Effect of Blockage Ratio on Steady Incompressible 2 …...the flow inside heat exchangers can be basically modelled as the flow past a square cylinder confined in a channel. On this

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International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 13, December 2018, pp. 477–487, Article ID: IJMET_09_13_049 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

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EFFECT OF BLOCKAGE RATIO ON STEADY INCOMPRESSIBLE 2-D FLOW AROUND A

SQUARE CYLINDER CONFINED IN A CHANNEL

Ercan Erturk Bahcesehir University, Mechatronics Engineering Department

Besiktas, Istanbul, Turkey

Orhan Gokcol Bahcesehir University, Computer Education and Instructional Technologies Department

Besiktas, Istanbul, Turkey

ABSTRACT The effect of the blockage ratio on the steady incompressible viscous flow past a

square cylinder confined in a channel is investigated numerically. In terms of the channel height with respect to the length of the square cylinder 1/4, 1/6, 1/8 and 1/10 blockage ratios are considered. For each of the considered blockage ratio the flow past a square cylinder confined in a channel is simulated up to very high Reynolds numbers. The numerical solutions of different channel blockage ratios are compared with each other and detailed results are presented.

Key words: Flow around confined square cylinder, effect of blockage ratio, high Reynolds numbers

Cite this Article: Ercan Erturk and Orhan Gokcol, 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, 9(13), 2018, pp. 477–487. http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=9&IType=13

1. INTRODUCTION In many fluids engineering applications flow over bluff bodies are encountered. As an example, the flow inside heat exchangers can be basically modelled as the flow past a square cylinder confined in a channel. On this subject, the flow around a square cylinder confined in a channel, it is possible to find many studies in the literature [1-5]. Among these, Tezel, Yapici and Uludag [1], Kumar, Dass and Dewan [2] and Breuer, Bernsdorf, Zeiser and Durst [3] presented numerical simulation of the steady incompressible 2-D flow over a square cylinder confined in a channel. However in these studies [1-3] the considered Reynolds numbers are very low. Also

Ercan Erturk and Orhan Gokcol


in these studies only one blockage ratio is considered. The channel height is an important parameter in this fluid flow problem and the solution of the flow over a confined square cylinder is affected by the channel height, i.e. by the blockage ratio. In the literature Dhiman, Chhabra and Eswaran [4] investigated the effect of the blockage ratio as well as the effect of the power-law on the steady flow across a confined square cylinder. Also, Dhiman, Sharma and Kumar [5] studied the effects of blockage ratio on the combined free and forced convection from a long heated square obstacle confined in a horizontal channel.

In this study we investigate the effect of the blockage ratio on the 2-D steady incompressible flow around a square cylinder. Using a very efficient finite difference numerical method presented by Erturk, Corke and Gökçöl [6], Erturk, Haddad and Corke [7] and Erturk [8] we solve the governing Navier-Stokes equations for high Reynolds numbers and also for channels that have different blockage ratios. We also tabulate detailed numerical results for future references.

2. PROBLEM FORMULATION AND NUMERICAL METHOD The steady incompressible 2-D flow around a square cylinder confined in a channel is governed by the Navier-Stokes equations and we consider the streamfunction ( ) and vorticity ( ) formulation. These equations are given as the following

(1)

(2)

In these equations, the and denote the Cartesian coordinates and is the Reynolds number. 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. Since in normalization, the length of the square cylinder ( ) and the maximum velocity ( ) is used, the length of the square cylinder and also the maximum velocity is equal to 1 after nondimensionalization.

The blockage ratio of the channel is defined as and in this study we consider channels with blockage ratios of 1/4, 1/6, 1/8 and 1/10. The schematic view of the flow over a square cylinder confined in a channel is given in Figure 1. We note that, for the channel blockage ratios of 1/4, 1/6 and 1/8, as shown in Figure 1-a, we have used uniform grids in

-direction. For 1/4, 1/6 and 1/8 blockage ratios, the height of the upper half of the channel above the symmetry line is equal to , and respectively and also in these we used 200, 300 and 400 uniform grid points respectively in the upper half of the channel. For the channel blockage ratio of 1/10, as shown in Figure 1-b, in the upper half of the channel above the symmetry line we used uniform grid points in the vicinity of the square cylinder up to and above we used stretched grid points with

. We used total of 400 grid points in -direction for the channel with blockage ratios of 1/10.

In all of the considered channel blockage ratio cases, as shown in Figure 1-a and 1-b, we used uniform grid points in -direction in the vicinity of the square cylinder, where

. In the upstream and downstream of the square cylinder, we have used stretched grids in order to have the inflow and outflow boundaries located far away from the square cylinder. We consider that is equal to and also is equal to

. In the computational domain, in -direction from the inflow boundary to the outflow boundary, we used total of 1300 grid points.

