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Wu J.-L.,China Aerodynamics Research And Development Center | Wu J.-L.,National Laboratory for Computational Fluid Dynamics | Li Z.-H.,China Aerodynamics Research And Development Center | Li Z.-H.,National Laboratory for Computational Fluid Dynamics | And 2 more authors.
Computers and Fluids | Year: 2016

The unsteady process of rarefied jet flows expanding into a vacuum is numerically solved by the Gas-Kinetic Unified Algorithm (GKUA) based on the Boltzmann model equations. The discrete velocity ordinate method (DVOM) is adopted to discretize the velocity space of the molecular velocity distribution function, while the corresponding numerical integral technique is developed to evaluate macroscopic flow variables, including number density, flow velocity, temperature and so on. The time-explicit finite difference scheme is constructed to capture the unsteady evolution of the discrete velocity distribution functions at each DVO point by using the unsteady time-splitting method. The multi-block docking grid technique is designed for different regions of flow field in physical space with self-adaptive adjustment. Then, the supersonic planar jet flows with the wide range of Knudsen numbers, from 100 to 0.1, are solved numerically and discussed, including startup to steady state and shutting down. The GKUA results of free-molecule jet flows are analyzed and compared with the Maxwellian analytical solutions for collisionless plume flows, in which good agreement shows the validity and accuracy of the present numerical method. When taking the collision effect into account, the unsteady jet flows with different Knudsen numbers are computed by the present GKUA method. It is shown that the collision term of the Boltzmann model equation plays an important role in this rarefied gas diffusion into the back flow when Kn ≤ 10. However, the colliding relaxation term have a little influence on the kernel region in the startup process for Kn ≥ 0.1. In general, the convective transport term of the Boltzmann model equation dominates the kernel region of the jet flow during the unsteady process of gas expanding into a vacuum. The numerical experience indicates that the present GKUA can provide a vital tool for solving unsteady flow problems covering various flow regimes by directly tracing the time evolution of the Boltzmann-type velocity distribution function. © 2016.


Kwang-Hua C.R.,BeiHang Training Center | Kwang-Hua C.R.,Distribution Center | Li Z.,China Aerodynamics Research And Development Center | Li Z.,National Laboratory for Computational Fluid Dynamics
Fluid Phase Equilibria | Year: 2015

Transport of the solid helium with imbedded defects in confined domains has been recently reported. Here we used the verified absolute-reaction theory which originates from the quantum chemistry approach together with the boundary perturbation approach to calculate the temperature-dependent viscosity of flowing solid He-4 in the bulk. Both effects of the activation energy and activation volume on the temperature-dependent (shear) viscosity of flowing solid He-4 were illustrated. We found that the temperature-dependent (shear) viscosity which is a ratio of the shear stress and the shear strain rate increases a little as the temperature decreases within the selected temperature regime. © 2015 Elsevier B.V.


Li Z.H.,China Aerodynamics Research And Development Center | Li Z.H.,National Laboratory for Computational Fluid Dynamics | Fang M.,China Aerodynamics Research And Development Center | Jiang X.,China Aerodynamics Research And Development Center | And 2 more authors.
Science China: Physics, Mechanics and Astronomy | Year: 2013

This paper investigates the convergence proof of the Direct Simulation Monte Carlo (DSMC) method and the Gas-Kinetic Unified Algorithm in simulating the Boltzmann equation. It can be shown that the particle velocity distribution function obtained by the DSMC method converges to a modified form of the Boltzmann equation, which is the equation of the gas-kinetic unified algorithm to directly solve the molecular velocity distribution function. Their convergence is derived through mathematical treatment. The collision frequency is presented using various molecular models and the local equilibrium distribution function is obtained by Enskog expansion using the converged equation of the DSMC method. These two expressions agree with those used in the unified algorithm. Numerical validation of the converging consistency between these two approaches is illustrated by simulating the pressure driven Poiseuille flow in the slip transition flow regime and the two-dimensional and three-dimensional flows around a circular cylinder and spherical-cone reentry body covering the whole flow regimes from low speed micro-channel flow to high speed non-equilibrium aerothermodynamics. © Science China Press and Springer-Verlag 2013.


