Choice of units in lattice Boltzmann simulations

Lattice Boltzmann (LB) simulations are supposed to represent the physics of an actually existing, real system. During implementation, the question invariably pops up how to chose the units of the simulated quantities, the “lattice variables”. Two constraints determine the choice of units. First, the simulation should be equivalent, in a well defined sense, to the physical system. Second, the parameters should be fine-tuned in order to reach the required accuracy, i.e. the grid should be sufficiently resolved, the discrete time step sufficiently small and so on. The present text assumes that your final aim is to solve a macroscopic fluid equation, such as the incompressible Navier-Stokes equation. Therefore, discretization of the system is discussed in terms of macroscopic variables, as it is common in computational fluid dynamics. A different approach should be chosen if you investigate a fluid from a microscopic point of view, that is, if you explicitly wish to solve the continuum Boltzmann equation, to represent for example high Knudsen number flows. The approach presented here consists of two steps. A physical system is first converted into a dimensionless one, which is independent of the original physical scales, but also independent of simulation parameters. In a second step, the dimensionless system is converted into a discrete simulation. The correspondence between these three systems (the physical one (P), the dimensionless one (D), and the discrete one (LB) ) is made through dimensionless, or scale-independent numbers. The solutions to the incompressible Navier-Stokes equations for example depend only on one dimensionless parameter, which is the Reynolds number (Re). Thus, the three systems (P), (D), and (LB) are defined so as to have the same Reynolds number. The transition from (P) to (D) is made through the choice of a characteristic length scale l0 and time scale t0, and the transition from (D) to (LB) through the choice of a discrete space step δx and time step δt. Here’s a graphical representation of this relationship: