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

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

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

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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. 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?

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lFrom 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 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. 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

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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. 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. Frisch, Pomean, Nie X, Doolen, Succi, Lee, Qian, Chen, Luo, Yong, Guo, Yan, Zhong, Wang etc. 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. 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 Beam, Pullin, Macrossan, Chen etc.: KFVS-type lBGK equation provides an effective and tractable ways:

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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. 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 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. 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. Chu, Shakhov, Morinishi, Chung, Yang, Aoki, Tcheremissine, Mieussens, Kolobov, Aristov, Titarev etc.

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Specially, Gas-Kinetic Unified Algorithm (GKUA) since 1999: lBoltzmann model equation can be described by modifying the BGK model from two sides on molecular collision relaxing parameter and local equilibrium distribution function lDiscrete velocity ordinate method (DVOM) and numerical integration techniques are developed to dynamically evaluate macroscopic flow variables lGas-kinetic numerical schemes are constructed to directly capture the evolution and update of velocity distribution function lUnified 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. 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. lRecently, a unified gas-kinetic scheme is developed from the combination of BGK and DVOM by Xu and Huang, Chen etc.

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How to solve BE models involving thermodynamics non-equilibrium effect ? Relaxation kinetic models involving rotational degrees of freedom Relaxation kinetic models involving rotational degrees of freedom Morse, Rykov etc. Morse, Rykov etc. lInelastic 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. C.S.Wang-Chang, G.E.Uhlenbeck etc. lRykov model is applied to simulate hypersonic flow around plate Titarev etc. 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.

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2. Unified Boltzmann Model Equation in Nonequilibrium Effect lBased on Rykov model, relaxation effect of rotational degrees of freedom is considered into the evolution and update of VDF lThe 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 lInternal energy is taken as the independent variables of VDF lInternal energy is equally distributed in each degree of freedom by introducing energy model partition function.

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

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

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lThe 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: lThe unified and reduced VDF equations in non-equilibrium effect is obtained for various flow regimes. lAll flow variables are evaluated and updated by the reduced non-equilibrium VDFs of 、 over the velocity space.

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3. Development of Discrete Velocity Ordinate Method uThe VDFs remain with probability density distribution on the principle of probability statistics, VDF possesses the property of exponential function exp(-c 2 ), not Maxwellian distribution. uVDF is confined to the finite region Bimodaldistribution

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uDiscrete 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. uThe 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. uBoltzmann’s H-theorem and conservation condition are guaranteed at each of DVO points with self-adaption.

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

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

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

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

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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: Bell-type Gauss quadrature formula: The macroscopic flow variables can be evaluated by the integrating summation with the weight function

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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. 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. 6. Gas-Kinetic Boundary Conditions and Numerical Procedures for the Velocity Distribution Function lMolecules hitting surface must be reflected back to the gas, the reflected VDFs are

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lThe number density of molecules diffusing from the surface are determined from mass balance condition on the surface. lIf, 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.

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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 decompose as subspace in block-layout manner

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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 Variables map to processors,distribute in accordance with processor arrays domain decomposition is carried by three-dimensional way:

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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

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Parallel speed-up goes up as near-linearity with high parallel efficiency. 256~512CPU 64~1024CPU Speed-up ratio Parallel efficiency Speed-up ratio Parallel efficiency 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.

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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. 4950~20625CPU 1024~8000CPU

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

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

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

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

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

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

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

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

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Continuum flow around tine bicone with Kn=0.0001

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The GKUA for re-entry hypersonic flows of spacecraft has been developed for the whole range of flow regimes by 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

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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. Thanks for organizing committee and Sino-German Center, specially to Prof. Jiequan Li and Song Jiang etc. Thank you for your kind attention! Thank you for your kind attention! Wish you have a good time in Beijing ！ Wish you have a good time in Beijing ！

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