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Gaskinetic unified algorithm for computable modeling of Boltzmann equation and application to aerothermodynamics for falling disintegration of uncontrolled TiangongNo.1 spacecraft
Advances in Aerodynamics volume 1, Article number: 4 (2019)
Abstract
How to solve the hypersonic aerothermodynamics around largescale uncontrolled spacecraft during falling disintegrated process from outer space to earth, is the key to resolve the problems of the uncontrolled TiangongNo.1 spacecraft reentry crash. To study aerodynamics of spacecraft reentry covering various flow regimes, a GasKinetic Unified Algorithm (GKUA) has been presented by computable modeling of the collision integral of the Boltzmann equation over tens of years. On this basis, the rotational and vibrational energy modes are considered as the independent variables of the gas molecular velocity distribution function, a kind of Boltzmann model equation involving in internal energy excitation is presented by decomposing the collision term of the Boltzmann equation into elastic and inelastic collision terms. Then, the gaskinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions by developing the discrete velocity ordinate method and numerical quadrature technique. The unified algorithm of the Boltzmann model equation involving thermodynamics nonequilibrium effect is presented for the whole range of flow regimes. The gaskinetic massive parallel computing strategy is developed to solve the hypersonic aerothermodynamics with the processor cores 500~45,000 at least 80% parallel efficiency. To validate the accuracy of the GKUA, the hypersonic flows are simulated including the reentry Tiangong1 spacecraft shape with the wide range of Knudsen numbers of 220~0.00005 by the comparison of the related results from the DSMC and NS coupled methods, and the lowdensity tunnel experiment etc. For uncontrolling spacecraft falling problem, the finiteelement algorithm for dynamic thermalforce coupling response is presented, and the unified simulation of the thermal structural response and the hypersonic flow field is tested on the Tiangong1 shape under reentry aerodynamic environment. Then, the forecasting analysis platform of endoflife largescale spacecraft flying track is established on the basis of ballistic computation combined with reentry aerothermodynamics and deformation failure/disintegration.
1 Introduction
Largescale spacecraft in low orbit of 300 km~ 500 km faces the problems of deorbiting fall around the end of life, and disintegrates during reentering back to the earth because they suffer tremendous aerothermodynamics environment and overloads [1]. The hightemperature thermochemical nonequilibrium gas flows produced by spacecraft reentering atmospheric surrounding will create cumulative effect on the metal truss softening, melting and the pyrolysis ablation of embedded composite. Under this circumstance, hypersonic aerothermodynamics problems need to be solved and meanwhile various fluidstructure interaction approaches are studied [2]. Aerothermodynamics induces structural deformation, softening, melting, ablation and disintegration. Figure 1 shows the schematic diagram of disintegrated flight path during reentry of largescale spacecraft fall from the outer space. How to forecast the flight trajectory of uncontrolled spacecraft, disintegrated components and debris ahead of time is of world problem in the field of spaceflight. The precision of reentry forecast for disintegrated flight path mainly depends on the integrated simulation of aerothermodynamics and structural deformation failure/disintegration covering various flow regimes [3,4,5,6,7].
To simulate the gas flows from various regimes, the traditional way is to deal with them with different methods. On the one hand, the methods related to rarefied gas flow have been developed, such as the microscopic molecularbased Direct Simulation MonteCarlo (DSMC) method [8, 9]. On the other hand, also the methods adapted to continuum flows have been well developed, such as the NavierStokes equation solvers of macroscopic fluid dynamics. However, both methods are totally different in nature, and the computational results are difficult to link up smoothly with various flow regimes. Engineering development of current or intending spaceflight projects is closely concerned with complex gas dynamic problems of lowdensity flows in the intermediate range of Knudsen numbers, especially in the rarefied transition and in the nearcontinuum flow regimes. In fact, the Boltzmann equation [10] depicts the evolutionary process of the molecular velocity distribution function (VDF) from nonequilibrium to equilibrium state at arbitrary time in gases. It can describe the molecular transport phenomenon covering continuum flow to freemolecular flow regimes. The NavierStokes equation based on continuum medium assumption can be obtained according to the firstorder ChapmanEnskog expansion of the Boltzmann equation. And one can prove that, for simple gases, when the number of simulated molecules approaches to infinite, the solution of the DSMC method, which has been widely using in rarefied gas dynamics, will be converged to that of the Boltzmann equation [11, 12]. Due to the complexity and uncertainties [10, 13] of the nonlinear multidimensional integraldifferential properties of the Boltzmann equation, it is very difficult directly and exactly to solve the Boltzmann equation.
To take full advantage of collision relaxation and transport characteristics of the Boltzmann equation, several approximate solution methods have been developed commendably, such as the linearized Boltzmann equation [14, 15], moment method [16], MonteCarlo finite difference method [17], and model equation methods [18,19,20,21]. In these techniques, the model equation methods have been widely used due to its simple frame and common numerical techniques [22,23,24,25]. By using a simplified collision relaxation model to replace the collisional integral of the Boltzmann equation, the kinetic model equation retains the basic properties of the Boltzmann equation, such as the Htheorem and conservation invariant conditions. The most famous kinetic model is the BGK model [18] for monatomic gases. However, the BGK model leads to the Prandtl number Pr = 1, while its correct value is about 2/3 for monatomic gases. In order to get correct Pr, some modified model equations such as the ES model by Holway [19] and the Shakhov model [20] were introduced. During the last two decades, the Lattice Boltzmann Method (LBM) [26, 27], the KFVS [28], a series of gas kinetic BGKtype schemes and direct modeling methods such as GKS, UGKS etc. [29,30,31,32,33,34], have been developed.
