**4.1 Analyses of the grid refinement**

66 Computational Simulations and Applications

1 2

*n*

2 2 *n+ n n+ n+*

7.5*d*

*n+*

1

Corrector step:

**4. Problem description** 

step used in all simulations is 0.0001 s.

Fig. 1. Schematic illustration of the calculation domain.

16.5*d*

the inlet: *u eV* 0

<sup>0</sup> *<sup>u</sup> x x* 

outlet and in the lateral boundaries: *p* 0 .

2

1 1

 

*i i n n*

Stability analysis of the second order spatial centered scheme with the time discretization schemes is performed by two-dimensional simulations of incompressible flows around a stationary circular cylinder. The rectangular domain is chosen to be 15 x *d* 30 with the *d* cylinder located at 16 cylinder diameters from the inlet as .5 illustrated in Fig. (1). The time

The flow develops from the bottom to top and the boundary conditions for velocity are : in

15*d*

. For the pressure, the boundary conditions used are: in the inlet: 0 *<sup>p</sup>*

*y y* 

and in the lateral boundaries:

*y*

, in the

, in the outlet: 0 *<sup>u</sup>*

*u u <sup>=</sup> <sup>f</sup> u ,u P +F <sup>Δ</sup><sup>t</sup>* 

*i j i i*

*i j i i*

(10)

(11)

30*d*

*d*

*i i nn n n*

*u u <sup>=</sup> <sup>f</sup> u ,u P +F <sup>Δ</sup><sup>t</sup>*

For these simulations three grids are used, which are shown in the Tab. (1), along with the three time discretization schemes. It is observed through the mean values of drag coefficients (Table 2), the similarity of results when different time discretization methods were considered and the same grid refinement. Considering the various refinements, it is noted that with the coarser grid the destabilization of the flow occurs more slowly. With the grid refinement, which filters the instabilities of high frequency, the transition of the flow is faster. It is also observed that with the grid refinement from the grid 2 to grid 3, the mean values of drag coefficients are approximately the same, which leads to the independence of the results for finer mesh than 125x250. The Sthouhal number, obtained by Fast Fourier Transform (FFT) of the lift coefficient signal is also shown in Tab. (2) for Reynolds number 100.


Table 1. Grids used for the three time discretization schemes, Re=100.


Table 2. Mean values of drag coefficients and Strouhal number for the three time discretization methods and different grids.

Note that the mean values of the drag coefficient decreases with grid refinement for the three methods. No significant difference is observed when passing from the intermediate to the most refined grid, as mentioned previously. These results are also visualized through the time evolution of the drag coefficient, Fig. (2), which presents the different grid refinement for each of the time discretization methods.
