ALONZO-GARCíA A.， GUTIéRREZ-TORRES C. del C.， JIMéNEZ BERNAL J. A.，MOLLINEDO-PONCE de LEóN H. R.， MARTINEZ-DELGADILLO S. A.，BARBOSA-SALDA?A J. G.

1. Instituto Politécnico Nacional， SEPI ESIME Zacatenco， LABINTHAP U. P. Adolfo López Mateos Edif. 5 3er. Piso， México， D. F. México， E-mail： alejandro_1980@hotmail.com

2. Instituto Politécnico Nacional， UPIITA， Av. IPN 2580， C.P. 07340， México， D. F. México

3. Universidad Autónoma Metropolitana-Azcapotzalco， Depto. Ciencias Básicas， Av. San Pablo 180，Azcapotzalco C.P 07740， México， D. F. México

RANS simulations of the U and V grooves effect in the subcritical flow over four rotated circular cylinders*

ALONZO-GARCíA A.1， GUTIéRREZ-TORRES C. del C.1， JIMéNEZ BERNAL J. A.1，MOLLINEDO-PONCE de LEóN H. R.2， MARTINEZ-DELGADILLO S. A.3，BARBOSA-SALDA?A J. G.1

1. Instituto Politécnico Nacional， SEPI ESIME Zacatenco， LABINTHAP U. P. Adolfo López Mateos Edif. 5 3er. Piso， México， D. F. México， E-mail： alejandro_1980@hotmail.com

2. Instituto Politécnico Nacional， UPIITA， Av. IPN 2580， C.P. 07340， México， D. F. México

3. Universidad Autónoma Metropolitana-Azcapotzalco， Depto. Ciencias Básicas， Av. San Pablo 180，Azcapotzalco C.P 07740， México， D. F. México

This paper presents a CFD study about the effect of the V and U grooves in the flow over four cylinders in diamond shape configuration at subcritical flow conditions （Re =41000）. The k-εRealizable turbulence model was implemented to fully structured hexahedral grids with near-wall refinements. Results showed that the numerical model was able to reproduce the impinging flow pattern and the repulsive forces present in the lateral cylinders of the smooth cylinder array. As a consequence of the flow alignment induced by the grooves， a jet-flow is formed between the lateral cylinders， which could cause an important vortex induced vibration effect especially in the rear cylinder. The magnitudes of the shear stresses at the valleys and peaks for the V grooved cylinders were lower than those of the U grooved cylinders， but the separation points were delayed due the U grooves presence. It is discussed the presence of a blowing effect caused by counter-rotating eddies located near the grooves peaks that cause a decrease of the shear stresses in the valleys， and promote them at the peaks.

RANS model， grooved cylinder， four cylinders， subcritical flow， impinging flow

Introduction

Similarly to the flow over a single cylinder， the flow over an array of cylinders exhibits for each cylinder， distinctive characteristics that define its individual behavior such as flow separation， shear layers which evolve forming periodic wakes that could also be fully laminar or composed （laminar and turbulent），and whose interactions cause the apparition of fluctuating forces with different frequencies and magnitudes. The knowledge of the manner in which these characteristics and their interactions affect the flow behavior and the forces formed on the cylinders， could be reflected in performance improvements of mechanical and thermal devices such as heat exchangers， radiators， offshore structures， among others.

Along the years， the problem has been studied in both laminar and turbulent flow regime， using from experimental techniques such as LIF， PIV， LDA of the research of Lam and Lo［1］， Lam et al.［2，3］， Lam and Zou［4］and Wang et al.［5］， to numerical techniques such as the Boundary Element Method applied by Farrant et al.［6］， and the Finite Volume method applied to both two-dimensional and three-dimensional domains of the works of Lam et al.［7］， Zou et al.［8］to mention some examples. From these studies， it is evident that the flow patterns in a square array of 4 cylinders for both laminar and turbulent regimes， strongly rely on 3 parameters， which of course can be affected by the experimental conditions （for example， blockage， the boundary layer of the lateral walls， turbulence intensity of the flow， end effects， alignment， etc.） those are： The Reynolds number （Re=DU/ν）， the spacing between cylinders （pitch ratio） “Lc”， and the rotationangle of the array defined by “α”.

