Journal of Mechanical Science and Technology 29 (4) (2015) 1649~1656 www.springerlink.com/content/1738-494x OI 10.1007/s12206-015-0336-2 Experimental characterization of flow field around a square prism with a small triangular prism Ki-eok Ro * epartment of Mechanical System Engineering Institute of Marine Industry, Gyeonsang National University, Gyeongnam, 650-160, Korea (Manuscript Received September 15, 2014; Revised ecember 11, 2014; Accepted January 5, 2015) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract The characteristics of the flow field around a square prism containing a small triangular prism upstream were investigated by measuring lift and drag and visualizing the flow field through particle image velocimetry. Experimental parameters included triangular prism-tosquare prism width ratio (H/B, H and B are side lengths of the triangular and square prisms, respectively) and triangular prism-to-square prism gap ratio (G/B, G is the gap distance between the triangular and square prisms). The drag reduction rate of the square prism increased and then decreased with increasing G/B at a constant H/B but increased with increasing H/B at a constant G/B. The maximum drag reduction rate was 78.5% at H/B = 0.6 and G/B = 1.5. The width and gap ratios also minimally affected the lift reduction rate of the square prism, with an average value of 52.4%. Stagnation regions were further observed upstream and downstream of the square prism. Keywords: Fluid force reduction; PIV; Separated flow; Unsteady flow; Square prism; Flow control ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction The flow within a uniform flow around a square prism can be characterized by stagnation, separation, reconnection, reseparation, and a periodic Karman s vortex in the wake region. In particular, the vortex shedding around a square prism in the wake region affects the dynamic stabilization of large structures. Thus, the flow field around a square prism has been a subject of interest to many researchers who aimed to improve dynamic stabilization and drag reduction. Existing studies of drag reduction are discussed as follows. Igarashi [1] reported that the drag of the prism was significantly reduced by installing a small circular rod upstream of a square prism; a boundary gap was also found, in which the drag between the controlling rod and the prism significantly reduced. Khalighi et al. [1] reduced the drag by installing four horizontal plates in the wake to restrict the sudden deflection of the intensity of turbulence and wake flow. Tamura and Miyagi [2] also experimentally investigated the characteristics of the flow field by machining the shape of the four edges of a square cylinder through three methods. They found that a square-edged cylinder exhibited separation, but a round-edged cylinder demonstrated separation and then reconnection; the latter case showed higher lift and drag. Ro et al. [3] reduced the fluid force on a square prism by attaching various fences * Corresponding author. Tel.: +82 55 772 9103, Fax.: +82 55 772 9109 E-mail address: rokid@gnu.ac.kr Recommended by Associate Editor Simon Song KSME & Springer 2015 on the edges. They revealed that attaching vertical fences at the trailing edges behind the square prism provided the most effective drag reduction effect, with an average of approximately 6.8% at each Reynolds number. In addition, the separated flow at the leading edge was reconnected at the trailing edge; this reconnection formed circulation and resulted in reduced separation size as confirmed through visualization experiment [4]. Park [5] attached a splitter plate to the center of the rear side of a square prism and calculated the effect of the plate on drag reduction through vortex tracing method. The attached plate restricted the swirl in the wake of the square prism, the drag coefficient decreased as the length of the plate increased, and the reduction rate was higher than that when the length of the splitter plate decreased. Ali et al. [6] also numerically showed that the length of a splitter plate attached to the center of the rear side of the square prism can substantially influence the flow structure at a Reynolds number Re = 150. oolan [7] reduced the drag of a square prism by installing a detached splitter plate, not an attached splitter plate, in a wake with a low Reynolds number (Re = 150). The drag coefficient of the prism, amplitude of the lift coefficient, and Strouhal number were also decreased by installing the detached splitter plate. Furthermore, Ali et al. [8] numerically investigated the change in the flow field according to the gap by installing a detached splitter plate with a length equal to the length of one side of the square prism; the plate was installed at the back center of the square prism at Re = 150, and they observed two regions of the flow at a critical gap G c = 2.3 B.