To study aerodynamic problems covering various flow regimes, the unified computable modeling has been actualized on the collision integral of the Boltzmann equation, in which the molecular collision relaxing parameter and the local equilibrium distribution function can be integrated with the macroscopic flow variables, the gas viscosity transport coefficient, the thermodynamic effect, the molecular power law, molecular models, and the flow state controlling parameter from various flow regimes [24, 35,36,37,38], and the gaskinetic unified algorithm (GKUA) has been presented and used to simulate the gas flows from highly rarefied freemolecular flow to continuum flow regimes with the whole range of Knudsen numbers [37,38,39]. In the past, the GKUA has been successfully applied to solve hypersonic reentry aerothermodynamics around kinds of space vehicles and microscale flows involved in MEMS devices [6, 35,36,37,38,39,40]. Because the VDF (f) is a multispace and multidimensional function on velocity space and physical space, especially for the threedimensional Boltzmann model equation, which is a sixdimensional (x, y, z, V_{x}, V_{y}, V_{z}) function, large computational memory is consumed while directly solving the kinetic model equation, and it is very important how to improve the efficiency of numerical computation for threedimensional hypersonic flows around complex bodies. The computational time step of explicit schemes is determined by the stability conditions of numerical schemes and would be very small for hypersonic flows around complex irregular vehicles due to metric coefficients of grid system and singularity of irregular objects, especially at continuum and near continuum flow regimes of nearspace flying surroundings. So, it is necessary to construct the gaskinetic implicit schemes [6, 7] in order to shorten the computing convergence time and improve the computational efficiency.
On the other hand, to simulate the thermal response destructive behavior, a finite element algorithm (FEA) has been developed [5] for the dynamic thermoelasticity coupling problem of materials incorporated with the GKUA for solving the exterior hypersonic reentry aerothermodynamic environment, in which the Newmark implicit scheme is employed to discretize the dynamic thermoelasticity equation and the CrankNicolson scheme is used to solve the heatconduction equation [41]. In engineering problems, the determination of thermal stresses is usually carried out in two steps. First, the temperature distribution is obtained from the Fourier’s heatconduction equation, the stresses are then calculated by the equations of thermoelasticity, including the temperature terms in the stressstrain relations. When subjected to the strong aerodynamic force and rigorous aerodynamic heating, the material deformation will significantly affect the temperature distribution in the interior of the material. The thermalinduced vibration should also not be neglected, and the heat propagation should be viewed as a wave phenomenon rather than a diffusion phenomenon. For this reason, a unified consideration and analysis to the coupled system of heat transfer, deformation and stresses on material and structure [42] is needed to simulate the real physical and mechanical problem. Thus, the dynamic thermoelasticity coupling response behaviors including material internal temperature distribution, structural deformation, and thermal damage are simulated in real time by applying the FEA + GKUA in the field of reentering hypersonic aerothermodynamics.
It is key and necessary to solve multibody flow interference once spacecraft disintegration is appeared during its falling and reentering back to the earth. The LUSGS implicit scheme and the cellcentered finite volume method are constructed to solve the Boltzmann model equation under the frame of the GKUA. In this work, to solve the multibody aerodynamic problems including two and three sidebyside cylinders and irregular bodies with different gap ratio covering highly rarefied to nearcontinuum flow regimes, a multiblock patched grid technique is built for irregular multibody flows, in which the grid points on both sides of joint interface are completely patched by one to one, and the discrete distribution functions and macroscopic flow variables for computation are transferred by the interfaces between blocks. On the other hand, the GKUA will be further extended and developed to solve the thermodynamic nonequilibrium hypersonic flows during the reentry disintegration of endoflife spacecraft, in which a kind of Boltzmann model equation considering the excitation of vibrational energy is constructed.
To establish the forecasting platform of flying track for the uncontrolled spacecraft falling from outer space to earth as the first attempt, the DSMC for hypersonic reentry thermochemical nonequilibrium flow, NS/DSMC, slip NS as verifying tools, and the computational methods of thermal environment and structural heat transfer/composite material pyrolysis, and disassembly and separation have been developed [7, 9, 43, 44] by taking the coupling simulation with trajectory and aerothermodynamic calculation as the main line combined with statistical analysis and 3D scene visualization, the forecasting analysis platform of flying track for the endoflife largescale spacecraft is established for the unified computation of reentry aerothermodynamics, deformation failure/ablation/disintegration with engineering treatment. The remaining parts of this paper are organized as follows: The unified Boltzmann model equations in thermodynamic nonequilibrium effect and implicit numerical scheme are constructed in Section 2. The dynamic coupled thermoelasticity equations and the finiteelement algorithm are proposed in Section 3. The unified simulation of structural deformation and hypersonic aerothermodynamics is presented in Section 4. The numerical simulation and analysis of aerothermodynamics for disintegrated spacecraft is in Section 5, followed by conclusions and the expectation of the future work in Section 6.
