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A Discrete Flux Scheme for Aerodynamic and Hydrodynamic Flows

Published online by Cambridge University Press:  20 August 2015

S. C. Fu ,
R. M. C. So  and
W. W. F. Leung
S. C. Fu*
Affiliation:
Mechanical Engineering Department, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
R. M. C. So*
Affiliation:
Mechanical Engineering Department, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088, USA
W. W. F. Leung*
Affiliation:
Mechanical Engineering Department, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Research Institute of Innovative Products & Technologies, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
*
Corresponding author.Email:mm.scfu@polyu.edu.hk
Email:mmmcso@polyu.edu.hk
Email:mmwleung@polyu.edu.hk
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Abstract

The objective of this paper is to seek an alternative to the numerical simulation of the Navier-Stokes equations by a method similar to solving the BGK-type modeled lattice Boltzmann equation. The proposed method is valid for both gas and liquid flows. A discrete flux scheme (DFS) is used to derive the governing equations for two distribution functions; one for mass and another for thermal energy. These equations are derived by considering an infinitesimally small control volume with a velocity lattice representation for the distribution functions. The zero-order moment equation of the mass distribution function is used to recover the continuity equation, while the first-order moment equation recovers the linear momentum equation. The recovered equations are correct to the first order of the Knudsen number (Kn); thus, satisfying the continuum assumption. Similarly, the zero-order moment equation of the thermal energy distribution function is used to recover the thermal energy equation. For aerodynamic flows, it is shown that the finite difference solution of the DFS is equivalent to solving the lattice Boltzmann equation (LBE) with a BGK-type model and a specified equation of state. Thus formulated, the DFS can be used to simulate a variety of aerodynamic and hydrodynamic flows. Examples of classical aeroacoustics, compressible flow with shocks, incompressible isothermal and non-isothermal Couette flows, stratified flow in a cavity, and double diffusive flow inside a rectangle are used to demonstrate the validity and extent of the DFS. Very good to excellent agreement with known analytical and/or numerical solutions is obtained; thus lending evidence to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid flow simulations.