Peng A.-P.,China Aerodynamics Research And Development Center | Peng A.-P.,National Laboratory for Computational Fluid Dynamics | Li Z.-H.,China Aerodynamics Research And Development Center | Li Z.-H.,National Laboratory for Computational Fluid Dynamics | And 2 more authors.
Journal of Computational Physics | Year: 2016

Based on the previous researches of the Gas-Kinetic Unified Algorithm (GKUA) for flows from highly rarefied free-molecule transition to continuum, a new implicit scheme of cell-centered finite volume method is presented for directly solving the unified Boltzmann model equation covering various flow regimes. In view of the difficulty in generating the single-block grid system with high quality for complex irregular bodies, a multi-block docking grid generation method is designed on the basis of data transmission between blocks, and the data structure is constructed for processing arbitrary connection relations between blocks with high efficiency and reliability. As a result, the gas-kinetic unified algorithm with the implicit scheme and multi-block docking grid has been firstly established and used to solve the reentry flow problems around the multi-bodies covering all flow regimes with the whole range of Knudsen numbers from 10 to 3.7E−6. The implicit and explicit schemes are applied to computing and analyzing the supersonic flows in near-continuum and continuum regimes around a circular cylinder with careful comparison each other. It is shown that the present algorithm and modelling possess much higher computational efficiency and faster converging properties. The flow problems including two and three side-by-side cylinders are simulated from highly rarefied to near-continuum flow regimes, and the present computed results are found in good agreement with the related DSMC simulation and theoretical analysis solutions, which verify the good accuracy and reliability of the present method. It is observed that the spacing of the multi-body is smaller, the cylindrical throat obstruction is greater with the flow field of single-body asymmetrical more obviously and the normal force coefficient bigger. While in the near-continuum transitional flow regime of near-space flying surroundings, the spacing of the multi-body increases to six times of the diameter of the single-body, the interference effects of the multi-bodies tend to be negligible. The computing practice has confirmed that it is feasible for the present method to compute the aerodynamics and reveal flow mechanism around complex multi-body vehicles covering all flow regimes from the gas-kinetic point of view of solving the unified Boltzmann model velocity distribution function equation. © 2016 The Authors


Xiang Q.,National Laboratory for Computational Fluid Dynamics | Wu S.,National Laboratory for Computational Fluid Dynamics
Lecture Notes in Electrical Engineering | Year: 2012

Lots of problems focus on large positive definite linear systems in science and engineering. It is crucial to solve the systems stably and efficiently. There are two categories of iterative methodologies. One is based on the splitting of coefficient matrices, such as the Jacobi, Gauss-Seidel and successive over relaxation (SOR) iterations. © 2012 Springer-Verlag London Limited.


Ma Q.,China Aerodynamics Research And Development Center | Ma Q.,National Laboratory for Computational Fluid Dynamics | Cui J.,CAS Academy of Mathematics and Systems Science | Li Z.,China Aerodynamics Research And Development Center | Li Z.,National Laboratory for Computational Fluid Dynamics
International Journal of Solids and Structures | Year: 2016

A new second-order two-scale (SOTS) analysis finite element algorithm is developed for the axisymmetric and spherical symmetric elastic problems with small periodic configurations. The axisymmetric structure considered is periodic in both radial and axial directions and homogeneous in circumferential direction, and the spherical symmetric structure is only periodic in radial direction and homogeneous in other two directions. The SOTS asymptotic expansions for the space problem, plane axisymmetric problem, and spherical symmetric problem are presented, and the main feature is that the anisotropic material is obtained by the homogenization. The analytical expressions of the cell functions and homogenized solutions for plane axisymmetric and spherical symmetric problems are obtained, and the error estimations of the expansions are established. The second-order asymptotic analysis finite-element algorithm is presented and the numerical examples are solved including the hollow cylinder, rotating disk and hollow sphere composed of periodic composite materials. The computational results demonstrate the effectiveness and accuracy of the SOTS asymptotic analysis algorithm, and the converging behavior of the asymptotic analysis algorithm agrees well with the theoretical prediction. It is also indicated that the stress distributions can be correctly computed only by adding the second-order correctors. © 2015 Elsevier Ltd.


