# Extrapolation scheme boundary conditions

Dear all

I have some doubts regarding the implementation of the extrapolation scheme proposed by Chen, Martinez and Mei. (On boundary conditions in lattice Boltzmann methods, 1996.)

1. According to these authors the collision is the standard relaxation type for all nodes except the wall nodes where the velocity (or pressure) boundary conditions are enforced for equilibrium distribution functions. Therefore, if no collision takes place at the wall nodes the macroscopic properties, i.e. velocity and pressure, cannot be computed through the moments of the distribution functions but have to be explicitly assign to the desired values. For wall the velocity is easy to assign but what about the pressure? I have to extrapolate (or interpolate) it to same value right?
In the same paper and regarding the use of this scheme to pressure boundary conditions the authors say that one must set the pressure to the desired values but do not set any value for the velocities. So how velocity is computed at these pressure boundary condition regions?
Since no collision takes places the distribution function moments cannot be used so I have to assign this value somehow!

2. In Succi’s book (The lattice Boltzmann equation for fluid dynamics and beyond) on page 94 the extrapolation scheme is also discussed. However, in this book it can be read that the walls undergo the same collisional step as the fluid nodes, with the difference that in this case the equilibrium distribution functions are explicitly tuned on the desired wall speed. Therefore, I can compute the macroscopic fluid properties from distribution function moments the problem is according to some test that I did the wall velocity is never zero!

In conclusion: There is a contradiction on the way to implement the extrapolation scheme. Moreover, if the right procedure is the one described originally by Chen et al. the way the fluid properties are assign at boundaries is also not clear to me.

Can anyone please shed some light on this issue to me?

Thank you.

Goncalo

Hello Goncalo,

Best,
Timm

Hi Timm

Thank you for the suggestion but unfortunately that article does not discuss the BC which I have doubts
(i.e. the Chen and Martinez extrapolation scheme)

The BC’s discussed there are: Inamuro, Zou/He, Regularized, FD and non-linear FD. I could use some of the information present there and adapt it to the extrapolation scheme but I would like to implement the original version. However, the paper describing it is not too clear and to worsen the situation there are other manuscripts that even contradict this paper. So now I am really confused!

Nevertheless thanks for the help

Goncalo