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Wake flow behind two side-by-side square cylinders
International Journal of Heat and Fluid Flow 32 (2011) 41–51
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Wake flow behind two side-by-side square cylinders Shun C. Yen ⇑, Jung H. Liu Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung 202, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 9 November 2009 Received in revised form 17 September 2010 Accepted 20 September 2010 Available online 14 October 2010 Keywords: Side-by-side square cylinders Surface pressure Gap-flow
a b s t r a c t This study investigates the flow structures, form drag coefficients and vortex-shedding characteristics behind a single-square cylinder and two side-by-side cylinders in an open-loop wind tunnel. The Reynolds number (Re) and gap ratio (g) are 2262 < Re < 28,000 and 0 6 g 6 12, respectively. The flow patterns around the square cylinders are determined using the smoke-wire scheme. Experimental results indicate that the flow structures behind two side-by-side square cylinders are classified into three modes – single mode, gap-flow mode and couple vortex-shedding. The gap-flow mode displays anti-phase vortex shedding induced from the interference between the two square cylinders. However, the couple vortex-shedding mode exhibits in-phase vortex shedding that is caused by the independent flow behavior behind each square cylinder. The surface-pressure profile, form drag coefficient for each square cylinder (C D ) and vortex-shedding frequency were measured and calculated using a pressure transducer and a hot-wire anemometer. For two side-by-side cylinder configurations, the maximum C D of 2.24 occurs in the single mode, while the minimum C D of 1.68 occurs in the gap-flow mode. Additionally, the C D in the coupled vortex-shedding mode is intermediate and approximately equal to that of a single (isolated) square cylinder. Moreover, the single mode has the highest Strouhal number (St) and the gap-flow mode has the lowest St. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction The flow around a bluff body has attracted considerable interest. Cylinders with a square/rectangular cross-section are commonly used in such architectural structures as buildings, bridge decks, monuments, cross-flow heat exchangers and others. A cylinder body induces various instantaneous unsteady effects such as separation, reattachment bubble, shear-layer instability, and vortex shedding when the flow moves around it. Numerous studies have employed flow visualization schemes and hot wire measurement to measure flow patterns behind the cylinders. Roshko (1954), Lyn et al. (1995), Williamson (1996), and Luo et al. (2003) performed pioneering work on irregular vortex shedding and turbulent properties. Many studies have focused on a single-square cylinder and elucidated its properties, such as vortex shedding in the wake, the drag coefficient (CD), flow patterns and other characteristics. Okajima (1982) analyzed the vortex-shedding frequency behind the rectangular cylinders by varying the width-to-height ratio and Reynolds number (Re) of 70 < Re < 2 104. He concluded that the Strouhal number (St) is 0.13 for 104 < Re < 2 104 while the ⇑ Corresponding author. Address: Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, No. 2, Pei-Ning Road, Keelung, Taiwan, ROC. Tel.: +886 2 2462 2192x3215; fax: +886 2 2462 0836. E-mail address: [email protected] (S.C. Yen). 0142-727X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2010.09.005
width-to-height ratio = 1. Robichaux et al. (1999) numerically solved the flow transition of a square cylinder and performed two-dimensional (2D) simulations at 70 < Re < 300. They identified three possible time-averaged mean streamline patterns in the 2D shedding state – (a) no flow separation on the side surfaces, (b) separation and reattachment on the side surfaces, and (c) separation at the leading edges but no reattachment on the side surfaces. Many studies have applied the two-cylinder flow-control mechanism to reduce the flow drag. Zdravkovich (1977) reviewed the flow interference between two circular cylinders placed in tandem, side by side and in a staggered arrangement to examine flow patterns, lift/drag ratio, pressure distribution, velocity profiles and vortex shedding. Williamson (1985) adopted various flow visualization methods to elucidate the flow field behind two side-by-side cylinders. He found that the vortex-shedding synchronizes at 1.0 < g < 5.0, where g is the gap ratio, which is the ratio of the spacing between the cylinder surfaces to the diameter of the cylinder. The vortex-shedding synchronization generates two parallel vortex streets in counter phase of which one is in-phase and the other is anti-phase. When g < 1.0, the certain harmonic modes of vortex-shedding existed behind the two circular cylinders. In the harmonic modes, the vortex pairs from the gap and these vortices are squeezed and merged with the dominant outer vortices. Kolar et al. (1997) probed the ensemble average characteristics of a near-wake flow around two side-by-side identical square cylinders at Re 23,100. They identified the enhanced vortex motion and
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Nomenclature a b CD CD C D;rms Cp
lateral spacing between the successive vortices in the same train transverse spacing between two vortex trains form drag coefficient average form drag coefficient for each square cylinder (=(sum CD)/2) average root-mean-square form drag coefficient for each square cylinder (=(sum CD,rms)/2) surface-pressure coefficient ¼ ðp pfs Þ 0:5 qu21
Cp,rms
root-mean-square surface-pressure qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ2 0:5 qu21 ¼ ðp p
f g g* p p pfs
vortex-shedding frequency (Hz) gap spacing between two cylinder surfaces gap ratio (=g/w) local static pressure on the cylinder surface average static pressure on the cylinder surface static pressure of free stream
coefficient
found that the average vortex speed in the base region was significantly high, even in the eventual near-wake equilibrium state in which the vortex speed was low. The Strouhal number slightly exceeded that associated with a single-square cylinder. However, other studies have indicated that St for a single-square cylinder equals that for two identical square cylinders. Inoue et al. (2006) employed a finite difference method to solve 2D unsteady compressible Navier–Stokes equations for two side-by-side square cylinders. They identified six wake patterns (non-synchronized, anti-phase, in-phase synchronized, flip-flopping, single bluff body and steady) using various spacing ratios. This work concentrates on the characteristic flow fields and flow behaviors around two side-by-side identical square cylinders. The effects of Re and gap spacing between two square cylinders were examined. The behaviors and flow patterns were also elucidated using the smoke-wire scheme. Additionally, the surface pressure and vortex-shedding frequency were probed using a pressure transducer and a hot-wire anemometer. Moreover, the form drag coefficient was determined using the surface-pressure profile. This study has the following objectives: (1) to elucidate the interaction between two identical side-by-side square cylinders, (2) to evaluate the form drag coefficient using the measured surfacepressure profile, and (3) to probe the vortex-shedding frequency behind the square cylinders.
2. Experiment 2.1. Experimental setup The experiments were conducted in an open-loop wind tunnel, as schematically depicted in Fig. 1. Dimensions of the test section were 50 50 120 cm (width height length), with a freestream turbulent intensity of <0.4% for 0.56 < u1 < 45 m/s, where u1 is the free-stream velocity. The free-stream velocity was monitored using a Pitot tube that was connected to a U-tube manometer. The non-uniformity of average-velocity profile across the test section was <0.5%. In the test section, a polished aluminum-alloy plate was set as the test-section floor and three highly transparent acrylic panels were installed as the ceiling and side walls for photography and visualization.
Re St TI u1 U V u0
v0 w x y
q m
Reynolds number (=u1w/m) Strouhal number (=f w/u1) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu02 þv 02 Þ=2 turbulence intensity ¼ pffiffiffiffiffiffiffiffiffiffiffi U 2 þV 2 free-stream velocity x-component of mean velocity y-component of mean velocity x-component of velocity fluctuation y-component of velocity fluctuation width of square cylinder, 2 cm streamwise coordinate spanwise coordinate air density kinetic viscosity of air
Subscripts Single single mode Gap gap-flow mode Couple couple vortex-shedding mode
2.2. Square-cylinder model The square cylinders were manufactured from an acrylic bar. The dimensions of square cylinder were 50 2 2 cm (span width height). The aspect ratio was therefore 25, referencing the results of West and Apelt (1982) and Szepessy and Bearman (1992). These two identical square cylinders were arranged as displayed in the inset in Fig. 1. The flow visualization plane was located at the mid-plane of the acrylic square cylinder bar to reduce the three-dimensional (3D) flow effects. 2.3. Smoke-wire scheme The flow patterns were visualized using the smoke-wire scheme, which was adopted by Yen and Hsu (2007). A zigzag tungsten wire with a diameter of 0.3 mm was placed in front of the square cylinder at a distance of x/w = 0.1. The zigzag structure was used to control the thickness and spacing of smoke streaks. A thin layer of mineral oil was brush-coated on the tungsten-wire surface, and then electrically heated to form a plain smoke-streak field. The Reynolds number, based on the tungsten-wire diameter was maintained below 40 to prevent vortex shedding behind the wire (Mueller, 2000). The oil-aerosol diameter measured using a Malvern 1600C Laser Particle Analyzer was 1.7 ± 0.3 lm. Therefore, for 1.8 < u1 < 9.8 m/s, the Stokes number (Hinds, 1982) is from 3.24 104 to 1.76 103, which is much lower than 1. The Stokes number utilized in this investigation is defined as Stk = su1/w, where s is the relaxation time for oil aerosols (s = 3.6 106 s) (Hinds, 1982) and w is the width of square cylinder. Consequently, the smoke streaks were considered to follow the flow properly. Finally, these smoke streaks were illuminated by a 0.5 mm-thick laser-light sheet which was focused on the mid-plane of the square cylinder bar. The 2D streak images were recorded on a computer at a frame rate of 30 frames/s (fps) using a CCD camera and a highquality image grabber. 2.4. Pressure transducer Thirty-two pressure taps were distributed around the midplane of each square cylinder. The diameter of pressure tap is 1 mm and the spacing between two adjacent taps is 0.8 mm. A
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short stainless steel tube was tightly inserted into the hole of each tap from the inside surface of the hollow square cylinder and then connected to a plastic polyester tube. A pressure transducer (Setra
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Co., Model 264) with the maximum measurable pressure of ±1245 N/m2 was applied to scan the surface pressure. A computer was utilized to control the pressure scanner to detect the distribution
Fig. 1. Experimental setup.
