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Journal of Environmental Accounting and Management
António Mendes Lopes (editor), Jiazhong Zhang(editor)
António Mendes Lopes (editor)

University of Porto, Portugal


Jiazhong Zhang (editor)

School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi Province 710049, China

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Gas-Kinetic Unified Algorithm for Aerodynamics Covering Various Flow Regimes Using Computable Modeling of Boltzmann Equation

Journal of Environmental Accounting and Management 11(3) (2023) 243--269 | DOI:10.5890/JEAM.2023.09.001

Zhihui Li$^{1,3}$, Zheng Han$^1$, Aoping Peng$^{2,3}$, Junlin Wu${}^{2,3}$, Xuguo Li${}^{2,3}$

$^1$ National Laboratory for Computational Fluid Dynamics, Beihang University (BUAA), Beijing 100191, China

$^2$ Laboratory of Aerodynamics in Multiple Flow Regimes, CARDC, Mianyang, SiChuan, 621000, China

$^3$ China Aerodynamics Research and Development Center, P.O.Box 211, Mianyang 621000, China

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The gas-kinetic unified algorithm solving the Boltzmann modeling velocity distribution function equation is developed and employed to study gas dynamic problems covering various flow regimes. Based on the computable modeling of Boltzmann equation, the modeled velocity distribution function equation covering to various flow regimes is presented. The discrete velocity ordinate method is developed and applied to remove the velocity space dependency of the distribution function. Based on the uncoupling technique on molecular movement and collision in the DSMC method, the gas-kinetic finite difference scheme is constructed to directly solve the discrete velocity distribution functions by using the unsteady time-splitting method from computational fluid dynamics. The discrete velocity numerical integration method with the Gauss-type weight function is developed to evaluate the macroscopic flow variables at each point in the physical space. The parallel implementation for the gas-kinetic numerical method is investigated to solve the three-dimensional complex flows. To validate the accuracy and feasibility of the present numerical method, one-dimensional shock wave problems, supersonic flows past two-dimensional circular cylinder and three-dimensional hypersonic flows past sphere and spacecraft shape covering various flow regimes are simulated with different Knudsen numbers and Mach numbers. The computational results are found in good agreement with the related theoretical, DSMC, N-S, and experimental data. The computing practice has confirmed that the present gas-kinetic algorithm probably provides a promising approach to resolve the hypersonic aerothermodynamics problems with the complete spectrum of flow regimes from the gas-kinetic point of view of solving the unified Boltzmann model equation.


