6.3 Inviscid-compressible-flow results
6.3.2 Quasi-1D flow through a converging-diverging nozzle
In quasi-1D flow all the flow parameters as well as the cross-sectional area are functions of the axial coordinate x. The geometric and free-stream details for quasi-1D flow through a converging-diverging nozzle are listed in table 6.1. Under steady state, a standing-normal shock exists in the diverging portion of the nozzle at the assigned back pressure. This problem also offers the advantage of possessing
an analytical solution.
Table. 6.1. The geometric and flow parameters for quasi-1D converging-diverging nozzle-flow.
Parameter Value
Length of the nozzle (L) 3 m
Cross-sectional area of the nozzle A(x) A(x) = 1 + 2.2 (x−1.5)2 Reservoir pressure (p0) 101325 N/m2
Reservoir temperature (T0) 300 K Back pressure (pb) 0.7 (p0)
(a) (b)
Fig. 6.5. Quasi-1D flow problem: (a) Variation of density along the nozzle and (b) The shock-switch profile.
The density variation and the shock-switch profile along the nozzle for the quasi-1D-flow problem are plotted in figures 6.5(a)-(b). The number of cells in the computational domain is 60 for this computation. Figure 6.5(a) shows that the ADSAH scheme resolves the standing-normal shock with much less smearing like the AUSM and Radespiel-Kroll Hybrid schemes and much better than the DRLLF scheme. It can be seen from figure 6.5(b) that the shock switch assumes high value only in the vicinity of the shock and does not get activated in smooth regions. In conformity with equations 6.1-6.8, the switch approaches unity near the upstream end of the shock since there is a sharp change of the Mach-number
gradient in that location. Within the shock the gradient does not change abruptly across neighbouring cell-faces and therefore the value of the shock switch drops.
Near the downstream end of the shock the change in the Mach-number gradient is milder compared to that in the upstream end. This fact gets reflected in the unequal peaks in figure 6.5(b). Owing to this reason the ADSAH scheme acts close to the AUSM scheme near the end-points of the shock and close to the DRLLF scheme within the shock as given by equation 6.9.
6.3.3 2D flow through a converging-diverging nozzle
(a) (b)
(c) (d)
Fig. 6.6. Pressure contours for 2D flow through a converging-diverging nozzle using (a) AUSM, (b) DRLLF, (c) Radespiel-Kroll Hybrid and (d) ADSAH schemes.
The same problem mentioned in table 6.1 is computed by using a 2D formulation.
The 2D Euler equations governing the flow are solved on a 120×51 grid. Figure 6.6(b) shows that the DRLLF scheme resolves the normal shock poorly. But the ADSAH scheme captures the normal shock with less smearing comparable with the AUSM and Radespiel-Kroll Hybrid schemes as seen in figures 6.6(a), 6.6(c) and 6.6(d). The centreline pressure and Mach-number variations with the AUSM,
DRLLF, Radespiel-Kroll Hybrid and ADSAH schemes are shown in figures 6.7(a)- (b). Expectedly the DRLLF scheme captures a more smeared shock compared with the AUSM scheme. On the other hand the present hybrid scheme resolves the shock almost as crisply as the AUSM and Radespiel-Kroll Hybrid schemes. In addition the ADSAH scheme is free from the sonic-glitch problem exhibited by the AUSM scheme as shown by the encircled region in figures 6.7(a)-(b).
(a) (b)
Fig. 6.7. Variations along the axis for 2D flow through a converging-diverging nozzle: (a) pressure and (b) Mach number.
(a) (b)
Fig. 6.8. The shock-switch profile with the ADSAH scheme for the 2D flow through a converging-diverging nozzle: (a) contours and (b) variation along the nozzle axis.
Figures 6.8(a)-(b) show the shock-switch contours and the variation of the shock switch along the nozzle axis, respectively, for the ADSAH scheme. Similarly to the quasi-1D flow problem the switch attains a value close to unity at the end- points of the standing-normal shock. In addition the switch does not shoot up in the smooth regions including the cells contained within the shock and approaches zero. As a result the ADSAH scheme acts close to the AUSM scheme near the end- points of the standing-normal shock and like the DRLLF scheme in the smooth regions including the cells contained within the shock.
6.3.4 Oblique-shock-reflection problem
The geometric and free-stream parameters for the oblique-shock-reflection problem are shown in table 6.2. A 120×90 grid is used for all the computations. Figures 6.9(a)-(d) compare the pressure contours for the oblique-shock-reflection problem that reveals the improved performance of the ADSAH scheme. The DRLLF scheme resolves the oblique shock much better than the AUSM scheme. The ADSAH scheme outperforms the AUSM and Radespiel-Kroll Hybrid schemes in capturing the incident and reflected shocks and resolves them almost as good as the DRLLF scheme. Figures 6.10(a)-(b) compare the pressure and Mach-number variations at a height of 1/3 m from the plate obtained by the four schemes. It reaffirms the superior performance of the ADSAH scheme over the AUSM and Radespiel-Kroll Hybrid schemes in resolving weak shocks.
Table. 6.2. The geometric and flow parameters for oblique-shock reflection.
Parameter Value
Length of the plate (L) 3 free-stream Mach number (M∞) 3
free-stream pressure (p∞) 101325 N/m2 free-stream temperature (T∞) 300 K
Incident wave angle (β) 300
(a) (b)
(c) (d)
Fig. 6.9. Pressure contours for the oblique-shock-reflection problem using (a) AUSM, (b) DRLLF, (c) Radespiel-Kroll Hybrid and (d) ADSAH schemes.
(a) (b)
Fig. 6.10. Variations at a height of 1/3 m from the plate for the oblique-shock- reflection problem: (a) pressure and (b) Mach number.
It is desirable that across the cell-interfaces intercepted by weak-oblique shocks, the shock switch should assume smaller values compared with those crossed by stronger shocks. Figures 6.11(a)-(b) show the shock-switch contours and the vari- ation of the shock switch at a height of 1/3 m from the plate. The zoomed region in figure 6.11(a) shows that the maximum value of the shock switch within the
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.