Li Z.,China Aerodynamics Research And Development Center | Li Z.,National Laboratory for Computational Fluid Dynamics | Jiang X.,China Aerodynamics Research And Development Center | Wu J.,China Aerodynamics Research And Development Center | And 2 more authors.
Lixue Xuebao/Chinese Journal of Theoretical and Applied Mechanics | Year: 2014

Based on the gas-kinetic unified algorithm (GKUA) for flows from rarefied transition to continuum, the Effect of rotational non-equilibrium is investigated involving the kinetic Rykov model with relaxation property of rotational degrees of freedom. 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, then the unified Boltzmann model equation involving rotational non-equilibrium Effect is presented for various flow regimes. The molecular velocity distribution function is integrated by the weight factor on the energy of rotational motion, and the closed system of two kinetic controlling equations is obtained with inelastic and elastic collisions. The discrete velocity ordinate technique and numerical quadrature methods are applied to discretize the velocity space, and the gas-kinetic finite-difference numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions. The gas-kinetic boundary conditions in rotational non-equilibrium and numerical procedures are studied and implemented by directly acting on the velocity distribution function, and then the unified algorithm of the Boltzmann kinetic model equation involving rotational non-equilibrium Effect is presented for the whole range of flow regimes. As the applications of the GKUA, the hypersonic flows of diatomic gas involving rotational non-equilibrium Effect are numerically simulated including the inner flows of shock wave structures in nitrogen with different Mach numbers of 1.5-25, the two-dimensional planar Ramp flow with the whole range of Knudsen numbers of 9 × 10-4-10 and the three-dimensional re-entering hypersonic flows around sphere, tine double-cone and spacecraft body. The computed results match the relevant experimental data, DSMC results, and the solutions of the generalized Boltzmann equation (GBE) and ellipsoidal statistical (ES) model equation well. It is tested and validated from this study that the GKUA solving the Boltzmann model equation in rotational nonequilibrium can simulate the complex hypersonic flow problems and flow mechanisms from high rarefied free-molecule flow to continuum flow regimes with good reliability and precision.


Li Z.-H.,National Laboratory for Computational Fluid Dynamics | Li Z.-H.,China Aerodynamics Research And Development Center | Bi L.,China Aerodynamics Research And Development Center | Zhang H.-X.,National Laboratory for Computational Fluid Dynamics | Li L.,National Laboratory for Computational Fluid Dynamics
Computers and Mathematics with Applications | Year: 2011

The Boltzmann simplified velocity distribution function equation, as adapted to various flow regimes, is described on the basis of the BoltzmannShakhov model from the kinetic theory of gases in this study. The discrete velocity ordinate method of gas-kinetic theory is studied and applied to simulate complex multi-scale flows. On the basis of using the uncoupling technique on molecular movements and collisions in the DSMC method, the gas-kinetic finite difference scheme is constructed by extending and applying the unsteady time-splitting method from computational fluid dynamics, which directly solves the discrete velocity distribution functions. The Gauss-type discrete velocity numerical quadrature technique for flows with different Mach numbers is developed to evaluate the macroscopic flow parameters in the physical space. As a result, the gas-kinetic numerical algorithm is established for studying the three-dimensional complex flows with high Mach numbers from rarefied transition to continuum regimes. On the basis of the parallel characteristics of the respective independent discrete velocity points in the discretized velocity space, a parallel strategy suitable for the gas-kinetic numerical method is investigated and, then, the HPF (High Performance Fortran) parallel programming software is developed for simulating gas dynamical problems covering the full spectrum of flow regimes. To illustrate the feasibility of the present gas-kinetic numerical method and simulate gas transport phenomena covering various flow regimes, the gas flows around three-dimensional spheres and spacecraft-like shapes with different Knudsen numbers and Mach numbers are investigated to validate the accuracy of the numerical methods through HPF parallel computing. The computational results determine the flow fields in high resolution and agree well with the theoretical and experimental data. This computing, in practice, has confirmed that the present gas-kinetic algorithm probably provides a promising approach for resolving hypersonic aerothermodynamic problems with the complete spectrum of flow regimes from the gas-kinetic point of view for solving the mesoscopic Boltzmann model equation. © 2011 Elsevier Ltd. All rights reserved.