Analyzing only the effect of the rotation angle“α” in the flow patterns， it could be said based on the research of Lam et al.［2，3］and Wang et al.［5］， that although there are two limit cases geometrically speaking， which cause the formation of flow structures with significant differences， given by the angles α= 0oand α=45o， in general， any angular variation induces flow structures， which although share some similarities with the two mentioned cases，are different among themselves.

For the case of the array of cylinders with α= 45o， although it does not exist a well-established classification for the flow patterns， Zou et al.［8］numerically studied the phenomenon for the laminar regime（Re =200）， varying the spacing Lc/D from 1.2 to 5. The authors reported that for small values of Lc/D＜ 1.5， it is possible to distinguish between the cylinders a gap flow， which causes the appearance of repulsive forces between all cylinders. Also， the resulting wake formed behind the array， behaves similar to the present on the single cylinder flow， as is shown in Fig.1（a）.

For intermediate pitch ratios （1.5＜Lc/D＜3），the flow between Cylinders 1 and 3 forms flow patterns similar to the shielding regime， however， the shear layer of the Cylinder 1 is sandwiched by the shear layers of the Cylinders 2 and 4， and it becomes narrower，（see Fig.1（b））. Finally， for pitch ratios larger than Lc/ D＞3， （shown in Fig.1（c））， the oscillating wake of the Cylinder 1， causes the apparition of well-defined vortices， who impinge directly on Cylinder 3， forming a flow pattern similar to the described above as the impinging vortex regime.， and the flows over the Cylinders 2 and 4， appear to be fairly similar to the flow over a single cylinder.

Regarding the flow structures for intermediate angular positions， although information about the topic is scarce， it could be mentioned the work of Lam et al.［3］， performed with the LIF technique for the Re =200and 800，Lc/D =4covering the angles from α=0oto α=180oat 15ointervals， and the work of Wang et al.［5］carried out using the PIV technique for Re =8000， varying the Lc/D from 2 to 5，and covering the range of α=0oto α=45oat 7.5ointervals. In these studies the presence of jet flows between the lateral cylinders is reported. This jet flow structures contain different oscillation frequencies product of the shear layer mixing between the lateral cylinders that could endanger the overall structure （i.e.，“impingement” regime Lam et al.［3］）.

On the other hand， based on the reports of Walsh［9］， about the drag reducing effect present in flat plates covered with riblets， and the subsequent studies of Ko et al.［10］and Leung and Ko［11］， which extended the study of the drag reduction phenomenon in a flat plate to the study of the riblets effect in a circular cylinder in subcritical regime， Lim and Lee［12，13］， conducted comparative experiments about the flow over a smooth cylinder versus the flow over a V-grooved cylinder and the flow over a U-grooved cylinder， for Reynolds numbers ranging from 8×103to 1.4×105，using techniques such as HWA， flow visualization by tracer particles， PIV， and piezoelectric load cells， reporting that in the case of the U-grooved cylinder， the drag decreased by 18.6% compared to the smooth cylinder flow， being that for the “V” grooved cylinder，the decrease was 2.5%. In this regard， the authors pointed out changes in the wake structures associated with drag reduction， promoted remarkably by the U grooves， attributing this effect to its sharp edges.

Fig.2 Overall dimensions of the computational domain for the three studied cases

Table 1 Grid sizes of the case AR-A mesh independence study

Fig.1 Flow regimes of four cylinders in diamond arrangement

Table 2 Non-dimensionalu velocity component for the case AR-A grid Independence

Based on the aforementioned and as an attempt to provide new insight （the topic has not been explored yet）， related to the effects of the V and U grooves in the flow structures of the “impinging” flow regime，this paper presents a comparative analysis of the effect of the V and U grooves on the flow over four cylinders arranged in a diamond shape （α=45o）， at Re= 41 000 with Lc/D=3.5. The analysis was performed using the turbulence model “k-εRealizable” for its capability of presenting mean information of the main flow parameters at a relatively low computational cost. For simplicity in the analysis， the flow over the array of smooth cylinders was named case AR-A， the flow over the array of V-grooved cylinder was named case AR-B， and the flow over the array of U-grooved cylinders was named case AR-C.