1650 K.-. Ro / Journal of Mechanical Science and Technology 29 (4) (2015) 1649~1656 Fig. 1. Experimental model. Ro et al. [9] obtained a maximum drag reduction rate of 19.9% at a gap ratio G/B = 0.75 by installing a detached splitter plate on the downstream side of the square prism at a high Reynolds number (Re = 10000). They further employed an advanced vortex method to calculate the velocity and pressure field fluctuations around the square prism with and without the splitter plate [10]. Igarashi [11], Sakamoto et al. [12], and Zhou et al. [13] also significantly reduced the drag on the square prism by installing a small circular prism and a vertical plate separately on the upstream side of the square prism. They reported that the stagnation region in the wake of the prism or the vertical plate reduced the drag. On the contrary, the sharp upstream side of the triangular prism produced a relatively larger stagnation region and lower drag coefficient than that of the square prism only. In this research, a triangular prism was installed in the place of a circular prism or a vertical plate; the side length of the triangular prism and the distance between the front of the square and triangular prisms were set as variables. The drag and lift reduction characteristics of the square prism were investigated through fluid force measurements. The gap ratio resulting in a relatively high drag reduction rate was also analyzed through visualization using particle image velocimetry (PIV). 2. Apparatus and experimental method 2.1 Experimental model The experimental model is shown in Fig. 1. A square prism with a side length B was placed in a uniform flow U, and a triangular prism with a side length H was placed at a specific distance G from the front side of the square prism. At this point, the drag on the square prism is the fluid force element that functions in the same direction as the uniform flow, whereas the lift L is the fluid force element that acts vertically upward against the uniform flow. L and, which function on the prism, were measured by visualizing the flow field with PIV as G/B increased from 0 to 3.0 in increments of 0.5 for three types of triangular prisms with H/B = 0.2, 0.4, and 0.6 at a constant Reynolds number. 2.2 Experimental measurement of lift and drag Fig. 2 shows a schematic of the apparatus fabricated for lift and drag measurement. The apparatus consisted of a triangular Fig. 2. Schematic diagram of the experimental device (Units: mm). prism, fixed board, square prism, and measuring axis. Three triangular prisms were constructed using transparent acrylic plates with a thickness of 2 mm, total length of 255 mm, and side lengths of 10, 20, and 30 mm. The upper part of the prism was fixed to the board as shown in Fig. 2. The fixed board was attached to the support of a circulating water tank, which was designed to move along the tank channel. The size of the square prism was 50 mm 50 mm 180 mm with 3 mmthick side planes and 10 mm-thick top and bottom planes. The square prism was made of acrylic panels. At the center of the top and bottom planes, a 12 mm-diameter hole was drilled and a fixing nut was attached for easy installation. The measuring axis consisted of an aluminum rod with 8 mm diameter and 345 mm length. A nut and bolt were also fixed to the square prism with 190 mm screws from the bottom of the axis. As shown in the figure, the upper part of the axis contained evenly spaced cuts at the front, back, right, and left sides; these cuts were 3.4 mm thick and 15 mm long and used to easily separate and measure the lift and drag in the uniform flow. Two strain gauges were then attached to each side (for a total of four gauges), and a bridge circuit was formed to measure the lift and drag according to the amount of strain. The output waves of each of the two channels of the bridge circuit were transmitted through a strain amplifier and an A converter; the waves were then transferred to a personal computer, which calculated the output of the lift and drag by using precalibrated coefficients. The velocity of the uniform flow in the measurement area was U = 0.182 m/s. The Reynolds number calculated using the square prism side length as the characteristic length was Re = 1.0 10 4. Fluid force was also determined at a fixed Reynolds number with a gap ratio G/B ranging from 0.25 to 3.0 at regular intervals for triangular prisms with width ratios of H/B = 0.2, 0.4, or 0.6. In particular, the triangular prism was positioned as illustrated in Fig. 1. rag and lift were measured by moving the triangular prism toward