Li Z.-H.,China Aerodynamics Research And Development Center | Li Z.-H.,National Laboratory for Computational Fluid Dynamics | Ma Q.,China Aerodynamics Research And Development Center | Ma Q.,National Laboratory for Computational Fluid Dynamics | Cui J.,CAS Academy of Mathematics and Systems Science
Journal of Computational Physics | Year: 2016

The new second-order two-scale (SOTS) finite element algorithm is developed for the dynamic thermo-mechanical coupling problems in axisymmetric and spherical symmetric structures made of composite materials. The axisymmetric structure considered is periodic in both radial and axial directions and homogeneous in circumferential direction. The spherical symmetric structure is only periodic in radial direction. The dynamic thermo-mechanical coupling model is presented and the equivalent compact form is derived. Then, the cell problems, effective material coefficients and the homogenized thermo-mechanical coupling problem are obtained successively by the second-order asymptotic expansion of the temperature increment and displacement. The homogenized material obtained is manifested with the anisotropic property in the circumferential direction. The explicit expressions of the homogenized coefficients in the plane axisymmetric and spherical symmetric cases are given and both the derivation of the analytical solutions of the cell functions and the quasi-static thermoelasticity problems are discussed. Based on the SOTS method, the corresponding finite-element procedure is presented and the unconditionally stable implicit algorithm is established. Some numerical examples are solved and the mutual interaction between the temperature and displacement field is studied under the condition of structural vibration. The computational results demonstrate that the second-order asymptotic analysis finite-element algorithm is feasible and effective in simulating and predicting the dynamic thermo-mechanical behaviors of the composite materials with small periodic configurations in axisymmetric and spherical symmetric structures. This may provide a vital computational tool for analyzing composite material internal temperature distribution and structural deformation induced by the dynamic thermo-mechanical coupling response under strong aerothermodynamic environment. © 2016 The Authors.


Wu J.,China Aerodynamics Research And Development Center | Li Z.,China Aerodynamics Research And Development Center | Li Z.,National Laboratory for Computational Fluid Dynamics | Jiang X.,National Laboratory for Computational Fluid Dynamics
Jisuan Wuli/Chinese Journal of Computational Physics | Year: 2013

To investigate effect of rotational degree of freedom on gas flows covering various flow regimes, Boltzmann-Rykov model is studied and reduced velocity distribution functions are introduced with quadrature to rotational degree of freedom of molecular velocity distribution function. A single gas-kinetic model equation is translated into simultaneous equations of three reduced velocity distribution functions at discrete velocity ordinate points with discrete velocity ordinate method and numerical integration technique. One- and two-dimensional Boltzmann-Rykov model equations for diatomic gases are computed with finite-difference method of computational fluid dynamics. One-dimensional shock-tube and the two-dimensional flow past erect plate are analyzed in the whole range of Knudsen numbers. Reliability of gas-kinetic unified algorithm (GKUA) is validated in solving one-and two-dimensional flows from free molecular flow to continuum regimes. It is indicated that the gas rarefaction degree and molecular inner degree of freedom affect flow field greatly. And rarefied gas flows with high Knudsen numbers take serious non-equilibrium effect.

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