25.5°, the pattern II

Aappears. This suggests that the angle C

A1lies somewhere

between 25.2° and 25.5°. Also, pattern IIA

persists up to CA2

which lies somewherebetween 40° and 45°. The effect of the gap ratio N/d is obvious here. In Section 3.4, it


Fig. 8. Pressure coefficients Cp

versus h for different b, on cylinders A (pressure patterns IA

and IIA) and

B (pressure patterns IB, II

Band III

B) for Re"2.2]105, N/d"1.5.

Fig. 9. The bistable biased pressure distributions on both cylinders for Re"2.2]105, N/d"1.5 andb"80°.

is shown that the appearance of IIA

is possible only when the Reynolds numberincreases from 2.2]105 to 3.3]105. However in the present case, when the gap ratioN/d is reduced from 1.7 to 1.5, the pressure pattern II

Ais observed even for

Re"2.2]105, although CA1

is much larger than CB1

.In the side-by-side arrangement (b"90°), the C

pversus h curves for cylinders A and

B show that their back pressures C18

are not equal. From established results given bymany investigators, it means that a biased flow exists and switching of the flow fromone side to another is possible. One interesting result is that, at b"80° in the presentcase, on the C

pversus h curves the values of the C

18of cylinders A and B are also not

equal and they interchange in another set of tests (Fig. 9). It indicates that the


switching process of bistable biased flow for the two cylinders appears not only in theside-by-side arrangement (b"90°), but also in a staggered arrangement in theneighbourhood of the side-by-side arrangement. These different C

18for cylinders

A and B in the neighbourhood of b"90°, happen in most cases in the present tests.For the case of Re"3.3]105 (the pressure distributions are not given here) the

patterns IIB

and IIA

first appear at the same angle b, i.e. CB1"C

A1"9°—10°. The

pattern IIB

disappears at CB2

which lies between 23° and 24° and the pattern IIA

atCA2

between 45°and 60°. Remarks about the process of disappearance of the patternII

Ain the case of N/d"1.7 apply also to this case.

3.6. Remarks on the switching process of pressure patterns

It should be emphasized that the change of pressure pattern from one to another isnot just an abrupt change but also a “switching process” which may be randomlychanged back and forth at the critical angle. The time interval for the switch variesgreatly for different cases. In one case, it could observed within 60 s which was thetime required for data logging for pressure scanning and recording for one test run.The recorded pressure data is a composite of two patterns (e.g. N/d"1.5,Re"2.2]105, C

B2"16.5°). In other cases, the switch has to be observed by intro-

ducing some kind of disturbance into the stream, such as increasing the test speeda few meters per second and then decreasing it to the original one (e.g. N/d"1.7,Re"3.3]105, C

B1"9.75°). In the situations above, the critical angles are the angles

at which the switch has been observed. Quite often, the change of these patterns can beobserved only by trails within a small range of b which are also taken as the boundaryvalues of the critical angle C. Due to the limited testing time, in fact, many values ofC presented here have not been satisfactorily determined, but only as a range of b’s,especially C

A2.

On the other hand, for some values of N/d, the switching process occurs not just atone angle b but within a small range of b’s. For example, in the case of N/d"2.0 andRe"2.2]105, the value of C

B1extends from 7.4° to 7.6° and for C

B2from 14° to 14.3°.

Two pressure patterns IB, and II

Bat b"7.4° are given in Fig. 10a and pressure

patterns IIB

and IIIB

at b"14.3° in Fig. 10b, respectively. It may be noted that inFig. 10b, the pressure distribution on cylinder B for run 1 is in the process of switchingfrom pattern II

Bto pattern III

B. For the same case but different runs, the pressure

distribution on cylinder A kept the type of pattern IA

but the back pressure couldchange a lot, as shown in Fig. 10a.

Comparing this with the results of the flow visualization, it is believed that themechanism of the switching process is mainly due to the geometry of the gap betweenthe cylinders and the basic instability that exists between the shear layers separatedfrom the upstream cylinder.

3.7. Velocity profiles and spectra in the gap area of two cylinders: N/d"2.0

Velocity surveys across the gap area between the two cylinders were made forcertain cases by means of a hot-wire anemometer. Fig. 11 gives the results for the case


Fig. 10. In the critical region, two pressure patterns exist for Re"2.2]105, N/d"2.0 (a) pressure patternsIB

and IIB

at b"7.4° and (b) pressure patterns IIB

and IIIB

at b"14.3°.