K.-. Ro / Journal of Mechanical Science and Technology 29 (4) (2015) 1649~1656 1651 the wake at 25 mm intervals from the center of the front side of the square prism. 2.3 Visualization experiment using PIV The size of the square prism used in the visualization experiment was 50 mm 50 mm 170 mm and made of 2 mmthick transparent acrylic plates. The triangular prisms were also constructed using transparent acrylic plates, which were 2 mm thick, 150 mm long, and side lengths of 10, 20, or 30 mm each. A narrow support, which was 30 mm wide and 80 mm long, was attached to the bottom of each triangular prism. Two 2 mm-diameter holes were drilled behind the support to allow the gap ratio to change by moving the support and fixing it to the board with screws. The schematic of the apparatus for flow visualization experiment using PIV is shown in Fig. 3. A square prism and a splitter plate prepared before the experiment were installed in the circulating water tank to visualize the flow field. After installing each experimental model inside the tank, lighting was vertically and horizontally applied to the sheet by a continuous wave laser outside the tank to photograph the flow field with a high-speed camera. In this experiment, traction particles were circular polyvinyl chloride with an average of 100 µm diameter and a specific gravity of 1.02. The velocity of the uniform flow in the measurement region of the circulating water tank was U = 0.182 m/s, and the Reynolds number converted with the length of one side of the prism as the representative length was Re = 1.0 10 4. Velocity distribution was estimated every 50 mm in the visible part of the circulating water channel and appeared at any section within ± 2% of the average value. In addition, the turbulence intensity in this region was approximately 0.012 0.06. Each parameter was continuously photographed for 16.37 s. A total of 2,048 image frames were obtained at intervals of 1/125 s, and 1,024 sheets of the vorticity field, velocity distribution, and velocity vector field were traced from two continuous frames at intervals of 1/62.5 s. The average flow field (Explained below) was the average time for 1024 sheets. Table 1 shows the main specifications of the PIV system. 3. Results and discussion 3.1 Characteristics of lift and drag The lift coefficient C L and drag coefficient C, which characterize the square prism hydrodynamics, are defined as follows: CL C L 1 2 ru S 2 1 2 ru S 2 = (1) = (2) Table 1. Main specifications of the PIV system. Item Image board Light source Specification Fast Cam-X panel link board drive 8 W contivuos wave laser Sheet light Cylindrical lens: f 3.8 11.4 mm Resolution 1280 1024 pixel Software CACTUS 3.2 Error vector(%) Average: about 0.1% Fig. 3. Schematic of the structure of the experimental device. Fig. 4. Time variations of the lift and drag coefficients (H/B = 0.4, G/B = 1.5). where the lift L and drag are the elements of the vertical and horizontal forces applied to the prism, respectively. ρ is the density of the fluid and S represents the projected area of the square prism under water. Fig. 4 shows the time variations of the lift coefficient C L and drag coefficient C for the square prism with a width ratio H/B = 0.4 and a gap ratio G/B = 1.5 of the triangular prism. The Ut/B value of the horizontal axis is the distance travelled by the uniform flow divided by the prism width and represents a non-dimensional time. This figure was constructed by connecting the points after collecting 1024 values for each coefficient for 10.24 s. The time variations of the drag coefficient C considered that the dotted line (with the triangular prism) is smaller than that of the solid line (without the triangular prism). However, the fluctuation in the amplitude is large and the periodicity appears clearly, which could be attributed to
1652 K.-. Ro / Journal of Mechanical Science and Technology 29 (4) (2015) 1649~1656 Table 2. Optimal average drag coefficients and drag reduction rates with H/B. H/B G/B C and R at the optimum G/B C R 0.2 1.0 0.81 57.8 0.4 1.0 0.65 66.8 0.6 1.5 0.44 78.5 Fig. 5. Average drag coefficients and drag reduction rates for various G/B and H/B values. the vortices shed from the triangular prism. In the observation on lift coefficient variation, the solid and dotted lines oscillate around 0 and the dotted line exhibits lower amplitude than the solid line. Hence, the drag and amplitude of the lift can be reduced by placing the triangular prism on the upstream side of the square prism to improve dynamic stability. Fig. 5 shows the average drag coefficient C and the average drag reduction rate R with respect to the gap ratio G/B and width ratio H/B of the different triangular prisms. Each point of the average drag coefficient C is the average value of 1024 C values in Fig. 4. In addition, the average drag reduction rate R is obtained by dividing the average drag coefficient reduction (in the presence of the triangular prism) by the average drag coefficient of the prism (without the triangular prism). The figure also shows the variation in the