Research Article  Open Access
Qi Li, Rui Zhao, Sijun Zhang, Wei Rao, Haogong Wei, "Study on Dynamic Characteristics of Mars Entry Module in Transonic and Supersonic Speeds", Space: Science & Technology, vol. 2022, Article ID 9753286, 15 pages, 2022. https://doi.org/10.34133/2022/9753286
Study on Dynamic Characteristics of Mars Entry Module in Transonic and Supersonic Speeds
Abstract
The aerodynamic configuration of the Tianwen1 Mars entry module that adopts a bluntnosed and short body shape has obvious dynamic instability from transonic to supersonic speeds, which may bring risk to parachute deployment. The unsteady detached eddy of the entry module cannot be accurately simulated by the ReynoldsAveraged NavierStokes (RANS) model, while the computational cost for direct numerical simulation (DNS) and large eddy simulation (LES) is huge. It is difficult to implement these methods in the coupled engineering calculation of unsteady flow and motion. This paper proposes the integrated numerical simulation method of computational fluid dynamics and rigid body dynamics (CFD/RBD) based on detached eddy simulation (DES) and calculates and studies the dynamic characteristics of attitude oscillation of the Mars entry module in free flight from transonic to supersonic speeds with one degree of freedom (1DOF) at small releasing angle of attack. In addition, the unstable range of Mach number and angle of attack are determined, and the effect of different afterbody shapes on dynamic stability is analyzed.
1. Introduction
The Tianwen1 probe successfully landed on the predetermined landing area of the southern Utopia Planaria on May 15, 2021, opening a new era of China’s landing and exploring on Mars.
Mars is the planet with the most Earthlike environment that has ever been detected, attracting the most interest from human race. The history of using space probes to explore Mars almost runs through the whole aerospace history of human beings. Since the 1960s, nearly 50 Mars exploration missions have been launched by the Soviet Union, the United States, Japan, Russia, India, Europe, and other countries, but more than half failed [1]. Only ten probes from the United States and China succeeded in the exploration missions of the surface of Mars.
The atmosphere of Mars is thin and its atmospheric density on the surface is only 1%~10% of that of the Earth [2, 3]. Therefore, the process of Mars entry, descent, and landing (EDL) requires the combined action of aerodynamic shape, parachute, and other deceleration methods to ensure the safe landing of a probe and carry out the next step of work. Tianwen1 adopts three deceleration methods, namely, the aerodynamic shape, supersonic parachute, and thrust reverser, which makes the probe decelerate to 0 m/s after entering the Martian atmosphere at a speed of 4.8 km/s and achieve a soft landing on the surface [1] of Mars. Therefore, whether the multiple deceleration methods can be safely connected is the key to the success of the mission of Mars EDL.
Due to the rarefied atmosphere of Mars, for the higher efficiency of aerodynamic deceleration, the aerodynamic configuration of a Mars entry module is generally designed as a bluntbody with large angle for higher drag [4–7]. However, such bluntbody presents dynamic instability when it decelerates to below Mach 3.5 and becomes more unstable with the decrease in the Mach number. In extreme cases, there may even be a risk that the parachute cannot be safely opened due to the attitude oscillation caused by the dynamic instability [8]. Therefore, accurately predicting the dynamic characteristics of the entry module in free flight in transonic and supersonic speeds and determining the range of Mach number, the angle of attack, the dynamic derivative limit, and the limit amplitudes during dynamic instability are the important prerequisites for determining whether the attitude control system of the entry module can effectively restrain the attitude oscillation in transonic and supersonic speeds and ensure safe parachuteopening.
The largescale unsteady detached eddy in the afterbody flow field of the entry module with a bluntnosed and short body cannot be accurately simulated by the ReynoldsAveraged NavierStokes (RANS) model. Direct numerical simulation (DNS) and large eddy simulation (LES) can accurately simulate the pulsation and eddy motion of various scales in the turbulent flow field, but the computational cost for both is huge, making it difficult to calculate the dynamic characteristics of unsteady coupling motion. In recent years, detached eddy simulation (DES) has adopted the RANS turbulence model in the boundary layer to save the calculation resources. For the separation zone far away from the object surface, the subgrid model is adopted for the smallscale eddy, and LES is employed for the largescale eddy, which can accurately and efficiently simulate the separated flow of the entry module afterbody and the corresponding flow stability [9].
