Research Article | Open Access
Lulu Jiang, He Jia, Xin Xu, Wei Rong, Wei Jiang, Qi Wang, Gang Chen, Xiaopeng Xue, "Numerical Study on Aerodynamic Performance of Mars Parachute Models with Geometric Porosities", Space: Science & Technology, vol. 2022, Article ID 9851982, 13 pages, 2022. https://doi.org/10.34133/2022/9851982
Numerical Study on Aerodynamic Performance of Mars Parachute Models with Geometric Porosities
The supersonic flows around rigid parachute-like two-body configurations are numerically simulated at Mach number of 1.978 by solving three-dimensional compressible Navier-Stokes equations, where the two-body model consists of a capsule and a canopy, and a geometric structure (i.e., gap) is located on the canopy surface. The objective of this study is to investigate the effects of gaps with different porosities and positions on the aerodynamic performance of supersonic parachute. The complicated periodic aerodynamic interactions between the capsule wake and canopy shock occur around these two-body models. From the formation of canopy shock and drag coefficient variation, the cycled flow structures can be divided into three types:(1) narrow wake period, (2) open wake period, and (3) middle wake period. In addition, it was found that the geometric gaps have no obvious influences on the flow modes. However, compared with models with different gap positions, the two-body model with an upper gap (gap is close to the canopy vent, UG model) has a smaller drag coefficient fluctuation and better lateral stability. On the other side, the increase of porosity has a more significant impact on UG models.
Parachute-like two-body configurations are widely used for decelerating the spacecraft into the atmospheres during the deep space exploration missions . During the Mars exploration mission, a supersonic parachute is always employed to undertake dual tasks: slowing the falling speed and providing stability for the Mars rover . To date, only the disk-gap-band (DGB) parachute has been used successfully to provide good aerodynamic performances in the Martian atmosphere . Maynard firstly conducted the supersonic wind tunnel tests to explore the supersonic aerodynamic performance of parachutes  and found that the drag coefficient of the parachute mainly depends on the canopy porosity, Mach number, capsule size, and distance between the capsule and canopy [3, 4]. In the recent Mars Science Laboratory mission, it was further found that in supersonic environments, the shock formed ahead of the canopy interacts with the wake from the capsule; these highly unsteady flow fields cause severe drag coefficient fluctuation and structure instability , which mainly depends on the Mach number, Reynolds number, angle of attack, canopy-to-capsule size and proximity, and material properties of the canopy and cables [1, 5–8].
However, among the dependent parameters of the aerodynamic interactions around a supersonic parachute-like two-body system, canopy porosity has not always been fully considered , especially in numerical simulations solving supersonic parachute problems. Canopy porosity consists of two main components: the fabric permeability and the geometric porosity. Canopy fabric permeability is defined as the volumetric flow rate of air per unit fabric area under a specified differential pressure; the fabric permeability dynamically changes with the unsteady flow fields and the corresponding pressure difference between the canopy inner and outer surfaces. Hence, the determination of fabric permeability in parachute experiments is very complicated. Heinrich  proposed that the fabric permeability can be transformed to an effective porosity to determine its contribution to the total porosity of a parachute. The expected relationship between the aerodynamic parameters and total porosity of a parachute has been confirmed in wind tunnel tests . The geometric porosity generally refers to the ratio of the open area in the canopy material to the total material area . It was reported that in a low-density environment such as Martian atmosphere, the geometric porosity might contribute significantly to the aerodynamic performances of the parachutes, and the effect of the fabric permeability on the aerodynamic behavior was smaller .