Fig. 2. Profile-view of smoke-streak flow patterns around square cylinders at Re = 2262. Exposure time: 1/2000s.
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of surface pressure. The detected voltages were transformed to the pressure signal using a data logger (Model KUSB-3116, Keithley Co.) which provides the maximum sampling rates of 500 k Samples/s. The pressure acquisition system started at the 10th second of the through-time and elapsed for 10 s. In order to justify the dynamic pressure measurements, the sampling rate and record length were set at 4000 Hz and thereby 40,000 samples was recorded. For the highest vortex-shedding frequency of 210 Hz occurring in this investigation, this sampling rate (i.e., 4000 Hz) satisfies the Nyquist criterion.
2.5. Hot-wire anemometry The vortex-shedding frequency and velocity properties were examined using a hot-wire sensor (TSI 1210-T1.5) and X-type hot-wire sensor (cross type, TSI 1240-T1.5), respectively. Furthermore, the output signals of hot-wire sensor were imported into a FFT analyzer (ONO SOKKI Co., Model CF-920). The diameter and length of the hot wire were 5 lm and 1.5 mm, respectively, ensuring the dynamic response between 15 and 25 kHz. The hot-wireanemometer signals were fed simultaneously into an FFT analyzer and a high-speed PC-based data-acquisition system.
2.6. Error analysis The measurement accuracy of free-stream velocity was affected primarily by the alignment of the Pitot tube and calibration of the pressure transducer. With applying a synchronized micro-pressure calibration system and careful alignment of the Pitot tube, the uncertainty of the u1 is estimated to be <3%. The bias error of square-cylinder dimensions and gap spacing is within ±0.025 mm measured using a precise caliper. Moreover, the accuracy of pressure transducer with an accuracy = ±1% was utilized to scan the surface pressure on the square cylinders. The pressure acquisition started at the 10th second of the through-time and elapsed for 10 s. The sampling rate and record length are 4000 Hz and 40,000 samples, respectively. Accordingly, the accuracies of pressure coefficient and form drag coefficient are <4% and <5%, respectively. Additionally, the accuracy of vortex-shedding frequency depended on the sampling rate of FFT analyzer. The sampling frequency is 16 kHz and the error of the vortex-shedding frequency is estimated to be <0.75% in this investigation.
2.7. Nomenclature In this work, the measurements of velocity are presented nondimensionally in the form of Reynolds number which is defined as Re = u1w/m, where m is the kinematic viscosity of air and w is the width of square cylinder. The gap ratio is defined as g = g/w, where g represents the spacing between the square-cylinder surfaces. Moreover, the vortex-shedding frequency in each flow pattern is characterized using the non-dimensional parameter – Strouhal number, St = fw/u1, where f is vortex-shedding frequency behind the square cylinders.
Fig. 3. Schematic sketches of smoke-streak flow patterns.
Fig. 4. Distribution of characteristic flow modes.
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3. Results and discussion 3.1. Smoke-streak flow patterns The smoke-wire scheme was utilized to visualize the flow configurations. Additionally, the flow separation and the vortex evolution were visualized using a high-speed photography system and a video recorder. The flow structures around the square cylinders were classified by varying the Re and the g of these two square cylinders. In the experiments, g was varied from 0 to 12 and 2262 < Re < 12,560 (corresponding free-stream velocity: 1.8 m/ s < u1 < 9.8 m/s). The instantaneous streaklines were recorded from the smokewire technique. Fig. 2 depicts the instantaneous smoke-streak flow patterns behind the square cylinders at Re = 2262 with exposure time of 1/2000 s. Moreover, Fig. 3 delineates the schematic sketches of surface-flow structures by referencing the smokestreak patterns shown in Fig. 2. Figs. 2a and 3a present the typical flow patterns behind a single-square cylinder, which is similar to
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those obtained by Okajima (1982) and Bearman and Trueman (1972). Fig. 2a reveals that no reattachment occurred on the lateral surfaces of the cylinder. Furthermore, the flow separated at the leading edges, and then a flow envelope was formed in the wake. Incidentally, the Kármán-type vortices shed alternatively downstream. The lateral spacing (a) between the successive vortices in the same train is 1.83 and the transverse spacing (b) between two vortex trains is 0.48. Consequently, the spacing ratio of b/a is 0.262. von Kármán (1956) determined that the theoretical b/a ratio for a circular cylinder is 0.281. Moreover, the time-averaged images were visualized using the particle tracking flow visualization (PTFV) scheme (Liu, 2008). Liu indicated that the flow separated near the front vertices of square cylinder and a pair of vortices was generated behind the square cylinder. Furthermore, an off-axis four-way saddle occurred in the wake. In Figs. 2b and 3b, two identical square cylinders were installed side by side (i.e. g = 0) to form a rectangular bulk. The flow separated from the two leading edges of rectangular bulk. A shedding vortex was formed behind the rectangular bulk and no reattachment
Fig. 5. Distribution of (a) the normalized streamwise velocity (u=u1 ), (b) the normalized transverse velocity (v =u1 ), (c) the normalized streamwise normal stress u0 u0 u21 , (d) the normalized transverse normal stress v 0 v 0 u21 , (e) the normalized shear stress u0 v 0 u21 , and (f) the turbulence intensity (TI).
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occurred on the lateral surfaces of rectangular bulk. The flow pattern was analogous to that presented in Fig. 2a. However, the shedding frequency behind a rectangular bulk was lower than that behind a single-cylinder because a broad low-pressure region was formed behind the rectangular bulk. Fig. 2c displays that the flow structure at g = 0.05 is similar to that shown in Fig. 2b. This flow pattern is called the single mode for low gap ratios. Figs. 2d and 3c depict the case with a jet moving through the gap. No separation occurred on the interfacial cylinder surfaces
at g = 0.5. The jet flow could not maintain its straight path-line, that is, the jet flow was deflected alternately by the Coanda effect (Coanda, 1936; Newman, 1961). Moreover, the flow structure exhibits anti-phase vortex shedding and a four-way saddle exists at (x/w, y/w) = (4.2, 0). The flow pattern is called the gap-flow mode. Fig. 2e displays that the flow structure at g = 1.5 is similar to that presented in Fig. 2d. Figs. 2f and 3d present the flow patterns behind the cylinders at g = 6.0. The flow structure behind each square cylinder is independent
Fig. 6. Distributions of pressure coefficient (Cp) on the peripheral of square cylinders.
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of the other at high g. The flow structure indicates that in-phase vortex shedding occurred behind the square cylinders. This flow pattern is defined as the couple vortex-shedding mode. Fig. 2g displays that the flow structure at g = 10.0 is similar to that presented in Fig. 2f. Fig. 4 depicts the distribution of characteristic flow patterns obtained by altering g and Re. The wake-flow patterns were classified into three modes – single mode, gap-flow mode, and couple vortex-shedding mode. At low gap ratio (g 6 0.1), the flow structures were located in the single mode which assembles the flow patterns behind a single-square cylinder. At 0.1 6 g 6 5.5, the gap-flow mode exhibits anti-phase vortex shedding, which is
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caused by the interference between two identical square cylinders. At high gap ratio (g P 5.5), the couple vortex-shedding mode exhibits in-phase vortex shedding because the flows move independently behind the square cylinders. Additionally, the borders that separate the characteristic flow regimes bear some uncertainty. The maximum uncertainties are ±0.02 for g and ±50 for Re. 3.2. Velocity properties Fig. 5 shows the velocity properties obtained by calculating the velocity fluctuation measured using X-type hot-wire sensor (cross type) against g* for Re = 1.3 104, 1.7 104, and 2.1 104. For
Fig. 7. Distributions of root-mean-square pressure coefficient (Cp,rms) on the peripheral of square cylinders.
