numerical heat transfer, part a: applications · et al. [8] found that for s=d < 0.3 at re ¼...

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Numerical Heat Transfer, Part A:ApplicationsAn International Journal of Computation andMethodologyPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713657973

Flow Characteristics Behind Rectangular CylinderPlaced Near a WallM. Kumaran a; S. Vengadesan aa Department of Applied Mechanics, IIT Madras, Chennai, India

Online Publication Date: 01 January 2007To cite this Article: Kumaran, M. and Vengadesan, S. (2007) 'Flow CharacteristicsBehind Rectangular Cylinder Placed Near a Wall', Numerical Heat Transfer, Part A:

Applications, 52:7, 643 - 660To link to this article: DOI: 10.1080/00397910601150015URL: http://dx.doi.org/10.1080/00397910601150015

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FLOW CHARACTERISTICS BEHIND RECTANGULARCYLINDER PLACED NEAR A WALL

M. Kumaran and S. VengadesanDepartment of Applied Mechanics, IIT Madras, Chennai, India

Three-dimensional unsteady flow over a bluff body located parallel to a wall, kept at differ-

ent gap height from the wall, has been studied numerically. The bluff body considered is a

rectangular cylinder with two different aspect ratios, B/D ¼ 1 and B/D ¼ 2, where B and D

are the width and height of the cylinder, respectively. The flow is considered as a laminar

flow, and the Reynolds number based on the height of the cylinder cross section and

oncoming reference velocity is 450. Numerical study is carried out by varying the distance

of the cylinder from the wall, and the development of the vortex shedding phenomenon under

the influence of the wall is investigated. From previous experiments, it is observed that as

the distance between the wall and the cylinder decreases, the wake behind the cylinder

becomes stationary and the vortex shedding is suppressed. The present numerical study

confirms a similar trend. Periodic activity in the downstream of the flow is disturbed

completely with decreasing gap between the wall and the cylinder.

1. INTRODUCTION

Periodic vortex shedding due to the flow over a bluff body occurs in many engin-eering and environmental problems. Flow around bluff bodies in a uniform stream hasbeen studied extensively, by both experiment and computation. Bluff bodies can besquare or circular or any nonregular shape. This vortex shedding causes dynamic load-ing on the bluff body and enhances mixing in the wake. Estimation of vortex sheddingfrequency and prediction of flow phenomena in such flows is of great practical impor-tance. The wake behavior depends on different aspects of the flow field, such as the endconditions, the blockage ratio of the flow passage, and the aspect ratio of the bluffbody. The shear effect due to the presence of a wall or ground in the flow makesthe velocity of the approaching stream vary in a direction normal to the bluff body.Then the mean structure and dynamic behavior of the wake change significantly asa function of the gap height. Vibrations of pipelines lying on the sea bottom underthe effects of sea currents, pipelines and bridges under the effect of the wind, andthe aerodynamics of a vehicle to reduce the drag and down force are typical examples.In such examples, vortex shedding may cause a noticeable erosion of the supports ofthese structures. In previously reported literature studies, circular and square crosssections have been considered as the bluff body and their wakes analyzed.

Received 28 March 2006; accepted 13 November 2006.

Address correspondence to S. Vengadesan, Department of Applied Mechanics, IIT Madras,

Chennai-36, India. E-mail: [email protected]

643

Numerical Heat Transfer, Part A, 52: 643–660, 2007

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/00397910601150015

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Elliptic cylinders have characteristics between those of a cylinder and a flatplate, depending on the ratio of major to minor axes. Choi and Lee [1] performeddetailed experiments on the ground effects on flow past an elliptic cylinder placedin a turbulent boundary layer. The flow around a circular cylinder in the presenceof a wall has been reported by conducting experiments at high Reynolds number[2, 3]. It is well known that for a circular cylinder the separation point is oscillating,but in the case of a rectangular cylinder, it is fixed. This difference brings consi-derable change in the flow behavior in the gap between wall and the bottom ofthe cylinder.