With the advancement of exploration missions, the disk-gap-band parachute will be difficult to use for future missions due to its size limit . One option considered was a modified supersonic ringsail parachute . The geometric porosity of the disk-gap-band parachute is mainly obtained by a single gap, while the porosity of the ringsail parachute is evenly distributed on the entire canopy surface through two gaps and many seams. In previous studies, it was suggested that a parachute with lower fabric permeability is easier to inflate at low-density conditions . However, in NASA’s Low-Density Supersonic Decelerator (LDSD) program tests, the supersonic ringsail (SSRS) parachute and supersonic disksail (SSDS) parachute with lower fabric permeability and higher geometric porosity failed to inflate normally. On the contrary, the disk-gap-band parachute with higher fabric permeability and lower geometric porosity (similar total porosity with SSRS or SSDS) succeeded in the Advanced Supersonic Parachute Inflation Research Experiments (ASPIRE) project . The failure of the SSRS or SSDS parachute may be closely related to its complicated geometric porous structures such as gaps and seams. However, there are very few researches on the detail investigations of unsteady flow fields and aerodynamics induced by the geometric porous structures of SSRS or SSDS parachute. This sets up the motivation of this numerical study. The present paper is aimed at studying the effects of geometric porosity on the flow structures and aerodynamic characteristics of the derivative model of SSDS parachute in the supersonic environment.
2. Parachute-Like Two-Body Models
The rigid parachute-like two-body models consist of a conical capsule with a diameter and a hemispherical canopy with a diameter ; the thickness of canopy is defined as . The capsule model is consistent with the Mars Science Laboratory probe model . The schematic diagram of the original parachute-like two-body model used in this study is shown in Figure 1, in which the canopy model (see Figure 2) is designed from the SSDS model in the NASA’s LDSD flight tests , and its size is shown in Table 1. In this study, is the axial distance from the capsule frontal surface to the inlet of the canopy, is the diameter of the frontal surface, and is the trailing distance from the frontal surface to the inlet of the canopy, and the diameter ratio was 0.2045. The porosity generally refers to the ratio of the open area in the canopy material to the total material area and is also called the “geometric porosity” . For 2D parachute-like two-body models, the canopy porosity refers to the ratio of the arc length of the gap to the total arc length of the canopy. The vent porosity () is 4.88%, and is the whole geometric porosity of the canopy including the vent and gap. SSDS has a designed geometric porosity of 13.69%.
Notably, because the SSDS or SSRS parachute used in the LDSD program  has a very complex geometrical porosity structure, including the vent, two gaps, and many seams, the effects of all the geometric porous structures would be much complicated; as the first step, this paper will focus on the investigation of the different position and geometric porosity of the gap on the aerodynamic performances of parachute-like two-body configurations. There are two gaps with different geometric porosity on the SSDS canopy surface, which are located at approximately 1/3 and 2/3 of the height of the canopy (from the mouth to top), respectively. In this study, a single gap with different porosity and position will be designed and investigated, a gap located 2/3 of the height of the canopy is named as upper gap (UG) model, and a gap located 1/3 of the height of the canopy is named as lower gap (LG) model. In addition, the gap is designed to be 5%, 10%, ad 15% of the arc length of the rear body, named porosity types 5, 10, to 15, respectively.
Previous studies on parachute models suggest that the flow patterns and aerodynamic performance of complicated geometric configurations can be computed relatively quickly and accurately with the assumption of a rigid, 2D geometry. To preliminarily study the relationship between geometrical porosity and aerodynamic characteristics of supersonic parachute system, this study was carried out using a 2D model and here verified using a 3D model. The drag coefficient is used as Equation (1). is the standard deviation (Equation (3)), means median absolute deviation (Equation (4)), and these two quantities can be used to describe the degree of data dispersion. For standard deviation, the square of the distance from the data to the mean value is used. A larger deviation has a larger weight, and the influence of outliers on the result cannot be ignored. For , a small number of outliers do not affect the results of the experiment.
It can be seen from the comparison results (Table 2) that the differences of the drag coefficient and its variation of 2D model and 3D model are reasonable, and the difference between them is 7.68%. And the flow mode and instantaneous flow fields are consistent with each other. Therefore, the 2D model can be employed for the further detailed study in this paper.