  1. [1]  Davies, M. A., Pratt, J. R., Dutterer, B., and Burns, T. J. (2002), Stability prediction for low radial immersion milling, Journal of Manufacturing Science and Engineering, 124, 217. % %
  2. [2]  Szalai, R., St{e}p{a}n, G., and Hogan, S. J. (2004), Global dynamics of low immersion high-speed milling, Chaos, 14(4), 1069. % %
  3. [3]  Balachandran, B. (2001), Nonlinear dynamics of milling process, Philosophical Transactions of the Royal Society A, 359, 793-819. % %
  4. [4]  Yang, B. and Suh, C.S. (2003), Interpretation of crack induced nonlinear response using instantaneous frequency, Mechanical Systems and Signal Processing, 18(3), 491-513. % %
  5. [5]  Liu, M.-K. and Suh, C.S. (2012), Temporal and spectral responses of a softening duffing oscillator undergoing route-to-chaos, Communications in Nonlinear Science and Numerical Simulations, 17(6), 2539-2550.
  6. [6]  Chapmann, S. and Cowling, T.G. (1990), The Mathematical Theory of Non-Uniform Gases (3rd ed.), Cambridge Press.
  7. [7]  Riedi, P.C. (1976), Thermal Physics: An Introduction to Thermodynamics, Statistical Mechanics and Kinetic Theory, The Macmillan Press Ltd., London.
  8. [8]  Bhatnagar, P.L., Gross, E.P., and Krook, M. (1954), A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Physical Review, 94(3), 511-525.
  9. [9]  Kogan, M.N. (1958), On the equations of motion of a rarefied gas, Applied Mathematics and Mechanics, 22, 597-607.
  10. [10]  Vincenti, W.G. and Kruger, C.H. (1965), Introduction to Physical Gas Dynamics, Wiley, New York.
  11. [11]  Cercignani, C. (1990), Mathematical Methods in Kinetic Theory (2nd ed.), Plenum Press, New York.
  12. [12]  Holway, L.H. (1966), New statistical models for kinetic theory, methods of construction, The Physics of Fluids, 9(9), 1658-1673.
  13. [13]  Shakhov, E.M. (1968), Generalization of the Krook Kinetic Relaxation Equation, Fluid Dynamics, 3(1), 158-161.
  14. [14]  Abe, T. and Oguchi, H. (1977), A hierarchy kinetic model and its applications, Progress in Astronautics and Aeronautics, 51, 781-793.
  15. [15]  Chu, C.K. (1965), Kinetic-theoretic description of the formation of a shock wave, The Physics of Fluids, 8, 12-22.
  16. [16]  Shakhov, E.M. (1984), Kinetic model equations and numerical results, in: Proc. of 14th International Symposium on Rarefied Gas Dynamics, edited by H. Oguchi, 1, University of Tokyo Press, Tsukubva, Japan, 137-148.
  17. [17]  Morinishi, K. and Oguchi, H. (1984), A computational method and its application to analyses of rarefied gas flows, in: Proc. of 14th International Symposium on Rarefied Gas Dynamics, edited by H. Oguchi, 1, University of Tokyo Press, Tsukubva, Japan, 149-158.
  18. [18]  Chung, C.H., Jeng, D.R., De Witt, K.J., and Keith, T.G. (1992), Numerical simulation of rarefied gas flow through a slit, Journal of Thermophysics and Heat Transfer, 6(1), 27-34.
  19. [19]  Deng, Z.T., Liaw, G.S., and Chou, L.C. (1995), Numerical investigation of low density nozzle flow by solving the boltzmann model equation, in: Proc. of 33rd Aerospace Science Meeting and Exhibit, NASA TM 110492, 9-12.
  20. [20]  Yang, J.Y. and Huang, J.C. (1995), Rarefied flow computations using nonlinear model Boltzmann equations, Journal of Computational Physics, 120(2), 323-339.
  21. [21]  Aoki, K., Kanba, K., and Takata, S. (1997), Numerical analysis of a supersonic rarefied gas flow past a flat plate, Physics of Fluids, 9(4), 1144-1161.
  22. [22]  Mieussens, L. (2000), Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries, Journal of Computational Physics, 162(2), 429-466.
  23. [23]  Li, Z.H. (2001), Study on the Unified Algorithm from Rarefied Flow to Continuum, Ph. D. thesis, China Aerodynamics Researchment and Development Center.
  24. [24]  Li, Z.H. and Zhang, H.X. (2003), Numerical investigation from rarefied flow to continuum by solving the boltzmann model equation, International Journal for Numerical Methods in Fluids, 42(4), 361-382.
  25. [25]  Li, Z.H. and Zhang, H.X. (2004), Study on gas kinetic unified algorithm for flows from rarefied transition to continuum, Journal of Computational Physics, 193(2), 708-738.