When a two-dimensional cylinder is placed near a plane wall, regular vortexshedding is absent [4]. At a critical gap height between the cylinder and the wall, vor-tex formation disappears. Taniguchi et al. [5] found that this value is 0.5 times theside of a square cylinder. Experiments on the flow around a square cylinder placedat various heights above a wall at high Reynolds number were carried out by Duraoet al. [6], Bosch and Rodi [7], and Martinuzzi et al. [8]. Durao et al. [6] found thecritical value for the gap beyond which the vortex shedding occurs to be in the range(gap height S to cylinder height D) S=D ¼ 0.25 to 0.5 at Re ¼ 13,600. In the experi-ments of Bosch and Rodi [7], conducted at Re ¼ 22,000, steady flow was observedfor S=D� 0.25, while vortex shedding was observed for S=D� 0.5. Martinuzziet al. [8] found that for S=D < 0.3 at Re ¼ 18,900, vortex shedding was completelysuppressed in the near-wake region, and for S=D>0.9 the flow and vortex sheddingstrength are similar to that of the no-wall case.

Davis and Moore [9] performed numerical simulation of two-dimensionalfree-stream flow over rectangles at Re ¼ 100 to 2,800. They suggested that three-dimensional flow simulation with finer grids will give results closer to the experi-mental data. Arnal et al. [10] have investigated three cases, (1) flow past a squarecylinder in a free stream, (2) flow past a square cylinder on a fixed wall, and (3) flowpast a square cylinder on a sliding wall, for Re ¼ 100 to 1,000. They reported thatthe presence of the wall strongly affected the stability of the flow and the vortexshedding. In the case of two-dimensional laminar boundary-layer flow, Hwangand Yao [11], from their numerical investigation, reported that the presence of a wallin a thick boundary layer (i.e., d=D ¼ 5, where d is the boundary-layer thickness) dis-torts the vortices, which in turn leads to a decrease in Strouhal number and averagedrag coefficient. Lee et al. [12] performed numerical investigations of the wall influ-ence on the vortex shedding and heat transfer characteristics for square cylinders.Their study was limited to two dimensions and performed at low Reynolds number.Korchi and Oufer [13], Younis et al. [14], and Sharma and Eswaran [15] have carried

NOMENCLATURE

B width of the cylinder

Cd coefficient of drag

Cl coefficient of lift

Cp coefficient of pressure

D height of the cylinder

f frequency of vortex shedding

Re Reynolds number

S gap height of cylinder from the wall

t non-dimensional time

U1 free-stream velocity

d boundary-layer thickness

q density of the fluid

644 M. KUMARAN AND S. VENGADESAN

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out similar studies. More recently, Bhattacharyya et al. [16] performed numericalsimulations to study the influence of buoyancy on vortex shedding and heat transferfrom a square cylinder in proximity to a wall.

From the previous literature, it is observed that experimental studies have beenreported for flow around both square and rectangular cylinders with ground effect,at high Reynolds number. In the case of numerical study, the flow at high Reynoldsnumber can be predicted by large-eddy simulation (LES) techniques, but this methodis computationally expensive. Few previous researchers reported numerical results atlow Reynolds number, and those were only two-dimensional and steady calcula-tions. However, Sohankar et al. [17] and Saha et al. [18] report that the flow becomesthree-dimensional and unsteady. Hence, to capture proper wake characteristicsunder the influence of a wall and associated changes in pressure field and hence influ-ence on heat transfer characteristics, it is essential to carry out unsteady three-dimensional simulations.

To understand the fundamental phenomenon associated with this flow further,in the present study, the flow past a rectangular cylinder placed near a wall has beenconsidered. To investigate the effect of geometry, rectangular cylinders having aspectratios (width B to height D) of 1 and 2 are taken. Four different gap ratios are con-sidered in this study. The Reynolds number defined based on height of the cylinder(D) and oncoming reference velocity is 450. Results are obtained by solving unsteadythree-dimensional Navier-Stokes equations using the finite-difference method.

2. FORMULATION OF THE PROBLEM

The physical problem considered in this study is a three-dimensional flow of anincompressible fluid around a bluff body. Figure 1 shows the geometry and coordi-nates chosen for the problem. The origin of the chosen Cartesian coordinates system(x, y, z) is aligned on the upstream face of the cylinder. In this, the x axis is aligned

Figure 1. Sketch of computational domain and boundary conditions.

FLOW CHARACTERISTICS BEHIND RECTANGULAR CYLINDERS 645

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along the inlet flow direction (streamwise direction), the y axis is parallel to thecylinder axis (spanwise direction), and the z axis is perpendicular to both thesedirections (cross-stream direction).

