Flow past a square prism with an upstream control rod at incidence to uniform stream

Ocean Engineering 108 (2015) 504–518

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Flow past a square prism with an upstream control rod at incidence to uniform stream Erhan Fırat a, Yahya Erkan Akansu b,n, Huseyin Akilli a a b

Department of Mechanical Engineering, Çukurova University, 01330 Adana, Turkey Department of Mechanical Engineering, Niğde University, 51240 Niğde, Turkey

art ic l e i nf o

a b s t r a c t

Article history: Received 15 October 2014 Received in revised form 15 June 2015 Accepted 24 August 2015

In the numerical study, it was mainly intended to test the capability of a control rod to reduce the drag and to suppress the fluctuating forces acting on the rod-square (total) system for various angles of incidence (α) and center-to-center spacing ratios (L/D). The Reynolds numbers (Re) based upon the diameter of control rod and the side length of the square prism are 50 and 200, respectively, for the control rod and the square prism. Seven distinct flow patterns were observed and it was demonstrated that the cavity flow pattern is the most effective in terms of simultaneous reduction of the time-averaged and RMS values of fluctuating force coefficients for both control rod and square prism. As the control rod located 2D or 3D upstream of the square prism at zero angle of incidence, the time-averaged drag coefficient of the total system i.e. the sum of the time-averaged drag coefficients of the control rod and the square prism is about 74% that of the square prism alone. Furthermore, the maximum reductions in RMS values of the fluctuating lift acting on the total system are 53% and 60%, respectively, for 2D and 3D. On the other hand, the effectiveness of control rod in reducing the time-averaged drag coefficient of the total system was generally diminished with increasing α. Instantaneous and time-averaged flow fields were also presented in order to identify the flow patterns around the rod-square system. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Control rod Square prism Angle of incidence Flow pattern Low Re number

1. Introduction It is an effective way to utilize simplified and small-scale models in order to determine the flow characteristics around complex and large-scale structures or equipments (buildings, towers, bridges, offshore structures etc). In addition to this, to associate the flow control methods with the above-mentioned models helps to minimize, as much as possible, the negative or undesired effects of these characteristics on the models. Roughly two flow control methods (active and passive) were known with respect to energy expenditure (Gad-el-Hak, 2000). In active methods, an external energy input to the system is required. Surface heating, blowing and/or suction, injection of micro-bubbles or particles, periodic rotation or oscillation of the body, wall motion, and electromagnetic forces could be given as examples. Passive methods are concerned with the geometrical modification of the flow system without any additional energy input. Splitter plate, axial slit, trip wire, rounded edges, surface roughness, and control rod could be given as examples. More information related

n

Corresponding author. Tel.: þ 90 388 2252250; fax: þ90 388 2250112. E-mail address: [email protected] (Y.E. Akansu).

http://dx.doi.org/10.1016/j.oceaneng.2015.08.041 0029-8018/& 2015 Elsevier Ltd. All rights reserved.

to the flow control methods and their major effects on the flow characteristics of the system could be found in the literature. As a passive control method, control rod has a great significance due to its capabilities such as, reducing the drag and suppressing the fluctuating forces acting on a system, enhancing the heat transfer performance of a system and so on. In order to reveal the effects of an upstream control rod on both reducing the drag and enhancing the heat transfer performance of a square prism, Tsutsui et al. (2001) conducted experimental study for the Reynolds numbers range from 5300 to 32,000. It was found that, the maximum total drag reduction is about 80%. Under these conditions, the overall heat transfer enhancement is 30% compared to those of the square prism without a rod. Likewise, the optimum enhancement in the overall heat transfer is 34% and the total drag reduction is 70%, compared to those of the square prism without a rod. Three flow patterns were observed by flow visualization by means of smoke-wire and oil-flow methods. Lee et al. (2004) investigated the effects of installing a small control rod upstream of a circular cylinder experimentally with a focus on the drag characteristics and the wake structure behind the cylinder. In the study, Reynolds number based on the diameter of the main cylinder was about 20,000. It was reported that the total drag coefficient of entire system, including the main cylinder and the control rod, was reduced by about 25% when compared to the

E. Fırat et al. / Ocean Engineering 108 (2015) 504–518

Nomenclature A C d D f F h H l L L/D ΔP P Re St T u, v U x, y

Δt

area, (m2) coefficient diameter of the control rod, (m) side length of the square prism, (m) vortex shedding frequency, (Hz) force, (N) vertical distance between the centers of the rod and square prism, (m) height or width, (m) horizontal distance between the centers of the rod and square prism, (m) distance between the centers of the rod and square prism, (m) center-to-center spacing ratio, spacing ratio difference between the surface and freestream static pressures, (Pa) local pressure, (Pa) Reynolds number Strouhal number vortex shedding period, (s) streamwise and cross-stream components of the velocity, (m/s) velocity, (m/s) streamwise and cross-stream coordinate directions, (m) time step size, (s)

main cylinder without rod. The flow was visualized by smoke-wire method and two different flow patterns were observed between the control rod and the main cylinder when pitch ratio was varied. Igarashi (1997) performed experimental study on drag reduction of a square prism by an upstream control rod at a Reynolds number of 32,000. Main result of study was that the system drag coefficient was reduced about 70% when compared to the single square prism. It was expressed that two flow patterns with and without vortex shedding from the rod occurred according to the longitudinal spacing (L/D) and the rod diameter (d/D). As a numerical study at low Reynolds number (Re¼200), Zhang et al. (2006) investigated the mechanism of the formation and the convection of vortices shedding from the cylinder with an upstream rod. The main parameters in this study are normalized control rod diameter (d/D) and center-to-center spacing ratio (L/D). It was obtained that, in optimum conditions, the drag coefficient of entire system is reduced about 35% and root-meansquare (RMS) value of fluctuating lift coefficient of the bare cylinder is reduced by up to 73%. It was expressed that two flow modes appeared with the variation of the spacing ratio. The cross-section of the control body can be different from a circle. Prasad and Williamson (1997) utilized a small flat plate placed upstream of and parallel to the cylinder as a drag reduction device. The Reynolds number based on the cylinder diameter was approximately 50,000 in the experimental study. It was found that the optimal geometrical configuration consisting of a plate height one-third the cylinder diameter placed 1.5 diameters upstream of the cylinder reduced the system drag by 62% compared to the drag of the bare cylinder. As the gap width increased, the existence of two distinct flow modes of the flow was suggested by aid of smoke-wire flow visualization. Zhou et al. (2005) placed a control plate upstream in order to control the flow around square cylinder in a two-dimensional (2-D) channel at Re¼ 250 based on the square width and the maximum incoming flow velocity. In

505

Operators o..4

the average over time of the enclosed quantity

Greek symbols

α ρ υ

angle of incidence, (°) density of the fluid, (kg/m3) kinematic viscosity of the fluid, (m2/s)

Subscripts Cr D L 0 P R RMS S T 1

critical drag lift single, isolated body pressure control rod root mean square value of a quantity square prism total (rod-square) system freestream