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examining the unsteady wake flow behind the square cylinders, the X-type hot-wire probe was placed at (x/w, y/w) = (3.0, 0) where a four-way saddle point occurred. Figs. 5a and 5b reveal the variations of normalized streamwise (u=u1 ) and transverse (v =u1 ) velocity against g*. In the single mode (for low g*), the velocity components were insignificant because the velocity at the fourway saddle is theoretically zero. Figs. 5c and 5d display the variations of normalized streamwise normal stress u0 u0 u21 and 2 transverse normal stress v 0 v 0 u1 against g*. In the gap-flow mode, the jet flow moved through the gap between the square cylinders and the jet flow could not maintain its straight path-line. Namely, the Coanda effect induced the jet flow deflected alternately. Additionally, the maximum normal stresses occurs at g* = 1.5. Fig. 5e shows the variation of shear stress u0 v 0 u21 versus g*. In the couple vortex-shedding mode, the flow structure behind each square cylinder is independent of the other. Moreover, the flow structure reveals an in-phase vortex shedding occurring behind the cylinders. The flow patterns were independent of gap ratio. Consequently, the shear stress approached zero while g* > 5.5. Fig. 5f shows the variations of turbulence intensity (TI) against g*, where TI is based on the local mea flow velocity. In the single mode, TI is around 7.8% because these two identical square cylinders were installed side by side. Then, a broad low-pressure wake was formed behind the rectangular bulk. Additionally, in the gapflow mode, the jet flow could not maintain its straight path-line and the jet flow deflected alternately. Consequently, the turbulence intensity in the gap-flow mode is higher than that in the single mode. Specifically, the maximum TI of around 19.2% occurred at g* = 1.5. In the couple vortex-shedding mode (at high g*), the flow pattern behind each square cylinder is independent of the other and TI remains constant TI 1.5% while g* > 5.5.
is the local static pressure on the cylinder surface; pfs is the static pressure of free stream, and q is the air density. Fig. 6a presents the Cp profile on the single-square cylinder. On Face A1B1, Cp is symmetrical about the stagnation point, where Cp is at its maximum of 1.0. Cp is at its minimum of 0.72 at Vertices A1 and B1. On Face B1C1, Cp declines from 1.1 (at Vertex B1) to 1.2 (at Vertex C1). On Face C1D1, Cp is symmetrical about the center of Face C1D1. The Cp curves on Faces B1C1 and D1A1 were similar to each other because of geometrical symmetry. Moreover, Figs. 6b and 6c depict the Cp curves on the two side-by-side square cylinders. Figs. 6b and 6c display the Cp profile in the single mode (g* = 0). The Cp distribution on Face B1C1 is similar to that for a singlesquare cylinder. The Cp profile on Face B2C2 is similar to that on Face D1A1 for a single-square cylinder. Figs. 6d and 6e show the Cp profile in the single mode (g* = 0.05). On Face D1A1, Cp increased from 0.8 (at Vertex D1) to 0.27 (at Vertex A1), unlike in Fig. 6b and c at g* = 0. Figs. 6f and 6g present the Cp profile in the gap-flow mode at g* = 0.5. The Cp profile on Face D1A1 is similar to that on Face D2A2. Additionally, Cp decreased from 0.5 (at Vertex D1) to 1.45 (at Vertex A1) on Face D1A1. Figs. 6h and 6i show the Cp profile in the gap-flow mode (g* = 1.5). The Cp profile on Faces D1A1 and D2A2 differed from those in Figs. 6f and 6g. Figs. 6j and 6k present
3.3. Surface pressure and form drag 3.3.1. Distribution of surface pressure The distributions of time-averaged surface pressure on the peripheries of square cylinder – on Faces A1B1, B1C1, C1D1 and D1A1 – were probed using a 32-tap pressure transducer (Setra Co., Model 264, accuracy = ±1.0%). Fig. 6 plots the curves of surface-pressure coefficient (Cp) along each face at Re =2.1 104. The definition of Cp herein is C p ðp pfs Þ 0:5 qu21 , where p
Fig. 8. Variation of form drag coefficient (CD) against Reynolds number (Re) behind two side-by-side square cylinders.
* Fig. 9. Variation of (a) average form drag coefficient
C D versus gap ratio (g ), and (b) root-mean-square form drag coefficient C D;rms versus gap ratio (g*).
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the Cp profile in couple vortex-shedding mode at g* = 10. The Cp profiles on the peripheries are similar to that on a single-square cylinder, as presented in Fig. 6a. Fig. 7 shows the distributions of root-mean-square pressure coefficient (Cp,rms) on the peripheries of square cylinder at Re = 2.1 104. The definition of Cp,rms herein is C p;rms qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ2 is the average surface pressure ðp p 0:5 qu21 , where p at each detecting point. Fig. 7a presents the Cp,rms profile on the single-square cylinder. On Face A1B1, the distribution of Cp,rms is symmetrical about the stagnation point, where the minimum Cp,rms of 0.058 occurs. Moreover, the maximum Cp,rms of 0.178 occurs at Vertices A1 and B1. On Face B1C1, Cp,rms declines from 0.588 (at Vertex B1) to 0.52 (at Vertex C1). However, on Face C1D1, Cp,rms is symmetrical about the center of Face C1D1. The Cp,rms curves on Faces B1C1 and D1A1 are similar because of the geometrical symmetry. Furthermore, the current results are similar to those conducted by Noda and Nakayama (2003) who worked on the single-square cylinder model. Figs. 7b–7k depict the Cp,rms curves on the two side-by-side square cylinders. Figs. 7b and 7c display the Cp,rms profile for the single mode (g* = 0). On Face A1B1, the minimum Cp,rms of 0.048 occurs at the stagnation point (Vertex A). Furthermore, the maximum Cp,rms of 0.187 occurs at Vertex B1. Additionally, the Cp,rms curves on Faces B1C1 and B2C2 are similar to those on Faces B1C1 and D1A1 for a single-square cylinder (Fig. 7a). Figs. 7d and 7e show the Cp,rms profile of the single mode (g* = 0.05). On Face D1A1, Cp,rms increased from 0.062 (at Vertex D1) to 0.162 (at Vertex A1). Figs. 7f and 7g present the Cp,rms profile in the gap-flow mode while g* = 0.5. The
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Cp,rms profile on Face D1A1 is similar to that on Face D2A2. Additionally, Cp,rms increased from 0.15 (at Vertex D1) to 0.4 (at Vertex A1) on Face D1A1. Figs. 7h and 7i show the Cp,rms profile in the gap-flow mode as g* = 1.5. The Cp,rms profile on Faces A1B1 and Face A2D2 indicates that the minimum Cp,rms of 0.058 occurs at the stagnation point and this stagnation point moves toward the center of Face A1B1. Furthermore, Figs. 7j and 7k present the Cp,rms profile of the couple vortex-shedding mode for g* = 10. The Cp,rms profiles for the couple vortex-shedding mode are similar to those for the single-square cylinder, as presented in Fig. 7a. 3.3.2. Form drag The form drag coefficient (CD) was calculated using the detected profile of surface pressure by varying Re and g (Zdravkovich, 1997). Fig. 8 plots CD as a function of g for various Re. In Fig. 8, CD differs slightly between Cylinder-I and II in each mode and at each gap ratio because of the flow interaction behind the two side-by-side cylinders (Fig. 2). Additionally, CD is close to a constant in each mode at Re > 1.7 104 . Fig. 9 plots the variations of average form drag coefficient (C D ) and the average root-mean-square form drag coefficient C D;rms on each square cylinder for the two-cylinder configurations while Re = 2.1 104. Fig. 9a shows the C D for a single-square cylinder is approximately 2.06, which is consistent with those (C D ¼ 2:07) obtained by Bearman and Trueman (1972), (C D ¼ 2:05) determined by Lee (1975), and (C D ¼ 2:04) conducted by Naudascher et al. (1981). Moreover, the experimental result also shows that the maximum C D 2:24 occurs in the single mode (g = 0). Namely, the maximum C D is similar to that obtained by Inoue et al.