2.1. Governing Equations

The governing equations for the incompressible flow considered are written as

qui

qxi¼ 0 ð1Þ

qui

qtþ quiuj

qxj¼ � qp

qxiþRe�1 q2ui

qxjqxjð2Þ

where u and p are the velocity and the pressure, respectively. The Reynolds number isdefined as Re ¼ qU1D=m; m is the dynamic viscosity of the fluid, and U1 is the free-stream velocity. All the characteristic lengths are normalized with D, velocities withU1, and physical times with D=U1.

2.2. Numerical Details

In the governing equations, viscous terms and pressure terms are discretized bya second-order-accurate central differencing scheme. Staggered arrangement of vari-ables is used. Convective terms are discretized by a third-order upwind-biasedscheme. Time advancing is done by the successive over relaxation (SOR) iterativemethod. Pressure is solved by the HSMAC (Highly Simplified Marker And Cellmethod) algorithm [19]. The following boundary conditions are applied, that is, non-slip boundary conditions on the cylinder surface, slip conditions on the top, nonslipconditions at the bottom boundaries, periodic boundary condition in the spanwisedirection, and convective boundary condition at the exit boundary. To avoid theboundary-layer effect in the near-wall region, the body is placed at 2.5D fromthe upstream boundary and uniform flow is forced at the upstream boundary. Inthe present work, the computational fluid dynamics (CFD) code and the numericalstrategies used are the same as detailed by Nakayama and Vengadesan [20]. In orderto assess the accuracy of the numerical method, we carried out flow past an isolatedsquare cylinder at Re ¼ 100 and 500. Simulation at Re ¼ 100 was carried out with atwo-dimensional grid, and that at Re ¼ 500 was carried out with a three-dimensionalgrid. Two different spanwise lengths, 4D and 8D, each with two-grid resolution, areconsidered. Results in terms of bulk parameters are compared with available litera-ture in Table 1. In addition, mean vorticity contour for the case of Re ¼ 500 per-formed with grid G28 is also compared in Figure 2. It is to be noted that theresults of Shoankar et al. [17] are by numerical simulation with spanlength 6D for3-D computation, and that of Robichaux et al. [21] is by an analytical approach.Our results are in agreement with all these, and spanlength 4D was chosen for furthercomputations in the present investigation.


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To study the influence of the grid resolution within the framework of afford-able mesh size, two different grid sizes 150� 157� 21, defined as grid G1, and217� 211� 21, defined as grid G2, are chosen for computations. The first pointfrom the wall is placed at 0.04D in the case of grid G1, 0.02D in the case of gridG2. The grid is finer near the surface of the cylinder in both the x and z directionsand the bottom wall, to resolve the gradient better. A uniform grid is used in the gapbetween the wall and the cylinder, and the rest of the domain uses nonuniform grids.

3. RESULTS AND DISCUSSION

Numerical investigation of three-dimensional flow past a rectangular cylinderwith ground effect has been carried out. The gap height ratios considered are

Table 1. Comparison of bulk parametersa

St Cd

(a) Re ¼ 100

G21 0.153 1.44

Sohankar et al. [17] 0.146 1.414

Robichaux et al. [21] 0.154 1.53

St Cl Cd Cl rms Cd rms

(b) Re ¼ 500

G21 0.13733 �0.00705 1.91716 1.16002 0.16037

G28 0.13733 0.00260 1.91039 1.13828 0.15243

G41 0.13733 0.00289 1.91250 1.15007 0.16368

G54 0.13733 0.00569 1.93408 1.10659 0.14237

Sohankar et al. [17] 0.126 — 1.87 1.23 —

aSt, Strohaul number; Cl and Cd, mean lift and drag coefficients; Cl rms and Cd rms, root-mean-square of

lift and drag coefficients. G21 ¼ 183� 133� 21, spanlength ¼ 4D, G28 ¼ 183� 133� 28, spanlength ¼ 4D,

G41 ¼ 183� 133� 41, spanlength ¼ 8D, G54 ¼ 183� 54� 133, spanlength ¼ 8D.

Figure 2. Comparison of mean vorticity contour-left, Sohankar et al. [17]; right, present by G28 grid.


