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Combined immersed boundary method and multiple-relaxation-time lattice Boltzmann flux solver for numerical simulations of incompressible flows

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Abstract

method combining the immersed boundary technique and a multi-relaxation-time (MRT) lattice Boltzmann flux solver (LBFS) is presented for numerical simulation of incompressible flows over circular and elliptic cylinders and NACA 0012 Airfoil. The method uses a simple Cartesian mesh to simulate flows past immersed complicated bodies. With the Chapman-Enskog expansion analysis, a transform is performed between the Navier-Stokes and lattice Boltzmann equations (LBEs). The LBFS is used to discretize the macroscopic differential equations with a finite volume method and evaluate the interface fluxes through local reconstruction of the lattice Boltzmann solution. The immersed boundary technique is used to correct the intermediate velocity around the solid boundary to satisfy the no-slip boundary condition. Agreement of simulation results with the data found in the literature shows reliability of the proposed method in simulating laminar flows on a Cartesian mesh.

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References

  1. Chen, H., Chen, S., and Matthaeus, H. Recovery of the Navier-Stokes equation using a lattice-Boltzmann method. Physical Review A, 45, 5339–5342 (1992)

    Article  Google Scholar 

  2. Qian, Y., D’humi`eres, D., and Lallemand, P. Lattice BGK models for Navier-Stokes equations. Europhysics Letters, 17, 479–484 (1992)

    Article  MATH  Google Scholar 

  3. Shan, X. and Chen, H. Lattice Boltzmann model for simulating flows with multiple phases and components. Physical Review E, 47, 1815–1819 (1993)

    Article  Google Scholar 

  4. Suryanarayanan, S., Singh, S., and Ansumali, S. Extended BGK Boltzmann for dense gases. Communications in Computational Physics, 13, 629–648 (2013)

    Article  Google Scholar 

  5. Yu, D., Mei, R., Luo, L., and Shyy, W. Various flow computations with the method of lattice Boltzmann equation. Progress in Aerospace Sciences, 39, 329–367 (2003)

    Article  Google Scholar 

  6. Aidun, C. and Clausen, J. Lattice Boltzmann method for complex flows. Annual Review of Fluid Mechanics, 42, 439–472 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Yu, H., Girimaji, S. S., and Luo, L. S. DNS and LES of decaying isotropic turbulence with and without frame rotation using lattice Boltzmann method. Journal of Computational Physics, 209, 599–616 (2005)

    Article  MATH  Google Scholar 

  8. Shu, C., Peng, Y., and Zhou, C. F. Application of Taylor series expansion and least-squares-based lattice Boltzmann method to simulate turbulent flows. Journal of Turbulence, 7(37), 1–12 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Li, K., Zhong, C. W., Zhuo, C. S., and Jun, C. Non-body-fitted Cartesian-mesh simulation of highly turbulent flows using multi-relaxation-time lattice Boltzmann method. Computational Physics, 204, 265–291 (2005)

    Article  MATH  Google Scholar 

  10. Shu, C., Wang, Y., Teo, C. J., and Wu, J. Development of lattice Boltzmann flux solver for simulation of incompressible flows. Advances in Applied Mathematics and Mechanics, 6, 436–460 (2014)

    Article  MathSciNet  Google Scholar 

  11. Wang, Y., Shu, C., and Teo, C. J. Thermal lattice Boltzmann flux solver and its application for simulation of incompressible thermal flows. Computers and Fluids, 94, 98–111 (2014)

    Article  MathSciNet  Google Scholar 

  12. Cao, Y. H. Variable property-based lattice Boltzmann flux solver for thermal flows in the low Mach number limit. International Journal of Heat and Mass Transfer, 103, 254–264 (2016)

    Article  Google Scholar 

  13. Wang, Y., Shu, C., Huang, H. B., and Teo, C. J. Multiphase lattice Boltzmann flux solver for incompressible multiphase flows with large density ratio. Journal of Computational Physics, 280, 404–423 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lallemand, P. and Luo, L. S. Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance and stability. Physical Review E, 61(6), 6546–6562 (2002)

    Article  MathSciNet  Google Scholar 

  15. D’Humi`eres, D., Ginzburg, I., Krafczyk, M., Lallemand, P., and Luo, L. S. Multiple-relaxation-time lattice Boltzmann models in three dimensions. Philosophical Transactions of the Royal Society, Mathematical, Physical, and Engineering sciences, 360, 437–451 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guo, X. X., Zhong, C. W., Zhuo, C. S., and Jun, C. Multiple-relaxation-time lattice Boltzmann method for study of two-lid-driven cavity flow solution multiplicity. Theoretical and Computational Fluid Dynamics, 28, 215–231 (2004)

    Article  Google Scholar 

  17. Yu, H. D., Luo, L. S., and Girimaji, S. S. LES of turbulent square jet flow using an MRT lattice Boltzmann model. Computers and Fluids, 35, 957–965 (2006)

