Zou and He BC: from D2Q9 to D3Q19

Hello all,

I am interested in implementating Zou and He boundary conditions to impose a given velocity at inlet and outlet. I have used Sukop’s book for this purpose. Sukop’s book explores the determination of the unknown distributions for D2Q9 model.

I am trying to do the same for D3Q19 model (using only the information available in the section book), but I have had problems to determine all unknown distributions. I don’t have much experience with respect to the implementation of this kind of boundary condition, so I would like to know if the derivation of the unknown distributions for D3Q19 model is direct (like for D2Q9) or if I need more constraints than those used in D2Q9.

I mean, considering each boundary of the domain, do I need use just one time the bounceback for the non-equilibrium part of the particle distribution normal to the boundary or I need something else?

I would be grateful for any contribution.



Dear Fabiano

The extension of Zou He BC for the D3Q19 is a bit more envolving than the D2Q9 case. Recently Hecht and Harting have plublished an article pointing out (quite thoroughly indeed) what modifications should you perform in order to apply this BC in the D3Q19 case. Please have a read to their article:


I hope it helps you :wink:

Good luck


Dear Goncalo,

thanks a lot for this information.



The Hecht&Harting article certainly gave me more insight into the working of the Zou&He condition. I’m wondering if there is a general formulation of this boundary condition that applies to every lattice and not just to D2Q9 or D3Q19.

Anyone has suggestions?



Dear Elrohir,

Don’t know if I’m right, but at least it looks like Inamuro boundary conditions may be your choice (never worked with them though). Plus BB works everywhere, i.e. there are modifications for velocity boundaries and pressure boundaries (pressure anti bounce back).


Thanks Alex, I’ll look into that.