The aerodynamic characteristic analysis of MOVE-III involves two configurations: before and after the deployment of the sail. Both DSMC and FMF were employed in this analysis, and an assessment was conducted to compare their agreement. The suitability of FMF in predicting the aerodynamic effect under LEO conditions was also evaluated.
4.1. Numerical Setup
To evaluate the impact of the drag sail on the aerodynamic characteristics of the CubeSat, it is imperative to construct computational models of two corresponding MOVE-III configurations, namely, pre- and post-sail-deployment.
For pre-sail-deployment scenarios, MOVE-III exhibits an identical configuration to that used when conducting scientific endeavors in orbit. Nevertheless, the actual complete assembly is too intricate to be incorporated into the SPARTA kernel of the DSMC method. As a result, the assembly needs to be simplified (see
Figure 1 in
Section 2). In this study, the surface details of the CubeSat are disregarded, while substantial elements, such as the deployable antenna and solar panel, are preserved. The panel construction of the CubeSat without a sail is shown in
Figure 5.
The main frame of MOVE-III has a size of 6U. Prior to its launch into orbit, the CubeSat was housed within the rocket’s payload adapter, and this configuration is referred to as the launch envelope, which denotes the size of MOVE-III’s frame. Once fully deployed in orbit, the movable components are extended to form the orbit envelope [
24].
In the case of post-sail-deployment scenarios, the scale of the characteristic length is determined by the span of the sail. Consequently, the body structure can be simplified to flat boxes. In the simulation cases presented in this study, the thickness of the sail has a negligible impact on the results. Therefore, to avoid introducing multi-dimensional issues and to ensure the fitness of the mesh, a sail thickness of 1 mm is assumed. It should be noted that the asymmetry stemming from the deployed solar panel is also taken into account. Panel constructions of the CubeSat with both sail sizes are shown in
Figure 6.
The dimensions of MOVE-III before and after sail deployment are shown in
Table 1. Note that
,
, and
are the
x-,
y-, and
z-dimensions, respectively;
is the number of vertices; and
is the number of panels.
At the orbital altitudes of interest, free-stream velocities are assumed to be the perfect-circle orbital velocity (see
Table 2),
, where
is the gravitational constant,
is the mass of the Earth,
is the average radius of the Earth, and
h is the orbital altitude. Static temperatures, densities, number densities, and compositions of the incident flow are taken from the US Standard Atmosphere (1976) [
34]. The GSI condition is assumed to be 100% diffuse.
To examine the effect of attitude on
, this study sets the angle of attack
and angle of sideslip
to 0, 30, and 60 degrees. Thus, a total of nine attitudes are produced and labeled in
Table 3. A case number consists of two digits. The former indicates
, and the latter indicates
. The digits 0, 3, and 6 represent 0, 30, and 60 degrees, respectively. In this study, the surfaces of the test component are kept fixed, while free streams flow in with varying velocity vectors.
For MOVE-III without a deployed sail, the surface areas of the main structure and the deployed antenna have magnitudes of 100 and 1 mm, respectively. The simulation domain is 1 m × 1 m × 1 m, and the mesh size is 4.5 × 10
−3 m (see
Figure 7). For MOVE-III with a deployed drag sail, the simulation domain is 6 m × 6 m × 6 m for 4 m
2 sail cases and 8 m × 8 m × 8 m for 6.25 m
2 sail cases. The mesh size is 4 × 10
−2 m. In both scenarios, the surface temperature of the CubeSat is 350 K [
35].
To confirm the flow regime, Knudsen numbers of the flow for MOVE-III with and without the drag sail are examined. The characteristic length of MOVE-III before deploying the sail is 0.2 m, which increases to 2.087 m after deploying the 4 m
2 sail and 2.609 m after deploying the 6.25 m
2 sail. The Knudsen numbers are then computed and presented in
Table 4.
4.2. Simulation Results and Aerodynamic Coefficients
DSMC simulations were performed based on the geometry and atmospheric data discussed in
Section 2 and
Section 4.1. The velocity fields of three selected attitudes at 125 km are shown in
Figure 8 and
Figure 9.
At orbital altitudes of 185 km and 300 km, cases of MOVE-III without deploying the sail and after deploying two sail sizes of 4 m
2 and 6.25 m
2 were simulated, respectively. The attitude of the CubeSat was set to be
and
. The effect of utilizing different sail sizes on the drag force and
is shown in
Table 5.