Superscripts ′ *

frontal planform

numerical study, lattice Boltzmann method (LBM) was applied to simulate isothermal channel flow. The main parameters were plate height (h/D) and perpendicular distance between the control plate and the square cylinder (s/D). It was stated that, negative drag on square cylinder was achieved, total drag coefficient, i.e. the sum of the drag coefficient of the square cylinder and the control plate, was decreased markedly and the amplitude of the fluctuating lift acting on the square cylinder was well suppressed. Rosales et al. (2000) performed numerical simulations to investigate the flow and the heat transfer characteristics of single and tandem pair of the square prisms in a channel with a fully developed inlet velocity profile at Re¼500 based on side length of the square prism. The side length of the eddy-promoting upstream square cylinder is one-half of the downstream heated one. It was reported that the existence of the upstream eddy-promoting cylinder slightly increased the overall heat transfer and reduced the drag of the downstream heated cylinder. It is clear from the literature that upstream control body has significant effects on aerodynamic and heat transfer characteristics of the total system in tandem arrangement. But, the total system subjected to cross-flow is not axisymmetric such like a single circular cylinder. Therefore, it is also important to know that how will these characteristics be affected by staggered arrangement which is also encountered in practical applications. Zhang et al. (2005) investigated the effects of upstream rod in staggered arrangement on drag reduction of fixed square cylinder at a Reynolds number of 82,000. It was reported that, for tandem arrangement, the drag coefficient of square cylinder can be reduced up to 10% that of the single square cylinder. It was stated that mean drag coefficient of the downstream square cylinder kept a low value for α o3°–5°, increased steadily up to α ¼20° and remained unchanged for α 420°. Besides that, six different flow modes were identified and showed via flow visualization by using hydrogen bubble technique. Wang et al. (2006) studied on drag

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reduction of a circular cylinder by using an upstream control rod at a Reynolds number of 82,000. The effects of rod diameter (d), staggered angle (α) and center-to-center distance between the rod and the cylinder (L) on the mean loading on the downstream cylinder were investigated based on the pressure measurements. It was reported that the drag of the cylinder can be reduced up to 2.34% that of a single cylinder at α ¼0°. It was stated that, the resultant force on the cylinder can be reduced by upstream rod for α o5° but α 410°. The five flow modes were identified. Sakamoto and Haniu (1994) investigated the suppression of the fluid forces acting on a main circular cylinder by introducing a very small control cylinder (d/D E0.061). Experiments were carried out for the different gaps between main and control cylinders and the angles of incidence of the control cylinder. The Reynolds number of the study was 65,000. It was expressed that, the time-averaged total drag can be reduced to 40% that of the single circular cylinder as the control cylinder was located near α ¼ 60° where the transition from laminar to turbulent boundary layer was detected along the surface of the main cylinder. The largest reductions in the RMS values of fluctuating lift and the drag were occurred at α ¼60° and α ¼120°. Five distinct flow patterns due to the different positions of the control cylinder were observed and depicted by both the smoke patterns and the schematic drawing. Zhao et al. (2005) simulated the viscous flow past two cylinders of different diameters at Re¼ 500 by using the finite element method (FEM) as discretization technique. The diameter of the small cylinder is onefourth that of the large one. In the numerical study, the effects of the position angle (from α/π ¼0 to 1) and the relatively small gap ratio between the cylinders (from G/D¼ 0.05 to 1) on various flow and aerodynamic characteristics were investigated. It was showed that, the total mean drag coefficient of two-cylinder system could be reduced up to 30% that of the single cylinder. It was also showed that, as G/D was increased, RMS values of both fluctuating drag and lift coefficients of two-cylinder system considerably reduced compared with the single cylinder case. Moreover, flow modes just behind the two cylinders were classified into three types. A lot of experimental studies were conducted on passive control of the flow by means of using a control rod but numerical. Besides, most of the experimental studies carried out at moderate Re. By performing this study, at α ¼0°, flow structures around the square prism with upstream rod at relatively low Re number could be compared with those of the relatively high Re numbers (Igarashi, 1997; Sarioglu et al., 2005; Zhang et al., 2005; Yen and Wu, 2012). To the authors' knowledge, for rod-square system, this is the first article that discusses the effects of angles of incidence on the flow and aerodynamic characteristics. In the present numerical study, the flow characteristics of the total system (consists of a square prism and an upstream control rod) and each component were tested with respect to center-tocenter spacing ratio (L/D) and angles of incidence (α) by

comparing with the corresponding single-component (isolated) cases. Then, the effects of mentioned parameters on aerodynamic characteristics i.e. time-averaged and RMS values of fluctuating force coefficients, were demonstrated. Furthermore, the distinct flow patterns depending upon the geometrical arrangement of the rod and the square prism were identified. The unsteady twodimensional (2-D) laminar equations were used to simulate the external flow around rod-square system at Reynolds number of 200 based on the side length of square prism and the freestream velocity even though the small scale three-dimensional (3-D) instabilities were observed in the wake of the square prism starting from Re¼1607 2 (Luo et al., 2007; Tong et al., 2008). Because, at relatively low Re values with low turbulence level, 2-D incompressible flows could be predicted accurately enough by employing numerical methods.

2. Governing equations and computational model The analytical form of the governing equations of unsteady two-dimensional (2-D) laminar flow of viscous incompressible fluid in cartesian coordinates can be expressed as follows:

∂u ∂v + =0 ∂x ∂y

(1)

⎛ ∂ 2u ∂u ∂u 1 ∂P ∂u ∂ 2u ⎞ ⎟ +u +v = − + υ⎜ 2 + ∂t ∂x ∂y ρ ∂x ⎝ ∂x ∂y2 ⎠

(2)

⎛ ∂ 2v ∂v ∂v ∂v 1 ∂P ∂ 2v ⎞ ⎟ +u +v = − + υ⎜ 2 + ∂t ∂x ∂y ρ ∂y ⎝ ∂x ∂y2 ⎠

(3)

The first equation represents continuity; the others represent transport of linear x-momentum and y-momentum, respectively. u and v are the velocity components, x and y are the streamwise and cross-stream directions, P is the pressure, υ is the kinematic viscosity and ρ is the density of the fluid. The schematic of the flow domain and the boundary conditions were given within the Fig. 1a. The main geometric parameters of the simulation were depicted in Fig. 1b. The center of the square prism was located 5.5D upstream from the center of circular flow domain that is 40D in diameter. Velocity inlet boundary condition was applied to the left half of the circumference of flow domain and pressure outlet boundary condition was applied to the right half of circumference of it. In velocity inlet boundary condition, only x component of velocity has a value (u¼ U1, v ¼0). In addition to that, the freestream velocity is steady and uniform. Pressure outlet boundary condition which requires the specification of a static (gauge) pressure at the outlet boundary was arranged to “0”. No-slip boundary condition

Fig. 1. Schematic of flow domain: (a) boundary conditions, (b) geometric parameters (d/D ¼ 0.25).