Effect of Blockage Ratio on Steady Incompressible 2-D Flow Around a Square Cylinder Confined in a Channel


a) schematic for 1/4, 1/6 and 1/8

channel blockage ratios b) schematic for 1/10 channel blockage ratio

Figure 1 Schematic view of the flow over a square cylinder confined in a channel

Figure 2 shows the computational domain and the mesh used in the present study for the different considered channel blockage ratios. In Figure 2, in -direction only 11 grid points and also in -direction only 261 grid points are plotted for a clear view. As seen in Figure 2 the center of the square cylinder is located at and . The width of the channel around the square cylinder is increased as the blockage ratio decreases. In our numerical simulation, following Erturk [8] the location of the inflow boundary and also the location of the outflow boundary are chosen away from the square cylinder in order for the boundary conditions used in these boundaries not to affect the numerical solution inside the computational domain. For this purpose, the inflow boundary and outflow boundary are located 50 and 200 times the size of the square cylinder upstream and downstream of the cylinder respectively for all considered channel blockage ratios as shown in Figure 2. Therefore the total length of the channel is equal to 250 times the size of the square cylinder. Such a long channel ensures that the location of the inflow and outflow boundaries do not have any influence on the interior numerical solution for any chosen channel blockage ratios.

a) 1/4 blockage ratio channel

b) 1/6 blockage ratio channel

c) 1/8 blockage ratio channel

d) 1/10 blockage ratio channel

Figure 2 Computational mesh used in the numerical solution (in y-direction 11 grid points, in x-direction 261 grid points are shown)

2.1. Boundary Conditions The boundary conditions used in our numerical simulation of the flow over a confined square cylinder is schematically shown in Figure 3. The boundary conditions are described briefly below.

uniform grids

L2

symmetry

line

symmetry

line

infl

ow b

ound

ary

outf

low

bou

ndar

y

wall boundary

stretched grids

L1

stretched grids

L3

unifo

rmgr

ids

L4y

x

parabolicinflow

profile

width ofsquare cylinder

Lsqu are

chan

nel h

eigh

tL

chan

nel

bloc

kage

rat

io

B=

L sq

uar

e

Lch

an

ne

l

Um a x unif

orm

grid

s

L4

uniform grids

L2

symmetry

line

symmetry

line

infl

ow b

ound

ary

outf

low

bou

ndar

y

wall boundary

stretched grids

L1

stretched grids

L3

stre

tche

d gr

ids

L5

y

x

parabolicinflow

profile

width ofsquare cylinder

Lsqu are

chan

nel h

eigh

tL

chan

nel

bloc

kage

rat

io

B=

L sq

uar

e

Lch

an

ne

l

Um ax



Figure 3 Schematic view of the boundary conditions used for the flow around a square cylinder confined in a channel

2.1.1. Boundary conditions at the inflow boundary We assume that at the inflow boundary the flow is fully developed channel flow between

parallel plates which is also known as the Plane Poiseuille Flow. Therefore at the inflow boundary the -velocity distribution has a parabolic profile. Using the parabolic -velocity profile, the streamfunction ( ) and vorticity ( ) distributions at the inflow boundary are given as the following

(3)

(4)

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. (5) (6)

2.1.3. Boundary conditions at the outflow boundary At the outflow boundary we used a non-reflecting boundary condition such that any wave

generated in the computational domain could pass through the exit boundary and leave without any reflection back into the computational domain [9], [10], [11]. At the outflow boundary we solve the streamfunction and vorticity equations (1) and (2) without the elliptic terms, i.e.

, such as

(7)

(8)

The outflow boundary conditions and used in the present study have been used in various similar studies [7], [8] successfully and this approach increases the accuracy of the numerical solution inside the computational domain.



2.1.3. Boundary conditions at solid walls At the solid channel walls, using the equation (3) the streamfunction value at the solid

channel wall is defined as the following

(9)

At the solid walls of the square cylinder, the streamfunction value is the same with the streamfunction value at the symmetry line, such that (10)

For the numerical values of the vorticity on the solid channel wall and also on the solid walls of the square cylinder, we used Thom’s formula. The vorticity value is calculated as

(11)

where 0 refers to points on the wall, 1 refers to the grid points adjacent to the wall and refers to the grid spacing.

3. RESULTS AND DISCUSSIONS Using the efficient numerical method described briefly in Erturk, Corke and Gökçöl [6], Erturk, Haddad and Corke [7] and Erturk[8], the governing flow equations are solved numerically. In our numerical solutions, we assume that the convergence is achieved when the maximum residual of both the streamfunction and the vorticity finite difference equations at every grid point inside the computational domain is less than .