average drag coefficient C according to the gap ratio G/B, which decreases as the gaps increase at each width ratio. Near G/B = 1.0 1.5, C reaches the minimum value and gradually increases as the gap ratio increases. The average drag reduction rate R increases with increasing gap ratio and reaches the maximum value at all width ratios within the range of G/B = 1.0 1.5; by contrast, R decreases as the gap ratio increases. Furthermore, for the same gap ratio, the average drag coefficient increases with increasing width ratio of the triangular prism. Table 2 shows the average drag coefficient C and the average drag reduction rate R at the optimal G/B for the H/B value of each triangular prism. The average drag reduction rate is highest when G/B = 1.0 at H/B = 0.2, 0.4 and G/B = 1.5 at H/B = 0.6. In addition, the average drag reduction rate increases as the width ratio of the triangular prism increases and reaches the maximum drag reduction rate of 78.5% at H/B = 0.6. Fig. 6 presents the average of the absolute lift coefficient C L according to the gap ratio G/B at various triangular prism width ratios H/B and average lift reduction rate R L as a function of C L. Each point of the average value of the absolute lift coefficient C in Fig. 6 is the average of the L Fig. 6. Average lift coefficients and lift reduction rates with G/B and H/B. 1024 C L values in Fig. 4. When time averaged, the lift coefficient reaches a value close to zero because it oscillates near zero. As a result, the absolute lift coefficient was averaged to determine the amplitude of the square prism caused by the lift, that is, its dynamic stability. Furthermore, the average lift reduction rate R L represents the percentage obtained by dividing the absolute lift coefficient reduction (in the presence of the triangular prism) by the average absolute lift coefficient of the square prism (without the triangular prism). Fig. 6 illustrates the variation in the average lift coefficient C L and average lift reduction rate R L as a function of the gap ratio G/B. The figure shows that none of the coefficients changes as the gap ratio increases at all three width ratios. The oscillations in the wake region of the triangular prism upstream of the square prism affect its lift. These results imply that the size of the triangular prism and the distance between the two prisms do not affect the lift. The mean values of the average lift coefficients and reduction rates at all three width ratios are C L = 0.10 and R L = 52.4%, respectively. Hence, the triangular prism upstream of the square prism not only reduces the drag force of the square prism and the lift amplitude, but also enhances the dynamic stability. 3.2 Characteristics of fluid fields Fig. 8 shows the results (a) without the triangular prism and (b) with the triangular prism with H/B = 0.4 and G/B = 1.5. The velocity V was measured at a position 3.0B in the downstream direction from the center of the square prism, as shown
K.-. Ro / Journal of Mechanical Science and Technology 29 (4) (2015) 1649~1656 1653 (a) Vt Fig. 7. Pickup point of the velocity V. (b) V0 (a) Without triangle cylinder (c) Vb (i) Without triangular prism (ii) H/B = 0.4, G/B = 1.5 Fig. 9. Instantaneous velocity vectors around the square prism at V t, V 0, and V b points of Fig. 8. (b) H/B = 0.4, G/B = 1.5 Fig. 8. Time variations of the velocity V at the wake region of the square prism. in Fig. 7. Figs. 8(a) and (b) show the clear periodic changes in the velocity at intervals of 2.3 and 1.2 s, respectively. The amplitude presented in Fig. 8(b) is lower than that in Fig. 8(a), indicating that installing a triangular prism slightly changes the amplitude of V but the frequency almost doubled. Fig. 9 shows the velocity vectors from the V t, V 0, and V b points in Fig. 8. In Figs. 9(a), V t, (b) V 0, and (c) V b represent the velocity vectors of the maximum, 0, and minimum points, respectively, of the V value from V p in Fig. 8. espite their similar location, the flow patterns around the square prism differ depending on the presence of the triangular prism. Without the triangular prism (Fig. 9(i)), large oscillations in the wake appear and the stagnation region cannot be detected. The flow period can also be analyzed because Figs. 9(a) and (c) show the wake patterns rotating in opposite directions. When the triangular prism is present (Fig. 9(ii)), a stagnation region appears between the square and triangular prisms. A uniform-sized stagnation region is also detected on the rear side of the square prism, regardless of time. Although the wake behind the stagnation region remains oscillating, its amplitude is lower than that shown in Fig. 9(i). These changes in the flow pattern, which depend on the presence of the triangular prism, may cause variations in the frequency and amplitude of the square prism (Fig. 