Derivation of the incompressible Navier--Stokes equations from the lattice Boltzmann equation: Difference between the Chapman--Enskog expansion and the Sone expansion

We derive the incompressible Navier--Stokes equations from the lattice Boltzmann equation using the Chapman--Enskog expansion and the Sone expansion and clarify the differences between the two approaches. In the Chapman--Enskog expansion, we first derive the compressible Navier—Stokes equations on the multiple time scales (the acoustic and diffusive time scales). Then the incompressible Navier--Stokes equations are derived under the conditions of low Mach number flows and small density variations on the diffusive time scale. If the acoustic time scale remains in the analysis of the derived macroscopic equations, the incompressible Navier--Stokes equations are recovered with only first-order spatial accuracy. On the other hand, in the Sone expansion we can derive the incompressible Navier--Stokes equations under the condition of low Mach number flows on the diffusive time scale. Despite some differences between the two approaches, we obtain the same result that the flow velocity and the pressure satisfy the incompressible Navier--Stokes equations with the second-order spatial accuracy in low Mach number flows on the diffusive time scale. The accuracy is verified through simulating a generalized Taylor--Green problem.