May 21--26, 2014 Beijing  China

1. Sino-German Symposium on Modern Numerical Methods for Compressible Fluid Flows and Related Problems Gas-Kinetic Unified Algorithm for Reentry Hypersonic Flows Covering Various Flow Regimes Solving Boltzmann Model Equation in Thermodynamic Nonequilibrium Zhi-Hui Li Hypervelocity Aerodynamics Institute , National Lab. for CFD ( China Aerodynamics Research & Development Center ) May 21--26, 2014 Beijing  China

2. 1 Introduction Outline 2Unified Boltzmann Model Equation in Nonequilibrium 3 Development of Discrete Velocity Ordinate Method 8 Numerical Study of Three-dimensional Complex Flows Covering Various Flow Regimes 4 Construct Gas-Kinetic Numerical Scheme for Solving Velocity Distribution Function 5 Development of Discrete Velocity Quadrature Methods for Macroscopic Flow Variables 6 Gas-Kinetic Boundary Conditions and Numerical Procedures for the Velocity Distribution Function 7 Gas-Kinetic Parallel Algorithm for 3D Complex Flows 9Concluding Remarks

3. 1. INTRODUCTION • To study the aerodynamics from various flow regimes, the traditional way is to deal with different methods. • Rarefied flow: DSMC • Continuum flow: Euler, Navier-Stokes • Two methods are totally different, and the computed results are difficult to be linked up smoothly with altitude. Slip flow Continuum Rarefied transition Free molecule

4. Problem:Engineering development of current or intending spaceflight projects is closely concerned with complex aerothermodynamics of hypersonic flows in the intermediate range of Knudsen numbers, especially in the rarefied transition and in the near-continuum flow regimes. • Challenge: How to solve multi-scale non-equilibrium flows over the whole flow regimes during spacecraft re-entry?

5. Boltzmann Equation(BE):describe molecular transport phenomena from full spectrum of flow regimes. The difficulties encountered in solving BE are associated with nonlinear multidimensional integral nature of collision term • From the kinetic theory of gases, numerous statistical or relaxation kinetic model equations resembling to the original BE have been put forward • BGK, ES, Shakhov, polynomial, hierarchy kinetic, and McCormack • Enlighten:Solve the nonlinear kinetic models simplified by BE and probably finds a more economical and efficient approach to deal with complex gas flows.

6. Way 1: Lattice Boltzmann Method (LBM) Applying Lattice Gas Cellular Automata (LGCA) and DVM, LBM methods have been developed for solving fluid dynamic problems, such as continuum or near-continuum flow in low speed, turbulence, microflow or porous medium models. Frisch, Pomean, Nie X, Doolen, Succi, Lee, Qian, Chen, Luo, Yong, Guo, Yan, Zhong, Wang etc. • BGK equation provides an effective and tractable ways: • Way 2: Gas-kinetic KFVS-, BGK-type schemes • The Maxwellian distribution function is translated into macroscopic flow variables in equilibrium, some gas-kinetic methods are developed to solve inviscid gas dynamics. • Beam, Pullin, Macrossan, Chen etc.: KFVS-type

7. Applying the asymptotic expansion of velocity distribution function to Maxwellian distribution on flux conservation at cell interface, BGK-type schemes have been presented, such as BGK-Euler,BGK-NS,BGK-Burnett,BGK-Super-Burnett. • Prendergast, Kun Xu, Kim C, Tang, Li, Zhong etc.: BGK-type • Way 3: Gas-kinetic numerical algorithm by directly solving the velocity distribution function • Applying discrete ordinate technique and reduced velocity distribution functions, finite difference explicit and implicit methods, and discrete-velocity models of conservation and dissipation of entropy for solving one- and two-dimensional BGK-Boltzmann model equations have been set forth for high Mach flows. • Chu, Shakhov, Morinishi, Chung, Yang, Aoki, Tcheremissine, Mieussens, Kolobov, Aristov , Titarev etc.

