FLOW PAST A SQUARE CYLINDER CONFINED IN A CHANNEL WITH INCIDENCE ANGLE

International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 13, December 2018, pp. 1642 1652, Article ID: IJMET_09_13_166 Available online at http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=9&itype=13 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 IAEME Publication Scopus Indexed FLOW PAST A SQUARE CYLINDER CONFINED IN A CHANNEL WITH INCIDENCE ANGLE Ercan Erturk Bahcesehir University, Mechatronics Engineering Department Besiktas, Istanbul, Turkey Orhan Gokcol Bahcesehir University, Computer Education and Instructional Technologies Department Besiktas, Istanbul, Turkey ABSTRACT In the present study, the effect of the incidence angle on the steady incompressible viscous flow past a square cylinder confined in a channel is numerically investigated. A blockage ratio of 1/6 which is defined as the ratio of the frontal area to the channel height is considered. The square cylinder is considered to have both 0 and 45 incidence angles to the incoming flow. The steady incompressible solutions are obtained for high Reynolds numbers. The solutions of the square cylinder with 45 incidence angle is compared with 0 incidence angle square cylinder in order to demonstrate the effect of the incidence angle on the flow problem. Detailed results are presented. Key words: Flow past confined square cylinder, effect of incidence angle Cite this Article: Ercan Erturk and Orhan Gokcol, Flow Past a Square Cylinder Confined in a Channel with Incidence Angle, International Journal of Mechanical Engineering and Technology, 9(13), 2018, pp. 1642 1652. http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=9&itype=13 1. INTRODUCTION Flow around square cylinders is encountered in many engineering applications such as cooling of electronic components or wind flow over buildings. The flow inside heat exchangers can also be counted as an example. The subject of the flow over a square cylinder is studied extensively in the literature. For example, Tezel, Yapici and Uludag [1], Kumar, Dass and Dewan [2] and Breuer, Bernsdorf, Zeiser and Durst [3] numerically studied the steady incompressible 2-D flow over a square cylinder confined in a channel. Recently Erturk and Gokcol [4] performed numerical simulations on the flow over a square cylinder confined in a channel and on the external flow past a square cylinder in [5]. Erturk and Gokcol [6] also considered different blockage ratio channels and investigated the effect of the blockage ratio on the internal flow around a square cylinder confined in a channel. In these studies [1-6], the angle of incidence of the square cylinder is 0 with respect to the incoming flow. As for example studies on the http://www.iaeme.com/ijmet/index.asp 1642 editor@iaeme.com

Flow Past a Square Cylinder Confined in a Channel with Incidence Angle subject of the flow around a square cylinder with an incidence angle, Ranjan, Dalal and Biswas [7] numerically investigate the flow and heat transfer around a square cylinder at various incidence angles. Dayem and Bayomi [8] performed experiments on flow around square cylinders with different incidence angles. They [8] also performed numerical simulations which are in agreement with the experimental results. Dutta, Panigrahi and Muralidhar [9,10] investigated the flow past a square cylinder placed at an angle to the incoming flow experimentally using particle image velocimetry, hot wire anemometry and flow visualization. Sheard [11,12] conducted a linear stability analysis on the flow past a square cylinder inclined at an angle to an oncoming flow. Yoon, Yang and Choi [13] studied the characteristics of flow past a square cylinder inclined with respect to the main flow in the laminar flow regime. Jamshidi, Farhadi and Sedighi [14] performed numerical simulation on turbulent flow over square cylinder with two different angles of incidence in the free stream. Sohankar, Norberg and Davidson [15] presented calculations of unsteady, incompressible 2D flow around a square cylinder at incidence and for low Reynolds numbers. In these studies, [7-15] the considered flows are unsteady. In this study we investigate the effect of the incidence angle on the 2-D steady incompressible flow around a square cylinder confined in a channel. We numerically solve the governing Navier-Stokes equations using the numerical method presented in [16,17,18] for high Reynolds numbers. We consider that the square cavity has two different incidence angles; 0 and 45. We compare the numerical results of 0 and 45 incidence angle results with each other to present the effect of the incidence angle of the square cylinder on the flow problem. We also tabulate detailed numerical results for future references. 2. PROBLEM FORMULATION AND NUMERICAL METHOD For the numerical simulation of the 2-D steady incompressible flow over a square cylinder confined in a channel, we consider the streamfunction () and vorticity () formulation of the Navier-Stokes equations. These equations are given as the following In these equations, the and denote the Cartesian coordinates and is the Reynolds number. We note that when the square cylinder is placed with an incidence angle in the channel, depending on the incidence angle the blockage ratio will change. If in normalization one side of the square cylinder is used as a characteristic length, then for different incidence angles the blockage ratio in the channel will be different for any chosen Reynolds number. Erturk and Gokcol [6] have showed that as the blockage ratio changes the flow characteristics are affected. Instead of using one side length of the square cylinder, Sheard [11,12] used the projected height of the cylinder facing the oncoming flow as the characteristic length and used this characteristic length in the definition of the Reynolds number. Following Sheard [11,12], we will use the projected frontal total height as the characteristic length in the definition of the Reynolds number. Therefore, as shown in Figure 1-a, when the incidence angle is 0 the Reynolds number is defined as where is the length of one side of the square cylinder and is the maximum velocity in the incoming parabolic velocity profile and also is the kinematic coefficient of viscosity. However, when the incidence angle is 45 the Reynolds number is defined as where is the diagonal height of the square cylinder as also shown in Figure 1-b. We note that with using the projected frontal total height as the characteristic length, for any Reynolds number the blockage ratio is exactly http://www.iaeme.com/ijmet/index.asp 1643 editor@iaeme.com

