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Boundary and Initial Conditions

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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 familiar with the basics of lattice Boltzmann boundary conditions. After also having read Chap. 3, you will be able to implement fluid flow problems with various types of grid-aligned boundaries, representing both no-slip and open surfaces. From the boundary condition theory explained in this chapter together with the theory given in Chap. 4, you will be familiar with the basic theoretical tools used to analyse numerical lattice Boltzmann solutions. Additionally, you will understand how the details of the initial state of a simulation can be important and you will know how to compute a good initial simulation state.

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Notes

  1. 1.

    Although no mathematical proof exists yet, this book takes for granted that solutions to the NSEs exist, are unique and vary smoothly with changes to initial conditions.

  2. 2.

    Chapter 11 extends this subject to the case of boundary conditions in complex geometries. This topic is further addressed in Sect. 12.4, where we discuss a special kind of boundary conditions designed to let sound waves exit the system smoothly.

  3. 3.

    The extension to more advanced collision operators is discussed in Chap. 10

  4. 4.

    Moving boundaries are also possible and are discussed in Sect. 5.3.3.

  5. 5.

    Exceptional cases of time-dependent problems where initial conditions have an immaterial role are discussed in Sect. 5.5.

  6. 6.

    The understanding of (F2) is also relevant in the context of LB. By exploiting the similarity between LB and finite difference schemes, a procedure similar to (F2) was proposed as a way to prescribe boundary conditions in the LBM. The original suggestion from Chen and co-workers [8] consists of adding virtual lattice sites. More refined versions of this method were later developed based on an improved understanding of LB theory, e.g. [9–12].

  7. 7.

    Although more complex boundary shapes and/or orientations can also be handled with the techniques discussed here, the resulting geometry will represent a “staircase” approximation of the true boundary. The implications are discussed in Sect. 11.1 Methods to treat smooth complex surfaces usually come at the price of increased complexity and/or the need for data from neighboring nodes, making them non-local schemes. While exceptions of local schemes for curved boundaries exist [13, 14], they are considerably more difficult to implement than comparable non-local techniques.

  8. 8.

    The same kind of “non-uniqueness” problem affects the specification of hydrodynamically consistent initial conditions in LBM (cf. Sect. 5.5).

  9. 9.

    We obtained this number by performing a Web of Science search for articles with the words “lattice Boltzmann boundary condition” in the title.

  10. 10.

    We say “approximately” because in link-wise schemes the exact boundary location is not fixed. Rather, standard link-wise boundaries have a “second-order” dependence on the relaxation rates of the LB collision scheme. For example, with the BGK model this defect leads to a dependency of the no-slip wall location with the fluid viscosity [15, 21, 23]. We will discuss this issue in Sect. 5.3.3 using numerical examples and in Sect. 5.4.1 with a theoretical analyses. Such a “second-order” artefact is also disturbed by anisotropic effects, meaning the wall location will change according to the orientation of the boundary with respect to the lattice.

  11. 11.

    The computational convenience of considering these extra layers of nodes in multithreading implementations of LB algorithms is given in Sect. 13.4.1

  12. 12.

    In an attempt to make this explanation of the bounce-back method more intuitive, we will often refer to the parameter f i as “particles” instead of “particle distributions” or “populations”, as it should be called more rigorously. We have to remember that LBM is not a true particle method, such as those presented in Chap. 2. Rather, LBM deals with discretised forms of continuous fields (cf. Chap. 3).

  13. 13.

    For fullway, we only need to check whether we are on a solid node or not. Algorithmically, this is one if-statement per lattice node. For half-way, we have to check where each population propagates, i.e. whether it will finish on a solid node (which implies bounce-back), or it will reach a fluid or boundary node (which implies normal streaming). For example, with the D3Q19 model this boils down to evaluating 18 if-statements!

  14. 14.

    Possible differences in the steady-state performance of fullway and halfway bounce-back schemes are related to grid-scaled artefacts called “staggered invariants” [13, 22, 30]. Staggered invariants manifest as a velocity oscillation between two constant values over two successive time steps [22]. Their magnitude is usually small, yet the precise value depends on several factors, such as the initial conditions, the mesh size and the parity of the number of grid nodes. Staggered invariants are conserved by halfway bounce-back. For example, a typical channel flow along the x-axis, if initialised with u y (t = 0) = u y, 0 and using an odd number of nodes along the y-axis, will conserve this constant transverse velocity u y, 0 throughout the channel width as a staggered invariant if halfway bounce-back is used [30]. However, with an even number of lattice nodes, this artefact vanishes. One way to correct this halfway bounce-back defect is by averaging the solution between two successive time steps as suggested in [22]. The fullway bounceback does not produce this artefacts since it delays the exchange of information between successive time steps. However, the conservation and stability properties of this scheme may be worse [30].