Fig. 10. Variations of vortex-shedding frequency (f) against free-stream velocity (u1) at (x/w, y/w) = (a) (3.0, –w–g/2), (b) (3.0, –g/2), (c) (3.0, g/2) and (d) (3.0, w + g/2).
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(2006) for a single rectangular cylinder. Furthermore, C D decreases with g in the single mode. Additionally, for two side-by-side square cylinders, the minimum C D of 1.68 occurs in the gap-flow mode because a jet flow moves through the cylinder gap. Fig. 9a also indicates that C D increases with g in the gap-flow mode. The C D for the two side-by-side square cylinder is less than that for a single-square cylinder. Moreover, the flow structure exhibits an anti-phase vortex shedding. In the couple vortex-shedding mode, C D is almost constant, that is, independent of g. Specifically,
the C D of 2.08 is approximately equal to that (C D ¼ 2:06) of a single-square cylinder. Additionally, the flow structure exhibits inphase vortex shedding, which occurs behind two side-by-side square cylinders. Fig. 9b shows the variation of the average CD,rms C D;rms against g. The definition of C D;rms is that sum-CD,rms divided by 2 for finding the rms-CD on each square cylinder. Fig 9b shows the CD,rms for a single-square cylinder is approximately 0.242. The maximum C D;rms of 0.263 occurs in the gap-flow mode (g = 0.5) which is about 8.6% higher than that of a single-square cylinder. Finally, the C D;rms of 0.241 occurring in the couple vortex-shedding mode is close to 0.242 for the single-square cylinder. 3.4. Vortex-shedding frequency The vortex-shedding frequency (f) behind two side-by-side square cylinders was probed using various free-stream velocities and gap ratios. Accordingly, vortex-shedding frequency in each flow pattern was characterized using the non-dimensional parameter – Strouhal number (St = f w/u1). Fig. 10 plots the variations of vortex-shedding frequency versus free-stream velocity using different g. The vortex-shedding frequency was determined at (x/w, y/w) = (3.0, –w–g/2), (3.0, –g/2), (3.0, g/2) and (3.0, w + g/2) behind a single-square cylinder and two side-by-side square cylinders. Fig. 10 reveals that in single mode, the vortex-shedding frequency at position (i) is close to that at position (iv); the value of f at position (ii) and (iii) was no prominent frequency. However, f at positions (i) and (iv) is only slightly lower than that for a single-square model. Fig. 10 also indicates that the vortex-shedding frequencies at these four probed positions are similar in gap-flow mode. However, these four probed frequencies are not equal that for a single-square cylinder. In the couple vortex-shedding mode, the vortex-shedding frequencies at these four probed positions are similar and close to that in the single-square cylinder. Fig. 11 plots the variations of St against Re behind a singlesquare cylinder and two side-by-side square cylinders using various gap ratios. Fig. 11a demonstrates that Strouhal number for a single-square cylinder (St) is close to 0.132 which is near the value obtained by Davis and Moore (1982), Okajima (1982), and Norberg (1993). Incidentally, the spacing ratio b/a is 0.262, which satisfies the condition b/a = 2St (for a circular cylinder) proven by Levi (1983), Roshko (1954), and Bearman (1967). Fig. 11a also reveals that, the maximum St for each Re for two side-by-side square cylinder is in the single mode. However, the minimum St occurs in the gap-flow mode. Eq. (1) gives the regressive relation between St and Re at g = 1.5.
St ¼ 0:145
Fig. 11. Variations of (a) Strouhal number (St) vs. gap ratio (g*) and (b) Strouhal number (St) vs. Reynolds number (Re).
0:34 ; Re
at g ¼ 1:5:
ð1Þ
The flow structure exhibits the anti-phase vortex shedding in the gap-flow mode. Fig. 11a displays that the weak Re-effect occurred while the gap ratio is 0.5. However, the strong Re-effect occurs at high gap ratio of 1.5. The reason is that the flow patterns at higher Re-effect (namely, higher inertia force) are independent of the Coanda effect. Therefore, the St approaches to the data occurring in the couple-vortex-shedding mode. Moreover, the shedding
Table 1 Correlations of CD and St for different flow modes. Single-square-cylinder model
Form drag coefficient Vortex shedding
Side-by-side square cylinders Single mode
Gap-flow mode
Couple vortex-shedding mode
C D Single Maximum
C D Gap
C D Couple
C D 2:06 St* St* 0.132
StSingle Maximum
C D Gap < C D Couple StGap Minimum
C D Couple C D StCouple StCouple St*
C D
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frequency is high at high Re and is low at low Re while g = 1.5. In couple vortex-shedding mode, St is close to a constant (St 0.13) at high Re. This St of 0.13 for two side-by-side square cylinders is close to St = 0.132 for a single-cylinder. Moreover, the flow structure exhibits in-phase vortex shedding behind the two side-byside square cylinders. Fig. 11b plots the curves of St versus g at various Re. In the single mode, St decreases as g increases. However, in the gap-flow mode, St increases with g. In the couple vortex-shedding mode, St becomes a constant. 4. Conclusions This study examined the characteristics of boundary-layer flow around one single-square cylinder and two side-by-side square cylinders using various Reynolds numbers and gap ratios. The flow behaviors and flow patterns were identified using the smoke-wire scheme. Furthermore, the smoke-streak patterns behind two sideby-side square cylinders are classified into three modes – single mode, gap-flow mode, and couple vortex-shedding. The surface-pressure profile, form drag coefficient, and vortex-shedding frequency were determined using a pressure transducer and a hot-wire anemometer. The gap-flow mode exhibits the anti-phase vortex shedding and the couple vortex-shedding mode exhibits the in-phase vortex shedding. Moreover, Table 1 presents the correlations of C D and St with changing the characteristic modes. The C D in various flow modes followed the order of C DGap < C DCouple , and furthermore the maximum C D occurs in the single mode. Specifically, the C DCouple of 2.08 is approximately equal to that (C D ¼ 2:06) of a single-square cylinder. Moreover, the gap-flow mode has the lowest St, whose magnitudes in the various modes followed the order of StGap < St StCouple < StSingle. Acknowledgment This research was supported by the National Science Council of the Republic of China, under Grant No. NSC 97-2221-E-019-039. References Bearman, P.B., 1967. On vortex street wakes. J. Fluid Mech. 28, 625–641. Bearman, P.B., Trueman, D.M., 1972. An investigation of the flow around rectangular cylinders. Aeronaut. Quart. 23, 229–237. Coanda, H., 1936. Device for Deflecting a Stream of Elastic Fluid Projected into an Elastic Fluid. United States Patent Office, No. 2052869.
