I would like to know how to calculate the real pressure for the isothermal case.
I know that the pressure for the isothermal case at LBM is equal to row/3.
If no isothermal assumption is made ( I calculate the temperature field), how than can I calculate the real pressure?
D2Q9 LBE does not have a consistent temperature. In fact, it may be better described as athermal than isothermal because it does not have a sensible energy equation. You will need a bigger lattice if you want want an enery equation
In some multiphase models, for example (like Lee-Fischer), a force term is included which removes the ideal pressure and introduces a new equation of state.
If you have temperature as a passive scalar and are using a bousinesq approx. it is ok to use the ideal equation of state rho/3.
You might also want to take a look at the recent post about fluctuating LBE.
Thanks for answering.
It makes sense to me now, the relations between real pressure, and the LBM one.
The reference to the recent post about fluctuating LBE seemed so relevant to my
question and helped me as well and was complementary answer to yours.