3. Freestream Conditions and Numerical Methods
3.1. Freestream Conditions
The Martian atmospheric condition used in this study can be computed by Equation (5) from NASA [14–18]. The velocity and altitude refer to the initial values of stable descent stage. Martian atmospheric conditions
is the height from the Martian surface, and , , and are the freestream pressure, density, and temperature, respectively. In addition, because the Martian atmosphere consists mainly of CO2, is employed for Martian environment cases. The freestream values are shown in Table 3.
3.2. Numerical Methods
The supersonic flows around the parachute-like two-body model are solved numerically by the compressible N-S equations based on the finite volume method. The HLLC Riemann solver is used to evaluate second-order inviscid flux terms , and the viscous terms are calculated by the 2nd-order central differencing scheme. The coefficient of viscosity is computed according to Sutherland’s law. The time advancement was performed by a second-order point-implicit solver with dual time stepping . The dimension time step was set to be to capture the unsteady flow fields. For the boundary conditions, because the rear body (canopy) is regarded as a rigid wall without fabric permeability in this study, the body surface is set as an adiabatic wall with no-slip, the freestream values are set as the far field conditions, and the outlet flow field parameters are obtained by center extrapolation of the computational domain. In addition, the laminar flow model was used in previous studies [17, 23–25], where the simulation results were in good agreement with that from the wind tunnel tests, and the laminar model is still used for the present numerical simulations.
Supersonic wind tunnel test data of SSDS parachutes is lacking, and the flight tests also failed. There are few studies on the aerodynamic characteristics of the SSDS or SSRS parachute under the Martian environment. In the previous study , the same numerical methods have been used and validated, where the disk-gap-band parachute of Mars Science Laboratory (, ) is employed under the supersonic conditions; the simulation results were in good agreement with those obtained from the reference data . This shows that the numerical methods are effective and reliable.
Figure 3 shows the parachute-like two-body grid, the capsule and canopy parts are considered as the adiabatic wall with no-slip, where Lindgren Boundary Layer profiles are considered, and the wall is set as wall function. Coarser, medium, and finer meshes were selected to conduct the validation of the grid independence. The comparisons of canopy drag coefficients are shown in Figure 4.
As shown in Figure 4, the variations of canopy drag coefficient of the three grids present obvious periodicity. It is obvious that the resolution of the coarse grid is not enough to capture some unsteady flows. By and large, the time histories of canopy drag coefficients are almost identical in terms of drag coefficient amplitude and time period. However, it seems that the resolution of the coarse grid is insufficient to capture the appropriate unsteady flow. The results of fine mesh and medium mesh are in reasonable agreement.
4. Results and Discussions
The previous studies show that different trailing distance ratios () and the capsule/canopy diameter ratios () cause different flow modes of the two-body system [1, 23]. With the increasing of trailing distance ratio, the flow modes would go through pulsation mode, oscillation mode, and wake/shock interaction mode [1, 23]. In this study, the flow mode is wake/shock interaction mode for the given and .
When the canopy has no gap (NG model), the time variation of canopy pressure is shown in Figure 5. According to the position of the high-pressure region and the flow field characteristics around the parachute system (Figure 6), the unsteady flow fields around the two-body configuration in a time period can be divided into four stages. (1) Stage from time to : the shock wave ahead of the canopy is weak, the canopy shock interacts with the open wake from the capsule, and the pressure inside the canopy is lower at time ; when the wake becomes narrow at time , the pressure inside the canopy becomes higher again, and the canopy shock starts to move back to the canopy mouth. (2) Stage from time to : the canopy shock becomes stronger, which interacts with the narrow wake, and this leads to the higher pressure inside the canopy. (3) Stage from time to F: the canopy shock becomes much stronger, the canopy shock/capsule wake also becomes stronger, and the aerodynamic interaction moves laterally and backward to the upstream due to the higher pressure inside the canopy. (4) Stage of time : the aerodynamic interaction of canopy shock and wake moves far away from the canopy mouth and becomes weaker, and the pressure inside the canopy also decreases significantly.