Fig. 11. Profiles of mean-velocity »/»=

and of turbulence intensity I6

along the line BB between twocylinders which corresponds to pressure patterns I

B(b"7°) and II

B(b"8°) prevailing on cylinder

B respectively (Re"2.2]105, N/d"2.0). The line BB is perpendicular to the direction of oncoming flowand is located 10 mm (i.e. 0.064d) in front of cylinder B (at h"0°), which is the origin of ½-axis, positiveupward.

of N/d"2.0 and Re"2.2]105. Here the profiles of velocity »/»=

and of turbulenceintensity I

6along line BB, when pressure patterns I

B(b"7°) and II

B(b"8°) prevail,

are given. The line BB is perpendicular to the direction of the oncoming flow and islocated 10 mm (i.e. 0.064d) in front of cylinder B (at h"0°), which is the origin ofY-axis, positive upward. The measurement shows that for pattern I

B, the time-mean

velocity »/»=

decreases from the maximum value of 0.8 near ½/d"0.2—0.4 at½/d"!0.4, while the turbulence intensity I

6increases from 30% to 48%. When


Fig. 12. Profiles of mean-velocity »/»=

and of turbulence intensity I6

along the line CC between twocylinders which corresponds to pressure patterns II

B(b"27°) prevailing on cylinder A (Re"2.2]105,

N/d"1.5). The line CC is perpendicular to the direction of oncoming flow and is located midway betweencylinders A and B.

pattern IIBprevails, flow with high time-mean velocity of a magnitude of about 1.5»

=appears at the front part of cylinder B for a certain region, say 0.1 or 0.2d wide. Thenthe time-mean velocity decreases rather rapidly as ½ decreases. The reverse is true forthe turbulence intensity I

6.

On the other hand, in the case of N/d"1.5 and Re"2.2]105, when patterns IIA

and IIIB

appear on cylinders A and B respectively (see Section 3.5), a much wider flowregion with velocity greater than »

=occurs in the gap (see Fig. 12, the line CC is

located at the midway between cylinders A and B). Meanwhile, the turbulenceintensity I

6is small up to ½/d"!0.5 and then increases suddenly to 40% or larger

where the wake region of cylinder A is supposed to exist.Several examples of power spectra of the flow velocity in the gap (located at the

midpoint of the center line between cylinders) are shown in Fig. 13, all for N/d"2.0and Re"2.2]105. At b"7.3° (before C

B1), there is a peak frequency of 18.5 Hz,

corresponding to a Strouhal number S5of 0.15 which is less than that of an isolated

cylinder. At b"7.7° and 13°, when a large suction area is present on the gap-side ofcylinder B (pattern II

B), the peak frequency increases to a much larger value of 55 Hz,

corresponding to S5"0.43. And at b"15°, the large suction area disappears (III

B)

and no dominant peak values are found.A section of the recording of the variation with time of the velocity measured with

the hot wire anemometer in the gap region during the switching process from IIB

toIII

Bfor N/d"2.0 and Re"2.2]105 at b"14.3° is presented in Fig. 14. It shows

that there are two types of velocity fluctuation and that they change intermittently atirregular time intervals. One type of signal is a harmonic oscillation which givesa strong peak value at a certain frequency in the power spectrum (corresponding tothe high Strouhal number of S

5"0.43), relevant to the large suction area on cylinder


Fig. 13. Several examples of power spectra of flow velocity in the gap region for Re"2.2]105, N/d"2.0(a) At b"7.3° (pressure pattern I

Bon cylinder B), S

5"0.15 which is less than that of single cylinder. (b) At

b"7.3° and b"13° (pressure pattern IIB

on cylinder B) S5"0.43 which is much greater than that of

a single cylinder. And (d) at b"15° (pressure pattern IIIB

on cylinder B), no dominant peak value can befound.

Fig. 14. A piece of recording of the velocity varying with time measured in the gap region during theswitching process of pressure patterns II

Band III

Bon cylinder B for Re"2.2]105, N/d"2.0 at b"14.3°

(h: the part of harmonic oscillation signal which gives strong peak value in power spectra analysiscorresponding to S

5"0.43).


B (pattern IIB). The other type appears to be a random signal, giving no dominant

peak frequency and corresponding to the presence of pressure pattern IIIB.