1. Materials and methods

1.1Solution method

The flow fields were obtained using the segregated solver of the Ansys Fluent V.13.0 which applies the finite volume method to approximate the solutions to the Navier-Stokes equations in their stationary and incompressible forms. With the aim to optimize the solution algorithms， and to decrease the numerical diffusivity problem， all grids were composed of hexahedral cells arranged in a structured way. For the discretization of the momentum equations as well as the equations of production and dissipation of turbulent energy “k ” and “ε”， were employed the second order upwind finite differences scheme. The coupling of the pressure and velocity in the equations of continuity and momentum was performed using the PISO scheme.

1.2Computational domain

Based on the work of Lam et al.［7］， Lam and Zou［4］，Zou et al.［8］， which were focused on the study of the flow over an array of smooth cylinders， the computational domain constructed for all cases presented rectangle shape， being its overall dimensions 34D×20D×2.143D， as is shown in Fig.2.

Fig.3 Locations of the measured points used for themesh independence study for the cases AR-A， AR-B and AR-C

Fig.4 Isometric view of the mesh ACL-2 nodal distribution

In all numerical simulations of the AR-B and AR-C cases， were placed 25 grooves equally distributed along the domain with height and spacing of h= 0.003 m and s =0.003mfor the V grooves， and h= 0.015 m and s=0.003mfor the U grooves， as those implemented in the works of Lim and Lee［12，13］. Forthe smooth cylinders array， the grid independence study was performed with the model “k-εRealizable”， being the nodal distribution varied only in the“x” and “y” coordinates.

Table 3 Grid sizes for the cases AR-B and AR-C　grid independence study

Considering that in an array of cylinders， the interaction between the different shear layers turns out to be a distinctive feature that could be related to changes in the momentum exchange in the “z” coordinate， 100 elements were equally distributed along the domain depth Lzinstead of the 64 elements suggested by Breuer［14］， and the 64 elements of the work of Zou et al.［8］.

Also all cylinders were discretized with 200 cells equally spaced along the cylinders periphery. Table 1 shows the sizes of for the meshes constructed in order to evaluate the grid independence for the case AR-A，named ACL-1， ACL-2， ACL-3 and ACL-4， which sizes ranged from 5.07×106to 7.275×106cells.

With the objective of evaluating the performance of the 4 constructed meshes， Table 2 shows the values of the non-dimensional “u ” velocity for the positions x/D =1and x/D=2， measured from the center of each cylinder as is shown in Fig.3.

From the results， it can be seen that between the ACL-2 and ACL3 meshes， the maximum difference was approximately 6.5%， being that between the ACL-3 and ACL-4 meshes was less than 4.5%. For this reason and considering the impact on the computational cost of the implementation of an unnecessarily fine mesh， it was considered the ACL-2 mesh for the final run， and post-processing tasks. Figure 4 shows an isometric view of the ACL-2 mesh， where it can observed the refinement of the cells located in the near-wall regions.

Table 3 shows the overall dimensions of the grids constructed to study the mesh independence of the cases AR-B and AR-C which were based on the nodal“x-y” configuration of the ACL-2 mesh， with the same 200 elements equally distributed along the cylinders periphery. In order to study theLznodal influence， the distribution of the cells between grooves was varied from 8 to 20 in intervals of 4， representing between 200 and 500 cells equally distributed in the“z” coordinate. The maximum and minimum sizes of the constructed grids for both kinds of grooves were of 4.3×107and 1.72×107cells respectively.

Fig.5 Partial view of the nodal distribution for the “2d” template implemented for the cases AR-B and AR-C grid independence

Table 4 Characteristics of the grids used for the final runs of cases AR-A， AR-B and AR-C

It is important to point out that the number of elements in the plane “x-y ” turned out to be the samefor all grids （i.e.， A-V1 and A-U1）. This was achieved by the use of a subroutine implemented in the software Gambit V.2.2.30， in which a 2d nodal template was used as a common base over which were generated the 25 grooves V or U， avoiding additional time consumption in the grid generation process. A zoom view of the nodal distribution of the “x-y” 2d template used for the grid generation process in the mesh independence of AR-B and AR-C cases is shown in Fig.5.

