# Ramjet Nozzle Analysis for Transport Aircraft Configuration for Sustained Hypersonic Flight
## 4 CFD Analysis of Ramjet Nozzles
### Bell Nozzle  
Contrary to conical nozzle, the bell nozzle optimises the performance by producing a higher effective axial thrust with a lower nozzle length (H1), in this case of 3.55 m. The expansion fan is readily visible in Figure 18, which accelerates the flow to comparatively higher Mach numbers. In addition, the thrust is far more axial as compared to the radial thrust produced by the conical nozzle previously. Additionally, internal shocks originating from the sharp corner of the throat forms a region of compression that decelerates the flow in the near wall region. Nonetheless, the contoured bell nozzle produces a relatively high exit velocity, as presented in Figure 18.  
Despite having a smaller exit radius and lower nozzle length, the nozzle is capable of producing a higher gross thrust value of 1.61 \(\times\) 10\({}^{6}\) N prior to the divergence calculation, as mentioned in Section 3.3.2. The divergence factor (\(\lambda\)) obtained was 0.98, generating a resultant axial thrust of 1.58 \(\times\) 10\({}^{6}\) N while resulting in correspondingly a lighter overall nozzle mass. The internal shock emanating from the nozzle contour can be confirmed by utilising the vector contour plots (Figure 19) as the flow can be seen turning inwards towards the nozzle axis as the jet exits the nozzle. Similarly, the compression waves arising from the nozzle lip combine into a surface shock which changes the direction and magnitude of the flow from the radial gradient zone. The two shock waves interact and merge into a stronger shock capable of slowing down the high velocity fluid flowing through the kernel region and redirecting the flow towards the nozzle axis. As depicted in Figure 20 (right), the pressure increases considerably along the y-direction away from the nozzle axis, in agreement with the literature discussed in Section 3.3.2. In the subsonic region, in order to efficiently shorten the effective nozzle length, the curvature radius for the contour is small but still delivers a flow over the throat. Axially downstream of the throat, where the flow is sonic, the high slope gradient allows considerable expansion. The internal shock originates from the sharp corner of the throat which distinguishes the kernel from the radial gradient zone. The nozzle selected has a design characteristic similar to that of thrust-optimised contour (TOC) nozzles, with larger curve inflection, and is capable of delivering high expansion ratios along the nozzle length [19], validating the nozzle design point. In Figures 19 and 20 the region below the shock is characterised by high momentum kernel flow which allows a higher amount of flow to  
Figure 18: **(Top)** Pressure contour and (**Bottom**) Mach contour thrust = 1.58 \(\times\) 106 N, _NPR_\(\approx\) 90, H1 = 3.55, V1 = 2, R1 = 6.  
be expanded and fully accelerated at the exit. The radial gradient region above the internal shock presents a gradual degradation of axial velocity towards the wall boundary. Pressure and Mach plots confirm the presence of oblique shocks where the static pressure increases while the gas decelerates dramatically downstream of the inlet. Comparison between the two nozzles of same dimensions (Figures 21 and 22) in terms of nozzle length and height indicate a higher Mach at the nozzle exit despite having the presence of an internal shock. This results in the bell nozzle producing a higher effective axial thrust instead of radial thrust. The flow in the kernel zone for the bell nozzle drops considerably, which accelerates the flow at a higher rate than the bell nozzle, leading to a higher production of momentum thrust. The pressure expansion is depicted in Figure 23 as the exhaust gas keeps expanding further into the nozzle boundaries, whereas in the conical nozzle the pressure plateaus. Figure 19: Evidence of shock utilising velocity vector contours. Figure 20: CFD result compared to a TOC nozzle flow pattern. Figure 23: Pressure across the divergent section. Figure 21: Pressure contour of bell nozzle (**top**) and conical nozzle (**bottom**).