4.1 Freezing point position
Numerical results of the different scale nozzles are now presented. The centerline evolution of the vibrational temperature TV and translational temperature T are plotted in Fig.5. Not far downstream of the nozzle throat, the larger-scale nozzle has a slightly higher translational temperature. The vibrational temperature modifies the translational temperature, as deduced from the kinetic energy in Eq. (10). Compared with the translational temperature, the vibrational temperature is more affected by the absolute scale. The larger the scale, the further away from the throat the freezing point, and the lower the vibrational temperature. The vibrational temperature of 0.5NS, NS, 2NS, 5NS is 2613, 2298, 2036, 1755, respectively. In this initial condition, the absolute scale of the nozzle is increased tenfold and the vibrational temperature is reduced by 858K, as shown in Fig.5. The freezing point moves down the nozzle throat as the nozzle scale increases. At the centerline vibrational temperature in the different nozzle scales (from 0.5NS to 5NS), the distances xf from the freezing point to the throat are 451mm, 1241mm, 3091mm, and 9712mm. The freezing point diameters df of their corresponding nozzles are 15.7mm, 39.3mm, 92.2mm, and 276.1mm. The ratio df/dt of the freezing point diameter to the throat diameter is 2.86, 3.57, 4.19, and 5.02, respectively. The thermochemical non-equilibrium scale effects are also suitable for the nozzle of FD-21. Fig.7 is a comparison of 0.5NS and NS flow field temperatures. At the centerline vibrational temperature in the 0.5NS and NS nozzles, the distances xf are 281mm and 875mm. The freezing point diameters df of their corresponding nozzles are 98.1mm and 254.1mm, and the ratios df/dt are 9.81 and 12.71, respectively. The absolute scale not only affects the translational temperature and vibrational temperature of the nozzle flow field but also the freezing point. Those phenomena can be explained in the following aspects.
(1) As the nozzle scale increases, the pressure gradient around the throat decreases (see Fig.6). The pressure downstream of the throat increases, which shortens TVERT, as deduced from Eq. (28). Additionally, the decrease of pressure gradient makes the flow velocity decrease and the flow time increase. Both lengthen the transition from equilibrium to freezing, and the flow is close to the equilibrium flow.
(2) Since the nozzles use the same inlet conditions, the velocity u at the nozzle exit is almost identical25, which is consistent with h≈u2/2. Because of the freezing of vibrational energy, the increase of the kinetic energy of the airflow in nozzle mainly comes from the translational energy, which leads to the decrease of translational temperature T.
As can be seen from the above conclusion, the flow is frozen not far downstream of the high enthalpy flow throat. Cases 1,3 and 4 in table 1 investigate the thermochemical nonequilibrium scale effects by changing the diameter of the throat alone. Fig.8 shows the distribution of vibrational temperature and translational temperature along the centerline of nozzle. For the throat radius dt are 20mm, 40mm, and 60mm, the freezing temperatures are 2442K, 2207K, and 2140K, respectively. The distances xf from the freezing point to the throat are 0.717m, 1.568m, and 3.030m. The freezing point diameters df of their corresponding nozzles are 0.254m, 0.536m, and 1.018m. The ratio df/dt of the freezing point diameter to the throat diameter is 12.7, 13.4, and 16.9, respectively. As the throat radius increases, the freezing point moves away from the throat. They can be explained in the following aspects.
(1) The effects of the throat diameter are the same as the scale effects of the nozzle. As the nozzle diameter increases, the pressure gradient around the throat decreases (see Fig.9)，and the flow velocity around the throat decrease(See Fig.10) and the flow time increase. Both lengthen the transition from equilibrium to freezing, and the flow is close to the equilibrium flow.
(2) In thermal equilibrium, the increase of the kinetic energy of the gas comes from both molecular translational energy and molecular vibrational energy. Therefore, thermal equilibrium causes the translational temperature and vibrational temperature to decrease. When the vibrational energy freezes, the increase in the kinetic energy of the gas comes only from the translational kinetic energy. The nozzle outlet speed is almost unchanged, which makes the reduction of translational energy deeper and the translational temperature lower.
4.2 Flowfield parameters
Fig.11 shows the Mach number distribution along the centerline. Before the freezing point, the value and evolution of the Mach number are the same. After the freezing point, this phenomenon has changed. It can be explained by the freezing of the vibrational levels. As the nozzle scale increases, the vibrational temperature decreases and the translational temperature increase. With the increase of the nozzle scale, the flow gradually approaches the thermodynamic equilibrium state. As the translational temperature increases, the local sound speed increases (see Table 4a), resulting in the decrease in the Mach number. Fig. 12 can also prove this. The larger the size, the larger the ratio of the exit uniform area radius to the exit radius, which will make the exit Mach number smaller. This is in contradiction with the assumption that the thicker the boundary wall of the nozzle wall and the smaller the Mach number under the same area ratio. It can be known by combining Fig.11 and Fig.12. The influence of thermochemical non-equilibrium scale effects on Mach number is more important than the viscous scale effects.
Cases 1, 3 and 4 in Table 1b are employed to study the effect of the variation throat diameter on the Mach number, as shown in Fig.13. When the throat diameter increases, the flow is close to the equilibrium state. More vibrational energy is transmitted to the translational energy, which makes the translational temperature rise, as deduced from the kinetic energy in Eq. (10). The change in Mach number due to ER and non-equilibrium effects is much more pronounced than the change in Mach number caused by ER alone.
Figs. 14 and 15 show the changes of O2, O and NO in the axial direction based on the HEG nozzle, respectively. The positions where these components change are the upstream of the freezing point, which is consistent with the changing trend of the vibrational temperature. The flow parameters depend on the dissociation of non-equilibrium before the freezing point. Tables 5a and 5b show the main flowfield parameters in the centerline of the nozzles exit, and the variation trend of the flowfield parameters can be seen. The nozzle flowfield parameters are monotonous to the absolute scale and throat diameter, such as pressure, temperature, sound velocity, Mach number, and species mass fraction. The larger the nozzle scale and throat diameter, the larger the mass fraction of the species N2 and O2, but the smaller the mass fraction of the species NO, O, N, NO+. The species N and NO+ are trace species, which are not listed here.
As can be seen from Table 5a, when the nozzle scale increases, the degree of the thermal non-equilibrium decreases and the frozen vibrational energy decreases, so that the kinetic energy of the airflow, namely the speed u∞, increases slightly.