Ercan Erturk and Orhan Gokcol the same for both 0 and 45 incidence angles. Also, since in normalization the maximum velocity ( ) is used, the maximum velocity at the inflow boundary is equal to 1 after nondimensionalization. In the present study we consider that the blockage ratio of the channel is 1/6 such that for the 0 incidence angle case and for the 45 incidence angle case. As shown in Figure 1, we have used 300 uniform grids in the upper half of the channel in -direction and the height of the upper half of the channel above the symmetry line is equal to or depending on the incidence angle. In -direction near the vicinity of the square cylinder, we used uniform fine grids for accuracy purposes where or depending on the incidence angle. Also, in order to have the location of the inflow and outflow boundaries far away from square cylinder, we used stretched grids upstream and downstream of the cylinder, where or and or depending on the incidence angle. In -direction from the inflow boundary to the outflow boundary, we used total of 1300 grid points in the computational domain. wall boundary wall boundary stretched grids stretched grids stretched grids stretched grids L1 uniform grids L3 L1 uniform grids L3 y L2 y L2 x Um ax symmetry line height of square cylinder symmetry line x Um ax symmetry line diagonal of square cylinder symmetry line Lsquare Dsquare parabolic inflow profile parabolic inflow profile a) square cylinder at 0 incidence angle b) square cylinder at 45 incidence angle Figure 1 Schematic view of the flow over a square cylinder confined in a channel In order to generate stretched grids, we used Robert's stretching transformation of the original uniform grid (Anderson, Tannehill and Pletcher [19]). The transformation is given as where represents the original uniformly spaced grid points, represents the stretched grid points and is the stretching parameter. a) square cylinder at 0 incidence angle b) square cylinder at 45 incidence angle Figure 2 Computational mesh used in the numerical solution (in x-direction 261 grid points, in y-direction 11 grid points are shown) In Figure 2, the computational grid used in the present study is given both for 0 and 45 incidence angle cases. In Figure 2, only 261 grid points and 11 grid points are plotted in - and -directions respectively for a clear view. As seen in Figure 2, the computational grid is exactly http://www.iaeme.com/ijmet/index.asp 1644 editor@iaeme.com

Flow Past a Square Cylinder Confined in a Channel with Incidence Angle the same for both 0 and 45 incidence angles except the detail near the vicinity of the square cylinder as encapsulated with a red square around the square cylinder which is located at and. The area in the red square shown in Figure 2 is given in a zoomed view in Figure 3. Figure 3 shows the square cylinder for 0 and 45 incidence angles. As seen in Figure 3, the total height, i.e. the projected frontal height, of the cylinder is equal to 1 for both 0 and 45 incidence angles which results in identical blockage ratios. a) square cylinder at 0 incidence angle b) square cylinder at 45 incidence angle Figure 3 Enlarged view of the computational grid near the square cylinder (1 out of 5 grid points are shown in both x- and y-directions) 2.1. Boundary Conditions Figure 4 summarizes the boundary conditions used in our numerical simulation of the channel flow over a square cylinder at an incidence schematically. Figure 4 Schematic view of the boundary conditions 2.1.1. Boundary conditions at the inflow boundary At the inflow boundary, the flow is fully developed channel flow between parallel plates which is also known as the Plane Poiseuille Flow. Using the Plane Poiseuille Flow parabolic -velocity profile, the following distribution is used for the streamfunction () and vorticity () variables at the inflow boundary http://www.iaeme.com/ijmet/index.asp 1645 editor@iaeme.com

