6.3 Inviscid-compressible-flow results
6.3.5 Supersonic flow through a ramped channel
captured shock is less than 0.4. Figure 6.11(b) reaffirms this fact.
(a) (b)
Fig. 6.11. The shock-switch profile for the oblique-shock-reflection problem: (a) contours and (b) variation along a line at a height of 1/3 m from the plate.
The AUSM and Radespiel-Kroll Hybrid schemes resolve the oblique shocks poorly compared with the DRLLF scheme. However, the ADSAH scheme succeeds in resolving these flow features comparable with the DRLLF scheme. Figures 6.12(a)-(d) corroborate this fact.
(a) (b)
Fig. 6.13. Variations at a height of 0.88 m for the ramp-in-a-channel problem:
(a) pressure and (b) Mach number.
The variations of pressure and Mach number along a line at a height of 0.88 m with all these schemes are compared in figures 6.13(a)-(b). The crisp resolution of the weak shocks offered by the ADSAH scheme is evident from the figure. It may be noted that the ADSAH scheme creates less numerical diffusion than the AUSM and marginally more numerical diffusion than the DRLLF scheme in capturing weak shocks, as seen in figures 6.10 and 6.13. The same trend is also observed for moving normal shocks as seen in figures 6.3(a)-(b). In case of stationary strong shocks, the numerical diffusion of the ADSAH scheme assumes a smaller value than the DRLLF and slightly higher value than the AUSM scheme as evident from figures 6.5(a), 6.6 and 6.7. Thus the numerical diffusion of the ADSAH scheme gets self-adjusted based on the type of shocks. It can also be seen from figures 6.10(a) and 6.13(a) that as the numerical diffusion increases the computed pressure downstream of the shock decreases. Figures 6.13(a) and 6.13(b) show that the ADSAH scheme also eliminates the numerical oscillation produced by the DRLLF scheme near the shock emanating from the compression-corner, which is shown by the encircled regions in these figures. It is important to note from figure 6.13(b) that the gradient of the Mach number reverses sign at the point of
intersection between the centred expansion wave and the reflected shock wave.
Figures 6.14(a)-(b) show the contours of the shock switch and the variation of the shock-switch values at a height of 0.88 m. The shock switch peaks near the end-points of the oblique shocks. At the intersection point of the reflected shock and the centred expansion wave the switch reaches unity owing to the large change in the contravariant-Mach-number gradient.
(a) (b)
Fig. 6.14. The shock-switch profile for the ramp-in-a-channel problem: (a) con- tours and (b) variation along a line at a height of 0.88 m.
6.3.6 2D Riemann problem
Schulz-Rinne et al. [116] demonstrated the importance of multidimensional Rie- mann problems for testing the efficiency of numerical schemes. The initial con- ditions were explicitly specified by Brio et al. [115] and have been used as test cases in the recent literature on multidimensional Riemann solvers [49–52]. The performance of the ADSAH scheme in computing a 2D Riemann problem is tested and compared with the AUSM, DRLLF and Radespiel-Kroll Hybrid schemes. The initial conditions in the four quadrants are specified in table 6.3. A computational domain of [−1,1]×[−1,1] is considered. A 400×400 grid is used for the compu- tations with a CFL number of 0.9. The solutions are obtained at time t= 1.1.
Table. 6.3. The initial conditions for the 2D Riemann problem.
x- and y- locations Parameters
x >0, y >0 p= 1.5,ρ= 1.5,u= 0, v = 0 x <0, y >0 p= 0.3, ρ= 0.5323, u= 1.206, v = 0 x <0, y <0 p= 0.029, ρ= 0.1379, u= 1.206, v = 1.206 x >0, y <0 p= 0.3, ρ= 0.5323, u= 0, v = 1.206
(The symbols p, ρ, u and v in the table denote the pressure, density, x-velocity and y-velocity, respectively.)
(a) (b)
(c) (d)
Fig. 6.15. Density contours for the 2D Riemann problem with (a) AUSM, (b) DRLLF, (c) Radespiel-Kroll Hybrid and (d) ADSAH schemes in the lower-left quadrant using 15 contour levels in the interval [0.2, 1.6].
For the given initial conditions the 2D Riemann problem results in simultane- ous interaction of four shocks with double-Mach reflection and shock-propagation at 450 to the mesh. Figures 6.15(a)-(d) compare the density contours in the lower- left quadrant computed with the first-order-accurate AUSM, DRLLF, Radespiel- Kroll Hybrid and ADSAH schemes. Owing to the presence of the Mach-stems and strong-shock interactions the DRLLF scheme exhibits more diffusive nature than the other schemes. The Radespiel-Kroll Hybrid scheme resolves the non-grid aligned strong shock better than the other schemes. This is due to the robustness contributed by the van Leer’s FVS part of the hybrid scheme, which imparts higher accuracy for strong shocks [18]. However, both ADSAH and AUSM schemes cap- ture the non-grid aligned weak shocks better than the Radespiel-Kroll’s scheme.
The slip lines using the ADSAH scheme are less diffused than those of the DR- LLF and Radespiel-Kroll’s schemes, and are comparable to those of the AUSM scheme. Thus it is observed that the ADSAH scheme adjusts itself close to either the AUSM or the DRLLF scheme, whichever performs more accurately for any particular flow-situation.
(a) (b)
Fig. 6.16. The shock-switch variation of the ADSAH scheme for the 2D Rie- mann problem: (a) contours in the lower-left quadrant using 10 contour levels in the range [0.05, 1.0] and (b) variations along the x-direction at two different y-locations.
Figure 6.16(a) shows the shock-switch contours in the lower-left quadrant and figure 6.16(b) presents the shock-switch profiles along thex-direction aty=−0.75
and y = 0. Expectedly the strong shock inclined at 450 to the gridlines gives the highest value of unity.