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Three-Dimensional Lattice Boltzmann Flux Solver and Its Applications to Incompressible Isothermal and Thermal Flows

Part of: Incompressible viscous fluids

Published online by Cambridge University Press:  14 September 2015

Yan Wang ,
Chang Shu ,
Chiang Juay Teo ,
Jie Wu  and
Liming Yang
Yan Wang
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
Chang Shu*
Affiliation:
Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Chiang Juay Teo
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
Jie Wu
Affiliation:
Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Liming Yang
Affiliation:
Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
*
*Corresponding author. Email addresses: [email protected] (Y. Wang), [email protected] (C. Shu), [email protected] (C. J. Teo), [email protected] (J. Wu), [email protected] (L.Yang)
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Abstract

A three-dimensional (3D) lattice Boltzmann flux solver (LBFS) is presented in this paper for the simulation of both isothermal and thermal flows. The present solver combines the advantages of conventional Navier-Stokes (N-S) solvers and lattice Boltzmann equation (LBE) solvers. It applies the finite volume method (FVM) to solve the N-S equations. Different from the conventional N-S solvers, its viscous and inviscid fluxes at the cell interface are evaluated simultaneously by local reconstruction of LBE solution. As compared to the conventional LBE solvers, which apply the lattice Boltzmann method (LBM) globally in the whole computational domain, it only applies LBM locally at each cell interface, and flow variables at cell centers are given from the solution of N-S equations. Since LBM is only applied locally in the 3D LBFS, the drawbacks of the conventional LBM, such as limitation to uniform mesh, tie-up of mesh spacing and time step, tedious implementation of boundary conditions, are completely removed. The accuracy, efficiency and stability of the proposed solver are examined in detail by simulating plane Poiseuille flow, lid-driven cavity flow and natural convection. Numerical results show that the LBFS has a second order of accuracy in space. The efficiency of the LBFS is lower than LBM on the same grids. However, the LBFS needs very less non-uniform grids to get grid-independence results and its efficiency can be greatly improved and even much higher than LBM. In addition, the LBFS is more stable and robust.

Keywords

Lattice Boltzmann flux solverlattice Boltzmann methodNavier-Stokes equationlid-driven cavity flowsnatural convection