4.1. Aerodynamics of LG5 Model
“LG5” represents a canopy with a lower gap and a geometric porosity of 5%. The time variation of the axis force (i.e., direction) of the LG5 model is shown in Figure 7, where eight periods are chosen here for the detailed studies. Notably, the change trend of the flow mode of LG5 two-body model is basically consistent with that of the NG (no gap) model. However, due to the geometric porosity caused by the lower gap, some differences mainly occur in the second stage; i.e., the stronger canopy shock interacts with the narrow capsule wake. These eight periods can be divided into three types according to the time variations of canopy pressure (shown in Figure 7). (1)“Narrow wake” type: in this type, the capsule narrow wake ()/the canopy shock interaction is observed (refer to Figure 8). T5 and T6 belong to this type. In this period type, the symmetric bow shock waves in front of the canopy are stronger, which interact with the much more narrow wake; this leads to the local high-pressure area inside the canopy. Comparing the time variations of drag coefficient of the NG model (Figure 5), it can be seen that during the whole cycle, the curves are characterized by higher pressure at peaks , , and and lower pressure at peaks and (Figure 9). Notably, the pressure difference between the higher and lower peaks is big, and the decrease is approximately 20%(2)“Open wake” type: in this type, the capsule wake has no close state in the period () (Figure 10). The weak canopy bow shock wave interacts with the open wake; the pressure fluctuation inside the canopy decreases significantly (Figure 11). However, the aerodynamic interaction zone is relatively close to parachute-like two-body model; the obvious higher pressure peak occurs at times and (3)“Middle wake” type: in this type, the width of the capsule wake is approximately equal to the diameter of the capsule (Figure 12). There may be multiple shock waves generated in front of the canopy, while the shock wave intensity is stronger than that in the “open wake” type. Seeing Figure 13, the canopy pressure is generally higher at the early time of the period (i.e., time ). The largest pressure in the period occurs at time , which reveals that the aerodynamic interaction of the canopy shock and capsule wake is much stronger at this time, and the high pressure inside the canopy forms the quickest; however, it also causes the aerodynamic interaction zone move upstream and expand laterally at an earlier time
Finally, the mean value and fluctuation of the drag coefficient and the lateral force coefficient of the canopy in different periods are compared in Figure 14, the mean value of canopy drag coefficient of “narrow wake” periods (i.e., T5 and T6) is higher, and their fluctuation range is a little larger. The drag performance of “open wake” period (T8) is poor, and its fluctuation is the smallest. However, for the “middle wake” periods, the mean canopy drag coefficient of T3 and T7 with wider wake is lower, and its fluctuation is also small. On the contrary, the mean drag coefficient and its fluctuation of T1, T2, and T4 periods are much higher. Among all the period types, the drag performance of T5 and T6 is the best. On the other hand, seeing Figure 14(b), the “middle wake” periods (T1-T4) show the better lateral force performance.
Figure 15 shows the distribution histogram of the drag coefficients of the canopy in different periods, which are computed by Equation (6). The median of the “narrow wake” periods is right-biased to the center, while that of the “open wake” period is left-biased. The median drag coefficient of “middle wake” periods is slightly close to the right of the median, and the degree of right deviation increases with the increase of wake closure.
4.2. Effects of Geometric Porosity and Position
In order to analyze the influence of porosity and gap positions of different structures on the drag and stability performance of canopies, five types of porosity models NG (no geometric porosity), UG5 (upper gap model, 5% geometric porosity), UG10 (upper gap model, 10% geometric porosity), UG15 (upper gap model, 15% geometric porosity), LG5 (lower gap model, 5% geometric porosity) LG10 (lower gap model, 10% geometric porosity), and LG15 (lower gap model, 15% geometric porosity) are designed and investigated.