2 Gaskinetic unified algorithm for thermodynamic nonequilibrium Boltzmann model equation
The Boltzmann equation [10] depicts the evolutionary process of the VDF from nonequilibrium to equilibrium state at arbitrary time in monatomic gas. It connects the microscopic molecular dynamics and the macroscopic fluid mechanics by the probability statistical distribution function, and can describe the transport phenomena covering continuum flow to freemolecular flow regimes. Gas flow approaches to steady state and obeys the fact that the VDF goes to the local equilibrium distribution taken by macroscopic flow variables in the locality. So, the computable modelling of the Boltzmann equation can be realized [6, 24, 35,36,37,38,39,40, 45] by using the gas molecular collision relaxing parameter and the local equilibrium distribution function. Based on the processing mode of continuous energy level, the rotational and vibrational energy in the quantum states can be introduced as the independent variables of VDF. The equilibrium distribution functions involving in the energy modes and the effective temperature of translation & rotation, or translation, rotation & vibration can be deduced in computable forms, and the threetemperature unified Boltzmann model equation in nonequilibrium effect with vibrational energy is presented in the framework of GKUA, in which the relaxation process of gas molecules is simplified as translation, translationrotation, and translationrotationvibration energy relaxation.
Where, i=1, 2, 3, and,
All flow variables are evaluated and updated by three reduced nonequilibrium VDFs of f_{1}, f_{2} and f_{3} over the velocity space.
Here, m denotes the molecular mass, R is the gas constant, \( \overset{\rightharpoonup }{c} \) denotes the molecular thermal velocity, k is the Boltzmann constant, δ_{ij} is the Kronecker symbols, i, j=1, 2, 3.
All flow variables are evaluated and updated by the three reduced nonequilibrium VDFs of f_{1}, f_{2} and f_{3} over the velocity space, and can be nondimensionalized, in which the characteristic variables are referred to its freestream equilibrium values at infinity, such as number density n_{∞}, temperature T_{∞}, the most probable velocity \( {c}_{m\infty }=\sqrt{2{RT}_{\infty }} \), time t_{∞} = L/c_{m∞}, where L is a characteristic length of the problem, force \( {mn}_{\infty }{c}_{m\infty}^2/2 \), heat flux \( {mn}_{\infty }{c}_{m\infty}^3/2 \) and the distribution function \( {n}_{\infty }/{c}_{m\infty}^3 \). And the Knudsen number is defined as Kn = λ_{∞}/L, where λ_{∞} is the mean free path in the free stream flow.
The Boltzmann model Eq. (1) can be transferred into a group of partial differential equations with nonhomogeneous and hyperbolic conservation [24, 35, 37,38,39] on time and physical space at every DVO points.
with,
Where,
Under the frame of the GKUA, the LUSGS implicit schemes based on the cell centraltype finite volume method (FVM) [6] are constructed to directly solve the Boltzmann model Eq. (11). Using the FVM, in the cell centraltype control volume Ω_{IJ}, the integral equation can be got for the exemplification of 2D flows,
Here, \( \overset{\rightharpoonup }{F}=\left({V}_{x\sigma}{f}_{\sigma, \delta}\right)\overrightarrow{i}+\left({V}_{y\delta}{f}_{\sigma, \delta}\right)\overrightarrow{j} \), \( \overset{\rightharpoonup }{n} \) is the normal vector on boundaries of the control volume, \( {\overline{X}}_{IJ} \) is the average value of X_{IJ} on Ω_{IJ}.
The second term in the left hand side of Eq. (12) can be rewritten as:
After obtaining \( {f}_{IJ}^{n+1} \) in every control volume, all of the macroscopic flow variables can be evaluated and updated by the quadrature methods [35,36,37,38], such as the GaussLegendre numerical quadrature method. Then, the integration of the distribution function over the discrete velocity domain [V_{a}, V_{b}] can be solved by the following extended Gauss–Legendre formula,
The time step Δt in the implicit scheme is less strict than that of the explicit schemes. In this study, the time step is set as
Where, the CFL number is set to 0.95. l_{i} is the characteristic linear size of the ith grid element. The V_{xσ} and V_{yδ} are the discretized velocity components in V_{x}‐ and V_{y}‐ directions, respectively.
To resolve the difficulty of the vast computer memory required by the current algorithm in solving threedimensional complex flows and to well exploit massive power of parallel computers, the multiprocessing strategy and parallel implementation technique suitable for the GKUA are investigated by using the technique of domain decomposition. For the velocity domain Ω_{V} decomposition strategy, the amount of computation for the discrete velocity quadrature integrations for macroscopic quantities occupies about only one fifth of the total amount of the computation of the whole GKUA. Accordingly, the number of CPU processors can reach the number of DVO points of N_{σ} × N_{θ} × N_{δ}, then one can realize superparallel computing. Thus, it is suggested that the velocity space Ω_{V} decomposition technique to be adopted. The numerical experience indicates that in each CPU the amount of work and the memory required for handling the solution of the distribution function is about 1/N_{p} of serial computing, where the N_{p} is the number of processors participated in the parallel computing. This overcomes the restriction of memory problem in serial computation and is of crucial importance in computing complex 3D hypersonic rarefied flows. When the number of DVO points used is set and the way for the parallel distribution of discrete velocity space is generated, each processor will have its correspondingly individual integrality in solving the discrete VDFs assigned to it and the degree of parallelization is higher. This belongs to a coarsegrain high performance parallel computing scheme.