MSC classification

Secondary: 76D05: Navier-Stokes equations
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 3 , September 2015 , pp. 593 - 620
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Dennis, S.C.R., Quartapelle, L., Direct solution of the vorticity-stream function ordinary differential equations by a Chebyshev approximation, Journal of Computational Physics, 52 (1983) 448–463.CrossRefGoogle Scholar
[2]Ehrenstein, U., Peyret, R., A Chebyshev collocation method for the NavierStokes equations with application to double-diffusive convection, International Journal for Numerical Methods in Fluids, 9 (1989) 427–452.CrossRefGoogle Scholar
[3]Shu, C., Richards, B.E., Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, 15 (1992) 791–798.Google Scholar
[4]Chorin, A.J., Numerical solution of the Navier-Stokes equation, Mathematics of Computation, 22 (1968) 745–762.Google Scholar
[5]Ramshaw, J.D., Mousseau, V.A., Accelerated artificial compressibility method for steady-state incompressible flow calculations, Computers & Fluids, 18 (1990) 361–367.Google Scholar
[6]Tamamidis, P., Zhang, G., Assanis, D.N., Comparison of pressure-based and artificial compressibility methods for solving 3D steady incompressible viscous flows, Journal of Computational Physics, 124 (1996) 1–13.CrossRefGoogle Scholar
[7]Patankar, S.V., Spalding, D.B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, International Journal of Heat and Mass Transfer, 15 (1972) 1787–1806.CrossRefGoogle Scholar
[8]Brown, D.L., Cortez, R., Minion, M.L., Accurate projection methods for the incompressible NavierStokes equations, Journal of Computational Physics, 168 (2001) 464–499.Google Scholar
[9]Qian, Y.H., Humiéres, D.D., Lallemand, P., Lattice BGK models for Navier-Stokes equation, EPL (Europhysics Letters), 17 (1992) 479.CrossRefGoogle Scholar
[10]Aidun, C.K., Clausen, J.R., Lattice-Boltzmann method for complex flows, Annual Review of Fluid Mechanics, 42 (2010) 439–472.Google Scholar
[11]Guo, Z.L., Shu, C., Lattice Boltzmann Method and Its Applications in Engineering, World Scientific Publishing, (2013).Google Scholar
[12]Chen, S., Doolen, G.D., Lattice Boltzmann method for fluid flows, Annual Review of Fluid Mechanics, 30 (1998) 329–364.CrossRefGoogle Scholar
[13]He, X., Chen, S., Doolen, G.D., A novel thermal model for the lattice Boltzmann method in incompressible limit, Journal of Computational Physics, 146 (1998) 282–300.CrossRefGoogle Scholar
[14]Kim, J., Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, Journal of Computational Physics, 59 (1985) 308–323.Google Scholar
[15]van Kan, J., A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM Journal on Scientific and Statistical Computing, 7 (1986) 870–891.Google Scholar
[16]Bell, J.B., Colella, P., Glaz, H.M., A second-order projection method for the incompressible navier-stokes equations, Journal of Computational Physics, 85 (1989) 257–283.Google Scholar
[17]Charles, M., Time-accurate unsteady incompressible flow algorithms based on artificial compressibility, in: 8th Computational Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics, 1987.Google Scholar
[18]Lim, C.Y., Shu, C., Niu, X.D., Chew, Y.T., Application of lattice Boltzmann method to simulate microchannel flows, Physics of Fluids, 14 (2002) 2299–2308.Google Scholar
[19]Shu, C., Niu, X.D., Chew, Y.T., Taylor-series expansion and least-squares-based lattice Boltzmann method: Two-dimensional formulation and its applications, Physical Review E, 65 (2002) 036708.CrossRefGoogle ScholarPubMed
[20]Niu, X.D., Chew, Y.T., Shu, C., Simulation of flows around an impulsively started circular cylinder by Taylor series expansion- and least squares-based lattice Boltzmann method, Journal of Computational Physics, 188 (2003) 176–193.CrossRefGoogle Scholar
[21]Guo, Z., Zheng, C., Shi, B., Zhao, T.S., Thermal lattice Boltzmann equation for low Mach number flows: Decoupling model, Physical Review E, 75 (2007) 036704.CrossRefGoogle ScholarPubMed
[22]Guo, Z., Han, H., Shi, B., Zheng, C., Theory of the lattice Boltzmann equation: Lattice Boltzmann model for axisymmetric flows, Physical Review E, 79 (2009) 046708.Google Scholar
[23]Shan, X., Chen, H., Lattice Boltzmann model for simulating flows with multiple phases and components, Physical Review E, 47 (1993) 1815–1819.Google Scholar
[24]Peng, G., Xi, H., Duncan, C., Chou, S.-H., Lattice Boltzmann method on irregular meshes, Physical Review E, 58 (1998) R4124–R4127.Google Scholar
[25]Xi, H.W., Peng, G.W., Chou, S.H., Finite-volume lattice Boltzmann method, Physical Review E, 59 (1999) 6202–6205.Google Scholar
[26]Patil, D.V., Lakshmisha, K.N., Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, Journal of Computational Physics, 228 (2009) 5262–5279.Google Scholar
[27]Fakhari, A., Lee, T., Finite-difference lattice Boltzmann method with a block-structured adaptive-mesh-refinement technique, Physical Review E, 89 (2014) 033310.CrossRefGoogle ScholarPubMed
