# Non-dimensionalisation and Choice of Simulation Parameters

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## Abstract

After reading this chapter, you will be familiar with how the “lattice units” usually used in simulations and articles can be related to physical units through unit conversion or through dimensionless numbers such as the Reynolds number. Additionally, you will be able to make good choices of simulation parameters and simulation resolution. As these are aspects of the lattice Boltzmann method that many beginners find puzzling, care is taken in this chapter to include a number of illustrative examples.

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## Notes

1. 1.

Note that the star also denotes post-collision values. There is no danger of confusing both concepts in this chapter, though.

2. 2.

Note that the definition of the Reynolds number is not unique: on the one hand, some people may choose for the length, others for the width of the considered system. On the other hand, U may be the average velocity or the maximum velocity—depending on the person defining the Reynolds number. This has to be kept in mind when comparing Reynolds numbers from different sources.

3. 3.

This is true as long as we stick to base SI units. Mixing metres, kilometres and feet, for example, will generally lead to additional numerical prefactors. This is the beauty of using SI units!

4. 4.

Here it is important to remember that the physical density of an incompressible fluid is constant while the LB density can fluctuate. Therefore, we relate the physical density ρ to the average lattice density ρ 0 . The fluctuation of the LB density, ρ, is then related to the pressure, as shown in (7.16).

5. 5.

The expression “diffusive scaling” stems from the apparent similarity of Δ t ∝ Δ x 2 and the diffusion equation, but there is no physical relation between both. Such a relation between the spatial and temporal scales is no special feature of the LB algorithm. It can also be found in typical time-explicit centred finite difference schemes, such as the DuFort-Frankel scheme.

6. 6.

Some phenomena remain (inversely) proportional to Re, for example the drag coefficient. Thus, there is still a significant difference in the result of a simulation between Re = 10−3 and 10−6, although inertia may be irrelevant in both.

7. 7.

As we will cover in Chap. 12, treating c s as physically relevant usually leads to small values of τ or Δ x and Δ t, rendering LB an expensive scheme to simulate acoustic problems. The only way to decrease the lattice size is by reducing τ which in turn can cause stability issues, though these can be alleviated, at least in part, by using other collision models than BGK.

8. 8.

Round-off errors may become important if ρ 0 deviates too strongly from unity.

9. 9.

If there is no analytical solution for the problem, one may run a test simulation and extract $$\hat{u}^{\star }$$.

10. 10.

The numerical prefactor becomes 4 in a 3D Poiseuille flow with circular cross-section.

11. 11.

One may check for temporal convergence by comparing the velocity profiles at times t and t T and applying a suitable error norm.

12. 12.

This definition appreciates the fact that the droplet/bubble may be deformed and therefore have a more complex shape than a section of a sphere.

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