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A visualization study of cross-flow around four cylinders in a square configuration
Journal of Fluids and Structures (1992) 6, 109-131
A VISUALIZATION STUDY OF CROSS-FLOW AROUND FOUR CYLINDERS IN A SQUARE CONFIGURATION K.
LAM
AND
S. C.
Department of Mechanical and
Hong Kong Polytechnic Hong
FLOW-INDUCED
VIBRATIONS
IMPOSED
on
a single
bluff
body
or a bundle
of bluff-body
are of great engineering importance. Many examples of failure in engineering applications, as in power transmission lines, nuclear power station heat exchangers, suspension bridges, etc., due to this destructive phenomenon have been reported (Paidoussis 1980) and show the need for research to overcome the vibration problem. In the last two decades, a large number of studies,. development of mathematical models as well as experimental investigations, have been carried out worldwide for this specific purpose. Throughout this literature, it is found that generally there are three kinds of mechanisms responsible for the vibration. They are (a) fluidelastic instability, (b) turbulent buffeting and (c) vortex-shedding excitation. In the study of fluidelastic instability, authors like Chen, Paidoussis, Weaver et al. and others, through their theoretical ideas and experimental verifications, established the general guidelines for studying the problem. They reasoned that the instability is generally associated with high-flow speed, and the threshold depends largely on the arrangement of the bodies. Semi-empirical models, relying on the influence of the motion of the surrounding tubes like negative damping and fluid-stiffness terms (Chen 1983), or more theoretical models which treat each tube as independently unstable and modelled by the unsteady flow equations (Weaver & Lever 1982), have evolved lately and achieve reasonable predictions on the instability onset for certain arrays. In the study of turbulent buffeting, most of the studies make use of statistical ideas, and it is well accepted that the induced vibrations exist over a broad band of frequencies with small response amplitude; in some cases, the response varies linearly with flow speed, U, rather than the expected U2 (Pdidoussis ef al. 1987). At one time, structures
0889%9746/92/010109 + 23 sos.Gu
@ 1992Academic Press Limited
110
K. LAM AND S. C. LO
there arose confusion between turbulent buffeting and the wake-shedding mechanism; Owen (1965) considered that the sharp peak observed in the response spectrum could have been generated by one dominant frequency-band of turbulence. However, after several years of verifications, the existence of periodic and narrow-band vortexshedding excitation prevailed. Since the time that Aeolian tones were first noticed, vortex-induced vibration has attracted attention. Research workers have looked into many interesting aspects of the mechanism. The establishment of constant Strouhal number of O-2 for a single cylinder in a wide range of Reynolds number, 400 to nearly 2 x lo’, and the discovery of the “lock-in” effect whereby vortex shedding follows the natural vibration of the tube and leads to resonant response over a range of speed, are some highlights of the investigations. In 1915, Pannell et al. reported the force measurements on a pair of parallel “circular wires” in close proximity. They found that within a small angle of stagger, the combined drag force was even smaller than that of a single wire. The results aroused the interest of studying the interference effects of surrounding bodies on the wake structure. Zdravkovich (1977, 1987) summarized an enormous amount of information like pressure, lift and drag coefficients, and Strouhal number around two tubes or multi-tube array systems. It seems that each system of the tube array carries a unique interference characteristic that the principle of superposition cannot overcome for adequate prediction. Weaver & Yeung (1984) and Sayers (1987) both pointed out that there is a need of basic experimental data for various standard configurations. The present study is carried out in this direction. The effects of different how incident angles and the spacing ratio on the flow characteristics around four cylinders in square arrangements is examined closely in this study, as the lack of data of this type breaks the connection from the most basic two-cylinder system to multi-cylinder arrays. Previously Lam & Cheung (1988) examined the shedding frequency characteristics and flow interference of three cylinders in different equilateral arrangements by means of the same apparatus and technique as in this experiment. Based on their experience, the Reynolds number effect is weak and adds only minor discrepancies to the Strouhal number characteristics, at least, within the range 2.1 x lo3 < Re < 3.5 x 10”. The present experiment is conducted at a lower flow velocity, Re = 2-l x 103, to obtain better photographic evidence. The study was focussed on the flow characteristics of four cylinders arranged in a square, which constitutes another type of the simplest unit of a tube bank. As before, a dye injection visualization technique is employed to study the changes during the variation of spacing ratio, 5.96 to 1.28, and the angle of incidence, 0 to 45” such that all possible orientations of the square array are covered. The results are categorized according to the orientation of the model, i.e., angle of incidence. The flow fields around the cylinders are summarized in steps of spacing ratio and are demonstrated by photographs.
2. EXPERIMENTAL
DETAILS
The experiments were carried out in the water tunnel of the Mechanical & Marine Engineering Department, Hong Kong Polytechnic and the details of the tunnel are described by Lam & Chu (1986). In the experiment, the flow speed was chosen to be O-223 m/s, which gives a Reynolds number of 2100, a value right inside the lower subcritical range where the
CROSS-FLOW
AROUND
FOUR
CYLINDERS
111
Reynolds number effect is small and also provides a good condition for visualization. The corresponding turbulence level is within 1% . The cylinders are 9.4 mm in diameter and 200 mm in length; they span across the O-2 m x O-2 m x O-5 m (length)
v -30
1 -1
- 1
D
Y
D2
t i._
X
,”
0
- 4
Figure 1. General configuration of the model.
112
K. LAM AND S. C. LO
3. RESULTS During the experiment, several features of flow that were found in the 2-cylinder and 3-cylinder cases were also found in the 4-cylinder cases. The interference among the cylinders reduces to a minimum as the distances between them becomes large. But then, as L/D decreases, complicated influences would come into play in which the angle of the flow plays a dominant role in determining the flow pattern and, hence. the Strouhal number characteristics. In order to simplify the description, the cylinders are labelled as 1 to 4. as in Figure 1 throughout the paper. The angle of rotation is in an anti-clockwise sense, so the cylinder 2 will be in the highest position in Figure 1 if the angle of incidence is 4.5”. To provide a systematic and detailed description of the phenomena, the results are grouped in three distinct arrangements according to the range of angle of incidence. the results are interpreted in descending L/D order for each Furthermore, arrangement. The Strouhal numbers for each cylinder, plotted against the spacing ratio, are summarized in Figures 2 to 5, and the reader should note that the Strouhal number in the graphs are normalized by the free-stream speed and the cylinder diameter. At 8 = O”, an important feature is that the flow exhibits a bistable state of a wide and a narrow wake behind the downstream cylinders at and below L/D = 1.54. Similar to the results of Bearman & Wadcock (1973), the shedding frequency of the wide wake is low and the narrow wake corresponds to a high frequency. At flow-incidence angles other than O”, it is certain that the wide wake is biased to cylinder 3 which is now situated in the aft-most position. The rest of the cylinders, on the other hand, possess a smaller size of wake than that behind a single isolated cylinder, and the implication is that higher values of Strouhal number would result, which agrees well with those obtained in the experiment. The trend for the Strouhal number characteristics is that they are rather insensitive to the change of angle, and the irregularities are largely reduced when compared to the results of three cylinder clusters (Lam & Cheung 1988). At 8 = 45”, the arrangement is geometrically symmetrical in the flow direction just like the in-line arrangement; however, the bistable state of wakes around the cylinders of the latter case is completely replaced by a symmetric pattern when L/D approaches unity. The wake length of the foremost cylinder (1) is incredibly cut down, such that tiny vortices are formed before impacting on the front face of the downstream cylinder (3). 3.1. IN-LINESQUARE ARRANGEMENT 3.1.1.
Vortex-shedding
charucteristics
( 0 = 00)
The Strouhal number (hereafter denoted by S) characteristics at 0 = 0” are shown in Figure 2. At large spacing, 5.96 < LID < 4.47, the vortex-shedding frequency values of the four cylinders are close to that of a single cylinder. Although S for each cylinder does not exactly have the same value, the small variation between them does not reveal any trend and remains within an 8.5% maximum deviation from that obtained for a single cylinder (S = 0.2) at LID = 5.96. However, at LID = 3.94, S of the downstream cylinders decreases, and the difference from the upstream ones, which are now only 3% away from 0.2, rises to about 12% of the single cylinder value. Further reducing the spacing ratio, it is observed that the regular shedding behind the upstream cylinders is “suppressed”. It is important to realize that although the
113
CROSS-FLOW AROUND FOUR CYLINDERS 1.0
2.0
3.0
4.0
6.0
5.0
0.7
0.6
0.5
“. 5
0.4
2 z 2 : arz
0.3
0.2
0.1
Cylinder 3
0.cb
1.0
I 2.0
I 3.0
,
I
I
4.0
5.0
6.0
Spacing ratio. L/D cylinder Figure 2. Strouhal number, S, plotted against spacing ratio, L/D, for f3=o”. --O-, -.-A-.-, cylinder 2; --V--, cylinder 3; -..a..--, cylinder 4; -O--, Fitz-Hugh’s results.
1;
spacing in between the cylinders does not allow normal vortex formation, the near-wake of the upstream cylinders is not sufficiently stable to continue providing shielding for the downstream ones. From time to time, the wake is seen to be deflected from the streamwise direction and only fragmented shedding periods can be recorded. The situation is reflected by the abrupt end of results for cylinders 1 and 2 at L/D = 3.94.