This paper proposes the integrated numerical simulation method of computational fluid dynamics and rigid body dynamics (CFD/RBD) based on SADES. It uses leastsquares method as an identification algorithm of dynamic derivatives to calculate and study the dynamic characteristics of attitude oscillation of the Mars entry module in 1DOF free flight in transonic and supersonic speeds and low release angle of attack.
2. Calculation Method
2.1. Fluid Flow Governing Equation and Its Algorithm
The fluid flow governing equation is a threedimensional unsteady compressible NavierStokes (NS) equation. In the generalized curvilinear coordinate, the conservative form of the dimensionless equation is where the characteristic length and the parameters of freestream, including the speed of sound of freestream , temperature , density , and viscosity coefficient , are taken as dimensionless parameters.
Equation (1) adopts the FDS discretion scheme of Roe in space and achieves the secondorder accuracy by MUSCL interpolation and the MINMOD restrictor. The unsteady timemarching methods include the dual timestepping LUSGS algorithm or dual timestepping subiterative approach.
2.2. Solution to RigidBody Dynamic Equation
The dynamic control equation of the entry module in free flight comprises a six degree of freedom rigidbody dynamic equation set and the related kinematic equation set. Among them, the dynamics equation set for the center of mass of the entry module in the inertial system can be expressed as
The dynamic equation of rotation around the center of mass under the bodyaxis coordinate system is where is the aerodynamic vector acting on the entry module and is the vector of momentum moment of the entry module relative to the center of mass. The above dynamic equation set and the related kinematic equation set can be coupled as a set of the nonlinear differential equation set with time as the independent variable.
In this paper, the fourthorder RungeKutta method is used to solve differential equations, and the attitude angle is calculated with the dualEuler method. Thus, the displacement and attitude angle of the entry module in three directions at the next moment can be obtained.
For the 1DOF free motion, with the pitching motion as an example, only static and dynamic derivatives are considered, and the highorder derivatives are ignored. The secondorder differential equation (dimensionless) of the 1DOF free vibration of the capsule is as follows: where is the dimensionless rotational inertia; is the dynamic derivative; and is the static derivative.
Let
Equation (4) can be rewritten as
The corresponding characteristic equation is
The characteristic root is .
In general, the capsule is in a state of static stability, namely, . Normally, the dimensionless rotational inertia is , and . Therefore, the root of the characteristic equation is a pair of conjugate complex roots, and
The special solution to Equation (6) is obtained from the initial conditions:
From Equation (9), the motion of the capsule is the vibration with the period , but different from that of the simple harmonic vibration, its amplitude changes exponentially with time . The positive or negative value of determines whether the motion of the capsule near the equilibrium angle of attack diverges or converges, which indicates that the sign of the dynamic derivative determines the dynamic stability of the capsule.
The timehistory curves of pitch angles can be obtained by calculation. Pitch angles and (corresponding to and and ) with one period part from each other are selected from the curves:
The following can be obtained by dividing the two equations:
Therefore, the static and dynamic derivatives of the capsule can be obtained.
2.3. SADES Method
The basic model of DES is the SpalartAllmaras (SA) model [10]. The differential equation for solving the viscosity coefficient of turbulent motion in this model is as follows:
In Equation (12), is the closest distance to the object surface, and the function is defined as
The first item on the right side of Equation (12) is the generation item. The second item is the dissipation item, and the rest are diffusion items. The variables of the generation item are defined as where is vorticity.
The DES method is to replace in the equation with , and the expression of is given as follows: where is a calibration constant equal to 0.65; is the distance to the wall surface; and Δ is the largest grid space in the area, defined as . In the nearwall region, , which is represented by the SA model and can be solved by the RANS method. In the flow separation region, , and the turbulent stress is solved by the LES subgrid model: where and is the deformation rate tensor of velocity.
