Results for Re =400 - Thesis Lid Driven

After increasing the Reynolds number from 100 to 400, the comparison of the horizontal velocities on the vertical centerline and the vertical velocities on the horizontal centerline with different differencing schemes are presented in Figures 4.10 and 4.11 and comparison of the results with different methods are illustrated in the Figures 4.12 and 4.13.

The first order accurate upwind scheme leads to adequately acceptable convergent results as shown in Figure 4.10 and 4.11. However these results are not as close to Ghia’s [8] results as the results obtained when the Reynolds number is 100.

58 y

0 0.25 0.5 0.75 1

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Ghia 257x257

SIMPLER 161x161 Upwind Scheme SIMPLER 161x161 Hybrid Scheme SIMPLER 161x161 Power Law Scheme

Figure 4.10 Vertical centerline u-velocity profiles for Re=400 with different schemes

0 0.25 0.5 0.75 1

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

Ghia 257x257

SIMPLER 161x161 Upwind Scheme Error=10-6 SIMPLER 161x161 Hybrid Scheme

SIMPLER 161x161 Power Law Scheme

Figure 4.11 Horizontal centerline v-velocity profiles for Re=400 with different schemes

SIMPLE 161x161 Power Law Scheme SIMPLE 161x161 Hybrid Scheme SIMPLER 161x161 Hybrid Scheme SIMPLER 161x161 Power Law Scheme

Figure 4.12 Vertical centerline u-velocity profiles for Re=400 with different algorithms

SIMPLE 161x161 Hybrid Scheme SIMPLE 161x161 Power Law Scheme SIMPLER 161x161 Hybrid Scheme SIMPLER 161x161 Power Law Scheme

Figure 4.13 Horizontal centerline v-velocity profiles for Re=400 with different algorithms

While the results of Ghia [8] was obtained using a 257x257 grid, the same results are achieved by SIMPLER method with a 161x161 uniform grid using both power law and hybrid schemes.

If the SIMPLE and SIMPLER methods are compared, using Figures 4.12 and 4.13, it clear that the SIMPLER method is more close to the Ghia [8] results although results of the SIMPLE method is not significantly different. On the other hand, if the locations of the vortices, which are tabulated in Table 4.4, are considered, the location of the primary and secondary bottom right vortices are close to each other with both SIMPLE and SIMPLER methods using power law or hybrid schemes, whereas only SIMPLE method could indicate the secondary bottom left vortex.

Also the results for the location of centers are in good agreement with the published results, as observed from Table 4.4.

Table 4.4 Location of the centers of the vortices for the lid-driven square cavity at Re=400

Grid Primary vortex Secondary

vortex Secondary vortex Re=400 Size Bottom right Bottom left

Location (x,y) Location (x,y) Location (x,y)

Ghia, Ghia and Shin [8] 257x257 (0.5547,06055) (0.8906,0.1250) (0.0508,0.0469) Schreiber and Keller [7] 141x141 (0.5571,06071) (0.8857,0.1143) (0.0500,0.0429) Vanka [28] 64x64 (0.5563,0.6000) (0.8875,0.1188) (0.0500,0.0500) Gupta and Kalita [4] 81x81 (0.5500,06125) (0.8875,0.1250) (0.0500,0.0500) Hou et. All [30] (0.5608,06078) (0.8902,0.1255) (0.0549,0.0510) SIMPLE Power law 161x161 (0.5590,0.6100) (0.8863,0.1239) (0.0487,0.0451) SIMPLE Hybrid 161x161 (0.5582,0.6100) (0.8854,0.1240) (0.0489,0.0453) SIMPLER Power law 161x161 (0.5551,0.6058) (0.8867,0.1226) - SIMPLER Hybrid 161x161 (0.5543,0.6051) (0.8845,0.1221)

Grid Tertiary vortex Tertiary vortex

Re=400 Size Bottom right Bottom left

Location (x,y) Location (x,y)

Ghia, Ghia and Shin [8] 257x257 (0.9922,0.0078) (0.0039,0.0039)

Ghia [8] found out tertiary bottom left and right vortices at a Reynolds number 400. However these vortices are not obtained by the other researchers.