To illustrate and test the feasibility and parallel efficiency of large scale 3D flow computation based on the present GKUA for solving the unified Boltzmann model equation, here we report our experience of such calculation on the Sunway BlueLight MPP parallel supercomputer with up to 8704 ShenWei SW1600 processors using the parallel strategies described above. The discrete velocity space MPI parallelism is designed and combined with corelevel Open ACC paralleling on dimension of location space. Recently, the hardware parallel speedup technique has been applied to accelerate computation of the GKUA. Figure 2a shows the speedup ratio of parallel calculation with numbers of CPU of (a). n=1440~7920CPU and (b). 4950~20625CPU, where the dashed line indicates the ideal speedup ratio and the solid line depicts the present one. A more recent largescale computation of the complex hypersonic flow over the Tiangong1 twocapsule vehicle based on the GKUA with sixdimensional phase space grid system of 101 × 61 × 31 × 120 × 90 × 70 is put in practice with 500~45,000 CPUs. The speedup ratio and corresponding parallel efficiency are shown in Fig. 3. It can be shown from Figs. 2 and 3 that the parallel speedup almost goes up as nearlinearity with the increase of the number of CPU. This indicates that the parallel computation of the GKUA possesses very high parallel efficiency and scalability with good load balance and data communication efficiency by the use of the present parallel computing scheme. From these tests on small, moderate and large scale parallel computing facilities for the domain decomposition strategy of the GKUA, the measured speedup ratio is very close to the theoretical one, and at least 80% parallel efficiency can be achieved for all various scale computations. This validates the present GKUA methods possess quite high parallel speedup and parallel efficiency. Good parallel speedup performance can ensure to enlarge the computational job scale by increasing the number of processors with high parallel efficiency and expandability. This enables us to perform extensive and large scale 3D hypersonic flow simulations for the uncontrolled Tiangong1 spacecraft at various flight trajectory points, although the computation is expensive and almost all the largest computing facilities in the HighPerformance Supercomputer Centers in China are utilized.
3 Finite element algorithm for structural deformation and thermal damage under hypersonic aerothermodynamic environment
Once the hypersonic aerothermodynamic environment is obtained around the largescale reentering spacecraft, the finiteelement methods to simulate the heat conduction and the dynamic deformation and destructive behavior of the metal truss structure need to be developed. In the threedimensional domain Ω ⊂ ℝ^{3}, the dynamic thermoelasticity equation [41, 42] is expressed, as follows
Where, u = [ u_{1} u_{2} u_{3}]^{T} is the displacement vector, \( \boldsymbol{f}={\left[\;{f}_1\kern0.5em {f}_2\kern0.5em {f}_3\right]}^T \) is the volume force and the superscript ^{T} denotes transposition. ρ is the density, τ is the damping coefficient and C_{ijkl} is the 4order elasticity tensor. For homogeneous and isotropic materials, C_{ijkl} is written as C_{ijkl} = λδ_{ij}δ_{kl} + G(δ_{ik}δ_{jl} + δ_{il}δ_{jk}). Here, δ_{ij} is the Kronecker symbol, λ and G are the Lamé constants, expressed by the Young’s modulus E and the Possion’s ration ν as.
θ = T − T_{0} is the temperature increment with T and T_{0} denoting the absolute temperature and reference temperature respectively and β_{ij} is the thermostress module, β_{ij} = C_{ijkl}α_{kl}, where α_{kl} is the thermal expansion coefficient. For homogeneous and isotropic materials, α_{kl} = α_{0}δ_{kl}, α_{0} is the thermal expansion constant, and then β_{ij} can be written as β_{ij} = 3γα_{0}δ_{ij}, here \( \gamma =\frac{E}{3\left(12\nu \right)} \) is the bulk modulus, which means that the increment or decrement of the temperature leads to the expansion or contraction of the body, effecting the stress distribution of the structure. The elastic strain ε_{ij} is
and the stress tensor σ_{ij} = C_{ijkl}ε_{kl} − β_{ij}θ.
Considering the deforming influence on the temperature field, the transient heat conductive equation is expressed as follows
Here, c is the specific heat, k_{ij} denotes the 2order conductivity tensor and h is the heat source. Considering also homogeneous and isotropic materials, from the expression of β_{ij}, the second term of Eq. (20) can be denoted by
in which ε_{ii} is the volumetric strain, Eq. (21) illustrates that the change rate of the volumetric strain is the positive correlation with the temperature field.
In summary, for the convenience of our derivation, the weak form of the dynamic thermoelasticity coupling equations is to find u and θ, such that
From the equations above, it is noted that the displacement u and temperature increment θ are dependent on each other, which have to be solved simultaneously in the finiteelement method. Equation Section (Next)
Adopting the Newmark implicit method [2, 5] to deal with the velocity and acceleration terms, we have the time stepping scheme
where the parameters 0 ≤ ω ≤ 1, 0 ≤ η ≤ 1/2, and usually when ω ≥ 0.5 and η ≥ 0.25(0.5 + ω)^{2}, the scheme above is unconditionally stable.
To ensure that the displacement and the temperature distribution of the inner structure of hypersonic flying body are continuously solved from Eq. (24), the unified simulation of structural deformation and hypersonic aerothermodynamics needs to be constructed with realtime trajectory updating, in which the values of the aerothermodynamic temperature and force computed by the GKUA as the boundary conditions for the coupled thermoelasticity problems. To combine the present FEA methods with the GKUA [6, 35, 37,38,39, 45] of aerothermodynamics covering various flow regimes for simplicity, we consider and verify the flowrounded infinite plate problem as an example of thermodynamic coupling response during spacecraft reentering Earth’s atmosphere, shown in Fig. 4. The size of the plate is 0.015m × 0.5m, the exterior of which is the reentry nearcontinuum transition flow, the inflowing Mach number, Knudsen number and ratio of specific heat are Ma_{∞} = 8.3666, Kn_{∞} = 0.01 and γ = 1.4, respectively. The exterior flow field of the plate with the mesh of 63 × 41 in the XoY coordinate is computed by the GKUA for solving the unified Boltzmann model equation. Because of the reentry nearcontinuum transition flow regime, the thick detached shock wave layer is formed around the thin plate shown in Fig. 4a, and the strong aerodynamic heating and force are imposed as the interface condition of the present thermodynamic coupling finiteelement algorithm on the plate surface. The high temperature and strong pressure make the plate expand, deform and damage.