2.4. Identification Method of Dynamic Derivatives
After the numerical simulation of free flight is carried out to obtain the oscillation curves of attitude and aerodynamic moment of the entry module within several periods, dynamic derivatives are identified according to the linearized dynamic derivative [11]. With the pitching direction as an example, according to the definition of linearization, there is a linear relationship between the pitching moment coefficient at the same angle of attack and the corresponding dimensionless angular velocity at each moment during the free motion of the entry module, as shown in the following equation. The slope is the value of the dynamic derivative, which can be obtained by the leastsquares method. where is the pitching moment coefficient at a certain moment; is the static pitching moment coefficient; is the pitching damping derivative; and is the dimensionless angular velocity at a certain moment.
3. Verification of Algorithm
The pitching attitude and dynamic derivative identification results in free flight tests within the supersonic ballistic range of Mars probe MER [11, 12], as shown in Figure 1, are used to verify the algorithm in this paper and evaluate the ability of the calculation method to predict the dynamic characteristics of the entry module with the bluntnosed and short body. MER is a Mars probe used by NASA in 2003 to implement a Mars exploration project. The main purpose is to send the probes named “Spirit” and “Opportunity” to the surface of Mars for detection. The aerodynamic configuration and dimensions of the ballistic target test model of MER are shown in Figure 2.
The altitude is zero, and the Mach numbers are 1.5, 2.5, and 3.5. The center of mass is fixed, and the initial release angle of attack is 20°. The model vibrates freely with only 1DOF under the above flow conditions.
Figure 3 shows the timehistory curves of pitch angles (angle of attack) of 1DOF vibration with three Mach numbers. The calculated results match well with the results from the reference [11]. From Figure 3, when the model is at Mach 1.5, the amplitude of the angle of attack increases gradually at the initial release angle of attack, while the trim angle of attack is 0°, which indicates that the model is dynamically unstable in this state. However, when the model is at Mach 2.5 and 3.5, the amplitude of the angle of attack decreases gradually at the initial release angle of attack, and the trim angle of attack converges to 0° in the end, which indicates that the model is dynamically stable at this time. Table 1 presents the comparison between the identified pitching static and dynamic derivatives and the values from the reference [11]. The difference between the pitching static derivative calculated by this paper and that from the reference is only 10%, and the dynamic derivative shows the consistent variation, with the error within 40%.

4. Calculation Model and State
The calculation model is the shape of the Tianwen1 entry module, as shown in Figure 4. Figures 4(a) and 4(b), respectively, show the configuration of the entry module with folded trimming tab and unfolded trimming tab. The maximum windward diameter of the entry module is approximately 3.4 m, and the total height of the entry module is about 2.6 m. The grid diagrams of the wall surface and symmetrical plane are shown in Figure 5. The mass characteristics of the entry module are shown in Table 2, and the calculation state of free flight is shown in Table 3. The parameters of the freestream of the entry module in transonic and supersonic speeds are shown in Table 4. The density, temperature, and pressure in Table 4 are derived from the Martian atmosphere model in reference [13]. According to the calculation method of the equivalent specific heat ratio of the Martian atmosphere [14, 15], the equivalent specific heat ratio across the transonic and supersonic speed region is approximately 1.29.



5. Analysis of Calculation Results
5.1. Flow Field Analysis
Figures 6 and 7, respectively, show the symmetrical plane pressure distribution nephogram of Mars entry module with the trim wing deployed and the trim wing retracted under typical conditions. It can be seen from the figures that with the decrease of Mach number, the distance of detached shock wave in front of the head of the capsule increases and the intensity decreases gradually. After the unfolded of the trim tab, the airflow compression appears near the tip of the windward side of the trim wing, while a large backflow low pressure area appears at the root of the wing. In addition, it can be seen from the figures that the wake region calculated based on DES model is quite large and unstable. Therefore, even at zero angle of attack, the pressure asymmetry caused by the unsteady effect of the afterbody separation vortex is very obvious. For the configuration with unfolded trimming tab, the distribution of the pressure field near the tab plate is very different Mach numbers and angles of attack, indicating that the trimming tab is sensitive to changes in the angle of attack and Mach number.