Considering that FFT analysis is carried out for the drag coefficient and lateral force coefficient of the canopy in multiple periods, the spectrum graph will be complicated and not convenient for analysis, so low-amplitude low-frequency signals are filtered. The FFT changes of the canopy drag coefficients filter the signal whose amplitude is less than 35% of the maximum value (i.e., the filtered amplitude is zero, refer to Figure 16(a)), and the FFT changes of the canopy lateral force coefficients filter the signal whose amplitude is less than 40% of the maximum value (i.e., the filtered amplitude is zero, refer to Figure 16(b)).
For the NG model (Figure 16), the drag coefficient presents an obvious periodicity (8 periods are taken for FFT analysis). It can be observed that the drag coefficient shows an obvious frequency (i.e., time period); however, the lateral force coefficient exhibits very complicated frequency distribution without an obvious periodic change, which would cause severe lateral instability.
The FFT analysis of drag and lateral force coefficient of different gap positions and porosity canopy models is shown in Figure 17 (8 periods are taken for FFT), and time variation of drag coefficient is shown in Figure 18. By comparing the spectrum diagram of drag coefficient, it can be seen that basically the drag coefficients of LG models change more cyclically with the larger amplitude. However, the amplitudes of LG models decrease with the increase of its geometric porosity, and the periodicity of LG models also becomes weak. On the contrary, the drag coefficients of UG models show more irregular change and become worse as the geometric porosity increases. In addition, comparing with the NG model, it can be observed that the periods of drag coefficient becomes more complicated and weaker. Consequently, the LG models have more significant periodical drag performance than UG models, and the increase of porosity could cause the periodic change of drag coefficient weaken for “gap” models.
By comparing the spectrum diagram of lateral force coefficient of different canopy porosity models, it can be seen that the lateral force of UG models has more complicated frequencies than that of LG models; as the porosity is increasing, the frequency distributions of UG models become less. This reveals that upper gaps have more effects on the lateral instability of parachute-like two-body model. In addition, compared with the NG model, it can be found that the lower frequency distribution becomes less for the “gap” models. Moreover, with the increase of porosity, the lateral force performance of canopy models is improved, and UG models can provide a bigger effect on improving the lateral force performance.
Figure 19 shows the pressure contours and Mach contours of different porosity models. It can be seen that with the increase of porosity, the canopy shock stand-off distance decreases and the shock comes closer to the canopy body, and the shock wave angle becomes bigger. Additionally, the increase of porosity makes the gap shock stronger, especially for the UG models, which causes the canopy wake become much more unstable; accordingly, the low pressure area behind the canopy becomes smaller with the increase of porosity, which leads to a larger drag coefficient. In addition, it was also observed that the low pressure in the canopy wake of UG models is much smaller than that of LG models, which also causes a larger drag coefficient. Furthermore, it can be seen that with the increase of porosity, especially for the UG models, the flow fields inside the canopy body becomes more unstable, which leads to poor stability performance, i.e., larger lateral coefficient.
As shown in Figure 20, the canopy drag and lateral coefficients of all porosity models are dimensionless by the corresponding amount of NG model. According to the analysis of drag coefficients of canopies with different porosity (Figure 20(a)), it can be seen that the increase of porosity has a more significant impact on UG models, and the drag coefficients reduce more obviously. The mean value and median absolute deviation of canopy drag coefficients of LG models decrease monotonically with the increase of porosity. These parameters of UG models with a geometric porosity of 10% show the lowest values. In the case of the smaller geometric porosity, the median absolute deviation and standard deviation of the canopy drag coefficient of UG models are both smaller than those of LG models; that is, the drag coefficient fluctuation of the UG model is smaller, but its skewness (Equation (7)) is larger than that of the LG model, which means the distribution of canopy drag in the periods is more asymmetric during the whole period time.
