# Ramjet Nozzle Analysis for Transport Aircraft Configuration for Sustained Hypersonic Flight
## 3 Conceptual Design of a Ramjet Nozzle
### Nozzle Geometric Configurations
#### 3.3.2 Bell Nozzle  
Bell-shaped nozzles are currently the most commonly utilised nozzle profile. Nozzle profiling in this case is tailored to yield certain desired pressure gradients to minimise flow separation, thus improving the performance of the simple cone nozzle.  
Figure 6: **(a) (Top)** Mostly axial thrust, (**Bottom**) mostly radial thrust; (**b**) and the divergence factor in relation to the nozzle geometry [8].  
Bell-shaped nozzles consist of two sections. The nozzle contours diverge at a relatively large angle near the throat and taper off further downstream. At the section near the nozzle exit the divergence angle is very small, almost parallel to the jet axis. This configuration allows a maximisation of the performance while minimising the weight, with the length of the nozzle being as much as 25% lower than the conical nozzle equivalent. This type of nozzle achieves the reduction of divergence by turning the flow to provide an axial flow [8]. Bell nozzles typically exhibit one design point where they are at their most efficient at a given altitude of operation.  
In addition, the bell nozzle, similarly to the conical nozzle, exhibits over- and under-expansion flow effects under different altitude and flow conditions. The principal regimes of over- and under-expansion for both nozzles are presented in Figure 7 below.  
The bell nozzle basic geometry is provided in Figure 8, with \(\alpha\) being the initial nozzle angle immediately downstream of the throat and \(\theta_{ex}\) the final nozzle exit angle.  
Figure 8: Bell nozzle divergence.  
Figure 7: Nozzle performance at different altitudes.  
The divergence loss is characterised by the divergence loss factor, \(\lambda\), in Equation (3), below:  
\[\lambda=\frac{1}{2}(1+cos\frac{(\alpha+\theta_{\pi\chi})}{2}) \tag{3}\]  
Truncated idealised contoured (TIC) nozzles (Figure 9, left) are a class of bell-shaped nozzle developed to produce a virtually unidirectional exhaust profile. The walls of the contour curve near the throat region and become nearly parallel to the nozzle axis at the exit plane. However, the nozzle length ought to be equal to 50 times the throat radius in theory. In cases where this requirement cannot be met, for shorter nozzles, the flow is forced to straighten out earlier, causing a decrease in the nozzle efficiency [10].  
Thrust-optimised contoured nozzles (TOC, Figure 9, right) are a step further from TIC nozzles where the near throat region has a shaped gradient of curve in the near throat region which corresponds to the greater expansion of the flow. The higher increase in the parabolic arc induces an internal shock that increases the pressure at the exit plane of the nozzle [10; 11].  
Bell nozzles can be characterised by the presence of weak compression waves, which lead to the formation of an internal shock induced by the sharp turn in the nozzle succeeding the throat area. The shock separates two regions of varying flow velocities, the kernel region and the radial gradient region. The kernel region is where the flow field depends on the expansion at the throat, involving expansion waves generated at the convex wall of the throat region and its reflection at the throat axis. The larger kernel regions lead to higher jet velocities at the nozzle exit. Radial gradient zones lie between the kernel zone and the wall of the nozzle where the radial pressure gradient is positive, i.e., the pressure increases moving away from the axis. Larger radial gradient zones, such as those in TIC nozzles, lead to separation within the nozzle wall boundaries and result in separation shocks. TOC nozzles, however, have smaller radial gradient zones and larger kernel zones, leading to higher flow velocities at the nozzle exit with a lower possibility of separation.