I am writing a Lattice Boltzmann code, and I have a problem dealing with nodes located on open boundaries (i.e. inlets/outlets).
Is it correct to stream populations towards inlet/outlet nodes or should we just skip the propagation step when the streaming direction is toward these nodes?
In general, the outlet and inlet was regarded as fluid node, after streaming, then solve the unknown populations.
The best way to deal with boundary nodes, in the framework of an LBM code is to have a buffer layer around your geometry. Then apply the collision-streaming step to all nodes (except buffers). After that step, nodes neighbouring your buffer nodes will have missing populations, you can fill the missing populations using the half-way bounce-back, modified half-way bounce-back or other BCs. You can find very good sample codes on the net, and also in Timm Krüger's book.
Hello,
I agree with the advice of Yutian Zhang, and to solve the unknown populations, non-equilibrium Extrapolation Boundary and Zou-he boundary (Q. Zou and X. He, On pressure and velocity boundary conditions for the lattice Boltzmann BGK model, Physics of Fluids, 9 (1997) 1591-1596) are both good choice.
Dear Honari,
One of the crude way to deal with inlet-outlet BC, is to have streaming for every node in the system and let there be periodic system i.e. the outlet outstreams be inlet inward unknown streams (vice versa) as it is easier to implement the periodicity e.g. using circshift in MATLAB.
Once it is done, you know your inlet incoming PDFs and outlet incoming PDFs are wrong. So then use the Zou and He or any BC you want to use to these nodes but make sure it should be done before any LB operations you're going to do in your code.
It is a crude but an efficient way to deal with the inlet/outlet boundary conditions.
However, as suggested by others like Seyed Ali Hosseini 's buffer layer advice is a better way to deal even when you're to work with multiphase flows.
For example, you are required to calculate the forces at each node in the domain with the information from its neighboring pseudopotential. Keeping a buffer layer at each side with a particular density (or extrapolated density from the known boundary densities) will help you out to find the interparticle forces at the boundary nodes. Such works have been demonstrated as ghost layer in this work: Preprint Numerical Simulation of Diffusion Process in T-shaped Microm...
Dear Siavash Honari
Depending on the different boundary conditions for your system, different actions can be taken.
1. If boundary nodes are solid, solid boundary conditions such as bounce-back and half-way bounce-back should be used.
2. If you wish to apply periodic boundary condition to boundary nodes, you do not need to take any specific action on the boundaries.( Note : The input nodes should be considered the Eastern neighbor of the output nodes and vice versa )
3. Finally, if you need to use specific boundary conditions, such as constant Pressure or constant Rate at boundary nodes, you can use the Zou and He method.
Thanks everybody, specially Debashis Panda and Seyed Ali Hosseini for your thorough answers.
Could you please explain the "buffer layer" method more?
Do we define the LBM populations for the nodes in the buffer layer? Is it possible to apply Zou/He method to fill the missing populations near the buffer layer?
Dear Honari,
I will explain step-wise as followed for the better understanding of the system:
(i) Collision: f* (post collision) we get from f(previous step streamed PDF), feq (you get from rho, u etc.)
(ii) Now the post collided stream, f* is going to propagate to the neighboring nodes by:
f(x+cx(k), y+cy(k))=f*(x,y); (Eqn (a))
Note: cx, cy are discrete velocity set, x,y are positions and k is the direction e.g. k=1:9 for D2Q9.
Now, understand eqn (a) carefully. If you are going to take the whole domain as your LB domain, then you will get a segmentation error as at the boundary nodes, the loop for eqn. (a) will go out of bound. For example, in the east boundary (x=nx), x+cx(1) will be greater than your domain boundary i.e. nx+1. In order to counteract these kind of errors, you can easily keep a buffer (dummy) layer adjacent to the LB field where the outgoing streams will be "dumped".
Now, once the outgoing streams at the boundaries are dumped, the incoming streams at the boundaries are not known. So now, you can initialize these incoming PDFs with your boundary conditions.
Keep in mind, half way bounce-back boundary conditions are implemented in this way. For full-way, you need to take flip-flop of streams in adjacent solids in the collision step and stream it up in your domain. But, halfway bounceback is beneficial.
The other way around as I said in my before reply, you can first keep your domain periodic e.g. ip=(i+nx%nx)?(1):(i+1) and so on. Now after it is done, you know your boundary incoming streams are wrong and hence, you can over-write these streams with the boundary condition you need to implement.
I hope you are able to understand.
Regards,
Debashis Panda
Thank you so much dear Debashis Panda . Your answer helped a lot.
Cheers,
Siavash Honari
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