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Forces

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The Lattice Boltzmann Method

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

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Abstract

After reading this chapter, you will be able to add forces to lattice Boltzmann simulations while retaining their accuracy. You will know how a forcing scheme can be derived by including forces in the derivation of the lattice Boltzmann equation, though you will also know that there are a number of other forcing schemes available. You will understand how to investigate forcing models and their errors through the Chapman-Enskog analysis, and how initial and boundary conditions can be affected by the presence of forces.

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Notes

  1. 1.

    Showing that these forcing schemes are equivalent to leading order is straightforward but involves lengthy calculations. We will not delve into details here and refer to [36] for a more qualitative analysis.

  2. 2.

    Also in Brinkman and Coriolis force models, where \({\boldsymbol F} \propto {\boldsymbol u}\), the error term \(\propto \nabla ^{2}{\boldsymbol F}\) is important [33, 34].

  3. 3.

    In certain cases, the forcing term requires a higher-order expansion. For example, for certain axisymmetric LB models [48, 49], the formal expansion of the forcing term is F i  = ε F i (1) +ε 2 F i (2).

  4. 4.

    Equation (6.47) results from omitting the time derivatives in equation (6.33a) based on the Chapman-Enskog analysis for steady flows discussed in Sect. 4.2.3.

  5. 5.

    The sign convention for the normal momentum correction is in line with the tangential case, cf. (5.43). If \({\boldsymbol n}\) and \({\boldsymbol t}\) denote the wall normal and the wall tangential vectors and if their positive sign coincides with the positive sign of the Cartesian axis, then the normal and tangential momentum corrections appear in the algorithm as \(f_{\bar{i}}^{\mathrm{neq}}({\boldsymbol x}_{\text{B}},t) = f_{i}^{\mathrm{neq}}({\boldsymbol x}_{\text{B}},t) - ({\boldsymbol n} \cdot {\boldsymbol c}_{i})N_{n} - ({\boldsymbol t} \cdot {\boldsymbol c}_{i})N_{t}\).

References

  1. E. Bodenschatz, W. Pesch, G.Ahlers, Annu. Rev. Fluid Mech. 32, 709 (2010)

    Google Scholar 

  2. S.I. Abarzhi, Phil. Trans. R. Soc. A 368, 1809 (2010)

    Article  ADS  Google Scholar 

  3. J. Lighthill, Waves in Fluids, 6th edn. (Cambridge University Press, Cambridge, 1979)

    MATH  Google Scholar 

  4. J.M. Buick, C.A. Greated, Phys. Rev. E 61 (5), 5307 (2000)

    Article  ADS  Google Scholar 

  5. G. Silva, V. Semiao, J. Fluid Mech. 698, 282 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  6. P.J. Dellar, Comput. Math. Appl. 65 (2), 129 (2013)

    Article  MathSciNet  Google Scholar 

  7. R. Salmon, J. Mar. Res. 57 (3), 847 (1999)

    Article  MathSciNet  Google Scholar 

  8. J. Wang, M. Wang, Z. Li, J. Colloid Interf. Sci. 296, 729 (2006)

    Article  Google Scholar 

  9. M. Wang, Q. Kang, J. Comput. Phys. 229, 728 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  10. T.Y. Lin, C.L. Chen, Appl. Math. Model. 37, 2816 (2013)

    Article  MathSciNet  Google Scholar 

  11. O. Shardt, S.K. Mitra, J.J. Derksen, Chem. Eng. J. 302, 314 (2016)

    Article  Google Scholar 

  12. S.H. Kim, H. Pitsch, Phys. Fluids 19, 108101 (2007)

    Article  ADS  Google Scholar 

  13. J. Zhang, D.Y. Kwok, Phys. Rev. E 73, 047702 (2006)

    Article  ADS  Google Scholar 

  14. L. Talon, D. Bauer, D. Gland, H. Auradou, I. Ginzburg, Water Resour. Res. 48, W04526 (2012)

    Article  ADS  Google Scholar 

  15. Z. Guo, C. Zheng, B. Shi, Phys. Rev. E 65, 46308 (2002)

    Article  ADS  Google Scholar 

  16. N.S. Martys, X. Shan, H. Chen, Phys. Rev. E 58 (5), 6855 (1998)

    Article  ADS  Google Scholar 

  17. X. Shan, X.F. Yuan, H. Chen, J. Fluid Mech. 550, 413 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  18. L.S. Luo, Phys. Rev. E 62 (4), 4982 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  19. P. Dellar, Phys. Rev. E 64 (3) (2001)

    Google Scholar 

  20. A.J.C. Ladd, R. Verberg, J. Stat. Phys. 104 (5–6), 1191 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  21. S. Ansumali, I.V. Karlin, H.C. Öttinger, Phys. Rev. Lett. 94, 080602 (2005)

    Article  ADS  Google Scholar 

  22. J.R. Clausen, Phys. Rev. E 87, 013309 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  23. G. Silva, V. Semiao, Physica A 390 (6), 1085 (2011)

    Article  ADS  Google Scholar 

  24. Z. Guo, C. Zheng, B. Shi, T. Zhao, Phys. Rev. E 75 (036704), 1 (2007)

    Google Scholar 

  25. R.W. Nash, R. Adhikari, M.E. Cates, Phys. Rev. E 77 (2), 026709 (2008)

    Article  ADS  Google Scholar 

  26. S. Ubertini, P. Asinari, S. Succi, Phys. Rev. E 81 (1), 016311 (2010)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  28. A. Kuzmin, Z. Guo, A. Mohamad, Phil. Trans. Royal Soc. A 369, 2219 (2011)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  30. S.D.C. Walsh, H. Burwinkle, M.O. Saar, Comput. Geosci. 35 (6), 1186 (2009)