8. Specially, Gas-Kinetic Unified Algorithm (GKUA) since 1999: Boltzmann model equation can be described by modifying the BGK model from two sides on molecular collision relaxing parameter and local equilibrium distribution function Discrete velocity ordinate method (DVOM) and numerical integration techniques are developed to dynamically evaluate macroscopic flow variables Gas-kinetic numerical schemes are constructed to directly capture the evolution and update of velocity distribution function Unified gas-surface interaction model is token by the velocity distribution function The GKUA has presented and applied from low speed to extremely hypersonic flows for MEMS and spacecraft reentry,see IJNMF2003; JCP2004; JCP2009; CMA2011; Adv. Space. Tech. 2011. Recently, a unified gas-kinetic scheme is developed from the combination of BGK and DVOM by Xu and Huang, Chen etc.

9. How to solve BE models involving thermodynamics non-equilibrium effect ? Relaxation kinetic models involving rotational degrees of freedom Morse, Rykov etc. Inelastic relaxing phenomena model is presented with experience treatment and the expansion of Chapman-Enskog with small disturbance C.S.Wang-Chang, G.E.Uhlenbeck etc. Rykov model is applied to simulate hypersonic flow around plate Titarev etc. This study is aimed at extending the GKUA to solve BE models involving thermodynamics non-equilibrium effect for possible engineering applications to spacecraft re-entry.

10. 2. Unified Boltzmann Model Equation in Nonequilibrium Effect • Based on Rykov model, relaxation effect of rotational degrees of freedom is considered into the evolution and update of VDF • The spin movement of diatomic molecule is described by moment of inertia, and the conservation of total angle momentum is taken as a new Boltzmann collision invariant • Internal energy is equally distributed in each degree of freedom by introducing energy model partition function. • Internal energy is taken as the independent variables of VDF

11. Boltzmann Model Eq. in non-equilibrium effect is presented in the framework of GKUA covering various flow regimes. • All macroscopic flow variables are determined by moments of the distribution function over the velocity space.

12. Focus:How to solve? Seven independent variables 、 、

13. The energy VDF is integrated by the weight factors of 1 and e on the internal energy. Two energy-level reduced distribution functions are introduced to remove the continuous dependence of Boltzmann Model Eq. on internal energy: • The unified and reduced VDF equations in non-equilibrium effect is obtained for various flow regimes. • All flow variables are evaluated and updated by the reduced non-equilibrium VDFs of 、 over the velocity space.

14. The VDFs remain with probability density distribution on the principle of probability statistics, VDF possesses the property of exponential function exp(-c2), not Maxwellian distribution. VDF is confined to the finite region 3. Development of Discrete Velocity Ordinate Method Bimodal distribution

15. Discrete Velocity Ordinate Method (DVOM) can be developed to discretize the finite velocity region removed from and to replace continuous dependency of VDF on velocity space. • The selection of DVO points is optimized and corresponded with the evaluation points and weight coefficients of the integration rule in a way that the approximation is exact. • Boltzmann’s H-theorem and conservation condition are guaranteed at each of DVO points with self-adaption.

16. The VDF Eq. is transformed into hyperbolic conservation laws with nonlinear source terms at each of

17. 4. Construct Gas-Kinetic Numerical Scheme for Solving Velocity Distribution Function • The finite-difference method from CFD can be extended to directly solve the discrete VDFs.

18. The finite difference second-order scheme for solving the discrete velocity distribution functions are constructed as • This can be split into four steps.

19. Second-order Runge-Kutta method solves non-linear colliding relaxation source term: • NND scheme with primitive variables solves convective term:

20. 5．Development of Discrete Velocity Quadrature Methods for Macroscopic Flow Variables • Once discrete VDFs are solved, macroscopic flow moments in the physical space are updated by the appropriate discrete velocity quadrature method. • The new Gauss-type integration methods and self-adaptive technique are presented to simulate hypersonic flows. • Bell-type Gauss quadrature formula: • The macroscopic flow variables can be evaluated by the integrating summation with the weight function

21. 6. Gas-Kinetic Boundary Conditions and Numerical Procedures for the Velocity Distribution Function • The gas-kinetic algorithm is based on time evolution of the VDF, the interaction of gas/surface and aerothermodynamics are expressed by directly acting on the VDF. • Escape the statistical fluctuation of DSMC. • Obviate the difficulties in expressing rarefied effect by macroscopic N-S solvers. • Molecules hitting surface must be reflected back to the gas, the reflected VDFs are

22. The number density of molecules diffusing from the surface are determined from mass balance condition on the surface. • If , molecules don’t strike on the surface, the discrete VDFs at wall cells are solved by using second-order upwind-difference approximations from adjacent grids in flow field.