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Davis, R.W., Moore, E.F., 1982. A numerical study of vortex shedding from rectangles. J. Fluid Mech. 116, 475–506. Hinds, W.C., 1982. Aerosol Technology. John Wiley, New York. p. 104–126. Inoue, O., Iwakami, W., Hatakeyama, N., 2006. Aeolian tones radiated from flow past two square cylinders in a side-by-side arrangement. Phys. Fluids 18, 046104. Kolar, V., Lyn, D.A., Rodi, W., 1997. Ensemble-average measurements in the turbulent near wake of the two side-by-side square cylinders. J. Fluid Mech. 346, 201–237. Lee, B.E., 1975. The effect of turbulence on the surface pressure field of a square prism. J. Fluid Mech. 69, 263–282. Levi, E., 1983. A universal Strouhal law. ASCE J. Eng. Mech. 109, 718–727. Liu, J.H, 2008. Characteristics of Fluid Behavior behind Two Aligned Square Cylinders at Low Reynolds Numbers. Master Thesis, National Taiwan Ocean University, Keelung, Taiwan, p. 85. Luo, S.C., Chew, Y.T., Ng, Y.T., 2003. Characteristics of square cylinder wake transition flows. Phys. Fluids 15, 2549–2559. Lyn, D.A., Einav, S., Rodi, W., Park, J.-H., 1995. A laser-Doppler velocimetry study of the ensemble-averaged characteristics of the turbulent near wake of a square cylinder. J. Fluid Mech. 304, 285–319. Mueller, T.J., 2000. Flow visualization by direct injection. In: Goldstein, R.J. (Ed.), Fluid Mechanics Measurements. Taylor & Francis, Washington DC, pp. 367–450. Naudascher, E., Weske, J.R., Fey, B., 1981. Exploratory study on damping of galloping vibrations. J.Wind Eng. Ind. Aerodynam. 8, 211–222. Newman, B.G., 1961. The deflection of plane jets by adjacent boundaries–Coanda effect. In: Lachmann, G.V. (Ed.), Boundary Layer and Flow Control: Its Principles and Application. Pergamon Press, New York, pp. 232–264. Noda, N., Nakayama, A., 2003. Free-stream turbulence effects on the instantaneous pressure and forces on cylinders of rectangular cross section. Exp. Fluids 34, 332–344. Norberg, C., 1993. Flow around rectangular cylinders: pressure forces and wake frequencies. J. Wind. Eng. Ind. Aerodynam. 49, 187–196. Okajima, A., 1982. Strouhal numbers of rectangular cylinders. J. Fluid Mech. 123, 379–398. Robichaux, J., Balachandar, S., Vanka, S.P., 1999. Three-dimensional floquet instability of the wake of square cylinder. Phys. Fluids 11, 560–578. Roshko, A., 1954. On the development of turbulent wakes from vortex streets. NACA Report 1191. Szepessy, S., Bearman, P.W., 1992. Aspect ratio and end plate effects on vortex shedding from a circular cylinder. J. Fluid Mech. 234, 191–217. von Kármán, T., 1956. Collected works of Theodore von Kármán. London 1, 331–338. West, G.S., Apelt, C.J., 1982. The effects of tunnel blockage and aspect ration on the mean flow past a circular cylinder with Reynolds number between 104 and 105. J. Fluid Mech. 114, 361–377. Williamson, C.H.K., 1985. Evolution of a single wake behind a pair of bluff bodies. J. Fluid Mech. 159, 1–18. Williamson, C.H.K., 1996. Vortex dynamics in the cylinder wake. Ann. Rev. Fluids Mech. 28, 477–539. Yen, S.C., Hsu, C.M., 2007. Flow patterns and wake structure of a swept-back wing. AIAA J. 45, 228–236. Zdravkovich, M.M., 1977. Review of flow interference between two circular cylinders in various arrangements. ASME J. Fluids Eng. 99, 618–633. Zdravkovich, M.M., 1997. Flow around Circular Cylinders: A Comprehensive Guide through Flow Phenomena, Experiments, Applications, Mathematical Models, and Computer Simulations. Fundamentals, vol. 1. Oxford University Press, New York.
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International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Wake flow behind two side-by-side square cylinders Shun C. Yen ⇑, Jung H. Liu Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung 202, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 9 November 2009 Received in revised form 17 September 2010 Accepted 20 September 2010 Available online 14 October 2010 Keywords: Side-by-side square cylinders Surface pressure Gap-flow
a b s t r a c t This study investigates the flow structures, form drag coefficients and vortex-shedding characteristics behind a single-square cylinder and two side-by-side cylinders in an open-loop wind tunnel. The Reynolds number (Re) and gap ratio (g) are 2262 < Re < 28,000 and 0 6 g 6 12, respectively. The flow patterns around the square cylinders are determined using the smoke-wire scheme. Experimental results indicate that the flow structures behind two side-by-side square cylinders are classified into three modes – single mode, gap-flow mode and couple vortex-shedding. The gap-flow mode displays anti-phase vortex shedding induced from the interference between the two square cylinders. However, the couple vortex-shedding mode exhibits in-phase vortex shedding that is caused by the independent flow behavior behind each square cylinder. The surface-pressure profile, form drag coefficient for each square cylinder (C D ) and vortex-shedding frequency were measured and calculated using a pressure transducer and a hot-wire anemometer. For two side-by-side cylinder configurations, the maximum C D of 2.24 occurs in the single mode, while the minimum C D of 1.68 occurs in the gap-flow mode. Additionally, the C D in the coupled vortex-shedding mode is intermediate and approximately equal to that of a single (isolated) square cylinder. Moreover, the single mode has the highest Strouhal number (St) and the gap-flow mode has the lowest St. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction The flow around a bluff body has attracted considerable interest. Cylinders with a square/rectangular cross-section are commonly used in such architectural structures as buildings, bridge decks, monuments, cross-flow heat exchangers and others. A cylinder body induces various instantaneous unsteady effects such as separation, reattachment bubble, shear-layer instability, and vortex shedding when the flow moves around it. Numerous studies have employed flow visualization schemes and hot wire measurement to measure flow patterns behind the cylinders. Roshko (1954), Lyn et al. (1995), Williamson (1996), and Luo et al. (2003) performed pioneering work on irregular vortex shedding and turbulent properties. Many studies have focused on a single-square cylinder and elucidated its properties, such as vortex shedding in the wake, the drag coefficient (CD), flow patterns and other characteristics. Okajima (1982) analyzed the vortex-shedding frequency behind the rectangular cylinders by varying the width-to-height ratio and Reynolds number (Re) of 70 < Re < 2 104. He concluded that the Strouhal number (St) is 0.13 for 104 < Re < 2 104 while the ⇑ Corresponding author. Address: Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, No. 2, Pei-Ning Road, Keelung, Taiwan, ROC. Tel.: +886 2 2462 2192x3215; fax: +886 2 2462 0836. E-mail address: [email protected] (S.C. Yen). 0142-727X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2010.09.005
width-to-height ratio = 1. Robichaux et al. (1999) numerically solved the flow transition of a square cylinder and performed two-dimensional (2D) simulations at 70 < Re < 300. They identified three possible time-averaged mean streamline patterns in the 2D shedding state – (a) no flow separation on the side surfaces, (b) separation and reattachment on the side surfaces, and (c) separation at the leading edges but no reattachment on the side surfaces. Many studies have applied the two-cylinder flow-control mechanism to reduce the flow drag. Zdravkovich (1977) reviewed the flow interference between two circular cylinders placed in tandem, side by side and in a staggered arrangement to examine flow patterns, lift/drag ratio, pressure distribution, velocity profiles and vortex shedding. Williamson (1985) adopted various flow visualization methods to elucidate the flow field behind two side-by-side cylinders. He found that the vortex-shedding synchronizes at 1.0 < g < 5.0, where g is the gap ratio, which is the ratio of the spacing between the cylinder surfaces to the diameter of the cylinder. The vortex-shedding synchronization generates two parallel vortex streets in counter phase of which one is in-phase and the other is anti-phase. When g < 1.0, the certain harmonic modes of vortex-shedding existed behind the two circular cylinders. In the harmonic modes, the vortex pairs from the gap and these vortices are squeezed and merged with the dominant outer vortices. Kolar et al. (1997) probed the ensemble average characteristics of a near-wake flow around two side-by-side identical square cylinders at Re 23,100. They identified the enhanced vortex motion and
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Nomenclature a b CD CD C D;rms Cp
lateral spacing between the successive vortices in the same train transverse spacing between two vortex trains form drag coefficient average form drag coefficient for each square cylinder (=(sum CD)/2) average root-mean-square form drag coefficient for each square cylinder (=(sum CD,rms)/2) surface-pressure coefficient ¼ ðp pfs Þ 0:5 qu21
Cp,rms
root-mean-square surface-pressure qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ2 0:5 qu21 ¼ ðp p
f g g* p p pfs
vortex-shedding frequency (Hz) gap spacing between two cylinder surfaces gap ratio (=g/w) local static pressure on the cylinder surface average static pressure on the cylinder surface static pressure of free stream
coefficient
found that the average vortex speed in the base region was significantly high, even in the eventual near-wake equilibrium state in which the vortex speed was low. The Strouhal number slightly exceeded that associated with a single-square cylinder. However, other studies have indicated that St for a single-square cylinder equals that for two identical square cylinders. Inoue et al. (2006) employed a finite difference method to solve 2D unsteady compressible Navier–Stokes equations for two side-by-side square cylinders. They identified six wake patterns (non-synchronized, anti-phase, in-phase synchronized, flip-flopping, single bluff body and steady) using various spacing ratios. This work concentrates on the characteristic flow fields and flow behaviors around two side-by-side identical square cylinders. The effects of Re and gap spacing between two square cylinders were examined. The behaviors and flow patterns were also elucidated using the smoke-wire scheme. Additionally, the surface pressure and vortex-shedding frequency were probed using a pressure transducer and a hot-wire anemometer. Moreover, the form drag coefficient was determined using the surface-pressure profile. This study has the following objectives: (1) to elucidate the interaction between two identical side-by-side square cylinders, (2) to evaluate the form drag coefficient using the measured surfacepressure profile, and (3) to probe the vortex-shedding frequency behind the square cylinders.