For the stretched grids in the channel, we used Robert's stretching transformation of the original uniform grid (Anderson, Tannehill and Pletcher [16]). The formula of the transformation is

(12)

where represents the original uniformly spaced grid points, represents the stretched grid points and is the stretching parameter.

We numerically solve the steady incompressible viscous flow around a square cylinder confined in channel that has 1/4, 1/6, 1/8 and 1/10 blockage ratios. For each of the considered channel blockage ratio, the solutions are obtained from =5 up to =400.

Figures 4, 5, 6 and 7 shows the streamfunction contours of the flow past a square cylinder confined in a channel at various Reynolds numbers for channel blockage ratios of 1/4, 1/6, 1/8 and 1/10 respectively. Figures 4, 5, 6 and 7 exhibit the formation of the recirculation eddies as the Reynolds number increases.

Re=50 . Re=100 .

Re=150 . Re=200 .

Re=250 . Re=300 .

Re=350 . Re=400 .

Figure 4 Streamfunction ( ) contours for 1/4 channel blockage ratio



Re=50 . Re=100 -

Re=150 - Re=200 -

Re=250 - Re=300 -

Re=350 - Re=400 -


Re=50 . Re=100 .

Re=150 . Re=200 .

Re=250 . Re=300 .

Re=350 . Re=400 .


Re=50 . Re=100 .

Re=150 . Re=200 .

Re=250 . Re=300 .

Re=350 . Re=400 .


As seen in Figures 4, 5, 6 and 7, as the blockage ratio of the channel changes the sizes of the recirculating eddies in the flow change. Figure 8 shows the recirculating eddies that appear in the flow as the Reynolds number increases schematically.



Figure 6 Schematic view of the primary and secondary eddies in the flow field

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, 2, 3 and 4 tabulate the results for 1/4, 1/6, 1/8 and 1/10 channel blockage ratios respectively.

Table 1 Values of the primary and secondary eddies for 1/4 channel blockage ratio

Re Primary Eddy Secondary Eddy

5 0.560 0.140 -0.00004085 0.08635149 - - - - 0.603 - - 10 0.660 0.190 -0.00115175 0.30739126 - - - - 0.820 - - 15 0.740 0.210 -0.00332168 0.47726378 - - - - 1.002 - - 20 0.810 0.220 -0.00585170 0.58853116 - - - - 1.172 - - 25 0.860 0.240 -0.00842107 0.69581250 - - - - 1.333 - - 30 0.920 0.240 -0.01090036 0.74185821 - - - - 1.488 - - 40 1.020 0.260 -0.01544554 0.86585526 - - - - 1.784 - - 50 1.110 0.270 -0.01942799 0.93365085 - - - - 2.063 - - 75 1.350 0.290 -0.02764305 1.06517079 - - - - 2.709 - - 100 1.614 0.300 -0.03477665 1.15015062 - - - - 3.314 - - 125 1.879 0.320 -0.04217278 1.25157680 - - - - 3.904 - - 150 2.202 0.330 -0.05071742 1.33750044 - - - - 4.489 - - 175 2.529 0.350 -0.06088539 1.44385189 - - - - 5.050 - - 200 2.884 0.360 -0.07274427 1.53822527 - - - - 5.535 - - 225 3.225 0.370 -0.08640804 1.65463763 7.467 1.920 1.33352681 -0.14470292 5.887 5.781 9.627 250 3.421 0.380 -0.10265000 1.75644437 7.965 1.820 1.33523357 -0.33434413 6.064 5.348 11.741 275 3.354 0.390 -0.12121349 1.80277611 8.332 1.740 1.33894065 -0.47371501 6.055 5.011 13.940 300 3.060 0.410 -0.13952302 1.79069988 8.495 1.670 1.34391669 -0.60716129 5.909 4.604 16.224 325 2.701 0.430 -0.15565524 1.76866315 8.716 1.630 1.34901032 -0.66520722 5.740 4.285 18.512 350 2.383 0.450 -0.17011235 1.78338700 8.886 1.590 1.35360880 -0.74637464 5.605 4.041 20.715 375 2.188 0.460 -0.18337908 1.83878056 9.176 1.570 1.35765816 -0.77753096 5.502 3.835 22.878 390 2.117 0.460 -0.19065043 1.87489086 9.355 1.560 1.35984264 -0.79461270 5.451 3.711 24.187 400 2.075 0.470 -0.19518845 1.89494324 9.475 1.550 1.36121680 -0.81970412 5.420 3.687 24.943

Examining the values listed in Table 1, 2, 3 and 4 and also looking at Figure 4, 5, 6 and 7 we see that for all of the considered channel blockage ratios, the size of the primary eddy increases as the Reynolds number increases. At a Reynolds number around ≈200, secondary eddies appear in the flow field at the top and bottom walls of the channel at all blockage ratios. As the Reynolds number increases further, the size of these secondary eddies increases. However after the appearance of the secondary eddies in the flow field, as the size of the secondary eddies increases the size of the primary eddy decreases with increasing the Reynolds number for any blockage ratio of the channel.