From Figures 4.12 and 4.13, it is clear that SIMPLE method is not in agreement to Ghia [8] at the extrema points and the difference is illustrated in Table 4.5. Also the extrema velocity values along the centerlines, which are calculated with the SIMPLE and SIMPLER methods and are given in the mentioned references, are shown in Table 4.5.

Table 4.5 Extrema of velocity profiles along centerlines for the lid-driven square cavity at Re=400

Grid Extrema of velocity profiles along the certerlines

Re=400 Size

Ghia, Ghia and Shin [8] 257x257 -0,3273 0,2813 0,3020 0,2266 -0,4499 0,8594

Soh [10] - -0,312 0,288 - - - -

SIMPLE Power law 161x161 -0,3009 0,2830 0,2780 0,2201 -0,4234 0,8616 SIMPLE Hybrid 161x161 -0,3040 0,2830 0,2803 0,2201 -0,4358 0,8616 SIMPLER Power law 161x161 -0,3217 0,2830 0,2987 0,2264 -0,4482 0,8616 SIMPLER Hybrid 161x161 -0,3264 0,2830 0,3025 0,2264 -0,4522 0,8616

In Figure 4.14, the streamlines obtained by SIMPLE method using power law differencing scheme on a 161x161 grid are presented. The primary vortex as well as the secondary vortices in the bottom corners of the cavity are observed. The secondary vortices dominate more space than these at a Reynolds number of 100 and these secondary vortices are shown in Figure 4.15. The streamlines of the other there calculations shown in the Figures 4.12 and 4.13 are in agreement with the streamlines shown in Figures 4.14 and 4.15. Since the location of the center of primary and secondary vortices are close to each other and it can be seen from Table 3.4 and 3.5, the size and shape of those vortices are also relevant to each other.

u

min

y

_min ^v_max ^x_max

v

_min

x

min

Figure 4.14 The streamlines for Re=400 by using the SIMPLE algorithm on a 161x161 mesh with the power law differencing scheme

Figure 4.15 The streamlines at bottom right and left corners for Re=400 by using the SIMPLE algorithm on a 161x61 mesh with the power law differencing scheme

Figures 4.16 and 4.17 show results of velocity profiles along the centerlines by SIMPLER method using 129x129 and 161x161 uniform and clustered meshes with the power law differencing scheme. 161x161 uniform mesh grid size gives slightly better results than 129x129 uniform meshes. The power law solution becomes grid independent when the grid size is around 161x161.

SIMPLER 129x129 Power Law Scheme SIMPLER 161x161 Power Law Scheme SIMPLER Clustered 129x129 Power Law Scheme SIMPLER Clustered 161x161 Power Law Scheme

Figure 4.16 Vertical centerline u-velocity profiles for Re=400 by using clustered mesh

SIMPLER 129x129 Power Law Scheme SIMPLER 161x161 Power Law Scheme SIMPLER Clustered 129x129 Power Law Scheme SIMPLER Clustered 161x161 Power Law Scheme

Figure 4.17 Horizontal centerline v-velocity profiles for Re=400 by using clustered mesh

If the results of clustered and uniform meshes are compared, it is observed that both of mesh types converge to the same solution with the same grid sizes.

However, the number of the iterations to reach a converged solution is considerably different. To get a converged solution by using clustered mesh requires significantly less iterations than that of uniform mesh as indicated in Table 4.6. Also, the same results are obtained by using both 161x161 and 129x129 clustered meshes, whereas 161x161 clustered mesh converges by making less iteration than the 129x129 clustered mesh. For this reason, it is effective to use clustered mesh in the regions with high flow gradients, such as; boundary layers, corners where the secondary or tertiary vortices are located.

Table 4.6 Number of Iterations with SIMPLER algorithm on uniform and clustered meshes using hybrid and power law schemes for Re=400

Grid Number

Re=400 Size of

Iterations

SIMPLER Power law 129x129 5381

SIMPLER Power law 161x161 6871

SIMPLER Hybrid 161x161 6496

SIMPLER Power law Clustered 129x129 2491 SIMPLER Power law Clustered 161x161 1073

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