2. Experiment 2.1. Experimental setup The experiments were conducted in an open-loop wind tunnel, as schematically depicted in Fig. 1. Dimensions of the test section were 50 50 120 cm (width height length), with a freestream turbulent intensity of <0.4% for 0.56 < u1 < 45 m/s, where u1 is the free-stream velocity. The free-stream velocity was monitored using a Pitot tube that was connected to a U-tube manometer. The non-uniformity of average-velocity profile across the test section was <0.5%. In the test section, a polished aluminum-alloy plate was set as the test-section floor and three highly transparent acrylic panels were installed as the ceiling and side walls for photography and visualization.
Re St TI u1 U V u0
v0 w x y
q m
Reynolds number (=u1w/m) Strouhal number (=f w/u1) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu02 þv 02 Þ=2 turbulence intensity ¼ pffiffiffiffiffiffiffiffiffiffiffi U 2 þV 2 free-stream velocity x-component of mean velocity y-component of mean velocity x-component of velocity fluctuation y-component of velocity fluctuation width of square cylinder, 2 cm streamwise coordinate spanwise coordinate air density kinetic viscosity of air
Subscripts Single single mode Gap gap-flow mode Couple couple vortex-shedding mode
2.2. Square-cylinder model The square cylinders were manufactured from an acrylic bar. The dimensions of square cylinder were 50 2 2 cm (span width height). The aspect ratio was therefore 25, referencing the results of West and Apelt (1982) and Szepessy and Bearman (1992). These two identical square cylinders were arranged as displayed in the inset in Fig. 1. The flow visualization plane was located at the mid-plane of the acrylic square cylinder bar to reduce the three-dimensional (3D) flow effects. 2.3. Smoke-wire scheme The flow patterns were visualized using the smoke-wire scheme, which was adopted by Yen and Hsu (2007). A zigzag tungsten wire with a diameter of 0.3 mm was placed in front of the square cylinder at a distance of x/w = 0.1. The zigzag structure was used to control the thickness and spacing of smoke streaks. A thin layer of mineral oil was brush-coated on the tungsten-wire surface, and then electrically heated to form a plain smoke-streak field. The Reynolds number, based on the tungsten-wire diameter was maintained below 40 to prevent vortex shedding behind the wire (Mueller, 2000). The oil-aerosol diameter measured using a Malvern 1600C Laser Particle Analyzer was 1.7 ± 0.3 lm. Therefore, for 1.8 < u1 < 9.8 m/s, the Stokes number (Hinds, 1982) is from 3.24 104 to 1.76 103, which is much lower than 1. The Stokes number utilized in this investigation is defined as Stk = su1/w, where s is the relaxation time for oil aerosols (s = 3.6 106 s) (Hinds, 1982) and w is the width of square cylinder. Consequently, the smoke streaks were considered to follow the flow properly. Finally, these smoke streaks were illuminated by a 0.5 mm-thick laser-light sheet which was focused on the mid-plane of the square cylinder bar. The 2D streak images were recorded on a computer at a frame rate of 30 frames/s (fps) using a CCD camera and a highquality image grabber. 2.4. Pressure transducer Thirty-two pressure taps were distributed around the midplane of each square cylinder. The diameter of pressure tap is 1 mm and the spacing between two adjacent taps is 0.8 mm. A
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short stainless steel tube was tightly inserted into the hole of each tap from the inside surface of the hollow square cylinder and then connected to a plastic polyester tube. A pressure transducer (Setra
43
Co., Model 264) with the maximum measurable pressure of ±1245 N/m2 was applied to scan the surface pressure. A computer was utilized to control the pressure scanner to detect the distribution
Fig. 1. Experimental setup.
Fig. 2. Profile-view of smoke-streak flow patterns around square cylinders at Re = 2262. Exposure time: 1/2000s.
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of surface pressure. The detected voltages were transformed to the pressure signal using a data logger (Model KUSB-3116, Keithley Co.) which provides the maximum sampling rates of 500 k Samples/s. The pressure acquisition system started at the 10th second of the through-time and elapsed for 10 s. In order to justify the dynamic pressure measurements, the sampling rate and record length were set at 4000 Hz and thereby 40,000 samples was recorded. For the highest vortex-shedding frequency of 210 Hz occurring in this investigation, this sampling rate (i.e., 4000 Hz) satisfies the Nyquist criterion.
2.5. Hot-wire anemometry The vortex-shedding frequency and velocity properties were examined using a hot-wire sensor (TSI 1210-T1.5) and X-type hot-wire sensor (cross type, TSI 1240-T1.5), respectively. Furthermore, the output signals of hot-wire sensor were imported into a FFT analyzer (ONO SOKKI Co., Model CF-920). The diameter and length of the hot wire were 5 lm and 1.5 mm, respectively, ensuring the dynamic response between 15 and 25 kHz. The hot-wireanemometer signals were fed simultaneously into an FFT analyzer and a high-speed PC-based data-acquisition system.
2.6. Error analysis The measurement accuracy of free-stream velocity was affected primarily by the alignment of the Pitot tube and calibration of the pressure transducer. With applying a synchronized micro-pressure calibration system and careful alignment of the Pitot tube, the uncertainty of the u1 is estimated to be <3%. The bias error of square-cylinder dimensions and gap spacing is within ±0.025 mm measured using a precise caliper. Moreover, the accuracy of pressure transducer with an accuracy = ±1% was utilized to scan the surface pressure on the square cylinders. The pressure acquisition started at the 10th second of the through-time and elapsed for 10 s. The sampling rate and record length are 4000 Hz and 40,000 samples, respectively. Accordingly, the accuracies of pressure coefficient and form drag coefficient are <4% and <5%, respectively. Additionally, the accuracy of vortex-shedding frequency depended on the sampling rate of FFT analyzer. The sampling frequency is 16 kHz and the error of the vortex-shedding frequency is estimated to be <0.75% in this investigation.
2.7. Nomenclature In this work, the measurements of velocity are presented nondimensionally in the form of Reynolds number which is defined as Re = u1w/m, where m is the kinematic viscosity of air and w is the width of square cylinder. The gap ratio is defined as g = g/w, where g represents the spacing between the square-cylinder surfaces. Moreover, the vortex-shedding frequency in each flow pattern is characterized using the non-dimensional parameter – Strouhal number, St = fw/u1, where f is vortex-shedding frequency behind the square cylinders.
Fig. 3. Schematic sketches of smoke-streak flow patterns.
Fig. 4. Distribution of characteristic flow modes.
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3. Results and discussion 3.1. Smoke-streak flow patterns The smoke-wire scheme was utilized to visualize the flow configurations. Additionally, the flow separation and the vortex evolution were visualized using a high-speed photography system and a video recorder. The flow structures around the square cylinders were classified by varying the Re and the g of these two square cylinders. In the experiments, g was varied from 0 to 12 and 2262 < Re < 12,560 (corresponding free-stream velocity: 1.8 m/ s < u1 < 9.8 m/s). The instantaneous streaklines were recorded from the smokewire technique. Fig. 2 depicts the instantaneous smoke-streak flow patterns behind the square cylinders at Re = 2262 with exposure time of 1/2000 s. Moreover, Fig. 3 delineates the schematic sketches of surface-flow structures by referencing the smokestreak patterns shown in Fig. 2. Figs. 2a and 3a present the typical flow patterns behind a single-square cylinder, which is similar to
45
those obtained by Okajima (1982) and Bearman and Trueman (1972). Fig. 2a reveals that no reattachment occurred on the lateral surfaces of the cylinder. Furthermore, the flow separated at the leading edges, and then a flow envelope was formed in the wake. Incidentally, the Kármán-type vortices shed alternatively downstream. The lateral spacing (a) between the successive vortices in the same train is 1.83 and the transverse spacing (b) between two vortex trains is 0.48. Consequently, the spacing ratio of b/a is 0.262. von Kármán (1956) determined that the theoretical b/a ratio for a circular cylinder is 0.281. Moreover, the time-averaged images were visualized using the particle tracking flow visualization (PTFV) scheme (Liu, 2008). Liu indicated that the flow separated near the front vertices of square cylinder and a pair of vortices was generated behind the square cylinder. Furthermore, an off-axis four-way saddle occurred in the wake. In Figs. 2b and 3b, two identical square cylinders were installed side by side (i.e. g = 0) to form a rectangular bulk. The flow separated from the two leading edges of rectangular bulk. A shedding vortex was formed behind the rectangular bulk and no reattachment
Fig. 5. Distribution of (a) the normalized streamwise velocity (u=u1 ), (b) the normalized transverse velocity (v =u1 ), (c) the normalized streamwise normal stress u0 u0 u21 , (d) the normalized transverse normal stress v 0 v 0 u21 , (e) the normalized shear stress u0 v 0 u21 , and (f) the turbulence intensity (TI).