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S=D ¼ 1.5, 1.0, 0.6, 0.4, and 0.2, where S is the gap height and D is the size of thecylinder. After an initial development of flow, which is observed by the time devel-opment of the lift coefficient, results are obtained by averaging over 15 vortex shed-ding cycles. Initially some transient flow is observed; after that, periodic flow isobtained. Figure 3 shows the averaged streamwise velocity component at differentlocations obtained by both grids G1 and G2. It shows that the differences for thosecases are less than 2%. Therefore, results presented in this study are considered to begrid-independent. First, results for the case of aspect ratio of 1, that is, the squarecylinder, are discussed, and then the results for aspect ratio 2 are discussed.

3.1. Square Cylinder (B/D ¼ 1)

3.1.1. Streamlines pattern. Figure 4 shows streamlines of the flow patternsfor different gap ratios. From the streamline contours, it is observed that, forS=D ¼ 1.5 and 1.0, the recirculation zone is absent, but for S=D ¼ 0.6, the recircula-tion zone is formed behind the cylinder on the ground. This zone is altered by theshear layer separated from the lower side of the cylinder. Reattachment length ofthe shear layer, i.e., the distance from the back face of the cylinder to the pointon the wake centerline where the shear layer separated from the lower side crosses,is almost the same for both S=D ¼ 1.0 and 1.5 (Figures 4a and 3b). From Figures 4cand 3d, it can be observed that the separated shear layer from the lower side isstretched and the reattachment of the shear layer is nearly at x ¼ 4D. At this point,

Figure 3. Streamwise velocity at different x=D——G1 grid, ——G2 grid: (a) S=D ¼ 1.0; (b) S=D ¼ 0.6;

(c) S=D ¼ 0.4; (d) S=D ¼ 0.2.


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flow is reversed. When the gap height decreases, the recirculation zone behind thecylinder becomes larger. It affects the further development of the vortex shedding.Figure 4e shows that the wake behind the cylinder up to x ¼ 4D is stationary, thereis no interaction of shear layers, and vortex shedding is prohibited. In addition, onemore stationary recirculation zone is formed on the upstream side of the cylindernearer to the wall, which may be due to the stagnant flow. It is also clearly noticedthat the flow between the wall and the cylinder is accelerated as the gap ratio isreduced.

3.1.2. Streamwise velocity. Figure 5 shows the time-averaged streamwisevelocity along the wake centerline. For both S=D ¼ 1.0 and 1.5, a similar trend isobserved. For S=D ¼ 0.6 and 0.4, the flow reaches peak velocity nearly at x ¼ 4D;this means that the separated shear layer from the lower side has reached the wakecenterline within this distance, as was observed in Figures 4c and 4d. For S=D ¼ 0.2,the maximum velocity is reached at nearly x ¼ 3D. However, the peak value isdecreased, which may be due to the stagnant flow. It is clearly observed inFigure 4e, where the flow below the wake centerline is nearly similar.

Figure 4. Streamline pattern for different gap ratios.


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3.1.3. Pressure coefficient and spectral analysis. Figure 6 shows the time-averaged pressure coefficient in the wake centerline. A similar trend is observed forboth S=D ¼ 1.0 and 1.5. For S=D ¼ 0.6, 0.4, and 0.2, pressure values are nearly con-stant within the stationary recirculation zone. Figure 7 shows the Cp distributionover the surface of the cylinder. For S=D� 0.6, Cp on the front face of the cylinderis not affected due to the gap height. The stagnation point occurs at the center of thefront face. For S=D� 0.6, pressure distribution on the bottom of the cylinderchanges slightly with the gap height. When the gap height is reduced, the locationof the stagnation point is shifted slightly downward. For S=D ¼ 0.2, drastic vari-ation in the pressure distribution on the side surface between the gap entry andthe trailing edge of the cylinder (3–4) is observed.

Figure 8 shows the pressure distributions along the bottom cylinder face andthe wall in the gap region. For S=D>0.6, Cp on the bottom face of the cylinder islower than that along the wall, indicating positive streamline curvature. These trendswere observed by Bailey et al. [22]. For S=D ¼ 0.4, the bottom face pressure is

Figure 5. Time-averaged streamwise velocities along the centerline.

Figure 6. Time-averaged pressure coefficients along the centerline.


