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Numerical Simulations of Unsteady Flows From Rarefied Transition to Continuum Using Gas-Kinetic Unified Algorithm

Published online by Cambridge University Press:  21 July 2015

Junlin Wu ,
Zhihui Li ,
Aoping Peng  and
Xinyu Jiang
Junlin Wu
Affiliation:
Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China
Zhihui Li*
Affiliation:
Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China National Laboratory for Computational Fluid Dynamics, Beijing 100191, China
Aoping Peng
Affiliation:
Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China
Xinyu Jiang
Affiliation:
Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China
*
*Corresponding author. Email: zhli0097@x263.net (Z. H. Li)
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Abstract

Numerical simulations of unsteady gas flows are studied on the basis of Gas-Kinetic Unified Algorithm (GKUA) from rarefied transition to continuum flow regimes. Several typical examples are adopted. An unsteady flow solver is developed by solving the Boltzmann model equations, including the Shakhov model and the Rykov model etc. The Rykov kinetic equation involving the effect of rotational energy can be transformed into two kinetic governing equations with inelastic and elastic collisions by integrating the molecular velocity distribution function with the weight factor on the energy of rotational motion. Then, the reduced velocity distribution functions are devised to further simplify the governing equation for one- and two-dimensional flows. The simultaneous equations are numerically solved by the discrete velocity ordinate (DVO) method in velocity space and the finite-difference schemes in physical space. The time-explicit operator-splitting scheme is constructed, and numerical stability conditions to ascertain the time step are discussed. As the application of the newly developed GKUA, several unsteady varying processes of one- and two-dimensional flows with different Knudsen number are simulated, and the unsteady transport phenomena and rarefied effects are revealed and analyzed. It is validated that the GKUA solver is competent for simulations of unsteady gas dynamics covering various flow regimes.

Keywords

82C4074H15Unsteady flowcovering various flow regimeskinetic theory of gasesBoltzmann model equationgas-kinetic unified algorithmdiscrete velocity ordinate methodShakov kinetic modelRykov kinetic model