Repeated experiments were carried out at small angles of incidence to ensure that this is not due to a misalignment of the angle. It is believed that the “deflection” is not due to the turbulence level of the water tunnel, but rather to the presence of the two
114
K. LAM AND S. C. LO
nearby tandem cylinders; the results of two cylinders in a tandem arrangement in the same tunnel and same Reynolds number show clearly that the upstream cylinder near-wake is stable and shields the downstream cylinder throughout the whole range of spacing. For the downstream cylinders, the shedding frequency decreases slightly at L/D = 3.51, but then starts to climb back slowly to 0.2 at L/D = 1*70. There is neither a discontinuity nor a jump phenomenon in S as suggested by Kiya et al. (1980), Baxendale & Barnes (1986), and Lam & Cheung (1988). All these results for two and three cylinders conclude that, at the spacing that the upstream cylinder near-wake starts to shield the downstream cylinder, due to modification of the streamlines so that now the two cylinders appear more as one streamlined body, the Strouhal number found behind the latter will jump and then follow a trend to a value much higher than that of a single cylinder as the spacing decreases. In this case, the tendency of streamlining is largely affected by the fact that the near-wake of the upstream cylinder deflects, which is indicated by the proximity interference of the other, adjacent tandem pair of cylinders. When L/D drops below 1.70, a bistable feature evolves behind the downstream cylinders and dominates the flow characteristics. A narrow wake of high shedding frequency and a wide wake of low frequency are recorded as in the results of Bearman & Wadcock (1973) and Lam & Cheung (1988). The high S value reaches O-36 at L/D = l-28, while the low S drops to a minimum of 0.11 at L/D = l-41 and then rises slightly back to O-12 at L/D = 1~28. The intermittent change of the two wakes is also observed. But for simplicity of presentation, the high frequency is assigned to cylinder 3 and the low one to cylinder 4. 3.1.2. Observations on fiow patterns (0 = 0’) At large spacing, 5.96 < L/D < 3.94, as shown in Plate l(a) the wake formation regions of the downstream cylinders are much shorter than the upstream ones. This is due to the fact that vortices are shed by the upstream cylinders, creating a more turbulent flow behind, accompanied by an early transition from a laminar to a turbulent shear layer and hence the shrinkage of the formation region. The four cylinders, in this spacing range, can be thought of as a pair of identical tandem as no prominent “proximity” effect is detected, but the “wake arrangements, interference” is obvious. Whenever the upstream vortex of either sign comes close to the downstream cylinder, it provokes the formation of a vortex of the same sense. The phase difference between the upstream and downstream cylinders is dependent on the distance between them. At L/D = 3.94, the vortices shed from behind cylinders 1 and 2 are so close to the front face of the cylinders 4 and 3, respectively, that occasionally the two shear filaments of the former cylinders do not have enough time to grow into a mature vortex; hence, a regular mature vortex shedding is no longer distinct, although regular shedding is still recorded most of the time. The flattening and lengthening of the wake lasts only for a short time before the recovery of the curling up of shear layers into vortices. During that short period of suppression of vortex formation, the downstream cylinders are being immersed in the wake of their respective upstream cylinders. However, the changing of states apparently does not affect the shedding frequency of the downstream cylinders. The intermittent changes become more and more pronounced as the spacing ratio decreases, the period of regular shedding is largely shortened which is revealed by the
CROSS-FLOW
AROUND
FOUR
CYLINDERS
115
stoppage of the curves for cylinders 1 and 2 at L/D = 3.94. In fact, below this spacing, the shedding found must be accompanied by a deflection of the shear layer as illustrated by the dotted lines in Figure g(b). The lower shear layer of cylinder 2 bends sharply and follows the contour of the downstream cylinder to the upper side when the spacing is below 240. The vortex formed in this way is small and the shear layer on that side of the respective downstream cylinder seems to be affected and leaves the cylinder surface at some angle from the X-axis, which explains the momentary occurrence of a wide wake near the downstream cylinders. But what is usually found is that the straight shear filament extends streamwise from cylinders 1 and 2, the wake shields the downstream cylinders 4 and 3 and no vortex is observed. Below L/D = 1.70, Plate l(c) shows that the bistable state of the wide and narrow wake which is attached behind cylinders 3 and 4. A narrow gap flow is biased to the narrow wake just as in the case of side-by-side cylinders. The initiation of the asymmetric mode is originated from an outward deflection of the outer shear layer of cylinders 1 or 2, which rolls up beside cylinders 4 or 3 without the presence of the inner one which, at this time, does not bend sharply outward but appears straight. It may attach at about 50 to 60” clockwise from the foremost point of the downstream cylinder or may just extend behind it. This condition promotes the outward shear layer of cylinder 4 or 3 to separate at an even wider angle. Once the wide wake is established, the gap flow between the two rows of cylinders is pushed to another side where the narrow wake is formed.
3.2.
SLIGHTLY
STAGGERED
3.2.1. Vortex-shedding
ARRANGEMENT
characteristics
(8 = lo”, 20”)
Figure 3 shows that the Strouhal number characteristics of the cylinders at 8 = 10” are largely different from the in-line arrangement. Only in the range L/D = 5-96 to L/D = 4.47 does S have a behaviour similar to the in-line situation. The value of S consistently remains in the range O-18 to 0.21, because the spacings are large enough to prevent any interference between the cylinders. For L/D < 4.47, the individual S values of the four cylinders diverge from one another. The S values of cylinders 1, 2 and 4 increase moderately from O-2 for up to L/D = 3.09, before rising sharply; on the other hand, the S value of cylinder 3 decreases in a stepwise fashion. For cylinders 1, 2 and 4, the S increase for cylinder 1 is largest, followed closely by that of cylinder 2 and then cylinder 4. The difference in S values between cylinders 1 and 4 is about 0.04 at L/D = 3.09; for L/D = 2.40, however, the difference is double. For L/D < 2.40, the shedding from cylinder 1 is weak and the visualization method is not capable of demonstrating the vorticity cloud that may have formed, because it diffuses quickly into the downstream wake as the shear layers almost touch the cylinder surface. Since no distinct vortex can be seen, the S value could only be determined down to L/D = 2.4 and, in like fashion, the same behaviour occurs for cylinder 2 at L/D = l-70. Returning to cylinder 3, it can be seen that the decrease of S begins at L/D = 4.47. It starts initially at a value of 0.17 between L/D = 3.94 and 3-51, then decreases to O-13 between L/D = 3.09 and 2.40 and finally reaches about O-09 in the last few values of L/D ratio. The S behavior is not related to the suppression of vortices from both cylinders 1 and 2, as seems to be evident in the figure; the L/D value at which the shielding occurs does not correspond to the L/D value at which the step changes of S
116
K. LAM AND S. C. LO
0.7
0.6
0.5
m r b 2 z -z 5 e + Lo
0.4
0.3
0.2
0.1
0.0 1.0
24
3.0 Spacing ratio.
I
,
4.0
5.0
6.0
L/D
Figure 3. Strouhal number, S, plotted against spacing ratio, L/D, for 0 = 10”; (see Figure 2 for symbol legend).
take place on cylinder 3. It merely indicates that, within a certain of cylinder 3 is insensitive to the influence of the others.
range,
the near-wake
If we look ahead at Figures 4 and 5, we note that the increase of angle of the arrangement has the effect of smoothing the kinks of the S behavior for cylinder 3. It may be reasoned that, at small angles, the flow around cylinder 3 is restricted by the wake of cylinder 2 and, at the same time, the vorticity of this wake also distrubs the shedding of the upper shear layer. The influence decreases as the angle increases, and the wake of cylinder 3 is mainly affected by the two more regular flows around cylinder 1. (The reader is also referred to Section 3.4.)
CROSS-FLOW
0.7
14.I
2.0 I
AROUND
3.0 I
FOUR CYLINDERS
4.0 1
117 5.0
6.0 ,
0.6
0.5
m Ei .
0.4
z z z z s
0.3
0.2
:’ 0. I
.
,V-t--*
\
T
Cylinder 3
OJJ/
1.0
2.0
34
3.0
5.0
6.0
Spacing ratio, L/D Figure 4. Strouhal number, S, plotted against spacing ratio, L/D, for 0 = 30”; (see Figure 2 for symbol legend).
3.2.2. Observations on j7ow patterns (8 = lo”, 20”) As L/D decreases to 4.47 and 4.89 at 8 = 10” and 20”, respectively, the flow patterns are analogous to that in Plate l(a), although the upstream cylinders are not directly in front of the downstream ones. The near wake of the former remains long and the latter remains short. Plate 2(a) shows the flow field at L/D = 3-W for 0 = 10”. It is typical of those found in the range from 3.94 to l-88. The wake of the upstream cylinders is relatively long and the shear layers that emanate from the separation points appear straight but roll up and “cloud” adjacent to the upper surface of the downstream cylinders. In doing so,
118
K. LAM AND
Cylinder
S. C. LO
2
s.0
4.11
Spacing Figure
5. Strouhal
number,
S, plotted
against
spacing legend)
ratio. ratio,
Lit>
L/D, for 0 = 45”. (see Figure
2 for symbol
the size of the wake is smaller as the undisturbed flow above maintains the upper layer in approximately the streamwise direction. The consequences of the fluctuation near the upper separation points for cylinder 3 are to cause the respective shear layer to roll up into small clouds more readily, which then convect downstream quickly and catch up to those shed previously. The clouds also entrain those vortices from cylinder 2 in the wake and merge together to form a
CROSS-FLOW AROUND
FOUR CYLINDERS
119
bigger vortex street. The lower shear layer becomes straight and long before curling up; the reason is that it is free from the direct disturbances of the wake from upstream. The above discussion can also be applied to cylinder 4, except for the wide wake which is replaced by a much narrower wake. The pattern shown in Plate 2(b) can be obtained at both L/D = 1.70 and 1.54; there are no fundamental differences from the previous picture and yet the wakes of cylinders 1 and 2 are now largely reduced in size and the lower shear layer curves more than 90” to recover the position on the upper side of the downstream cylinders. The wake of cylinder 1 is in between the shielding and the sharp bending of the filament to the other side. The situation changes again as soon as L/D decreases. In Plate 2(c), the abrupt curvature of the shear layer is absent, the lower shear layer of cylinder 2 is now extended and passes beneath the downstream cylinder and the upper shear layer from cylinder 1 is shifted down and attaches to the front face of cylinder 4. Plate 2(d, e, f) show the flow tield at 8 = 20”. Based on a careful comparison with the previous pictures, it is noticed that the wake of cylinder 3 is getting wider and those in the front a little narrower. Moreover, the angle opens up for the flow from beneath and encourages the curling up of the lower shear layers of the front cylinders which in turn delays the suppression of vortex formation.
3.3. LARGE ANGLE
STAGGERED
ARRANGEMENT
3.3.1. Vortex-shedding characteristics (8 = 30”, 40”, 45”) As the angle of incidence is increased, the wake interference of cylinder 1 on cylinder 4 decreases, but this wake now affects cylinder 3. The upstream influence of cylinder 1 comes much earlier than that in the slightly rotated or in-line square arrangement, as can be seen in Figures 4 and 5. There is a change in the S value between cylinders 1, 2 and 4, which possess higher values, and cylinder 3, which has a lower value, even at the largest L/D ratio in the experiment. At L/D = 5.96, the largest difference in S values for this range of incidence angles remains as high as 14 to 19% of the S value of an isolated cylinder. Both figures show that the S value of the front three cylinders increases sharply as L/D drops below 2.65, especially for cylinder 1. The larger angle of incidence shows the greatest variation in S between cylinder 3 and the rest, but the trend for each cylinder is the same as in the previous section. At 0 = 30”, S for cylinder 1 rises from 0.34 at L/D = 2.65 to 0.57 at L/D = 1.41; at 8 = 45”, it rises from 0.34 to 064. On the other hand, at 8 = 45”, S for cylinder 3 reduces gradually to about O-075 at L/D = 1.70 to 1.88 and increases slowly to about 0.08 at L/D = 1.28. It should be noticed that the suppression of vortex formation only happens on cylinder 1 and also that the L/D ratio at which it takes place is the smallest tested, i.e., 1.28. Comparing the two figures, it is noticeable that, at many spacing ratios, the S value of cylinder 2 in the case of smaller angles is higher. In Figure 4, S for cylinder 4 is generally lower than that of cylinder 2, then catches up with it as the angle increases. At 8 = 45”, as shown in Figure 5, the S values for the two cylinders are close, which is expected, because the geometric positions of the two cylinders are now identical. Moreover, as can be seen in the figures, no bistable state is ever observed throughout the whole range of spacing.