Table 5 Experimental and numerical data of theandcoefficients for the flow over 4 cylinders in diamond shape configuration

Table 5 Experimental and numerical data of theandcoefficients for the flow over 4 cylinders in diamond shape configuration

Re /LD1DC2DC3DC4DC1LC2LC3LC4LC Sayers （Exp.）［17］30 000 3.5 1.05 1.233 0.41 1.233 0　0.015 0 -0.015 Lam and Fang （Exp.）［18］12 800 3.58 1 1.31 0.65 1.31 0　0.025 0 -0.025 Lam et al.（Exp.）［3］41 000 3.41 1.11 1.42 0.545 1.42 0.025　0.055 0 -0.055 Wang et al. （Exp.）［5］8 000 3.5 1.24 1.42 0.89 1.34 0.026 -0.037 0.041　0.038 CASE AR-A 41 000 3.5 1.03 1.21 0.68 1.19 0　0.024 0 -0.024 CASE AR-B 41 000 3.5 1.21 1.38 0.80 1.28 0　0.070 0 -0.067 CASE AR-C 41 000 3.5 0.99 1.03 0.63 1.02 0　0.041 0 -0.033

Similar to that was performed for the grid independence analysis of case AR-A， for cases AR-B and AR-C， the chosen grids to conduct the final simulations were the AV-3 and AU-3.

These selections were made taking into consideration minimum and maximum differences of the mean velocity component of 2.5% and 9.3% between the AV-3 and AV-4 grids， and 3.7% and 6.9% for AU-3 and AU-4 grids. Finally， in all the final runs， additional refinements were implemented in the nearwall cells in order to obtain values of y+＜5. Table 4 shows the total number of cells and the y+of the final grids used to simulate the three studied cases.

1.3Boundary conditions

The implemented boundary conditions for all flow cases were the same. At the inlet of the domain it was prescribed a uniform velocity profile defined by（u =1，v =0，w=0）， in which random velocity fluctuations in the order of 10% were imposed， taking as characteristic length one cylinder diameter. At the outlet of the domain， the “outflow” boundary condition defined by ?φ/?t+Uconv?φ/?x=0， where “Uconv” is the velocity component normal to the domain exit， and“φ” the transported physical quantity in its convective form was implemented.

The upper and lower walls were treated as zero stress walls， in order to avoid the resolution of the boundary layer in such regions， and in the lateral walls the translational periodicity was imposed. In all cylinders was implemented the non-slip boundary conditions. Considering that the near-wall cells of the cylinders were located in the inner regions of the viscous sub-layer （y+＜5）， for the final runs of the AR-A， AR-B and AR-C cases （shown in the Table 4）， the enhanced wall treatment was applied.

Such treatment employs the Wolfstein equation［15］in the viscous sub-layer and a combination of the model of Chen and Patel［16］with a hyperbolic equation in order to estimate the velocities within the buffer layer. Finally， for the case AR-A， the residuals were below 10-8for the continuity， 10-9for the momentum and 10-7for the equations of the production and dissipation of turbulent energy “k ” and “ε”. The residuals for the cases AR-B and AR-C were of 10-7for the continuity， 10-8for the momentum and 10-6for the “k ” and “ε” equations.

2. Results and discussion

2.1Drag and lift coefficients

Due the lack of experimental and numerical data about the flow parameters related to the flow over a cylinder array covered with U and V grooves， the validations of the results for the cases AR-A， AR-B and AR-C， was performed comparing the predicted drag and lift coefficients for the case AR-A versus the experimental studies shown in Table 5， all focused on the study of the flow over a smooth circular cylinder array at an incidence angle α=45o.

It can be seen that there is a significant dispersion among the values of the drag and lift coefficients obtained for similar Reynolds numbers and spacing.In order to quantify such discrepancies， the Table 6 presents a comparison of the distinct reported drag coefficients and the numerical results for the three studiedcases. From the two first rows of Table 6， it is evident that there are differences of about 19% between Cylinders 1， 2 and 4， being noticeable the 27% found for the Cylinder 3， which is the cylinder that receives the mixed shear layers of the frontal and lateral cylinders. Although these differences of the coefficients could be attributed firstly to the differences in the Reynolds numbers and spacings of the few experiments reported in the literature， other factors are the experimental methodology， indirect measurement techniques as those implemented to estimate the coefficients， as the integration of the pressure distribution， and the nature of the various wind tunnels. In this regard， just to mention one example， Sayers［1］， in order to eliminate the effects of blockage， conducted his experiment in an open jet wind tunnel， while Lam et al.［3］obtained their results using a low speed closed wind tunnel， with a blockage ratio of around 11%， which is near the limit required for a two-dimensional flow experiment. The latter is a clear suggestion that the flow over a square array of cylinders is quite sensitive to external disturbances， being that its intrinsic nature due to the complex interactions of the several oscillating shear layers is noticeably complex.