E. Fırat et al. / Ocean Engineering 108 (2015) 504–518

was applied to the remaining boundaries, including surfaces of the rod and square prism (u ¼0, v ¼0). The numerical simulations were carried out by the commercial CFD package-FLUENTs which uses finite volume method (FVM) as discretization technique. Discretized governing equations at each grid were linearized then solved by using a point implicit Gauss– Seidel linear equation solver in conjunction with an algebraic multigrid method (Fluent Inc., 2006). Second-order accuracy was used for the viscous terms. The temporal discretization was also second-order accurate. Second-order upwind discretization scheme was employed for the convection terms. SIMPLE (SemiImplicit Method for the Pressure-Linked Equation) was adopted as pressure–velocity coupling algorithm. The time step was obtained from the quotient of approximate vortex shedding period of the single square prism at Re¼200 by 150 (Δt¼0.007 s) in order to avoid any unphysical oscillation and obtain a smooth time-independent solution (Zhang et al., 2006). As the time step of 0.007 s was used, solutions were found stable in time and time integration errors were found insignificant. The residuals which are useful indicators of solution convergence were less than 10  5 for all concerned equation variables. Triangular mesh (grid) that is easy to apply regardless of the flow geometry was preferred. Triangular mesh was generated by using size function to smoothly control the growth in mesh size over the flow geometry (Fig. 2). There are several methods to assess mesh quality. But among these, the most important metrics are the equiangle skew and the size change. In this study, the values of the equiangle skew and the size change were lower than 0.55 and 2, respectively, for each cell in the computational domain. Besides that, boundary layer mesh was created on the surfaces of both the square prism and the control rod for improving the accuracy of the flow simulation. The thickness of the first boundary layer cell adjacent to the square surface was 0.0002D. The triangular mesh was first generated as coarse and then the structure of it was refined by means of mesh adaption considering the effect of the mean velocity magnitude (Fig. 2). Before the adaption, the total number of the cells which includes both boundary layer and triangular cells was approximately 30,000 whereas it was increased up to 90,000 after the adaption. The main purpose of using mesh adaption is to capture accurately the important features of the flow by improving the mesh resolution around the bodies and the regions beneath them. A mesh independence test was also conducted for the predetermined flow configuration (α ¼0° and L/D ¼3). The number of cells for the considered configuration was increased two times by applying a second mesh adaptation (increased more than four times especially just around the control rod and square prism). Simulation results of original and fine meshes were compared both for control rod and square prism. For square prism, the discrepancies between the values of St and 〈CD〉 are obtained as about 1.7% and 0.8%, respectively. On the other hand, discrepancies obtained as about 0.01% and 0.17%, respectively, for control rod. As

507

one can see from the results, there are no noticeable discrepancies therefore it was concluded that the predetermined simulations can be sufficiently represented by using the original mesh. The normalized spacing ratios were determined as L/D¼ 1, 2, 3, 4 and the angles of incidence that increase in counter-clockwise rotation were determined as α ¼ 0°, 2°, 4°, 6°, 8°, 10°, 12.5°, 15°, 20°, 30°, 45°, 60°, 75°, and 90° for this investigation.

3. Numerical verification In order to verify the accuracy of computational work, two geometrical configurations were examined. A single isothermal cylinder which subjected to cross-flow at Re¼ 100 and 200 were simulated and results were compared with other experimental and numerical studies that available in literature (Table 1 and 2). Moreover, same comparisons conducted for a single square prism that subjected to cross-flow at Re ¼100 (Table 3). The boundary conditions and mesh generation techniques which applied on single cylinder and prism are the same with that of rod-square system. In numerical verification, after the mesh adaption the number of cells was increased from 30,000–35,000 to 60,000– 65,000. Our results are generally in accord with previously published data. They fell within the range of numerical values, even some experimental values (Roshko, 1954; Norberg, 2001, 2003). As seen from the tables, there are, of course, slight differences between the experimental and present results. The multitude of those differences were arisen from the experimental uncertainties, threedimensional (3-D) effects (as mentioned at the end of the introduction section), using various boundary conditions and grid Table 1 Comparison of the flow characteristics for a single circular cylinder at Re¼ 100. Source

St

〈CD〉

CL

Computational Braza et al. (1986) Sa and Chang (1991) Meneghini (1993) Liu et al. (1998) Saltara (1999) Meneghini et al. (2001) Ding et al. (2007) Mahir and Altaç (2008) Lam and Lin (2009) Present study

0.160 0.155 0.162 0.164 0.160 0.165 0.166 0.172 0.164 0.167

1.364 1.230 1.520 1.350 1.330 1.370 1.356 1.368 1.340 1.347

7 0.250

Experimental Roshko (1954) Tritton (1959) Norberg (2003)

7 0.339

7 0.287 7 0.343 7 0.302

0.160–0.170 1.25 0.168

7 0.18–0.54

Fig. 2. A portion of the mesh near the upper surface of the cylinder (L/D¼ 3). The rectangular boundary layer mesh, the growth in the triangular mesh size and adapted mesh can be seen in sufficient detail. A view of the mesh around the rod-square (total) system at L/D¼ 3.

508

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Table 2 Comparison of the flow characteristics for a single circular cylinder at Re¼ 200. Source

St

〈CD〉

Computational Borthwick (1986) Braza et al. (1986) Meneghini (1993) Arkell (1995) Belov et al. (1995) Liu et al. (1998) Saltara (1999) Zhao et al. (2005) Zhang et al. (2006) Ding et al. (2007) Mahir and Altaç (2008) Present study

0.188 0.200 0.196 0.196 0.193 0.192 0.190 0.204 0.200 0.196 0.192 0.200

1.02 1.40 1.25 1.30 1.19 1.31 1.25 0.140 1.355 1.348 1.376 1.351

Experimental Norberg (2001)

0.190–0.210

1.300–1.350

Single Square

CL

7 0.750

L/D=1

7 0.640 7 0.690 7 0.700

L/D=2

7 0.659 7 0.698 7 0.658

L/D=3

Table 3 Comparison of the flow characteristics for a single square prism at Re¼100. Source

St

〈CD〉

Computational Davis and Moore (1982) Davis et al. (1984) Franke et al. (1990) Wang (1996) Sohankar et al. (1998) Robichaux et al. (1999) Cheng et al. (2005) Sen and Mittal (2011) Sen et al. (2011) Lam et al. (2012) Present study

0.150 0.160 0.154 0.150 0.146 0.154 0.145 0.145 0.145 0.144 0.146

1.630 1.660 1.610 1.520 1.460 1.530 1.440 1.529 1.529 1.493 1.481

Experimental Okajima (1982) Luo et al. (2007)

0.140 0.142–0.145

CL

L/D=4 RMS

0.139 0.151 0.192 0.193 0.185 0.181

structures, the effects of blockage, etc. Consequently, it can be said that present results are reasonable and the adopted numerical scheme is suitable for employing.

4. Numerical results and discussion 4.1. Flow patterns and Strouhal numbers 4.1.1. Tandem arrangement It can be noticeable that, flow patterns that observed in rodsquare (total) system both in tandem and staggered arrangements showed complete or partial similarities to those of the following experimental and numerical studies (Sumner et al., 2000; Wang et al., 2006; Zhang et al., 2005, 2006, Malekzadeh and Sohankar, 2012). The critical Reynolds numbers (Kármán vortex street first appeared beyond the body) for the single circular cylinder and the single square prism are 44 and 70, respectively (Zdravkovich, 1997; Okajima, 1982). However, these values may vary from one researcher to another but usually close to one another (ReCr of the single cylinder was determined as 47 by Norberg (2003)). In this study, the Reynolds number of the control rod and the square prism are 50 and 200, respectively, therefore Kármán vortex street forms behind both bodies under normal circumstances (single body, without any interference and isothermal conditions).

Fig. 3. Instantaneous vorticity contours of the single square prism and the rodsquare system in tandem arrangements. The shear layers separating from the control rod changed from reattachment from the front surface of the square prism to roll up, forming vortices, at a critical distance between the bodies. The transition from the CFP to the VIFP can be openly seen as a function of spacing ratio.

