In the previous section we defined the geometry of the LTMCC, at times using some “rather arbitrary” rules of thumb. This was acceptable because the subsonic portion of the LTMCC is relatively tolerant to gradual changes in geometry. This is not the case in the supersonic portion of the LTMCC and nozzle downstream of the throat. In this region we must turn to a more rigorous approach to minimize separation and turbulent losses. [38]
One such approach is to use a conical design, as shown below in Figure 5.2.1. Conical nozzles expand linearly from the the throat with a half-angle $\alpha$ generally ranging between 12-18 degrees. While these nozzles are relatively easy to manufacture, they have several downsides. At the nozzle exit plane a component of flow velocity is not aligned with the vehicle’s thrust vector. This component is skewed by an angle of $\alpha$, which decreases the overall efficiency of engine. This efficiency, $\lambda$, is plotted below. A typical 15-degree conical nozzle has an efficiency of about 98%. [38]
Figure 5.2.1 [58]
The length of the nozzle from the throat to the exit plane must be considered as well. Prescribing the expansion ratio $\epsilon$ and and half-angle $\alpha$ gives the following expression for nozzle length:[38]
The radii $R_u$ and $R_t$ are defined as shown below in Figure 5.2.2. Typical values of $R_u$ are on the order of 0.5-1.5 times $R_t$.[38] In the case of the SSME, $R_u=0.494R_t$ (see Figure 5.1.1).
Figure 5.2.2 [58]
An engineer by the name of G.V.R. Rao came up with a design to improve upon the conical nozzle.[59], [60] The resulting wall contour is based on a “skewed parabola.” The length of a Rao nozzle is typically defined to be 80% of the length of a 15-degree conical nozzle with the same expansion ratio. This results in a shorter and lighter nozzle. The parabolic contour reduces the flow exit angle, $\alpha$ as compared to a conical nozzle. This helps to align the flow exit velocity with the vehicle’s thrust vector and increases nozzle efficiency. Figure 5.2.3 highlights the differences between these two designs.
Figure 5.2.3 [58]
The wall coordinates of the Rao curve used on the SSME nozzle can be represented by the following equations. Reference [61] contains an excellent discussion on how these equations are derived. The coordinate system used for the nozzle wall contour is consistent with that established for the wall coordinates of the LTMCC in Section 5.1. In both cases $x$ is defined to be 0 inches at the injector face. In the case of the LTMCC, the piecewise equation for wall coordinates stopped at 16.1 inches (see Table 5.1.2 and Figure 5.1.12 ). The following equation is valid for $x$ values greater than 16.1 inches all the way up to the nozzle exit plane at $x=$ 135.5 inches. The x-coordinate of the nozzle exit plane was calulated by determining the length of an equivalent 80% conical nozzle with Equations 5.2.1 and 5.2.6 (120.25 inches), then adding the length of the LTMCC between the throat and injector face (15.2 inches, see Figure 5.1.10).
It is important to remember that although the wall coordinate equations transition at $x=$16.1 inches, this is not the location where the nozzle mates with the LTMCC. The nozzle attach flange is physically located at $x=$24.29 inches. 16.1 inches is the location where the wall contour of the LTMCC and nozzle become identical and can be described by a single equation for both pieces of hardware. Point “D” in Figure 5.1.10 illustrates this concept.
In order to use these equations we must begin with several known radii: $R_t$, $R_d$, $R_e$,and $R_u$ . All of these values are tabulated in Figure 5.1.10 We also must know the wall angle immediately departing the throat ($\theta_d$) and the nozzle exit angle $\alpha$. Both of these angles are tabulated in Figure 5.1.11 as well. If these angles are not known a priori, they can be estimated from the following chart as a function of nozzle length and expansion ratio. In the case of the SSME, read the angles associated with the 80% conical nozzle and an expansion ratio of 69.5. The data shown below in Figure 5.2.4 lines up fairly well with the actual SSME data tabulated in Figure 5.1.11.
Figure 5.2.4 [58]
We are now finally positioned to plot the SSME wall contour all the way from the injector face to the nozzle exit plane at $x=135.5$ inches. The colors displayed below correspond to physical hardware in the SSME. Blue represents the walls of the LTMCC and orange represents the nozzle. These components are mated at the nozzle attach flange ($x=$25.19 inches). Both pieces of hardware utilize the same wall contour equation from $x=16.1$ inches onward.
Figure 5.2.5