Why no "temperature" out of the original LBM?

Hello guys,

I’m trying to understand, why you cant get the temperature of the fluid as a macroscopic variable out of the seccond moment of the distribution function?
Why I have to introduce a new distribution equation like Guo et. al. in “A couple lattice BGK model for the Boussinesq equations”?
Are there any thermodynamic reasons?


Hello Fankhay,

With some LB models, it is actually possible to calculate the temperature as a second-order velocity moment of the distribution function (e.g. with the D3Q39 model described here: http://wiki.palabos.org/literature:shan_06). For this, the requirement is that your LB model actually simulates temperature properly, i.e. it must be able to solve the full Navier-Stokes equations for a compressible fluid, including the energy equation.

Simplified LB models like D3Q19 can’t do this. For the sake of computational efficiency, they just simulate a “slightly compressible fluid”, which recover some of the aspects of a compressible fluid (such as propagation of acoustic waves), but can’t represent the time evolution of the temperature.

In some cases like thermal convection in buildings, the energy equation is reasonably approximated by an advection-diffusion equation for the temperature, which is coupled into the momentum equation through a buoyancy term. This is what the Guo e.a. article does. It’s still cheaper than switching to a lattice with extended neighborhood, and good enough for many applications.


Hello Jonas,

thank you very much for your answer.
You argued, that is it not possible to calculate the temperature out of the seccond moment of the distribution function, since the LB model doenst solve the full Navier-Stokes-equations for incompressible fluids.
Is that the reason why I have no change to get the temperature on this way with the original LBM? Because the original LBM solves only the incompressible Navier-Stokes-equations.

What other options are there still there for solving the energy equation with my LB model for a incompressible fluild? Can you recommend literature on this subject?