Fig. 3 presents the typical flow structures around the square prism with and without a control rod. As one can see from the figure, there is no vortex shedding from the control rod at any spacing ratio except L/D ¼4. The flow pattern that no periodic vortex shedding observed from the upstream control body and separated shear layers from the control body generally attached to the front or side surfaces of the downstream bare body was broadly referred to as cavity flow pattern (CFP), (Prasad and Williamson, 1997; Lee et al., 2004; Zhang et al., 2005; Wang et al., 2006; Zhang et al., 2006). In this flow pattern, the wake of upstream body shields the downstream body from the direct interaction with approaching flow (Figs. 3 and 4). A pair of counter-rotating vortices (one on the top of another and there is no shedding) between and adjacent to the rear surface of the control body and front surface of bare prism was also considered as another characteristic of CFP (Fig. 4). Strouhal numbers (St) of the single square prism and the square prisms with upstream control rod at spacing ratios of 1, 2 and 3 are 0.144, 0.177, 0.187 and 0.180, respectively. It is clear that the upstream interference of the control rod caused an increase in the vortex shedding frequency of the square prism (fs). This increase can also be predicted qualitatively from the Fig. 3, as follows: when the control rod added to the flow system, the number of the vortices that appeared behind the square prism increased from 7 to 8 for L/D¼1 and 3, to 9 for L/D ¼2, in the same wake region length. The effects of control rod on the near wake flow structure of square prism can be explained as follows: in the uncontrolled case, the time-averaged values of the streamwise and cross-stream velocity components near the leading edges of the single square prism are close to each other and shear layer separation was observed at the leading edges. When control rod was added to the flow system, the time-averaged cross-stream velocity reduced by half in the same region. Thus, the separation tendency of the shear layers from the leading edges of the square prism was diminished

E. Fırat et al. / Ocean Engineering 108 (2015) 504–518

509

Fig. 4. Time-averaged streamlines of the flow between the control rod and the square prism for different spacing ratios. This figure is important for the identification of the reattachment point on the square prism and to determine the critical spacing value for the onset of the vortex shedding from the control rod.

considerably. The separation point was shifted from the leading edge to trailing edge of the square prism owing to the flow attachments/reattachments to the side surfaces (Fig. 5). As a result of that, the length and width of recirculation region in the nearwake of the prism became narrower when compared with the single square prism (the values of length and width of timeaveraged recirculation regions are 1.58D and 1.35D for the single square prism, 1.47D and 1.12D for L/D¼1, 1.40D and 1.03D for L/D¼ 2, 1.52D and 1.04D for L/D¼3, 1.56D and 1.10D for L/D¼ 4). Moreover, the time-averaged recirculation regions observed near the side surfaces of the square prism were almost disappeared at L/D¼ 1 and completely disappeared at L/D¼ 2 and 3. It is obvious from the Fig. 4 that a pair of counter-rotating vortices existed between control rod and square prism in the case of L/D ¼1. As the spacing ratio increased from 1 to 2, the pair of counter-rotating vortices elongated in the downstream direction. For L/D¼ 3, two pairs of counter-rotating vortices (each elongated vortex in L/D ¼2 arrangement splitted into two smaller vortices) that two upper rotate in clockwise direction and the others rotate in counter-clockwise direction were observed. The upstream pair of the vortices located just behind the control rod, the other pair of the vortices located near the front surface of the square prism. This splitting process may be associated with onset of the instabilities in the shear layer of the rod. As the spacing ratio reached or passed a critical value, the shear layers from the upstream rod begun to periodically roll up into Kármán vortices and then impinged on the square prism (Fig. 3). Therefore, this flow pattern was referred to as vortex impingement (VIFP), (Sumner et al., 2000). By the way, it generally referred to as wake impingement (Prasad and Williamson, 1997; Lee et al., 2004; Zhang et al., 2005, 2006; Wang et al., 2006). After the critical value of the spacing ratio, square prism has lost its effect on the nearwake of the control rod. Therefore the vortex shedding from the control rod cannot be suppressed anymore, that is to say, CFM is

no longer observed. This can also be seen clearly in Fig. 4. In the figure, the downstream pair of the counter-rotating vortices disappeared and upstream pair of the counter-rotating vortices shortened as spacing ratio increased to 4. This also shows that the separated shear layers from the upstream control rod cannot reattach on the front surface of the square prism anymore. The upper vortex from the control rod and upper shear layer of the square prism combined to form like-sign vortex in the near wake of the square prism. The lower vortex from the control rod and lower shear layer of the square prism combined in a similar fashion to form like-sign vortex. As a result of this, two dominant frequencies were obtained from the fast Fourier transform (FFT) analysis of lift coefficient-time history of the square prism at L/ D¼ 4 (Fig. 6). In here, the higher peak represents the fS and the lower one represents the fR. Briefly, the fR can also be obtained from the FFT analysis of lift coefficient-time history of the downstream square prism if the regular vortex impingements to the walls of the square prism are available. 4.1.2. Staggered arrangement The asymmetrical flow structure around rod-square (total) system was completely arisen from the non-zero angle of incidence, therefore, will be observed except zero angle of incidence. In addition to that, as α increased, five new distinct flow patterns which are also consistent with the literature were observed. The largest spacing ratio, L/D¼ 4, was preferred first so as to explain the important features of this arrangement. Fig. 7 depicts the instantaneous vorticity contours of the total system for different angles of incidence at L/D¼ 4. The VIFP can be seen clearly at α ¼0°. The vortices shed from the rod still alternately kept going through the upper and lower side of the square prism at α ¼2°. However, the sizes and magnitudes of the vortices shed from the upper and lower sides of the prism showed some differences when compared to the case of α ¼ 0° (α ¼2° case was not given in

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E. Fırat et al. / Ocean Engineering 108 (2015) 504–518

Fig. 5. Comparison of the time-averaged streamlines in the near-wake of the square prism with and without an upstream rod for the tandem configuration.

0.04

CLS

0.03

2

PSD (N/N) /Hz

CLS

0.03

2

PSD (N/N) /Hz

0.04

0.02

t (s)

fS

0.01

0.02

t (s)

fS

0.01

fR

0.00

0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

f (Hz)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

f (Hz)

Fig. 6. Time histories of the lift coefficients of the square prisms at L/D¼ 3 (left) and L/D ¼4 (right) and FFT of them in the form of power spectral density (PSD). The shape of the lift coefficient signal at L/D¼ 3 is very similar to those obtained at L/D¼ 1 and 2 (not given) but at L/D¼ 4.

Fig. 7). It indicates that asymmetric flow in the near wake of the square prism was started to develop. The VIFP predominant on the system for the angle of incidence range of 0°–2°, because upper vortices from the upper shear layer of the control rod still merged with the upper shear layer of square prism. At α ¼4°, both rows of vortices from the control rod started to merge with the lower shear layer of the square prism in order to form the lower vortex of the square prism. As a result of this, a large-scale vortex when compared to the upper vortex of square prism was formed. That one-sided combination suggests the existence of a new flow pattern. It was named as vortex merging (VMFP). This flow pattern also called as wake merging (Zhang et al., 2005; Wang et al., 2006). The VMFP is predominant in the incidence angle range from 4° to 15° for L/D ¼4. Another prominent feature of the VMFP is that the

vortices shed from the square prism started to split into two rows at a further downstream and the splitting point (a virtual point that occurrence of two vortices rows from one is recognizable) moved near downstream as angle of incidence increased. The vortices from the single square prism started to split into two rows at α ¼12.5°. At α ¼20°, direct combination of the vortices from the control rod with the lower shear layer of the square prism was no longer observed. The Kármán vortex street behind the upstream rod was partly disturbed, especially during the lower vortex formation process of the square prism. As explained before, when Reo ReCr, no Kármán vortex street can be seen in the wake of the concerning body. The mutual interaction of the shear layers of the control rod and the square prism at α ¼30° is weaker than those observed from α ¼ 4° to

E. Fırat et al. / Ocean Engineering 108 (2015) 504–518

α = 0˚

α = 4˚

α = 12.5˚

α = 20˚

α = 30˚

α = 45˚

α = 75˚

α = 90˚

511

Fig. 7. Flow visualization around the rod-square system at L/D ¼ 4 by means of instantaneous vorticity contours for various incidence angles.