120
K. LAM AND S.C. LO
3.3.2. Observations on flow patterns (0 = 30”, 40”, 45”) In Plate 3(a), the pattern shown is typical of those in the range of L/D > 2-40; the wakes of the front cylinders remain regular and lie in the flow direction as before, but that of cylinder 3 already indicates the onset of a wide wake. The wake of cylinder 1 is comparatively narrow due to the gap flow on each side rushing into the central region and the natural expansion of the shear layer in the transverse direction after separation is restricted. Despite the wake being sandwiched between the gap flow, it is not always stable in staying in place in the middle, but sometimes may be biased upward or downward randomly and remains there for a short time before changing again. However, the random motion is small in amplitude. Plate 3(b,c) represents the flow for 1.28 < L/D < 240. At an angle 8 > 30”, cylinder 1 divides the frontal entry of the flow between cylinders 2 and 4. The gap flow passing each side of cylinder 1 suppresses its near wake into a very narrow region. The narrowing of the distance between the two shear layers enhances the conditions for rolling up and lessens the longitudinal distance for vortex shedding, causing a delayed suppression of vortex shedding. At the exit of the gap flows on each side of cylinder 4, there is a wake divergence which depends on an angle which, in turn, depends on the L/D ratio of the model. The smaller the L/D value, the larger the angle of the flow. It should be noted that the formation regions of cylinders 2 and 4 are also largely reduced in length and width and, more importantly, they are strongly affected by the exiting gap flow which rotated the wake axis by up to nearly 30”. However, as the wake proceeds further out, it loses the momentum to maintain that direction and the free stream overtakes the gap flow position and pushes the wake back in the streamwise direction. At the same time, the gap flow from each side of cylinder 1 also leads to the development of straight shear layers which are separated from cylinder 3. The layers then meet those from cylinders 2 and 4 some distance downstream and mix together to form a low frequency wake oscillation. Plate 3(d) shows the flow field of the smallest L/D, i.e., 1.28, which is similar to the other situations at 8 > 30”. The most impressive point is that the weak vortex street behind cylinder 1 disappears, but is replaced by a narrow stagnant region as the short distance suppresses the curling of the shear layer before striking the front face of the downstream cylinder. Consequently, the layers follow the contour of the cylinder face, and the two points of contact on the cylinder are close to each other; if measured from the centre of the cylinder, the angle subtended is less than 40”. 3.4. STROUHAL NUMBERS
FOR CYLINDERS
1 AND
3 FOR ALL ANGLES
To obtain an overall view on the vortex-shedding characteristics of a particular cylinder, the variations of Strouhal number S with spacing ratio L/D for cylinders 1 and 3 at all tested angles of incidence were plotted in Figure 6. The results clearly illustrate that cylinders 1 and 3 possess distinctly different characteristics. All the S values of cylinder 1 decrease smoothly from O-6 to O-2 as the spacing ratio increases while the S values of cylinder 3 rise from 0.08 to 0.2 in a somewhat stepwise manner especially when the spacing ratio is small. 4. DISCUSSION 4.1. CLUSTER EFFECTON FAR WAKE It is evident, in Plates l(c), 2(e,f) and in fact for any angle orientation that, as long as L/D < 1.70, a distinct oscillating motion of wake exists far downstream. These sinuous
CROSS-FLOW 2.0
0.7 I.0
AROUND
FOUR
4.0 I
3.0 I
121
CYLINOER~ 5.0
6.0 I
i---0.6
!
Cylinder
1: -
Cylinder
3: - - - - -
L
I -
I_
i-o Spacing Figure
6. Strouhal
number,
S, plotted
against
4.0
ratio.
spacing ratio, incidence.
L/D L/D,
for cylinders
1 and 3 for all angles
of
oscillations result from the merging of the far wake of the four cylinders. Actually, these results can also be found in the 3-cylinder case in various flow speeds (Lam & Cheung 1988; Zdravkovich 1968). It is interesting to note that the period of this wake oscillation coincides with that of the wide wake of cylinder 3 at all angle arrangements except 0”. Plates l(c) and 2(e) show clearly the domination of the wide wake: the weak vortex streets of the rest are just being entrained into the central region. It was suggested by Zdravkovich that this oscillating motion is due to the curling of the outermost shear layer while the inner layer vorticity coalesces and dies out. In such a way, the cluster of whatever arrangement could then be imagined as a large body and the wake oscillation as a new vortex street. An evaluation of S according to this idea is
122
K.
LAM AND S. c‘
LO
TABLE I Strouhal
number
for wake oscillation, normalized values of spacing
Angle of incidence,
by D,, for small
H
Spacin ratio, L.PD
0”
IO”
20”
30”
40”
45”
1.28 1.41 1.54 1.70
0.23 O-24 0.23 0.28
0.22 0.23 0.23 0.30
0.22 0.23 0.23 0.26
0.23 0.22 0.24 0.25
0.24 0.23 0.24 0.25
0.24 0.24 0.24 0.26
examined and listed in Table 1. The diameter, L),, , is the largest normal distance of the cluster to the flow at the current angle. From the table, we can see that, between L/i) = 1.28 and 1.54, S falls consistently into the range of O-22 to 0.24 but diverges in an inconclusive manner if L/D increases beyond 1.70.
4.2.
DIMENSIONLESSWAKE WIDTH AND STROUHAI. NUMBER
It has been concluded by Gerrard (1966) that, in the case of a single cylinder. the entrainment of fluid from the interior of the formation region and its replenishment by reversed flow is fundamental in determining the wake size. The two characteristic lengths of the wake, namely the scale of the formation region and the width to which the free shear layers diffuse, determine the frequency of vortex shedding. In this experiment and in some other multi-cylinder cases, the results show that the interference between cylinders may largely affect the size of the wake without altering the shedding frequency. For example, in Plate l(a), the downstream cylinder wake encounters a very turbulent flow from the upstream wake and the length of the downstream wake is largely reduced; yet, the frequency of shedding remains the same as for the upstream cylinders. But for the latter characteristic length. just as the well-known result that a wide wake always contains low frequency and the narrow wake high frequency in the two-cylinder case, certain relationships between shedding frequency and wake width may exist. In our case, measurements of the wake width, W, or the thickness between the two shear layers at the end of the formation region where the layer is drawn across the wake (nondimensionalized by the cylinder diameter, W* = W/D), corresponding to the S values obtained earlier, show that. no matter what the orientation or spacing ratio of the model, the points he within a certain region. This region is bounded on its upper and lower sides by two curves as in Figure 7; their respective equations are S = 0.356 W*-O‘xy,
S = 0.156 W*-“+“.
The variation of S with W * for the 45” orientation corresponds to the middle full line which can be approximated by S = O-264 W*-“‘. However, the Strouhal number and width relation for the 0” orientation does not follow the same trend. Instead, the values of S scatter around O-2 for O-8 -=cW* -=c2. This may be due to the “tandem effect” of the cylinders, such that the wake widths of the cylinders can neither be too wide nor too narrow in most cases.
CROSS-FLOWAROUND FOUR CYLINDERS
123
I_
(1/
Wake width. Figure
7. Strouhal
number,
S. plotted against A, f3=30”;
s
3
2
I
W*
wake width, W*. Angle: x, 0=40”; V, %=45”.
0.
0 = 0”; +, 0 = 10”; 0,
0 = 20”;
4.3. APPLICABILKY OF RESULTS TO MULTI-TUBE HEAT EXCHANGERS A cluster of four cylinders in a square configuration may be considered as an element of an in-line (0 = 0”) and rotated square (0 = 45”) tube bank in heat exchangers. The present visualization study may provide some clarifications and additional information on vortex shedding and flow interference due to change of spacing ratio and flow direction that may occur in heat exchangers. However, in attempting to draw any conclusions, the following points should be noted. The visualization study was chosen at Re = 2100 because, for the present set-up, the flow visualization and photography are the best at such a Reynolds number. Investigations carried out at a slightly higher Reynolds number (Re = 3500) indicated that the vortex-shedding characteristics and flow pattern are basically the same. Nevertheless, Re = 2100-3500 are still within the lower subcritical range of Reynolds numbers. It is well known that for a single cylinder, at Re = 2100, the drag coefficient (C, = 0.9) is lower and the length of the vortex formation region (-3.50) is longer than that which would occur in the upper subcritical range of Reynolds numbers. Most heat exchanger tubes operate in the upper subcritical range of Reynolds numbers (-104) where the drag coefficient for an isolated cylinder is 1.2, the vortex formation region is short (-0.5D to lD), separated shear layers are tubulent and vortex shedding is strong and regular. Our immediately after separation, observations at Re = 3500 also revealed a shortening of the formation region, especially at large spacing ratios, where the effects of interference among the cylinders are small. However, for small spacing ratios (LID < 3.5), there are small changes in the size and shape of the wakes as the Reynolds number increases, because there is little space for the lengthening of the shear layers in the downstream direction. The shear layers would bend to form different wake patterns due to the interference of the
124
K. LAM
AND
S. (‘
LC)
J Figure 8. Typical Row pattern in various arrangements (a) H = 0”. 3.94 K L/D; (b) H = 0”. I .70 c L/D c L/D < 1.70; (d) O= 10”. 3.94~ L/D: (c) f)= IO”. I-XX< L/D -- 3.Y4; (f) H = IO”. 3.94; (c) f3=0”, L/D < 1.88; (g) 0 = 20”. 2.40 < L/D; (h) 19= 20”. 1.2X c. L/D < 2.40; (i) B = 20”. L/D = 1.28.
neighboring cylinders. We believe that the wake patterns for small spacing ratios would remain as we observed, even if the Reynolds number is increased to the upper subcritical range. We also believe that the frequency of vortex shedding is mainly determined by the width of the wake and not by the length of vortex formation region. Hence, we would expect the Strouhal number characteristics to remain the same even if the Reynolds number is increased to the upper subcritical range. However, as the turbulence level in the shear layer is very high at Re = 104, we were unable to verify these points at the upper subcritical range of Re by the present visualization technique. Thus, an important point remains to be clarified: whether or not the phenomena of flow interference and vortex-shedding characteristics would still be the same as that in the lower subcritical range of Reynolds number. Attention should also be paid to the difference in flow patterns between a cylinder cluster and a multi-tube array. The effect of shielding of the downstream cylinder by the upstream cylinders for the in-line (0 = 0”) arrangement for a small spacing ratio is also observed in heat exchangers (Weaver & Abd-Rabbo 1985). For the rotated square (0 = 45”) arrangement, the extremely narrow wake between cylinders 1 and 3 at small spacing ratios results in a potential-like flow distribution around cylinder 1. This may provide some favorable physical justifications for some researchers using inviscid modelling for such a cylinder array. However, the existence of a wide wake and the low frequency of vortex shedding behind cylinder 3 is doubtful in a multi-tube array, where there is further interference from the remaining cylinders. Some lower Strouhal frequencies were detected by Pdidoussis et al. (1989), the exact nature of which is not known.
