1. Grid Studies:

• Fine Grid: (81x51x2). All Pe/P0i runs were performed for this grid.

Figure 1. Fine grid used in the calculations

Coarse Grid 1: This grid is obtained by reducing the number of fine grid points in the I and J directions by a factor of 2. The grid sequencing option of GASP has been used to obtain the grid. Dimensions: (41x26x2). All Pe/P0i runs were performed for this grid.

• Coarse Grid 2: This grid is obtained by modifying coarse grid 1. New J points (except j=1, 2, and jmax-1) are located at  half distance to  j and j+1 of the coarse grid 1. Since x and y values of the grid points at the top wall boundary are also  calculated by averaging, a small error in the boundary values (y values at the  wall) may also be introduced (especially at the start of the converging section following the constant area section). All Pe/P0i runs were performed for this grid. Dimensions:(42x26x2)

• Coarse Grid 3. The same strategy as in the previous case has been used to obtain this grid. However, the  y values at the top wall boundary are calculated by using the analytical equation defining this surface. Therefore, no error is introduced through the boundary conditions. Dimensions:(43x26x2).

Figure 4. Coarse grid 3 superimposed on coarse grid 1.

Figure 5. The boundaries of coarse grid 1,2, and 3.

• 4 additional runs were performed between Pe/P0i=0.78 and 0.79: These pressure ratios are 0.78125, 0.78375, 0.78625, and 0.78875. 

• Nozzle Efficiency:

 

Figure 6.  Nozzle efficiencies for all Pe/P0i ratios.
 

Figure 7. Nozzle efficiencies for the Pe/P0i ratios between 0.75 and 0.79.

P0/P0i ratios at the duct centerline: P0/P0i ratios on  the duct centerline were calculated for two x/ht stations: x/ht=-1.5 (upstream of the shock) and x/ht=3.4 (downstream of the shock). At both stations, the values are obtained by linear interpolation.

Figure 8. P0/P0i ratios at different x/ht stations for all Pe/P0i ratios.

Figure 9. P0/P0i ratios at different x/ht stations for Pe/P0i ratios between 0.75 and 0.79.

Shock Locations:

Figure 10. Shock locations for all Pe/P0i ratios.

Figure 11. Shock locations for Pe/P0i ratios between 0.75 and 0.79.

 

Table 1. Nozzle efficiencies and shock locations for all Pe/P0i ratios:

• Errors in nozzle efficiency and shock location:

 

Figure 12. Nozzle efficiency error distribution for all Pe/P0i.

Figure 13. Nozzle Efficiency error distribution for Pe/P0i between 0.75 and 0.79.

Figure14. Histogram for the nozzle efficiency error (Coarse grid 1)

Figure 15. Shock location error distribution for all Pe/P0i ratios. (Coarse grid 1)
 
 

Figure 16. Shock location error distribution for Pe/P0i ratios between 0.75 and 0.79. (Coarse grid 1)

Figure 17. Histogram for the shock location error. (Coarse grid 1)

• Mach convergence check at two different Pe/P0i ratios:

Figure 18. Comparison of the mach number convergence at x/ht=2.1176, y/ht=1.054. (Coarse grid 1)