APPLICATIONS TOWARDS DESIGN AND OPTIMISATION
7.3 Design of optimal nozzle for supersonic flows
We also quantitatively verify the 1D solutions from the Low-fidelity framework by comparing those obtained using the IB-FV solver as shown in Table 7.4. It can be observed that the quantities of interest at the isolator agree well from both the LF and HF framework. This is not entirely surprising as the HF framework is based on the inviscid dynamics of the flow. We have observed some difference with [152], which is likely because of the differentMe/M∞ value and flow-deflection angle correction. Raj and Venkattasubbaiah [152] attribute the difference between their LF and HF solu- tions to the two-dimensionality effects however, in contrary to their report, we have noticed an excellent agreement between the solutions form LF and HF framework.
It is also worth reporting that this difference between LF and HF is not due to the flow-deflection correction, as has been seen in our numerical investigations (not shown here for brevity).
From the studies carried out in this section, we see that the scramjet inlet captures full inlet air-stream, generates sufficiently high compression ratio unlike reported liter- ature. The scramjet inlet configurations generates high TPR which were not strongly sensitive on freestream Mach number. Moreover, the deflection angle correction en- ables uniform flow in the isolator of the scramjet inlet. The computations performed using the IB-FV solver clearly highlight the utility of the design framework based on the one-dimensional approach. Nevertheless, one can make use of the multi-fidelity framework which uses the optimal shape shown herein as an initial guess and carries out a shape optimisation to optimise one or more cost functions. This is however beyond the scope of the present study and may be taken up in the future.
area nozzle (Figure 7.16), we can arrive at an useful “area-Mach number relation” as given in equation [1],
A A∗
2
= 1 M2
2 γ + 1
1 + γ−1 2 M2
γ−1γ+1
(7.17) where, γ is the ratio of specific heats. Eq. 7.17 is referred to as the Area-Mach number relation and is used to determine the flow properties across any cross-section inside the nozzle. This idea is now employed as shown in the algorithm detailed in Table 7.5.
A∗ u∗=a∗ M∗= 1
M A
u
Figure 7.16: Variable area nozzle Table 7.5: Low-fidelity flow solver Algorithm: LF Flow solver
1 Fix the inner and outer radius ri, ro of the nozzle, respectively (Figure 7.17) 2 Fix the inlet Mach numberM and ratio of specific heats γ
3 Parameterise the geometry using cubic Bezier curve, (shown in section 7.1) 4 Evaluate the RHS of Eq. 7.17
5 Increment the Mach number M by small value (10−6).
6 If difference between LHS and RHS≤ Terminate. Or repeat steps 4-5 above We now incorporate the IB-FV solver, as also done in Section 7.2, to evaluate the design obtained using the quasi one-dimensional approach. As opposed to study in Section 7.2 which involves planar scramjet intakes, we reiterate that studies herein are necessarily involving axisymmetric geometries and axisymmetric flow.
l ri
ro
Figure 7.17: Schematic of the nozzle configuration
We choose an initial shape interms of a conical frustum and carryout optimisation to minimise the radial velocity at the exit of the nozzle defined as,vr/Vt, which ensures that the flow is parallel at the exit of the nozzle, necessary for maximising the positive thrust. We start by fixing the inlet radius and outer radius as ri=0.02 and ro=0.045 and the length of the nozzlel is taken as l=0.1. The optimisation process (carried out using steepest descent, as elaborated previously in section 7.1) is terminated after 200 iterations or when the difference between the cost functions becomes lesser than 10−3, whichever happens earlier. The inlet Mach number is fixed at 1.001 (small positive perturbation above the Mach number of unity grantees that the flow accelerates to supersonic flow at the exit).
Optimisation cycles vr/Vt
0 20 40 60 80 100
10-4 10-3 10-2 10-1 100 101 102
Convergence, LFF
Figure 7.18: Convergence history
x
y
0 0.02 0.04 0.06 0.08 0.1
0 0.01 0.02 0.03 0.04 0.05
Initial shape Optimal shape
Figure 7.19: Optimal nozzle configuration obtained from LFF
The convergence plot in Figure 7.18 obtained using the LFF shows monotonic decrease in the cost function (i.e vr/Vt). Figure 7.19 shows the optimal nozzle con- figuration obtained from LFF and its comparison with the initial shape. It can be seen that the optimal nozzle shape resembles a “bell-shaped” contoured nozzle with seamless transition from the leading edge to the trailing edge, allowing for reduction in the radial velocity at the exit. This reduction can yet again be confirmed Figure 7.18 where the cost function reduces by nearly 4 orders. This study is in contrast to the remarks in [1] where they mention that the utility of the quasi one-dimensional approach is only restricted to the determination of axial distribution of flow properties.
Figure 7.20: Mach contour obtained from IB-FV flow solver for the optimal nozzle configuration (Min: 0.05, ∆:0.0493, Max: 3.45)
x (m)
Machnumber,M
0 0.02 0.04 0.06 0.08
0 0.5 1 1.5 2 2.5 3 3.5 4
HF flow-solver LF flow-solver
Figure 7.21: Comparison of Mach number obtained from both flow solvers We now carry out studies by immersing this optimal nozzle shape in a Cartesian grid that employs 150×50 control volumes along with AUSM flux splitting scheme to compute the inviscid flow within the nozzle. In Figure 7.20 Mach contour are shown and Figure 7.21 shows the Mach number distribution over the nozzle wall wherein small differences between the low-fidelity approach and IB-FV can be seen. However, the exit Mach number obtained from low-fidelity flow solver is 3.19 while the IB-FV flow solver gives an average exit Mach number 3.35. One can see that the quantita- tive agreement for the average exit Mach number is good, similar to what has been observed in the scramjet intake studies as well (see Table7.4).