CROSS-FLOW
AROUND
FOUR
CYLINDERS
125
The characteristics of vortex shedding with spacing ratio are very similar to those reported by Chen (1977) and Fitz-Hugh (1973). For comparison, Fitz-Hugh’s results for both the in-line and rotated square arrays were incorporated in Figures 2 and 5.
4.4.
FURTHER
STUDIES
The study of force coefficients on cylinder clusters has always been related to the flow pattern. At present, most of the forces acting on two-cylinder groups have been measured extensively, in all kinds of arrangement (Bearman & Wadcock 1973; Igarashi 1981; Zdravkovich 1987; the magnitude and direction of the forces are also explained qualitatively by means of some plausible basic mechanisms. Unfortunately, for the cases of other three- or four-cylinder configurations, either regular or irregular, few experimental data are published; limited information on the force field of these arrangements can be used to compare the different mechanisms assumed to be operative in the two-cylinder case. It is felt that further measurements of the force coefficients on the cylinders of standard configurations are required, for the sake of verification of the basic mechanisms, as well as investigation of the relationships between the excitation forces and the instability threshold on a simple single array. In a visualization study, it is very important to obtain clarification on whether the effect of flow interference in the upper subcritical range of Reynolds numbers is the same as that in the lower subcritical range. It is suggested that a study in a larger water tunnel system with a better dye injection technique should be carried out, so that the phenomena at Re = lo4 can be observed. The length of the formation region should also be measured, as it is essential for the proper location of the hot wire. It was not feasible to do this measurement in this study. 5. CONCLUSIONS The flow patterns of various angle arrangements at different L/D values are summarized in Figure 8, with the dotted lines indicating the possible positions that the wake may have for some time. The results presented in this paper can be summarized as follows. (1). In the in-line arrangement, distinct vortex shedding of the upstream cylinders was suppressed below L/D = 3.94. The shear layers were not stable in attaching to the downstream cylinders and, instead, the wake deflected sharply from the streamwise direction toward one side. Intermittent and weak vortex shedding is observed, but the frequency is not recorded due to the rapid diffusion into the wake of the respective downstream vortex. Equally, the wake is also observed to extend sometimes to shield downstream cylinders. The Strouhal number, S, for L/D down to l-70 remains about the same as those obtained for the large spacing, i.e. -0.18-0.2 (Figure 2). When L/D decreases to below l-70, a bistable state starts to evolve behind the downstream cylinders. A wide and a narrow wake are found attached to the cylinders, which intermittently swap positions. The S value of the narrow wake attains a value of O-3 at L/D = 1.28 while the wide wake value drops to a minimum of 0.112 at L/D = l-41. The asymmetry is extremely sensitive to the flow incidence angle, and the wide or the narrow wake will easily be biased to one side, without any more interchanging of the wake pattern, as long as small angle is introduced to the incident flow.
126
K. LAM AND S. ('.1.0
(2). In the staggered arrangements, as L/D decreases, the formation region of the front three cylinders shrinks consistently, but the shedding frequency is increased inversely to the wake size, especially for the large incidence angles. A critical feature to all staggered positions is that the wide wake is biased behind cylinder 3 and is wider and appears earlier for the larger angle. The wide wake which is associated with low frequency is observed at L/D = 3.51 and 3.94 at 8 = lo” and 20” respectively; in the case of 30” to 45”, this is noticed up to the largest tested L/D ratio. For angles less than 20”, the near-wake behind the upstream cylinders bends sharply towards the side where very weak vortices are shed. until the L/D ratio is diminished to below 2.13. Then the shielding, or attachment, of a shear layer starts to occur, which is characterized by a region of stagnant fluid connecting the upstream and downstream cylinders. For the case of large angles of incidence up to 4.5”, cylinder 4 is free from direct wake interference from cylinder 1. The wakes of cylinder 2 and 4 deflect away from the central region for L/D < 2.40; this is likely forced by the two gap-flow exits from the model. The wake forms of the two cylinders (2 and 4) become increasingly alike as the angle increases; in fact, it is symmetrical at 8 = 45”. The wake of cylinder I is also controlled by the gap flow, and it is pressed into a very small region which enhances the formation of vortices. Clear, distinct vortices, though small in size, can be detected down to L/D reaches 1.28. The shear layers then attach onto the front portion of cylinder 3, forming a very narrow stagnant region between cylinders 1 and 3. The trend of the S values for all staggered positions indicates that it is not very sensitive to the angle of incidence. Despite the kinks in the experimental data, cylinder 1 always possesses the largest S value and increases to 0.64 at L/d = 1.41 and 8 = 45”. and cylinder 3 has the lowest value of S and drops to S = 0.08 at L/D = l-41 at 8 = 30”. The trend for the rest is similar to that of cylinder 1, but with lower values of S. The S values of cylinder 4 catch up with those of cylinder 2 as 0 reaches 45”. (3). An oscillating motion of the resultant wake can be found far downstream of a cylinder cluster of arbitrary number and orientation. as long as the cylinders are closely packed. The frequency of the oscillation. if nondimensionalized by the largest normal distance of the cluster to the flow. falls between 0.22 to 0.24 below L/D = 1.70. (4). The Strouhal number, S, representing the shedding frequency, appears to decrease with the increasing width of wake, W*, by an exponent of about -0.90. Although the data points were taken from different spacing ratios L/D, and angle of incidence 8, they are scattered within a certain region, with boundaries which can be described by the two equations given in Section 4. This indicates that S may be solely W*-dependent, while W* is a complicated function of L/D and 0.
6. ACKNOWLEDGMENTS The authors wish to thank the Research the Head of Department of Mechanical
Committee of the Hong and Marine Engineering
Kong Polytechnic for their support.
and
REFERENCES BALL, D. J. & HALL, C. D. 1980.Drag of yawed pile groups at low Reynolds numbers. Journal of the Waterways, Ports, Coastal and Ocean Division 106, 229-238. BAXENDALE, A. J.,GRANT, I. & BARNES, F. H. 1985. The flow past two cylinders different diameters. The Aeronautical Journal 89. 12S--134.
AXE having
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AROUND
FOUR
CYLINDERS
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BEARMAN, P. W. & WADCOCK, A. J. 1973. The interaction between a pair of circular cylinders normal to a stream. Journal of Fluid Mechanics 61, 499-511. CHEN, 3. S. 1983 Instability mechanisms and stability criteria of a group of circular cylinders subjected to cross-flow; Part 1: Theory. ASME Journal of Vibration, Acoustics, Stress and Reliability in Desgn 105,51-58. CHEN, Y. N. 1977 The sensitive tube spacing region of tube bank heat exchangers for fluid-elastic coupling in cross-flow. In Fluid-Structure Interaction Phenomena in Pressure Vessels and Piping Systems (eds M. K. Au-Yang & S. J. Brown), pp. l-18 ASME: New York. FITZ-HUGH, J. S. 1973 Flow-induced Vibration in Heat Exchangers Oxford University Report RS57 ( AERE-P7238). FITZPATRICK, J. A., DONALDSON, I. S. & MCKNIGHT, W. 1988 Strouhal numbers for flows in deep tube array models. Journal of Fluids and Structures 2, 145-160. GERRARD, J. H. 1966 The mechanics of formation region of vortices behind bluff bodies. Journal of Fluid Mechanics 25, 401-413. HILL, R. S., SHIM, K. C. & LEWIS, R. I. 1986 Sources of excitation in tube banks due to vortex shedding. Proceedings of the Institution of Mechanical Engineers 200, No. C4, 293-301. IGARASHI, T. 1981 Characteristics of flow around two circular cylinders arranged in tandem. Bulletin of the Japan Society of Mechanical Engineers 24, 323-331. KIYA, M., ARIE, M., TAMURA, H. & MORI, H. 1980 Vortex shedding from two circular cylinders in staggered arrangement. ASME Journal of Fluids Engineering 102, 166-173. LAKSHAMANA GOWDA, B. H. & PRABHU, D. R. 1987 Interference effects on the flow-induced vibrations of a circular cylinder. Journal of Sound and Vibration 112, 487-502. LAKSHMANA GOWDA, B. H. & DESHKLJLKANI, K. P. 1988 Interference effects on the flow-induced vibrations of a circular cylinder in side-by-side and staggered arrangement. Journal of Fluid Mechanics 122, 465-478. LAM, K. & CHU, K. H. 1986 Construction and performance of a recirculating water-tunnel system. Internal Report No. MME-T-6, Department of Mechanical and Marine Engineering, Hong Kong Polytechnic. LAM, K. & CHEUNG, W. C. 1988 Phenomena of vortex shedding and flow interference of three cylinders in different equilateral arrangements. Journal of Fluid Mechanics 196, l-26. OWEN, P. R. 1965 Buffeting excitation of boiler tube vibration. I. Mech. E. Journal of Mechanical Engineering Science 7, 43 l-439. PA~DOUSSIS, M. P. 1980 Flow-induced vibrations in nuclear reactors and heat exchangers: practical experiences and state of knowledge. In Practical Experiences with Flow-induced Vibrations (eds E. Naudascher & D. Rockwell), pp. l-81. Berlin: Springer-Verlag. PA~DOUSSIS, M. P. 1983 A review of flow-induced vibrations in reactors and reactor components. Nuclear Egnineering and Design 74, 31-60. PA~DOUSSIS, M. P. & PRICE, S. J. 1984 The aerodynamic forces acting on groups of two and three circular cylinders when subjected to a cross flow. Journal of Wind Engineering and Industrial Aerodynamics 17, 329-347. PA~DOUSSIS, M. P., PRICE, S. J., MACDONALD, R. & MARK, B. 1987 The flow-induced vibration of a single flexible cylinder in a rotated square array of rigid cylinders with pitch-to-diameter ratio of 2.12. Journal of Fluids and Structures 1,359-378. PA’~DOUSSIS,M. P., PRICE, S. J. & MARK, B. 1986 An experimental stability analysis of a single flexible cylinder positioned in an array of rigid cylinders and subject to cross-flow. ASME Journal of Pressure Vessel Technology 108,62-72. PA~DOUSSIS, M. P., PRICE, S. J., NAKAMURA, T., MARK, B. & NJUKI MUREITHI, W. 1989 Flow-induced vibrations and instabilities in a rotated-square cylinder array in cross-flow. Journal of Fluiak and Structures 3, 229-254. PANNELL, J. R.. GRIFFITHS, E. A., & COALES, J. D., 1915 Experiments on the interference between pairs of aeroplane wires of circular and lenticular cross section. (British) Advistory Committee for Aeronautics, Reports and Memoranda No. 208, Annual Reports for 191551916, 7, 219-221. PE~TIGREW, M. J., TROMP, J. H. & MASTORKOS, J. 1985. Vibration of tube bundles subjected to two-phase cross-flow. ASME Journal of Pressure Vessel Technology 107,335-343. SAYERS, A. T. 1987 Flow interference between three equi-spaced cylinders when subjected to a cross flow. Journal of Wind Engineering and Industrial Aerodynamics 26, 1-19. TANAKA, H. & TAKAHARA, S. 1981 Fluid elastic vibration of tube arrays in cross flow. Journal of Sound and Vibration 77, 19-37.