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slightly higher than the wall pressure at the trailing edge, indicating the concavecurvature (negative curvature) nature of the streamlines. For S=D ¼ 0.2 (Figure 8e),the pressure difference between the wall and the bottom face vanishes from the mid-dle to the trailing edge, indicating that the streamlines become parallel in the gap.Figure 9 shows powerspectra for the all gap heights obtained from the instantaneouslift coefficient. It is clearly seen that the dominant peak is reduced as the gap heightdecreases. For S=D ¼ 0.2, the peak is not seen at all, which indicates that vortexshedding is completely suppressed at this gap height.

3.2. Rectangular Cylinder (B/D ¼ 2)

To study the influence of the geometry aspect ratio in association with walleffects in the wake region, a rectangular cylinder with aspect ratio B=D ¼ 2 is con-sidered. From the square cylinder case it is observed that for gap heights S=D ¼ 1.5and S=D ¼ 1.0, a more or less similar trend is observed. Hence, in this case(B=D ¼ 2), only four gap heights, S=D ¼ 1.0, 0.6, 0.4, and 0.2, are considered.

3.2.1. Streamlines pattern. The streamlines of the flow patterns for differ-ent gap ratios are plotted in Figure 10. It is observed that, unlike the case of thesquare cylinder, even at S=D ¼ 1.0 and 0.6, a recirculation zone is formed behindthe cylinder on the ground. It is altered by the separated shear layer from the lowerside of the cylinder. The reattachment length is almost the same for both S=D ¼ 1.0and 0.6 (Figures 10a and 10b). For S=D� 0.6, the size of the recirculation zonebehind the cylinder is nearly the same for both aspect ratios. Due to this similarityin recirculation zone, mean drag attains the same value for S=D� 0.6 as shown later,in Figure 18. It is observed that the reattachment length is nearly equal for bothaspect ratios. From Figure 10c, it can be observed that the separated shear layerfrom the lower side of the cylinder is stretched, and reattachment of the shear layer

Figure 7. Time-averaged pressure coefficients around the square cylinder.


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occurs nearly at around x ¼ 5D. Similar to the square cylinder case, when the gapheight is decreased, the recirculation zone behind the cylinder becomes larger. Figure11d, corresponding to S=D ¼ 0.2, shows that the wake behind cylinder is stationaryand vortex shedding is completely prohibited. In addition, one more stationary recir-culation zone is formed on the upstream side of the cylinder nearer to the wall, andalso the flow between the wall and the cylinder is accelerated due to the presence ofthe wall.

3.2.2. Streamwise velocity. Figure 11 shows the time-averaged streamwisevelocity along the wake centerline. For both S=D ¼ 1.0 and 0.4, the flow reachesits peak velocity nearly at x ¼ 5D; this means that the separated shear layer fromthe lower side crosses the wake centerline within this distance. For S=D ¼ 0.6, peakvelocity is attained nearly at x ¼ 4D, and the same trend is observed in Figure 10cfor the square cylinder case. For S=D ¼ 0.2, it occurs nearly at x ¼ 4D, but the peakvalue is decreased, which could be due to the stagnant flow.

Figure 8. Time-averaged surface pressure coefficient distribution in gap: open square, wall; filled

diamond, cylinder.


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3.2.3. Pressure coefficient and spectral analysis. Figure 12 shows thetime-averaged pressure coefficient along the wake centerline. A similar trend isattained for both S=D ¼ 1.0 and 0.6. However, for S=D ¼ 0.4 and 0.2, pressure

Figure 9. Power spectra for all the gap ratios.

Figure 10. Streamline pattern for different gap ratios.


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values are constant in the stationary recirculation zone. Figure 13 shows the Cp

distribution over the surface of the cylinder. For S=D� 0.6, Cp on the front faceof the cylinder is not affected due to the gap height and the stagnation point occursat the center of the front face. For S=D < 0.6, the stagnation point is slightly shiftedtoward the lower side. For all gap ratios, pressure distribution on the bottom face ofthe cylinder is similar to as that of square cylinder. From Figure 14, for S=D>0.6,wecan observe that Cp on the bottom face of the cylinder is lower than that along thewall. This indicates that there exists a positive streamline curvature. However, forS=D ¼ 0.4 and 0.2, the pressure difference between the wall and the bottom facenearly vanishes from x ¼ 0.5D, indicating that the streamlines becomes parallel inthe gap. In the case of the square cylinder, this was observed only at S=D ¼ 0.2.The power spectra are obtained from the instantaneous lift coefficient for the allgap heights, and they are shown in Figure 15. It is clearly seen that the dominantpeak is reduced as the gap height decreases. For S=D ¼ 0.2, the peak has

Figure 11. Time-averaged streamwise velocities along the centerline.

