Skip to main content
SpringerLink
Log in

Lattice Boltzmann for Advection-Diffusion Problems

  • Chapter
  • First Online:
The Lattice Boltzmann Method

Part of the book series: Graduate Texts in Physics ((GTP))

Abstract

After reading this chapter, you will understand how the lattice Boltzmann equation can be adapted from flow problems to advection-diffusion problems with only small changes. These problems include thermal flows, and you will know how to simulate these as two interlinked lattice Boltzmann simulations, one for the flow and one for the thermal advection-diffusion. You will understand how advection-diffusion problems require different boundary conditions from flow problems, and how these boundary conditions may be implemented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 99.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Of course “conserved” quantities are only conserved in the absence of source terms.

  2. 2.

    Note that D2Q5 and D3Q7 are not sufficient for problems involving anisotropic diffusion with non-zero off-diagonal coefficients [8].

  3. 3.

    D3Q21 is the D3Q15 lattice with six additional vectors (±2, 0, 0)⊤, (0, ±2, 0)⊤ and (0, 0, ±2)⊤.

References

  1. R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, 2nd edn. (Wiley, New York, 1960)

    Google Scholar 

  2. D. Wolf-Gladrow, J. Stat. Phys. 79 (5–6), 1023 (1995)

    Article  ADS  Google Scholar 

  3. E.G. Flekkoy, Phys. Rev. E 47 (6), 4247 (1993)

    Article  ADS  Google Scholar 

  4. T. Inamuro, M. Yoshino, H. Inoue, R. Mizuno, F. Ogino, J. Comp. Phys. 179 (1), 201 (2002)

    Article  ADS  Google Scholar 

  5. X. Shan, Phys. Rev. E 55 (3), 2780 (1997)

    Article  ADS  Google Scholar 

  6. X. He, S. Chen, G.D. Doolen, J. Comput. Phys. 146 (1), 282 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  7. T. Toffoli, N.H. Margolus, Physica D 45 (1-3), 229 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  8. I. Ginzburg, Adv. Water Resour. 28 (11), 1171 (2005)

    Article  ADS  Google Scholar 

  9. S. Suga, Int. J. Mod. Phys. C 17 (11), 1563 (2006)

    Article  ADS  Google Scholar 

  10. I. Ginzburg, D. d’Humières, A. Kuzmin, J. Stat. Phys. 139 (6), 1090 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  11. L. Li, C. Chen, R. Mei, J.F. Klausner, Phys. Rev. E 89 (4), 043308 (2014)

    Article  ADS  Google Scholar 

  12. H.B. Huang, X.Y. Lu, M.C. Sukop, J. Phys. A: Math. Theor. 44 (5), 055001 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  13. B. Chopard, J.L. Falcone, J. Latt, Eur. Phys. J. Spec. Top. 171, 245 (2009)

    Article  Google Scholar 

  14. Z. Chai, T.S. Zhao, Phys. Rev. E 87 (6), 063309 (2013)

    Article  ADS  Google Scholar 

  15. I.V. Karlin, D. Sichau, S.S. Chikatamarla, Phys. Rev. E 88 (6), 063310 (2013)

    Article  ADS  Google Scholar 

  16. Q. Kang, P.C. Lichtner, D. Zhang, J. Geophys. Res. 111 (B5), B05203 (2006)

    Article  ADS  Google Scholar 

  17. S.G. Ayodele, F. Varnik, D. Raabe, Phys. Rev. E 83 (1), 016702 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  18. J. Zhang, G. Yan, Comput. Math. Appl. 69 (3), 157 (2015)

    Article  MathSciNet  Google Scholar 

  19. X. Yang, B. Shi, Z. Chai, Comput. Math. Appl. 68 (12, Part A), 1653 (2014)

    Google Scholar 

  20. J. Kang, N.I. Prasianakis, J. Mantzaras, Phys. Rev. E 89 (6), 063310 (2014)

    Article  ADS  Google Scholar 

  21. T. Seta, Phys. Rev. E 87 (6), 063304 (2013)

    Article  ADS  Google Scholar 

  22. B. Shi, B. Deng, R. Du, X. Chen, Comput. Math. Appl. 55 (7), 1568 (2008)

    Article  MathSciNet  Google Scholar 

  23. I. Ginzburg, Adv. Water Resour. 28 (11), 1196 (2005)

    Article  ADS  Google Scholar 

  24. S. Suga, Int. J. Mod. Phys. C 20, 633 (2009)

    Article  ADS  Google Scholar 

  25. H. Yoshida, M. Nagaoka, J. Comput. Phys. 229 (20), 7774 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  26. I. Ginzburg, Commun. Comput. Phys. 11, 1439 (2012)

    Article  Google Scholar 

  27. A. Mohamad, Lattice Boltzmann Method: Fundamentals and Engineering Applications with Computer Codes, 1st edn. (Springer, New York, 2011)