Table 6 Percentage differences between thecoefficients of similar experiments for the case of the flow over an array of 4 cylinders in diamond shape configuration

Table 6 Percentage differences between thecoefficients of similar experiments for the case of the flow over an array of 4 cylinders in diamond shape configuration

（% Diff.）1DC（% Diff.）2DC（% Diff.）3DC（% Diff.）4DC Sayers［17］vs Lam et al.［3］5.40 13.17 24.8 13.17 Lam and Fang［18］vs Wang et al.［5］19.35 7.75 26.97 2.23 CASE AR-A vs Lam et al.［3］7.21 14.8 24.77 16.2 CASE AR-A vs Sayers［17］1.90 1.87 65.9 3.48

Table 7 Effects of the grooves in the drag coefficient of the different cylinders for the 3 studied cases

Regarding the differences between the CFD results and the experiment of Lam et al.［3］quantified for thein the third and fourth row of Table 6， it is important to be noted that the slight change of spacing from 3.41 to 3.5 is not expected to cause important modifications in the overallandcoefficients（see Figs.19 and 20 from Lam et al.［3］）， so it is evident that the physics of the steady RANS model deliver under-predicted values， especially for the rear and lateral cylinders. This is due to the inability of the model to capture the temporal evolution of the shear layers composed of eddies with different frequencies formed by the interactions of the frontal cylinder with the lateral ones. However， even if the obtainedandvalues were under-predicted， they were within the experimental scatter of the few experiments conducted under similar flow conditions， and they were considered valid at least from the qualitative point of view.

Further， the simulations were capable of replicating the repulsive forces on Cylinders 2 and 4， which is evident inCL≠0， as well as the trends found experimentally for the drag coefficients values， that is

In relation to the overall effects of the V and U grooves regarding the increase or decrease of the drag forces acting on each cylinder， Table 7 shows the comparison obtained from the AR-A， AR-B and ARC cases. It can be seen that the presence of the grooves caused opposite effects. While the U grooves caused a global decrease in the drag forces acting on each cylinder， the V grooves caused an increase in the same parameter.

Fig.6 Non-dimensional velocity magnitude contours obtained in the central x-yplane， for the case AR-A

It is noteworthy that the turbulence model k-ε Realizable was originally developed to provide infor-mation about a fully developed turbulent flow， and it is expected some deficiencies in the transitional and near wall zones. Therefore， the results here presented consist of our best attempt to improve the understanding of the phenomenon， by implementing a viable numerical tool in terms of computational time costs.

Fig.7 Contour map of the non-dimensional velocity obtained for the case AR-B

Fig.8 Non-dimensional velocity magnitude contours for the case AR-C

2.2Velocity magnitude in the x-ycentral plane

Figures 6， 7 and 8 show the maps of the non-dimensional velocity magnitudefor the cases AR-A， AR-B and AR-C respectively， obtained from thex-y plane located in z/D≈1.07（mid-span of the cylinders）.

In general， the three contours， show high velocity regions in the upper and lower zones of each cylinder，where the flow reaches values up to 1.5 times the free stream velocity magnitude. Also， in all the rear regions it can be seen a low velocity region， which form well-defined recirculation bubbles. This is especially important with respect to the interstitial space between the Cylinders 1 and 3， because it suggests that the numerical code was able to replicate the flow pattern called “vortex impingement”.

In relation to the Cylinders 2 and 4， although the flow structures were quite similar to those present in a single cylinder flow， if it is observed carefully， it is visible a small deflection of the wakes （upwards for Cylinder 2， and downwards for Cylinder 4）. This deflection is commonly attributed to the presence of repulsive forces caused by the wake of the Cylinder 1，quantified in the≠0values shown in Table 5.

Fig.9（u，v）vector components for the case AR-B， superimposed on the velocity magnitude contour map

When each figure is compared， it is possible to notice an important increase in the recirculation length induced by the V and U grooves for the cases AR-B and AR-C respectively. Also， an increase of the high velocity regions between the Cylinders 2 and 3， and 4 and 3， for the cases AR-B and AR-C is visible， suggesting the presence of jet flows. According to Lam et al.［3］and Wang et al.［5］， such jet flow structures contain different oscillation frequencies， product of the mixing of the shear layers of the Cylinders 1， 2 and 4，which are commonly associated to vibrations especially on the Cylinder 3. A detail of such jet structures is illustrated in the vector map of Fig.9， shown for simplicity just for the AR-B case （V-grooved cylinders array）.

