MIT Benchmark

Results and discussion

Several comparisons have been made to see the difference from the other references. In [4] it is mentioned that the $Q_2 P_{1}$ element with coarse mesh (27 x 121) performs poorly in the sense that the results show too low amplitudes for velocity and temperature at point 1 (0.00542 and 0.00442). In contrast, we observe good results even with the level 2 mesh (16 x 88). They also calculated Nusselt numbers which are slightly different from the reference, see Le Quéré [6]. In fact, we produce the same results (2R, 3R, 4R), but only as soon as we introduce local refinement near the wall, the Nusselt number improves strongly even with the level 2 mesh. It is obvious that the Nusselt number calculated on level 3 and 4 (3R and 4R) can be improved by using the level 2 mesh with local refinement (2R_a1 and 2R_a5).

Results of the MIT Benchmark 2001 simulations
Author $u_1$ $\Theta_1$/span> -Nu Period
Turek 0.0572 0.2647 4.5791 3.422
Davis 0.0563 0.2655 4.5796 3.412
Gresho 0.05665 0.26547 4.5825 3.4259
Le Quéré 0.056356 0.26548 4.57946 3.4115
2R 0.058139 0.26539 4.66245 3.4000
2R_a1 0.057674 0.26538 4.59295 3.4214
2R_a5 0.057490 0.26540 4.57941 3.4214
3R 0.056787 0.26548 4.59318 3.4214
3R_a1 0.056665 0.26546 4.58155 3.4214
3R_a4 0.056591 0.26549 4.57967 3.4214
4R 0.056451 0.26549 4.58158 3.4200
4R_a1 0.056394 0.26546 4.57994 3.4154
4R_a3 0.056372 0.26546 4.57969 3.4214

Note: The point datas value for each mesh can be found here.

We believe that without local grid refinement we might have to use level 5 or higher to produce nearly the same Nusselt numbers as the one produced by Le Quéré. This information shows us the expected result that local grid refinement helps a lot for this test configuration. The time step is not an issue as long as we put enough time steps over one period. 20 up to 40 time steps are already sufficient to produce excellent results for this problem, and no specific gain/loss in the quality of the Nusselt number has been observed if we increase/decrease the number of time steps in one period (see [7]). Summarizing, we observe differences from the reference result with 0.02% for velocity $u_1$ and with 0.003% for temperature $\Theta_1$; and we are very close with 0.004% difference for the Nusselt number.

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