23. Computing space decompose as physical space and velocity space 7． Gas-Kinetic Parallel Algorithm for Three-Dimensional Complex Flows • For three-dimensional flow, GKUA computing space relates to discrete velocity, physical and energy-level multi-dimensional space, and the GKUA requires six-dimensional arrays to access discrete VDFs at all discrete ordinate points. • Parallel domain decomposition of discrete velocity space • Data from sub-space map to corresponding processors decompose as subspace in block-layout manner

24. Evaluate data traffic by two fork tree parallel reduction: • Variable dependency relations of GKUA: For domain decomposition, complete parallelization without data communication arise from velocity space during solving the discrete VDFs. Data communication arise in the sum-reduction computation during evaluating macroscopic flow variables. • Data communication analysis • domain decomposition is carried by three-dimensional way: • Variables map to processors ,distribute in accordance with processor arrays

25. The computation load produced from parallel reduction sum of discrete velocity numerical integration only account for of total workload of the algorithm. The number of parallel processors can reach up to the maximum so that the parallel scalability can be effectively enlarged. • To get smaller ,take , , • For spacecraft reentry with high Mach numbers, decomposition strategy is suitable for many DVO points • parallel scalability

26. For spacecraft simulation with Mach 25, the DVO points is up to , the algorithm to solve the BE model can realize high-performance computation with thousands and tens of thousands, even more massive parallelism. 64~1024CPU 256~512CPU Speed-up ratio Parallel efficiency • Parallel speed-up goes up as near-linearity with high parallel efficiency.

27. 1024~8000CPU 4950~20625CPU • The gas-kinetic parallel algorithm possesses quite high parallel efficiency and expansibility with good load balance, which makes it possible to solve three-dimensional complex hypersonic problems covering various flow regimes.

28. 8．Numerical Study of Three-dimensional Complex Flows Covering Various Flow Regimes • Inner flows of shock wave structures in nitrogen • Computed profiles agree with experimental data and solutions of GBEwell

29. Sphere flows in perfect gas and nonequilibrium effect • The computation spends 47 seconds/step for perfect gas, however 78s/step for non-equilibrium model, the computational load and memory increases two times

30. The front shock for thermodynamic non-equilibrium effect is closer 14% to the stagnation surface than that of perfect gas

31. Bicone reentry flows from high rarefied to continuum flow regimes Stagnation line density profiles • Computed results match solutions of GBE and theoretic analysis well

32. Density and pressure contours around tine bicone with different Kn=1, 0.1

33. Mach number contours and streamlines around tine bicone with Kn=1, 0.1

34. Effect of thermodynamic non-equilibrium on translational and rotational temperature contours around tine bicone with Kn=1, 0.1

35. Overall temperature and heat flux contours around tine bicone with Kn=1, 0.1

36. Continuum flow around tine bicone with Kn=0.0001

38. Continuum flow around tine bicone with Kn=0.0001

40. The GKUA for re-entry hypersonic flows of spacecraft has been developed for the whole range of flow regimesby solving Boltzmann model equation involving non-equilibrium effect. The computations of hypersonic flows and aerodynamic phenomena around sphere, double-cone and spacecraft covering various flow regimes have indicated both high resolution of the flow fields and good agreement with the relevant theoretical, DSMC and experimental results. The GKUA provides a feasible way to simulate spacecraft re-entry hypersonic aerothermodynamics with high-performance massively parallel computation. Further investigation on cargo re-entry capsule and large spacecraft involving real-gas effect with internal energy. 9. Concluding Remarks

41. Acknowledgements • This work are supported by NSFC under Grants Nos. 91130018, 11325212, and National Key Basic Research Program (2014CB744100) • Massively parallel computations are run by National Supercomputer Center in Tianjin and Jinan, and National Parallel Computing Center in Beijing • Joint work with my postgraduates Aoping Peng, Junlin Wu, Xinyu Jiang, Ming Fang and Qiang Ma. • Thanks for organizing committee and Sino-German Center, specially to Prof. Jiequan Li and Song Jiang etc. Thank you for your kind attention! Wish you have a good time in Beijing！