Figure 12. Time-averaged pressure coefficients long the centerline.


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disappeared. It seems that vortex shedding is completely diminished at this gapheight. A similar phenomenon was obtained in the square cylinder case.

3.3. Lift and Drag Coefficient

The mean lift and mean drag coefficients as a function of gap height for bothsquare and rectangular cylinders are plotted in Figure 16. As the gap height

Figure 14. Time-averaged surface pressure coefficient distribution in gap: filled diamond, cylinder; open

square, wall.

Figure 13. Time-averaged pressure coefficients around the rectangular cylinder.


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Figure 15. Power spectra for all the gap ratios.

Figure 16. Lift and drag coefficient as function of gap height: filled diamond, square cylinder; filled

square, rectangular cylinder.


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increases, mean lift attains smaller values and approaches zero. This probably resultsfrom the recovery to periodic vortex shedding from the no-vortex-shedding situ-ation. This was also observed in the power spectra analyses (Figures 9 and 15). Meanlift of the square cylinder is less than that of the rectangular cylinder for lower gapratio. Mean drag coefficient for both the square and the rectangular cylinder isaround the same value when the body is placed within the range of S=D ¼ 0.6. Thismay be due to the formation of a stationary recirculation zone behind the cylinder.For large gap heights, S=D>0.6, the square cylinder has more drag compared to therectangular cylinder.

Figure 17. Pressure fluctuation time series at x=D ¼ 0.5 on the upper cylinder face, bottom cylinder face,

and wall for square cylinder case.


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3.4. Pressure Fluctuation

Figures 17 and 18 show the pressure time series at three different places, theupper cylinder face, the lower cylinder face, and the wall. The normalized pressuretime series is obtained as

CPðtÞ � Cp

C0pð3Þ

where CpðtÞ ¼ instantaneous pressure coefficientCp ¼ mean pressure coefficientC0P ¼ root-mean-square pressure coefficient.

Figure 18. Pressure fluctuation time series at x=D ¼ 1.0 on the upper cylinder face, bottom cylinder face,

and wall for rectangular cylinder case.


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This quantity is obtained after the initial development of the flow. Results areshown for two gap ratios, S=D ¼ 1.0 and S=D ¼ 0.2. Streamwise locations ofx ¼ 0.5D and 1.0D are chosen for the square and rectangular cylinder cases, respect-ively. From the figures, we observe that for S=D ¼ 1.0, regularity of pressure con-tours is present. However, for S=D ¼ 0.2, the surface pressures are not coherentand regularity is absent. Contours are disrupted. The effect is slightly less on thetop surface. Thus the pressures generated on the cylinder by the two shear layersare out of phase and those separated shear layers are not coupled. However, it isobserved that they are in phase, when the vortices are about to be shed. This isobserved for both the square and the rectangular cylinders.

4. CONCLUSIONS

Vortex shedding under the influence of a wall for laminar flow over a rectangu-lar cylinder has been investigated. Cylinders were placed at different gap ratios andnumerical investigations were carried out. From the results it is found that forS=D� 0.6, both cylinders have same mean drag, while the mean lift is different. Simi-larity in reattachment length of the shear layer is observed with varying geometryaspect ratio with respect to the gap height. It is observed that behind the cylindera recirculation zone is created for S=D� 0.6, due to the presence of the wall. Forthe lower gap ratio, the wake behind the cylinder is stationary, the periodic activityis disturbed, and vortex shedding is fully suppressed. From the pressure fluctuationtime series for S=D ¼ 0.2, it is observed that pressure contours are irregular and theseparated shear layers are out of phase. Due to the straightening of the shear layerthat is separated from the lower side, coupling with the shear layer that is separatedfrom the upper side is affected and hence suppression of the vortex shedding occurs.

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numerical heat transfer, part a: applications · et al. [8] found that for s=d < 0.3 at re ¼...

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