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occurred on the lateral surfaces of rectangular bulk. The flow pattern was analogous to that presented in Fig. 2a. However, the shedding frequency behind a rectangular bulk was lower than that behind a single-cylinder because a broad low-pressure region was formed behind the rectangular bulk. Fig. 2c displays that the flow structure at g = 0.05 is similar to that shown in Fig. 2b. This flow pattern is called the single mode for low gap ratios. Figs. 2d and 3c depict the case with a jet moving through the gap. No separation occurred on the interfacial cylinder surfaces
at g = 0.5. The jet flow could not maintain its straight path-line, that is, the jet flow was deflected alternately by the Coanda effect (Coanda, 1936; Newman, 1961). Moreover, the flow structure exhibits anti-phase vortex shedding and a four-way saddle exists at (x/w, y/w) = (4.2, 0). The flow pattern is called the gap-flow mode. Fig. 2e displays that the flow structure at g = 1.5 is similar to that presented in Fig. 2d. Figs. 2f and 3d present the flow patterns behind the cylinders at g = 6.0. The flow structure behind each square cylinder is independent
Fig. 6. Distributions of pressure coefficient (Cp) on the peripheral of square cylinders.
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of the other at high g. The flow structure indicates that in-phase vortex shedding occurred behind the square cylinders. This flow pattern is defined as the couple vortex-shedding mode. Fig. 2g displays that the flow structure at g = 10.0 is similar to that presented in Fig. 2f. Fig. 4 depicts the distribution of characteristic flow patterns obtained by altering g and Re. The wake-flow patterns were classified into three modes – single mode, gap-flow mode, and couple vortex-shedding mode. At low gap ratio (g 6 0.1), the flow structures were located in the single mode which assembles the flow patterns behind a single-square cylinder. At 0.1 6 g 6 5.5, the gap-flow mode exhibits anti-phase vortex shedding, which is
47
caused by the interference between two identical square cylinders. At high gap ratio (g P 5.5), the couple vortex-shedding mode exhibits in-phase vortex shedding because the flows move independently behind the square cylinders. Additionally, the borders that separate the characteristic flow regimes bear some uncertainty. The maximum uncertainties are ±0.02 for g and ±50 for Re. 3.2. Velocity properties Fig. 5 shows the velocity properties obtained by calculating the velocity fluctuation measured using X-type hot-wire sensor (cross type) against g* for Re = 1.3 104, 1.7 104, and 2.1 104. For
Fig. 7. Distributions of root-mean-square pressure coefficient (Cp,rms) on the peripheral of square cylinders.
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examining the unsteady wake flow behind the square cylinders, the X-type hot-wire probe was placed at (x/w, y/w) = (3.0, 0) where a four-way saddle point occurred. Figs. 5a and 5b reveal the variations of normalized streamwise (u=u1 ) and transverse (v =u1 ) velocity against g*. In the single mode (for low g*), the velocity components were insignificant because the velocity at the fourway saddle is theoretically zero. Figs. 5c and 5d display the variations of normalized streamwise normal stress u0 u0 u21 and 2 transverse normal stress v 0 v 0 u1 against g*. In the gap-flow mode, the jet flow moved through the gap between the square cylinders and the jet flow could not maintain its straight path-line. Namely, the Coanda effect induced the jet flow deflected alternately. Additionally, the maximum normal stresses occurs at g* = 1.5. Fig. 5e shows the variation of shear stress u0 v 0 u21 versus g*. In the couple vortex-shedding mode, the flow structure behind each square cylinder is independent of the other. Moreover, the flow structure reveals an in-phase vortex shedding occurring behind the cylinders. The flow patterns were independent of gap ratio. Consequently, the shear stress approached zero while g* > 5.5. Fig. 5f shows the variations of turbulence intensity (TI) against g*, where TI is based on the local mea flow velocity. In the single mode, TI is around 7.8% because these two identical square cylinders were installed side by side. Then, a broad low-pressure wake was formed behind the rectangular bulk. Additionally, in the gapflow mode, the jet flow could not maintain its straight path-line and the jet flow deflected alternately. Consequently, the turbulence intensity in the gap-flow mode is higher than that in the single mode. Specifically, the maximum TI of around 19.2% occurred at g* = 1.5. In the couple vortex-shedding mode (at high g*), the flow pattern behind each square cylinder is independent of the other and TI remains constant TI 1.5% while g* > 5.5.
is the local static pressure on the cylinder surface; pfs is the static pressure of free stream, and q is the air density. Fig. 6a presents the Cp profile on the single-square cylinder. On Face A1B1, Cp is symmetrical about the stagnation point, where Cp is at its maximum of 1.0. Cp is at its minimum of 0.72 at Vertices A1 and B1. On Face B1C1, Cp declines from 1.1 (at Vertex B1) to 1.2 (at Vertex C1). On Face C1D1, Cp is symmetrical about the center of Face C1D1. The Cp curves on Faces B1C1 and D1A1 were similar to each other because of geometrical symmetry. Moreover, Figs. 6b and 6c depict the Cp curves on the two side-by-side square cylinders. Figs. 6b and 6c display the Cp profile in the single mode (g* = 0). The Cp distribution on Face B1C1 is similar to that for a singlesquare cylinder. The Cp profile on Face B2C2 is similar to that on Face D1A1 for a single-square cylinder. Figs. 6d and 6e show the Cp profile in the single mode (g* = 0.05). On Face D1A1, Cp increased from 0.8 (at Vertex D1) to 0.27 (at Vertex A1), unlike in Fig. 6b and c at g* = 0. Figs. 6f and 6g present the Cp profile in the gap-flow mode at g* = 0.5. The Cp profile on Face D1A1 is similar to that on Face D2A2. Additionally, Cp decreased from 0.5 (at Vertex D1) to 1.45 (at Vertex A1) on Face D1A1. Figs. 6h and 6i show the Cp profile in the gap-flow mode (g* = 1.5). The Cp profile on Faces D1A1 and D2A2 differed from those in Figs. 6f and 6g. Figs. 6j and 6k present
3.3. Surface pressure and form drag 3.3.1. Distribution of surface pressure The distributions of time-averaged surface pressure on the peripheries of square cylinder – on Faces A1B1, B1C1, C1D1 and D1A1 – were probed using a 32-tap pressure transducer (Setra Co., Model 264, accuracy = ±1.0%). Fig. 6 plots the curves of surface-pressure coefficient (Cp) along each face at Re =2.1 104. The definition of Cp herein is C p ðp pfs Þ 0:5 qu21 , where p
Fig. 8. Variation of form drag coefficient (CD) against Reynolds number (Re) behind two side-by-side square cylinders.
* Fig. 9. Variation of (a) average form drag coefficient
C D versus gap ratio (g ), and (b) root-mean-square form drag coefficient C D;rms versus gap ratio (g*).