It is shown that the presence of pressure pattern IIB

or IIA

is associated not onlywith the high-speed gap flow with low turbulence intensity, but also with regularvortex shedding at a rather high frequency.

3.8. Other cases of N/d, CD

and CL

In the present investigation, the gap ratio between the two cylinders varied from 1.1to 3.5 and the tested Reynolds numbers were 2.2]105 and 3.3]105. Not all thecomposite cases of N/d and Re were carried out due to the limited testing time.

For a larger gap ratio such as N/d"2.2, almost the same thing happened on bothcylinders at these two Reynolds numbers. That is, for Re"2.2]105, pressure pat-terns I

B, II

Band III

Bappear on cylinder B, whereas only one pressure pattern of

IA

appears on cylinder A, and for Re"3.3]105, all five patterns appear on bothcylinders. The difference is that, for Re"2.2]105, C

B1and C

B2are located at 7.3° and

between 30° and 45°, respectively. For Re"3.3]105, CB1

lies between 8° and 8.5°,and C

B2between 15° and 30°; C

A1lies somewhere between 30° and 45°, and C

A2near

60°. In the side-by-side arrangement, as well as in the neighbourhood of this arrange-ment, biased flow was no longer observed.

The observed C are given in Table 1. It is obvious that the appearance of thepattern II

Aon the upstream cylinder occurred only for the cases of smaller N/d and

relatively higher Reynolds numbers, but the switching process is quite common on thedownstream cylinder. In general, for the same N/d and Re, C

A2is equal to or larger

than CB2

when observed. For the same N/d, values of CB2

increase, i.e. the largesuction area on cylinder B is retained longer at higher Reynolds numbers. Otherwise,no definite conclusion can be reached for the variation of the C with N/d.

To show the integrated effect of pressure-pattern changes on the aerodynamicbehaviors of the cylinders, for the typical case of a gap ratio N/d"1.7 andRe"2.2]105, the drag and lift force coefficients C

Dand C

Lper unit length of

cylinders, together with the locations of CB1

and CB2

and the ranges of the pressurepattern I

B, II

Band III

Bon the downstream cylinder, are given in Fig. 15. It is found

that the CD

and CL

of cylinder A change more or less smoothly. The drag coefficientC

Dof cylinder B is negative at smaller b’s and then rises up quickly to near unity for

b from 15° to 30°. After that, it does not change much. The lift coefficient CL, on the

other hand, changes abruptly by a large amount at CB1

and CB2

due to the appearanceand disappearance of the large suction areas. The resulting abrupt change in lift forceat C

B1agrees well with the previous work reported by Suzuki et al. [5], in which the

total forces on the downstream cylinder were given. The switching process may resultin serious vibrations of cylinder B.

In order to present the overview of the total effects of interference on both cylindersmore clearly, the drag and lift force coefficients C

Dand C

Lfor the N/d tested in the

present experiments are shown in the form of a surface drawing in Fig. 16.The drag forces on cylinder B reduce a great deal in the tandem arrangement and

even become negative in the small region for N/d less than 2. The drag forces on


Table 1Observed values of C at which the switching process occurred

N/d Re"2.2]105 Re"3.3]105

CB1

CB2

CA1

CA2

CB1

CB2

CA1

CA2

1.1 7.5°—7.8° 10°—15° — — 0° 11°—11.5° 0° 25°—30°1.15 4°—5° 12° 4°—5° 20°—30° 0° 12°—13° 0° 20°—30°1.2 4°—5° 10.5°—11° 4°—5° 20°—30° 0° 12.8°—13.5° 0° 20°—30°1.5 8.5°—8.7° 16.5° 25.2°—25.5° 40°—45° 9°—10° 20.8°—21.5° 9°—10° 45°—60°1.7 9.65°—9.7° 15°—16° — — 9.75° 23°—24° 9.75° 55°—58°2.0 7.4°—7.6° 14°—14.3° — —2.2 7.3° 10°—15° — — 8°—8.5° 15°—30° 30°—45° 60°—75°3.0 7.2° 16°—25° — —

Unstable3.5 10°—20° 20°—20° — —

Note: “—” denoting no discontinuity observed.“Blank” denoting no test being done.

Fig. 15. Drag and lift force coefficients CD

and CL

on cylinders A and B in staggered arrangement(Re"2.2]105, N/d"1.7), together with the locations of C

B1and C

B2and the ranges of the pressure

patterns IB, II

Band III

Bon the downstream cylinder.

cylinder A are also reduced as the gap ratio decreases. However, the greater reductionhappens in the side-by-side arrangement at very small gap ratios, and in the staggeredarrangements at small angles of b.