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Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 5 , October 2015 , pp. 569 - 596
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Bergh, H., Tijdeman, H. and Zwaan, R. J., Theory and Experiment in Unsteady Aerodynamics, Aeroelastic Colloquium, Gottingen, N82-17128, 1980.Google Scholar
[2]Thorley, A. R. D. and Tiley, C. H., Unsteady and transient flow of compressible fluids in pipelinesła review of theoretical and some experimental studies, Int. J. Heat Fluid Flow, 8 (1987), pp. 3–15.Google Scholar
[3]El Chebair, R., Misra, A. K. and Paidoussis, M. P., Theoretical study of the effect of unsteady viscous forces on inner- and annular-flow-induced instabilities of cylindrical shells, J. Sound Vibration, 138 (1990), pp. 457–478.Google Scholar
[4]Azuma, A. and Okamoto, M., Theoretical study on two-dimensional aerodynamic characteristics of unsteady wings, J. Theoretical Biology, 234 (2005), pp. 67–78.Google Scholar
[5]Jiang, Z., Takayama, K., Babinsky, H. and Meguro, T., Transient shock wave flows in tubes with a sudden change in cross section, Shock Waves, 7 (1997), pp. 151–162.Google Scholar
[6]Kelder, J. D. H., Dijkers, R. J. H., Esch, B. P. M. and Kruyt, N. P., Experimental and theoretical study of the flow in the volute of a low specific-speed pump, Fluid Dynamics Research, 28 (2001), pp. 267–280.Google Scholar
[7]Mamou, M., Tahi, A., Benmeddour, A., Cooper, K. R., Abdallah, I., Khalid, M. and Fitzsimmons, J., Computational fluid dynamics simulations and wind tunnel measurements of unsteady wind loads on a scaled model of a very large optical telescope: A comparative study, J. Wind Eng. Indust. Aerodyn., 96 (2008), pp. 257–288.Google Scholar
[8]Mladin, E. C. and Padet, J., Unsteady planar stagnation flow on a heated plate, Int. J. Thermal Sci., 40 (2001), pp. 638–648.Google Scholar
[9]Fernie, R. M. and Babinsky, H., Unsteady shock motion on a NACA0012 aerofoil at low reduced frequencies, AIAA, 2004-49.Google Scholar
[10]Thomas, JEFFREY P. and Custer, CHAD H., Unsteady flow computation using a harmonic balance approach implemented about the overflow 2 flow solver, AIAA, 2009-4270.Google Scholar
[11]Al-Mdallal, Q. M., A numerical study of initial flow past a circular cylinder with combined streamwise and transverse oscillations, Comput. Fluids, 63 (2012), pp. 174–183.Google Scholar
[12]Ohwada, T., Sone, Y. and Aoki, K., Numerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall on the basis of the linearized Boltzmann equation for hard-sphere molecules, Phys. Fluids A, 1 (1989), pp. 1588–1599.Google Scholar
[13]Austin, J. V. and Goldstein, D. B., Rarefied gas model of Io’s sublimation-driven atmosphere, Icarus, 148 (2000), pp. 370–383.Google Scholar
[14]Apte, S. and Yang, V., Unsteady flow evolution and combustion dynamics of homogeneous solid propellant in a rocket motor, Combus. Flame, 131 (2002), pp. 110–131.Google Scholar
[15]Bird, G. A., Approach to translational equilibrium in a rigid sphere gas, Phys. Fluids, 6 (1963), pp. 1518–1519.CrossRefGoogle Scholar
[16]Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, 1994.CrossRefGoogle Scholar
[17]Li, Z. L., Liang, J. and Xie, Y. R., DSMC analysis of normal shock wave structure in gas mixtures involving internal energy transfer, Proc. Second ASIA Workshop on Computational Fluid Dynamics, Tokyo, Japan, December 19-23, (1996), pp. 233–238.Google Scholar
[18]Alexeenko, A. A., Fedosov, D. A., Gimelshein, S. F. and Levin, D. A., Transient heat transfer and gas flow in a MEMS-based thruster, J. Microelectromech. Syst., 15 (2006), pp. 181–194.Google Scholar
[19]Goldsworthy, MARK, Macrossan, MICHAEL and Abdel-Jawad, MADHAT, Simulation of unsteady flows by the DSMC macroscopic chemistry method, J. Comput. Phys., 228 (2009), pp. 976–982.Google Scholar
[20]Lim, C. W., Smith, M. R., Cave, H. M., Jermy, M. C., Wu, J. S. and Krumdieck, S. P., The direction decoupled quiet direct simulation method for rapid simulation of axisymmetric inviscid unsteady flow in pulsed pressure chemical vapour deposition, Comput. Fluids, 86 (2013), pp. 14–27.Google Scholar
[21]Sharipov, Felix, Transient flow of rarefied gas through a short tube, Vacuum, 90 (2013), pp. 25–30.CrossRefGoogle Scholar
[22]Hwang, Young-Kyu, Heo, Joong-Sik and Kwon, Myoung-Keun, Unsteady computations of rarefied gas flows induced by a rotor-stator interaction in a disk-type drag pump, Rarefied Gas Dynamics, 23rd International Symposium, 2003.Google Scholar
[23]Hadjiconstantinou, N., Garcia, A., Bazant, M. and He, G., Statistical error in particle simulations of hydrodynamic phenomena, J. Comput. Phys., 187 (2003), pp. 274–297.Google Scholar
[24]Chigullapalli, Sruti and Alexeenko, Alina, Unsteady 3D rarefied flow solver based on Boltzmann-ESBGK model kinetic equations, AIAA, 2011-3993.Google Scholar
[25]Cercignani, C., Kinetic Theories and the Boltzmann Equation, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1984.Google Scholar
[26]Luo, L. S., Krafczyk, M. and Liu, Y., Mesoscopic methods in engineering and science, Int. Comput. Fluid Dyn., 20 (2006), pp. 359–360.Google Scholar
[27]Li, Z. H. and H X, H. X.ZHANG, Gas-kinetic numerical method solving mesoscopic velocity distribution function equation, Acta Mech. Sinica, 23 (2007), pp. 21–132.Google Scholar
[28]Mieussens, L., Discrete velocity model and implicit scheme for the bgk equation of rarefied gas dynamics, Math. Models Meth. Appl. Sci., (2000).Google Scholar
[29]Wagner, Alexander, Theory and Applications of the Lattice Boltzmann Method, University of Oxford, 1997.Google Scholar