Keywords

76M2576D05Aerodynamicshydrodynamicslattice Boltzmann equation
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 5 , May 2011 , pp. 1257 - 1283
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]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(3) (1954), 511–525.Google Scholar
[2]Martinez, D. O., Matthaeus, W. H., Chen, S., and Montgomery, D. C., Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics, Phys. Fluids., 6 (1994), 1285–1298.Google Scholar
[3]Chen, S., and Doolen, G. D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid. Mech., 30 (1998), 329–364.CrossRefGoogle Scholar
[4]Mei, R., Shyy, W., Yu, D., and Luo, L. S., Lattice Boltzmann method for 3-D flows with curved boundary, J. Comput. Phys., 161 (2000), 680–699.Google Scholar
[5]Fu, S. C., Leung, W. W., and So, R. M. C., A lattice Boltzmann based numerical scheme for microchannel flows, J. Fluids. Eng., 131 (2009), 081401.CrossRefGoogle Scholar
[6]McNamara, G., and Alder, B., Analysis of the lattice Boltzmann treatment of hydrodynamics, Phys. A., 194 (1993), 218–228.CrossRefGoogle Scholar
[7]He, X., Chen, S., and Doolen, G. D., A novel thermal model for the lattice Boltzmann method in incompressible limit, J. Comput. Phys., 146 (1998), 282–300.Google Scholar
[8]Li, X. M., Leung, R. C. K., and So, R. M. C., One-step aeroacoustics simulation using lattice Boltzmann method, AIAA J., 44(1) (2006), 78–89.Google Scholar
[9]Fu, S. C., So, R. M. C., and Leung, R. C. K., Modeled Boltzmann equation and its application to direct aeroacoustics simulation, AIAA J., 46(7) (2008), 1651–1662.Google Scholar
[10]Kam, E. W. S., So, R. M. C., and Leung, R. C. K., Acoustic scattering by a localized thermal disturbance, AIAA J., 47(9) (2009), 2039–2052.Google Scholar
[11]Kam, E. W. S., So, R. M. C., Fu, S. C., and Leung, R. C. K., Finite difference lattice Boltzmann method applied to acoustic-scattering problems, AIAA J., 48(2) (2010), 354–371.CrossRefGoogle Scholar
[12]So, R. M. C., Leung, R. C. K., and Fu, S. C., Modeled Boltzmann equation and its application to shock capturing simulation, AIAA J., 46(12) (2008), 3038–3048.Google Scholar
[13]So, R. M. C., Fu, S. C., and Leung, R. C. K., Finite difference lattice Boltzmann method for compressible thermal fluids, AIAA J., 48(6) (2010), 1059–1071.Google Scholar
[14]Rowlinson, J. S., and Widom, B., Molecular Theory of Capillarity, Clarendon Press, 1982.Google Scholar
[15]Shan, X., and Chen, H., Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E., 47 (1993), 1815–1820.Google Scholar
[16]Shan, X., and Chen, H., Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation, Phys. Rev. E., 49 (1994), 2941–2948.Google Scholar
[17]Swift, M. R., Osborn, W. R., and Yeomans, J. M., Lattice Boltzmann simulation of nonideal fluids, Phys. Rev. Lett., 75 (1995), 830–834.Google Scholar
[18]Chin, J., Boek, E. S., and Coveney, P. V., Lattice Boltzmann simulation of the flow of binary immiscible fluids with different viscosities using the Shan-Chen microscopic interaction model, Phil. Trans. Royal. Soc. London. A., 360 (2002), 547–558.CrossRefGoogle ScholarPubMed
[19]Shan, X., Yuan, X. F., and Chen, H., Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation, J. Fluid. Mech., 550 (2006), 413–441.CrossRefGoogle Scholar
[20]Phillips, O. M., On flows induced by diffusion in a stably stratified fluid, Deep. Sea. Res., 17 (1970), 435–443.Google Scholar
[21]Chen, C. F., Double-diffusive convection in an inclined slot, J. Fluid. Mech., 72 (1975), 721– 729.Google Scholar
[22]Fernando, H. J. S., The formation of a layered structure when a stable salinity gradient is heated from below, J. Fluid. Mech., 182 (1987), 525–541.Google Scholar
[23]He, X., and Luo, L. S., Lattice Boltzmann model for the incompressible Navier-Stokes equation, J. Stat. Phys., 88 (1997), 927–944.Google Scholar
[24]Panton, R. L., Incompressible Flow, 2nd ed., Chapter 10, Wiley-Interscience, New York, 1996.Google Scholar
[25]Fu, S. C., and So, R. M. C., Modeled lattice Boltzmann equation and the constant density assumption, AIAA J., 47(12) (2009), 3038–3042.Google Scholar
[26]Guo, Z., Shi, B., and Zheng, C., A coupled lattice BGK model for the Boussinesq equations, Int. J. Numer. Methods. Fluids., 39 (2002), 325–342.Google Scholar
[27]Li, Q., He, Y. L., Wang, Y., and Tao, W. Q., Coupled double-distribution-function lattice Boltz-mann method for the compressible Navier-Stokes equations, Phys. Rev. E., 76 (2007), 056705.Google Scholar
[28]Fu, S. C., Numerical simulation of blood flow in stenotic arteries, PhD Thesis, Mechanical Engineering Department, Hong Kong Polytechnic University, Hong Kong, 2011.Google Scholar
[29]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 2nd ed., Chapter 15, Springer-Verlag Berlin Heidelberg, New York, 1999.Google Scholar
[30]Mittal, R., and Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid. Mech., 37 (2005), 239–261.Google Scholar
[31]Patankar, S. V., and Spalding, D. B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat. Mass. Trans., 15 (1972), 1787–1806.CrossRefGoogle Scholar
[32]Kantha, L. H., and Clayson, C. A., Small Scale Processes in Geophysical Fluid Flows, Academic Press, 2000.Google Scholar
[33]Brenner, H., Kinematics of volume transport, Phys. A., 349 (2005), 11–59.CrossRefGoogle Scholar
[34]Brenner, H., Navier-Stokes revisited, Phys. A., 349 (2005), 60–132.CrossRefGoogle Scholar
[35]Greenshields, C. J., and Reese, J. M., The structure of shock waves as a test of Brenner’s modifications to the Navier-Stokes equations, J. Fluid. Mech., 580 (2007), 407–429.Google Scholar
[36]Alsmeyer, H., Density profiles in Argon and Nitrogen shock waves measured by the absorption of an electron beam, J. Fluid. Mech., 74 (1976), 497–513.Google Scholar
[37]Lee, J. W., and Hyun, J. M., Double-diffusive convection in a rectangle with opposing horizontal temperature and concentration gradients, Int. J. Heat. Mass. Trans., 33(8) (1990), 1619– 1632.Google Scholar
[38]Bejan, A., A note on Gill’s solution for free convection in a vertical enclosure, J. Fluid. Mech., 90 (1979), 561–568.Google Scholar