4 Results and discussion
To verify the accuracy and reliability of the present computable modeling of Boltzmann equation and GKUA in solving thermodynamic nonequilibrium flows with vibrational energy excitation, the cylinder flows of Nitrogen gas with Kn_{∞} = 0.01, Ma_{∞} = 5, n_{∞} = 1.4966E20/m^{3}, T_{∞} = T_{w} = 500k are solved by the present GKUA and the DSMC, respectively. Figure 5 shows macroscopic flow variables including (a) pressure, (b) Mach number, (c) overall temperature, (d) translational temperature, (e) rotational temperature and (f) vibrational temperature with the comparison of the DSMC results, in which the two results agree well, especially, the contours of pressure and overall temperature distribution are almost completely identical. It is indicated from the translational, rotational, vibrational and overall temperature distribution that the translational temperature is the highest, the rotational temperature is the second and the vibrational temperature is the lowest in the interior stagnation region after the detached shock wave, and the vibrational temperature computed by the present GKUA with better resolution is slightly higher than the DSMC results.
In addition, the present GKUA uses the structured mesh with refinement near surface and slightly wider grid in the location of detached shock wave, however, the DSMC uses selfadaptive unstructured mesh and the space mesh is refined to better capture shock wave where the macroscopic flow gradient is large, so that there is a slight difference in the shock location between the two methods. Figure 6 shows the surface pressure and heat flux distribution with different nonequilibrium effects including the GKUA results with the Shakhov model of simple gas, the ES model of rotational excitation polyatomic gas and Nitrogen vibrational energy excitation, respectively, where the “DSMCtranslation”, “DSMCrotation” and “DSMCvibration” denote the DSMC results of the Nitrogen as a simple gas without excitation of internal energy, the Nitrogen gas with rotational energy and the Nitrogen gas with vibrational energy excitation, respectively, and Fig. 6b and d correspond to the local enlarged profiles of surface pressure and heat flux near the stagnation point. It is indicated that the surface pressure distribution near the stagnation point is the lowest for the “DSMCtranslation”, the next for the “GKUAShakhov”, and other results almost coincide. The surface heat flux distribution shown in Fig. 6c and d for the “DSMCtranslation” and “DSMCrotation” near the stagnation point is obviously higher than the other results and exists serious statistical fluctuation. On the whole, the results obtained by these methods are in very close agreement.
In order to study the influence of vibration energy excitation on the aerodynamics of Tiangongtype spacecraft with solar array, Fig. 7 shows (a) translational temperature, (b) rotational temperature, (c) vibrational temperature, (d) Overall Temp. (e) Mach No. and (f) streamline structure around the Tiangongtype spacecraft with H = 90km, Ma_{∞} = 8 in thermodynamic nonequilibrium effect of vibrational energy excitation solved by the present GKUA in massively parallel computing. It is indicated that the vortex structure is formed between the solar array and the module, and the axial force and normal force coefficients are respectively obtained as Ca = 9.635 and Cn = 1.525 for thermodynamic nonequilibrium gas, and Ca = 9.535 and Cn = 1.795 for completely simple gas with the relative error of 1.04% and 17.7%, in which the complex nonequilibrium flow between the solar panels and the modules has a great influence on the normal force around the Tiangongtype spacecraft.
To simulate metaltruss structure deformation and damage under the strong aerothermodynamic environment during the reentry falling process of the largescale loworbit vehicle with solar array from H = 250 km, Fig. 8 shows the pressure contours around Tiangongtype spacecraft with Kn_{∞} = 220.34, Ma_{∞} = 22.75, and the temperature, normal stress and deformation distribution of Tiangongtype spacecraft structure varied with falling height from H = 120 km to H = 90 km during the first disintegration.
To compare and verify the correctness of different algorithm models and experiments in solving the common hypersonic flows around the twocapsule body of Tiangongtype spacecraft after disintegration for the first time, Fig. 9a, b and c show the pressure and temperature contours of flow field and the structural temperature of twocapsule vehicle of Tiangong1 spacecraft computed by the GKUA at 801 s during the reentry, and Fig. 9df show the aerodynamics solved by the GKUA, DSMC [9], NS/DSMC [43], Slip NS and lowdensity windtunnel experiments [44] for hypersonic flows around the twocapsule of Tiangongtype spacecraft, in which good agreement exists and the GKUA results are more identical to the experimental data and the DSMC results, in which the GKUA has been verified in highprecision and strong simulation ability in solving the aerodynamics of irregular largescale spacecraft covering various flow regimes.
Largescale spacecraft such as the Tiangong1 in uncontrolled falling will disintegrate into multibodies in nearspace flying surrounding during its reentry passing through the atmosphere. By developing multiblock patched grid generation technique, a computational platform based on the GKUA with implicit scheme has been established to solve the multibody flow problems covering various flow regimes. Figure 10 shows the multibody interference and flow phenomena around different multibodies of the irregular disintegration/debris simulated by the GKUA. When H = 6 L, interference become very weak. When H = 10 L, the Mach stem disappears, and the two oblique shock waves are formed by head shock waves, the separation vortex disappear.
