Gravity in a lattice Boltzmann model

In this paper we consider the introduction of a body force, in the incompre ssible limit, into the lattice Boltzmann model. A number of methods are con sidered and their suitability to our objectives determined. When there is n o density variation across the fluid, gravity can be introduced in the form of an altered pressure gradient. This method correctly satisfies the Navie r-Stokes equation; however, if there is a non-negligible density variation present (produced by the body force or otherwise) this method becomes less accurate as the density variation increases and the constant density approx imation becomes less valid. Three other methods are also considered for app lication when there is a non-negligible density variation. The equations of motion satisfied by these models are found up to second order in the Knuds en number and it is seen that only one of these methods satisfies the true Navier-Stokes equation. Numerical simulations are performed to compare the different models and to assess the range of application of each.