Also comparing different channel blockage ratio results given in Table 1, 2, 3 and 4, we see that for a given Reynolds number, as the blockage ratio of the channel decrease from 1/4 to 1/10, both the sizes of the primary and secondary eddies increase. In order to see the effect of the blockage ratio of the channel on the primary and secondary eddies, in Figure 7 we plot the length of the primary and secondary eddies ( , , ) of different channel blockage ratios as a function of the Reynolds .





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



5 0.610 0.180 -0.00016704 0.09642860 - - - - 0.709 - - 10 0.740 0.210 -0.00159576 0.22687470 - - - - 0.998 - - 15 0.840 0.240 -0.00395222 0.33845255 - - - - 1.276 - - 20 0.940 0.260 -0.00679182 0.42570472 - - - - 1.559 - - 25 1.030 0.280 -0.00985422 0.49941358 - - - - 1.845 - - 30 1.120 0.290 -0.01299990 0.54618260 - - - - 2.130 - - 40 1.300 0.320 -0.01922152 0.64288649 - - - - 2.691 - - 50 1.490 0.340 -0.02522430 0.70467697 - - - - 3.234 - - 75 2.021 0.390 -0.03962692 0.82153454 - - - - 4.527 - - 100 2.648 0.430 -0.05449186 0.88645809 - - - - 5.766 - - 125 3.376 0.470 -0.07088608 0.93743768 - - - - 6.963 - - 150 4.189 0.500 -0.08915920 0.97787800 - - - - 8.071 - - 175 5.028 0.530 -0.10977275 1.04112134 - - - - 9.003 - - 200 5.616 0.550 -0.13425905 1.08672885 - - - - 9.682 - - 225 5.801 0.580 -0.16279553 1.11091873 13.125 3.840 2.66703012 -0.09489472 10.054 9.877 17.170 250 5.543 0.620 -0.19221903 1.09894070 13.721 3.710 2.66908126 -0.16500128 10.133 9.087 20.715 275 5.061 0.650 -0.22012315 1.09249338 14.163 3.600 2.67272296 -0.22787592 10.020 8.522 24.038 300 4.620 0.680 -0.24643416 1.10836496 14.527 3.520 2.67710742 -0.27042631 9.826 7.990 27.516 325 4.272 0.710 -0.27086630 1.13766621 14.805 3.460 2.68156984 -0.30140906 9.613 7.539 30.518 350 4.028 0.720 -0.29276659 1.16964295 15.184 3.410 2.68576223 -0.33058531 9.410 7.160 34.020 375 3.873 0.740 -0.31191282 1.19723176 15.475 3.370 2.68954678 -0.35409690 9.231 6.798 37.213 390 3.798 0.740 -0.32215540 1.21210478 15.771 3.360 2.69161900 -0.35661022 9.137 6.711 38.795 400 3.773 0.750 -0.32853011 1.22158058 15.971 3.350 2.69291783 -0.36287675 9.079 6.539 40.433