In order to establish more engineering processing approaches for the flow interference among disintegrated multibodies, three sidebyside square cylinder flows are simulated by the present GKUA for Kn_{∞} = 0.001, Ma_{∞} = 3 with zero attack angle. As previous, H is defined as the distance between the centers of the upper and middle square debris, and L is the square cylinder side length. The flow state cases with different gap spacing of H = 2 L, 4 L, 6 L, 8 L and 10 L are simulated, respectively. Figures 11, 12, 13 and 14 show the number density, temperature, Mach number contours and the corresponding flow streamlines relative to the four cases of H = 2 L, 6 L, 8 L and 10 L, respectively. It can be seen that similar flow phenomena to the two sidebyside square cylinder flows are revealed in Figs. 11 and 12 for H = 2 L and 6 L, respectively. In addition, when H = 2L, a series of interference shock waves appear in the zone after the throat, disturb the tail flow, and lead to a very complex flow field. As H increases, the interference shock waves move backward, when H = 6L, the flow interference interaction become very weak with saddletype bow wave. When H increases to 8 L, a Mach stem wave system is formed by the interaction of the bow head shock waves. Behind the Mach stem, two oblique shock waves are formed and act on the tail flow, resulting in a series of shock waves and rarefaction waves. When H = 10L, the Mach stem disappears, and two oblique shock waves are formed by the detached head shock waves. It is indicated from the flow streamline structures of Figs. 11d, 12d and 13d that trailing vortices appear behind each square cylinder with the increase of gap spacing distance from H = 2 L to H = 8 L, respectively, and because of flow interference among three sidebyside cylinders, the trailing vortices after the upper square cylinder are not symmetrical. The trailing vortices become large by increasing H. When H increases to 10 L (Fig. 14d), because of the shock interaction, a new separation vortex generates and appears at x = 7 behind the middle square cylinder. At this time, the flow interference between the sidebyside square cylinders becomes very small, and the trailing vortexes behind the upper square cylinder are almost symmetrical. The separated vortex would disappear when H increases to a certain value. Figures 11, 12, 13 and 14 reveal the varying features of multibody flow in the continuum flow regime with different gap spacing and the influential range of the flow interference among the multibody sidebyside square debris.
Combined by developing the DSMC for hypersonic reentry thermochemical nonequilibrium flow, the NS/DSMC, the slip NS, and the computational methods of thermal environment, structural heat transfer/composite material pyrolysis in embedded layer or special device, and disassembly and separation with multibody flow interference, the forecasting software for largescale spacecraft falling from outer space has been established [5,6,7, 9, 43, 44] with the unified simulation on ballistic trajectory, rapid engineering calculation modified by the present numerical algorithm results for aerothermodynamics covering various flow regimes, ablation, deformation failure and disintegration. Figure 15 shows the forecast of flight trajectory and falling area of disintegrated wreckage and debris from the Tiangong1 spacecraft falling into the atmosphere with the comparison of the monitoring results afterwards from the map calibration of NASATv’s post website (www.apolo11.com) announcement in good agreement and compatibility. It is indicated from the falling reentry forecast that the uncontrolled Tiangong1 will be disintegrated firstly at the range of 110105 km, secondly at 10095 km, specially, the main bearingcone platform and trajectory controlling engines will be disintegrated at 83 km56 km and so on. The present numerical forecasting platform obtained the scope of falling area distribution of longitudinal length 1200 km and lateral width 100 km from the first disintegration to debris falling to the South Pacific Ocean. These results on the multiple disintegration, falling area distribution and trajectory calculation coupled with aerothermodynamics for the uncontrolled Tiangong1 spacecraft affirm the accuracy and reliability of the unified modeling and typical computation of structural deformation failure from thermodynamic response and hypersonic aerothermodynamics for falling disintegration along ballistic trajectory with different flying heights, Mach numbers covering various flow regimes from outer space to earth.
5 Conclusion
In this paper, the collision term of the Boltzmann equation is divided into elastic and inelastic collision terms. The inelastic collision is characterized into translationalrotational energy relaxation and translationalrotationalvibrational energy relaxation according to certain relaxation rates in realtime computation. Then, a kind of Boltzmann model equation considering the excitation of vibrational energy is constructed, and an implicit gaskinetic unified algorithm has been presented to directly solve the unified Boltzmann model equation. The massively parallel MPI and OpenACC technique is built and applied to the irregular largescale and multibody reentry flows from freemolecular to continuum flow regimes.
To simulate the disintegration of metal truss structure for spacecraft during uncontrolled falling back to the atmosphere, the dynamic thermomechanical coupling model has been derived by the variational principle on the basis of the dynamic thermoelasticity and the heat conductive equations, then the finiteelement implicit schemes are constructed, as a result, the corresponding finiteelement algorithm and computational procedures of FEA + GKUA have been established to simulate the transient thermal and mechanical damage behaviors of the structures under reentering aerothermodynamic environment as a new researching direction.
Integrated by developing the other simulation prediction methods as verification and supplement of reliable modeling for hypersonic reentry aerothermodynamics covering various flow regimes, the forecasting analysis platform of spacecraft falling flight track has been being established for the unified computation to reentry aerothermodynamics and structural deformation failure/ablation/disintegration, and has been applied to the numerical forecast of the flight track of the uncontrolled Tiangong1 spacecraft, in which authenticates the correctness and validity of the computable modeling of Boltzmanntype velocity distribution function equation with internal energy excitation and the gaskinetic massively parallel algorithm for hypersonic nonequilibrium aerothermodynamics during falling disintegration of the uncontrolled Tiangong1 spacecraft.