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WEAVER, D. S. & ABD-RABBO, A. 1985 A flow visualization study of a square array of tubes in water crossflow. ASME Journal of Fluids Engineering 107,354-363. WEAVER, D. S. & FITZPATRICK, J. A. 1988 A review of cross-flow induced vibrations in heat exchanger tube arrays. Journal of Fluids and Structures 2,73-93. WEAVER, D. S. & LEVER, J. H. 1982 A theoretical model for fluid-elastic instability in heat exchanger tube bundles. ASME Journal of Pressure Vessel Technology 104,147-158. WEAVER, D. S. & YEUNG, H. C. 1984 The effect of tube mass on the flow induced response of various tube arrays in water. Journal of Sound and Vibration 93, 409425. WILLIAMSON, C. H. K. 1985 Evolution of a single wake behind a pair of bluff bodies. Journal of Fluid Mechanics 159, 1-18. WILLIAMSON, C. H. K. & ROSHKO, A. 1988 Vortex formation in the wake of an oscillating cylinder. Journal of Fluids and Structures 2, 355-381. ZDRAVKOVICH, M. M. 1968 Smoke observation of the wake of a group of three cylinders at low Reynolds number. Journal of Fluid Mechanics 32, 339-351. ZDRAVKOVICH, M. M. 1977 Review of flow interference between two circular cylinders in various arrangements. ASME Journal of Fluids Engineering 99, 618-633. ZDRAVKOVICH, M. M. 1987 The effects of interference between circular cylinders in cross flow. Journal of Fluids and Structures 1,235-261.
CROSS-FLOW
AROUND
FOUR
CYLINDERS
Plate 1. Flow visualization for (a) 0 = o”, L/D = 5.32; (b) 0 = O”, L/D = 2.65; (c) 0 = O”, L/D = 1.54.
129
a)
f) plate 2. Flow visualization for (a) 0 = lO”, L/D = 3.09; (b) 0 = 10”. L/D = 1.70: (C) H = (d) 0 = 20”. L/D = 349: (e) 0 = 20”. L/D = 1.70: (f) H = 20”. L/D = 1.28.
10”.
!
=
1.3
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131
a)
b)
d) Plate 3. Flow visualization
for (a) 0 = 45”, L/D = 5.32; (b) 0 = 45”, L/D = 2.13; (c) 0 = 45”, L/D = 1.54; (d) 0 = 45”. L/D = 1.28.
A VISUALIZATION STUDY OF CROSS-FLOW AROUND FOUR CYLINDERS IN A SQUARE CONFIGURATION K.
LAM
AND
S. C.
Department of Mechanical and
Hong Kong Polytechnic Hong
FLOW-INDUCED
VIBRATIONS
IMPOSED
on
a single
bluff
body
or a bundle
of bluff-body
are of great engineering importance. Many examples of failure in engineering applications, as in power transmission lines, nuclear power station heat exchangers, suspension bridges, etc., due to this destructive phenomenon have been reported (Paidoussis 1980) and show the need for research to overcome the vibration problem. In the last two decades, a large number of studies,. development of mathematical models as well as experimental investigations, have been carried out worldwide for this specific purpose. Throughout this literature, it is found that generally there are three kinds of mechanisms responsible for the vibration. They are (a) fluidelastic instability, (b) turbulent buffeting and (c) vortex-shedding excitation. In the study of fluidelastic instability, authors like Chen, Paidoussis, Weaver et al. and others, through their theoretical ideas and experimental verifications, established the general guidelines for studying the problem. They reasoned that the instability is generally associated with high-flow speed, and the threshold depends largely on the arrangement of the bodies. Semi-empirical models, relying on the influence of the motion of the surrounding tubes like negative damping and fluid-stiffness terms (Chen 1983), or more theoretical models which treat each tube as independently unstable and modelled by the unsteady flow equations (Weaver & Lever 1982), have evolved lately and achieve reasonable predictions on the instability onset for certain arrays. In the study of turbulent buffeting, most of the studies make use of statistical ideas, and it is well accepted that the induced vibrations exist over a broad band of frequencies with small response amplitude; in some cases, the response varies linearly with flow speed, U, rather than the expected U2 (Pdidoussis ef al. 1987). At one time, structures
0889%9746/92/010109 + 23 sos.Gu
@ 1992Academic Press Limited
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K. LAM AND S. C. LO
there arose confusion between turbulent buffeting and the wake-shedding mechanism; Owen (1965) considered that the sharp peak observed in the response spectrum could have been generated by one dominant frequency-band of turbulence. However, after several years of verifications, the existence of periodic and narrow-band vortexshedding excitation prevailed. Since the time that Aeolian tones were first noticed, vortex-induced vibration has attracted attention. Research workers have looked into many interesting aspects of the mechanism. The establishment of constant Strouhal number of O-2 for a single cylinder in a wide range of Reynolds number, 400 to nearly 2 x lo’, and the discovery of the “lock-in” effect whereby vortex shedding follows the natural vibration of the tube and leads to resonant response over a range of speed, are some highlights of the investigations. In 1915, Pannell et al. reported the force measurements on a pair of parallel “circular wires” in close proximity. They found that within a small angle of stagger, the combined drag force was even smaller than that of a single wire. The results aroused the interest of studying the interference effects of surrounding bodies on the wake structure. Zdravkovich (1977, 1987) summarized an enormous amount of information like pressure, lift and drag coefficients, and Strouhal number around two tubes or multi-tube array systems. It seems that each system of the tube array carries a unique interference characteristic that the principle of superposition cannot overcome for adequate prediction. Weaver & Yeung (1984) and Sayers (1987) both pointed out that there is a need of basic experimental data for various standard configurations. The present study is carried out in this direction. The effects of different how incident angles and the spacing ratio on the flow characteristics around four cylinders in square arrangements is examined closely in this study, as the lack of data of this type breaks the connection from the most basic two-cylinder system to multi-cylinder arrays. Previously Lam & Cheung (1988) examined the shedding frequency characteristics and flow interference of three cylinders in different equilateral arrangements by means of the same apparatus and technique as in this experiment. Based on their experience, the Reynolds number effect is weak and adds only minor discrepancies to the Strouhal number characteristics, at least, within the range 2.1 x lo3 < Re < 3.5 x 10”. The present experiment is conducted at a lower flow velocity, Re = 2-l x 103, to obtain better photographic evidence. The study was focussed on the flow characteristics of four cylinders arranged in a square, which constitutes another type of the simplest unit of a tube bank. As before, a dye injection visualization technique is employed to study the changes during the variation of spacing ratio, 5.96 to 1.28, and the angle of incidence, 0 to 45” such that all possible orientations of the square array are covered. The results are categorized according to the orientation of the model, i.e., angle of incidence. The flow fields around the cylinders are summarized in steps of spacing ratio and are demonstrated by photographs.
2. EXPERIMENTAL
DETAILS
The experiments were carried out in the water tunnel of the Mechanical & Marine Engineering Department, Hong Kong Polytechnic and the details of the tunnel are described by Lam & Chu (1986). In the experiment, the flow speed was chosen to be O-223 m/s, which gives a Reynolds number of 2100, a value right inside the lower subcritical range where the
CROSS-FLOW
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111
Reynolds number effect is small and also provides a good condition for visualization. The corresponding turbulence level is within 1% . The cylinders are 9.4 mm in diameter and 200 mm in length; they span across the O-2 m x O-2 m x O-5 m (length)
v -30
1 -1
- 1
D
Y
D2
t i._
X
,”
0
- 4
Figure 1. General configuration of the model.
112
K. LAM AND S. C. LO
3. RESULTS During the experiment, several features of flow that were found in the 2-cylinder and 3-cylinder cases were also found in the 4-cylinder cases. The interference among the cylinders reduces to a minimum as the distances between them becomes large. But then, as L/D decreases, complicated influences would come into play in which the angle of the flow plays a dominant role in determining the flow pattern and, hence. the Strouhal number characteristics. In order to simplify the description, the cylinders are labelled as 1 to 4. as in Figure 1 throughout the paper. The angle of rotation is in an anti-clockwise sense, so the cylinder 2 will be in the highest position in Figure 1 if the angle of incidence is 4.5”. To provide a systematic and detailed description of the phenomena, the results are grouped in three distinct arrangements according to the range of angle of incidence. the results are interpreted in descending L/D order for each Furthermore, arrangement. The Strouhal numbers for each cylinder, plotted against the spacing ratio, are summarized in Figures 2 to 5, and the reader should note that the Strouhal number in the graphs are normalized by the free-stream speed and the cylinder diameter. At 8 = O”, an important feature is that the flow exhibits a bistable state of a wide and a narrow wake behind the downstream cylinders at and below L/D = 1.54. Similar to the results of Bearman & Wadcock (1973), the shedding frequency of the wide wake is low and the narrow wake corresponds to a high frequency. At flow-incidence angles other than O”, it is certain that the wide wake is biased to cylinder 3 which is now situated in the aft-most position. The rest of the cylinders, on the other hand, possess a smaller size of wake than that behind a single isolated cylinder, and the implication is that higher values of Strouhal number would result, which agrees well with those obtained in the experiment. The trend for the Strouhal number characteristics is that they are rather insensitive to the change of angle, and the irregularities are largely reduced when compared to the results of three cylinder clusters (Lam & Cheung 1988). At 8 = 45”, the arrangement is geometrically symmetrical in the flow direction just like the in-line arrangement; however, the bistable state of wakes around the cylinders of the latter case is completely replaced by a symmetric pattern when L/D approaches unity. The wake length of the foremost cylinder (1) is incredibly cut down, such that tiny vortices are formed before impacting on the front face of the downstream cylinder (3). 3.1. IN-LINESQUARE ARRANGEMENT 3.1.1.