5.2. Static Aerodynamic Force Calculation and Analysis
Figure 8 shows the comparison of calculated aerodynamic coefficients of the Mars entry module with the trim tab unfolded in the air environment and the Martian atmosphere environment. It can be seen that at the same Mach number, the axial force coefficient of the entry module in Martian atmosphere is slightly larger than that in air environment. The linearity of pitching moment coefficient changing with angle of attack is good, and the trim angle of attack is all around 1 degree, and there is little difference between the two environments.
5.3. Dynamic Stability Analysis before and after Unfolding Trimming Tab
Figure 9 compares the characteristic curves of free oscillation of the pitching attitude with 1DOF when the entry module is released at an initial angle of attack of −2° in the states of Mach 1.5 and 2.0 with folded and unfolded trimming tab. From the figure, the divergence trend of attitude oscillation of the shape with unfolded trimming tab is more obvious within the range of than that with folded trimming tab. In this Mach range, the pitching oscillation of folded trimming tab at a small angle of attack tends to be stable, while unfolded trimming tab tends to oscillate and diverge, and the oscillation period is slightly less than that with folded trimming tab. Therefore, the dynamic stability of the shape with unfolded trimming tab is worse than that with folded trimming tab within the supersonic range of .
Figure 10 compares the characteristic curves of the free oscillation of the pitching attitude with 1DOF when the entry module is released at an initial angle of attack of 5° in the states of Ma2.5 and Ma3.0 with folded and unfolded trimming tab. The convergence trend of attitude oscillation of the shape with unfolded trimming tab is more evident than that with folded trimming tab in the same state; namely, the dynamic stability of the shape with unfolded trimming tab is better than that with folded trimming tab within the supersonic range of .
5.4. Attitude Oscillations at Different Release Angles of Attack
Figure 11 shows the curves of the angle of attack for the pitching attitude oscillation with 1DOF when folding and unfolding the trimming tab of the entry module at different release angles of attack under characteristic Mach number. When the absolute value of the release angle of attack increases, the oscillation of pitching attitude tends to converge; namely, the initial attitude will affect the oscillation characteristics of the entry module. Therefore, it can be inferred that the angle of attack causing the dynamic instability of the entry module should be limited to the small angle of attack.
5.5. Dynamic Instability of the Entry Module in Transonic and Supersonic Speeds
The dynamic derivative of the entry module is identified by the leastsquares method to obtain the dynamic derivative at a small angle of attack for the shape with folded and unfolded trimming tab under each characteristic state, as shown in Tables 5 and 6, respectively. The entry module with the folded trimming tab only shows the dynamic instability within the angle of attack (−1°, 1°), and the maximum dynamic derivative does not exceed 0.5. However, when the trimming tab are unfolded, the entry module is dynamically unstable within the angle of attack (−4°, 4°), and the maximum dynamic derivative is close to 1.5, as shown in Table 7. When , the dynamic instability derivative of the shape with unfolded trimming tab is larger than that with folded trimming tab. When , the dynamic instability of the shape with folded trimming tab is stronger than that with unfolded trimming tab.



5.6. Comparison of Dynamic Instability between Different Afterbody Shapes
Afterbody shapes will affect the dynamic instability of the entry module in transonic and supersonic speeds by affecting the configuration of the separated vortex. To compare the influences of different afterbody shapes on dynamic stability, this paper calculates and compares the pitching derivatives in free flight at a small angle of attack and a transonicsupersonic speed between the entry module with the tricone afterbody shape of Mars Science Laboratory (MSL) [16] and that with the spherecone afterbody of Tianwen1. The comparison between the shapes of tricone and spherecone afterbodies is shown in Figure 12. They both have the same windward base and the first rear cone angle, and the maximum windward envelope, the center of mass, and the moment of inertia are also consistent. The shapes for calculation all have the unfolded trimming tab. The layout and size parameters of trimming tab for the two afterbody shapes of the entry module are the same. The grid distribution of the wall surface and symmetry plane is shown in Figure 13, and the grid topology of the two afterbody shapes is consistent.