  15. 15.

    According to [20], instability is often triggered by the boundary condition rather than the bulk algorithm when LBM is running close to τ = Δ t∕2.

  16. 16.

    Conversion between simulation and physical units will be discussed in Chap. 7

  17. 17.

    The numerical values for the correction terms come from 2w i ∕c s 2 = 1∕(6c 2), with c s 2 = (1∕3)c 2 and using the D2Q9 lattice weights w i given in Table 3.1

  18. 18.

    Since this Poiseuille flow is driven by the pressure difference Δ p, the incompressible LB model of Sect. 4.3.2 is appropriate. Recall that the standard LB equilibrium uses an isothermal equation of state where pressure and density relate linearly. Therefore, a pressure gradient inevitably leads to a gradient of density, which is incompatible with incompressible hydrodynamics. With the incompressible model this compressibility error can be avoided (at least in steady flows).

  19. 19.

    We will cover closure relations for boundary conditions in Sect. 5.4.1.

  20. 20.

    Although the LBM may reproduce the continuity condition in two different forms, depending on the equilibrium model adopted, i.e. \(\partial _{t}\rho + \nabla \cdot (\rho {\boldsymbol u}) = 0\) for the standard compressible equilibrium or \(c_{\mathrm{s}}^{-2}\partial _{t}p +\rho _{0}\nabla \cdot {\boldsymbol u} = 0\) for the incompressible equilibrium, the procedure described here remains applicable in both cases.

  21. 21.

    Recall that f i neq is related to the spatial derivatives of the flow field. Consequently, f i neq = 0 corresponds to a spatially uniform solution.

  22. 22.

    Obviously, the use of other equilibrium distributions (e.g. the full standard equilibrium) will not lead to improvements as the problem identified here comes from neglecting the non-equilibrium part of the boundary populations.

  23. 23.

    Here we will exclude cases where this requirement is violated. An example is the lid-driven cavity flow [62, 75], an often used benchmark problem with a velocity discontinuity at corners.

  24. 24.

    A possible numerical approximation procedure is using the formula given in Sect. 5.3.5 to find \({\boldsymbol u}_{\text{w}}\) in the anti-bounce-back approach, but now applied to ρ w.

  25. 25.

    Remember that c s 2 = c 2∕3 = Δ x 2∕(3Δ t 2).

  26. 26.

    Here, and throughout this discussion, we will assume no mass source is present.

  27. 27.

    Depending on the definition of mass balance, as explained in [55], a third case may be considered: the presence of tangential density gradients along the wall surface. Here, a mass flux difference, proportional to the density gradient, may exist.

  28. 28.

    Note that we have used the no-slip condition ∂ x u x  = ∂ x u y  = 0 at the wall to simplify (5.77).

  29. 29.

    The MEA also works for curved boundaries as we will discuss in Sect. 11.2

  30. 30.

    Note that the number of identified links varies with the chosen lattice. For example, there will be more links for the same geometry when D3Q27 rather than D3Q15 is used. This does not affect the validity of the MEA, though.

  31. 31.

    One possible way to implement this is to run over all solid nodes and identify all neighbouring boundary nodes [80, 83].

  32. 32.

    Wet-node formulations that replace all populations are simpler to implement in 3D. Examples are the equilibrium scheme [23], the non-equilibrium extrapolation method [9], the finite-difference velocity gradient method [18] and the regularised method [18]. As they are based on reconstructing all populations, they are not sensitive to the number of populations. The downside of these approaches is that they either decrease the accuracy or increase the complexity of implementation, e.g. by making the scheme non-local [18].

  33. 33.

    One obtains this equation by computing the divergence of the incompressible NSE.

  34. 34.

    A more accurate discussion of how to split the time step is found in recent works by Dellar [99] and Schiller [100].

  35. 35.

    For this initialisation method, it is important to employ the incompressible LB algorithm. The standard equilibrium leads to large initial pressure errors.

  36. 36.

    Using the incompressible equilibrium for the actual simulations does not result in significantly different results. This is no surprise since the incompressible model is only formally more accurate for steady flows.

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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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Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., Viggen, E.M. (2017). Boundary and Initial Conditions. In: The Lattice Boltzmann Method. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-44649-3_5

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