By comparing the data in
Table 5, it is observed that the shape of the CubeSat with the deployed drag sail is the dominant factor, while the influence of sail size is relatively minor. However, it is evident that a 56% increase in sail size results in a 55% increase in drag force, leading to a greater deceleration of the CubeSat with the larger sail. Upon the installation of the 4 m
2 sail, the aerodynamic drag force acting on MOVE-III increases by a factor of 57 at an orbital altitude of 185 km and by a factor of 85 at 300 km compared to the case without the sail, leading to a significant reduction in deorbiting time. A further quantitative analysis of the effect of the drag sail on the deorbiting time is presented in
Section 5.
A further analysis is presented to juxtapose the drag force generated by the drag sail with the thrust force of widely utilized electric thrusters, as introduced in
Section 1. Given that electric thrusters represent an alternative approach to the drag sail technique, this study undertook a comparative evaluation of the drag forces derived for the two higher-altitude scenarios examined herein against the thrust forces produced by standard ion thrusters [
11]. The 4 m
2 drag sail is capable of generating drag forces of 5.1 mN at an orbital altitude of 300 km and 0.4 mN at 450 km. Considering that vacuum arc, pulsed plasma, and microwave thrusters typically yield thrust forces of 0.01, 0.04, and 3 mN with system sizes of approximately 0.1, 1, and 5 U and weights of 0.3, 2, and 26 kg, respectively [
36], the deorbiting capability of the 4 m
2 sail is evidently more effective than that of electric thrusters with similar dimensions and weights.
4.3. Comparison to FMF Theory
The FMF analysis of MOVE-III with and without the 4 m
2 drag sail for all nine attitudes listed in
Table 3 is based on the parameters set in
Section 4.1. Upon collecting simulation result data, the errors of FMF based on the DSMC method were readily computed (see
Table 6). According to
Figure 10, FMF shows good consistency for vertical incidence cases with DSMC results for altitudes larger than or equal to 185 km. However, for large-incident-angle cases, considerable deviations from zero-angle cases are observed. Note that Cases 00 and 63 at an altitude of 240 km were simulated as a complement. Atmospheric data were also obtained from the US Standard Atmosphere (1976) [
34]. Analyzing
Table 6, the FMF and DSMC results at 185 km and 300 km coincide well for Cases 00, 03, 06, and 60. The errors for these cases are all no more than 5%. Cases 30 and 33 possess errors of less than 10%. Large deviations (
) are found for Cases 36, 63, and 66. By cross-checking the “angle-swap” pairs, Case 03 with 30, Case 06 with 60, and Case 36 with 63, the
of each case group is found to be different due to the asymmetric geometry of MOVE-III. Hence, the accuracy of FMF also relies on the configuration of the test part’s surface.
GSI models of FMF only account for one-time particle–surface collisions and neglect multi-time particle–surface collisions and inter-molecular collisions. By inspecting the flow field streamlines near the surface (see
Figure 8 and
Figure 9), it is discovered that the translating directions of particles are changed near the surface and vary at different locations according to the geometry. As the incident angle increases, the bottom and lateral surfaces, which are shaded in zero-incident-angle cases, become exposed to the incoming flow. The normal vectors of the surface elements now have three non-zero components in the aerodynamic reference frame, thereby increasing the geometry concavity influence. Inter-molecular collisions should exist, and the basic assumption of GSI models no longer applies. Consequently, this leads to discrepancies in the FMF predictions.
Furthermore, FMF is a valuable tool for estimating the
of LEO spacecraft under small free-stream incident angles. This method relies on analytically solving equations that are obtained through appropriate simplification assumptions. A computer only requires the input of relevant parameters (e.g., atmospheric and solar conditions) to calculate the values, which makes this a relatively fast and resource-efficient process. In contrast, no general solution to the Boltzmann Equation has been developed thus far. The DSMC method involves a reduction of real-world physical phenomena and a step-by-step reproduction of them. This process consumes a significant amount of computational resources and time. Researchers have acknowledged the efficiency advantage of the FMF method and the accuracy advantage of the DSMC method and are currently investigating ways to overcome the limitations of the DSMC method and to combine the advantages of both approaches [
37].