L/D=1 L/D=2 L/D=3 L/D=4 Single Cylinder (Re=50)

0.16 0.14 0.12

0.24

L/D=1 L/D=2 L/D=3 L/D=4 Single Square (Re=200)

0.22

0.1235

0.20

StR

0.10

StS

Vortex shedding is available

0.08 0.06

0.18

0.16

0.04

0.14 0.02 0

20

40

60

80

100

α (°)

0

20

40

60

80

100

α (°)

Fig. 8. Strouhal number distributions of the control rod and the square prism for different incidence angles and spacing ratios.

α ¼15°. But, the streamwise component of approaching flow to control rod was slightly affected from the angular position of the rod-square system, thus Re of the control rod was dropped below the Rec. As a result of that, vortex shedding from the control rod suppressed at α ¼ 30°, 45° and 60° (Figs. 7 and 8). On the other hand, the shear layers of the control rod still have an influence on the lower shear layer of the square prism as well as the wake region. This new flow pattern that weak mutual interaction is sensible was referred to as weak interaction (WIFP). The suppression of the vortex shedding behind the control rod was stopped at α ¼ 75° but the effect of square prism on the near wake of the rod continued. For α ¼ 90°, the vertical distance between the bluff bodies are large enough so that mutual interference between the separating shear layers of the bodies are

negligible. This is the so-called negligible interaction flow pattern (NIFP) and dominant at α ¼ 90° for L/D ¼4. In here, only flow visualizations are inadequate to separate the NIFP from the WIFP easily, thus, additional attention must be paid to quantitative data. In this pattern, the StR and St values become closer to the values obtained in the single cases (Fig. 8). This situation is also valid for the time-averaged and RMS values of the fluctuating force coefficients of both control rod and square prism (this will be illustrated in the Section 4.2). So far, the distinct flow patterns around the rod-square system at L/D ¼4 were presented. The flow patterns observed at L/D ¼1 will be demonstrated hereafter. Flow patterns that encountered at L/D¼2 and 3 have already existed in at least one of the spacing ratio of 1 or 4 therefore will not be revised again. Before the

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L/D=1

L/D=4

Fig. 9. The comparison of the shear layer merging and the vortex merging flow patterns at α ¼ 10°.

explanation of the shear layer merging flow pattern (SLMFP) it will be beneficial to mention the reason behind the transition from CFP to VIFP once again. As the spacing between the bodies reached a critical value at a specific angle of incidence, Kármán vortex street became visible beyond the control rod i.e. separated shear layers from the sides of the control rod roll up into discrete vortices. After critical spacing, aforesaid discrete vortices instead of shear layers started to combine with the growing shear layers of the square prism. The same situation is valid for the transition from SLMFP and VMFP. This means that, the shear layers issued from the control rod combined with the lower shear layer of the square prism in SLMFP but formed vortices issued from the control rod combined with the lower shear layer of the square prism in VMFP (Fig. 9). Some features of the SLMFP close to those of VMFP. For example, either shear layers or vortices from the control rod directly interfere with growing lower shear layer of the square prism. It was also observed in SLMFP that the vortices shed from the square prism started to split into two rows at a further downstream and the splitting point approached to the prism as α increased. But, the splitting tendency of the vortices in the wake in SLMFP is lower than that of VMFP at identical incidence angles. Generally, Kármán vortex street beyond the square prism in SLMFP was less distorted than that of the VMFP, at identical incidence angles (Fig. 9). Besides that, Kármán vortex street beyond the square prism was started to deform as α or L/D started to increase within the boundary of SLMFP. The last flow pattern was detected in relatively low spacing ratio (L/D ¼1) and high angles of incidence (α ¼75° and 90°). This flow pattern was named as gap vortex enveloping (GVEFP). In this flow pattern, the vortex shedding from the control rod was suppressed by means of square prism owing to the proximity effect, such as observed in the CFP. The formation of shear layers of opposite sign across the gap resulted in that the upper shear layer of the control rod was encompassed by the lower shear layer of control rod and the lower shear layer of square prism. At a further downstream of the gap between the rod and prism, the lower shear layer of the square prism enveloped by the upper and lower shear layers of the control rod, respectively, while curling towards the near-wake of the prism. The lower shear layer of square prism broke off the rear half parts of the elongated shear layers of the rod shortly afterwards. The coalescence of the mentioned shear layers then caused a cumbersome vortex (vorticity magnitude is lower and distorted in shape) when compared with the vortex shed from the upper half of the square prism. The growth of the shear layers and the vortices were illustrated in a series of equally spaced instants of time within the one period of vortex shedding (Fig. 10). Fig. 11 represents the stairs and the approximate boundaries of flow patterns for the different configurations of the rod-square system subjected to cross-flow at Re¼200. In Fig. 11a, the symbol type which surrounded by a hollow rectangle represents the possible transition between the flow patterns of the neighbor symbols in the same spacing ratio. As seen from the Fig. 11, GVEFP is peculiar to the rod-square system with a spacing ratio of 1.