8). Fig. 10 shows the velocity profile at V t, V 0, and V b points in Fig. 8. When the velocity profiles around the square prism with and without the triangular prism are compared, the wake region velocity profile of the square prism alone significantly oscillates. By contrast, when the triangular prism is present, stagnation regions of similar sizes and shapes exist on the upstream and downstream sides of the square prism (Fig. 9). (a) Vt (b) V0 (c) Vb (i) Without triangular prism (ii) H/B = 0.4, G/B = 1.5 Fig. 10. Instantaneous velocity profiles around the square prism at V t, V o, and V b points of Fig. 8. Furthermore, the velocity profile, which vertically passes through the uniform flow from the center of the square prism, was compared. The boundary layer is thinner in all three cases with the triangular than that without the triangular prism. The upper and lower boundary layers of the square prism are possibly supplied with flow energy because of the wake of the triangular prism, resulting in thin boundary layers. Fig. 11 shows the Strouhal number according to the gap ratio G/B when the width ratio of the triangular prism is H/B = 0.4. The representative frequency in this case is the estimated value from V p in Fig. 7. The Strouhal number S t is a dimensionless value obtained by dividing the multiplied value of the representative frequency along the length of one side of the square prism by the uniform flow value. As shown in Fig. 11, the Strouhal number increases as the gap ratio increases, reaches the maximum value within the G/B range = 1.0 1.5,
1654 K.-. Ro / Journal of Mechanical Science and Technology 29 (4) (2015) 1649~1656 Fig. 11. Strouhal number with G/B (H/B = 0.4). (a) G/B = 0.5 (b) G/B = 1.5 (c) G/B = 2.5 (i) Path liver (ii) Velocity profiles Fig. 12. Average flow patterns around the square prism with G/B at H/B = 0.4. and then decreases as the gap ratio continues to increase. Similar behavior of the average drag reduction rate is observed in Fig. 5 because the wake frequency of the triangular prism is highest for G/B = 1.0 1.5, as shown in Fig. 10; hence, it energizes the upper and lower streams, resulting in a high drag reduction rate in this gap ratio. Fig. 12 indicates the time-averaged flow pattern around the square prism with the gap ratio G/B when the width ratio of the triangular prism is H/B = 0.4. An examination of the path lines in Fig. 12(i) reveals a stagnation region between the square and triangular prisms, regardless of the gap ratio G/B. Moreover, vortices are induced in the upper and lower regions behind the square prism. As observed in the velocity profile in Fig. 12(ii), the size of the stagnation region in the wake of the square prism is largest at G/B = 0.5 and reduces at G/B of 2.5 and 1.5. The average drag coefficient changes depending on the size of the stagnation region, as observed in Fig. 5. Fig. 13 illustrates the time-averaged flow pattern around the square prism with the H/B value of the triangular prism at G/B = 1.5. The path lines in Fig. 13(i) show that vortices appear in the upper and lower sides of the wake region of the square prism in the four cases. The velocity vectors in Fig. 13(ii) demonstrate that the vortex on the upper side rotates clockwise, whereas that on the lower side rotates counterclockwise. The velocity profile in Fig. 13(iii) clearly presents the stagnation region in the upstream and downstream regions of the square prism in the presence of the triangular prism. The stagnation region in the upstream region expands as the width ratio of the triangular prism increases, whereas that in the downstream region decreases. As a result, the size difference between upstream and downstream stagnation regions decreases as the width ratio of the triangular prism increases, (a) Without triangular prism (b) H/B = 0.2 (c ) H/B = 0.4 (d) H/B = 0.6 (i) Path lines (ii) Velocity vectors Fig. 13. Average flow patterns around the square prism with H/B at G/B = 1.5. (iii) Velocity profiles
K.-. Ro / Journal of Mechanical Science and Technology 29 (4) (2015) 1649~1656 1655 thus reducing the pressure difference. Therefore, the average drag reduction rate increases as the width ratio of the triangular prism increases, as shown in Fig. 5. 