Fig.10 Non-dimensionalvelocity magnitude obtained in thex-z plane located at y/D=0， for the case AR-A

2.3Velocity magnitude in the x-zcentral plane

Regarding the changes in the velocity gradients induced by the grooves presence， Figs.10， 11 and 12 show the non-dimensional velocity magnitudefor the three studied cases， obtained from the horizontal x-z plane， located at y/D=0. As a general behavior， the figures show high velocity regions near thefront part of the Cylinder 1， which are related with the free stream velocity， while in the rear parts of the Cylinders 1 and 3， there are slow velocity zones corresponding to the recirculation region.

Fig.11 Non-dimensional velocity magnitude contours for the case AR-B

Fig.12 Non-dimensional velocity magnitude for the case AR-C

When comparing the three figures with each other， it can be appreciated that the grooves caused an important elongation in the recirculation length of the Cylinder 1， as this increase from Lr≈1.1Dfor the case AR-A， toLr≈2.25Dand Lr≈1.9Dfor the cases AR-B and AR-C respectively.

Table 8 Recirculation lengths predicted for the 4 cylinders of the 3 studied arrays of cylinders

In order to evaluate the grooves influence on this parameter， Table 8 shows the values of the recirculation lengths obtained for the three studied cases. Comparing the values of Lrpredicted for the various cylinders， it can be seen that such increases in the recirculation lengths for the cases AR-B and AR-C， were not present just for the Cylinder 1， since all the other cylinders suffered elongations in their re-circulations zones， being interesting the fact that the predictedLrof the Cylinder 3 in the V-grooved array， presented a value of Lr≈6.05D， which is more than twice the value predicted for the case of the smooth cylinders array.

Fig.13 Skin friction coefficient scaled with the Reynolds number for the 4 cylinders of the case AR-A

2.4The skin friction coefficient along the cylinders periphery

The mean skin friction coefficient scaled with the Reynolds number predicted in the periphery the case AR-A cylinders is shown in Fig.13. Although the predicted values for Cylinders 1， 2 and 4 were almost identical， there is a slight asymmetry between the curves of the Cylinders 2 and 4， which is attributed to the repulsion forces induced in the interstitial region between the cylinder 1 and 3， and especially noticeable in the regions defined by （30o＜θ＜70o）and （290o＜θ＜330o）. Also， it can be seen that due the shielding effect induced by the Cylinder 1， the shear stresses formed in the Cylinder 3 periphery presented important decreases. Regarding the separation points obtained from the angular positions whereCf≈0， from the curves， it can be seen that for the four cylinders， the flow separates atθ≈103o.

Fig.14 Scaled skin friction coefficient of the Cylinder 2 of the case AR-A， versus the peaks and valleys distribution of the same cylinder， for cases AR-B and AR-C

In relation to the influence of the V and U grooves in the shear stresses acting on the cylinders， Fig.14 shows a comparison of the mean scaled skin friction coefficient of the valleys and peaks of the Cylinder 2 for the cases AR-B and AR-C， versus case AR-A. Forreasons of simplicity in the analysis， the data of the Cylinders 1， 3 and 4 were omitted because they were found to be quite similar to those obtained for the Cylinder 2.

It is evident that the shear stresses were higher at the grooves peaks with respect to the valleys. Also， at the peaks the shear stresses were higher on comparison to the stresses predicted to the smooth cylinder array， while in the opposite way， in the valleys， such stresses were lower.

When comparing the stresses at the peaks and valleys of the V and U grooves， in the frontal zones defined in the ranges （0o＜θ＜90o）and （270o＜θ＜360o）， it can be seen that U grooves induced the higher shear stresses. Ostensibly， the sharp edges and concave valleys of their semicircular transversal sections， are capable to enhance the fluid friction in their walls.

Furthermore， it is visible that the separation points of the fluid obtained in the valleys of the grooves，were different from those presented in the tips. In the valleys of the V-grooves， the flow was separated at θ≈70o， and at the peaks， separation occurred at θ≈100o， while for the U-grooves， the separation points were delayed， as the flow was separated atθ≈92oin the valleys， and atθ≈118oin the peaks. Apparently， the manner in which the flow is separated is a unique and distinctive feature， inherent to the transversal section of the grooves.