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the Cp profile in couple vortex-shedding mode at g* = 10. The Cp profiles on the peripheries are similar to that on a single-square cylinder, as presented in Fig. 6a. Fig. 7 shows the distributions of root-mean-square pressure coefficient (Cp,rms) on the peripheries of square cylinder at Re = 2.1 104. The definition of Cp,rms herein is C p;rms qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ2 is the average surface pressure ðp p 0:5 qu21 , where p at each detecting point. Fig. 7a presents the Cp,rms profile on the single-square cylinder. On Face A1B1, the distribution of Cp,rms is symmetrical about the stagnation point, where the minimum Cp,rms of 0.058 occurs. Moreover, the maximum Cp,rms of 0.178 occurs at Vertices A1 and B1. On Face B1C1, Cp,rms declines from 0.588 (at Vertex B1) to 0.52 (at Vertex C1). However, on Face C1D1, Cp,rms is symmetrical about the center of Face C1D1. The Cp,rms curves on Faces B1C1 and D1A1 are similar because of the geometrical symmetry. Furthermore, the current results are similar to those conducted by Noda and Nakayama (2003) who worked on the single-square cylinder model. Figs. 7b–7k depict the Cp,rms curves on the two side-by-side square cylinders. Figs. 7b and 7c display the Cp,rms profile for the single mode (g* = 0). On Face A1B1, the minimum Cp,rms of 0.048 occurs at the stagnation point (Vertex A). Furthermore, the maximum Cp,rms of 0.187 occurs at Vertex B1. Additionally, the Cp,rms curves on Faces B1C1 and B2C2 are similar to those on Faces B1C1 and D1A1 for a single-square cylinder (Fig. 7a). Figs. 7d and 7e show the Cp,rms profile of the single mode (g* = 0.05). On Face D1A1, Cp,rms increased from 0.062 (at Vertex D1) to 0.162 (at Vertex A1). Figs. 7f and 7g present the Cp,rms profile in the gap-flow mode while g* = 0.5. The
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Cp,rms profile on Face D1A1 is similar to that on Face D2A2. Additionally, Cp,rms increased from 0.15 (at Vertex D1) to 0.4 (at Vertex A1) on Face D1A1. Figs. 7h and 7i show the Cp,rms profile in the gap-flow mode as g* = 1.5. The Cp,rms profile on Faces A1B1 and Face A2D2 indicates that the minimum Cp,rms of 0.058 occurs at the stagnation point and this stagnation point moves toward the center of Face A1B1. Furthermore, Figs. 7j and 7k present the Cp,rms profile of the couple vortex-shedding mode for g* = 10. The Cp,rms profiles for the couple vortex-shedding mode are similar to those for the single-square cylinder, as presented in Fig. 7a. 3.3.2. Form drag The form drag coefficient (CD) was calculated using the detected profile of surface pressure by varying Re and g (Zdravkovich, 1997). Fig. 8 plots CD as a function of g for various Re. In Fig. 8, CD differs slightly between Cylinder-I and II in each mode and at each gap ratio because of the flow interaction behind the two side-by-side cylinders (Fig. 2). Additionally, CD is close to a constant in each mode at Re > 1.7 104 . Fig. 9 plots the variations of average form drag coefficient (C D ) and the average root-mean-square form drag coefficient C D;rms on each square cylinder for the two-cylinder configurations while Re = 2.1 104. Fig. 9a shows the C D for a single-square cylinder is approximately 2.06, which is consistent with those (C D ¼ 2:07) obtained by Bearman and Trueman (1972), (C D ¼ 2:05) determined by Lee (1975), and (C D ¼ 2:04) conducted by Naudascher et al. (1981). Moreover, the experimental result also shows that the maximum C D 2:24 occurs in the single mode (g = 0). Namely, the maximum C D is similar to that obtained by Inoue et al.
Fig. 10. Variations of vortex-shedding frequency (f) against free-stream velocity (u1) at (x/w, y/w) = (a) (3.0, –w–g/2), (b) (3.0, –g/2), (c) (3.0, g/2) and (d) (3.0, w + g/2).
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(2006) for a single rectangular cylinder. Furthermore, C D decreases with g in the single mode. Additionally, for two side-by-side square cylinders, the minimum C D of 1.68 occurs in the gap-flow mode because a jet flow moves through the cylinder gap. Fig. 9a also indicates that C D increases with g in the gap-flow mode. The C D for the two side-by-side square cylinder is less than that for a single-square cylinder. Moreover, the flow structure exhibits an anti-phase vortex shedding. In the couple vortex-shedding mode, C D is almost constant, that is, independent of g. Specifically,
the C D of 2.08 is approximately equal to that (C D ¼ 2:06) of a single-square cylinder. Additionally, the flow structure exhibits inphase vortex shedding, which occurs behind two side-by-side square cylinders. Fig. 9b shows the variation of the average CD,rms C D;rms against g. The definition of C D;rms is that sum-CD,rms divided by 2 for finding the rms-CD on each square cylinder. Fig 9b shows the CD,rms for a single-square cylinder is approximately 0.242. The maximum C D;rms of 0.263 occurs in the gap-flow mode (g = 0.5) which is about 8.6% higher than that of a single-square cylinder. Finally, the C D;rms of 0.241 occurring in the couple vortex-shedding mode is close to 0.242 for the single-square cylinder. 3.4. Vortex-shedding frequency The vortex-shedding frequency (f) behind two side-by-side square cylinders was probed using various free-stream velocities and gap ratios. Accordingly, vortex-shedding frequency in each flow pattern was characterized using the non-dimensional parameter – Strouhal number (St = f w/u1). Fig. 10 plots the variations of vortex-shedding frequency versus free-stream velocity using different g. The vortex-shedding frequency was determined at (x/w, y/w) = (3.0, –w–g/2), (3.0, –g/2), (3.0, g/2) and (3.0, w + g/2) behind a single-square cylinder and two side-by-side square cylinders. Fig. 10 reveals that in single mode, the vortex-shedding frequency at position (i) is close to that at position (iv); the value of f at position (ii) and (iii) was no prominent frequency. However, f at positions (i) and (iv) is only slightly lower than that for a single-square model. Fig. 10 also indicates that the vortex-shedding frequencies at these four probed positions are similar in gap-flow mode. However, these four probed frequencies are not equal that for a single-square cylinder. In the couple vortex-shedding mode, the vortex-shedding frequencies at these four probed positions are similar and close to that in the single-square cylinder. Fig. 11 plots the variations of St against Re behind a singlesquare cylinder and two side-by-side square cylinders using various gap ratios. Fig. 11a demonstrates that Strouhal number for a single-square cylinder (St) is close to 0.132 which is near the value obtained by Davis and Moore (1982), Okajima (1982), and Norberg (1993). Incidentally, the spacing ratio b/a is 0.262, which satisfies the condition b/a = 2St (for a circular cylinder) proven by Levi (1983), Roshko (1954), and Bearman (1967). Fig. 11a also reveals that, the maximum St for each Re for two side-by-side square cylinder is in the single mode. However, the minimum St occurs in the gap-flow mode. Eq. (1) gives the regressive relation between St and Re at g = 1.5.
St ¼ 0:145
Fig. 11. Variations of (a) Strouhal number (St) vs. gap ratio (g*) and (b) Strouhal number (St) vs. Reynolds number (Re).
0:34 ; Re
at g ¼ 1:5:
ð1Þ
The flow structure exhibits the anti-phase vortex shedding in the gap-flow mode. Fig. 11a displays that the weak Re-effect occurred while the gap ratio is 0.5. However, the strong Re-effect occurs at high gap ratio of 1.5. The reason is that the flow patterns at higher Re-effect (namely, higher inertia force) are independent of the Coanda effect. Therefore, the St approaches to the data occurring in the couple-vortex-shedding mode. Moreover, the shedding
Table 1 Correlations of CD and St for different flow modes. Single-square-cylinder model
Form drag coefficient Vortex shedding
Side-by-side square cylinders Single mode
Gap-flow mode
Couple vortex-shedding mode
C D Single Maximum
C D Gap
C D Couple
C D 2:06 St* St* 0.132
StSingle Maximum
C D Gap < C D Couple StGap Minimum
C D Couple C D StCouple StCouple St*
C D
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frequency is high at high Re and is low at low Re while g = 1.5. In couple vortex-shedding mode, St is close to a constant (St 0.13) at high Re. This St of 0.13 for two side-by-side square cylinders is close to St = 0.132 for a single-cylinder. Moreover, the flow structure exhibits in-phase vortex shedding behind the two side-byside square cylinders. Fig. 11b plots the curves of St versus g at various Re. In the single mode, St decreases as g increases. However, in the gap-flow mode, St increases with g. In the couple vortex-shedding mode, St becomes a constant. 4. Conclusions This study examined the characteristics of boundary-layer flow around one single-square cylinder and two side-by-side square cylinders using various Reynolds numbers and gap ratios. The flow behaviors and flow patterns were identified using the smoke-wire scheme. Furthermore, the smoke-streak patterns behind two sideby-side square cylinders are classified into three modes – single mode, gap-flow mode, and couple vortex-shedding. The surface-pressure profile, form drag coefficient, and vortex-shedding frequency were determined using a pressure transducer and a hot-wire anemometer. The gap-flow mode exhibits the anti-phase vortex shedding and the couple vortex-shedding mode exhibits the in-phase vortex shedding. Moreover, Table 1 presents the correlations of C D and St with changing the characteristic modes. The C D in various flow modes followed the order of C DGap < C DCouple , and furthermore the maximum C D occurs in the single mode. Specifically, the C DCouple of 2.08 is approximately equal to that (C D ¼ 2:06) of a single-square cylinder. Moreover, the gap-flow mode has the lowest St, whose magnitudes in the various modes followed the order of StGap < St StCouple < StSingle. Acknowledgment This research was supported by the National Science Council of the Republic of China, under Grant No. NSC 97-2221-E-019-039. References Bearman, P.B., 1967. On vortex street wakes. J. Fluid Mech. 28, 625–641. Bearman, P.B., Trueman, D.M., 1972. An investigation of the flow around rectangular cylinders. Aeronaut. Quart. 23, 229–237. Coanda, H., 1936. Device for Deflecting a Stream of Elastic Fluid Projected into an Elastic Fluid. United States Patent Office, No. 2052869.
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