[30]Buick, James Maxwell, Lattice Boltzmann Methods in Interfacial Wave Modelling, Philosophy: The University of Edinburgh, 1997.Google Scholar
[31]Nie, X., Doolen, G. D. and Chen, S., Lattice Boltzmann simulation of fluid flows in MEMS, J. Statist. Phys., 107 (2002), 279.Google Scholar
[32]Ricot, Denis, Maillard, Virginie and Bailly, Christophe, Numerical simulation of unsteady cavity flow using lattice Boltzmann method, AIAA, 2002-2532.Google Scholar
[33]Xu, K., A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys., 171 (2001), pp. 289–335.Google Scholar
[34]Xu, K. and Li, Z. H., Microchannel flow in the slip regime: gas-kinetic BGK-Burnett solutions, J. Fluid Mech., 513 (2004), pp. 87–110Google Scholar
[35]Li, Z. H. and Zhang, H. X., Study on gas kinetic unified algorithm for flows from rarefied transition to continuum, J. Comput. Phys., 193 (2004), pp. 708–738.Google Scholar
[36]Chen, S. Z., Xu, K., Lee, C. B. and Cai, Q. D., A unified gas kinetic scheme with moving mesh and velocity space adaptation, J. Comput. Phys., 231 (2012), pp. 6643–6664.Google Scholar
[37]Guo, Z. L. and Zheng, C. G., Theory and Applications of Lattice Boltzmann Method, Beijing: Science Press, 2009.Google Scholar
[38]Huang, H. B., Lu, X. Y. and Krafczyk, M., Numerical simulation of unsteady flows in C-zochralski crystal growth by lattice Boltzmann methods, Int. J. Heat Mass Trans., 74 (2014), pp. 156–163.Google Scholar
[39]Hejranfar, K. and Ezzatneshan, E., Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates, J. Comput. Phys., 267 (2014), pp. 28–49.Google Scholar
[40]Bhatnagar, P. L., Gross, E. P. and Krook, M., A model for collision processes in gases: i. small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), pp. 511–525.Google Scholar
[41]Jr.Holway, L. H., New statistical models for kinetic theory: methods of construction, Phys. Fluids, 9 (1966), pp. 1658–1673.Google Scholar
[42]Shakhov, E. M., Generalization of the krook kinetic relaxation equation, Fluid Dyn., 3 (1968), pp. 142–145.Google Scholar
[43]Li, Z. H. and Zhang, H. X., Gas-kinetic description of shock wave structures by Boltzmann model equation, Int. J. Comput. Fluid Dyn., 22 (2008), pp. 623–638.Google Scholar
[44]Wilson, CHRISTOPHER D. and Agarwal, R. K., Computation of hypersonic flow of a diatomic gas in rotational nonequilibrium past 3d blunt bodies using the generalized Boltzmann equation, AIAA, 2009-3836.CrossRefGoogle Scholar
[45]Lihnaropoulos, John and Valougeorgis, Dimitris, Unsteady vacuum gas flow in cylindrical tubes, Fusion Eng. Design., 86 (2011), pp. 2139–2142.CrossRefGoogle Scholar
[46]Chigullapalli, Sruti and Alexeenko, Alina, Unsteady 3D rarefied flow solver based on Boltzmann-ESBGK model kinetic equations, AIAA, 2011-3993.Google Scholar
[47]Polikarpov, A. PH. and Graur, I., Unsteady rarefied gas flow through a slit, Vacuum, 101 (2014), pp. 79–85.Google Scholar
[48]Cai, Chunpei and Wang, L., Gas kinetic analytical and numerical studies on rarefied unsteady planar jet flows, Phys. Scripta, 88 (2013), pp. 1–8.Google Scholar
[49]Cai, C., Sun, Q. and Vanderwyst, A., Analytical exact solutions for unsteady collisionless plume flows in a vacuum, Acta Astronautica, 91 (2013), pp. 218–227.Google Scholar
[50]Li, Z. H., Study on Gas-Kinetic Unified Algorithm from Rarefied Transition to Continuum, Mianyang in Sichuan: China Aerodynamics Research and Development Center, 2001.Google Scholar
[51]Li, Z. H. and Zhang, H. X., Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation, Int. J. Numer. Methods Fluids, 42 (2003), pp. 361–382.Google Scholar
[52]Li, Z. H. and Zhang, H. X., A gas kinetic algorithm for flows in microchannel, Int. J. Nonlinear. Sci. Numer. Simulation, 6 (2005), pp. 261–270.Google Scholar
[53]Li, Z. H. and Zhang, H. X., Gas-kinetic numerical studies of three-dimensional complex flows on spacecraft re-entry, J. Comput. Phys., 228 (2009), pp. 1116–1138.Google Scholar
[54]Rykov, V. A., Model kinetic equation of a gas with rotational degrees of freedom, Fluid Dyn., 10 (1975), pp. 959–966.Google Scholar
[55]Shakhov, E. M., Approximate kinetic equations in rarefied gas theory, Fluid Dyn., 3 (1968), pp. 156–161.Google Scholar
[56]Rykov, V. A., Titarev, V. A. and Shakhov, E. M., Numerical study of the transverse supersonic flow of a diatomic rarefied gas past a plate, Comput. Math. Math. Phys., 47 (2007), pp. 136–150.CrossRefGoogle Scholar
[57]Rykov, V. A., Titarev, V. A. and Shakhov, E. M., Shock wave structure in a diatomic gas based on a kinetic model, Fluid Dyn., 43 (2008), pp. 316–326.Google Scholar
[58]Zhang, Hanxin, Numerical problems for unsteady flows, 6th Countrywide Hydrodynamics Conference, Shanghai, 2001, 23-31.Google Scholar
[59]Zhang, Hanxin and Shen, Mengyu, Computational Fluid Dynamics-Fundamentals and Applications of Finite Difference Methods, Beijing National Defence Industry Press, 2003.Google Scholar
[60]Li, Z. H., Fang, M., Jiang, X. Y. and Wu, J. C., Convergence proof of the DSMC method and the gas-kinetic unified algorithm for the Boltzmann equation, Sci. China Phys. Mech. Astronomy, 56 (2013), pp. 404–417.Google Scholar
[61]Yang, J. Y., Rarefied flow computations using nonlinear model Boltzmann equations, J. Comput. Phys., 120 (1995), pp. 323–339.Google Scholar
[62]Yang, J. Y. and Huang, J. C., Computations of kinetic model equations for gases with internal degrees of freedom, AIAA, 95-2315.Google Scholar
[63]Yang, J. Y. and Huang, J. C., High resolution schemes for rarefied gas dynamics using kinetic model equations, AIAA, 95-1678-CP.Google Scholar
[64]Yang, J. Y., Nonoscillatory schemes for kinetic model equations for gases with internal energy states, AIAA J., 34 (1996).Google Scholar