[28]Peng, Y., Shu, C., Chew, Y.T., A 3D incompressible thermal lattice Boltzmann model and its application to simulate natural convection in a cubic cavity, Journal of Computational Physics, 193 (2004) 260–274.Google Scholar
[29]Peng, Y., Shu, C., Chew, Y.T., Simplified thermal lattice Boltzmann model for incompressible thermal flows, Physical Review E, 68 (2003) 026701.CrossRefGoogle ScholarPubMed
[30]Wang, J., Wang, M., Li, Z., Lattice PoissonBoltzmann simulations of electro-osmotic flows in microchannels, Journal of Colloid and Interface Science, 296 (2006) 729–736.CrossRefGoogle ScholarPubMed
[31]Huang, H., Lee, T.S., Shu, C., Hybrid lattice Boltzmann finite-difference simulation of axisymmetric swirling and rotating flows, International Journal for Numerical Methods in Fluids, 53 (2007) 1707–1726.CrossRefGoogle Scholar
[32]Lee, T., Liu, L., Lattice Boltzmann simulations of micron-scale drop impact on dry surfaces, Journal of Computational Physics, 229 (2010) 8045–8063.CrossRefGoogle Scholar
[33]Lee, T., Lin, C.-L., A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio, Journal of Computational Physics, 206 (2005) 16–47.Google Scholar
[34]Peng, Y., Shu, C., Chew, Y.T., Qiu, J., Numerical investigation of flows in Czochralski crystal growth by an axisymmetric lattice Boltzmann method, Journal of Computational Physics, 186 (2003) 295–307.CrossRefGoogle Scholar
[35]Monaco, E., Luo, K., Brenner, G., Multiple Relaxation Time Lattice Boltzmann Simulation of Binary Droplet Collisions, in: Tromeur-Dervout, D., Brenner, G., Emerson, D.R., Erhel, J. (Eds.) Parallel Computational Fluid Dynamics 2008, Springer Berlin Heidelberg, 2010, pp. 257–264.CrossRefGoogle Scholar
[36]Wu, J., Shu, C., Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications, Journal of Computational Physics, 228 (2009) 1963–1979.Google Scholar
[37]Martin, D.F., Colella, P., Graves, D., A cell-centered adaptive projection method for the incompressible Navier-Stokes equations in three dimensions, Journal of Computational Physics, 227 (2008) 1863–1886.Google Scholar
[38]San, O., Staples, A.E., A coarse-grid projection method for accelerating incompressible flow computations, Journal of Computational Physics, 233 (2013) 480–508.Google Scholar
[39]Shapiro, E., Drikakis, D., Artificial compressibility, characteristics-based schemes for variable-density, incompressible, multispecies flows: Part II. Multigrid implementation and numerical tests, Journal of Computational Physics, 210 (2005) 608–631.CrossRefGoogle Scholar
[40]He, X., Luo, L.-S., Dembo, M., Some progress in lattice Boltzmann method. Part I. Nonuniform mesh grids, Journal of Computational Physics, 129 (1996) 357–363.Google Scholar
[41]Shu, C., Wang, Y., Teo, C.J., Wu, J., Development of lattice Boltzmann flux solver for simulation of incompressible flows, Advances in Applied Mathematics and Mechanics, 6 (2014) 436–460.Google Scholar
[42]Wang, Y., Shu, C., Teo, C.J., Development of LBGK and incompressible LBGK-based lattice Boltzmann flux solvers for simulation of incompressible flows, International Journal for Numerical Methods in Fluids, 75 (2014) 344–364.Google Scholar
[43]Wang, Y., Shu, C., Teo, C.J., Thermal lattice Boltzmann flux solver and its application for simulation of incompressible thermal flows, Computers & Fluids, 94 (2014) 98–111.Google Scholar
[44]Shu, C., Wang, Y., Yang, L.M., Wu, J., Lattice Boltzmann flux solver: An efficient approach for numerical simulation of fluid flows, Transactions of Nanjing University of Aeronautics and Astronautics, 31 (2014) 1–15.Google Scholar
[45]Wang, Y., Shu, C., Teo, C.J., A fractional step axisymmetric lattice Boltzmann flux solver for incompressible swirling and rotating flows, Computers & Fluids, 96 (2014) 204–214.Google Scholar
[46]Ku, H.C., Hirsh, R.S., Taylor, T.D., A pseudospectral method for solution of the three-dimensional incompressible Navier-Stokes equations, Journal of Computational Physics, 70 (1987) 439–462.Google Scholar
[47]Ding, H., Shu, C., Yeo, K.S., Xu, D., Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method, Computer Methods in Applied Mechanics and Engineering, 195 (2006) 516–533.Google Scholar
[48]Fusegi, T., Hyun, J.M., Kuwahara, K., Farouk, B., A numerical study of three-dimensional natural convection in a differentially heated cubical enclosure, International Journal of Heat and Mass Transfer, 34 (1991) 1543–1557.Google Scholar
[49]Ha, M.Y., Jung, M.J., A numerical study on three-dimensional conjugate heat transfer of natural convection and conduction in a differentially heated cubic enclosure with a heat-generating cubic conducting body, International Journal of Heat and Mass Transfer, 43 (2000) 4229–4248.Google Scholar
[50]Asinari, P., Ohwada, T., Chiavazzo, W. and Rienzo, A.F.D., Link-wise artificial compressibility method, Journal of Computational Physics, 231 (2012) 5019–5143.Google Scholar