    Article  ADS  Google Scholar 

  31. Z. Guo, T.S. Zhao, Phys. Rev. E 66, 036304 (2002)

    Article  ADS  Google Scholar 

  32. X. Nie, N.S. Martys, Phys. Fluids 19, 011702 (2007)

    Article  ADS  Google Scholar 

  33. I. Ginzburg, Phys. Rev. E 77, 066704 (2008)

    Article  ADS  Google Scholar 

  34. I. Ginzburg, G. Silva, L. Talon, Phys. Rev. E 91, 023307 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  35. R. Salmon, J. Mar. Res. 57 (3), 503 (1999)

    Article  MathSciNet  Google Scholar 

  36. H. Huang, M. Krafczyk, X. Lu, Phys. Rev. E 84 (4), 046710 (2011)

    Article  ADS  Google Scholar 

  37. I. Ginzburg, F. Verhaeghe, D. d’Humières, Commun. Comput. Phys. 3, 427 (2008)

    MathSciNet  Google Scholar 

  38. X. Shan, H. Chen, Phys. Rev. E 47 (3), 1815 (1993)

    Article  ADS  Google Scholar 

  39. X. He, X. Shan, G. Doolen, Phys. Rev. E. Rapid Comm. 57 (1), 13 (1998)

    Article  ADS  Google Scholar 

  40. A. Kupershtokh, D. Medvedev, D. Karpov, Comput. Math. Appl. 58 (5), 965 (2009)

    Article  MathSciNet  Google Scholar 

  41. D. Lycett-Brown, K.H. Luo, Phys. Rev. E 91, 023305 (2015)

    Article  ADS  Google Scholar 

  42. A. Kupershtokh, in Proc. 5th International EHD Workshop, University of Poitiers, Poitiers, France (2004), p. 241–246

    Google Scholar 

  43. X. He, Q. Zou, L.S. Luo, M. Dembo, J. Stat. Phys. 87 (1–2), 115 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  44. L.S. Luo, Phys. Rev. Lett. 81 (8), 1618 (1998)

    Article  ADS  Google Scholar 

  45. I. Ginzbourg, P.M. Adler, J. Phys. II France 4 (2), 191 (1994)

    Article  Google Scholar 

  46. A.J.C. Ladd, J. Fluid Mech. 271, 285 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  47. Z. Guo, C. Zheng, B. Shi, Phys. Rev. E 83, 036707 (2011)

    Article  ADS  Google Scholar 

  48. I. Halliday, L.A. Hammond, C.M. Care, K. Good, A. Stevens, Phys. Rev. E 64, 011208 (2001)

    Article  ADS  Google Scholar 

  49. T. Reis, T.N. Phillips, Phys. Rev. E 75, 056703 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  50. I. Ginzburg, F. Verhaeghe, D. d’Humières, Commun. Comput. Phys. 3, 519 (2008)

    MathSciNet  Google Scholar 

  51. M. Gross, N. Moradi, G. Zikos, F. Varnik, Phys. Rev. E 83 (1), 017701 (2011)

    Article  ADS  Google Scholar 

  52. Q. Zou, X. He, Phys. Fluids 9, 1591 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  53. A. D’Orazio, S. Succi, Future Generation Comput. Syst. 20, 935 (2004)

    Article  Google Scholar 

  54. A. Markus, G. Hazi, Phys. Rev. E 83, 046705 (2011)

    Article  ADS  Google Scholar 

  55. A. Karimipour, A.H. Nezhad, A. D’Orazio, E. Shirani, J. Theor. Appl. Mech. 51, 447 (2013)

    Google Scholar 

  56. D.R. Noble, Chen, J.G. Georgiadis, R.O. Buckius, Phys. Fluids 7, 203 (1995)

    Google Scholar 

  57. M. Bouzidi, M. Firdaouss, P. Lallemand, Phys. Fluids 13, 3452 (2001)

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

  59. M. Junk, A. Klar, L.S. Luo, J. Comput. Phys. 210, 676 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  60. T. Krüger, F. Varnik, D. Raabe, Phys. Rev. E 79 (4), 046704 (2009)

    Article  ADS  Google Scholar 

  61. C.M. Pooley, K. Furtado, Phys. Rev. E 77, 046702 (2008)

    Article  ADS  Google Scholar 

  62. I. Ginzburg, D. d’Humières, Phys. Rev. E 68, 066614 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  63. K. Premnath, J. Abraham, J. Comput. Phys. 224, 539 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  64. J. Latt, Hydrodynamic limit of lattice Boltzmann equations. Ph.D. thesis, University of Geneva (2007)

    Google Scholar 

  65. M. Rohde, D. Kandhai, J.J. Derksen, H.E.A. Van den Akker, Phys. Rev. 67, 066703 (2003)

    Google Scholar 

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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

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  1. Timm Krüger
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  2. Halim Kusumaatmaja
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  3. Alexandr Kuzmin
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  4. Orest Shardt
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  5. Goncalo Silva
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  6. Erlend Magnus Viggen
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Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., Viggen, E.M. (2017). Forces. In: The Lattice Boltzmann Method. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-44649-3_6

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