As this work is only the beginning of numerical forecast for uncontrolling spacecraft falling from outer space, further investigations on the threedimensional irregular multibody flows with real gas effects involving internal energy excitation around disintegrating debris of uncontrolled spacecraft, need to be studied in the future.
References
Reyhanoglu M, Alvarado J (2013) Estimation of debris dispersion due to a space vehicle breakup during reentry. Acta Astronautica 86:211–218
Balakrishnan D, Kurian J (2014) Material thermal degradation under reentry aerodynamic heating. J Spacecr Rocket:1–10. https://doi.org/10.2514/1.A32712
Wu ZN, Hu RF, Qu X et al (2011) Space debris reentry analysis methods and tools. Chin J Aeronaut 24(4):387–395
Caggiano A, Etse G (2015) Coupled thermomechanical interface model for concrete failure analysis under high temperature. Comput Methods Appl Mech Eng 289:498–516
Li ZH, Ma Q, Cui JZ (2016) Finite element algorithm for dynamic thermoelasticity coupling problems and application to transient response of structure with strong aerothermodynamic environment. Commun Comput Phys 20(3):773–810
Peng AP, Li ZH, Wu JL, Jiang XY (2016) Implicit gaskinetic unified algorithm based on multiblock docking grid for multibody reentry flows covering all flow regimes. J Comput Phys 327:919–942
Li ZH, Peng AP, Wu JL, Ma Q, Tang XW, Liang J, GasKinetic Unified Algorithm for Computable Modeling of Boltzmann Equation for Aerothermodynamics during Falling Disintegration of Tiangongtype Spacecraft, Proc. of 31^{st} Intern. Symposium on Rarefied Gas Dynamics, Glasgow, U.K., Jul.23–27, 2018
Bird GA (1994) Molecular gas dynamics and direct simulation of gas flows. Oxford University Press, Oxford
Liang J, Li ZH, Li XG, Shi WB (2018) Monte Carlo Simulation of Spacecraft Reentry Aerothermodynamics and Analysis for Ablating Disintegration. Commun Comput Phys 23(4):1037–1051
Cercignani C (1988) The Boltzmann equation and its applications. SpringerVerlag, New York
Wagner W (1992) A convergence proof for Bird’s direct simulation Monte Carlo method for Boltzmann equation. J Stat Phys 66:1011–1044
Li ZH, Fang M, Jiang XY, Wu JL (2013) Convergence proof of the DSMC method and the gaskinetic unified algorithm for the Boltzmann equation. Sci ChinaPhys Mech Astron 56(2):404–417
Bobylev AV, Rjasanow S (1999) Fast deterministic method of solving the Boltzmann equation for hard spheres. Eur J Mech B Fluids 18(5):869–887
Tipton EL, Tompson RV, Loyalka SK (2009) Chapmann–Enskog solutions to arbitrary order in Sonine polynomials II: viscosity in a binary, rigidsphere, gas mixture. Eur J Mech B Fluids 28:335–352
Sone Y, Takata S, Ohwada T (1990) Numerical analysis of the plane Couette flow of a rarefied gas on the basis of the linearized Boltzmann equation for hardsphere molecules. Eur J Mech B Fluids 9(3):449–456
Sheng Q, Tang GH, Gu XJ, Emerson DR, Zhang YH (2014) Simulation of thermal transpiration flow using a highorder moment method. Int J Mod Phys C 25(11):1450061
Tcheremissine FG (2006) Solution of the Boltzmann kinetic equation for high speed flows. Comput Math Math Phys 46:315–329
Bhatnagar PL, Gross EP, Krook M (1954) A Model Collision Processes in Gases: I. Small Amplitude Processes is Charged and Neutral OneComponent System. Phys Rev 94:511–525
Holway LH Jr (1966) New statistical models for kinetic theory: methods of construction. Phys Fluids 9(9):1658–1673
Shakhov EM (1968) Generalization of the Krook kinetic relaxation equation. Fluid Dynamics 3(5):95–96
Rykov VA (1975) Model kinetic equation of a gas with rotational degrees of freedom. Fluid Dynamics 10:959–966
Yang JY, Huang JC (1995) Rarefied flow computations using nonlinear model Boltzmann equations. J Comput Phys 120:323–339
Titarev VA, Shakhov EM (2002) Heat transfer and evaporation from a plane surface into a halfspace upon a sudden increase in body temperature. Fluid Dynamics 37(1):126–137
Li ZH, Zhang HX (2003) Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation. Int J Numer Methods Fluids 42:361–382
Kudryavtsev AN, Shershnev AA (2013) A numerical method for simulation of microflows by solving directly kinetic equations with WENO schemes. J Sci Comput 57:42–73
Nie X, Doolen GD, Chen S (2002) Lattice Boltzmann simulation of fluid flows in MEMS. J Stat Phys 107:279
Meng JP, Zhang YH, Hadjiconstantinou NG, Radtke GA, Shan XW (2013) Lattice ellipsoidal statistical BGK model for thermal nonequilibrium flows. J Fluid Mech 718:347–370
Mandal JC, Deshpande SM (1994) Kinetic flux vector splitting for euler equations. Comput Fluids 23(2):447
Xu K (2001) A gaskinetic BGK scheme for the Navierstokes equations and its connection with artificial dissipation and Godunov method. J Comput Phys 171(1):289–335
Xu K, Li ZH (2004) Microchannel flow in the slip regime: gaskinetic BGKBurnett solutions. J Fluid Mech 513:87–110
Xu K, Huang JC (2010) A unified gaskinetic scheme for continuum and rarefied flows. J Comput Phys 229:7747–7764
Liu S, Yu PB, Xu K, Zhong ZW (2014) Unified gas kinetic scheme for diatomic molecular simulations in all flow regimes. J Comput Phys 259:96–113
