## Real Nozzles

In a real nozzle the flow is really two-dimensional, but axisymmetric. For simple single nozzle shapes the temperatures and velocities are not uniform over any one section and are usually higher in the central region and lower near the periphery. For example, the surface where the Mach number is one is a plane at the throat for an ideal nozzle; for two-dimensional flow it is typically a slightly curved surface somewhat downstream of the throat. If the velocity distribution is known, the average value of v2 can be determined for an axisymmetric nozzle as a function of the radius r.

The 11 assumptions and simplifications listed in Section 1 of this chapter are only approximations that allow relatively simple algorithms and simple mathematical solutions to the analysis of real rocket nozzle phenomena. For most of these assumptions it is possible either (1) to use an empirical correction factor (based on experimental data) or (2) to develop or use a more accurate algorithm, which involves more detailed understanding and simulation of energy losses, the physical or chemical phenomena, and also often a more complex theoretical analysis and mathematical treatment. Some of these approaches are mentioned briefly in this section.

Compared to an ideal nozzle, the real nozzle has energy losses and energy that is unavailable for conversion into kinetic energy of the exhaust gas. The principal losses are listed below and several of these are discussed in more detail.

1. The divergence of the flow in the nozzle exit sections causes a loss, which varies as a function of the cosine of the divergence angle as shown by Eq. 3-34 and Table 3-3 for conical nozzles. The losses can be reduced for bell-shaped nozzle contours.

2. Small chamber or port area cross sections relative to the throat area or low nozzle contraction ratios Ax/A, cause pressure losses in the chamber and reduce the thrust and exhaust velocity slightly. See Table 3-2.

3. Lower flow velocity in the boundary layer or wall friction can reduce the effective exhaust velocity by 0.5 to 1.5%.

4. Solid particles or liquid roplets in the gas can cause losses up to 5%, as described below.

5. Unsteady combustion and oscillating flow can account for a small loss.

6. Chemical reactions in nozzle flow change gas properties and gas temperatures, giving typically a 0.5% loss. See Chapter 5.

7. There is lower performance during transient pressure operation, for example during start, stop, or pulsing.

8. For uncooled nozzle materials, such as fiber reinforced plastics or carbon, the gradual erosion of the throat region increases the throat diameter by perhaps 1 to 6% during operation. In turn this will reduce the chamber pressure and thrust by about 1 to 6% near the end of the operation and cause a slight reduction in specific impulse of less than 0.7%.

9. Non-uniform gas composition can reduce performance (due to incomplete mixing, turbulence, or incomplete combustion regions).

10. Using real gas properties can at times change the gas composition, the value of k and 93Í, and this can cause a small loss in performance, say 0.2 to 0.7%.

11. Operation at non-optimum nozzle expansion area ratio can reduce thrust and specific impulse. There is no loss if the vehicle always flies at the altitude for optimum nozzle expansion (p2 = Pi)- If it Aies with a fixed nozzle area ratio at higher or lower altitudes, then there is a loss (during a portion of the flight) by up to 15% in thrust compared to a nozzle with altitude compensation, as can be seen in Figs. 3-7 and 3-8. It also reduces performance by 1 to 5%.

### Boundary Layer

Real nozzles have a viscous boundary layer next to the nozzle walls, where the gas velocities are much lower than the free-stream velocities in the inviscid flow regions. An enlarged schematic view of a boundary layer is shown in Fig. 3-16. Immediately next to the wall the flow velocity is zero and then the boundary layer can be considered as being built up of successive annular-shaped thin layers of increasing velocity until the free-stream velocity is reached. The low-velocity flow close to the wall is laminar and subsonic, but in the higher-velocity regions of the boundary layer the flow is supersonic and can become turbulent. The local temperature in part of the boundary layer can be substantially higher than the free-stream temperature because of the conversion of kinetic energy into thermal energy as the local velocity is slowed down and as heat is created by viscous friction. The layer right next to the wall will be cooler because of heat transfer to the wall. The gaseous

Subsonic flow can bend up to 180°

Nozzle exit lip Nozzle wail

Boundary layer thickness

Typical steam line

Subsonic flow can bend up to 180°

Nozzle exit lip Nozzle wail

Boundary layer thickness

Typical steam line

Subsonic portion of boundary layer

' Supersonic portion of boundary layer Nozzle wall

Wall thickness

Boundary layer thickness

Subsonic portion of boundary layer

' Supersonic portion of boundary layer Nozzle wall

Wall thickness

Boundary layer thickness

Velocity profile

FIGURE 3-16. Flow conditions at a nozzle exit lip at high altitude, showing streamlines, boundary layer, velocity and temperature profiles.