Figure 20(b) shows the analysis of the canopy lateral force coefficient of the different porosity models. It can be seen that the standard deviation, median absolute deviation, and skewness of UG and LG models all decrease with the increase of porosity, which reveals that the increase of porosity will lead to the decrease of the canopy lateral force fluctuation. The lateral force coefficients of LG and UG models increase with the increase of porosity, and the lateral force coefficients of LG models have a larger increase than those of the UG model at the same geometric porosity. Consequently, from the comparison of the lateral force fluctuation and deviation, UG models show better lateral stability than LG models.
The supersonic flow over parachute-like two-body models with different gaps under Martian atmospheric conditions was numerically studied at a freestream Mach number of 1.978. The effects of the geometric porosity and gap positions on the flow behavior and aerodynamic characteristic of the two-body models are investigated in detail. The results obtained in this study can be summarized as follows: (1)According to the time-varying curve of canopy drag coefficient of canopy models, the flow fields around two-body models can be divided into three types: (1) “narrow wake” period, (2) “open wake” period, and (3) “middle wake” period. The drag performance of “narrow wake” period is the best. On the other hand, the “middle wake” periods show the better lateral force performance(2)Compared with models at different gap positions, the drag coefficients of LG models show more significant periodical change than those of UG models; UG models have small fluctuation of canopy drag coefficient, but its distribution is more asymmetric. UG models can provide a bigger effect on improving the lateral force performance(3)The increase of porosity has a more significant impact on UG models (with upper gaps), and the drag coefficients reduce more obviously. The mean value and median absolute deviation of canopy drag coefficients of UG models with a geometric porosity of 10% show the lowest values. The mean value and median absolute deviation of canopy drag coefficients of LG models decrease monotonically with the increase of porosity. In addition, from the comparison of the lateral force fluctuation and deviation, the increase of porosity would lead to the decrease of the canopy lateral force fluctuation. UG models show better lateral stability than LG models
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
All authors declare no possible conflicts of interests.
L.L. Jiang performed the numerical simulations and the data analysis and accomplished writing of the manuscript; Xin Xu participated in the numerical simulations and data analysis; He Jia, Wei Rong, Wei Jiang, Qi Wang, and Gang Chen participated in the research design and guided the data analysis; Xiaopeng Xue guided the numerical simulations, data analysis, and writing of the manuscript.
This work was substantially supported by the National Natural Science Foundation of China (Grant No. 12072377 and No. 11702332) and the Natural Science Foundation of Hunan Province, China (Grant No. 2022JJ30678). This work was also partly supported by the Laboratory of Aerospace Entry, Descent and Landing Technology (Grant No. EDL19092126).
- X. P. Xue and C. Y. Wen, “Review of unsteady aerodynamics of supersonic parachutes,” Progress in Aerospace Sciences, vol. 125, p. 100728, 2021.
- J. Lingard and M. Darley, Simulation of parachute fluid structure interaction in supersonic flow, AIAA Paper, 2005.
- J. Maynard, “Aerodynamics of decelerators at supersonic speeds,” AIAA Proc. Recovery Space Vehicles Symp., pp. 48–54, 1960.
- J. Maynard, “Aerodynamic characteristics of parachute at Mach numbers from 1.6 to 3,” Technical Report TN-D-752, NASA, 1961.
- A. Sengupta, R. Kelsch, J. Roeder, M. Wernet, A. Witkowski, and M. Kandis, “Supersonic performance of Disk-Gap-Band parachutes constrained to a 0-degree trim angle,” Journal of Spacecraft and Rockets., vol. 46, no. 6, pp. 1155–1163, 2009.
- A. Sengupta, A. Steltzner, A. Witkowski, G. Candler, and C. Pantano, Findings from the supersonic qualification program of the Mars Science Laboratory parachute system, AIAA Paper, 2009.
- A. Sengupta, Fluid structure interaction of parachutes in supersonic planetary entry, AIAA Paper, 2011.
- A. Sengupta, M. Wernet, and L. Hall, Temporal characteristics of Disk-Gap-Band parachute from Mach 2 to 2.5, AIAA Paper, 2013.
