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The navigation modules are located on the slopes of the wedges (Fig.2). Based on the assumptions on simplifying the side section of waverider hypersonic vehicle to a wedge structure in Section 1, the half wedge angle
$\theta $ could be set as 0.1359 radians. Different wedge angle values can be assigned for the other waverider wedge structures.Figure 2. Load layout[28] and wedge-shaped structure of preliminary vehicle
Furthermore, the following parameters were used in the simulations: the altitude of the vehicle as 20 km, specific heat ratio of the gas
$K = 1.41$ , medium density of the approaching flow in the flight area${\;\rho _1} = 0.088\;91\;{\rm{kg}}/{{\rm{m}}^3}$ ; angle of attack angle is 0°; atmospheric pressure of 5529.3 Pa; air temperature of 216.65 K.Using Eq. (9) and Eq. (10), the shock wave angle,
$\;\beta $ , and the ratio of the density of the media above and under the shock wave surface,$\;{\rho _2}/\;{\rho _1}$ , at different Mach numbers ($Ma = 5.0,5.5,6.0,6.5,7.0,7.5,8.0$ ) were calculated. The results are summarized in Table 1.Ma $\;\beta$/rad ${\rho _2}/{\rho _1}$ 5.0 0.3039 1.8491 5.5 0.2861 1.9436 6.0 0.2716 2.0393 6.5 0.2596 2.1359 7.0 0.2495 2.2330 7.5 0.2410 2.3301 8.0 0.2336 2.4270 Table 1. Shock angles and density ratios at different Mach numbers
The beam deflection angle,
$\Delta \eta $ , when light traveled through shock waves at different angles of incidence ($10^\circ ,20^\circ ,30^\circ ,40^\circ ,50^\circ $ ) were also obtained in Table 2. The angle is generally measured in degrees or radians in design and manufacture of aircrafts or sensors. In the process of celestial observations, the angle of beam deflection is usually measured with arcseconds. Therefore, arcsecond was used as the unit of beam measurement in the tables and figures below.Ma 10° 20° 30° 40° 50° 5.0 0.6040 1.2468 1.9778 2.8745 4.0825 5.5 0.6713 1.3856 2.1979 3.1943 4.5368 6.0 0.7394 1.5262 2.4209 3.5184 4.9971 6.5 0.8081 1.6680 2.6459 3.8454 5.4615 7.0 0.8771 1.8105 2.8720 4.1740 5.9282 7.5 0.9462 1.9532 3.0982 4.5028 6.3952 8.0 1.0152 2.0955 3.3239 4.8308 6.8611 Table 2. Deflection angles due to the shock wave surfaces at different Mach numbers and angles of incidence (Unit: ("))
As can be seen in Table 1, as the Mach number increases, the compression effect of the shock waves and the density ratio increase and the shock wave angle reduces gradually. The results reported in Table 2 and Fig.3 reveal that beam deflection becomes more significant as the Mach number and angle of incidence increase. At Mach numbers of 5-8, the deflection angles can be up to 6.8".
Airborne celestial attitude determination systems could provide an attitude solution with accuracy better than 1.0" RMS[29]. And the beam deflection was needed to be corrected to achieve such accurate for observation.
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Under hypersonic conditions, the experimental observation of shock wave structure is very difficult. The researches on the shock wave of hypersonic vehicle were mainly carried out with computational fluid dynamics (CFD) technology, which did not rely on simplified physical models, but used numerical algorithms with the help of computers to directly solve fluid flow functions. However, CFD processing requires huge computing resources, and the real-time performance is not good enough[30].
The shock wave structure and the light deflection caused by it could be calculated by analytical methods as Section 2. If the CFD solutions were used as the actual results, the error analysis of the theoretical results could be carried out to improve the accuracy of the theoretical analytical model.
The deviations of the theoretically calculated shock wave angles from the actual values will cause the deflection angles to vary. Therefore, CFD simulations at Mach numbers of 5–8 were performed using the defined parameter values. The results demonstrated that stable shock wave structures formed over the wedge under all the aforementioned conditions (Fig. 4). However, the measured shock wave angles obtained through CFD simulations differed from the theoretically calculated ones.
The measured shock wave angles were extracted from the CFD raster data. Table 3 presents the deviations of the simulated shock wave angles from the theoretical values at different Mach numbers.