    Book  MATH  Google Scholar 

  28. R.G.M. Van der Sman, Phys. Rev. E 74, 026705 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  29. R. Huang, H. Wu, Phys. Rev. E 89 (4), 043303 (2014)

    Article  ADS  Google Scholar 

  30. Q. Li, Z. Chai, B. Shi, Comput. Math. Appl. 70 (4), 548 (2015)

    Article  MathSciNet  Google Scholar 

  31. J. Perko, R.A. Patel, Phys. Rev. E 89 (5), 053309 (2014)

    Article  ADS  Google Scholar 

  32. X. Yang, B. Shi, Z. Chai, Phys. Rev. E 90 (1), 013309 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  33. B. Servan-Camas, F. Tsai, Adv. Water Resour. 31, 1113 (2008)

    Article  ADS  Google Scholar 

  34. I. Ginzburg, Adv. Water Resour. 51, 381 (2013)

    Article  ADS  Google Scholar 

  35. Z. Chai, T.S. Zhao, Phys. Rev. E 90 (1), 013305 (2014)

    Article  ADS  Google Scholar 

  36. R. Huang, H. Wu, J. Comput. Phys. 274, 50 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  37. E.D. Siggia, Ann. Rev. Fluid Mech. 26, 137 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  38. W.H. Reid, D.L. Harris, Phys. Fluids 1 (2), 102 (1958)

    Article  ADS  MathSciNet  Google Scholar 

  39. F.J. Alexander, S. Chen, J.D. Sterling, Phys. Rev. E 47 (4), R2249 (1993)

    Article  ADS  Google Scholar 

  40. Y. Chen, H. Ohashi, M. Akiyama, Phys. Rev. E 50 (4), 2776 (1994)

    Article  ADS  Google Scholar 

  41. Y.H. Qian, J. Sci. Comput. 8 (3), 231 (1993)

    Article  MathSciNet  Google Scholar 

  42. G.R. McNamara, A.L. Garcia, B.J. Alder, J. Stat. Phys. 81 (1–2), 395 (1995)

    Article  ADS  Google Scholar 

  43. D.A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models (Springer, New York, 2005)

    MATH  Google Scholar 

  44. Y. Peng, C. Shu, Y.T. Chew, Phys. Rev. E 68 (2), 026701 (2003)

    Article  ADS  Google Scholar 

  45. Z. Guo, C. Zheng, B. Shi, T.S. Zhao, Phys. Rev. E 75 (3), 036704 (2007)

    Article  ADS  Google Scholar 

  46. S. Chen, K.H. Luo, C. Zheng, J. Comput. Phys. 231 (24) (2012)

    Google Scholar 

  47. P.A. Thompson, Compressible-Fluid Dynamics (McGraw-Hill, New York, 1972)

    MATH  Google Scholar 

  48. Z. Guo, B. Shi, C. Zheng, Int. J. Numer. Meth. Fluids 39 (4), 325 (2002)

    Article  MathSciNet  Google Scholar 

  49. T. Zhang, B. Shi, Z. Guo, Z. Chai, J. Lu, Phys. Rev. E 85 (016701), 1 (2012)

    Google Scholar 

  50. M. Yoshino, T. Inamuro, Int. J. Num. Meth. Fluids 43 (2), 183 (2003)

    Article  Google Scholar 

  51. A. Polyanin, A. Kutepov, A. Vyazmin, D. Kazenin, Hydrodynamics, Mass and Heat Transfer in Chemical Engineering (Taylor and Francis, London, 2002)

    Google Scholar 

  52. M. Ozisik, Heat Conduction (Wiley, New York, 1993)

    Google Scholar 

  53. A. Kuzmin, M. Januszewski, D. Eskin, F. Mostowfi, J. Derksen, Chem. Eng. J. 225, 580 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Engineering University of Edinburgh, Edinburgh, UK

    Timm Krüger

  2. Department of Physics, Durham University, Durham, UK

    Halim Kusumaatmaja

  3. Maya Heat Transfer Technologies, Westmount, Québec, Canada

    Alexandr Kuzmin

  4. Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, USA

    Orest Shardt

  5. IDMEC/IST, University of Lisbon, Lisbon, Portugal

    Goncalo Silva

  6. Acoustics Research Centre, SINTEF ICT, Trondheim, Norway

    Erlend Magnus Viggen

Authors
  1. Timm Krüger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Halim Kusumaatmaja
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexandr Kuzmin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Orest Shardt
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Goncalo Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Erlend Magnus Viggen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., Viggen, E.M. (2017). Lattice Boltzmann for Advection-Diffusion Problems. In: The Lattice Boltzmann Method. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-44649-3_8

Download citation

Publish with us

Policies and ethics

Search

Navigation