Yang LM, Shu C, Wu J (2014) A simple distribution functionbased gaskinetic scheme for simulation of viscous incompressible and compressible flows. J Comput Phys 274:611–632
Chen SZ, Xu K, Li ZH (2016) Gas kinetic scheme in Cartesian grid method for regular flows with different Mach numbers. J Comput Phys 326:862–877
Li ZH, Zhang HX (2004) Study on gas kinetic unified algorithm for flows from rarefied transition to continuum. J Comput Phys 193:708–738
Li ZH, Zhang HX (2008) Gaskinetic description of shock wave structures by solving Boltzmann model equation. Int J Comput Fluid Dyn 22(9):623638
Li ZH, Zhang HX (2009) Gaskinetic numerical studies of threedimensional complex flows on spacecraft reentry. J Comput Phys 228:1116–1138
Li ZH, Peng AP, Zhang HX, Yang JY (2015) Rarefied gas flow simulations using highorder gaskinetic unified algorithms for Boltzmann model equations. Prog Aerosp Sci 74:81–113
Li ZH, Zhang HX (2007) Gaskinetic numerical method solving mesoscopic velocity distribution function qeuation. Acta Mech Sinica 23(3):121–132
Li ZH, Zhang HX, Fu S (2005) Gas kinetic algorithm for flows in Poiseuillelike microchannels using Boltzmann model equation. Sci ChinaPhys Mech Astron 48(4):496–512
Gu W, Wang P (2014) A CrankNicolson difference scheme for solving a type of variable coefficient delay partial differential equations. J Appl Math ID560567:1–6
Hashiguchi K (2013) General description of elastoplastic deformation/sliding phenomena of solids in high accuracy and numerical efficiency: subloading surface concept. Arch Comput Methods Eng 20:361–417
Li ZH, Dang LN, Li ZH (2018) Study on NS/DSMC hybrid numerical method with chemical nonequilibrium for hypersonic flow. Acta Aeronautica et Astronautica Sinica 39(10):122229
Li ZH. Technical Summary on Numerical Forecast and Hazard Analysis for Falling Disintegration of Uncontrolled TiangongNo.1 Target Spacecraft, Tech. Rep. No. S2018.14, Hypersonic Aerodynamics Institute, China Aerodynamic Research and Development Center. 2018
Peng AP, Li ZH, Wu JL, Jiang XY (2017) Validation and Analysis of GasKinetic Unified Algorithm for Solving Boltzmann Model Equation with Vibrational Energy Excitation. Acta Phys Sin 66(20):204704
Acknowledgements
This work is supported by the National Key Basic Research and Development Program (2014CB744100), the National Outstanding Youth Fund(11325212), the Integrating Project of NSFC Great Researching Plan(91530319) of China, and the China Manned Space Engineering Office. The first author would like to thank his projects’ group including Jie Liang, Zhonghua Li, Weibo Shi, Xinyu Jiang, Xuguo Li, Geshi Tang, Zhiyong Bai, Dun Li, Boqiang Du, Yu Xiao, Junlin Wu, Ming Fang, Siyao Su for their helpful work. Parts of present computation were carried out by the National Supercomputing Centers in Wuxi, Tianjin and Jinan. The authors are thankful to the reviewers for their valuable comments to improve the quality of the manuscript.
Funding
The National Key Basic Research and Development Program (2014CB744100), and the National Natural Science Foundation of China (91530319 and 11325212) support the present researches in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript.
Availability of data and materials
Data supporting the results reported in the article can be found including where generated during the study. By data, the minimal dataset that would be necessary to interpret, replicate and build upon the findings reported in the article.

The datasets generated and/or analysed during the current study are not publicly available due to the Chinese manned space engineering but are available from the corresponding author on reasonable request.
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ZHL designed the computable modeling of Boltzmann equation, the gaskinetic unified algorithm for aerothermodynamics during falling disintegration of uncontrolled Tiangong1 spacecraft, and computational procedures of FEA+GKUA to solve the transient thermal and mechanical damage behaviors of the structures. APP developed the gaskinetic unified algorithm for the computable modeling of Boltzmann equation in solving the hypersonic nonequilibrium aerothermodynamics. QM designed the finiteelement implicit schemes and computation of structural deformation failure to couple the dynamic thermoelasticity and the heat conductive equations. LND developed the rapid engineering calculation of ballistic trajectory combined with aerodynamics. XWT designed the forecasting analysis platform of spacecraft falling flight track. XZS analyzed and collected to interpret the flying data during uncontrolled Tiangong1 spacecraft falling back to the atmosphere. All authors read and approved the final manuscript.
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Li, ZH., Peng, AP., Ma, Q. et al. Gaskinetic unified algorithm for computable modeling of Boltzmann equation and application to aerothermodynamics for falling disintegration of uncontrolled TiangongNo.1 spacecraft. Adv. Aerodyn. 1, 4 (2019). https://doi.org/10.1186/s4277401900094
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DOI: https://doi.org/10.1186/s4277401900094