Ercan Erturk and Orhan Gokcol 2.1.2. Boundary conditions at the symmetry line The symmetry line itself is a streamline and on this symmetry line both the streamfunction value and also the vorticity value is zero. 2.1.3. Boundary conditions at the outflow boundary In order to increase the numerical accuracy, we used a non-reflecting boundary condition at the outflow boundary [20,21,22]. At the outflow boundary we solve the streamfunction and vorticity equations (1) and (2) without the elliptic terms, i.e., such as At sufficiently far away downstream of the square cylinder, the flow is away from the influence of the square cylinder and therefore the 2-D steady incompressible Navier Stokes equations should approach to a fully developed channel flow between parallel plates, such that the velocity profile is parabolic and also the second -derivatives of streamfunction and vorticity variables should be equal to zero, i.e. and. The outflow boundary conditions used in the present study have been also used in various similar studies [17,18] and [4,5,6] successfully. 2.1.3. Boundary conditions at solid walls At the top channel wall, the streamfunction value is defined as the following At the solid walls of the square cylinder, the streamfunction value is defined as the following At the solid walls in order to find the vorticity values, we used Thom s formula. On the solid walls applying the no-slip no-penetration conditions such that and, at the top channel wall the vorticity is calculated as where ( ) refer to points on the wall, ( ) refer to the first grid points adjacent to the wall. On the solid walls of the square cylinder, for 0 incidence angle the vorticity is calculated as on the left, top and right walls of the square cylinder respectively. Here again, ( ) refers to points on the wall, ( ), ( ) and ( ) refer to the first grid points adjacent to the wall. Also for 45 incidence angle, the vorticity is calculated as on the left and right walls of the 45 square cylinder respectively. http://www.iaeme.com/ijmet/index.asp 1646 editor@iaeme.com

Flow Past a Square Cylinder Confined in a Channel with Incidence Angle 3. RESULTS AND DISCUSSIONS We solve the governing equations numerically using the numerical method described briefly in Erturk, Corke and Gökçöl [16], Erturk, Haddad and Corke [17] and Erturk[18]. We continue the numerical iterations until the maximum residual of both the streamfunction and the vorticity equations at every grid point inside the computational domain is less than. We obtain numerical solutions of steady incompressible viscous flow past a square cylinder confined in channel that has 1/6 blockage ratio for 0 and 45 incidence angles of the square cylinder. For 0 incidence angle of the square cylinder, the solutions are obtained from =5 up to =400 and also for 45 incidence angle of the square cylinder the solutions are obtained from =10 up to =450. Figure 5 shows the streamfunction and vorticity contours of the flow past a square cylinder with 0 incidence angle. In this figure, above the symmetry line the streamfunction contours and below the symmetry line the vorticity contours are plotted. In the figure the symmetry line is shown with a red line. In this figure the cyan color denotes the solid walls. Figure 5 exhibit the formation of the recirculating eddies as the Reynolds number increases. As seen in Figure 5, the separation bubble exists even at =5 when the square cylinder has 0 incidence angle. We can see that after =200 a secondary separation bubble appears in the flow field at the top channel wall and the size of this secondary eddy grows as the Reynolds number increases. We can also see that at Reynolds numbers lower than =100, for the primary eddy while the separation occurs at the upper right corner of the square cylinder, above =100 the separation point shifts to the upper left corner of the square cylinder. We note that we could obtain steady solutions only up to =400 for 0 incidence angle square cylinder. Re=5_ Re=75_ Re=10_ Re=100_ Re=15_ Re=150_ Re=20_ Re=200_ Re=25_ Re=250_ Re=30_ Re=300_ Re=40_ Re=350_ Re=50_ Re=400_ Figure 5 Streamfunction contours for square cylinder with 0 incidence angle Similarly Figure 6 shows the streamfunction and vorticity contours of the flow past a square cylinder with 45 incidence angle. Comparing with 0 incidence angle square cylinder, as seen in Figure 6 the flow is attached at =10 and there is no separation bubble downstream of the http://www.iaeme.com/ijmet/index.asp 1647 editor@iaeme.com