Furthermore, the minimum number of the flow patterns also observed at this spacing ratio. Based upon spacing ratio, rodsquare system in cross-flow appeared with one flow pattern between the cavity and the vortex impingement at α ¼0° and one flow pattern amongst gap vortex enveloping, negligible interaction and weak interaction at α ¼90°. As α increased from 0° to 90°, rodsquare system climbs the specific flow pattern stairs depending upon the spacing ratio (for example, for L/D¼ 2, flow pattern stairs are comprised of cavity, shear layer merging, weak interaction, respectively). Moreover, any previous flow pattern/patterns will not be encountered again as α increased. Another observation is that the flow pattern changed from cavity to vortex impingement (α r2°), from shear layer merging to vortex merging (2° o α r20°) and from shear layer merging to weak interaction (20° o α r75°) as spacing ratio was increased. In Fig. 11b, rough boundaries for the seven distinct flow patterns were drawn. The fields of detected transition patterns in Fig. 11a were divided in an appropriate manner between adjacent flow patterns in Fig. 11b. Apart from this, it must be noted that the boundaries do not indicate that a sudden transition occurs from one pattern to another. 4.2. Time-averaged and RMS values of fluctuating force coefficients 4.2.1. Control rod In this section, the time-averaged and RMS values of both fluctuating drag and lift coefficients of control rod will be presented for different configurations of rod-square system. Fig. 12a shows the time-averaged drag and lift coefficients of control rod at various spacing ratios and angles of incidence. It is apparent from the figure that the lowest 〈CDR〉 value was obtained at the lowest L/ D and α. 〈CDR〉 started to increase as the spacing between the bluff bodies increased at zero angle of incidence but still remained lower than that of the single cylinder. As seen from the Fig. 13, the significant pressure recovery in leeward surface of the control rod is responsible for the remarkable reduction in 〈CDR〉. The positive effects of the lower spacing ratios on time-averaged drag reduction ended at α ¼ 45° (Fig. 12a). The lower spacing ratios showed negative effect on reducing drag from α ¼45° to α ¼90°. In addition, for α 445°, the 〈CDR〉 values did not fall under the value of single cylinder regardless of spacing ratio. It may be noticeable that as the spacing was extended, time-averaged values of drag and lift coefficients of control rod usually approached to those of the single cylinder due to that square prism and rod interaction become weaker (Fig. 12a). For 〈CLR〉, the most tremendous changes occurred at L/D¼ 1 with increasing angle of incidence. For L/D¼1, 〈CLR〉 was started to increase in positive direction up to α ¼6° due to the pressure level differences between upper-right and lower-right quadrants of the cylinder. It can be explained as follows: it was seen from the timeaveraged streamlines (they were not given here) that the lower vortex behind the control rod was disappeared as α increased to 2°. In contrary to upper vortex, disappeared lower vortex brought about an increase in pressure level of the lower-right quadrant of the control rod. It must be remembered that, all the observations were, of course, confirmed by the time-averaged surface pressure distributions around the rod as well as square prism. The upper vortex commenced to move downward over the rear surface of control rod and diminished gradually as α increased from 8° to 12.5°. It disappeared completely at α ¼ 15°. Thus, positive lift force was balanced by an equal and opposite lift force. As a result of that, 〈CLR〉 value fell close to the value of the single cylinder, i.e. zero (Fig. 12a). For α 415°, 〈CLR〉 was increased in negative lift (downward) direction up to α ¼ 60°. As α increased from 15° to 60°, the stagnation point on the middle of the windward surface of the control rod slightly shifted in the clockwise direction. And besides,

E. Fırat et al. / Ocean Engineering 108 (2015) 504–518

513

T

5T/15

10T/15

T/15

6T/15

11T/15

2T/15

7T/15

12T/15

3T/15

8T/15

13T/15

4T/15

9T/15

14T/15

VMFP

VMFP

WIFP

WIFP

WIFP

SLMFP SLMFP

SLMFP SLMFP

WIFP

WIFP

WIFP

NIFP

SLMFP SLMFP

SLMFP Transition VMFP

WIFP

SLMFP SLMFP

WIFP

WIFP

WIFP

WIFP

WIFP

SLMFP

SLMFP SLMFP SLMFP

VMFP

SLMFP

SLMFP SLMFP SLMFP

VMFP

CFP

SLMFP

SLMFP

SLMFP

GVEFP

GVEFP

4

6

8

10

12.5

15

20

30

45

60

75

90

Vortex Merging Shear Layer Merging

L/D=1 L/D=2 L/D=3 L/D=4 0

20

40

Vortex Impingement Cavity

60

80

100

α (°)

NIFP

VMFP

CFP

CFP

0

3

WIFP

VMFP

VMFP

CFP

CFP

1

2

Gap Vortex Enveloping

VMFP

VIFP

VIFP

CFP

2

4

Weak Interaction

L/D

Flow pattern types

Negligible Interaction

Transition

Fig. 10. The instantaneous vorticity contours of the rod-square system at α¼ 90° and L/D ¼1.

α (°)

Fig. 11. (a) Flow patterns observed with respect to the incidence angles and the spacing ratios (b) the approximate demarcations of the flow patterns.

a pair of counter-rotating vortices came into existence over the lower-right quadrant in time-averaged near-wake of the rod due to the increased velocity of flow across the gap. Hence, a pressure

level difference between upper and lower surfaces of control rod was occurred and 〈CLR〉 increased in negative direction. It can be noted that, absolute value of negative peak of the 〈CLR〉 that was

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E. Fırat et al. / Ocean Engineering 108 (2015) 504–518

2.1

0.2 -0.00012

1.8

0.0





1.445

1.5

1.2 L/D=1 L/D=2 L/D=3 L/D=4 Single Cylinder (Re=50)

0.9

-0.2

-0.4

C

L/D=1 L/D=2 L/D=3 L/D=4 Single Cylinder (Re=50)

-0.6

-0.8

(s)

0.6 0

20

40

60

80

100

0

20

40

α (°) L/D=1 L/D=2 L/D=3 L/D=4 Single Cylinder (Re=50)

0.25

0.20

0.25

80

100

60

80

100

L/D=1 L/D=2 L/D=3 L/D=4 Single Cylinder (Re=50)

0.20

0.15

CLR RMS

0.15

CDR RMS

60

α (°)

0.10

0.10

0.05

0.05

0.033

0.00

0.00 0.00054 0

20

40

60

80

100

α (°)

0

20

40

α (°)

Fig. 12. Drag and lift coefficients distributions of the control rod: (a) Time-averaged values, (b) RMS values.

U

90

8

1.5

180

1.0

o

θ

o

0

o



0.5 0.0 -0.5 L/D=1 L/D=2 L/D=3 L/D=4 Single Cylinder (Re=50)

-1.0 -1.5 -2.0 0

90

180

270

360

θ (°) Fig. 13. Time-averaged pressure coefficient distribution around the control rod in tandem arrangement.

obtained at α ¼60° is 11 times more than the positive peak of it that was obtained at α ¼6°. For α 460°, the position of the stagnation point on the windward surface of the control rod almost remained unchanged but counter-rotating vortices over the lower-right quadrant slightly moved towards upper-right quadrant as α increased. As a consequence of this, pressure level difference between upper and lower surfaces of control rod decreased; the 〈CLR〉 slightly increased. It was also observed that the length of the recirculating region consists of counter-rotating vortices increased with increased α.

As seen from the Fig. 12b, RMS values of fluctuating drag coefficients of the upstream control rod (CDR RMS) are overwhelmingly greater than that of the single cylinder even at small angles of incidence. The graph lines of CDR RMS roughly resembled a dome regardless of spacing ratio; as the spacing ratio was increased, the height of the dome reduced. These reductions were mainly attributed to the weakness of interaction between bodies. From the Fig. 12b, it can be inferred that the RMS values of the fluctuating drag coefficients of the control rod will be overlapped with that of the single cylinder at relatively high spacing ratios. Apart from the L/D¼ 1 case, the CLR RMS values generally much lower than that of the single cylinder (Fig. 12b). For L/D¼ 2 and 3, the CLR RMS values can be reduced to 20% and 5% that of the single cylinder, respectively, in tandem arrangement. For L/D ¼3, as the incidence angle increased from 10° to 12.5° or as the flow pattern changed from the SLMFP to the VMFP, a sudden jump in CLR RMS was observed. The main reason underlying this finding was attributed to the appearance of vortex shedding from the control rod. On the other hand, for L/D ¼3 and 4, the change of the flow pattern from VMFP to WIFP accompanied to reductions in CLR RMS. And this time, the reason was attributed to the suppression of the vortex shedding from the control rod. 4.2.2. Square prism In this section, the time-averaged and RMS values of both fluctuating drag and lift coefficients of square prism will be presented for different configurations of rod-square system. In the case of single square prism, 〈CD0〉 was decreased up to α ¼4° then increased up to α ¼ 12.5° (Fig. 14a). The reduction in 〈CD0〉 from α ¼0° to α ¼ 4° related to that the increase in 〈FD0〉 is lower across the AT'. At α ¼6°, the lower side vortex and the lower vortex of the