The following table shows the dynamic derivative at a small angle of attack corresponding to the entry module with the two afterbody shapes in free flight with 1DOF under Mach 1.2 and 1.5, where all the release angles of attack are 2°. With the same forebody shape and the configuration of trimming tab, the angle of attack and the maximum dynamic derivative of the dynamic instability of the tricone afterbody in transonic and supersonic speeds are greater than those of the spherecone afterbody. In other words, the spherecone afterbody can improve the dynamic stability of the entry module at a small angle of attack and a transonicsupersonic speed.
In order to analyze the reason why the dynamic stability of the spherecone afterbody configuration is better than that of the tricone afterbody configuration, the entry module shape is divided into three parts, the forebody, the midpiece, and the backend. Figure 14 shows the subsection rules of the two configurations of the entry module.
The 1DOF free flight dynamic simulation under the condition of , was carried out for the entry module of the two afterbodies, respectively, and the pressure field on the surface of the module body obtained was pieceally integrated. The method in Section 2.4 was used to complete the identification of the dynamic derivatives of the pitch moment in each section. The piecewise contribution of the dynamic derivatives of typical states under the two afterbody configurations is shown in Table 8.

It can be seen from the table that the flow field of the forebody of the entry module is dynamically stable; thus, its contribution to the dynamic derivative is negative. Due to unstable vortex separation, the contribution of the midpiece to the dynamic derivative is positive. The separation vortex of the flow field in the backend of the two configurations is obviously different, so the contributions of the two backend shapes to the dynamic derivative have obvious differences. In detail, the poor flow structure stability of the tapered backend will increase the dynamic instability of the entry module, while the spherical backend can reduce the separation area of the afterbody. In other words, the spherical backend can reduce the instability of the separation vortex, so its contribution to the dynamic derivative of the entry module is negative.
6. Conclusions
An integrated numerical simulation method of computational fluid dynamics and rigid body dynamics (CFD/RBD) based on SADES is adopted in this paper to simulate the dynamic characteristics of the Tianwen1 Mars entry module before and after unfolding the trimming tab. The oscillation characteristics of pitching attitude and the identification results of dynamic derivatives are analyzed, and the effects of different angles of attack and afterbody shapes on dynamic stability are compared. Conclusions can be drawn as follows: (1)The entry module is generally dynamically unstable at a small angle of attack within the transonicsupersonic range of Mach 1.2–3.0. The unstable angle of attack is only (−1°, 1°) for the shape with folded trimming tab, and the maximum dynamic derivative is not greater than 0.5. The entry module is dynamically unstable within the angle of attack (−4°, 4°) after unfolding the trimming tab, and the maximum dynamic derivative is close to 1.5(2)The maximum dynamic instability of the shape with the folded trimming tab is in the vicinity of Ma2.5, and the maximum dynamic instability of the shape with the unfolded trimming tab occurs at about Mach 1.5. When the angle of attack for initial vibration increases, the convergence trend of attitude oscillation grows(3)With the same forebody shape, the spherecone afterbody can improve the dynamic stability of the entry module in transonic and supersonic speeds compared with the tricone afterbody, including reducing the angle of attack of dynamic instability and the extreme value of dynamic derivatives. It is because the spherical backend can reduce the separation area of the afterbody to reduce the instability of the separation vortex. Thus, the contribution of the spherical backend to the dynamic derivative is negative
Data Availability
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.
Authors’ Contributions
Li Qi is the main writer of this paper and has completed the data collation and analysis related to the paper. Zhao Rui is responsible for the research on the dynamic aerodynamic modeling method of Mars entry module. Zhang Sijun completed the numerical simulation of dynamic characteristics. Wei Haogong completed the identification and analysis of dynamic derivatives. Rao Wei completed the creation of relevant research ideas of the paper.
Acknowledgments
This work came from the Tianwen1 Mars exploration mission and was supported by the Natural Science Foundation of China (Grant No. 11902025).
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