5 0.630 0.180 -0.00021797 0.09763711 - - - - 0.741 - - 10 0.760 0.220 -0.00174026 0.22673181 - - - - 1.047 - - 15 0.870 0.240 -0.00420277 0.31897429 - - - - 1.352 - - 20 0.970 0.270 -0.00717854 0.41428289 - - - - 1.669 - - 25 1.080 0.280 -0.01040523 0.46841349 - - - - 1.993 - - 30 1.180 0.300 -0.01372310 0.52969702 - - - - 2.319 - - 40 1.380 0.330 -0.02032727 0.61699001 - - - - 2.966 - - 50 1.592 0.350 -0.02674947 0.66965784 - - - - 3.600 - - 75 2.217 0.410 -0.04252833 0.77772021 - - - - 5.135 - - 100 3.000 0.460 -0.05930085 0.82785907 - - - - 6.639 - - 125 3.976 0.510 -0.07820333 0.86467263 - - - - 8.114 - - 150 5.061 0.550 -0.09965026 0.89372640 - - - - 9.488 - - 175 6.150 0.580 -0.12473732 0.94001370 - - - - 10.644 - - 200 6.821 0.620 -0.15570731 0.97541267 - - - - 11.477 - - 225 6.910 0.660 -0.19114260 0.97518610 15.573 4.852 3.33353936 -0.06055778 11.920 12.119 19.952 250 6.519 0.710 -0.22630196 0.95699178 16.276 4.688 3.33526645 -0.12705387 12.016 11.013 24.184 275 5.992 0.750 -0.25946103 0.95726580 16.796 4.573 3.33863237 -0.16758745 11.901 10.265 28.194 300 5.543 0.790 -0.29085486 0.97546352 17.224 4.475 3.34285402 -0.20574268 11.687 9.627 31.838 325 5.194 0.820 -0.31994628 1.00150076 17.662 4.394 3.34726153 -0.23965284 11.435 9.087 35.691 350 4.963 0.840 -0.34619490 1.02664080 17.998 4.345 3.35149232 -0.25476616 11.181 8.631 39.489 375 4.773 0.860 -0.36943563 1.05043244 18.455 4.297 3.35534871 -0.27585384 10.947 8.251 43.639 390 4.680 0.860 -0.38199568 1.06348803 18.688 4.281 3.35747375 -0.28013762 10.820 8.093 45.462 400 4.650 0.870 -0.38983619 1.07205659 18.805 4.264 3.35880390 -0.28742501 10.741 7.938 46.802



Figure 7 Effect of the channel blockage ratio on the length of the primary and secondary eddies

When we compare the -locations of the primary eddy centers of different channel blockage ratios in Figure 8, we see that when the blockage ratio decreases from 1/4 to 1/10 the center location increases at all Reynolds numbers while they behave similar in increasing Reynolds number. As clear in Figure 8, after the appearance of the secondary eddy on the top and bottom walls of the channel, as the Reynolds number increases the location of the center of the primary eddy moves towards left and gets closer to the square cylinder. This behavior is similar in all of the considered channel ratios.

Figure 8 Variation of the -locations of the primary eddy centers for different channel blockage ratios

In Figure 9, we compare the vorticity values at the center of the primary vortex at different channel blockage ratios. In Figure 9 we see that the strength of the vorticity increases as the blockage ratio increases from 1/10 to 1/4. In all of the considered blockage ratios, after the



appearance of the secondary eddies, the strength of the primary vorticity weakens slightly at first and then starts to increase afterwards as the Reynolds number increases.

Figure 9 Effect of the channel blockage ratio on the vorticity at the center of the primary eddy

4. CONCLUSIONS In this study the steady incompressible viscous flow around a square cylinder confined in a channel is numerically simulated. In the simulations various channel heights are considered in order to investigate the effect of the blockage ratio on the flow around a square cylinder. We find that the flow parameters of different blockage ratios behave similar in Reynolds number. Our results indicate that the length of the primary and secondary eddies increases as the blockage ratio decreases. Also the strength of the primary eddy increases as the channel blockage ratio increases.

REFERENCES [1] Dhiman AK, Chhabra RP, Eswaran V. “Steady Flow Across a Confined Square Cylinder:

Effects of Power-Law Index and Blockage Ratio”, Journal of Non-Newtonian Fluid Mechanics 2008; 148:141-150

[2] Dhiman AK, Sharma N, Kumar S. “Wall Effects on the Cross-Buoyancy Around a Square Cylinder in the Steady Regime”, Brazilian Journal of Chemical Engineering 2012; 29:253-264

[3] 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

[4] 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

[5] 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

ω

ω

ω

ω

ω



[6] 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

[7] Erturk E, Haddad OM, Corke TC. “Laminar Incompressible Flow Past Parabolic Bodies at Angles of Attack”. AIAA Journal 2004; 42:2254-2265

[8] Erturk E. “Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions”. Computers & Fluids 2008; 37:633-655

[9] Engquist B, Majda A. “Absorbing boundary conditions for the numerical simulation of waves”. Mathematics of Computation 1977;31:629-651.

[10] Jin G, Braza M. “A non-reflecting outlet boundary condition for incompressible unsteady Navier–Stokes calculations”. Journal of Computational Physics 1993;107:239-253

[11] Liu C, Lin Z. “High order finite difference and multigrid methods for spatially-evolving instability”. Journal of Computational Physics 1993;106:92–100.

[12] Anderson DA, Tannehill JC, Pletcher RH. “Computational Fluid Mechanics and Heat Transfer”. McGraw-Hill, 1984

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Effect of Blockage Ratio on Steady Incompressible 2 …...the flow inside heat exchangers can be basically modelled as the flow past a square cylinder confined in a channel. On this