4. Conclusions The characteristics of the flow field around a square prism with a detached small triangular prism on its upstream side were investigated by measuring the lift and drag on the square prism and visualizing the flow field by using PIV. The characteristics of the lift/drag reduction and flow field that act on the square prism were investigated with a fixed Reynolds number Re = 1.0 10 4, H/B = 0.2, 0.4, and 0.6, and G/B = 0.0 3.0 in increments of 0.5. The experimental results can be summarized as follows: (1) Measurements of the time variations of the lift/drag showed that the presence of a detached triangular prism in the upstream direction considerably reduced the drag size and lift amplitude compared with those of the initial square prism without the triangular prism. (2) At a fixed width ratio H/B of the triangular prism, the drag reduction rate of the square prism increased and then decreased as the gap ratio G/B increased. (3) At a constant gap ratio G/B, the drag reduction rate was higher for a higher width ratio H/B of the triangular prism and a maximum value of 78.5% was obtained when G/B = 1.5 and H/B = 0.6. (4) The width and gap ratios of the triangular prism minimally affected the lift reduction rate of the square prism, with an average lift reduction rate of 52.4%. (5) The Strouhal number for the square prism increased and then decreased as the gap ratio continued to increase. (6) Stagnation regions appeared in the upstream and downstream regions of the square prism in the presence of the triangular prism; their size increased in the upstream region and decreased in the downstream region with increasing width ratio. Nomenclature------------------------------------------------------------------------ B : Length of a side of the square prism C : rag coefficient C L : Lift coefficient C L : Absolute value of the average lift coefficient : rag acting on the square prism G : Gap distance between the triangular and the square prisms H : Length of a side of the triangular prism R : rag reduction rate R L : Lift reduction rate S : Projected area of the square prism under water S t : Strouhal number t : Time U : Uniform flow V : Vertical velocity with respect to the uniform flow V : Zero point of the V value 0 V b V t Greek letters r Subscripts : Bottom point of the V value : Top point of the V value : ensity of fluid : rag L : Lift 0 : Zero point b : Bottom point p : Point t : Top point Superscripts : Average References [1] B. Khalighi, S. Zang, C. Korokilas and S. R Balkanyi, Experimental and computational study of unsteady wake flow behind a bluff body with a drag reduction device, Society of Automotive Engineers (2001) 2001-01-1042. [2] T. Tamura and T. Miyagi, The effect of Turbulence on aerodynamic forces on a square cylinder with various corner shapes, Journal of Wind Engineering and Industrial Aerodynamics, 83 (1999) 135-145. [3] K.-. Ro and K.-S. Kim, Fluid force reduction characteristics of a square prism having fences on the corner, Journal of the Korean Society of Marine Engineers, 30 (3) (2006) 389-395. [4] K.-. Ro, K.-S. Kim and S.-K. Oh, The visualization of the flow-field around square prism having fences using the PIV, Journal of The Korean Society of Marine Engineering, 32 (1) (2008) 94-99. [5] W.-C. Park, Effect of the length of a splitter plate on drag reduction, Translation of the Korean Society of Mechanical Engineers B, 17 (11) (1993) 2809-2815. [6] M. S. M. Ali, C. J. oolan and V. Wheatley, Low Reynolds number flow over a square cylinder with a splitter plate, Physics of Fluids, 23 (3) (2011) 033602-1-033602-12. [7] C. J. oolan, Flat-plate interaction with the near wake of a Square Cylinder, The American Institute of Aeronautics and Astronautics Journal, 47 (2) (2009) 475-478. [8] M. S. M. Ali, C. J. oolan and V. Wheatley, Low Reynolds number flow over a square cylinder with a detached flat plate, International Journal of Heat Fluid Flow, 36 (2012) 133-141. [9] K.-. Ro, Experimental characterization of the flow field of square prism with a detached splitter plate at high reynolds number, Journal of Mechanical Science and Technology, 28 (7) (2014) 2651-2657. [10] K.-. Ro, Characteristic calculation of flowfield around a
1656 K.-. Ro / Journal of Mechanical Science and Technology 29 (4) (2015) 1649~1656 square prism having a detached splitter plate using vortex method, Journal of the Korean Society of Marine Engineering, 37 (2) (2013) 156-162. [11] T. Igarashi, rag reduction of a square prism by flow control using a small rod, Journal of Wind Engineering and Industrial Aerodynamics, 69 (71) (1997) 141-153. H. Sakamoto, K. Tan, N. Takeuchi and H. Haniu, Suppression of fluid forces acting on a square prism by passive control, Journal of Fluids Engineering, 119 (1997) 506-511. [12] L. Zhou, M. Cheng and K.C. Hung, Suppression of fluid force on a square cylinder by flow control, Journal of Fluids and Structures, 21 (2005) 151-167. Ki-eok Ro received his B.S. degree in Marine Engineering from Pukyong National University, Korea in 1977. He then received his M.S. and Ph.. degrees from Kobe University, Japan, in 1986 and 1989, respectively. r. Ro is currently a Professor at the epartment of Mechanical Engineering at Gyeongsang National University in Gyeongnam, Korea. He serves as a Vice President of the Journal of the Korean Society of Marine Engineering. r. Ro s research interests include fluid mechanics, CF, and vortex method.