5.0 5.5 6.0 6.5 7.0 7.5 8.0 $\Delta \beta $ 0.0357 0.0806 0.0328 0.0431 0.0048 0.0606 0.0193 Table 3. Measurement errors of shock wave angles at different Mach numbers (Unit: ("))
Under the same conditions, the measurement error,
$\Delta \;\beta $ , does not increase as the Mach number increases. Hence, further investigations are required to determine whether the simulation errors are random. -
According to the conclusion given in Section 4, when the measurement errors,
$\Delta \;\beta $ , and the angle of incidence,${\eta _1}$ , are smaller than${10^{ - 1}}$ ,$\Delta \;\beta $ has a negative and linear relation with the deflection angle.However, this is based on the assumption that
$Q' \approx Q$ ;$Q'$ and$Q$ were solved separately at different$\Delta \;\beta $ values from 0° to 5.0° at$Ma = 3,5,8$ . The similarity between$Q'$ and$Q$ was determined using$\left| {(Q' - Q)/Q} \right|$ , and the results are presented in Table 4.Ma 5.0 6.0 7.0 8.0 0° 0.000 0.0000 0.0000 0.0000 0.5° 0.1890 0.2231 0.2528 0.2777 1° 0.3759 0.4427 0.5002 0.5479 1.5° 0.5606 0.6585 0.7420 0.8103 2.0° 0.7428 0.8703 0.9779 1.0648 2.5° 0.9225 1.0781 1.2080 1.3114 3.0° 1.0995 1.2816 1.4320 1.5500 3.5° 1.2737 1.4808 1.6499 1.7808 4.0° 1.4450 1.6756 1.8618 2.0038 4.5° 1.6134 1.8660 2.0676 2.2191 5.0° 1.7788 2.0519 2.2673 2.4269 Table 4. Process parameters
$\left| {(Q' - Q)/Q} \right|$ at different Mach numbers (${10^{ - 4}}$ )From the table, it can be seen that the values of
$\left| {(Q' - Q)/Q} \right|$ are in the order of${10^{ - 4}}$ under all conditions. Therefore, the assumption that$Q' \approx Q$ stated in Section 3 is valid. -
The shock wave structure and the light deflection caused by it could be calculated by analytical methods as Section 2. If the CFD solutions were used as the actual results, the error analysis of the theoretical results could be carried out to improve the accuracy of the theoretical analytical model.
In the range of
$\Delta \;\beta $ considered in this study, the error propagation,$\Delta {\eta _{\Delta \;\beta }}$ , can be calculated in different ways. Direct Numerically Calculation can be performed using Eq. (6), and Rapid Calculation Model from Eq. (19) can be used for rapid approximation results.From the figures below, it can be seen that
$\Delta {\eta _{\Delta \;\beta }}$ and$\Delta \;\beta $ generally have a negative linear relationship. When the angle of incidence is smaller than$50^\circ $ , the errors estimated via rapid approximation and direct numerical calculation are similar. However, as the angle of incidence,${\eta _1}$ , increases,${c_1} = \left(\dfrac{Q}{2} - \dfrac{{{Q^3}}}{2}\right)\eta _1^2 + \dfrac{{5{Q^3}}}{{18}}\eta _1^4 - Q$ in Eq. (17) could no longer be approximated to$ - Q$ . Therefore, the results obtained using Rapid Calculation Model started to deviate from those obtained through the Direct Numerically Calculation.If an equation similar to
$\Delta {\eta _{\Delta \;\beta }} \approx c + {c_1}\Delta \;\beta + {c_2}{(\Delta \;\beta )^2}$ in Eq. (17) is used to fit the relationship between$\Delta {\eta _{\Delta \;\beta }}$ and$\Delta \;\beta $ at a certain angle of incidence${\eta _1}$ , as shown in Fig.5(a), then the above expression becomes$\Delta \eta = a + {a_1}\Delta \;\beta + {a_2}{(\Delta \;\beta )^2}$ . Figure 6 shows the differences between the curve parameters used for the Direct Numerically Calculation solutions ($a,{a_1},{a_2}$ ) and those used Rapid Calculation Model ($c,{c_1},{c_2}$ ) for different${\eta _1}$ .The results indicate that when the angle of incidence is within
$50^\circ $ , there are small differences between${a_1}$ and${c_1}$ , which are the parameters that critically influence$\Delta \;\beta $ . The relationship between$\Delta \;\beta $ and$\Delta \eta $ can be expressed effectively using Rapid Calculation Model from Eq. (19). In contrast, when the angle of incidence exceeds$50^\circ $ , the two parameters differ remarkably, and the approximation of Eq. (19) is no longer suitable.