Abstract
After reading this chapter, you will have insight into a number of other fluid simulation methods and their advantages and disadvantages. These methods are divided into two categories. First, conventional numerical methods based on discretising the equations of fluid mechanics, such as finite difference, finite volume, and finite element methods. Second, methods that are based on microscopic, mesoscopic, or macroscopic particles, such as molecular dynamics, lattice gas models, and multi-particle collision dynamics. You will know where the particle-based lattice Boltzmann method fits in the landscape of fluid simulation methods, and you will have an understanding of the advantages and disadvantages of the lattice Boltzmann method compared to other methods.
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Notes
- 1.
Before electronic computers, numerical solutions were performed manually by people whose job title was “computer”!
- 2.
- 3.
The forward difference approximation corresponds to the forward Euler approximation for time discretisation, shown in (2.1).
- 4.
As a practical example, a deer can smell a hunter who is upwind of it, since the wind blows the hunter’s scent towards the deer.
- 5.
We here use the term “volume” in a general sense, where a 2D volume is an area and a 1D volume is a line segment.
- 6.
That is not to say that the FV method is limited to conservation equations; it can also be used to solve more general hyperbolic problems [6].
- 7.
For the internal surfaces between adjacent finite volumes, the surface integrals from the two volumes will cancel each other.
- 8.
In Fig. 2.5, S i is the triangular surface around each volume, and s j represents the straight-line faces of these triangles.
- 9.
This straightforward force approach scales with the number N of particles as \(\mathcal{O}(N^{2})\), though more efficient approaches that scale as \(\mathcal{O}(N)\) also exist [10].
- 10.
Most of them also conserve energy.
- 11.
While the kernel function can be e.g. Gaussian, it is advantageous to choose kernels that are zero for \(\vert {\boldsymbol x} -{\boldsymbol x}_{j}\vert> h\), so that only particles in the vicinity of \({\boldsymbol x}\) need be included in the sum. Additionally, the fact that h j can be particle-specific and varying allows adaptive resolution.
- 12.
For instance, molecular dynamics is tailored to simulating phenomena on an atomic and molecular level, and smoothed-particle hydrodynamics was invented to deal with the largely empty domains of astrophysical CFD.
- 13.
- 14.
This concise description is attributed to Sauro Succi.
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Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., Viggen, E.M. (2017). Numerical Methods for Fluids. In: The Lattice Boltzmann Method. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-44649-3_2
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