Ercan Erturk and Orhan Gokcol 45 square cylinder. For the primary eddy, the separation occurs between =10 and =15 and as seen in Figure 6 there occurs a separation bubble behind the cylinder at =15. For the 45 square cylinder we can see that between =150 and =200 the secondary eddy appears in the flow field at the top channel wall. We also note that, for 45 incidence angle square cylinder, we were able to obtain steady solutions up to =450 beyond which we did not attempt to continue solving for higher Reynolds numbers. Re=10_ Re=100_ Re=15_ Re=150_ Re=20_ Re=200_ Re=25_ Re=250_ Re=30_ Re=300_ Re=40_ Re=350_ Re=50_ Re=400_ Re=75_ Re=450_ Figure 6 Streamfunction contours for square cylinder with 45 incidence angle In Figure 7 we schematically show the recirculating eddies that appear in the flow field as the Reynolds number increases. In this figure important flow parameters are designated for the primary and secondary eddies. For future references, we tabulate - and -locations of the center of the primary and secondary eddies (, ) and also the streamfunction (, ) and vorticity (, ) values at these centers at various Reynolds numbers. We also tabulate the length of the primary and secondary eddies (,, ) as a function of the Reynolds number. Table 1 tabulates these values for square cylinder with 0 incidence angle and Table 2 tabulates the same values for square cylinder with 45 incidence angle. a) square cylinder at 0 incidence angle b) square cylinder at 45 incidence angle Figure 7 Schematic view of the primary and secondary eddies in the flow field http://www.iaeme.com/ijmet/index.asp 1648 editor@iaeme.com

Flow Past a Square Cylinder Confined in a Channel with Incidence Angle Table 1 Values of the primary and secondary eddies for 0 incidence angle case Primary Eddy Secondary Eddy Re 5 0.590 0.160-0.00009868 0.09177976 - - - - 0.660 - - 10 0.710 0.200-0.00136736 0.24553407 - - - - 0.921 - - 15 0.800 0.230-0.00355560 0.37140204 - - - - 1.157 - - 20 0.890 0.240-0.00615786 0.45060108 - - - - 1.389 - - 25 0.960 0.260-0.00893723 0.53030113 - - - - 1.619 - - 30 1.040 0.270-0.01174939 0.58651646 - - - - 1.845 - - 40 1.190 0.290-0.01723424 0.67782166 - - - - 2.285 - - 50 1.330 0.310-0.02242183 0.75160587 - - - - 2.708 - - 75 1.724 0.350-0.03441485 0.88718873 - - - - 3.701 - - 100 2.159 0.380-0.04629349 0.97293964 - - - - 4.642 - - 125 2.666 0.410-0.05922998 1.05245917 - - - - 5.552 - - 150 3.225 0.430-0.07371842 1.11180369 - - - - 6.407 - - 175 3.798 0.450-0.08980736 1.18185473 - - - - 7.147 - - 200 4.300 0.470-0.10803957 1.25936671 9.722 2.990 2.00000007-0.01721115 7.707 9.445 10.006 225 4.590 0.480-0.12945195 1.31974581 10.432 2.860 