Vortex-shedding
charucteristics
( 0 = 00)
The Strouhal number (hereafter denoted by S) characteristics at 0 = 0” are shown in Figure 2. At large spacing, 5.96 < LID < 4.47, the vortex-shedding frequency values of the four cylinders are close to that of a single cylinder. Although S for each cylinder does not exactly have the same value, the small variation between them does not reveal any trend and remains within an 8.5% maximum deviation from that obtained for a single cylinder (S = 0.2) at LID = 5.96. However, at LID = 3.94, S of the downstream cylinders decreases, and the difference from the upstream ones, which are now only 3% away from 0.2, rises to about 12% of the single cylinder value. Further reducing the spacing ratio, it is observed that the regular shedding behind the upstream cylinders is “suppressed”. It is important to realize that although the
113
CROSS-FLOW AROUND FOUR CYLINDERS 1.0
2.0
3.0
4.0
6.0
5.0
0.7
0.6
0.5
“. 5
0.4
2 z 2 : arz
0.3
0.2
0.1
Cylinder 3
0.cb
1.0
I 2.0
I 3.0
,
I
I
4.0
5.0
6.0
Spacing ratio. L/D cylinder Figure 2. Strouhal number, S, plotted against spacing ratio, L/D, for f3=o”. --O-, -.-A-.-, cylinder 2; --V--, cylinder 3; -..a..--, cylinder 4; -O--, Fitz-Hugh’s results.
1;
spacing in between the cylinders does not allow normal vortex formation, the near-wake of the upstream cylinders is not sufficiently stable to continue providing shielding for the downstream ones. From time to time, the wake is seen to be deflected from the streamwise direction and only fragmented shedding periods can be recorded. The situation is reflected by the abrupt end of results for cylinders 1 and 2 at L/D = 3.94.
Repeated experiments were carried out at small angles of incidence to ensure that this is not due to a misalignment of the angle. It is believed that the “deflection” is not due to the turbulence level of the water tunnel, but rather to the presence of the two
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nearby tandem cylinders; the results of two cylinders in a tandem arrangement in the same tunnel and same Reynolds number show clearly that the upstream cylinder near-wake is stable and shields the downstream cylinder throughout the whole range of spacing. For the downstream cylinders, the shedding frequency decreases slightly at L/D = 3.51, but then starts to climb back slowly to 0.2 at L/D = 1*70. There is neither a discontinuity nor a jump phenomenon in S as suggested by Kiya et al. (1980), Baxendale & Barnes (1986), and Lam & Cheung (1988). All these results for two and three cylinders conclude that, at the spacing that the upstream cylinder near-wake starts to shield the downstream cylinder, due to modification of the streamlines so that now the two cylinders appear more as one streamlined body, the Strouhal number found behind the latter will jump and then follow a trend to a value much higher than that of a single cylinder as the spacing decreases. In this case, the tendency of streamlining is largely affected by the fact that the near-wake of the upstream cylinder deflects, which is indicated by the proximity interference of the other, adjacent tandem pair of cylinders. When L/D drops below 1.70, a bistable feature evolves behind the downstream cylinders and dominates the flow characteristics. A narrow wake of high shedding frequency and a wide wake of low frequency are recorded as in the results of Bearman & Wadcock (1973) and Lam & Cheung (1988). The high S value reaches O-36 at L/D = l-28, while the low S drops to a minimum of 0.11 at L/D = l-41 and then rises slightly back to O-12 at L/D = 1~28. The intermittent change of the two wakes is also observed. But for simplicity of presentation, the high frequency is assigned to cylinder 3 and the low one to cylinder 4. 3.1.2. Observations on fiow patterns (0 = 0’) At large spacing, 5.96 < L/D < 3.94, as shown in Plate l(a) the wake formation regions of the downstream cylinders are much shorter than the upstream ones. This is due to the fact that vortices are shed by the upstream cylinders, creating a more turbulent flow behind, accompanied by an early transition from a laminar to a turbulent shear layer and hence the shrinkage of the formation region. The four cylinders, in this spacing range, can be thought of as a pair of identical tandem as no prominent “proximity” effect is detected, but the “wake arrangements, interference” is obvious. Whenever the upstream vortex of either sign comes close to the downstream cylinder, it provokes the formation of a vortex of the same sense. The phase difference between the upstream and downstream cylinders is dependent on the distance between them. At L/D = 3.94, the vortices shed from behind cylinders 1 and 2 are so close to the front face of the cylinders 4 and 3, respectively, that occasionally the two shear filaments of the former cylinders do not have enough time to grow into a mature vortex; hence, a regular mature vortex shedding is no longer distinct, although regular shedding is still recorded most of the time. The flattening and lengthening of the wake lasts only for a short time before the recovery of the curling up of shear layers into vortices. During that short period of suppression of vortex formation, the downstream cylinders are being immersed in the wake of their respective upstream cylinders. However, the changing of states apparently does not affect the shedding frequency of the downstream cylinders. The intermittent changes become more and more pronounced as the spacing ratio decreases, the period of regular shedding is largely shortened which is revealed by the
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CYLINDERS
115
stoppage of the curves for cylinders 1 and 2 at L/D = 3.94. In fact, below this spacing, the shedding found must be accompanied by a deflection of the shear layer as illustrated by the dotted lines in Figure g(b). The lower shear layer of cylinder 2 bends sharply and follows the contour of the downstream cylinder to the upper side when the spacing is below 240. The vortex formed in this way is small and the shear layer on that side of the respective downstream cylinder seems to be affected and leaves the cylinder surface at some angle from the X-axis, which explains the momentary occurrence of a wide wake near the downstream cylinders. But what is usually found is that the straight shear filament extends streamwise from cylinders 1 and 2, the wake shields the downstream cylinders 4 and 3 and no vortex is observed. Below L/D = 1.70, Plate l(c) shows that the bistable state of the wide and narrow wake which is attached behind cylinders 3 and 4. A narrow gap flow is biased to the narrow wake just as in the case of side-by-side cylinders. The initiation of the asymmetric mode is originated from an outward deflection of the outer shear layer of cylinders 1 or 2, which rolls up beside cylinders 4 or 3 without the presence of the inner one which, at this time, does not bend sharply outward but appears straight. It may attach at about 50 to 60” clockwise from the foremost point of the downstream cylinder or may just extend behind it. This condition promotes the outward shear layer of cylinder 4 or 3 to separate at an even wider angle. Once the wide wake is established, the gap flow between the two rows of cylinders is pushed to another side where the narrow wake is formed.
3.2.
SLIGHTLY
STAGGERED
3.2.1. Vortex-shedding
ARRANGEMENT
characteristics
(8 = lo”, 20”)
Figure 3 shows that the Strouhal number characteristics of the cylinders at 8 = 10” are largely different from the in-line arrangement. Only in the range L/D = 5-96 to L/D = 4.47 does S have a behaviour similar to the in-line situation. The value of S consistently remains in the range O-18 to 0.21, because the spacings are large enough to prevent any interference between the cylinders. For L/D < 4.47, the individual S values of the four cylinders diverge from one another. The S values of cylinders 1, 2 and 4 increase moderately from O-2 for up to L/D = 3.09, before rising sharply; on the other hand, the S value of cylinder 3 decreases in a stepwise fashion. For cylinders 1, 2 and 4, the S increase for cylinder 1 is largest, followed closely by that of cylinder 2 and then cylinder 4. The difference in S values between cylinders 1 and 4 is about 0.04 at L/D = 3.09; for L/D = 2.40, however, the difference is double. For L/D < 2.40, the shedding from cylinder 1 is weak and the visualization method is not capable of demonstrating the vorticity cloud that may have formed, because it diffuses quickly into the downstream wake as the shear layers almost touch the cylinder surface. Since no distinct vortex can be seen, the S value could only be determined down to L/D = 2.4 and, in like fashion, the same behaviour occurs for cylinder 2 at L/D = l-70. Returning to cylinder 3, it can be seen that the decrease of S begins at L/D = 4.47. It starts initially at a value of 0.17 between L/D = 3.94 and 3-51, then decreases to O-13 between L/D = 3.09 and 2.40 and finally reaches about O-09 in the last few values of L/D ratio. The S behavior is not related to the suppression of vortices from both cylinders 1 and 2, as seems to be evident in the figure; the L/D value at which the shielding occurs does not correspond to the L/D value at which the step changes of S
116
K. LAM AND S. C. LO
0.7
0.6
0.5
m r b 2 z -z 5 e + Lo
0.4
0.3
0.2
0.1
0.0 1.0
24
3.0 Spacing ratio.
I
,
4.0
5.0
6.0
L/D
Figure 3. Strouhal number, S, plotted against spacing ratio, L/D, for 0 = 10”; (see Figure 2 for symbol legend).
take place on cylinder 3. It merely indicates that, within a certain of cylinder 3 is insensitive to the influence of the others.
range,
the near-wake
If we look ahead at Figures 4 and 5, we note that the increase of angle of the arrangement has the effect of smoothing the kinks of the S behavior for cylinder 3. It may be reasoned that, at small angles, the flow around cylinder 3 is restricted by the wake of cylinder 2 and, at the same time, the vorticity of this wake also distrubs the shedding of the upper shear layer. The influence decreases as the angle increases, and the wake of cylinder 3 is mainly affected by the two more regular flows around cylinder 1. (The reader is also referred to Section 3.4.)
CROSS-FLOW
0.7
14.I
2.0 I
AROUND
3.0 I
FOUR CYLINDERS
4.0 1
117 5.0
6.0 ,
0.6
0.5
m Ei .
0.4
z z z z s
0.3
0.2
:’ 0. I
.
,V-t--*
\
T
Cylinder 3
OJJ/
1.0
2.0
34
3.0
5.0
6.0
Spacing ratio, L/D Figure 4. Strouhal number, S, plotted against spacing ratio, L/D, for 0 = 30”; (see Figure 2 for symbol legend).