  26. [26]  Li, Z.H. and Zhang, H.X. (2007), Gas-kinetic numerical method solving mesoscopic velocity distribution function equation, Acta Mechanica Sinica, 23(3), 121-132.
  27. [27]  Zhang, H.X. and Zhuang, F.G. (1992), NND schemes and their application to numerical simulation of two- and three-dimensional flows, Advances in Applied Mechanics, 29, 193-256.
  28. [28]  Bird, G.A. (1994), Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, London.
  29. [29]  Cercignani, C. (2000), Rarefied Gas Dynamics: from Basic Concepts to Actual Calculations, Cambridge University Press, Cambridge.
  30. [30]  Park, D.C. (1981), The Kinetic Theory of Gases with Applications in Rarefied Gas Dynamics, University of Strathclyde, Glasgow, Scotland.
  31. [31]  Li, Z.H. (2003), Applications of Gas Kinetic Unified Algorithm to Flows from Rarefied to Continuum, Post-doc. Dissertation, Tsinghua University, Beijing.
  32. [32]  Huang, A.B. and Giddens, D.P. (1967), The Discrete Ordinate Method for the Linearized Boundary Value Problems in Kinetic Theory of Gases, in: Proc. of 5th International Symposium on Rarefied Gas Dynamics, edited by C. L. Brundin, 1, Academic Press, New York, 481-486.
  33. [33]  Li, Z.H. and Xie, Y.R. (1996), Technique of molecular indexing applied to the direct simulation Monte Carlo method, in: Proc. of 20th International Symposium on Rarefied Gas Dynamics, edited by C. Shen, Peking University Press, Beijing, China, 205-211.
  34. [34]  Shizgal, B. (1981), A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems, Journal of Computational Physics, 41, 309-327.
  35. [35]  Golub, G.H. and Welsch, J. (1981), Calculation of Gauss Quadrature Rules, Tech. Rep. No. CS81, Computer Science Department, Stanford University.
  36. [36]  Alsmeyer, H. (1976), Density profiles in Argon and Nitrogen Shock waves measured by the absorption of an electron beam, Journal of Fluid Mechanics, 74(3), 497-513.
  37. [37]  Bird, G.A. (1970), Aspects of the Structure of Strong Shock Waves, Physics of Fluids, 13(5), 1172-1177.
  38. [38]  Gilbarg, D. and Paolucci, D. (1953), The structure of shock waves in the continuum theory of fluids, Journal of Rational Mechanics and Analysis, 2, 617-642.
  39. [39]  Vogenitz, F.W., Bird, G.A., Broadwell, J.E., and Rungaldier, H. (1968), Theoretical and experimental study of rarefied supersonic flows about several simple shapes, AIAA Journal, 6(12), 2388-2394.
  40. [40]  Li, Z.H., Peng, A.P., Zhang, H.X., and Yang J.Y. (2015), Rarefied gas flow simulations using high-order gas-kinetic unified algorithms for Boltzmann model equations, Progress in Aerospace Science, 74, 81-113.
  41. [41]  Zheng, Y., Garcia, A.L., and Alder, B.J. (2002), Comparison of kinetic theory and hydrodynamics for Poiseuille flow, Journal of Statistical Physics, 109, 495-505.
  42. [42]  Xu, K. and Li, Z.H. (2004), Microchannel flow in the slip regime: gas-kinetic BGK-Burnett solutions. Journal of Fluid Mechanics, 513, 87-110.
  43. [43]  Li, Z.H., Zhang, H.X., and Fu, S. (2005), Gas Kinetic Algorithm for Flows in Poiseuille-like Microchannels Using Boltzmann Model Equation, Science in China (Series G-Physics, Mechanics $\&$ Astronomy), 48(4), 496-512.
  44. [44]  Peter, P.W. and Harry, A. (1962), Wind tunnel measurements of sphere drag at supersonic speeds and low reynolds, Journal of Fluid Mechanics, 10, 550-560.
  45. [45]  Koppenwallner, G. and Legge, H. (1985), Drag of Bodies in Rarefied Hypersonic Flow. AIAA Paper 85-0998, Progress in Astronautics and Aeronautics: Thermophysical Aspects of Reentry Flows, New York, 103, 44-59.
  46. [46]  Han, Z., Li, Z., Bai, Z., Li, X., and Zhang, J. (2022), Study on numerical algorithm of the NS equation for multi-body flows around irregular disintegrations in near space, Aerospace, 9(7), 347-369.
  47. [47]  Li, X.G., Yang, Y.G., Li, Z.H., and Pi, X.C. (2013), Design methods of the small size strain gauge balance, Journal of Experiments in Fluid Mechanics, 27, 78-82.
  48. [48]  Dai, J.W., Yang, Y.G., and Li, X.G. (2003), Investigation of Rarefied Gas Aerodynamics of Spacecraft Shape in Low Density Wind Tunnel, Tech. Rep. No.S423.15, China Aerodynamic Research \& Development Center.