    Article  MATH  Google Scholar 

  18. Nie, D. M., Lin, J. Z., and Qiu, L. M. Direct numerical simulations of the decaying turbulence in rotating flows via the MRT lattice Boltzmann method. International Journal of Computational Fluid Dynamics Dynamics, 27(3), 173–183 (2013)

    Article  MathSciNet  Google Scholar 

  19. Geller, S., Uphoff, S., and Krafczyk, M. Turbulent jet computations based on MRT and cascaded lattice Boltzmann models. Computers and Mathematics with Applications, 65, 1956–1966 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mittal, R. and Iaccarino, G. Immersed boundary methods. Annual Review of Fluid Mechanics, 37, 239–261 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Peskin, C. Flow patterns around heart valves: a numerical method. Journal of Computational Physics, 10, 220–252 (1972)

    Article  MATH  Google Scholar 

  22. Fadlun, E., Verzicco, R., Orlandi, P., and Mohd-Yusof, J. Combined immersed-boundary/finite difference methods for three-dimensional complex flow simulations. Journal of Computational Physics, 161, 35–60 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wu, J. and Shu, C., Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications. Journal of Computational Physics, 228, 1963–1979 (2009)

    Article  MATH  Google Scholar 

  24. Wu, J. and Shu, C. An improved immersed boundary-lattice Boltzmann method for simulating three-dimensional incompressible flows. Journal of Computational Physics, 229, 5022–5042 (2010)

    Article  MATH  Google Scholar 

  25. Wang, Y., Shu, C., Teo, C. J., and Wu, J. An immersed boundary-lattice Boltzmann flux solver and its applications to fluid-structure interaction problems. Journal of Fluids and Structures, 54, 440–465 (2015)

    Article  Google Scholar 

  26. Qiu, Y. L., Shu, C., Wu, J., Sun, Y., Yang, L. M., and Guo, T. Q. A boundary condition-enforced immersed boundary method for compressible viscous flows. Computers and Fluids, 136, 104–113 (2016)

    Article  MathSciNet  Google Scholar 

  27. Ren, W. W., Shu, C., Wu, J., and Yang, W. M. Boundary condition-enforced immersed boundary method for thermal flow problems with Dirichlet temperature condition and its applications. Computers and Fluids, 57, 40–51 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shukla, R. K., Tatineni, M., and Zhong, X. Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier-Stokes equations. Journal of Computational Physics, 24, 1064–1094 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yuan, R., Zhong, C., and Zhang, H. An immersed-boundary method based on the gas kinetic BGK scheme for incompressible viscous flow. Journal of Computational Physics, 296, 184–208 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Calhoun, D. A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular region. Journal of Computational Physics, 176, 231–275 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Russell, D. and Wang, Z. J. A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow. Journal of Computational Physics, 191, 177–205 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Choi, J. I., Oberoi, R. C., Edwards, J. R., and Rosati, J. A. An immersed boundary method for complex incompressible flows. Journal of Computational Physics, 224, 757–784 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Badr, H. M. Laminar combined convection from a horizontal cylinder—parellel and contra flow regimes. International Journal of Heat and Mass Transfer, 27, 15–27 (1984)

    Article  MATH  Google Scholar 

  34. Badr, H. M. Mixed convection from a straight isothermal tube of elliptical cross-section. International Journal of Heat and Mass Transfer, 37(15), 2343–2365 (1994)

    Article  MATH  Google Scholar 

  35. Bhattacharyya, S. and Singh, A. K. Vortex shedding and heat transfer dependence on effective Reynolds number for mixed convection around a cylinder in cross flow. International Journal of Heat and Mass Transfer, 53, 3202–3212 (2010)

    Article  MATH  Google Scholar 

  36. Chang, K. S. and Sa, J. Y. The effect of buoyancy on vortex shedding in the near wake of a circular cylinder. Journal of Fluid Mechanics, 220, 253–266 (1990)

    Article  Google Scholar 

  37. Johnson, S. A., Thompson, M. C., and Hourigan, K. Flow past elliptical cylinders at low Reynolds numbers. 14th Australasian Fluid Mechanics Conference Adelaide University, Adelaide, Australia (2001)

    Google Scholar 

  38. Lockard, D. P., Luo, L. S., Milder, S. D., and Singer, B. A. Evaluation of power flow for aerodynamic applications. Journal of Statistical Physics, 107, 423–478 (2002)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Energy Science and Engineering, Harbin Institute of Technology, Harbin, 150001, China

    Xiaodi Wu, Fu Chen & Huaping Liu

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  1. Xiaodi Wu
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  2. Fu Chen
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  3. Huaping Liu
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Correspondence to Xiaodi Wu.

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Wu, X., Chen, F. & Liu, H. Combined immersed boundary method and multiple-relaxation-time lattice Boltzmann flux solver for numerical simulations of incompressible flows. Appl. Math. Mech.-Engl. Ed. 38, 1679–1696 (2017). https://doi.org/10.1007/s10483-017-2290-7

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  • DOI: https://doi.org/10.1007/s10483-017-2290-7

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

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