2.00046853-0.12733757 8.041 7.736 14.027 250 4.501 0.510-0.15309396 1.33277636 10.980 2.740 2.00270514-0.24133999 8.132 7.159 16.743 275 4.162 0.540-0.17627003 1.31977276 11.264 2.650 2.00658608-0.32073733 8.039 6.711 19.584 300 3.724 0.560-0.19770621 1.32088037 11.555 2.580 2.01118356-0.38174481 7.869 6.209 22.318 325 3.376 0.580-0.21733992 1.34699993 11.854 2.530 2.01573950-0.42379402 7.697 5.896 25.097 350 3.121 0.600-0.23492200 1.38482202 12.160 2.490 2.01992747-0.46072405 7.549 5.560 27.855 375 2.981 0.610-0.25031704 1.42080713 12.474 2.460 2.02367297-0.48856684 7.425 5.383 30.517 390 2.923 0.610-0.25855291 1.43987985 12.714 2.450 2.02571062-0.49531857 7.361 5.210 31.839 400 2.884 0.620-0.26367089 1.45195425 12.795 2.440 2.02699484-0.50460092 7.322 5.143 33.211 Table 2 Values of the primary and secondary eddies for 45 incidence angle case Primary Eddy Secondary Eddy Re 15 0.550 0.240-0.00631074 0.62288471 - - - - 0.993 - - 20 0.640 0.290-0.01649315 0.89285299 - - - - 1.383 - - 25 0.730 0.310-0.02662635 1.01672701 - - - - 1.744 - - 30 0.820 0.340-0.03595472 1.14645964 - - - - 2.087 - - 40 1.010 0.370-0.05173033 1.24394353 - - - - 2.722 - - 50 1.220 0.400-0.06423904 1.30494636 - - - - 3.289 - - 75 1.818 0.440-0.08693921 1.32576946 - - - - 4.459 - - 100 2.431 0.470-0.10346396 1.36001554 - - - - 5.353 - - 125 3.020 0.480-0.11760173 1.39898777 - - - - 6.033 - - 150 3.511 0.480-0.13158886 1.44978472 - - - - 6.537 - - 175 3.798 0.490-0.14660278 1.49029202 8.772 2.990 2.00000036-0.01112513 6.887 8.252 9.385 200 3.924 0.500-0.16235957 1.51466993 9.295 2.880 2.00026306-0.11238166 7.099 7.067 12.119 225 3.899 0.510-0.17773683 1.51977143 9.660 2.790 2.00135682-0.20104818 7.198 6.667 14.027 250 3.773 0.520-0.19192564 1.51778523 9.974 2.730 2.00321302-0.24985160 7.218 6.414 15.818 275 3.581 0.540-0.20489905 1.51470677 10.233 2.670 2.00551777-0.31195044 7.189 6.089 17.387 300 3.421 0.550-0.21691381 1.52982759 10.432 2.630 2.00799414-0.34714681 7.134 5.857 19.101 325 3.267 0.560-0.22814303 1.55332031 10.634 2.590 2.01046173-0.39033203 7.063 5.707 20.715 350 3.141 0.560-0.23865768 1.58271898 10.840 2.570 2.01283134-0.40267533 6.986 5.525 22.457 375 3.060 0.570-0.24847005 1.60906675 10.980 2.540 2.01505113-0.43814963 6.904 5.313 24.038 400 2.981 0.570-0.25758166 1.63645280 11.192 2.520 2.01710268-0.46015566 6.822 5.143 25.722 425 2.903 0.580-0.26610155 1.66336323 11.409 2.510 2.01899092-0.46687082 6.741 5.011 27.182 450 2.847 0.580-0.27403143 1.68852640 11.555 2.490 2.02071554-0.49338526 6.663 4.882 28.717 Figure 8 Primary and secondary eddy length for different incidence angles of the cylinder http://www.iaeme.com/ijmet/index.asp 1649 editor@iaeme.com