Fig.15 Non-dimensional mean vorticity map obtained from the z-y plane for the Cylinder 2 of case AR-B

Based on the above， it can be mentioned that the main effect of grooves in the shear stresses consisted primarily in a redistribution of the flow， so that in the valleys， the stresses were smaller than those present in the peaks， being conjectured that the intensity of such effect depends more on the shape and the sizes of the transversal sections. That is， the U-grooves， induced the higher shear stresses for both， valleys and peaks compared with the V-grooves， and this higher shear stresses are related to flow reattachments that also causes a delay in the separation point of each cylinder，all reflected in the drag reduction effect.

Fig.16 Non-dimensional mean vorticity map obtained from the z-ylane for the Cylinder 2 of case AR-C

As an effort to improve the understanding of what is conjectured as the major effect of the grooves in the shear stresses， the Figs.15 and 16 show the mean non-dimensional vorticity maps ωxh/U∞， obtained from thez-yplane located at the center of the Cylinder 2， atθ=90o， for the cases AR-B and AR-C respectively. This plane was chosen since the flow inidence is totally normal to the transversal section of the grooves as occurs in the flow over a flat plate covered by riblets. Moreover， in these regions， the grooved cylinders presented high shear stresses， as shown in Fig.14.

In both figures it is evident the high concentration of negative and positive vorticity near the grooves peaks. When the rotation direction of the vorticity is evaluated， it can be seen that for the positive regions denoted by red color， the flow packages revolve counterclockwise， while in the negative regions denoted by blue color， the rotation direction is inverted. This suggests the presence of a pair of eddies positioned near the grooves peaks， which when rotating， produce a blowing effect on the valleys. The positioning of this pair of eddies as well as the blowing effect， seems to explain the decreased shear stresses on the valleys and the increasing on the peaks.

This was also reported in the DNS of Choi et al.［19］， focused on the study of the turbulent flow over a flat plate covered with riblets. Even if the phenomenon has been described only in the normal plane， it could be generalized over the entire frontal area of the cylinders if it is considered that in such regions， the flow is not separated and even more， the accelerated flow is aligned as a consequence of the grooves lateral walls.

3. Conclusions

This paper presents the result of a comparative numerical study of the effect of the V and U grooves on the flow structures of an array of four cylinders in diamond shape configuration， with a spacing of Lc/ D =3.5and Re =41000. The “k-εRealizable” turbulence model was implemented and the grid sizes were of 5.98 million elements for the array of smooth cylinders， and 3.442×107elements for the arrays of cylinders covered by V and U grooves respectively.

In general， for the smooth cylinders array， the turbulence model was able to properly replicate some of the main characteristics reported experimentally as：the vortex impinging flow pattern between the Cylinders 1 and 3， the repulsive forces between the Cylinders 2 and 4 induced by the wake of the Cylinder 1，and a drag reduction in the Cylinder 3， caused by the shielding effect of the Cylinder 1.

For the arrays of grooved cylinders， the results suggest important changes in parameters such as velocities， recirculation lengths and separation points，being noticeable the fact that the U grooves increase the mixing capabilities of the flow， without causing increases in the drag forces. Furthermore， it is important to notice that due to the mixing between the shear layers of the Cylinders 1， 2 and 4， a high velocity interstitial jet flow is formed， which could have different oscillation frequencies， and cause vortex induced vibrations in the rear cylinder.

Finally， for the arrays of grooved cylinders， it was found high shear stresses in the peaks and low shear stresses on the valleys， being the magnitudes of such stresses， different on each cylinder. This is conjectured to be caused by a blowing effect， consequence of the formation of a pair of eddies positioned near the grooves peaks， that as rotate in opposite directions，cause an decrease of the shear stresses in the valleys，and an increase in the peaks.

Acknowledgements

The authors wish to thank CONACYT （Consejo Nacional de Cienciay Tecnología） and the laboratorio de Hidráulicay Térmica aplicada （LABINTHAP） del Instituto Politécnico Nacional-ESIME ZACATENCO for the support given to the accomplishment of this research.

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（September 21， 2014， Revised March 2， 2015）

* Biography： ALONZO-GARCíA A. （1980-）， Male， Ph. D.，Professor