Inlet boundary conditions for poiseuille flow

Hi there,

I am actually using the f_equilibrium boundary conditions on the inlet but I was told that it was technically wrong to use these kind of BC in the inlet. I would like to understand why is not correct, as I use it like this:

using D2Q9

  • I set a velocity profile Poiseuille on inlet nodes
  • I calculate the equilibrium distribution functions for f3, f6 and f7, the only ones that are not propagated and taking rho as local.

Moreover, I was told use the Ladd’s bounceback, shich it is easy to implement.
Any paper where they show the formula?

Thanks and regards



I can answer the first part of your question. On the inlet, you would ideally like to have the physically correct f_i^(0) distribution and the correct first order perturbation, f_i^(1).* The first contains the density and velocity, while the second contains the stress. Using an equilibrium BC like you describe, you get only the f_i^(0) distribution, with f_i^(1) = 0. Consequently, you are initialising a solution at the inlet which is not fully correct. This will probably negatively affect your accuracy in the domain.

I’m not familiar with Ladd’s bounceback, but you could have a look at this paper, which compares several BCs that you could use for the inlet:

  • If you’re not familiar with this terminology or the Chapman-Enskog expansion, I highly recommend that you take a look at this paper, which is the smoothest introduction that I am familiar with to that topic:

The bounce back boundary condition for 2D corners is not needed anymore. Not for corner or inlet/outlet.
I suggest you to read this paper:
On pressure and corner boundary conditions with two lattice Boltzmann construction approaches

Mathematics and Computers in Simulation
Volume 84, October 2012, Pages 26–41

which also available

Good luck

I strongly object to the above statement that bounce back boundary conditions are not needed anymore. The strongest statement we can make at present is in some special flows alternatives to bounce back work better, or just as well, as bounce back. There are many alternative methods, the first that I’m aware of being Noble et al 1994, but there may be even earlier contributions. The most popular alternative is Zou and He’s non-equilibrium bounce-back. The paper referenced above seems to combine Zou and He with a different corner implementation, and the corner treatment appears to be identical to the earlier work of Bennet (Bennett 2010, Bennett et al 2012, Reis and Dellar 2012). The “moment-method” of Bennett appears in his 2010 thesis (freely available on the internet) and extra details can be found in material given at and (it may also be worth noting that Bennett showed what conditions are needed for corners when Zou and He method is used at boundaries)

Besides the point of giving credit where it is due, it can not be said that these alternatives, in general, are better than bounce back. They can all eliminate the slip error of bounce back, but so can an MRT/TRT method (when used with bounce back). Both are equally simple, but bounce back is more stable. If there is to be an advantage of the method of Bennett, then it is its generality. Still, its weaker stability for 2D flows remains. Note also that all these on-node methods that I have mentioned, including the one referenced above, are currently limited to very simple geometries which have boundaries sitting on grid points. Bounce-back and interpolation methods, however, have a simple extension.

My apologies for the negative tone of the previous long paragraphs, but I feel compelled to summarise: it is not fair nor justified to make a sweeping statement about bounce back. At the very least, a comparison of the methods using a simple 2D flow (say lid driven cavity or flow over an obstacle) with identical parameters is needed. And even then, appropriate conditions are likely to be problem-dependent. No method is perfect, despite what the marketing and advertising might try to suggest, and you have to be aware of the pros and cons of your model.

Back to the original post. Erlend’s reply is completely correct: f^1 contains gradients of the conserved variables. With reference again to “alternative” boundary conditions, this time for inflow/outflow, Zou and He schemes , Bennett’s moment method, and the referenced, rely on us knowing what either the velocity profile or the pressure (density) is at the entrance and exit. This is often not know. However, Inlet/outlet based on f can interpolate/extrapolate to impose zero flux conditions (while being consistent with the Chapman-Enskog solution). I think the moment method could also be used to impose a no flow exit condition, but as far as know its success has not yet be assessed. Moreover, none of the “alternatives” (ie all those other then inflow/outflow based on f and the Chapman-Enskog solution of f) take higher order contributions (from Chapman-Enskog/asymptotic analysis) and moments into consideration (they cannot - you have 3 unknowns but 9 moments) and are thus often unstable and/or inaccurate for all but this simplest flows).

People like Junk have done some very nice work on consistent inflow/outflow conditions which you may find useful.

Good luck!

Hi pleb01

The above mentioned paper (Mathematics and Computers in Simulation, Volume 84, October 2012, Pages 26–41) appears to test several combinations of boundary conditions, to find out which is better, among other comparisons, like the entropic and the Hermite-one.
If I’m not mistaken, those with higher accuracy give lower Reynolds numbers and the opposite. I guess that paper (Mathematics and Computers in Simulation, Volume 84, October 2012, Pages 26–41) is based on single relaxation time only as the entropic construction has been designed only for single relaxation time. I assume a comparison with one model with SRT and other with MRT would be not fair, right?.
You have a point that the above works are just only for 2D, but the original question was about “using D2Q9”.
Hence, for more general cases, like 2D/3D, the Zou and He’s non-equilibrium bounce-back rule seems more general.


Some quick points.

Firstly, the same corner conditions appears in earlier work by Sam Bennett (and co workers).

Secondly, although the paper cited by you is for a 2D model the tested flows are 1 dimensional (velocity has one component, which depends on one coordinate: u=(u,v)=(u(y),0)). The method is likely to have different numerical properties for other, more interesting and useful, flows. Take, for example, a flow with a 2D velocity solution and four boundaries such as lid-driven cavity (other tests are available but this is one of the best benchmarks). Find the converged LB solution using bounce back and an on-node local method, with identical collision operators (BGK) and identical parameters (physical and numerical). Compare the methods with the standard benchmark grid size of 257*257 and Ma of the order of 0.1. At a low Re (less than 1000) the test is easy. With Re of the order of 1000 you may well find that the on-node local (non bounce back) does not converge (note also that predictions of the streamfunction should be very accurate for this flow). In short, results for a simple 1D channel flow test are not sufficient to suddenly start making claims that bounce back in redundant and one particular method should be used. (Zou and He, for example, is far less stable than bounce back when both use BGK)

Finally, even if this corner method proves to be reasonable the best you can say is it can be used, not should be used. As mentioned above, people have used alternatives to bounce back for over 15 years, so it should not be claimed that only now do we have something different.

Hi pleb01

Do you know of a paper that deals with 2D AND 3D corners BC with NO interpolation and NO bounce back rule?
The above cited works seem to be for just 2D, as you well put it.


The moment method is general and can be applied to any lattice and be used to apply a variety of different boundary conditions for different flows. This is its advantage. The drawback is it appears to suffer from the same stability problems as all other local on-node methods. There are also 3D extensions to Zou and He (see Hect and Harting 2010, for example). Excuse me for repeating myself but the main issue is how well all these methods work for realistic flows compared with bounce back. Statements claiming that bounce back is, at present, “no longer needed” are simply misleading.


Even if this corner technique turns out to be reasonable, the best you can say is that it can, but not necessarily should, be employed. As previously said, people have been using alternatives to recover for over 15 years, so it should not be assumed that we have something new now.