Ercan Erturk and Orhan Gokcol In order to see the effect of the incidence angle of the square cylinder on the flow more clearly, we plot the length of the primary and secondary eddies (,, ) for 0 and 45 incidence angles of the cylinder in Figure 8. In Figure 8 we can see that in terms of length, as the Reynolds number increases the behavior of the primary and secondary eddy are very different when the incidence angle changes from 0 to 45. At low Reynolds numbers while we see that the length of the primary eddy for 45 incidence angle is higher than that for 0 incidence angle, at high Reynolds numbers the 0 incidence angle length is higher than that for 45 incidence angle. We see the same type of behavior in the -locations of the primary eddy center as shown in Figure 9. Figure 9 Primary eddy center -location for different incidence angles of the cylinder In Figure 10, we compare the vorticity values at the center of the primary vortex for 0 and 45 incidence angles of the cylinder. From Figure 10 we can see that when the incidence angle of the cylinder is 45, the vorticity at the primary eddy is much stronger compared to the cylinder with 0 incidence angle. ω ω ω Figure 10 Vorticity at the primary eddy center for different incidence angles of the cylinder http://www.iaeme.com/ijmet/index.asp 1650 editor@iaeme.com

Flow Past a Square Cylinder Confined in a Channel with Incidence Angle 4. CONCLUSIONS In this study the effect of the incidence angle of the cylinder on the steady incompressible viscous flow around a square cylinder confined in a channel is studied numerically. In the simulations 0 and 45 incidence angles of the square cylinder are considered. In both 0 and 45 incidence angles, the size of the square cylinders is considered to give the same blockage ratio in the channel. We present the effect of the incidence angle of the square cylinder on the internal flow in a channel. We find that the strength of the primary eddy for 45 incidence angle is higher than the strength of the primary eddy for 0 incidence angle. REFERENCES [1] Tezel GB, Yapici K, Uludag Y. Numerical and Experimental Investigation of Newtonian Flow Around a Confined Square Cylinder, Periodica Polytechnica Chemical Engineering 2019, 63;190-199 [2] Kumar DS, Dass AK, Dewan A. Numerical Simulation of Viscous Flow Over a Square Cylinder on Graded Cartesian Meshes Using Multigrid Method, Proceedings of the 37th International & 4th National Conference on Fluid Mechanics and Fluid Power FMFP2010, December 16-18, 2010, IIT Madras, Chennai, India [3] Breuer M, Bernsdorf J, Zeiser T, Durst F. Accurate Computations of the Laminar Flow Past a Square Cylinder Based on Two Different Methods: Lattice-Boltzmann and Finite- Volume, International Journal of Heat and Fluid Flow 2000, 21;186-196 [4] Erturk E, Gokcol O. High Reynolds Number Solutions of Steady Incompressible 2-D Flow around a Square Cylinder Confined in a Channel with 1/8 Blockage Ratio, International Journal of Mechanical Engineering and Technology 2018, 9;13:452-463 [5] Erturk E, Gokcol O. Steady Flow over a Square Cylinder at High Reynolds Numbers, International Journal of Mechanical Engineering and Technology 2018, 9;13:638-649 [6] Erturk E, Gokcol O. Effect of Blockage Ratio on Steady Incompressible 2-D Flow Around a Square Cylinder Confined in a Channel, International Journal of Mechanical Engineering and Technology 2018, 9;13:477-487 [7] Ranjan R, Dalal A, Biswas G. A Numerical Study of Fluid Flow and Heat Transfer around a Square Cylinder at Incidence using Unstructured Grids, Numerical Heat Transfer, Part A: Applications 2008, 54;890-913 [8] Dayem AMA, Bayomi NN. Experimental and Numerical Flow Visualization of a Single Square Cylinder, International Journal for Computational Methods in Engineering Science and Mechanics 2006, 7;113-127 [9] Dutta S, Panigrahi PK, Muralidhar K. Effect of orientation on the wake of a square cylinder at low Reynolds numbers, Indian Journal of Engineering and Materials Sciences 2004, 11;447-459 [10] Dutta S, Panigrahi PK, Muralidhar K. Experimental Investigation of Flow Past a Square Cylinder at an Angle of Incidence, Journal of Engineering Mechanics 2008, 134;9:788-803 [11] Sheard GJ. Wake stability features behind a square cylinder: Focus on small incidence angles, Journal of Fluids and Structures 2011, 27;734-742 [12] Sheard GJ. Flow past a Square Cylinder at Small Incidence Angles: Characteristics of Leading Three-Dimensional Instabilities, International Conference on Mechanical, Industrial, and Manufacturing Engineering (MIME 2011) 2011, ISBN: 978-0-9831693-1-4 [13] Yoon DH, Yang KS, Choi CB. Flow past a square cylinder with an angle of incidence, Physics of Fluids 2010, 22:043603 [14] Jamshidi N, Farhadi M, Sedighi K. Flow structure in the downstream of a square cylinder with different angles of incidence, International Journal of Multiphysics 2010, 4; 1:11-20 http://www.iaeme.com/ijmet/index.asp 1651 editor@iaeme.com

Ercan Erturk and Orhan Gokcol [15] Sohankar A, Norberg C, Davidson L. Low-Reynolds-Number Flow Around a Square Cylinder at Incidence: Study of Blockage, Onset of Vortex Shedding and Outlet Boundary Condition, International Journal for Numerical Methods in Fluids 1998, 26; 39-56 [16] Erturk E, Corke TC, Gökçöl C. Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. International Journal for Numerical Methods in Fluids 2005, 48;747-774 [17] Erturk E, Haddad OM, Corke TC. Laminar Incompressible Flow Past Parabolic Bodies at Angles of Attack. AIAA Journal 2004, 42;2254-2265 [18] Erturk E. Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions. Computers and Fluids 2008, 37;633-655 [19] Anderson DA, Tannehill JC, Pletcher RH. Computational Fluid Mechanics and Heat Transfer. McGraw-Hill, 1984 [20] Engquist B, Majda A. Absorbing boundary conditions for the numerical simulation of waves. Mathematics of Computation 1977, 31;629-651. [21] Jin G, Braza M. A non-reflecting outlet boundary condition for incompressible unsteady Navier Stokes calculations. Journal of Computational Physics 1993, 107;239-253 [22] Liu C, Lin Z. High order finite difference and multigrid methods for spatially-evolving instability. Journal of Computational Physics 1993, 106;92-100 http://www.iaeme.com/ijmet/index.asp 1652 editor@iaeme.com