3.2.2. Observations on j7ow patterns (8 = lo”, 20”) As L/D decreases to 4.47 and 4.89 at 8 = 10” and 20”, respectively, the flow patterns are analogous to that in Plate l(a), although the upstream cylinders are not directly in front of the downstream ones. The near wake of the former remains long and the latter remains short. Plate 2(a) shows the flow field at L/D = 3-W for 0 = 10”. It is typical of those found in the range from 3.94 to l-88. The wake of the upstream cylinders is relatively long and the shear layers that emanate from the separation points appear straight but roll up and “cloud” adjacent to the upper surface of the downstream cylinders. In doing so,
118
K. LAM AND
Cylinder
S. C. LO
2
s.0
4.11
Spacing Figure
5. Strouhal
number,
S, plotted
against
spacing legend)
ratio. ratio,
Lit>
L/D, for 0 = 45”. (see Figure
2 for symbol
the size of the wake is smaller as the undisturbed flow above maintains the upper layer in approximately the streamwise direction. The consequences of the fluctuation near the upper separation points for cylinder 3 are to cause the respective shear layer to roll up into small clouds more readily, which then convect downstream quickly and catch up to those shed previously. The clouds also entrain those vortices from cylinder 2 in the wake and merge together to form a
CROSS-FLOW AROUND
FOUR CYLINDERS
119
bigger vortex street. The lower shear layer becomes straight and long before curling up; the reason is that it is free from the direct disturbances of the wake from upstream. The above discussion can also be applied to cylinder 4, except for the wide wake which is replaced by a much narrower wake. The pattern shown in Plate 2(b) can be obtained at both L/D = 1.70 and 1.54; there are no fundamental differences from the previous picture and yet the wakes of cylinders 1 and 2 are now largely reduced in size and the lower shear layer curves more than 90” to recover the position on the upper side of the downstream cylinders. The wake of cylinder 1 is in between the shielding and the sharp bending of the filament to the other side. The situation changes again as soon as L/D decreases. In Plate 2(c), the abrupt curvature of the shear layer is absent, the lower shear layer of cylinder 2 is now extended and passes beneath the downstream cylinder and the upper shear layer from cylinder 1 is shifted down and attaches to the front face of cylinder 4. Plate 2(d, e, f) show the flow tield at 8 = 20”. Based on a careful comparison with the previous pictures, it is noticed that the wake of cylinder 3 is getting wider and those in the front a little narrower. Moreover, the angle opens up for the flow from beneath and encourages the curling up of the lower shear layers of the front cylinders which in turn delays the suppression of vortex formation.
3.3. LARGE ANGLE
STAGGERED
ARRANGEMENT
3.3.1. Vortex-shedding characteristics (8 = 30”, 40”, 45”) As the angle of incidence is increased, the wake interference of cylinder 1 on cylinder 4 decreases, but this wake now affects cylinder 3. The upstream influence of cylinder 1 comes much earlier than that in the slightly rotated or in-line square arrangement, as can be seen in Figures 4 and 5. There is a change in the S value between cylinders 1, 2 and 4, which possess higher values, and cylinder 3, which has a lower value, even at the largest L/D ratio in the experiment. At L/D = 5.96, the largest difference in S values for this range of incidence angles remains as high as 14 to 19% of the S value of an isolated cylinder. Both figures show that the S value of the front three cylinders increases sharply as L/D drops below 2.65, especially for cylinder 1. The larger angle of incidence shows the greatest variation in S between cylinder 3 and the rest, but the trend for each cylinder is the same as in the previous section. At 0 = 30”, S for cylinder 1 rises from 0.34 at L/D = 2.65 to 0.57 at L/D = 1.41; at 8 = 45”, it rises from 0.34 to 064. On the other hand, at 8 = 45”, S for cylinder 3 reduces gradually to about O-075 at L/D = 1.70 to 1.88 and increases slowly to about 0.08 at L/D = 1.28. It should be noticed that the suppression of vortex formation only happens on cylinder 1 and also that the L/D ratio at which it takes place is the smallest tested, i.e., 1.28. Comparing the two figures, it is noticeable that, at many spacing ratios, the S value of cylinder 2 in the case of smaller angles is higher. In Figure 4, S for cylinder 4 is generally lower than that of cylinder 2, then catches up with it as the angle increases. At 8 = 45”, as shown in Figure 5, the S values for the two cylinders are close, which is expected, because the geometric positions of the two cylinders are now identical. Moreover, as can be seen in the figures, no bistable state is ever observed throughout the whole range of spacing.
120
K. LAM AND S.C. LO
3.3.2. Observations on flow patterns (0 = 30”, 40”, 45”) In Plate 3(a), the pattern shown is typical of those in the range of L/D > 2-40; the wakes of the front cylinders remain regular and lie in the flow direction as before, but that of cylinder 3 already indicates the onset of a wide wake. The wake of cylinder 1 is comparatively narrow due to the gap flow on each side rushing into the central region and the natural expansion of the shear layer in the transverse direction after separation is restricted. Despite the wake being sandwiched between the gap flow, it is not always stable in staying in place in the middle, but sometimes may be biased upward or downward randomly and remains there for a short time before changing again. However, the random motion is small in amplitude. Plate 3(b,c) represents the flow for 1.28 < L/D < 240. At an angle 8 > 30”, cylinder 1 divides the frontal entry of the flow between cylinders 2 and 4. The gap flow passing each side of cylinder 1 suppresses its near wake into a very narrow region. The narrowing of the distance between the two shear layers enhances the conditions for rolling up and lessens the longitudinal distance for vortex shedding, causing a delayed suppression of vortex shedding. At the exit of the gap flows on each side of cylinder 4, there is a wake divergence which depends on an angle which, in turn, depends on the L/D ratio of the model. The smaller the L/D value, the larger the angle of the flow. It should be noted that the formation regions of cylinders 2 and 4 are also largely reduced in length and width and, more importantly, they are strongly affected by the exiting gap flow which rotated the wake axis by up to nearly 30”. However, as the wake proceeds further out, it loses the momentum to maintain that direction and the free stream overtakes the gap flow position and pushes the wake back in the streamwise direction. At the same time, the gap flow from each side of cylinder 1 also leads to the development of straight shear layers which are separated from cylinder 3. The layers then meet those from cylinders 2 and 4 some distance downstream and mix together to form a low frequency wake oscillation. Plate 3(d) shows the flow field of the smallest L/D, i.e., 1.28, which is similar to the other situations at 8 > 30”. The most impressive point is that the weak vortex street behind cylinder 1 disappears, but is replaced by a narrow stagnant region as the short distance suppresses the curling of the shear layer before striking the front face of the downstream cylinder. Consequently, the layers follow the contour of the cylinder face, and the two points of contact on the cylinder are close to each other; if measured from the centre of the cylinder, the angle subtended is less than 40”. 3.4. STROUHAL NUMBERS
FOR CYLINDERS
1 AND
3 FOR ALL ANGLES
To obtain an overall view on the vortex-shedding characteristics of a particular cylinder, the variations of Strouhal number S with spacing ratio L/D for cylinders 1 and 3 at all tested angles of incidence were plotted in Figure 6. The results clearly illustrate that cylinders 1 and 3 possess distinctly different characteristics. All the S values of cylinder 1 decrease smoothly from O-6 to O-2 as the spacing ratio increases while the S values of cylinder 3 rise from 0.08 to 0.2 in a somewhat stepwise manner especially when the spacing ratio is small. 4. DISCUSSION 4.1. CLUSTER EFFECTON FAR WAKE It is evident, in Plates l(c), 2(e,f) and in fact for any angle orientation that, as long as L/D < 1.70, a distinct oscillating motion of wake exists far downstream. These sinuous
CROSS-FLOW 2.0
0.7 I.0
AROUND
FOUR
4.0 I
3.0 I
121
CYLINOER~ 5.0
6.0 I
i---0.6
!
Cylinder
1: -
Cylinder
3: - - - - -
L
I -
I_
i-o Spacing Figure
6. Strouhal
number,
S, plotted
against
4.0
ratio.
spacing ratio, incidence.
L/D L/D,
for cylinders
1 and 3 for all angles
of
oscillations result from the merging of the far wake of the four cylinders. Actually, these results can also be found in the 3-cylinder case in various flow speeds (Lam & Cheung 1988; Zdravkovich 1968). It is interesting to note that the period of this wake oscillation coincides with that of the wide wake of cylinder 3 at all angle arrangements except 0”. Plates l(c) and 2(e) show clearly the domination of the wide wake: the weak vortex streets of the rest are just being entrained into the central region. It was suggested by Zdravkovich that this oscillating motion is due to the curling of the outermost shear layer while the inner layer vorticity coalesces and dies out. In such a way, the cluster of whatever arrangement could then be imagined as a large body and the wake oscillation as a new vortex street. An evaluation of S according to this idea is
122
K.
LAM AND S. c‘
LO
TABLE I Strouhal
number
for wake oscillation, normalized values of spacing
Angle of incidence,
by D,, for small
H
Spacin ratio, L.PD
0”
IO”
20”
30”
40”
45”
1.28 1.41 1.54 1.70
0.23 O-24 0.23 0.28
0.22 0.23 0.23 0.30
0.22 0.23 0.23 0.26
0.23 0.22 0.24 0.25
0.24 0.23 0.24 0.25
0.24 0.24 0.24 0.26
examined and listed in Table 1. The diameter, L),, , is the largest normal distance of the cluster to the flow at the current angle. From the table, we can see that, between L/i) = 1.28 and 1.54, S falls consistently into the range of O-22 to 0.24 but diverges in an inconclusive manner if L/D increases beyond 1.70.
4.2.
DIMENSIONLESSWAKE WIDTH AND STROUHAI. NUMBER
It has been concluded by Gerrard (1966) that, in the case of a single cylinder. the entrainment of fluid from the interior of the formation region and its replenishment by reversed flow is fundamental in determining the wake size. The two characteristic lengths of the wake, namely the scale of the formation region and the width to which the free shear layers diffuse, determine the frequency of vortex shedding. In this experiment and in some other multi-cylinder cases, the results show that the interference between cylinders may largely affect the size of the wake without altering the shedding frequency. For example, in Plate l(a), the downstream cylinder wake encounters a very turbulent flow from the upstream wake and the length of the downstream wake is largely reduced; yet, the frequency of shedding remains the same as for the upstream cylinders. But for the latter characteristic length. just as the well-known result that a wide wake always contains low frequency and the narrow wake high frequency in the two-cylinder case, certain relationships between shedding frequency and wake width may exist. In our case, measurements of the wake width, W, or the thickness between the two shear layers at the end of the formation region where the layer is drawn across the wake (nondimensionalized by the cylinder diameter, W* = W/D), corresponding to the S values obtained earlier, show that. no matter what the orientation or spacing ratio of the model, the points he within a certain region. This region is bounded on its upper and lower sides by two curves as in Figure 7; their respective equations are S = 0.356 W*-O‘xy,
S = 0.156 W*-“+“.
The variation of S with W * for the 45” orientation corresponds to the middle full line which can be approximated by S = O-264 W*-“‘. However, the Strouhal number and width relation for the 0” orientation does not follow the same trend. Instead, the values of S scatter around O-2 for O-8 -=cW* -=c2. This may be due to the “tandem effect” of the cylinders, such that the wake widths of the cylinders can neither be too wide nor too narrow in most cases.
CROSS-FLOWAROUND FOUR CYLINDERS
123
I_
(1/
Wake width. Figure
7. Strouhal
number,
S. plotted against A, f3=30”;
s
3
2
I
W*
wake width, W*. Angle: x, 0=40”; V, %=45”.