E. Fırat et al. / Ocean Engineering 108 (2015) 504–518

0.4

2.0

0.3



1.8



L/D=1 L/D=2 L/D=3 L/D=4 Single Square (Re=200)

0.5

2.2

1.6 1.4

L/D=1 L/D=2 L/D=3 L/D=4 Single Square (Re=200)

1.2 1.0

20

40

60

80

0.2 0.1 0.0 -0.1 -0.2

0.8 0

515

100

0

20

40

α (°)

60

80

100

α (°) 1.0

0.30 0.25

0.8

CLS RMS

CDS RMS

0.20 0.15 0.10

0.4

L/D=1 L/D=2 L/D=3 L/D=4 Single Square (Re=200)

0.05 0.00 0

20

40

60

0.6

L/D=1 L/D=2 L/D=3 L/D=4 Single Square (Re=200)

0.2

80

100

0

α (°)

20

40

60

80

100

α (°)

Fig. 14. Drag and lift coefficients distributions of the square prism: (a) Time-averaged values, (b) RMS values.

α = 12.5˚

α = 15˚

Fig. 15. Instantaneous vorticity contours and time-averaged streamlines around the single square prism before and after the flow pattern transition.

counter-rotating vortices in the near-wake of single square prism were started to coalesce around the lower trailing edge. The second decrease in 〈CD0〉 that observed at α ¼ 15° was associated with the occurrence of the new flow pattern that one row of vortices beyond the single square prism started to split into two rows (Fig. 15). Strouhal number distribution of the single square prism also supports this idea (Fig. 8). In this flow pattern, lower vortex of counter-rotating vortices in the near-wake of square prism at

α ¼12.5° moved towards the vortex of lower edge and totally coalesced with it as α reached to 15° (Fig. 15). 〈CD0〉 was increased up to α ¼45°. It can be noted that the shapes of the 〈CD0〉 distributions from α ¼ 45° to α ¼0° and from α ¼45° to α ¼90° are identical. In other words, α ¼ 45° is a vertical symmetry axis for the 〈CD0〉 distribution. On the other hand, when control rod added to the flow, it caused great reductions in 〈CDS〉, especially at the lower range of the incidence angles. The maximum reduction in 〈CDS〉 is

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E. Fırat et al. / Ocean Engineering 108 (2015) 504–518

of an order of 46% (α ¼0° and L/D ¼3). It is apparent from the Fig. 14a that the minimum values of 〈CDS〉 were obtained in CFP. Owing to the fact that the front surface of the square prism was well shielded by the wake of the control rod from the direct impact of the incoming flow, the high pressure level, particularly concentrated around the middle of the front surface of the square prism, was lowered (Fig. 16). Not only the front surface but also the rear surface was affected in a positive manner by the presence of the upstream rod. The base pressure level of the square prism was increased (Fig. 16). On the other hand, in general it can be said that for L/D41 and α 412.5°, the control rod is not beneficial to the reduction in 〈CDS〉. At L/D ¼1 case, 〈CDS〉 are lower than 〈CD0〉 for values up to but not including α ¼75°. For α ¼ 90°, NIFP dominated on rod-square system except L/D ¼1 and 2 as demonstrated before, therefore, the values of aerodynamic characteristics of square prism at L/D¼3 and 4 approximated to those of the single square prism (Fig. 14). While angle of incidence reached to α ¼45°, 〈CD0〉 reached to its maximum value but 〈CL0〉 reached to zero again like in the case of α ¼0°. It can be noted that, the 〈CL0〉 distribution from α ¼ 45° to α ¼0° is identical to the negative 〈CL0〉 distribution from α ¼45° to α ¼90°. As seen from the Fig. 14a, 〈CL0〉 values at α ¼0°, 45° and 90° are equal to zero due to the symmetricity of solid square shape with respect to the axis that parallel to the freestream direction and pass throughout the center of the single square prism. At L/D ¼1, the 〈CLS〉 data scattered around the imaginary zero lift line, up to GVEFP (Fig. 14a). As the α of rod-square system entered into the range of GVEFP, a sharp increase in 〈CLS〉 was recorded at L/D¼ 1. It can be explained as follows: As L/D was reduced, the 1.5

L/D=1 L/D=2 L/D=3 L/D=4 Single Square (Re=200)

1.0 0.5

-0.5 -1.0

U

C 8



0.0

B

-1.5

A

-2.0 -2.5

D

B

A

D

C

Fig. 16. Time-averaged pressure coefficient distribution around the square prism in tandem arrangement. The time-averaged pressure coefficient distributions on the front and the rear surfaces play an important role in time-averaged drag minimization.

interaction between the inner shear layers of the control rod and square prism becomes stronger. This interaction produces a decrease in the vortex strength (circulation) of the inner shear layer of the square prism; thus, vortex-induced lift that acts in negative lift (downward) direction was diminished. On the other hand, the reduction in AT* is also responsible for this sharp increase. The root-mean-square values of the fluctuating lift coefficients of the square prism, CLS RMS, are obviously larger than that of the fluctuating drag coefficients of it, CDS RMS, due to the regular vortex shedding from the square prism (Fig. 14b). It can be seen clearly that root-mean-square values of the fluctuating lift coefficients of the square prism at L/D ¼1 are lower than those of the single square prism regardless of α. That is to say, regular vortex shedding from the square prism is interrupted. This is presumably due to that the control rod that located close to square prism generally disturbed the lower shear layer of the square prism, thus, regular vortex formation from the lower shear layer is interrupted. The CLS RMS can be reduced to 32% of and CDS RMS can be reduced to 31% of that of the single square prism by locating the control rod 3D upstream of the square prism at α ¼0°. The increase in α from 0° to 15° generally results in a decrease in the difference between CLS RMS and CL0 RMS values, irrespective of L/D. For the spacing ratios except L/D¼ 1, the RMS values of both fluctuating drag and lift coefficients of the square prism are higher than those of the single square prism at relatively high incidence angles, α ¼45°, α ¼ 60° and α ¼75°. 4.2.3. Rod-square system Fig. 17 shows the ratios of time-averaged drag coefficients and RMS of fluctuating lift coefficients of rod-square system to those of the single square prism. It is clear from the figure that both ratios sharply increased up to certain levels regardless of L/D with the increase in α from 0°. At very small incidence angles, 0° and 2°, control rod reduced the time-averaged drag of the total system very effectively, especially for L/D ¼2 and 3. The maximum reduction of the total drag coefficient (about 26%) is observed at center-to-center spacing ratios of L/D ¼2 and 3 for α ¼ 0°. For α ¼0°, although the total drag coefficient in the L/D ¼2 case is almost equal to that of L/D ¼3 case, the CLT RMS in the L/D ¼2 case is about 17.5% higher than that of the L/D¼ 3 case. The maximum reduction of the CLT RMS (α ¼0° and L/D ¼3) is 60% in comparison to the CL0 RMS. At the highest incidence angle, α ¼ 90°, fluctuating lift forces also dramatically suppressed irrespective to L/D, like in the α ¼0° and 2° cases. This suppression was mainly attributed to the mutual interaction that leads to a decrease in vorticity magnitude of vortices, especially inner ones. But, at α ¼ 90°, the same cannot 1.4

CLT RMS / CL0 RMS

/

1.2

1.1

1.0

0.9

L/D=1 L/D=2 L/D=3 L/D=4

0.8

20

40

60

α (°)

80

1.0

0.8

0.6

0.4

0.7 0

L/D=1 L/D=2 L/D=3 L/D=4

1.2

100

0

20

40

60

80

100

α (°)

Fig. 17. The control rod effectiveness in reducing the time-averaged drag and suppressing the fluctuating force coefficients of the rod-square system for various arrangements.