- H. G. Heinrich, “The effective porosity of parachute cloth, in: Zeitschrift Fur Flugwissenschaften,” vol. 11, Verlag Frier. Vieweg & Sohn, Braunschweig, 1963.
- R. Lafarge, J. Nelsen, and K. Gwinn, A novel CFD/structural analysis of a cross parachute, AIAA Paper, 1994.
- E. G. Ewing, H. W. Bixby, and T. W. Knacke, Recovery System Design Guide, AFFDL-TR78-151, 1978.
- C. L. Tanner, I. G. Clark, J. C. Gallon, and T. P. Rivellini, Aerodynamic characterization of new parachute configurations for low-density deceleration, AIAA Paper, 2013.
- J. C. Gallon, I. G. Clark, and A. Witkowski, “Parachute decelerator system performance during the low-density supersonic decelerator program’s first supersonic flight dynamics test,” in 23rd AIAA Aerodynamic Decelerator Systems Technology Conference, Daytona Beach, FL, United States: AIAA 2015-2130, 2015.
- I. G. Clark, M. Alder, and T. P. Rivellini, “Development and testing of a new family of supersonic decelerators,” in 22nd AIAA Aerodynamic Decelerator Systems Technology Conference, Daytona Beach, Florida, United States: AIAA 2013-1252, 2013.
- J. Gallon, A. Witkowski, I. G. Clark, T. Rivellini, and D. S. Adams, “Low density supersonic decelerator parachute decelerator system,” in AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar, Daytona Beach, Florida, 2013.
- L. L. Jiang, M. Y. Lin, X. P. Xue, J. He, W. Rong, and Q. Wang, “Numerical study on aerodynamic characteristic of supersonic parachute system under different atmospheric conditions,” Spacecraft Recovery & Remote Sensing, vol. 41, no. 6, pp. 77–89, 2020.
- X. P. Xue, H. Jia, W. Rong, Q. Wang, and C. Y. Wen, “Effect of Martian atmosphere on aerodynamic performance of supersonic parachute two-body systems,” Chinese Journal of Aeronautics, vol. 35, no. 4, pp. 45–54, 2022.
- G. H. Shen, Y. Q. Xia, and H. R. Sun, “A 6DOF mathematical model of parachute in Mars EDL,” Advances in Space Research, vol. 55, no. 7, pp. 1823–1831, 2015.
- L. Xu and K. J. Zhang, “Three dimensional compressible viscous flow computation based on HLLC Riemann solver and overset grids,” Chinese Journal of Applied Mechanics, vol. 6, pp. 1025–1030, 2015.
- M. Technologies, CFD++ User Manual, 2020.
- L. L. Jiang, Numerical study of porosity and aerodynamic characteristics of Mars supersonic disksail parachutes, Changsha: Central South University, 2021.
- M. Barnhardt, T. Drayna, S. Nompelis, G. Candler, and W. Garrard, “Detached eddy simulations of the MSL parachute at supersonic conditions,” in 19th AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar, Williamsburg, VA: AIAA 2007-2529, 2007.
- X. P. Xue, Y. Nishiyama, Y. Nakamura, K. Mori, and C. Y. Wen, “Parametric study on aerodynamic interaction of supersonic parachute system,” AIAA Journal, vol. 53, no. 9, pp. 2796–2801, 2015.
- X. P. Xue, Y. Nakamura, K. Mori, C. Y. Wen, and H. Jia, “Numerical investigation of effects of angle-of-attack on a parachute-like two- body system,” Aerospace Science and Technology, vol. 69, pp. 370–386, 2017.
- X. P. Xue, H. Koyama, and Y. Nakamura, “Numerical simulation of supersonic aerodynamic interaction of a parachute system,” Transactions of the Japan Society for Aeronautical and Space Sciences, Aerospace Technol Jpn, vol. 11, pp. 33–42, 2013.
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