0.
0 = 0”; +, 0 = 10”; 0,
0 = 20”;
4.3. APPLICABILKY OF RESULTS TO MULTI-TUBE HEAT EXCHANGERS A cluster of four cylinders in a square configuration may be considered as an element of an in-line (0 = 0”) and rotated square (0 = 45”) tube bank in heat exchangers. The present visualization study may provide some clarifications and additional information on vortex shedding and flow interference due to change of spacing ratio and flow direction that may occur in heat exchangers. However, in attempting to draw any conclusions, the following points should be noted. The visualization study was chosen at Re = 2100 because, for the present set-up, the flow visualization and photography are the best at such a Reynolds number. Investigations carried out at a slightly higher Reynolds number (Re = 3500) indicated that the vortex-shedding characteristics and flow pattern are basically the same. Nevertheless, Re = 2100-3500 are still within the lower subcritical range of Reynolds numbers. It is well known that for a single cylinder, at Re = 2100, the drag coefficient (C, = 0.9) is lower and the length of the vortex formation region (-3.50) is longer than that which would occur in the upper subcritical range of Reynolds numbers. Most heat exchanger tubes operate in the upper subcritical range of Reynolds numbers (-104) where the drag coefficient for an isolated cylinder is 1.2, the vortex formation region is short (-0.5D to lD), separated shear layers are tubulent and vortex shedding is strong and regular. Our immediately after separation, observations at Re = 3500 also revealed a shortening of the formation region, especially at large spacing ratios, where the effects of interference among the cylinders are small. However, for small spacing ratios (LID < 3.5), there are small changes in the size and shape of the wakes as the Reynolds number increases, because there is little space for the lengthening of the shear layers in the downstream direction. The shear layers would bend to form different wake patterns due to the interference of the
124
K. LAM
AND
S. (‘
LC)
J Figure 8. Typical Row pattern in various arrangements (a) H = 0”. 3.94 K L/D; (b) H = 0”. I .70 c L/D c L/D < 1.70; (d) O= 10”. 3.94~ L/D: (c) f)= IO”. I-XX< L/D -- 3.Y4; (f) H = IO”. 3.94; (c) f3=0”, L/D < 1.88; (g) 0 = 20”. 2.40 < L/D; (h) 19= 20”. 1.2X c. L/D < 2.40; (i) B = 20”. L/D = 1.28.
neighboring cylinders. We believe that the wake patterns for small spacing ratios would remain as we observed, even if the Reynolds number is increased to the upper subcritical range. We also believe that the frequency of vortex shedding is mainly determined by the width of the wake and not by the length of vortex formation region. Hence, we would expect the Strouhal number characteristics to remain the same even if the Reynolds number is increased to the upper subcritical range. However, as the turbulence level in the shear layer is very high at Re = 104, we were unable to verify these points at the upper subcritical range of Re by the present visualization technique. Thus, an important point remains to be clarified: whether or not the phenomena of flow interference and vortex-shedding characteristics would still be the same as that in the lower subcritical range of Reynolds number. Attention should also be paid to the difference in flow patterns between a cylinder cluster and a multi-tube array. The effect of shielding of the downstream cylinder by the upstream cylinders for the in-line (0 = 0”) arrangement for a small spacing ratio is also observed in heat exchangers (Weaver & Abd-Rabbo 1985). For the rotated square (0 = 45”) arrangement, the extremely narrow wake between cylinders 1 and 3 at small spacing ratios results in a potential-like flow distribution around cylinder 1. This may provide some favorable physical justifications for some researchers using inviscid modelling for such a cylinder array. However, the existence of a wide wake and the low frequency of vortex shedding behind cylinder 3 is doubtful in a multi-tube array, where there is further interference from the remaining cylinders. Some lower Strouhal frequencies were detected by Pdidoussis et al. (1989), the exact nature of which is not known.
CROSS-FLOW
AROUND
FOUR
CYLINDERS
125
The characteristics of vortex shedding with spacing ratio are very similar to those reported by Chen (1977) and Fitz-Hugh (1973). For comparison, Fitz-Hugh’s results for both the in-line and rotated square arrays were incorporated in Figures 2 and 5.
4.4.
FURTHER
STUDIES
The study of force coefficients on cylinder clusters has always been related to the flow pattern. At present, most of the forces acting on two-cylinder groups have been measured extensively, in all kinds of arrangement (Bearman & Wadcock 1973; Igarashi 1981; Zdravkovich 1987; the magnitude and direction of the forces are also explained qualitatively by means of some plausible basic mechanisms. Unfortunately, for the cases of other three- or four-cylinder configurations, either regular or irregular, few experimental data are published; limited information on the force field of these arrangements can be used to compare the different mechanisms assumed to be operative in the two-cylinder case. It is felt that further measurements of the force coefficients on the cylinders of standard configurations are required, for the sake of verification of the basic mechanisms, as well as investigation of the relationships between the excitation forces and the instability threshold on a simple single array. In a visualization study, it is very important to obtain clarification on whether the effect of flow interference in the upper subcritical range of Reynolds numbers is the same as that in the lower subcritical range. It is suggested that a study in a larger water tunnel system with a better dye injection technique should be carried out, so that the phenomena at Re = lo4 can be observed. The length of the formation region should also be measured, as it is essential for the proper location of the hot wire. It was not feasible to do this measurement in this study. 5. CONCLUSIONS The flow patterns of various angle arrangements at different L/D values are summarized in Figure 8, with the dotted lines indicating the possible positions that the wake may have for some time. The results presented in this paper can be summarized as follows. (1). In the in-line arrangement, distinct vortex shedding of the upstream cylinders was suppressed below L/D = 3.94. The shear layers were not stable in attaching to the downstream cylinders and, instead, the wake deflected sharply from the streamwise direction toward one side. Intermittent and weak vortex shedding is observed, but the frequency is not recorded due to the rapid diffusion into the wake of the respective downstream vortex. Equally, the wake is also observed to extend sometimes to shield downstream cylinders. The Strouhal number, S, for L/D down to l-70 remains about the same as those obtained for the large spacing, i.e. -0.18-0.2 (Figure 2). When L/D decreases to below l-70, a bistable state starts to evolve behind the downstream cylinders. A wide and a narrow wake are found attached to the cylinders, which intermittently swap positions. The S value of the narrow wake attains a value of O-3 at L/D = 1.28 while the wide wake value drops to a minimum of 0.112 at L/D = l-41. The asymmetry is extremely sensitive to the flow incidence angle, and the wide or the narrow wake will easily be biased to one side, without any more interchanging of the wake pattern, as long as small angle is introduced to the incident flow.
126
K. LAM AND S. ('.1.0
(2). In the staggered arrangements, as L/D decreases, the formation region of the front three cylinders shrinks consistently, but the shedding frequency is increased inversely to the wake size, especially for the large incidence angles. A critical feature to all staggered positions is that the wide wake is biased behind cylinder 3 and is wider and appears earlier for the larger angle. The wide wake which is associated with low frequency is observed at L/D = 3.51 and 3.94 at 8 = lo” and 20” respectively; in the case of 30” to 45”, this is noticed up to the largest tested L/D ratio. For angles less than 20”, the near-wake behind the upstream cylinders bends sharply towards the side where very weak vortices are shed. until the L/D ratio is diminished to below 2.13. Then the shielding, or attachment, of a shear layer starts to occur, which is characterized by a region of stagnant fluid connecting the upstream and downstream cylinders. For the case of large angles of incidence up to 4.5”, cylinder 4 is free from direct wake interference from cylinder 1. The wakes of cylinder 2 and 4 deflect away from the central region for L/D < 2.40; this is likely forced by the two gap-flow exits from the model. The wake forms of the two cylinders (2 and 4) become increasingly alike as the angle increases; in fact, it is symmetrical at 8 = 45”. The wake of cylinder I is also controlled by the gap flow, and it is pressed into a very small region which enhances the formation of vortices. Clear, distinct vortices, though small in size, can be detected down to L/D reaches 1.28. The shear layers then attach onto the front portion of cylinder 3, forming a very narrow stagnant region between cylinders 1 and 3. The trend of the S values for all staggered positions indicates that it is not very sensitive to the angle of incidence. Despite the kinks in the experimental data, cylinder 1 always possesses the largest S value and increases to 0.64 at L/d = 1.41 and 8 = 45”. and cylinder 3 has the lowest value of S and drops to S = 0.08 at L/D = l-41 at 8 = 30”. The trend for the rest is similar to that of cylinder 1, but with lower values of S. The S values of cylinder 4 catch up with those of cylinder 2 as 0 reaches 45”. (3). An oscillating motion of the resultant wake can be found far downstream of a cylinder cluster of arbitrary number and orientation. as long as the cylinders are closely packed. The frequency of the oscillation. if nondimensionalized by the largest normal distance of the cluster to the flow. falls between 0.22 to 0.24 below L/D = 1.70. (4). The Strouhal number, S, representing the shedding frequency, appears to decrease with the increasing width of wake, W*, by an exponent of about -0.90. Although the data points were taken from different spacing ratios L/D, and angle of incidence 8, they are scattered within a certain region, with boundaries which can be described by the two equations given in Section 4. This indicates that S may be solely W*-dependent, while W* is a complicated function of L/D and 0.
6. ACKNOWLEDGMENTS The authors wish to thank the Research the Head of Department of Mechanical
Committee of the Hong and Marine Engineering
Kong Polytechnic for their support.
and
REFERENCES BALL, D. J. & HALL, C. D. 1980.Drag of yawed pile groups at low Reynolds numbers. Journal of the Waterways, Ports, Coastal and Ocean Division 106, 229-238. BAXENDALE, A. J.,GRANT, I. & BARNES, F. H. 1985. The flow past two cylinders different diameters. The Aeronautical Journal 89. 12S--134.
AXE having
CROSS-FLOW
AROUND
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127
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CROSS-FLOW
AROUND
FOUR
CYLINDERS
Plate 1. Flow visualization for (a) 0 = o”, L/D = 5.32; (b) 0 = O”, L/D = 2.65; (c) 0 = O”, L/D = 1.54.
129
a)
f) plate 2. Flow visualization for (a) 0 = lO”, L/D = 3.09; (b) 0 = 10”. L/D = 1.70: (C) H = (d) 0 = 20”. L/D = 349: (e) 0 = 20”. L/D = 1.70: (f) H = 20”. L/D = 1.28.
10”.
!
=
1.3
CROSS-FLOW
AROUND
FOUR
CYLINDERS
131
a)
b)
d) Plate 3. Flow visualization
for (a) 0 = 45”, L/D = 5.32; (b) 0 = 45”, L/D = 2.13; (c) 0 = 45”, L/D = 1.54; (d) 0 = 45”. L/D = 1.28.