In comparison with the drag force, the lift force on cylinder B changes abruptly,which corresponds to the appearance of pattern II

Band the two-peak value areas. It,

in general, qualitatively agrees with the result obtained by Zdravkovich [11]. In hispaper two constant maximum C

Llines have been designated as the “inner” and

“outer” lift peaks. Nevertheless, it is difficult to express the switching process clearlyby this kind of plot. From consideration of the results of the existence of C

B1and

CB2

at various N/d, the locations where the downstream cylinder encounters the


Fig. 16. Surface drawing of drag and lift forces acting on cylinders for Re"2.2]105 (note, the value alongZ-axis for lift force on cylinder B is negative upward, i.e. !C

L).

extremely large lift force should be within certain areas bounded between CB1

andCB2

for certain N/d. On the other hand, the appearance of a large lift force oncylinder A at small gap ratios is evident. In general, for 0°(b(45° the lift forces oncylinder A and B are directed towards each other and for 45°(b(90°, they areaway from each other.

4. Conclusions

(1) The results of the interference of pressure distributions between two parallelidentical cylinders in staggered arrangements at high subcritical Reynolds numbers,as measured by means of wind-tunnel testing, have been presented. Close attention


was paid to the relevation of how and why the discontinuity in pressure distributionsor the switching process occurs on the downstream cylinder in certain conditions. Thegap ratio N/d tested varied from 1.1 to 4.0 and the Reynolds numbers were 2.2]105and 3.3]105. As an aid for analysis, results of flow visualizations by means ofsmoke-wire technique were also presented.

(2) In general, three different pressure-distribution patterns IB, II

Band III

Bon the

downstream cylinder and two switching processes were observed for the wind angleb varying from 0° (in tandem) to 90° (in side-by-side). The corresponding flow patternscan be classified, in essence, as wake, shear layer and neighbourhood interferencerespectively. The patterns from I

Bto II

Band from II

Bto III

Bchange abruptly, and may

switch back and forth at critical angles CB1

and CB2

, respectively.For smaller N/d or higher Reynolds number, two different patterns I

Aand II

Aalso

occur on the upstream cylinder and a similar switching process from IA

to IIA

andfrom II

Ato I

Ais observed at C

A1and C

A2, respectively. Nevertheless, the process of

change from pattern IIA

to IA

at CA2

is more gradual than that at CA1

.Pattern II

Band II

A, which are maintained for a certain range of b, give large areas of

rather low suction of the gap-side surface of cylinders and thus, result in large lift(lateral) forces. It may cause strong vibrations, especially during the switching process.

(3) The mechanism of the appearance of the large suction area on the gap-sidesurface of the downstream cylinder is caused not only by the “displacement” or“squeeze” of the wake behind the upstream cylinder, but also by the shear layerseparated from the gap side of upstream cylinder with its high speed side “reattachingonto” and “sweeping along” the gap-side surface of the downstream cylinder, as wellas by the “passage” of alternate vortices at high frequency and high speed. Neverthe-less, the mechanism of the two switching processes (i.e. from I

Bto II

Band from II

Bto

IIIB) is mostly due to the geometry of the gap and the basic instability that exists

between the shear layers separated from the upstream cylinder.On the other hand, the appearance of pressure distribution II

Aon cylinder A is

mostly due to the combined effect of the Reynolds number close to the critical regionand the group-effect. It is somewhat different from those of pattern II

B, which is

mostly caused by the group-effect.(4) The bistable biased flow between two cylinders, as studied by many researchers,

not only exists in the side-by-side arrangement for a certain range of N/d at highsubcritical Reynolds numbers, e.g. N/d(1.7 and Re"2.2]105, but also occurs inthe neighbourhood of the side-by-side arrangement up to a certain angle.

Acknowledgements

The work described in this paper was supported by the National Natural ScienceFoundation of China. The authors thank their colleague Professor R.S. Lin for hiscollaboration during the wind-tunnel testing and Professor G.J. Zhou for his valuablediscussions and great help in preparing the revised manuscript. Special thanks to Dr.Bruce R. White for his valuable advice and help during the first author’s stay in UCDavis preparing the manuscript.


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On interference between two circular cylinders in staggered arrangement at high subcritical Reynolds numbers