E. Fırat et al. / Ocean Engineering 108 (2015) 504–518

be said for the drag reduction. For L/D ¼1, the CLT RMS values are lower than CL0 RMS values, apart from slightly greater angles of incidence, α ¼ 10°, 12.5°, 15°, 20° and 30°. In GVEFP, the 〈CDT〉 values are higher than 〈CD0〉 values even though the CLT RMS values are quite lower than CL0 RMS values. Overall, CLT RMS values dropped below the CL0 RMS values regardless of L/D.

517

Acknowledgments The authors gratefully acknowledge the financial support from the Scientific and Technological Research Council of Turkey (TÜBİTAK) for this study through grant number of 105M241.

Appendix A 5. Conclusion The study herein focused on the flow around the rod-square (total) system at incidence to a uniform stream, that is, flow characteristics such as vortex shedding frequency, near-wake flow structure, suppression of vortex shedding, mutual interactions, shear layer separation and attachment, the size of time-averaged wake; aerodynamic characteristics such as time-averaged and RMS values of fluctuating drag and lift coefficients. Simulations were conducted for various center-to-center spacing ratios (L/D) and incidence angles of the total system (α). The Reynolds number based on the side length of the square prism and inlet flow velocity is 200. The concluding remarks can be summarized as follows: (1) Seven distinct flow patterns (named as cavity, vortex impingement, shear layer merging, vortex merging, gap vortex enveloping, weak interaction, and negligible interaction), depending on the geometrical arrangement of the circular rod and the square prism, were identified in the simulations. It was expressed that CFP is the most effective pattern in order to reduce the undesired effects of fluctuating aerodynamic forces acting on rod-square system and its each component. It was found that GVEFP is peculiar to the lowest spacing ratio tested. Furthermore, in GVEFP, the 〈CDT〉 values are higher than 〈CD0〉 values even though the CLT RMS values are quite lower than CL0 RMS values. In NIFP, the interaction between the control rod and square prism is quite weak. Hence, the values of the flow and aerodynamic characteristics of the mentioned bodies approached to those of the single-body cases. (2) As a passive flow control method, the control rod is effective not only in reducing of drag, but also in suppressing the unsteady aerodynamic force coefficients acting on a square prism. The time-averaged drag coefficient of the downstream square prism can be decreased up to 46%, CD RMS and CL RMS values can be decreased up to 69% and 68%, respectively (α ¼ 0° and L/D ¼3), when compared to the single square prism. On the other hand, the presence of the square prism in the wake of control rod also provides a reduction in the fluid forces acting on control rod. The maximum reductions in 〈CDR〉 and CLR RMS values are 55% and 95%, respectively, when compared to the single circular cylinder. (3) The results showed that the optimum geometry comprises a rod with a diameter one-fourth the side length of the square prism placed 3D upstream of the cylinder at α ¼ 0°. In this case, α ¼0° and L/D¼ 3, the time-averaged drag and RMS of fluctuating lift coefficients of the rod-square (total) system decreased by 26% and 60%, respectively, when compared to the single square prism. (4) There is no specific relation between the angle of incidence and 〈CDT〉. In general, at small angles of incidence, α ¼ 0° to 12.5°, control rod is largely effective on 〈CDT〉, regardless of L/D. At relatively higher angles of incidence, α ¼15° to 60°, the 〈CDT〉/ 〈CD0〉 ratios randomly varied between the 0.9 and 1.1. However, at high angles of incidence, α ¼ 75° and 90°, the values of 〈CDT〉 is slightly higher than that of the 〈CD0〉. (5) Overall, CLT RMS values dropped below the CL0 RMS values regardless of angles of incidence tested.

The parameters of the flow configuration were shown in Fig. 1b. First of all, the time-averaged total drag coefficient, 〈CDT〉, that includes the drag effects of both control rod and square prism can be defined in the following form:

(

2 FDR + FDS )/ 0. 5⋅ρ⋅U∞ ⋅AT‵

CDT =

(

CDT =

( C ⋅0. ( 0. 5⋅ρ⋅U DR

2 ⋅AR‵ 5⋅ρ⋅U∞

2 ∞ ⋅AT‵

CDT =

(

) )

2 + CDS ⋅0. 5⋅ρ⋅U∞ ⋅AS‵ /

)

CDR ⋅AR‵ + CDS ⋅AS‵ )/AT‵

(A.1)

In the above equation, the frontal area of the rod-square system (AT') can be calculated by the following procedure:

HS‵ = D (Sin α + Cos α )

(A.2)

HR‵ = d

(A.3)

h = L⋅Sin α

(A.4)

If h + HR‵ /2 ≤ HS‵ /2 then AT‵ = HS‵⋅1 m

(A.5)

If h − HR‵ /2 < HS‵ /2 < h + HR‵ /2 then AT‵ = ( HR‵ /2 + HS‵ /2 + h) (A.6)

⋅1 m

If HS‵ /2 ≤ h − HR‵ /2 then AT‵ = ( HR‵ + HS‵ )⋅1 m

(A.7)

In a similar manner, the time-averaged lift coefficient of the total system, 〈CLT〉, that includes the lift effects of both control rod and square prism and the planform area of the rod-square system (AT*) can be expressed as 2 * FLR + FLS )/(0.5⋅ρ⋅U∞ ⋅AT )

CLT =

(

CLT =

(C

LR

)

2 * 2 * ⋅0. 5⋅ρ⋅U∞ ⋅AR + CLS ⋅0. 5⋅ρ⋅U∞ ⋅AS /(0.5

2 * ⋅ρ⋅U∞ ⋅AT )

CLT =

(

)

CLR ⋅AR* + CLS ⋅AS* /AT*

HS* = D (Sin α + Cos α )

(A.8)

(A.9)

HR* = d

(A.10)

l = L⋅Cos α

(A.11)

If l + HR*/2 ≤ HS*/2 then AT* = HS*⋅1 m

(A.12)

If l − HR*/2 < HS*/2 < l + HR*/2 then AT* = (HR*/2 + HS*/2 + l) ⋅1 m If HS*/2 ≤ l − HR*/2 then AT* = (HR* + HS* )⋅1 m

(A.13) (A.14)

The root-mean-square value of the fluctuating lift coefficient can be expressed as follows:

518

CLT RMS =

E. Fırat et al. / Ocean Engineering 108 (2015) 504–518

1 n

n

∑ ( CLT , i −

CLT

)2

(A.15)

i=1

where n is the total number of samples. After a simple manipulation, we obtain

CLT RMS =

Δt (t f − ti + Δt )

tf

∑ ( CLT (t ) − t = ti

CLT

)2 (A.16)

where CLT (t ) is instantaneous lift coefficient, ti is the initial value of the time and tf is the final value of the time in the lift coefficienttime history. It must be noted that, ti and tf must be determined carefully in order to minimize the RMS errors arisen from the initial transient period, insufficient number of samples, etc.

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